diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzndme" "b/data_all_eng_slimpj/shuffled/split2/finalzzndme" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzndme" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\\label{sec:intro}\n\nUltra-luminous X-ray sources (or ULXs) are extra-galactic sources first discovered by \\cite{1989ARA&A..27...87F} that have been observed to have X-ray luminosities of $L_\\mathrm{X}\\gtrsim10^{39}$~erg~s$^{-1}$. X-ray variability suggests that ULXs are binaries with a non-degenerate star, referred to as a donor, transferring mass onto a compact object, the accretor \\citep{1976MNRAS.175..395B, 2009MNRAS.397.1061H}. These objects often dominate the total X-ray emission of their host galaxy. They are generally too bright to be low-mass X-ray binaries (LMXBs; for which $L_\\mathrm{X}\\lesssim10^{37}$~erg~s$^{-1}$), and too dim when compared to active galactic nuclei (AGNs; for which $L_\\mathrm{X}\\gtrsim10^{41}$~erg~s$^{-1}$) with their position being off-centre in their host galaxy \\cite[see][for a review on the observational properties of ULXs]{2017ARA&A..55..303K}.\n\nEddington luminosity is the limit above which any accretion onto the compact object is stopped by outgoing radiation. In general, ULX luminosities far exceed the Eddington limit of stellar-mass black holes (which is of the order of $10^{39}$~erg~s$^{-1}$) provided the emission is assumed to be isotropic. Initial suggestions as to the nature of the accretor include intermediate-mass black holes (IMBHs; with masses $\\gtrsim100$~M$_{\\odot}$) accreting at sub-Eddington rates \\citep{1999ApJ...519...89C,2001ApJ...562L..19E, 2006ASPC..352..121M, 2007Natur.445..183M}, or stellar-mass black holes (SMBHs) accreting at super-Eddington rates. For the latter, in order not to exceed the Eddington limit locally while still producing the observed luminosity, magnetic fields and geometric X-ray beaming effects have been proposed as an explanation \\citep[e.g.][]{2001ApJ...552L.109K,2014arXiv1411.5434C}. \n\nSince the first gravitational wave event, GW150914, detected by LIGO \\citep{2016PhRvD..93l2003A}, there has been vast renewed interest in studying the formation of double compact objects. Many of the proposed formation channels for these double compact objects predict that the binary will go through one or more phases of super-Eddington accretion onto a compact object. \\cite{2017A&A...604A..55M} explored how some observed ULXs might indeed be progenitors of coalescing double compact object binaries, specifically through the chemically homogeneous channel. In this type of binary evolution, the star would avoid large expansion of the envelope, and therefore the binary would avoid a common-envelope (CE) phase \\citep{2016MNRAS.458.2634M}. \\cite{2017MNRAS.472.3683F} compared observation trends in the number of ULXs per unit of star-formation rate and as a function of metallicity of the host galaxy to the merger rate of binary BHs. These latter authors found that the majority of ULXs could be progenitors of binary BH mergers. Therefore, studying the formation of ULXs could shed light on merging binary compact objects, and vice versa.\n\nIn recent decades, debate over the nature of ULXs has focused on whether they are SMBHs or IMBHs, and neutron stars (NSs) have not been considered. However, theoretical studies have extended ULX binary models to include NSs \\citep[e.g.][]{2001ApJ...552L.109K, 2009MNRAS.393L..41K}. That is until \\cite{2014Natur.514..202B} reported the first ever observations of X-ray pulsations in the M82 galaxy with a period of about 1.37~seconds and a 2.52-day sinusoidal modulation in the ULX~X-2. Detection of X-ray pulsations in ULXs suggests the presence of NSs as accretors instead of BHs, at least in a certain fraction of the ULX population where such pulses have been observed. This is because X-ray pulsations are characteristic of accreting NSs with radiation being emitted along their magnetic poles as they rotate about their axes. Therefore, the question of accreting NSs in binaries with relatively massive donors was raised. This discovery was followed by the detection of X-ray pulsations in several other ULXs \\citep{2017MNRAS.466L..48I, 2017Sci...355..817I, 2018MNRAS.476L..45C,2018NatAs...2..312B, 2019MNRAS.tmpL.104S,2019ApJ...879...61Z,2019arXiv190604791R}. \n\nThe two most prominent questions regarding the nature of NS ULXs refer to the emission mechanism and the formation channel.\nFor the first question, a number of mechanisms are invoked to explain super-Eddington luminosities. With the apparent extremely high mass-transfer rate, a lot of the transferred mass could be blown away by strong radiation outflows \\citep{1973A&A....24..337S}. \\cite{2006MNRAS.370..399B} applied this idea to SS433 where observations of massive outflows suggest that the source is a ULX seen from the side. The absorption features associated with these outflows have been observed in some but not all ULXs; see for example Holmberg IX X-1 and NGC 1313 X-1 \\citep{2012MNRAS.426..473W}. However, this does not invalidate the theory, as beamed X-ray emission would not be visible unless the observer had a direct line of sight to the accreting compact object, down the collimating funnel, in which direction the outflows are limited. \\cite{1973A&A....24..337S}, \\cite{2002ApJ...568L..97B}, and \\cite{2007MNRAS.377.1187P} explored the idea of the presence of strong optically thick outflows that blow away some part of the disc from where radiation can escape. \\cite{1973A&A....24..337S} and \\cite{2007MNRAS.377.1187P} also suggested the formation of geometrically thick accretion discs. This structure would cause the emission to be beamed, and therefore the observed isotropic-equivalent luminosity would be much higher than the intrinsic one. \\cite{2001ApJ...552L.109K} postulated that based on the assumption of mild beaming, intermediate- and high-mass X-ray binaries (IMXBs and HMXBs) undergoing mass transfer on a thermal timescale would be the best candidates for ULXs. The effect of beaming on the emission has, in general, been explored \\citep{2009MNRAS.393L..41K,2017MNRAS.468L..59K,2019ApJ...875...53W}. In addition to that, NSs with strong magnetic fields (around $10^{14}$~G) reduce the electron-scattering cross-section and could anchor the infalling matter to accretion columns above the magnetic poles and thereby produce sufficiently high luminosities \\citep{1976MNRAS.175..395B, 2018NatAs...2..312B}. \\cite{2017ApJ...845L...9T} carried out general relativistic radiation magnetohydrodynamics simulations of super-Eddington accretion onto a non-rotating, magnetised NS and found a spin-up rate of $\\sim -{10}^{-11}~{\\rm{s}}~{{\\rm{s}}}^{-1}$, which is consistent with observations. In contrast, \\cite{2019MNRAS.485.3588K} suggested that the observed ULX properties are explained by NSs with normal magnetic fields and not by the presence of magnetars. For the specific case of M82~X-2, \\cite{2014arXiv1410.8745L} suggested the presence of an optically thick accretion disc that acts as a curtain and shields some of the outgoing radiation, thus allowing for super-Eddington luminosities driven by Roche-lobe overflow (RLO). Finally, \\cite{2011ApJ...739...42W} studied accretion discs in weakly magnetised NSs, finding that at super-Eddington rates, the magnetic field has little effect on the accretion disc. \n\nDespite a lot of research done in the field, the question surrounding how ULXs attain super-Eddington luminosities is still open and an active field of research. Regarding a possible formation channel for NS ULXs, a lot of work has been carried out to address evolution and mass transfer in X-ray binaries. \\cite{1999A&A...350..928T} performed non-conservative mass-transfer calculations of low-mass X-ray binaries with a $1.3$~M$_{\\odot}$ NS and found that the binaries can undergo very high mass-transfer rates (super-Eddington by a factor of $\\sim 10^4$) for donors with deep convective envelopes (mass range $1.6$--$2.0$~M$_{\\odot}$). An important X-ray source in the context of high mass-transfer rates is Cygnus X-2, which is an X-ray binary containing a NS. Cygnus X-2 has a mass ratio of $\\approx 0.34~(=M_{\\mathrm{donor}}\/M_{\\mathrm{acc}})$, so for an estimated accretor mass of about $1.78$~M$_{\\odot}$ the donor has a mass of $0.6$~M$_{\\odot}$ \\citep{1998ApJ...493L..39C, 1999MNRAS.305..132O}. \\cite{1999MNRAS.309..253K} argued that the donor star lost a lot of mass ($\\sim 3.0$~M$_{\\odot}$) in an intense mass-transfer phase on a thermal timescale. \\cite{1999MNRAS.309..253K,2000ApJ...529..946P,2000MNRAS.317..438K} showed that Cygnus X-2 observations can be explained by case B mass transfer from a donor of mass $3.5$~M$_{\\odot}$; that is, the progenitor for the source was an IMXB. \n\nContrary to the general understanding of the stability of mass-transfer in X-ray binaries, there is similar evidence in the literature that IMXBs can undergo stable mass transfer with a NS and avoid CE. \\cite{2000ApJ...530L..93T} carried out numerical calculations of IMXBs with $2.0$--$6.0$~M$_\\odot$ donor and $1.3$~M$_\\odot$ accretor masses using an updated version of the Eggleton code \\citep{1971MNRAS.151..351E,1972MNRAS.156..361E,1998MNRAS.298..525P}. The authors studied the initial parameter space for producing binary millisecond pulsars with a heavy carbon--oxygen (CO) white dwarf companion, and demonstrated for the first time the full stability of IMXBs. \\cite{2009MNRAS.393L..41K} stated that NSs would have lower accretion rates for the same mass-transfer rates because NSs would have stronger beaming as their Eddington limit would be lower than that of BHs. Calculations similar to those of \\cite{2000ApJ...530L..93T} were carried out by \\cite{2012ApJ...756...85S} using the Eggleton code. These latter authors studied the initial parameter space for binary pulsars while considering orbital angular momentum losses from gravitational wave radiation, magnetic braking, and mass lost from the system. \\cite{2017ApJ...846..170T} compared the orbital evolution between IMXBs and HMXBs in studying the connection between ULXs and double NS systems and concluded that the orbital period evolution of IMXBs makes them more likely to be NS ULXs than HMXBs. \n\n\\cite{2015ApJ...802L...5F} studied the origin of the NS ULX M82~X-2 specifically, by combining parametric population synthesis calculations (using BSE; \\citealt{2002MNRAS.329..897H}) with detailed binary evolution calculations (using MESA; \\citealt{2011ApJS..192....3P,2013ApJS..208....4P}). Assuming highly non-conservative mass transfer and that a significant fraction of the mass lost from the binary carries the specific orbital angular momentum of the donor star, these latter authors found that the most probable parameters to form a NS ULX are donors with initial masses in the range of $3.0$--$8.0$~M$_{\\odot}$ , and initial orbital periods of $1.0$--$3.0$~days. \\cite{2015ApJ...802..131S} suggested that NS ULXs, in general, might represent a higher contribution to the general ULX population than BH ULXs, and a significant portion of those would be IMXBs. Similarly, \\cite{2015ApJ...810...20W,2017ApJ...846...17W}, using parametric population synthesis calculations, showed that ULXs are more likely to be NS ULXs than BH ULXs, especially in solar metallicity environments. These latter authors found a typical NS ULX to have a $\\sim 1.0$\n~M$_{\\odot}$ red giant. However, they found that extreme NS ULXs ($L_\\mathrm{X}\\gtrsim10^{42}$~erg~s$^{-1}$) typically have evolved, low-mass, stripped helium-star donors ($\\sim 2.0$~M$_{\\odot}$). \\cite{2019ApJ...875...53W} found that most BH ULXs emit X-rays isotropically, while NS ULXs are generally beamed. Therefore, even though NS ULXs might be intrinsically more numerous than BH ULXs, these latter authors predict that BH accretors dominate the observed ULX population.\n\nAnother discussed property of pulsating ULXs is the very high NS spin-up rate (i.e. the rate of change of spin period) in some observed pulsars. NGC~5907~ULX1 shows a change from a spin period of $1.43$~s to $1.13$~s in about 10 years \\citep{2017Sci...355..817I} with an inferred spin-up rate of $-8\\pm0.1\\times10^{-10}$s~s$^{-1}$. The spin period of NGC~300~ULX1 went from $31.71$~s to $31.54$~s in $4$~days \\citep{2018MNRAS.476L..45C} with a spin-up rate $\\sim-5.56\\times10^{-7}$s~s$^{-1}$, which is the highest rate observed so far for a NS ULX. However, the rate of change of spin period given by \n\\begin{equation}\n \\dot{P} = \\dot{\\bigg(\\frac{1}{{\\nu}}\\bigg)} = - \\frac{\\dot{\\nu}}{\\nu^2},\n\\end{equation}\nwhere $\\nu$ is the NS spin frequency, does not directly reflect the mass accretion rate. Rather, the rate of change of $\\nu$ is roughly proportional to the mass accretion rate \\citep[for instance see Eq.~(2) in][]{2017MNRAS.468L..59K}. Nevertheless, we continue to mention $\\dot{P}$ values when talking about observations (Table \\ref{tab:obs_data}) as they are the ones most often reported in the literature.\n\nHigh rates of frequency increase could suggest efficient spin up from very high accretion rates onto the NS. This is because high mass accretion provides the NS with enough torque to spin it up to such high rates \\citep{2001ASPC..229..423R, 2006csxs.book..623T}. However, there is a caveat in calculating accretion rates from spin-up frequency as they might not represent the secular average accretion rate over an evolutionary timescale of the donor star. In some cases, the extremely high spin-up rate would grossly overestimate the amount of matter accreted by the NS. For example, M82~X-2 showed a high spin-up rate of $-2\\times10^{-10}$s~s$^{-1}$ when X-ray pulses were first discovered \\citep{2014Natur.514..202B}. However, later, \\cite{2019arXiv190506423B} observed an average spin down of $-5\\times 10^{-11}$~Hz~s$^{-1}$ over a period of two years. These latter authors suggested that the source might be close to spin equilibrium and is alternating between phases of spin up and spin down. \\cite{2017MNRAS.468L..59K} estimated the spin-up timescale for three pulsating ULXs and concluded that we observe them close to equilibrium. \\cite{2019A&A...626A..18C} performed semi-analytical calculations for accretion onto a magnetised NS using the NS being close to spin equilibrium as a boundary condition. The reason for this is that even though $\\dot{\\nu}$ gives an estimate of the instantaneous accretion rate, it should not be compared directly to long-term average estimates that binary evolution models are giving, as many of these systems might be close to their spin equilibrium.\n\nIn our work, we do not investigate X-ray pulses and the super-Eddington emission mechanism. Instead, we focus on ULX formation and long-term evolution. Therefore, in the entirety of this study, we refer to LMXBs\/IMXBs with NS accretors that drive super-Eddington mass-transfer rates as NS ULXs. Pulsating ULXs are a subset of NS ULXs as pulsations are not a necessary outcome of super-Eddington mass transfer but rather they are a product of the presence of a relatively strongly magnetised NS. In this paper, we investigate how these NS ULXs could be formed and try to explain the physical properties involved using numerical computations. In doing so, we study how the stability of binaries is affected by spin-orbit coupling, and by a higher accretor mass ($2.0$~M$_\\odot$ instead of a typical NS mass of $\\sim 1.3$~M$_\\odot$). We also assume that there is no precession of the accretion disc or absorption of X-ray flux by optically thick material around the source that might cause the luminosity to vary. In the following section (Section~\\ref{sec:observations}), we summarise the properties of the currently observed pulsating ULX sample. In Section~\\ref{sec:numericaltools}, we discuss the numerical methods and physics employed for the simulations, while in Section~\\ref{sec:results} we present the results from our simulations, highlighting the allowed initial parameter space for NS ULX formation and the properties of the formed population. Section~\\ref{sec:discussion} discusses the observed NS ULXs in the context of our results and how the angular momentum exchange between spin and orbit affects the result through tides. Finally, we end with concluding remarks in Section~\\ref{sec:conclusion}.\n\n\\section{Currently observed pulsating ULX sample}\\label{sec:observations}\n\nIn this section, we discuss some of the observed and predicted parameters for the NS ULXs present so far in the literature. In most cases, the observables are not well constrained, and therefore large uncertainties are involved and assumptions about the physical properties have been made. In the entirety of this paper we refer to the mass of the donor as $M_{\\mathrm{donor}}$, that of the accretor as $M_{\\mathrm{acc}}$, the orbital separation as $a$, and the orbital period as $P_{\\mathrm{orb}}$. For our work, the most important pulsating ULX is M82~X-2 as it has the most well-constrained observational parameters, even though we comment on other pulsating ULXs observed as well. We summarise all the relevant properties of the currently known sample of pulsating ULXs in Table~\\ref{tab:obs_data}.\n\n\\begin{table*}[htp]\n\\begin{center}\n\\begin{threeparttable}\n\\begin{tabular}{lllllll}\n\\hline\nNS ULX & $L_\\mathrm{X}$ (erg~s$^{-1}$) & $M_{\\mathrm{donor}}$ (M$_{\\odot}$) & $P_\\mathrm{orb}$ (days) & $P_\\mathrm{spin}$ (s) & $\\dot{P}_\\mathrm{spin}$ (s s$^{-1}$) \\\\ \\hline\nM82~X-2\\tnote{1} & $1.8\\times 10^{40}$ & $\\gtrsim 5.2$ & $2.52$ & $1.37$ & $-2.0\\times 10^{-10}$ \\\\\nNGC~7793~P13\\tnote{2} & $5.0\\times 10^{39}$ & $18.0$--$23.0$\\tnote{3} & $64.0$\\tnote{4} & $0.417$ & $-3.5\\times 10^{-11}$ \\\\\nNGC~5907~ULX1\\tnote{5} & $\\sim 10^{41}$ & $2.0$--$6.0$ & $5.3^{+2.0}_{-0.9}$ & $1.137$ & $-8.1\\times 10^{-10}$ \\\\\nNGC~300~ULX1\\tnote{6} & $4.7\\times 10^{39}$ & $\\gtrsim 8.0$--$10.0$\\tnote{7} & $\\gtrsim 1.0$~yr\\tnote{8} &$31.6$ & $-5.56\\times 10^{-7}$ \\\\\nM51~ULX-8\\tnote{9} & $2.0\\times 10^{39}$ & - & - & - & - \\\\\nNGC~1313~X-2\\tnote{10} & $1.5\\times 10^{40}$ & $\\lesssim 12.0$\\tnote{11} & - & $1.5$ & $-1.2\\times 10^{-10}$ \\\\\nSwift~J0243.6+6124\\tnote{12} & $\\sim 10^{39}$ & - & $28.3$\\tnote{13} & $9.86$\\tnote{14} & $-2.2\\times 10^{-8}$ \\\\\nM51~ULX-7\\tnote{15} & $10^{39}$--$10^{40}$ & $\\gtrsim 8.0$ & 2.0 & $2.8$ & $-10^{-9}$ \\\\\n\\hline\n\\end{tabular}\n\\begin{tablenotes}\n\\item[1]\\cite{2014Natur.514..202B},\\item[2]\\cite{2017MNRAS.466L..48I, 2016ApJ...831L..14F},\\item[3]\\cite{2011AN....332..367M},\\item[4]\\cite{2014Natur.514..198M},\\item[5]\\cite{2017Sci...355..817I},\\item[6]\\cite{2018MNRAS.476L..45C},\\item[7]\\cite{2019arXiv190902171H},\\item[8]\\cite{2018A&A...620L..12V,2019ApJ...879..130R},\\item[9]\\cite{2018NatAs...2..312B},\\item[10]\\cite{2019MNRAS.tmpL.104S},\\item[11]\\cite{2008A&A...486..151G},\\item[12]\\cite{2019ApJ...879...61Z}, \\item[13]\\cite{2018A&A...613A..19D,2017ATel10907....1G},\\item[14]\\cite{2017ATel10809....1K,2017ATel10812....1J},\\item[15]\\cite{2019arXiv190604791R}.\n\\end{tablenotes}\n\\end{threeparttable}\n\\end{center}\n\\caption{Observed and inferred parameters of NS ULXs from the literature. $L_\\mathrm{X}$ is the X-ray luminosity, $M_{\\mathrm{donor}}$ the donor mass, $P_{\\mathrm{orb}}$ the orbital period, $P_{\\mathrm{spin}}$ the NS spin period, and $\\dot{P}_\\mathrm{spin}$ is the time derivative of the NS spin period.}\n\\label{tab:obs_data}\n\\end{table*}\n\n\\subsection{M82~X-2}\\label{m82x2}\nThis source is one of the most studied NS ULXs. It is observed in the core of M82 galaxy and was discovered to show X-ray pulsations by \\cite{2014Natur.514..202B}. It has a peak X-ray luminosity of $1.8\\times10^{40}$erg~s$^{-1}$. Using the observed orbital period of $2.52$~days and inclination of $<60\\degree$, the binary mass function for M82~X-2 was calculated as $2.1$~M$_{\\odot}$. Assuming that the NS mass is $1.4$~M$_{\\odot}$, the donor mass was estimated to be $\\gtrsim5.2$~M$_\\odot$.\n\n\\subsection{NGC~7793~P13}\\label{7793}\nThis ULX source appears to be a HMXB containing an accreting NS in galaxy NGC 7793, with a luminosity of about $5.0\\times 10^{39}$erg~s$^{-1}$. Earlier, \\cite{2011AN....332..367M} identified the donor in the then-not-discovered ULX system as a late B-type super-giant star with a mass of $18.0$~M$_{\\odot}10^{39}$erg~s$^{-1}$, confirming the source to be the first Galactic ULX with an NS accretor. Furthermore, these latter authors calculated a high spin-up rate of $-2.2\\times 10^{-8}$~s~s$^{-1}$. \\cite{2019ApJ...873...19T} reported on the spectral behaviour of this source and suggested that if it were located in an external galaxy it would have a similar appearance to the other pulsating ULXs observed.\n\n\\subsection{M51~ULX-7}\n\\cite{2019arXiv190604791R} discovered $2.8$~s X-ray pulses in observations of ULX-7 in galaxy M51, therefore finding another pulsating ULX following ULX-8 in the same galaxy. The secular spin-up rate was measured as $-10^{-9}$~s~s$^{-1}$ and a variable X-ray luminosity in the range of $10^{39}$--$10^{40}$~erg~s$^{-1}$. The authors used the projected semi-major axis of $28.0$~lt-s and an assumed accretor mass of $1.4$~M$_{\\odot}$ to infer a donor of mass $>8.0$--$13.0$~M$_{\\odot}$. \\cite{2020MNRAS.491.4949V} studied the X-ray light curves of the source and estimated a magnetic field of the NS (rotating near spin equilibrium) of $2.0$--$7.0\\times 10\n^{13}$~G. They also assumed that the NS is freely precessing and estimated the magnetic field to be $3.0$--$4.0\\times 10\n^{13}$~G, agreeing with their previous estimate.\n\n\n\\section{Numerical tools and calculations}\\label{sec:numericaltools}\n\nIn this section we discuss the numerical code used for the binary evolution calculations along with the adopted model parameters and the code modifications that we introduced. \n\n\\subsection{Numerical stellar evolution code and progenitor binary}\n\\label{code}\n\nTo simulate the evolution of the binaries we use MESA (version 10108; MESASDK version 20180127) which is a stellar structure and binary evolution code developed by \\cite{2011ApJS..192....3P,2013ApJS..208....4P,2015ApJS..220...15P,2018ApJS..234...34P,2019ApJS..243...10P}.\n\nAll LMXBs\/IMXBs are calculated as systems with an initially zero-age main-sequence (ZAMS) donor and a point-mass NS accretor. It is assumed that the NS was formed during a previous evolutionary stage which we do not study, and the binary survived a possible NS natal kick from the supernova explosion that formed the NS. We compute a grid of models spanning $0.92$--$8.0$~M$_{\\odot}$ in initial donor mass, and $0.5$--$100$~days in initial orbital period. For reference, initial parameters refer to the orbital parameters at the onset of RLO.\nThe calculation of the mass-transfer rate during RLO is done implicitly using the scheme proposed by \\cite{1990A&A...236..385K}.\n\nWe use $1.3$~M$_{\\odot}$ and $2.0$~M$_{\\odot}$ for the mass of the accretor; $1.3$~M$_{\\odot}$ is close to the post-supernova Chandrasekhar mass limit for the formation of a NS and $2.0$~M$_{\\odot}$ is on the high-mass end of the NS mass distribution \\citep{2010arXiv1012.3208L,2013Sci...340..448A, 2018ApJ...852L..25R}. The NS companion to the pulsar J0453+1559 has a mass of $1.174^{0.004}_{0.004}$~M$_{\\odot}$ \\citep{2015ApJ...812..143M}\\footnote{See \\cite{2019ApJ...886L..20T} for an alternative possibility} and therefore NS masses below $1.3$~M$_{\\odot}$ have indeed been observed. However, we take $1.3$~M$_{\\odot}$ as the standard NS mass.\n\nFor the radiative efficiency of the accretion onto the NS with initial mass $M^i_{\\rm acc}$ (i.e. the release of gravitational energy of the infalling material in the form of radiation; in units of rest-mass energy) we use,\n\n\\begin{equation}\\label{eta}\n \\eta = \\frac{G M^i_{\\rm acc}}{c^2 R_{\\rm acc}},\n\\end{equation}\nwhere, $c$ is the speed of light and $R_{\\rm acc}$ is the NS radius which we take as 11.0~km.\nFor the Eddington limit of an accretor with initial mass $M^i_{\\rm acc}$,\n\\begin{equation}\\label{Ledd}\n L_{\\rm Edd} = \\frac{4\\pi G M^i_{\\rm acc} c }{\\kappa},\n\\end{equation}\nwhere, $\\kappa$ is the opacity. Using Eq.~(\\ref{eta}) and simplifying Eq.~(\\ref{Ledd}) (for accretion of pure ionised hydrogen), we get\n\\begin{equation}\\label{Eq:Edd}\n \\dot{M}_\\mathrm{Edd}=1.5\\times 10^{-8}~\\bigg(\\frac{M^i_{\\mathrm{acc}}}{1.3~ \\mathrm{M}_{\\odot}}\\bigg) ~\\mathrm{M}_{\\odot}~\\mathrm{yr}^{-1}.\n\\end{equation}\n\nWe fix the Eddington limit and the radiative efficiency to initial values as they would not change significantly from the amount of mass that the NSs accrete in our grids. If the mass-transfer rate goes beyond this value, the radiation pressure will prevent any excess material from being accreted. Mass from the donor is transferred conservatively to the accretor and a fraction of this transferred mass is lost from the vicinity of the accretor as an isotropic fast wind or jet with the specific angular momentum of the accretor \\citep[see][for a detailed explanation]{2006csxs.book..623T}. The efficiency of accretion ($\\epsilon$) by the NS is $\\epsilon = 1 - (\\alpha + \\beta + \\delta)$, where $\\alpha$ is the fractional mass lost directly from the donor, $\\beta$ is the fractional mass lost from the vicinity of the accretor, and $\\delta$ is the fractional mass lost from a circumbinary toroid. We take the values, $\\alpha =\\delta= 0$, and $\\beta = {\\it max}~\\{0.7, 1-{\\dot{M}_{\\mathrm{Edd}}}\/{\\dot{M}_{\\mathrm{donor}}}\\}$. \n\nWe consider orbital angular momentum (J$_{\\mathrm{orb}}$) losses via gravitational wave radiation, spin-orbit coupling due to tidal effects (Section~\\ref{sync_time}), and mass lost from the system. We also include effects due to magnetic braking following the prescription by \\cite{1983ApJ...275..713R} for donor masses that develop an outer convective envelope at any point. We take the eccentricity to be negligible, as tidal forces would circularise the orbit of a semi-detached binary with a giant star on a relatively short timescale of $10^4$~years \\citep{1995A&A...296..709V}. This is orders of magnitude shorter than the main sequence lifetime of intermediate-mass stars which are of the order of $10^8$--$10^9$~years (main sequence lifetime of low-mass stars would be even longer). Furthermore, tidal forces aim to synchronise the stars with the orbit. We assume the orbit is synchronised by the time RLO begins since the tidal forces would cause the stars to synchronise with the orbit on a relatively short timescale \\citep{2000ScChA..43..331H}.\n\nWe assume solar metallicity, that is Z$_{\\odot}$=0.02, and that any layer in the donor interior is stable against convection if the Ledoux criteria for convection is fulfilled \\citep{1947ApJ...105..305L}. At the edges of convective zones, we account for overshooting because the convective material slightly enters non-convective zones due to inertia. To describe overshooting we follow the exponential overshooting efficiencies used in the MIST models for low- and intermediate-mass stars, which are $f_{\\mathrm{ov, core}} = 0.0160$ in the core calibrated from properties of the open cluster M67 and $f_{\\mathrm{ov, en}} = 0.0174$ in the envelope calibrated from solar properties \\citep{2016ApJS..222....8D, 2016ApJ...823..102C}. For stellar winds, we use the cool red giant branch wind scheme described by \\cite{1975psae.book..229R} with a scaling factor of 0.1. In cases where\nwe get a stripped helium (He) star, we use the prescription for Wolf-Rayet stars by \\cite{2000A&A...360..227N}, included in MESA under the Dutch hot wind scheme. \n\nThe mass-transfer calculations are carried out until one of the following conditions are met: \\textit{(i)} the donor forms a white dwarf (WD), \\textit{(ii)} the age of the donor star exceeds the Hubble time, \\textit{(iii)} the radius of the donor star extends so far beyond its Roche lobe that L$_{2}$ overflow is initiated and the system becomes dynamically unstable (see Section~\\ref{rl2}), \\textit{or (iv)} the number of computational steps exceeds a limit of 300,000 (this value was chosen based on previous grid runs). We include condition \\textit{(iv)} for those systems where MESA runs into converging problems and cannot find a solution. This happens for only two types of binaries in our numerical calculations. In the first case, the mass transfer cannot properly remove the last bit of the envelope from the donor. In the second case, MESA is not able to solve for the donor radius which extends quite far beyond the Roche lobe but not enough to trigger the condition of L$_2$ overflow. Both these cases occur at the end of the mass-transfer phase, and therefore we accept that the binary was stable until the end of its evolution.\n\n\\subsection{Super-Eddington accretion onto the NS}\\label{acc}\n\nFor non-conservative super-Eddington mass transfer the amount of mass that is accreted is less than that transferred per unit time ($0.3\\times $ mass transferred, as per our assumptions) until the Eddington limit prevents a further increase in accretion rate. Cygnus X-2 is an example of a system observed to have survived super-Eddington mass transfer while presumably having accreted comparatively less. This source suggested that high mass transfer onto a NS (or BH) could avoid a CE phase with the NS ejecting most of the mass that was transferred at super-Eddington rates \\citep{1999MNRAS.309..253K, 1999ApJ...519L.169K, 2000ApJ...529..946P, 2001ApJ...552L.109K}.\n\n\\cite{1973A&A....24..337S} studied the observational characteristics of accretion discs in sub-Eddington and super-Eddington regimes of mass transfer in the case of black-hole binaries. In this picture, as the mass transfer approaches the Eddington limit, the structure of the disc changes from a slim disc to a disc with an inner geometrically thick component and an outer thin component. As matter that is transferred to the accretion disc at super-Eddington rates moves radially inwards (transporting angular momentum outwards), strong outflows begin at a certain radius which remove a fraction of the matter, thereby also taking away excess angular momentum \\cite[see][for an application of the super-Eddington disc model]{2013AstBu..68..139V}. The spherisation radius is defined as the radius at which the accretion luminosity first reaches the Eddington limit and strong outflows begin, and one can approximate it as \n\\begin{equation}\n R_\\mathrm{sph}= \\dot{M}_{\\mathrm{donor}}\\frac{G M_\\mathrm{acc}}{L_\\mathrm{Edd}}.\n\\end{equation}\nOutside $R_\\mathrm{sph}$ the disc emits X-rays with luminosity $L_\\mathrm{Edd}$ which depends on the accretion rate following the equation,\n\\begin{equation}\n L_\\mathrm{Edd}= \\eta \\dot{M}_{\\mathrm{Edd}}c^2.\n\\end{equation}\n\nInside $R_\\mathrm{sph}$, the outflowing matter has a velocity which depends on the difference between inward gravity and outward radiation pressure. The velocity of the outflow increases inward which in turn decreases the mass-accretion rate at each radius. This keeps the disc locally Eddington limited. The mass-accretion rate within $R_{\\mathrm{sph}}$ at each point in the disc can be described by the following equation:\n\\begin{equation}\n \\dot{M}^{\\mathrm{local}}_{\\mathrm{acc}}(R) = \\dot{M}_{\\mathrm{donor}} \\frac{R}{R_\\mathrm{sph}}.\n\\end{equation}\nSince the radiation pressure is balanced by the gravitational pressure at each point, the accretion disc inside the spherisation radius has a thickness of the order of the distance from the accretor and emits X-rays with luminosity $L_\\mathrm{Edd}\\times\\ln{\\dot{m}}$ \\citep[where \n$\\dot{m} \\equiv \\dot{M}_\\mathrm{donor}\/\\dot{M}_\\mathrm{Edd}$; ][]{1973A&A....24..337S}. For $\\dot{m}>1$, the total radiated luminosity from the accretor can exceed the Eddington limit by\n\\begin{equation}\\label{edd_eq}\n L_{\\mathrm{acc}} = L_{\\mathrm{Edd}}(1 + \\ln{\\dot{m}}).\n\\end{equation}\n\nFor highly super-Eddington mass-transfer rates another effect could come into play because of the geometrically thick accretion disc.\nA narrow funnel forms along the rotation axis of the accretor from where radiation can escape as a collimated jet. The observed isotropic-equivalent accretion luminosity as described by \\cite{2001ApJ...552L.109K} and \\cite{2009MNRAS.393L..41K} is then as follows:\n\n\\begin{equation}\\label{new_edd}\n L^{\\mathrm{iso}}_{\\mathrm{acc}} = \\frac{L_{\\mathrm{Edd}}}{b}(1 + \\ln{\\dot{m}}),\n\\end{equation}\nwhere $b$ is the beaming factor describing the amount of collimation to the outgoing radiation. The approximated value of $b$ is\n\\begin{equation}\\label{eq:beaming}\n b= \n\\begin{dcases}\n \\frac{73}{\\dot{m}^2},& \\text{if } \\dot{m}> 8.5,\\\\\n 1, & \\text{otherwise}.\n\\end{dcases}\n\\end{equation}\nBecause the beaming factor is an approximation, we apply an upper limit ($\\sim 10^{42}$~erg~s$^{-1}$) in calculating the isotropic-equivalent accretion luminosities so that we do not get unphysically high values.\n\n\\subsection{Spin-orbit coupling and synchronisation timescales}\n\\label{sync_time}\n\nAs mentioned before, tidal forces synchronise the stars with the orbit on a timescale which is relatively short. Therefore, whenever there is a change in either the spin angular momentum or the orbital angular momentum, tides will work to synchronise the system again. This action of tides on the orbit may affect the stability and evolution of the binary \\citep{2001nsbh.conf..337T}.\n\nWhen mass is lost from the donor star during mass transfer it also removes spin angular momentum from the star which is supplied to the orbit. If the spin angular momentum removed from the donor and returned to the orbit is non-negligible, then it has a widening effect on the orbit. This is followed by mass loss from the accretor's vicinity, with the mass lost carrying away the specific angular momentum of the accretor. This competing effect tends to shrink the orbit. Mass that is leaving the system from the accretor's vicinity carries with it relatively high specific orbital angular momentum (when the accretor is the less massive binary component), having a shrinking effect on the orbit. The donor then becomes sub-synchronous with the orbit and angular momentum has to be transferred from the orbit to the star in order to spin it up, causing the orbit to shrink even further. The interplay between the orbital shrinking and widening effects can reveal how spin-orbit coupling affects the stability of mass transfer. In the presence of strong winds, there is additional loss of angular momentum from the donor. However, for stars with masses of $0.92$--$8.0$~M$_{\\odot}$, stellar winds are too weak to cause significant orbital change. \n\nThe tidal synchronisation timescale is defined as follows \\citep{1977A&A....57..383Z,1981A&A....99..126H}:\n\\begin{equation}\n \\frac{1}{T_\\mathrm{sync}} = 3 \\frac{K}{T} \\bigg(\\frac{q}{r_\\mathrm{g}}\\bigg)^2 \\bigg(\\frac{R_\\mathrm{donor}}{a}\\bigg)^6,\n\\end{equation}\nwhere $q$ is the mass ratio (we define $q\\equiv M_{\\mathrm{acc}}\/{M_{\\mathrm{donor}}}$) and $r_\\mathrm{g}$ is the gyration radius of the star. ${K}\/{T}$ is the spin-orbit coupling parameter and is described in two ways for a star with an outer radiative envelope as $({K}\/{T})_{\\mathrm{rad}}$, and a star with an outer convective envelope as $({K}\/{T})_{\\mathrm{conv}}$. For the former case we use,\n\\begin{align}\n \\bigg(\\frac{K}{T}\\bigg)_{\\mathrm{rad}} &= \\bigg(\\frac{GM_{\\mathrm{donor}}}{R_\\mathrm{donor}^3}\\bigg)^{1\/2}(1+q)^{5\/6}E_{2}\\bigg(\\frac{R_\\mathrm{donor}}{a}\\bigg)^{5\/2},\\\\\n E_2 &= 10^{-0.42}\\bigg(\\frac{R_\\mathrm{conv}}{R_\\mathrm{donor}}\\bigg)^{7.5},\n\\end{align}\nwhere $R_\\mathrm{conv}$ is the radius of the convective core. The value for $E_2$ is computed from fitting formulae for H-rich stars derived by \\citet{2018A&A...616A..28Q}. For $({K}\/{T})_{\\mathrm{conv}}$, the definition is taken from \\cite{2002MNRAS.329..897H} to be\n\\begin{equation}\n \\bigg(\\frac{K}{T}\\bigg)_{\\mathrm{conv}} = \\frac{2}{21}\\frac{f_\\mathrm{conv}}{\\tau_\\mathrm{conv}}\\frac{M_\\mathrm{env}}{M_{\\mathrm{donor}}} \\mathrm{yr}^{-1},\n\\end{equation}\nwhere $f_\\mathrm{conv} = {\\it min}~\\{1.0, (P_{\\rm tidal}\/(2\\tau_{\\rm conv}))^{2}\\}$ is a numerical factor and $P_{\\rm tidal}$ is the tidal pumping time-scale defined by $\\lvert 1\/P_{\\rm orb} - 1\/P_{\\rm spin}\\rvert$. Here, $\\tau_\\mathrm{conv}\\approx(MR^2\/L)^{1\/3}$ is the eddy turnover timescale in units of years \\citep{1996ApJ...470.1187R} and $M_\\mathrm{env}$ is the convective envelope mass. More details about our applied orbital angular momentum evolution, with spin-orbit coupling including tides, are discussed in Section~\\ref{ls-coup} and Appendix~\\ref{append1}.\n\n\n\\subsection{Angular momentum loss from the second Lagrangian point}{\\label{rl2}}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=\\linewidth]{countour.pdf}\n\\caption{Equipotential lines of the Roche potential for a binary consisting of stars with mass ratio $q=0.26$ and binary separation $a$. The equipotential lines passing through Lagrangian points L$_1$, L$_2$ , and L$_3$ are shown.}\n\\label{contour}\n\\end{figure} \n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=\\linewidth]{countour_f.pdf}\n\\caption{Equipotential lines of the Roche potential for a binary consisting of stars with mass ratio $q=1.85$ and binary separation $a$ to illustrate the swapping of the positions of L$_2$ and L$_3$ when $M_{\\mathrm{donor}}\\leq M_{\\mathrm{acc}}$ (compared to Fig.~\\ref{contour}).}\n\\label{contour_f}\n\\end{figure} \n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[scale=0.6]{L2_R2.pdf}\n\\caption{Our results form the volume integration of L$_2$ equipotential surfaces with respect to different mass ratios $q$. All distance values are presented in units of $R_{\\mathrm{L}_{1}}$. Here, $R_{\\mathrm{L}_2}$ is the radius of a sphere with volume equal to that of the L$_2$ equipotential surface (solid orange line). $D_{\\mathrm{L}_2}$ is the distance of the L$_2$ point from $M_{\\mathrm{donor}}$ (solid blue line). The dashed black line shows the orbital separation in units of $R_{\\mathrm{L}_{1}}$. Systems where the donor's radius exceeds any of these limiting radii are considered as undergoing dynamical instability.}\n\\label{L2_R2}\n\\end{figure*} \n\nLagrangian points are equilibrium points in space where the gravitational and centrifugal forces in the system cancel each other out. L$_1$, L$_2$, and L$_3$ are unstable equilibrium points from where a test particle, upon small displacement, would move further away. Figure~\\ref{contour} shows these unstable Lagrangian points for a system with a mass ratio of $q=0.26$.\n\nIn most cases of X-ray binaries, analysis has been done for mass transfer via L$_{1}$. L$_1$ lies in between the two stars (hence also known as the inner Lagrangian point) and the equipotential surface passing L$_1$ is known as the Roche lobe. When a star fills its Roche lobe, any material that crosses L$_1$ from one star will fall towards the other star. This transfer of matter either decreases the radius (for radiative envelopes) of the donor or increases it (for convective envelopes). In some cases of extreme binary mass ratio, the RLO might not be enough to provide efficient mass-transfer rate and the donor might extend far beyond its Roche lobe to reach the equipotential surface passing through L$_2$. However, contrary to when mass passes through L$_1$, the mass that crosses L$_2$ takes away a large amount of angular momentum from the binary. Once the outer layers of the donor reach L$_2$ (or the donor obtains a volume equivalent to that of the equipotential lobe passing through L$_2$; see below) it is expected that the binary orbit will shrink rapidly. We consider this the onset of dynamical instability. For illustration, the L$_2$ potential surface is shown in Fig.~\\ref{contour}. It is the peanut-shaped surface enclosing both binary mass components and passing through the point L$_2$. \n\n\\cite{1983ApJ...268..368E} calculated the stellar radius needed in order to initiate mass transfer from the inner Lagrangian point of a binary by calculating the radius of a sphere that will have the same volume as the Roche lobe. This radius for the donor star is referred to as the Roche-lobe radius ($R_{\\mathrm{donor},\\mathrm{L}_1}$). We take a similar approach to quantify overflow from L$_2$. In cases where mass transfer via the L$_1$ point is not sufficient to keep the donor star confined within its Roche lobe we assume that the expanding star needs to fill the entire volume enclosed by the L$_2$ equipotential surface before the onset of dynamical instability. $R_{\\mathrm{L}_2}$ is the volume equivalent radius for the equipotential surface passing through the L$_{2}$ point.\n\nAnother possibility of mass loss from L$_2$ point occurs when the radius of the donor star reaches the point L$_2$ before the donor volume overfills the L$_2$ potential surface. This case applies only when $q\\ge 1$ (i.e. $M_{\\mathrm{donor}}\\le M_{\\mathrm{acc}}$) as the point L$_2$ is much closer to the donor. We refer to the distance between the centre of the donor and L$_2$ as $D_{\\mathrm{L}_2}$. This case is illustrated in Fig.~\\ref{contour_f} which shows the equipotential surfaces with mass ratio, $q=1.85$. In our simulations we assume that binaries experience stable mass transfer when the donor star does not cross any of the two limits discussed above at any point during each evolution (i.e. for stable RLO: $R_{\\rm donor}< {\\it min}~\\{R_{\\mathrm{L}_2},~D_{\\mathrm{L}_2}\\}$ for the entire binary evolution).\n\nTo calculate the L$_2$ volume via numerical integration, we begin by finding the Lagrange points (L$_1$, L$_2$ and L$_3$) for a particular mass ratio. Along the axis joining the centres of the $M_{\\mathrm{donor}}$ and $M_{\\mathrm{acc}}$, the entire volume is assumed to be a summation of thin discs. The volume of each disc slice is calculated going from one boundary end to the other using the boundaries of the L$_2$ equipotential surface, and the subsequent disc volumes are added together to cover the entire volume. In these calculations we consider two mass ratio regimes, $q < 1$ and $q\\ge 1$.\n\nWe use both $R_{\\mathrm{L}_2}$ and $D_{\\mathrm{L}_2}$ normalised to R$_{\\mathrm{donor},\\mathrm{L}_1}$ in order to remove the dependence on the binary orbital separation. Once these radii are calculated we find a fit of ${R_{\\mathrm{L}_2}}\/{R_{\\mathrm{donor},\\mathrm{L}_1}}$ and ${D_{\\mathrm{L}_2}}\/{R_{\\mathrm{donor},\\mathrm{L}_1}}$ on the mass ratio ($q=M_{\\rm acc}\/M_{\\rm donor}$) of the binary. For $q < 1$, we find that the $R_{\\mathrm{L}_2}$ and $D_{\\mathrm{L}_2}$ follow monotonically increasing trends toward $q = 1$ (shown in Fig.~\\ref{L2_R2} in the left panel), which is fitted by the following functions,\n\\begin{equation}\\label{eq1}\n \\frac{R_{\\mathrm{L}_2} (q < 1)}{R_{\\mathrm{donor},\\mathrm{L}_1}} = 0.784~q^{1.05} e^{-0.188~q} + 1.004,\n\\end{equation}\n\\begin{align}\\label{eq2}\n \\frac{D_{\\mathrm{L}_2}(q < 1)}{R_{\\mathrm{donor},\\mathrm{L}_1}} = 3.334~q^{0.514} e^{-0.052~q} + 1.308.\n\\end{align}\nHere, $R_{\\mathrm{donor},\\mathrm{L}_1}$ is also calculated from the volume of the L$_{1}$ equipotential surface using the method described above and the result is consistent with the calculations from \\cite{1983ApJ...268..368E}. \n\nFor the second mass ratio regime, we calculate $R_{\\mathrm{L}_2}(q\\ge 1)$ using\n\n\\begin{equation}\n R_{\\mathrm{L}_2}(q) = R_{\\mathrm{L}_2}\\bigg(\\frac{1}{q}\\bigg) \\qquad \\forall\\; q>0.\n\\end{equation}\nWe find $D_{\\mathrm{L}_2}(q\\ge 1)$ using the distance between L$_2$ and $M_{\\mathrm{acc}}$ from the case $q < 1$. As seen in Fig.~\\ref{L2_R2} on the right, there is a sudden jump in the values of $D_{\\mathrm{L}_2}$: when crossing $q=1,$ it goes from around $4.45$ for $q=0.997$ to around $1.82$ for $q=1.003$. We fitted functions to the calculated $R_{\\mathrm{L}_2}$ and $D_{\\mathrm{L}_2}$ values as follows,\n\\begin{equation}\\label{eq3}\n \\frac{R_{\\mathrm{L}_2}(q \\ge 1)}{R_{\\mathrm{donor},\\mathrm{L}_1}} = 0.290~q^{0.829}~e^{-0.016~q} + 1.362,\n\\end{equation}\n\\begin{equation}\\label{eq4}\n \\frac{D_{\\mathrm{L}_2}(q \\ge 1)}{R_{\\mathrm{donor},\\mathrm{L}_1}} = -0.040~q^{0.866}~e^{-0.040~q} + 1.883.\n\\end{equation}\n\nThe relative errors between our calculations and the corresponding fits are less than 1\\%. \n\nVolume equivalent L$_{2}$ radii calculations have also been done by \\cite{2016A&A...588A..50M} for $q<1$ but with an entirely different approach as these latter authors considered the case of overcontact binaries. They calculated the L$_{2}$ volume assuming that both the stars expand and fill their respective L$_2$ sub-volume. In order to test our numerical volume-integrating scheme against that of \\cite{2016A&A...588A..50M}, we split the L$_2$ volume at the L$_1$ point and compared the resulting calculations to their $R_{\\mathrm{L}_{2}}$~radii, finding good agreement. However, we should stress again that the approach by \\cite{2016A&A...588A..50M} is only applicable to overcontact binaries. Our work was followed by \\cite{2020arXiv200600774G} who, in a different context, derived fits to the volume equivalent L$_{2}$ radius using a similar approach.\n\n\n\\section{Results}\\label{sec:results}\n\nWe explore the evolution of LMXBs\/IMXBs with different initial conditions, taking into account the physics described in the earlier sections. In both our grids (for NS masses of 1.3~M$_{\\odot}$ and 2.0~M$_{\\odot}$ respectively) the binaries that interacted via mass transfer can undergo either stable or unstable mass transfer, excluding the systems with P$^i_\\mathrm{orb} \\lesssim 0.50$~days where the donor star already overflows its Roche lobe at ZAMS which we do not further evolve. The superscript $i$ stands for initial values which corresponds to the orbital parameters at the onset of RLO. We flag systems as `stable' when either the donor has detached from its Roche lobe at the end of mass transfer or the hydrogen in the outer layer has been almost completely removed (remaining hydrogen in the outer layer $<0.005$~M$_\\odot$). Donors in stable binaries formed a WD at the end of the mass-transfer sequence resulting in a neutron star--white dwarf (NS--WD) binary. The `unstable' binaries underwent the onset of L$_{2}$ overflow.\n\n\\begin{figure}[t]\n \\vfill\n \\includegraphics[width=\\linewidth]{rlo_cases1.pdf}\n \\caption{Allowed initial parameter space for LMXBs\/IMXBs to undergo stable mass transfer with $1.3$~M$_\\odot$ accretors (orange squares) and $2.0$~M$_\\odot$ accretors (blue squares). Grey squares correspond to systems that encountered dynamical instability. The dark red squares towards the left edge of the figure correspond to systems that never initiated RLO. The lower dashed white line separates systems undergoing case A mass transfer from those undergoing case B mass transfer (same boundary in both grids). The middle dotted white line separates early case B (where the donor has yet to form a deep convective envelope) from late case B RLO (where donor has formed a deep convective envelope at onset of RLO). The upper dashed white curve encloses systems that undergo case C mass transfer (same boundary in both grids). Green stars correspond to those systems that undergo a second mass-transfer phase from a stripped Helium-giant star, i.e. case BB RLO. The three red crosses in case BB correspond to systems that terminated due to numerical issues.}\n \\label{rlo_cases1}\n\\end{figure}\n\nIn Fig.~\\ref{rlo_cases1} we present this allowed initial parameter space for LMXBs\/IMXBs to undergo stable mass transfer for grids containing $1.3$~M$_\\odot$ (orange squares) and $2.0$~M$_\\odot$ NS accretors (blue squares) along with the unstable sequences (grey squares). The dark red squares towards the left are the systems that never initiated RLO. The general shape of the stable region resembles the work done by \\cite{2000ApJ...530L..93T} where they explored the allowed parameter space to form binary millisecond pulsars while avoiding a CE phase. The different dashed white lines separate the grids based on the type of mass-transfer phase that the system underwent. Case A is when the donor is on the main sequence at the onset of mass loss, that is, it is burning hydrogen in the core. Case B is when the donor has exhausted H in its core and H-shell burning phase (post-main sequence). The threshold between the two cases A and B (lower dashed white line in figure) depends more on the initial orbital period than the donor mass; the limit for case A being in the range $2.0$--$2.8$~days for both grids. The middle dashed white line separates two subsets of case B RLO: early case B and late case B. This threshold depends almost linearly on both the initial orbital period and the initial donor mass. At RLO onset, if the donor has a radiative envelope it is termed early case B and if the donor has developed a convective envelope it is termed late case B RLO. The upper dashed white curve encloses systems that undergo case C RLO, which means the donor has exhausted He in its core at the onset of RLO. \n\nThe stability region increases for higher accretor mass to include higher donor masses and orbital periods. A similar effect of increase in the parameter space with increasing the accretor mass was obtained by \\cite{2012ApJ...756...85S} for the formation of recycled pulsars from IMXBs and LMXBs. The overall shape of the parameter space for both the grids depends a lot on the structure of the donor envelope and the response of the donor to mass loss. A radiative envelope would shrink on mass loss and contribute to the stability of the binary while a convective envelope would expand rapidly on mass loss and make the system increasingly unstable. \\cite{1999ApJ...519L.169K} showed that mass transfer on a thermal timescale would avoid the CE phase in a binary as long as the envelope is mostly radiative. Case A and early case B have radiative envelopes, with binaries of initial mass ratio greater than $\\sim 0.28,$ undergoing stable mass-transfer depending on the initial orbital period. In contrast, late case B and case C have convective envelopes at RLO with the stability region being defined by a fixed critical mass ratio which we find to be at around $\\sim 0.5$. \\cite{2014A&A...571A..45I} studied the binary evolution in close LMXBs and also found an increase in the depth of the convective envelope with increasing P$^i_\\mathrm{orb}$ (Fig.~4 in their paper).\n\nIn Fig.~\\ref{rlo_cases1}, green stars correspond to systems that go through case~BB RLO, which is when a binary, after having lost its hydrogen envelope in a case~B mass-transfer phase, detaches and evolves as a stripped helium star and initiates a second RLO phase during the helium-shell burning stage. This subset of case B occurs for only a small part of the parameter space (and only for the grid with a $2.0$~M$_{\\odot}$ accretor). This is because low-mass helium stars ($<0.8\\;M_\\odot$) do not expand by any significant amount \\citep[e.g.][]{heu16,2018MNRAS.481.1908K} and thus only donor stars $\\ga 5.8\\;M_\\odot$ leave behind stripped helium stars massive enough to eventually lead to case~BB RLO. The three red crosses in case BB correspond to systems that terminated due to numerical issues.\n\nDuring the RLO phase, all LMXBs\/IMXBs are expected to be bright X-ray binaries, and in many cases the mass-transfer rate from the donor star to the vicinity of the NS can exceed the Eddington limit ($L_{\\rm Edd}$) significantly. As described in Section~\\ref{acc}, we only allow accretion onto the NS up to the Eddington limit, but we do consider the transition from a thin accretion disc to an inner thick disc model when the mass-transfer rate supplied from the donor star exceeds the Eddington limit \\citep{1973A&A....24..337S}, as well as the geometric beaming model proposed by \\cite{2001ApJ...552L.109K}. \n\n\\begin{figure}[t]\n \\vfill\n \\centering\n \\includegraphics[width=\\linewidth]{mdot.pdf}\n \\caption{Estimated observed X-ray luminosity for a binary with M$^i_\\mathrm{donor} =2.77$~M$_{\\odot}$, M$^i_\\mathrm{acc} =1.30$~M$_{\\odot}$, and P$^i_\\mathrm{orb} =5.38$~days under different assumptions. The solid green curve is the accretion luminosity corresponding to the mass-accretion rate onto the NS, assuming a fixed radiative efficiency ($\\eta=0.1$). The solid orange curve is the accretion luminosity corresponding to a super-Eddington accretion disc following the \\cite{1973A&A....24..337S} disc model given by Eq.~(\\ref{edd_eq}). The solid blue curve is the isotropic-equivalent accretion luminosity corresponding to beamed super-Eddington emission following the \\cite{2001ApJ...552L.109K} geometric beaming model (limited at $10^{42}$~erg~s$^{-1}$) given by Eqs.~(\\ref{new_edd}) and (\\ref{eq:beaming}). Four reference luminosity values, corresponding to $L_{\\rm Edd}$ (solid black line), $10\\,L_{\\rm Edd}$ (dashed black line), $100\\,L_{\\rm Edd}$ (dotted black line), and $1000\\,L_{\\rm Edd}$ (dot-dashed black line), are also plotted. The initial properties of this binary are highlighted with a magenta star in Fig.~\\ref{peak_acc}.}\n \\label{mdot}\n\\end{figure}\n \nFigure~\\ref{mdot} shows an example of a stable system with M$^i_\\mathrm{donor} =2.77$~M$_{\\odot}$, M$^i_\\mathrm{acc} =1.30$~M$_{\\odot}$, P$^i_\\mathrm{orb} =5.38$~days, where we demonstrate the estimated observed X-ray luminosity under different assumptions. The solid green curve is the accretion luminosity corresponding to the amount of mass accreted by the NS (i.e. capped at exactly the Eddington limit), assuming a radiative efficiency following Eq.~(\\ref{eta}). The solid orange curve is the accretion luminosity corresponding to a super-Eddington accretion disc following Eq.~(\\ref{edd_eq}), where although the Eddington limit is locally satisfied at every point in the disc, the integrated luminosity of the disc can exceed the Eddington limit by a small factor. Finally, the solid blue curve is the isotropic-equivalent accretion luminosity corresponding to beamed super-Eddington emission following Eq.~(\\ref{new_edd}) where the estimated observed luminosity can exceed the Eddington limit by up to a few orders of magnitude. \nFor comparison, we mark with horizontal lines the luminosity of ULXs corresponding to $L_{\\rm Edd}$ (solid black line), $10\\,L_{\\rm Edd}$ (dashed black line), $100\\,L_{\\rm Edd}$ (dotted black line), and $1000\\,L_{\\rm Edd}$ (dot-dashed black line). For simplicity, we use Eq.~(\\ref{Eq:Edd}) which is fixed for an initial accretor mass, because the amount of mass accreted does not change the accretion luminosity significantly. We note that since the prescription used for beamed emission (Section~\\ref{acc}) is an approximation, we fix an upper limit for the calculated isotropic-equivalent accretion luminosities at $10^{42}$~erg~s$^{-1}$ in order to avoid unphysically high values. In any case, the time that our binaries spend at such high luminosities combined with the inferred very small beaming factors make binaries on that phase effectively non-detectable. In the remainder of the paper, we use the beamed model (i.e. equivalent to the blue curve based on Eqs.~(\\ref{new_edd}) and (\\ref{eq:beaming})) as our estimate of the observed X-ray luminosity of the binaries.\n\n\\begin{figure*}[t]\n\\centering\n \\vfill\n \\includegraphics[width=1.0\\textwidth]{accreted_integrated.pdf}\n \\caption{(Left) Time-averaged isotropic-equivalent accretion luminosities ($\\langle L_{\\rm acc}^{\\rm iso}\\rangle$) in LMXBs\/IMXBs with a $1.3$~M$_\\odot$ NS accretor that reached an instantaneous luminosity above $10\\,L_{\\rm Edd}$. These luminosities were calculated based on Eqs.~(\\ref{new_edd}) and (\\ref{eq:beaming}) for each binary and at each time-step, and averaged over the entire RLO lifetime. Grey colour denotes the sequences that never achieved an instantaneous luminosity above $10\\,L_{\\rm Edd}$. The systems that reach the highest accretion luminosities correspond to unstable systems, on the higher donor mass ends in both panels, before the onset of dynamical instability. The stable systems (enclosed by the solid black boundary) have a relatively low accretion luminosity on average. The magenta star corresponds to the binary shown in Fig.~\\ref{mdot}. (Right) Same as left but for LMXBs\/IMXBs with $2.0$~M$_\\odot$ NSs. The three red crosses correspond to systems that terminated due to numerical issues.}\n \\label{peak_acc}\n\\end{figure*}\n\nFigure~\\ref{peak_acc} shows the time-averaged isotropic-equivalent accretion luminosities with respect to the initial parameters in both the grids. The magenta star is the binary shown in Fig.~\\ref{mdot}. For accreting NSs of either $1.3\\;$M$_\\odot$ or $2.0\\;$M$_\\odot$, we find time-averaged isotropic-equivalent X-ray luminosities of $\\langle L_{\\rm acc}^{\\rm iso}\\rangle \\simeq 10^{36}-10^{41}\\;{\\rm erg\\,s}^{-1}$ (in some cases even up to $10^{42}\\;{\\rm erg\\,s}^{-1}$).\nThese luminosities were calculated following Eq.~(\\ref{new_edd}) and averaged over the entire RLO lifetime for systems that reached instantaneous luminosity above $10\\;L_{\\rm Edd}$. The stable systems are enclosed within the solid black boundary (for this and all subsequent figures). Comparing Fig.~\\ref{rlo_cases1} with Fig.~\\ref{peak_acc}, even the systems that undergo dynamical instability (L$_2$ overflow) are included. This is because, when using detailed binary evolution calculation, we are able to resolve the onset of the dynamical instability which is not instantaneous and is most often preceded by a short but intense phase of mass transfer. In fact, the systems that reach the highest luminosities correspond to the unstable systems on the higher donor-mass end in both panels of Fig.~\\ref{peak_acc}. Therefore, binaries that would suffer dynamical instabilities and coalesce (likely producing Thorne-\\.Zytkow objects hypothesised by \\cite{1977ApJ...212..832T}) could also be observed as NS ULXs earlier in their evolution. However, the lifetimes of these binaries as ULXs is very short, and so this would act against their detectability.\n\n\\begin{figure*}[t]\n \\vfill\n \\centering\n \\includegraphics[width=\\linewidth]{ulx_time1.pdf}\n \\caption{(Top row) ULX lifetime for LMXB\/IMXB systems with a NS accretor mass of $1.3$~M$_{\\odot}$. Left, middle, and right panels show the time that systems spent with $L_{\\rm acc}^{\\rm iso}$ above $10$, $100$, and $1000\\,L_{\\rm Edd}$, respectively. White arrows (middle panels) enclose the potential properties of M82~X-2 at the onset of RLO (see Sections~\\ref{m82x2} and \\ref{m82x2b}). (Bottom row) Same as the top row but for LMXB\/IMXB systems with a NS accretor mass of $1.3$~M$_{\\odot}$.}\n \\label{result_1}\n\\end{figure*}\n\nWe define three X-ray luminosity ranges >$10\\,L_{\\rm Edd}$, >$100\\,L_{\\rm Edd}$, and >$1000\\,L_{\\rm Edd}$, and calculate how long each system spends in each luminosity range. We refer to this time duration as ULX lifetime. The defined luminosity ranges are based on the observed pulsating ULX luminosities, which are in the range of $10$--$1000\\, L_{\\rm Edd}$ (Section \\ref{sec:observations}). We compare the isotropic-equivalent accretion luminosities with these luminosity ranges. Figure~\\ref{result_1} presents these results for LMXBs\/IMXBs with $1.3$~M$_{\\odot}$ and $2.0$~M$_{\\odot}$ accretors. In systems with $1.3$~M$_{\\odot}$ NSs, the longest ULX lifetime is $1.6\\times 10^6$~years, corresponding to $M^i_\\mathrm{donor} = 2.3$~M$_\\odot$ and $P^i_{\\mathrm{orb}}=2.16$~days, for observed luminosities of >$10\\,L_{\\rm Edd}$. In systems with $2.0$~M$_{\\odot}$ NSs, the longest ULX lifetime is $1.1\\times 10^6$~years, corresponding to $M^i_\\mathrm{donor}=2.77$~M$_\\odot$ and $P^i_{\\mathrm{orb}}= 2.16$~days, again for observed luminosities of >$10\\,L_{\\rm Edd}$. The upper limits to ULX lifetimes are comparable to the ULX age estimate of $\\sim~1$~Myr for NGC~1313~X-2 by \\cite{2008AIPC.1010..303P} (Section~\\ref{1313}). Looking at similar initial donor masses and initial orbital periods in both sets of LMXBs\/IMXBs (going from top to bottom row in Fig.~\\ref{result_1}), higher accretor mass corresponds to a much higher ULX lifetime as the stability area increases. As an example, for $M^i_\\mathrm{donor}=5.0$~M$_\\odot$ and $P^i_{\\mathrm{orb}}=1.0$~days, the ULX time increases from $3.0\\times 10^4$~years to $3.0\\times 10^5$~years (going from lower to higher NS accretor mass).\n\nIn Fig.~\\ref{result_1} (top row), from the leftmost panel to the rightmost (from >$10\\,L_{\\rm Edd}$ to >$1000\\,L_{\\rm Edd}$) the systems which initially have long ULX lifetimes ($M^i_\\mathrm{donor}\\sim 2.3$~M$_\\odot$ and $P^i_{\\mathrm{orb}}\\sim 2.0$~days) either no longer appear on the plot or have a smaller ULX lifetime. Their isotropic-equivalent luminosities barely reach the higher cutoff values. Similarly, in the bottom row of Fig.~\\ref{result_1}, most LMXBs\/IMXBs with long ULX lifetimes ($M^i_\\mathrm{donor}\\sim 2.77$~M$_\\odot$ and $P^i_{\\mathrm{orb}}\\sim 2.0$~days) for >$10\\,L_{\\rm Edd}$, do not reach luminosities >$1000\\,L_{\\rm Edd}$. This implies that binaries that are in the ULX phase for the longest time do not always achieve the highest luminosities. This effect is not seen in the unstable systems, which maintain their short ULX lifetime across the different ULX criteria. There is a part of the stable parameter space where the ULX lifetime decreases significantly, the ULX lifetime going from about $10^5$~years for systems outside this region to $10^1$--$10^2$~years (the boundary corresponding to $M^{i}_{\\rm donor}<3.0$~M$_{odot}$ and $P ^{i}_{\\rm orb}<2.0$). Their accretion luminosity is sub-Eddington for almost the entire mass-transfer phase. After a mass-transfer episode, these systems are left with a He core and thin H envelope which expands briefly during H-shell burning, which is enough to increase their luminosity to exceed the Eddington limit albeit for a very short period of time.\n\nMost NS ULXs observed so far have been in the higher luminosity range ($100$--$1000\\;L_{\\rm Edd}$) potentially due to selection effects as more luminous ULXs have a higher chance of being observed. On the other hand, the short lifespans of NS ULXs compared to the age of the universe ($\\sim 1.4\\times 10^{10}$~yr) implies that it is unlikely to observe these systems in large numbers, which is consistent with the existing small NS ULX sample. The extra physical argument that the emission is highly beamed, especially for very luminous ULXs, further decreases the chances of detecting a large number of NS ULXs.\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=\\linewidth]{likelihood.pdf}\n \\caption{(Top row) Relative likelihood ($\\mathscr{L}$) to observe a system as a ULX, as described by Eq.~(\\ref{beam_eq}), for LMXBs\/IMXBs with a $1.3$~M$_\\odot$ NS. The panels are arranged as in Fig.~\\ref{result_1}. Going from the leftmost to the rightmost panel ($>10$ to >$1000\\,L_{\\rm Edd}$), $\\mathscr{L}$ is higher for stable binaries with lower luminosities as their emission is not as strongly beamed. There is in addition some effect by the transition from case~A to case~B RLO. All values below $10^{-6}$ are shown in black as they correspond to insignificant likelihood. (Bottom row) Same as the top row but for LMXBs\/IMXBs with a $2.0$~M$_\\odot$ NS.}\n \\label{likelihood}\n\\end{figure*}\n\nAs alluded to above, the probability of observing any of these LMXB\/IMXB sources as ULXs depends on the beaming factor, the ULX lifetime, and the probability of the particular system being formed. We evaluate the first two factors in the following equation describing the likelihood of a ULX observation, which is the beaming factor integrated over the three defined ULX lifetimes:\n\\begin{equation}\\label{beam_eq}\n \\mathscr{L} = \\int_{\\mathrm{ULX}} b(t) dt.\n\\end{equation}\nCalculating the likelihood using this equation for both the grids and for all three ULX lifetime criteria, we obtain a measure of the relative chance of observing each system in Fig.~\\ref{likelihood} for the $1.3$~M$_\\odot$ NS (top row) and the $2.0$~M$_\\odot$ NS (bottom row) grids. All values below $10^{-6}$ are shown in black as they correspond to insignificant likelihood. The highest likelihood is for the stable systems with lower luminosities ($10\\;L_{\\rm Edd}$), where the mass-transfer rate never increases to extreme values and thus collimation is very small to almost negligible. Figure~\\ref{likelihood} shows that the chance of a system being observed as a ULX decreases at higher luminosities. This is due to the fact that for higher luminosities the mass-accretion rates (and the mass-transfer rates) are extremely high which causes the emission from the system to be highly beamed. Also, the probability depends on the structure of the donor at the onset of RLO. Case A systems have lower mass-transfer rates and more isotropic emission and thus slightly higher probabilities of being observed than case~B. As mentioned before, there is a part in the stable parameter space where the ULX lifetime drops by many orders of magnitude (Fig.\n~\\ref{result_1}). Looking at the same systems in Fig.~\\ref{likelihood}, the likelihood to observe them is negligible. These systems are super-Eddington for a very brief moment in time which is to the detriment of their detectability. Looking at Figs.~\\ref{result_1} and \\ref{likelihood}, it is evident that even though LMXBs are included in the initial parameter space, IMXBs (with donor masses $\\gtrsim 2.0$~M$_{\\odot}$) are better candidates for NS ULXs.\n\nOne caveat is that we assume that all systems have an equal probability of formation whereas in reality many systems might be formed at a higher rate than others. To account for these effects we would need to do population synthesis studies which would include exploring the formation probability of close NS + main sequence binaries and the distribution of their binary parameters. However, this is outside the scope of this study. We intend to study the formation rate of these systems in a future work.\n\nFurthermore, one should always be careful when directly comparing our results to the observed population as our models give predictions about the whole population of ULXs with NS accretors, and not only the pulsating ones. The conditions for a NS to produce coherent pulses while also reaching super-Eddington luminosities are still unclear. If for example a strong magnetic field is required then perhaps the pulses are only observable at the beginning of the mass-transfer phase before any significant amount of material is accreted onto the NS, which could bury the magnetic field.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\linewidth]{wd.pdf}\n\\caption{(Left) Resulting distribution of final WD masses for LMXBs\/IMXBs with $1.3$~M$_\\odot$ NSs. The He~WDs are the least massive, followed by hybrid WDs, and the CO~WDs are the most massive. The outermost black boundary encloses the stable systems and the inner boundaries differentiate between the different WD types. (Right) Same as left but for LMXBs\/IMXBs with $2.0$~M$_\\odot$ NSs.}\n\\label{wd}\n\\end{figure*}\n\nIn cases where the mass-transfer sequence is stable throughout the binary evolution, we expect a NS--WD binary to be formed. Figure~\\ref{wd} shows the type of white dwarf formed at the end and the final white dwarf masses for the $1.3$~M$_\\odot$ NS (left panel) and the $2.0$~M$_\\odot$ NS (right panel) grids. The superscript $f$ stands for final values. Overall the white dwarfs resulting from the $1.3$~M$_{\\odot}$ NS grid are in the mass range of $0.23$--$0.71$~M$_{\\odot}$. For systems with $2.0$~M$_\\odot$ NS accretors, the white dwarfs have an overall mass range of $0.25$--$0.95$~M$_{\\odot}$. We used final total mass fractions of carbon to distinguish between different WD types. We define WDs with $>95\\%$ carbon mass fraction as CO white dwarfs, $0.01$--$95\\%$ as hybrid white dwarfs, and $<0.01\\%$ as He white dwarfs. The initially higher donor masses and longer orbital periods result in a degenerate CO core with negligible helium on the surface. For some donor masses and orbital periods, the donor forms a degenerate CO core with a relatively large helium-rich envelope leading to hybrid WDs. In systems with low donor masses and short orbital periods, the donors end up as helium WD systems. The reason for the final fate of the latter class of systems is that due to deep envelope stripping early in the evolution of the donor, combined in some cases with the low initial mass of the donor, the final stripped helium cores are not massive enough to ignite helium.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\linewidth]{ns.pdf}\n\\caption{(Left) Mass accreted by the NS ($M_{\\rm acc}^{f}-M_{\\rm acc}^I$) for LMXBs\/IMXBs with a $1.3$~M$_\\odot$ NS and $\\beta = \\mathrm{max}[0.7, 1-{\\dot{M}_{\\mathrm{Edd}}}\/{\\dot{M}_{\\mathrm{donor}}}]$. Binaries producing the lowest WD masses show the highest accretion on the NS. (Right) Same as left but for LMXBs\/IMXBs with a $2.0$~M$_\\odot$ NS. The systems below the white boundary are those where the accretor has accreted enough mass to exceed the maximum mass limit for a NS, here assumed to be $2.17\\;M_\\odot$.}\n\\label{ns}\n\\end{figure*}\n\nFigure~\\ref{ns} shows the mass accreted by the NSs, for systems with $1.3$~M$_\\odot$ (left) and $2.0$~M$_{\\odot}$ NS (right) accretors. We should reiterate here our assumption of non-conservative mass transfer with $\\beta = \\mathrm{max}[0.7, 1-{\\dot{M}_{\\mathrm{Edd}}}\/{\\dot{M}_{\\mathrm{donor}}}]$ and that the accretion by the NS is Eddington limited. The $1.3$~M$_\\odot$ NSs end up accreting $0.004$--$0.415$~M$_{\\odot}$ of mass by the end of the mass-transfer phase, the systems with values around $M_{\\mathrm{donor}}^i = 2.0$~M$_\\odot$ and $P_{\\rm{orb}}^i = 0.87$~day accreting the most. The amount of mass accreted is not enough to collapse the NS. The $2.0$~M$_{\\odot}$ NSs accrete mass in the range of $0.002$--$0.585$~M$_{\\odot}$, with the systems with values around $M_{\\mathrm{donor}}^i = 3.0$~M$_\\odot$ and $P_{\\rm{orb}}^i = 3.1$~days accreting the highest amount. We compare our results to the maximum NS mass known. The most massive precisely measured NS is PSR~J0348+0432 with a mass of $2.01\\pm0.04\\;M_\\odot$ \\citep{2013Sci...340..448A}. However, recently, a candidate with a higher NS mass of $2.17^{+0.11}_{-0.10}\\;M_\\odot$ was announced \\citep[PSR~J0740+6620;][]{2019arXiv190406759C}. Although the error bar of the latter source is relatively large, we take $2.17\\;M_\\odot$ as our assumed upper limit. This value is also supported by constraints on GW170817 based on combined gravitational wave and electromagnetic observations \\citep{2017ApJ...850L..19M}.\n\nLooking at the final NS masses in Fig.~\\ref{ns} (right panel), we see a certain population enclosed by a white boundary. In these systems the accretor has accreted enough material to overcome the neutron degeneracy pressure that is supporting the star against gravitational collapse (assuming an upper NS mass limit of $2.17\\;M_\\odot$). Therefore, according to the maximum NS mass limit assumed, these NSs will collapse to form BHs. \n\nNet accretion on the NS can be broadly described by,\n\\begin{equation}\n \\Delta M_{\\mathrm{acc}} = \\langle\\dot{M}_{\\mathrm{acc}}\\rangle \\times \\Delta t_{\\mathrm{X}},\n\\end{equation}\nwhere $\\Delta t_{\\mathrm{X}}$ is the lifetime as an X-ray binary (or the overall mass-transfer phase) and $\\langle\\dot{M}_{\\mathrm{acc}}\\rangle$ is the average accretion rate onto the NS. A high amount of accreted mass is achieved with a combination of both a large accretion rate and a long time duration over which it occurs. \nFor LMXBs\/IMXBs that reach up to Eddington mass-transfer rates for only a small part of the mass-transfer phase, to zeroth order, $\\Delta M_{\\mathrm{acc}}\\simeq 0.3\\times \\Delta M_{\\rm donor}$. In contrast, for the brightest ULXs, where most of the mass-transfer happens at a highly super-Eddington rate, $\\langle\\dot{M}_{\\mathrm{acc}}\\rangle \\simeq \\dot{M}_{\\rm Edd}$ and $\\Delta M_{\\mathrm{acc}} << \\Delta M_{\\rm donor}$. Even if the accretion onto the NS was allowed to reach a few times the Eddington limit, the net amount of accreted material by the NS in the brightest ULXs would still be small.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\linewidth]{pd.pdf}\n\\caption{(Left) Final orbital periods $(P^{f}_{\\mathrm{orb}})$ for LMXBs\/IMXBs with a $1.3$~M$_\\odot$ NS. The longest final orbital periods result from initial periods $\\gtrsim 10.0$~days and donors with masses less than about 2.75~M$_{\\odot}$. The shortest final orbital periods are for systems that formed the heaviest WDs, originating from donors of $\\gtrsim 3.5$~M$_{\\odot}$. (Right) Same \nas left but for LMXBs\/IMXBs with $2.0$~M$_\\odot$ NSs. }\n\\label{pd}\n\\end{figure*}\n\nThe final orbits in most systems that underwent stable mass transfer are wide because during the mass-transfer phase, after the orbit shrinks initially, the donor evolves to become less massive than the accretor causing the system to widen significantly. These widened orbits can be seen in Fig.~\\ref{pd} which shows that the final orbital periods can be as large as $5$--$7$ times the initial orbital periods. Some of these binaries may be observable by Gaia. \\cite{2019ApJ...886...68A} postulate that Gaia can detect and measure the properties of hidden wide binaries with WD and NS components using astrometric observations. The comparison of NS--WD observations to final orbital parameters from our calculations might help in understanding which binaries have undergone a super-Eddington mass-transfer phase in the past and could have been observed as ULXs, as well as constrain binary evolution physics such as the accretion efficiency.\n\n\n\n\\section{Discussion}\\label{sec:discussion}\n\n\\subsection{Comparison to observations of M82~X-2}\\label{m82x2b}\nSince M82~X-2 has fairly well constrained parameters (see Section~\\ref{m82x2} and Table~\\ref{tab:obs_data}), we can compare it to our results and see if our results help to explain the observations. We look at the middle panels in both rows in Figs.~\\ref{result_1} and \\ref{likelihood}. Since the currently observed donor is more massive than the accretor, the orbit will shrink as mass is lost following the orbital angular momentum balance equation as described in Eq.~(\\ref{analytical}) for a non-conservative mass-transfer phase. Therefore, we use the observed parameters as lower limits of the initial binary configuration.\n\nWe only consider the luminosity range of $100\\,L_{\\rm Edd}$ because it is comparable to the observed luminosity of about $1.8\\times 10^{40}$~erg~s$^{-1}$. \\cite{2014Natur.514..202B} assumed an accretor of $1.4$~M$_{\\odot}$ and estimated the donor to be $\\gtrsim 5.2$~M$_{\\odot}$. For a $1.3$~M$_{\\odot}$ accretor mass, the donor mass is $\\gtrsim 5.1$~M$_{\\odot}$, using the same binary mass function. The initial parameter estimates are enclosed by the white arrows in Figs.~\\ref{result_1} and \\ref{likelihood} (middle panels in both rows). The potential initial systems have ULX lifetimes as long as $0.7\\times 10^4$ years and a relative peak likelihood of $0.7\\times 10^{-4}$. However, if we assume a higher accretor mass of $2.0$~M$_{\\odot}$, the donor mass is $\\gtrsim 5.83$~M$_{\\odot}$. The longest ULX lifetime in this case for the potential initial systems is about $1.1\\times 10^5$ years with a peak likelihood of $0.7\\times 10^{-2}$. Taking these numbers at face value, it is clear that an initially heavy NS is preferred in order to explain the currently observed properties of M82~X-2. This cannot be excluded as the $1.4$~M$_{\\odot}$ NS used by \\cite{2014Natur.514..202B} was an assumption. We should note, however, that for both NS masses, the peak of the relative likelihood does not lie very close to the observed limits we have for the current properties of M82~X-2. This is not necessarily problematic, as in order to calculate the actual probability density distribution of what the properties of NS ULXs ought to be, based on our model, we need to multiply the relative likelihood shown in Fig.~\\ref{likelihood} with the `prior' probability of forming a NS binary with these initial conditions. This convolution might significantly shift the peak of the resulting probability distribution. We leave this calculation for future work. \\cite{2015ApJ...802L...5F}, who followed such an approach, estimated the most probable initial donor mass of any NS ULX to be $3.0$--$8.0$~M$_{\\odot}$, and the initial orbital period to be $1.0$--$3.0$~days. This parameter space lies within our results.\n\n\\subsubsection{Comparison with high-mass X-ray binary ULXs}\n\nWith our work we aim to explore the possibility of an LMXB or IMXB origin for pulsating ULXs. However, there are at least three known NS ULXs which are HMXBs, namely NGC~7793~P13, M51~ULX-7, and more recently NGC 300 ULX1.\n\nNGC~7793~P13 has a luminosity of about $100\\;L_{\\rm Edd}$. With the presence of an observed high-mass donor ($>18\\;M_\\odot$, Section~\\ref{7793} and Table~\\ref{tab:obs_data}), this system cannot be explained by our LMXB\/IMXB models. In our NS ULXs, extreme mass ratios produce unstable binaries which reach the ULX luminosity observed by the source for a timescale of $\\sim 10^2$--$10^4 $~years before initiating L$_2$ overflow. However, this system is a HMXB with a donor of mass $18.0$~M$_{\\odot} \\mu_c$ \\cite{munoz}, which would make the color\/charge conserving vacuum metastable.\n In the presence of such CCB minimum or UFB direction, there are two\npoints to be clarified to make sure that the model is\nphenomenologically viable. One first needs a cosmological scenario\nwhich allows our universe to be settled down at the correct\nvacuum, not at the CCB minimum or UFB direction. The second is\nthat the color\/charge preserving vacuum should be stable enough\nagainst the tunnelling into CCB minimum or UFB direction. The\nfirst point might depend on the detailed history of the early\nuniverse. However in view of that squark\/sleptons get large\npositive mass-squares in the high temperature limit, it is a\nrather plausible assumption that squark\/sleptons are settled down\nat the color\/charge preserving minimum after the inflation\n\\cite{kuzenko}. As for the vacuum stability, it has been noticed\nthat the potential barrier between $\\phi=0$ and $\\phi\\gtrsim\n\\mu_c$ gives a tunnelling rate much less than the Hubble expansion\nrate as long as $\\mu_c\\gtrsim 10$ TeV \\cite{riotto}. In mirage\nmediation with positive $\\alpha={\\cal O}(1)$, once one requires a\nsuccessful electroweak symmetry breaking, $\\mu_c$ is higher than\n$10^3$ TeV, thus satisfies safely the stability condition. In\nthis paper, we do not take the existence of CCB minimum or UFB\ndirection at large squark\/slepton value $|\\phi|\\gg 10$ TeV as a\nserious problem of the model as long as a good color\/charge\npreserving and electroweak symmetry breaking vacuum exists, and\nfocus on the phenomenology of the model under the assumption that\nwe are living in the color\/charge preserving local minimum which\nis stable enough to have a lifetime longer than the age of the\nuniverse.\n\n\n\n\n\n\nThe organization of this paper is as follows. In section II, we\nreview the basic features of mirage mediation. In section III, we\nexamine the prospect of neutralino DM in intermediate scale\nmirage mediation scenario for several different choices of the\nmatter\/Higgs modular weights. In section IV, we extend the\nanalysis to general values of the mirage messenger scale. Section\nV is the conclusion, and Appendix A contains a summary of our\nconvention and notation.\n\n\n\\section{Mirage mediation}\n\n\nMirage mediation is a natural consequence of the KKLT-type moduli\nstabilization scenario satisfying the following two assumptions:\n(i) the modulus $T$ (or dilaton) which determines the standard\nmodel gauge couplings is stabilized by non-perturbative effects\nand (ii) SUSY is broken by a brane-localized source which is\nsequestered from the visible sector. A well known example of such\nset-up is the KKLT moduli stabilization \\cite{kklt} in type IIB\nstring theory\\footnote{As was noticed in \\cite{choi3,hebecker},\ndue to the effect of throat vector multiplet, the sequestering\nmight not be precise enough in the case of KKLT compactification\nof type IIB string theory. The size of non-sequestered soft scalar\nmass induced by the exchange of throat vector multiplet is quite\nsensitive to the unknown details of compactification, however\nthere exists a reasonable parameter limit in which the\nnon-sequestered effects can be safely ignored \\cite{choi3}.} A\nsimilar but simpler example would be 5D brane model with a flat\ninterval in one side and an warped interval in other side, in\nwhich SUSY breaking brane is introduced at the IR fixed point of\nthe warped interval.\n\nUnder these two assumptions, the 4D effective action of the\nvisible sector fields and the gauge coupling modulus $T$ is given\nby\n \\begin{eqnarray}\n \\label{superspace}\n \\int d^4\\theta \\left[-3CC^*e^{-K\/3}\n-C^2C^{*2}{\\cal P}_{\\rm\nlift}\\theta^2\\bar{\\theta}^2\\right]+\\left(\\int d^2\\theta \\left[\n\\frac{1}{4}f_aW^{a\\alpha}W^a_\\alpha+C^3W\\right]+{\\rm h.c.}\n\\right), \\end{eqnarray} where $C=C_0+F^C\\theta^2$ is the chiral compensator\nsuperfield, $f_a$ are the holomorphic gauge kinetic function of\nthe visible sector gauge fields, $K$ and $W$ are the effective\nK\\\"ahler potential and superpotential of the visible matter\nsuperfields $\\Phi_i$ and the gauge coupling modulus $T$, which\nwould be obtained by integrating out heavy moduli. As long as the\nSUSY breaking brane is sequestered from the visible gauge and\nmatter superfields, its low energy consequence can be described by\na simple spurion operator of the form ${\\cal P}_{\\rm\nlift}\\theta^2\\bar{\\theta}^2$, independently of the detailed\n dynamics on the SUSY-breaking brane. Assuming an axionic shift\nsymmetry: \\begin{eqnarray} \\label{shift} U(1)_T:\\quad {\\rm Im}(T)\\rightarrow\n{\\rm Im}(T)+\\mbox{real constant}\\end{eqnarray} which is broken by\nnon-perturbative term in the superpotential, the model is given by\n\\begin{eqnarray}\n\\label{u1tmodel}K&=&K_0(T+T^*)+Z_i(T+T^*)\\Phi^*_i\\Phi_i, \\nonumber \\\\\nW &=& w-Ae^{-aT}+\\frac{1}{6}\\lambda_{ijk} \\Phi_i\\Phi_j\\Phi_k,\n\\nonumber\n\\\\\nf_a&=& kT+\\Delta f_a, \\nonumber \\\\\n{\\cal P}_{\\rm lift}&=& {\\cal P}_{\\rm lift}(T+T^*), \\end{eqnarray} where $a$\nand $k$ are (discrete) real constants, while $\\Delta f_a, w, A$\nand $\\lambda_{ijk}$ are complex effective constants obtained\nafter heavy moduli are integrated out. The axionic symmetry\n(\\ref{shift}) ensures that $K_0$,\n $Z_i$ and ${\\cal P}_{\\rm lift}$ depend only on $T+T^*$, and $a$ and $k$ are real constant.\nWith these features, the resulting gaugino masses and trilinear\n$A$ parameters preserve CP as was pointed out in \\cite{susycp}.\n\nThere might be various ways to generate the modulus superpotential\n$W_0=w-Ae^{-aT}$ stabilizing $T$. Generically the\nnon-perturbative term $e^{-aT}$ can be induced by either a\ngaugino condensation of $T$-dependent hidden gauge interaction or\na stringy instanton whose Euclidean action is controlled by $T$.\nAs for the constant term $w$, it might be induced by a fine-tuned\nconfiguration of fluxes as in the original KKLT scenario\n\\cite{kklt}, or alternatively by $T$-independent non-perturbative\neffect whose strength is controlled by heavy moduli \\cite{luty}.\nAs we will see, the non-perturbative stabilization of $T$ by $W_0$\ngenerates a little hierarchy between the modulus mass and the\ngravitino mass: \\begin{eqnarray} \\frac{m_T}{m_{3\/2}}\\,\\sim\\, aT\\,\\sim\\,\n\\ln(M_{Pl}\/m_{3\/2}), \\end{eqnarray} which results in a little suppression of\nthe modulus $F$-component: \\begin{eqnarray} \\frac{F^T}{T} \\,\\sim\n\\,\\frac{m_{3\/2}^2}{m_T} \\,\\sim\\,\n\\frac{m_{3\/2}}{\\ln(M_{Pl}\/m_{3\/2})}.\\end{eqnarray} Then, $F^T\/T$ naturally\nhas a size comparable to the anomaly mediated soft mass of ${\\cal\nO}(m_{3\/2}\/4\\pi^2)$ for the gravitino mass around TeV. If the SUSY\nbreaking source is sequestered from the visible sector, the soft\nterms of visible fields are determined by the modulus mediation of\n${\\cal O}(F^T\/T)$ and the anomaly mediation of ${\\cal\nO}(m_{3\/2}\/4\\pi^2)$ which are comparable to each other. This leads\nto a mirage unification \\cite{Choi:2005uz} of soft masses at a\nscale hierarchically different from the gauge coupling unification\nscale $M_{GUT}$.\n\n\n\nIn the Einstein frame, the modulus potential from\n(\\ref{superspace}) is given by\n \\begin{eqnarray}\n \\label{moduluspotential}\n V_{\\rm TOT}\n=e^{K_0}\\left[(\\partial_T\\partial_{\\bar{T}}K_0)^{-1}|D_TW_{0}|^2-3|W_{0}|^2\\right]+V_{\\rm\nlift}, \\end{eqnarray} where\n\\begin{eqnarray}\n W_0=w-Ae^{-aT}, \\quad V_{\\rm lift}= e^{2K_0\/3}{\\cal P}_{\\rm\nlift}.\\end{eqnarray} The superspace lagrangian density (\\ref{superspace})\nalso determines the auxiliary components of $C$ and $T$ as\n\\begin{eqnarray}\n\\label{approx-F} \\frac{F^C}{C_0}&=\n&\\frac{1}{3}\\partial_TK_0F^T+m_{3\/2}^*,\n\\nonumber \\\\\nF^T&=\n&-e^{K_0\/2}\\left(\\partial_T\\partial_{T^*}K_0\\right)^{-1}\\left(D_T\nW_{0}\\right)^*, \\end{eqnarray} where $m_{3\/2}= e^{K_0\/2}W_0$. Note that one\ncan always make both $w$ and $A$ real by appropriate $U(1)_R$ and\n$U(1)_T$ transformations. In such field basis, the $U(1)_T$\ninvariance of $K_0$ assures that both $m_{3\/2}$ and $F^T$ are\nreal. In the following, we will use this field basis in which the\nCP invariance of soft parameters is easier to be recognized.\n\n\nTo stabilize $T$ at a reasonably large value while having\n$m_{3\/2}$ hierarchically smaller than $M_{Pl}$, one needs to\nassume that $w$ is hierarchically smaller than $A$ in the unit\nwith $M_{Pl}=1$. Since $w\\sim m_{3\/2}$ and one needs $m_{3\/2}\\sim\n10$ TeV to get the weak scale superparticle masses, $\\ln(A\/w)$\ntypically has a value of ${\\cal O}(4\\pi^2)$. It is then\nstraightforward to compute the vacuum values of $T$ and $F^T$ by\nminimizing the modulus potential (\\ref{moduluspotential}) under\nthe fine tuning condition $\\langle V_{\\rm TOT}\\rangle =0$. At\nleading order in $\\epsilon=1\/\\ln(A\/w)$,\none finds \\cite{choi1} \\begin{eqnarray} \\label{vev} a T &\\simeq &\\ln(A\/w),\\nonumber \\\\\n \\frac{F^T}{T+T^*}\n &\\simeq& \\frac{m_{3\/2}}{\\ln(A\/w)}\\left(1+\\frac{3\\partial_T\\ln({\\cal P}_{\\rm\nlift})}{2\\partial_TK_0}\n \\right)\\,,\\end{eqnarray}\nwhich shows that $F^T\/T$ is indeed of the order of\n$m_{3\/2}\/4\\pi^2$ for $\\ln(A\/w)\\sim 4\\pi^2$.\n\nLet us consider the soft terms of canonically normalized visible\nfields derived from the 4D effective action (\\ref{superspace}):\n\\begin{eqnarray}\n{\\cal L}_{\\rm\nsoft}&=&-\\frac{1}{2}M_a\\lambda^a\\lambda^a-\\frac{1}{2}m_i^2|\\phi_i|^2\n-\\frac{1}{6}A_{ijk}y_{ijk}\\phi_i\\phi_j\\phi_k+{\\rm h.c.},\n\\end{eqnarray}\nwhere $\\lambda^a$ are gauginos, $\\phi_i$ are the scalar component\nof $\\Phi_i$ and $y_{ijk}$ are the canonically normalized Yukawa\ncouplings: \\begin{eqnarray}\ny_{ijk}=\\frac{\\lambda_{ijk}}{\\sqrt{e^{-K_0}Z_iZ_jZ_k}}. \\end{eqnarray} For\n$F^T\/T\\sim m_{3\/2}\/4\\pi^2$,\n the soft parameters at energy scale just below $M_{GUT}$ are\n determined by the modulus-mediated and\n anomaly-mediated contributions which are comparable to each\n other.\n One then finds \\cite{choi1}\n\\begin{eqnarray}\n\\label{soft1} M_a&=& M_0 +\\frac{m_{3\/2}}{16\\pi^2}\\,b_ag_a^2,\n\\nonumber \\\\\nA_{ijk}&=&\\tilde{A}_{ijk}-\n\\frac{m_{3\/2}}{16\\pi^2}\\,(\\gamma_i+\\gamma_j+\\gamma_k),\n\\nonumber \\\\\nm_i^2&=& \\tilde{m}_i^2-\\frac{m_{3\/2}}{16\\pi^2}M_0\\,\\theta_i\n-\\left(\\frac{m_{3\/2}}{16\\pi^2}\\right)^2\\dot{\\gamma}_i\n\\label{eq:bc}\n\\end{eqnarray}\nwhere $M_0$, $\\tilde{A}_{ijk}$ and $\\tilde{m}_i$ are the pure\nmodulus-mediated gaugino mass, trilinear $A$-parameters and\nsfermion masses which are given by\n\\begin{eqnarray}\n\\label{tmediation} M_0&=&F^T\\partial_T\\ln{\\rm Re}(f_a),\n\\nonumber \\\\\n\\tilde{m}_i^2 &=& -F^TF^{T*}\\partial_T\\partial_{\\bar{T}}\n\\ln(e^{-K_0\/3}Z_i),\n\\nonumber \\\\\n\\tilde{A}_{ijk}&=& -F^T\\partial_T\\ln\n \\left(\\frac{\\lambda_{ijk}}{e^{-K_0}Z_iZ_jZ_k}\\right)\n \\,=\\,\n F^T\\partial_T\\ln(e^{-K_0}Z_iZ_jZ_k)\n\\nonumber \\\\\n &=&\\tilde{A}_i+\\tilde{A}_j+\\tilde{A}_k\n \\quad\\,\n \\mbox{for}\\quad\\,\\tilde{A}_i=F^T\\partial_T\\ln(e^{-K_0\/3}Z_i).\n\\end{eqnarray} Here we have used that the holomorphic Yukawa couplings\n$\\lambda_{ijk}$ are $T$-independent constants as a consequence of\nthe axionic shift symmetry $U(1)_T$, and the one-loop beta\nfunction coefficient $b_a$, the anomalous dimension $\\gamma_i$\nand its derivative $\\dot{\\gamma}_i$, and $\\theta_i$ are defined as\n\\begin{eqnarray}\nb_a&=&-3{\\rm tr}\\left(T_a^2({\\rm Adj})\\right)\n +\\sum_i {\\rm tr}\\left(T^2_a(\\phi_i)\\right),\n\\nonumber \\\\\n\\gamma_i&=&2\\sum_a g^2_a C^a_2(\\phi_i)-\\frac{1}{2}\\sum_{jk}|y_{ijk}|^2,\n\\nonumber \\\\\n\\dot{\\gamma}_i&=&8\\pi^2\\frac{d\\gamma_i}{d\\ln\\mu},\\nonumber \\\\\n\\theta_i&=& 4\\sum_a g^2_a C^a_2(\\phi_i)-\\sum_{jk}|y_{ijk}|^2\n\\frac{\\tilde{A}_{ijk}}{M_0},\n \\end{eqnarray} where the\nquadratic Casimir $C^a_2(\\phi_i)=(N^2-1)\/2N$ for a fundamental\nrepresentation $\\phi_i$ of the gauge group $SU(N)$,\n$C_2^a(\\phi_i)=q_i^2$ for the $U(1)$ charge $q_i$ of $\\phi_i$, and\n$\\omega_{ij}=\\sum_{kl}y_{ikl}y^*_{jkl}$ is assumed to be diagonal.\nIn Appendix A, we provide a summary of our convention and\nnotation.\n\nFor our later discussion, it is convenient to define\n\\begin{eqnarray}\n\\alpha\\,\\equiv\\,\\frac{m_{3\/2}}{M_0\\ln(M_{Pl}\/m_{3\/2})},\\quad\n a_i\\,\\equiv\\,\\frac{\\tilde{A}_{i}}{M_0}, \\quad\n c_i\\,\\equiv\\, \\frac{\\tilde{m}_i^2}{M_0^2},\n\\label{eq:def}\n \\end{eqnarray}\n where $\\alpha$ represents the anomaly to modulus mediation\n ratio, while $a_{i}$ and $c_i$ parameterize the pattern of the pure modulus mediated soft masses.\nThen the boundary values of soft parameters at $M_{GUT}$ are given\nby\n\\begin{eqnarray}\nM_a&=& M_0 \\Big[\\,1+\\frac{\\ln(M_{Pl}\/m_{3\/2})}{16\\pi^2} b_a\ng_a^2\\alpha\\,\\Big],\\nonumber \\\\\nA_{ijk}&=&M_0\\Big[\\,(a_i+a_j+a_k)\n-\\frac{\\ln(M_{Pl}\/m_{3\/2})}{16\\pi^2}(\\gamma_i+\\gamma_j+\\gamma_k)\\alpha\\,\\Big],\n\\nonumber \\\\\nm_i^2&=&M_0^2\\Big[\\,c_i-\\,\\frac{\\ln(M_{Pl}\/m_{3\/2})}{16\\pi^2}\n\\theta_i\\alpha-\\left(\\frac{\\ln(M_{Pl}\/m_{3\/2})}{16\\pi^2}\\right)^2\\dot{\\gamma}_i\\alpha^2\\,\\Big],\n\\label{eq:bc1}\n\\end{eqnarray}\nwhere \\begin{eqnarray}\n\\theta_i=4\\sum_a g^2_a C^a_2(\\phi_i)-\\sum_{jk}|y_{ijk}|^2(a_i+a_j+a_k).\\end{eqnarray}\nIn this prescription, generic mirage mediation is parameterized by\n\\begin{eqnarray} M_0,\\,\\, \\alpha,\\,\\, a_i,\\,\\, c_i,\\,\\, \\tan\\beta,\\end{eqnarray} where\nwe have replaced the Higgs mass parameters $\\mu$ and $B$ by\n$\\tan\\beta=\\langle H_u\\rangle\/\\langle H_d\\rangle$ and $M_Z$ as\nusual. As we will see, this parameterization of mirage mediation is\nparticularly convenient when one compute the mirage mediation\nparameters from underlying SUGRA model. For instance, $\\alpha$,\n$a_i$ and $c_i$ are given by simple rational numbers in minimal\nKKLT-type moduli stabilization.\n\n\n\n\nTaking into account the 1-loop RG evolution, the soft masses of\n(\\ref{soft1}) at $M_{GUT}$ leads to low energy soft masses which\ncan be described in terms of the mirage messenger scale: \\begin{eqnarray}\nM_{\\rm mir}=\\frac{M_{GUT}}{(M_{Pl}\/m_{3\/2})^{\\alpha\/2}}. \\end{eqnarray} For\ninstance, the low energy gaugino masses are given by\n\\cite{Choi:2005uz}\\begin{eqnarray} \\label{lowgaugino} M_a(\\mu)=M_0\\left[\\,\n1-\\frac{1}{8\\pi^2}b_ag_a^2(\\mu)\\ln\\left(\\frac{M_{\\rm\nmir}}{\\mu}\\right)\\,\\right] =\\frac{g_a^2(\\mu)}{g_a^2(M_{\\rm\nmir})}M_0, \\end{eqnarray} showing that the gaugino masses are unified at\n$M_{\\rm mir}$, while the gauge couplings are unified at $M_{GUT}$.\nThe low energy values of $A_{ijk}$ and $m_i^2$ generically depend\non the associated Yukawa couplings $y_{ijk}$. However if $y_{ijk}$\nare small enough or \\begin{eqnarray}\\label{con1} a_i+a_j+a_k=c_i+c_j+c_k=1\n\\quad \\mbox{for} \\quad y_{ijk}\\sim 1,\\end{eqnarray} their low energy values\nare given by \\cite{Choi:2005uz}\\begin{eqnarray} \\label{lowsfermion}\nA_{ijk}(\\mu)&=& M_0\\left[\\,a_i+a_j+a_k+\n\\frac{1}{8\\pi^2}(\\gamma_i(\\mu)\n+\\gamma_j(\\mu)+\\gamma_k(\\mu))\\ln\\left(\\frac{M_{\\rm\nmir}}{\\mu}\\right)\\,\\right], \\nonumber \\\\\nm_i^2(\\mu)&=&M_0^2\\left[\\,c_i-\\frac{1}{8\\pi^2}Y_i\\left(\n\\sum_jc_jY_j\\right)g^2_Y(\\mu)\\ln\\left(\\frac{M_{GUT}}{\\mu}\\right)\\right.\n\\nonumber \\\\\n&+&\\left.\\frac{1}{4\\pi^2}\\left\\{\n\\gamma_i(\\mu)-\\frac{1}{2}\\frac{d\\gamma_i(\\mu)}{d\\ln\\mu}\\ln\\left(\n\\frac{M_{\\rm mir}}{\\mu}\\right)\\right\\}\\ln\\left( \\frac{M_{\\rm\nmir}}{\\mu}\\right)\\,\\right], \\end{eqnarray} where $Y_i$ is the $U(1)_Y$\ncharge of $\\phi_i$. Quite often, the modulus-mediated squark and\nslepton masses have a common value, i.e.\n$c_{\\tilde{q}}=c_{\\tilde{\\ell}}$. Then, according to the above\nexpression of low energy sfermion mass, the 1st and 2nd generation\nsquark and slepton masses are unified again at the mirage\nmessenger scale $M_{\\rm mir}$.\n\n\n\nIn KKLT compactification of type IIB string theory \\cite{kklt},\n$T$ corresponds to the Calabi-Yau volume modulus, and the\nuplifting brane is located at the end of warped throat. In this\ncase, the uplifting operator is sequestered from $T$ \\cite{choi1}:\n\\begin{eqnarray} \\label{sequestered}\\partial_T {\\cal P}_{\\rm lift}=0. \\end{eqnarray}\nAlso, the minimal KKLT compactification of IIB theory gives \\begin{eqnarray}\n\\label{minimal} K_0&=&-3\\ln(T+T^*), \\quad\nZ_i\\,=\\,\\frac{1}{(T+T^*)^{n_i}}, \\nonumber \\\\\nf_a&=& kT, \\quad W_0=w-Ae^{-aT} \\quad (A\\,=\\, {\\cal O}(1)),\\end{eqnarray}\nwhere the modular weight $n_i$ is a rational number depending on\nthe origin of matter superfield $\\Phi_i$. Then from\n(\\ref{tmediation}) and (\\ref{vev}), one immediately finds \\begin{eqnarray}\n\\label{benchmark} \\alpha\\,=\\, 1,\\quad a_i\\,=\\, c_i\\,=\\, 1-n_i,\n\\end{eqnarray} giving an intermediate mirage messenger scale: \\begin{eqnarray} M_{\\rm\nmir}\\,\\sim\\, 3\\times 10^9 \\,\\, \\mbox{GeV}. \\end{eqnarray}\n If $\\Phi_i$ lives on the entire world-volume of $D7$ brane\nfrom which the visible gauge bosons originate, the corresponding\nmodular weight $n_i=0$. However, if $\\Phi_i$ is confined in the\nintersections of $D7$ branes, $n_i$ has a positive value, e.g.\n$n_i=1\/2$ or 1. In Fig.~\\ref{fig:rge}, we depict the RG evolution\nof gauge couplings and soft masses in intermediate scale mirage\nmediation scenario with $n_i=0$, i.e. $\\alpha=a_i=c_i=1$, which\nshows that indeed the gaugino masses and the 1st and 2nd\ngenerations of squarks and slepton masses are unified at $M_{\\rm\nmir}\\sim 3\\times 10^9$ GeV as indicated by (\\ref{lowgaugino}) and\n(\\ref{lowsfermion}). Note that still the gauge couplings are\nunified at the conventional GUT scale $M_{GUT}\\sim 2\\times\n10^{16}$ GeV. In view of its minimality, intermediate scale mirage\nmediation can be considered as a benchmark scenario, thus we\nperform a detailed analysis of neutralino DM in intermediate scale\nmirage mediation in the next section.\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=7cm,width=7cm]{gauge_tb10M0800t173dmkklt.eps}\n\\includegraphics[height=7cm,width=7cm]{kklt3_tb10M0800t173.eps}\n\\includegraphics[height=7cm,width=7cm]{kklt_tb10M0800t173.eps}\n\\includegraphics[height=7cm,width=7cm]{kklt2_tb10M0800t173.eps}\n\\end{center}\n\\vskip -0.5cm \\caption{RG evolution of (a) gauge couplings\n$\\alpha_i$, (b) gaugino masses $M_a$, (c) sfermion and Higgs masses\n$m_i$, (d) trilinear $A$ parameters in intermediate scale mirage\nmediation with $a_i=c_i=1$. Here we choose $M_0=800$ GeV and\n$\\tan\\beta=10$.} \\label{fig:rge}\n\\end{figure}\n\nIt is in fact easy to generalize the compactification to get\ndifferent values of the mirage mediation parameters $\\alpha$,\n$a_i$ and $c_i$. For instance, the compactification can be\ngeneralized to have a dilaton-modulus mixing in gauge kinetic\nfunctions \\cite{ck,lust} and\/or the non-perturbative\nsuperpotential \\cite{abe}. For the case of type IIB\ncompactification, a nonzero gauge flux on $D7$ branes can generate\nsuch dilaton-modulus mixing in $D7$ gauge kinetic functions\n\\cite{lust}, which would result in \\begin{eqnarray} f_a\\,=\\,kT+lS_0, \\quad\nW_0\\,=\\,w-Ae^{-(aT+bS_0)}\\quad (A={\\cal O}(1)),\\end{eqnarray} where $S_0$\ndenotes the vacuum value of the string dilaton $S$ which is\nassumed to get superheavy mass from RR and NS-NS 3-form fluxes,\nand $k,l,a$ and $b$ are real parameters. In such IIB\ncompactifications, the coefficients of $T$ in $D7$ gauge kinetic\nfunctions and non-perturbative superpotential, i.e. $k$ and $a$,\nare positive, however the coefficients of $S$, i.e. $l$ and $b$,\nmight have both signs under the conditions:\\begin{eqnarray}\n\\label{condition}k{\\rm Re}(T)+l{\\rm Re}(S_0)&\\simeq&\n\\frac{1}{g_{GUT}^2}\\,\\simeq\\, 2,\n\\nonumber \\\\\na{\\rm Re}(T)+b{\\rm Re}(S_0)&\\simeq& \\ln(M_{Pl}\/m_{3\/2})\\,\\simeq\\,\n4\\pi^2.\\end{eqnarray} Assuming that $e^{-K_0\/3}Z_i$ has the same form as the\nminimal model (\\ref{minimal}), it is then straightforward to find\nthat the mirage mediation\nparameters are given by \\begin{eqnarray}\\label{general} \\alpha&=&\\frac{1+R_1}{(1+R_2)(1+R_3)},\\nonumber \\\\\na_i&=& (1-n_i)(1+R_1),\\nonumber \\\\\nc_i&=& (1-n_i)(1+R_1)^2 \\end{eqnarray} where \\begin{eqnarray} R_1=\\frac{l{\\rm\nRe}(S_0)}{k{\\rm Re}(T)}, \\quad R_2=\\frac{b{\\rm Re}(S_0)}{a{\\rm\nRe}(T)},\\quad R_3=\\frac{3\\partial_T\\ln({\\cal P}_{\\rm\nlift})}{2\\partial_T K_0}.\\end{eqnarray}\n Again, for ${\\cal P}_{\\rm lift}$\ninduced by an uplifting brane at the end of warped throat,\n$R_3=0$. However, $1+R_{1}$ and $1+R_2$ can have a variety of\n(positive) values under the condition (\\ref{condition}). As a\nresult, the anomaly to modulus mediation ratio $\\alpha$ can easily\nhave any (positive) value within the range of order unity.\n\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=7cm,width=7cm]{mirage3_tb10M0800t173.eps}\n\\includegraphics[height=7cm,width=7cm]{mirage_tb10M0800t173.eps}\n\\end{center}\n\\vskip -0.5cm \\caption{RG evolution of (a) gaugino masses and (b)\nsfermion and Higgs masses in TeV scale mirage mediation. Here we\nfixed $M_0=800$ GeV, $a_{\\rm H}=c_{\\rm H}=0$ and $a_{\\rm M}=c_{\\rm\nM}=1\/2$.} \\label{fig:rge2}\n\\end{figure}\n\n\nA particularly interesting model is the TeV scale mirage mediation\nwith \\begin{eqnarray} \\alpha\\,=\\,2,\\quad a_{H_u}\\,=\\, c_{H_u}\\,=\\, 0, \\quad\na_{Q_3}+a_{U_3}=c_{Q_3}+c_{U_3}=1, \\end{eqnarray} where\n$Q_3$ and $U_3$ denote the left-handed and right handed top-quark\nsuperfields, respectively. This particular model has been claimed\nto minimize the fine tuning for the electroweak symmetry breaking\nin the MSSM \\cite{tevmirage}. In Fig.~\\ref{fig:rge2}, we depict\nthe RG evolution of soft masses in TeV scale mirage mediation\nmodel with $a_{\\rm H}=c_{\\rm H}=0$ and $a_{\\rm M}=c_{\\rm M}=1\/2$,\nwhere the subscripts H and M stands for the MSSM Higgs doublets\n$H_{u,d}$ and the quark\/lepton matter superfields, respectively.\n\nNote that the squark\/slepton mass-squares renormalized at high\nenergy scale, e.g. at a scale near $M_{GUT}$, are negative in\nthis model, while the values at low energy scale below $10^6$ GeV\nare positive.\n In fact,\nsuch tachyonic {\\it high energy} squark\/slepton mass-squares is a\ngeneric feature of mirage mediation for $\\alpha>\\alpha_c$ where\nthe precise value of $\\alpha_c$ depends on $a_i$ and $c_i$, but\nnot significantly bigger than 1 in most cases. As long as the low\nenergy squark\/slepton mass-squares are positive, the model has a\ncorrect color\/charge preserving (but electroweak symmetry\nbreaking) vacuum. For instance, the TeV scale mirage mediation\nmodel of Fig.~\\ref{fig:rge2} has a such vacuum which is a local\nminimum of the scalar potential over the squark\/slepton values\n$|\\phi|\\lesssim 10^6$ GeV. On the other hand, tachyonic squark\nmass-squares at the RG point $\\mu > 10^6$ GeV indicates that there\nmight be a deeper CCB minimum color\/charge breaking (CCB) or an\nunbounded from below (UFB) direction at $|\\phi|> 10^6$ GeV. One\nthen needs a cosmological scenario which allows our universe to be\nsettled down at the correct vacuum with $\\phi=0$. In view of that\nthe squarks and sleptons get large positive mass-squares in the\nhigh temperature limit, it is rather plausible assumption that\nsquark\/sleptons are settled down at the color\/charge preserving\nminimum after the inflation \\cite{kuzenko}. One still needs to\nconfirm that the color\/charge preserving vacuum is stable enough\nagainst the decay into CCB vacuum. It has been noticed that the\ncorresponding tunnelling rate is small enough, i.e. less than the\nHubble expansion rate, as long as the RG points of vanishing\nsquark\/slepton mass-squares are all higher than $10^4$ GeV\n\\cite{riotto,kuzenko}, which is satisfied safely by the TeV scale\nmirage mediation of Fig.~\\ref{fig:rge2}.\n\n\n\n\n\n\\section{Neutralino DM in intermediate scale mirage mediation}\n\n\nIn this section, we examine the prospect of neutralino DM in\nintermediate scale mirage mediation scenario. As was noticed in\nthe previous section, the minimal KKLT-type model (\\ref{minimal})\nwith a sequestered uplifting brane gives $\\alpha=1$, thus an\nintermediate mirage messenger scale \\begin{eqnarray} M_{\\rm mir}\\sim\nM_{GUT}(m_{3\/2}\/M_{Pl})^{1\/2}\\sim 3\\times 10^9\\,\\, {\\rm GeV}.\\end{eqnarray}\nIn this minimal set-up, the discrete parameters $a_i$ and $c_i$\ndescribing the modulus mediated $A$-parameters and sfermion masses\nare determined to be $a_i=c_i=1-n_i$, where $n_i$ denote the\nmodular weights of matter and Higgs superfields. Throughout this\npaper, we will assume $a_i=c_i$ and consider the following four\ndifferent cases: \\begin{eqnarray} \\label{4choices} (a_{\\rm H}=c_{\\rm H},\\,\na_{\\rm M}=c_{\\rm M})=(1,\\,1), \\quad (\\frac{1}{2},\\,\\frac{1}{2}),\n\\quad (0,\\,1), \\quad (0,\\,\\frac{1}{2}). \\end{eqnarray} We also choose the\nHiggsino mass parameter $\\mu > 0$ in light of the experimental\nvalue of the muon anomalous magnetic moment which favors positive\n$\\mu$ \\cite{g-2}, and treat $\\tan\\beta=\\langle H_u\\rangle\/\\langle\nH_d\\rangle$ as a free parameter without specifying the origin of\nthe corresponding $\\mu$ and $B$ parameters. We then obtain the\nparameter range of the model for which the LSP is the lightest\nneutralino as well as the relic neutralino DM abundance under the\nassumption of thermal production, and finally the direct and\nindirect detection rates of the neutralino LSP using the DarkSUSY\nroutine \\cite{darksusy}.\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=7cm,width=7cm]{t-m0.eps}\n\\hskip 1cm\n\\includegraphics[height=7cm,width=7cm]{t-m0-nq05-nh05.eps}\n\\vskip 1cm\n\\includegraphics[height=7cm,width=7cm]{t-m0-nq0-nh1.eps}\n\\hskip 1cm\n\\includegraphics[height=7cm,width=7cm]{t-m0-nq05-nh1.eps}\n\\end{center}\n\\vskip -0.4cm \\caption{ Parameter region of neutralino LSP and the\nthermal relic density depicted on the plane of $(\\tan\\beta, M_0)$\nin intermediate scale mirage mediation models with $(a_i,c_i)$\nspecified in Eq.~(3.2).\n}\n\\label{fig:density}\n\\end{figure}\n\n\\subsection{Parameter region of neutralino LSP and thermal relic density}\n\nIn Fig.~\\ref{fig:density}, we show the neutralino LSP region and\nthe thermal neutralino relic density in intermediate scale mirage\nmediation scenario on the ($\\rm{tan}\\beta$,$M_0$)-plane for the\nvalues of\n $a_i$ and $c_i$ specified in Eq.~(\\ref{4choices}).\nWe computed the sparticle mass spectrum at the electroweak scale\nby solving the RG equations with the boundary condition\n(\\ref{soft1}) at $M_{GUT}$. Our results show that in all cases\nthere is a large parameter region for which the LSP is given by\nthe lightest neutralino.\n\n\nFig.~\\ref{fig:density}.a is the result for the case in which\n$a_i=c_i=1$ for both matter and Higgs multiplets. In this case,\nlarge $\\rm{tan}\\beta >34$ (grey color) for which the tau Yukawa\ncouplings becomes sizable gives stau LSP (see also Fig.\n\\ref{fig:higgs}.a), while small $M_0\\lesssim0.5~ \\rm TeV$ (green color)\ngives stop LSP. In the remaining region, the LSP is the lightest\nneutralino which turns out to be Bino-like. Small\n$\\tan\\beta\\lesssim 3$ is excluded by the Higgs mass limit $m_h >\n114$ GeV.\nUnder the assumption that the DM neutralinos are produced purely\nby the conventional thermal production mechanism, the magenta stripe corresponds to\nthe parameter region giving a relic neutralino density consistent with the recent WMAP observation\n\\cite{wmap}:\n\\begin{eqnarray}\n0.085 < \\Omega_{\\rm DM} h^2 < 0.119 ~~(2\\sigma~ \\rm level).\n\\label{eq:wmap}\n\\end{eqnarray}\nIn the region below the magenta stripe, $\\Omega_\\chi h^2 < 0.085$,\nwhile $\\Omega_\\chi h^2 > 0.119$ for the upper region. Thus the\n(cyan) region below the magenta stripe (but above the stop LSP\nregion) can be phenomenologically viable if additional DM\nneutralinos were produced by non-thermal mechanism such as the\ndecays of flaton in thermal inflation \\cite{stewart}.\n\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=7cm,width=7cm]{mass.eps}\n\\hskip 1cm\n\\includegraphics[height=7cm,width=7cm]{nzmw-alpha1-mass.eps}\n\\vskip 1cm\n\\includegraphics[height=7cm,width=7cm]{nzmw2-alpha1-mass.eps}\n\\hskip 1cm\n\\includegraphics[height=7cm,width=7cm]{nzmw3-alpha1-mass.eps}\n\\end{center}\n\\vskip -0.5cm \\caption{ Particle masses as a function of $\\tan\n\\beta$ in intermediate scale mirage mediation models with\n$(a_i,c_i)$ specified in Eq.~(3.2).\nHere, we fixed $M_0=800$ GeV.}\n\\label{fig:higgs}\n\\end{figure}\n\n\n\n\n\nAlthough the neutralino LSP is Bino-like in this particular\nintermediate scale mirage mediation, the WMAP mass density is\nobtained for a rather heavy neutralino mass $m_{\\chi^0}\\gtrsim\n450$ GeV. This can be understood by Fig.~\\ref{fig:higgs}.a which\nshows the masses of the lightest neutralino, lighter stop and\nstau, and also the pseudo-scalar Higgs boson as a function of\n$\\tan\\beta$ for the model with $\\alpha=1$, $a_i=c_i=1$ and\n$M_0=800$ GeV. Around $\\tan\\beta\\sim 20$, the pseudoscalar Higgs\nmass becomes same as $2m_{\\chi^0}$, leading to a resonant\nenhancement of neutralino annihilation through the s-channel\npseudo-scalar Higgs exchange. For other values of $\\tan\\beta$, the\nneutralino mass is somewhat close to the stop mass (or to the\nstau mass at $\\tan\\beta\\sim 34$), making the stop-neutralino or\nstau-neutralino coannihilation process becomes efficient. In\nFig.~\\ref{fig:coan}, we depicted $\\Omega_\\chi h^2$ as a function\nof $\\rm tan\\beta$ for $M_0 = 800$ GeV, which shows clearly the\neffect of Higgs resonance at $\\tan\\beta\\sim 20$ and also the\neffect of stop\/stau coannihilation effects for other values of\n$\\tan\\beta$. On the plot, the dotted line corresponds to the relic\ndensity computed without including coannihilation effects. It\nindicates that the stop\/stau-neutralino coannihilation plays a\ncrucial role for the relic neutralino density to have the WMAP\nvalue (\\ref{eq:wmap}) for $M_0=700\\sim 800$ GeV and $\\tan\\beta$\noutside the Higgs resonance region. Note that in intermediate\nscale mirage mediation with $a_i=c_i=1$, the pseudoscalar Higgs\nresonance condition $m_A \\simeq 2 m_\\chi$ is satisfied for smaller\nvalue of $\\rm tan\\beta$ compared to the mSUGRA case. This can be\nunderstood by noting that the low energy gaugino masses in mirage\nmediation are more compressed compared to mSUGRA, e.g. $M_3 \/M_1\n\\sim 2.3$ in the intermediate scale mirage mediation with $\\alpha\n= 1$, while $M_3 \/M_1 \\sim 6$ in mSUGRA. For a given value of\n$M_1$, smaller $M_3$ gives smaller $\\mu$ and $m_A^2 \\sim m_{H_d}^2\n+ \\mu^2$ at the electroweak scale, thus the pseudoscalar Higgs\nresonance appears at smaller value of $\\rm tan\\beta$ compared to\nmSUGRA case.\n\n\n\\vskip 0.8cm\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=7cm,width=7cm]{oh2.eps}\n\\end{center}\n\\vskip -0.5cm \\caption{$\\Omega_\\chi h^2$ as a function of\n$\\tan\\beta$ in intermediate scale mirage mediation with\n$a_i=c_i=1$ and $M_0=800$ GeV. Here the dotted line corresponds to\nthe result computed without including the stop\/stau coannihilation\neffects.} \\label{fig:coan}\n\\end{figure}\n\n\n\n\n\n\nSo far, we have been focusing on the specific intermediate scale\nmirage mediation model with $a_i=c_i=1$ which might be obtained\nwhen all modular weights $n_i=0$. However as anticipated in the\nprevious section, $a_i=c_i=1$ is not necessarily a more favored\nchoice than the other values of $(a_i,c_i)$ in\nEq.~(\\ref{4choices}). Different values of $a_i$ and $c_i$, e.g.\nsmaller but still non-negative values, are also equally plausible.\nObviously, for a fixed value of $M_0$, the gaugino masses are not\naffected by changing $a_i$ and $c_i$.\nHowever the low energy stop, stau and Higgs masses are somewhat\nsensitive to the values of $a_i$ and $c_i$. They depend on $a_i$\nand $c_i$ either through their boundary values at $M_{GUT}$, or\nthrough their RG evolutions, or through the mass-mixing induced by\nthe low energy $A$-parameters.\n\nThe effects of changing $a_i$ and $c_i$ on the RG evolution can be\nread off from the following one-loop RG equations for the Higgs\nand third generation sfermion mass-squares:\n\\begin{eqnarray}\n\\label{rgequation} 16\\pi^2{d\\over dt} m_{H_u}^2 &=& 3 X_t -6 g_2^2\n|M_2|^2\n- {6\\over 5} g_1^2 |M_1|^2 ,\\nonumber \\\\\n16\\pi^2{d\\over dt} m_{H_d}^2 &=& 3 X_b + X_\\tau-6 g_2^2 |M_2|^2\n- {6\\over 5} g_1^2 |M_1|^2 ,\\nonumber \\\\\n16\\pi^2{d\\over dt} m_{Q_3}^2 &=& X_t+X_b-{32\\over 3} g_3^2 |M_3|^2\n-6 g_2^2 |M_2|^2-{2\\over 15} g_1^2 |M_1|^2, \\nonumber \\\\\n16\\pi^2{d\\over dt} m_{U_3}^2 &=& 2 X_t-{32\\over 3} g_3^2 |M_3|^2\n-{32\\over 15} g_1^2 |M_1|^2, \\nonumber \\\\\n16\\pi^2{d\\over dt} m_{L_3}^2 &=& X_\\tau\n-6 g_2^2 |M_2|^2-{3\\over 5} g_1^2 |M_1|^2, \\nonumber \\\\\n16\\pi^2{d\\over dt} m_{E_3}^2 &=& 2 X_\\tau -{24\\over 5} g_1^2\n|M_1|^2,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nX_t &=& 2 y_t^2 (m_{H_u}^2+m_{Q_3}^2+m_{U_3}^2+ A_{H_uQ_3U_3}^2), \\nonumber \\\\\nX_b &=& 2 y_b^2 (m_{H_d}^2+m_{Q_3}^2+m_{D_3}^2+ A_{H_dQ_3D_3}^2), \\nonumber \\\\\nX_\\tau &=& 2 y_\\tau^2 (m_{H_d}^2+m_{L_3}^2+m_{E_3}^2+\nA_{H_dL_3E_3}^2).\n\\end{eqnarray}\nThese RG equations show that smaller $X_I$ ($I=t,b,\\tau$) increase\nthe low energy soft mass-squares.\n Since $a_i$ and $c_i$ determine\n the modulus-mediated trilinear $A$\nparameters and soft mass-squares at $M_{GUT}$ as\n$\\tilde{A}_{ijk}=(a_i+a_j+a_k)M_0$ and $\\tilde{m}_i^2=c_iM_0^2$,\nsmaller $a_{\\rm H}=c_{\\rm H}$ give smaller $X_I$ without\naffecting the boundary values of squark and slepton masses at\n$M_{GUT}$, eventually making the stop and stau masses at TeV scale\nlarger. On the other hand, the consequence of smaller $a_{\\rm\nM}=c_{\\rm M}$ is more complicate as it depends on the relative\nimportance of the Yukawa-induced RG evolution. It turns out that\nchanging $a_{\\rm M}=c_{\\rm M}$ to smaller value makes the stop\nmass larger, while the stau mass smaller.\n\nIn Figs.~\\ref{fig:density}.b and \\ref{fig:higgs}.b, we depict the\nresults for the case in which $a_i=c_i=1\/2$ for both the matter\nand Higgs multiplets. As can be understood from the above\ndiscussion, this intermediate scale mirage mediation\ndoes not contain any parameter region of stop LSP, while having a\nlarger parameter region of stau LSP (see Fig.~\\ref{fig:higgs}.b).\nAnother important feature is that the weak scale value of\n$|\\,m_{H_u}^2|$ becomes smaller compared to the case of\n$a_i=c_i=1$, which is mainly due to smaller $X_t$.\n This results in smaller $\\mu$ and $m_A$. Smaller $\\mu$ makes the\nneutralino LSP have a sizable Higgsino component, while smaller\n$m_A$ makes the pseudo-scalar resonance region disappear. Again\nthe magenta region in Fig. \\ref{fig:density}.b corresponds to the\nparameter region giving the WMAP DM density (\\ref{eq:wmap}) under\nthe assumption of pure thermal production. In this case, the\nneutralino pair annihilation into gauge boson pair becomes\nefficient due to the enhanced Higgsino component of neutralino\nLSP. Finally, the brown region is excluded by giving the Br($b \\to\ns \\gamma$) smaller than the allowed range.\n\nFigs.~\\ref{fig:density}.c and \\ref{fig:higgs}.c are the result for\nthe case with $a_{\\rm M}=c_{\\rm M}=1$ and $a_{\\rm H}=c_{\\rm H}=0$,\nwhile Figs.~\\ref{fig:density}.d and \\ref{fig:higgs}.d are for the\ncase with $a_{\\rm M}=c_{\\rm M}=1\/2$ and $a_{\\rm H}=c_{\\rm H}=0$.\nThe case of $a_{\\rm M}=c_{\\rm M}=1\/2$ and $a_{\\rm H}=c_{\\rm H}=0$\nis quite similar to the case of $a_i=c_i=1\/2$: LSP is the lightest\nneutralino with a sizable Higgsino component for\n$\\tan\\beta\\lesssim 20$ (see Figs.~\\ref{fig:higgs}.b and\n\\ref{fig:higgs}.d). On the other hand, the case of $a_{\\rm\nM}=c_{\\rm M}=1$ and $a_{\\rm H}=c_{\\rm H}=0$ is somewhat\ndistinctive since there is no parameter region of stop or stau LSP\nand the WMAP DM density is obtained for a light neutralino mass\n$m_{\\chi^0}\\sim 250$ GeV, while in other cases the WMAP DM density\nis obtained for heavier $m_{\\chi^0}\\gtrsim 350$ GeV. Again, the\nbrown region is excluded by giving the Br($b \\to s \\gamma$)\nsmaller than the allowed range.\n\n\n\n\n\\subsection{Dark matter detections}\n\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=7cm,width=7cm]{mn-sigsi.eps}\n\\hskip 1cm\n\\includegraphics[height=7cm,width=7cm]{nzmw-alpha1-si.eps}\n\\vskip 1.2cm\n\\includegraphics[height=7cm,width=7cm]{nzmw2-alpha1-si.eps}\n\\hskip 1cm\n\\includegraphics[height=7cm,width=7cm]{nzmw3-alpha1-si.eps}\n\\end{center}\n\\vskip -0.4cm \\caption{Scatter plot of the spin-independent\nneutralino-proton scattering cross section vs. $m_\\chi$ in\nintermediate scale mirage mediation with $(a_i,c_i)$ specified in\nEq.~(3.2).\n}\n\\label{fig:sigsip}\n\\end{figure}\n\nIf neutralino LSP is the main component of the matter budget in\nthe Milky Way, it might be detected through the elastic scattering\nwith terrestrial nuclear target \\cite{goodman,lspdm}. In the MSSM,\n$t$-channel Higgs boson and $s$-channel squark exchange processes\ncontribute to the spin-independent (scalar) scattering between\nneutralino and nuclei. In many cases, dominant contribution to the\nscalar cross section comes from the Higgs exchange process which\nbecomes bigger for larger $\\tan\\beta$, smaller Higgs masses, and\nmixed Bino-Higgsino LSP.\n\n\n\n\nIn the specific intermediate scale mirage mediation model with\n$a_i=c_i=1$, the neutralino LSP is Bino-like and the mass of heavy\nCP-even Higgs boson is rather large when we require the neutralino\nto be LSP. It is thus expected that the elastic scattering cross\nsection between neutralino DM and nuclei is rather small. In\nFig.~\\ref{fig:sigsip}.a, we depict spin-independent (scalar) cross\nsection $\\sigma_{SI}$ of neutralino-proton scattering as a\nfunction of the LSP neutralino mass in this specific intermediate\nscale mirage mediation. Here we have imposed the experimental\nbounds on the Higgs\/sparticle masses and $b \\rightarrow s \\gamma$\nbranching ratio, and required that the lightest neutralino is the\nLSP. Red points in the figure correspond to the parameter values\ngiving the WMAP DM density (\\ref{eq:wmap}) under the assumption of\npure thermal production, while the cyan points represent the\nparameter values for which the thermal production mechanism gives\na smaller relic density. As expected, the cross section in the\ncase of $a_i=c_i=1$ is quite small: $\\sigma_{SI}\\lesssim 5 \\times\n10^{-9}$ pb, which is much smaller than the current experimental\nupper bound. It is even smaller than the sensitivity of future\nexperiment such as SuperCDMS \\cite{supercdms} which would reach\nnear $10^{-9}$ pb level. On the other hand, intermediate scale\nmirage mediations with different values of $(a_i,c_i)$ have a\nquite better prospect for direct detection. As can be seen from\nFig.~\\ref{fig:sigsip}, most of the (red) WMAP points are above the\nsensitivity of SuperCDMS for the other three cases of different\n$(a_i,c_i)$. This is mainly due to the enhanced Higgsino component\nof the neutralino LSP and the reduced Higgs mass.\n\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=7cm,width=7cm]{mn-gamma.eps}\\hskip 1cm\n\\includegraphics[height=7cm,width=7cm]{nzmw-alpha1-gamma.eps}\n\\vskip 1cm\n\\includegraphics[height=7cm,width=7cm]{nzmw2-alpha1-gamma.eps}\n\\hskip 1cm\n\\includegraphics[height=7cm,width=7cm]{nzmw3-alpha1-gamma.eps}\n\\end{center}\n\\vskip -0.5cm \\caption{Scatter plot of the continuum gamma ray\nflux vs. $m_\\chi$ in intermediate scale mirage mediation with\n$(a_i,c_i)$ specified in Eq.~(3.2).\n}\n\\label{fig:gamma}\n\\end{figure}\n\n\nLet us now examine gamma ray signals from DM annihilation in the\ngalactic center, providing another feasible but indirect\ndetection method for dark matter. The integrated gamma ray flux\ndepends on the quantity $\\bar{J} (\\Delta\\Omega)$, which is a\nmeasure of the cuspiness of the galactic halo density profile over\na spherical region of solid angle $\\Delta\\Omega$. In this paper,\nwe use a conservative galactic halo model (isothermal halo density\nprofile) which gives $\\bar{J} \\sim 30$ with the detector angular\nresolution $\\Delta\\Omega = 10^{-3}$ sr and set $E_{thr} = 1$ GeV\nfor gamma ray energy threshold. Fig.~\\ref{fig:gamma} shows\ncontinuum gamma ray flux from the galactic center in intermediate\nscale mirage mediation scenarios under consideration, where red\npoints give the WMAP value (\\ref{eq:wmap}) of the relic DM\ndensity. Here the four different choices of $a_i=c_i$ do not lead\nto a dramatic difference in the gamma ray flux. The maximal value\nof flux given by the most favored WMAP (red) points is about ${\\rm\nfew}\\times 10^{-11} \\rm{cm^{-2} s^{-1}}$ which is somewhat below\nthe expected reach ($\\sim 10^{-10}\\rm{cm^{-2} s^{-1}}$) of GLAST,\nalthough the (cyan) points giving smaller relic density can give a\nlarger flux around $10^{-10}\\rm{cm^{-2} s^{-1}}$. However, it\nshould be noticed that our calculation for the gamma ray flux is\nbased on a conservative halo density profile. If one uses an\nextreme halo model like the spiked profile \\cite{moore}, the\nresulting gamma ray flux increases by a factor of $\\sim 10^4$. In\nthis case, the gamma ray signals can be detected for a significant\nportion of the parameter space. A caveat is that the continuum\ngamma ray signals suffer from unknown astrophysical background.\nRecent observations of a bright gamma ray source in the direction\nof galaxy center by the Air Cherenkov Telescopes such as H.E.S.S.\n\\cite{hess} might be explained by an astrophysical process rather\nthan the dark matter annihilation \\cite{hooper}.\n\nWe finally notice an interesting enhancement of the gamma ray flux\ndue to the Higgs resonance effect. Fig.~\\ref{fig:gammas} shows the\ngamma ray flux from the galactic center as a function of $\\rm\ntan\\beta$ in the specific intermediate scale mirage mediation with\n$a_i=c_i=1$ and $M_0=800$ GeV. One can see a clear enhancement of\nthe flux around $\\rm tan\\beta \\sim 22$ for which $m_A \\sim 2\nm_\\chi$. In this case, neutralino annihilation to heavy quarks is\ndominated and the subsequent quark hadronization produces many\ngamma rays.\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=7cm,width=7cm]{gamma.eps}\n\\end{center}\n\\vskip -0.5cm \\caption{Continuum gamma ray flux as a function of\n$\\tan\\beta$ in the intermediate scale mirage mediation with\n$a_i=c_i=1$ and $M_0=800$ GeV. Note the resonant peak due to the\npsuedo-scalar Higgs resonance.} \\label{fig:gammas}\n\\end{figure}\n\n\n\n\n\\section{Neutralino DM for generic mirage messenger scale}\n\n\nIn the previous section, we have examined the prospect of\nneutralino DM in intermediate scale mirage mediation models\n($\\alpha=1$). As was discussed in section 2, in string\ncompactifications with non-trivial dilaton-modulus mixing, the\nanomaly to modulus mediation ratio $\\alpha$ can have a more\nvariety of values. In fact, the nature of neutralino LSP is\nsomewhat sensitive to the value of $\\alpha$, typically it changes\nfrom Bino-like to Higgsino-like via Bino-Higgsino mixing region\nwhen $\\alpha$ is increased from zero to a value of order unity.\nThis feature is essentially due to the following behavior of the\ngaugino masses as a function of $\\alpha$: \\begin{eqnarray} M_3:M_2:M_1\\simeq\n (1-0.3\\alpha)g_3^2:(1+0.1\\alpha)g_2^2:(1+0.66\\alpha)g_1^2,\n \\end{eqnarray}\nIf $\\alpha$ increases from zero, the gluino mass decreases as\n$M_3\\propto (1-0.3\\alpha)$. Smaller $M_3$ then weakens the\nradiative electroweak symmetry breaking mechanism as it gives a\nsmaller stop mass-square, thus leads to smaller $|m_{H_u}|^2$ and\n$|\\mu|$ at the weak scale. On the other hand,\n the Bino mass increases as $M_1\\propto\n(1+0.66\\alpha)$, thus the lightest neutralino changes from\nBino-like to Higgsino-like when $\\alpha$ is varying from zero to a\npositive value of order unity. If $\\alpha$ is further increased,\neventually the model does not allow electroweak symmetry breaking.\nIn this section, we extend the analysis of the previous section to\nthe range of $\\alpha$ from zero to the value at which the\nelectroweak symmetry starts to be restored.\n\n\\subsection{Parameter region of neutralino LSP and thermal relic\nDensity}\n\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=7cm,width=7cm]{tb10-mass.eps}\\hskip 0.5cm\n\\includegraphics[height=7cm,width=7cm]{tb35-mass.eps}\n\\vskip 0.5cm\n\\includegraphics[height=7cm,width=7cm]{tb10.eps}\\hskip 0.5cm\n\\includegraphics[height=7cm,width=7cm]{tb35.eps}\n\\end{center}\n\\vskip -0.4cm \\caption{Sparticle masses vs. $\\alpha$ for\n$\\tan\\beta=10$ and $\\rm tan\\beta=35$ in case with $a_i=c_i=1$. The\nlower figures show the parameter space of neutralino LSP and its\nthermal relic density on the plane of ($\\alpha$, $M_0$).}\n\\label{fig:density-tb35}\n\\end{figure}\n\n\nAgain, let us first consider the case with $a_i=c_i=1$. We will\ntreat $M_0$ and $\\alpha$ as free parameters, while focusing on\n$\\tan\\beta=10$ and 35. Figs.~\\ref{fig:density-tb35}.a and\n\\ref{fig:density-tb35}.b show how some of the superparticle\nmasses vary as a function of $\\alpha$ for a fixed $M_0 = 800$ GeV.\nFor $\\alpha\\lesssim 1$, the LSP is the lightest neutralino which\nis mostly Bino, and thus its mass varies as $m_{\\chi^0}\\simeq\nM_1\\propto (1+0.66\\alpha)$. In the range of\n$1\\lesssim\\alpha\\lesssim 1.8$, stau or stop becomes the LSP. For\n$1.8\\lesssim \\alpha\\lesssim 2$, the lightest neutralino which is\nnow mostly Higgsino becomes the LSP. If $\\alpha$ increases\nfurther, the model does not allow electroweak symmetry breaking.\n\nIn Figs.~\\ref{fig:density-tb35}.c and \\ref{fig:density-tb35}.d,\nthe two distinct magenta regions seperated by stop\/stau LSP\nregions give the WMAP DM density, $0.085<\\Omega_{DM} h^2<0.119$,\nunder the assumption that all neutralino DMs are produced by the\nconventional thermal production mechanism. Below (above) these\nmagenta regions, $\\Omega_\\chi h^2 < 0.085$ ($> 0.119$). In the\nBino-like LSP region, stop-neutralino coannihialtion plays a\ncrucial role to get the WMAP DM density for $\\rm tan\\beta=10$,\nwhile stau-neutralino coannihilation or pseudoscalar Higgs\nresonance processes are important for $\\rm tan\\beta=35$. For\nHiggsino-like LSP, the charged Higgsino $\\chi_1^\\pm$ and two\nneutral Higgsinos $\\chi_1^0, \\chi_2^0$ are nearly degenerate. Then\nthe dominant annihilation processes are the neutralino pair\nannihilation into gauge bosons, and the neutralino-chargino\nco-annihilation into fermion pair \\cite{coanil}. These\nannihilations of Higgsino-like LSP are very efficient, so that the\nrelic mass density is too small unless $m_\\chi^0$ is quite heavy.\nIndeed, from Figs.~\\ref{fig:density-tb35}.c and\n\\ref{fig:density-tb35}.d, we can see that the WMAP DM density is\nobtained only for $M_0\\gtrsim 2.2$ TeV in the Higgsino LSP region\naround $\\alpha\\sim 1.8$. However it should be stressed that the\ncyan regions of Figs.~\\ref{fig:density-tb35}.c and\n\\ref{fig:density-tb35}.d can be allowed if some part of DM were\nproduced by non-thermal mechanism. Such parameter region contains\n$\\alpha\\sim 1.8$ and $M_0\\sim 1$ TeV for which the neutral\nHiggsino with $m_{\\chi^0}\\sim 200$ GeV is the LSP and the stop is\nrather light as $m_{\\tilde{t}_1}\\sim 250$ GeV. \nThe brown regions are excluded by the $b \\to s \\gamma$ constraint. On the brown\nregion in Fig. 9.c, the chargino loop\ncontribution to $b\\to s\\gamma$ dominates, which results in Br$(b\n\\to s \\gamma)$ smaller than the experimentally allowed range. \n\n\\vskip 0.8cm\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=7cm,width=7cm]{nzmw-tb10-mass.eps}\\hskip 1cm\n\\includegraphics[height=7cm,width=7cm]{nzmw-tb35-mass.eps}\n\\vskip 1cm\n\\includegraphics[height=7cm,width=7cm]{nzmw-tb10.eps}\\hskip 1cm\n\\includegraphics[height=7cm,width=7cm]{nzmw-tb35.eps}\n\\end{center}\n\\vskip -0.4cm\n\\caption{The results for the case in which $a_i=c_i=1\/2$ for both\nthe Higgs and matter multiplets.} \\label{fig:density-nzero}\n\\end{figure}\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=7cm,width=7cm]{nzmw2-tb10-mass.eps}\\hskip 1cm\n\\includegraphics[height=7cm,width=7cm]{nzmw2-tb35-mass.eps}\\vskip\n1cm\n\\includegraphics[height=7cm,width=7cm]{nzmw2-tb10.eps}\\hskip 1cm\n\\includegraphics[height=7cm,width=7cm]{nzmw2-tb35.eps}\n\\end{center}\n\\vskip -0.4cm \\caption{The results for the case in which $a_{\\rm\nH}=c_{\\rm H}=0$ and $a_{\\rm M}=c_{\\rm M}=1$.}\n\\label{fig:density-nzero2}\n\\end{figure}\n\nLet us now consider the case with $a_i=c_i=1\/2$. Obviously,\nsmaller $(a_i,c_i)$ give smaller stop\/stau mass-squares at\n$M_{GUT}$. However, as was anticipated in the previous section,\n$X_I$ ($I=t,b,\\tau$) which govern the RG evolution of stop\/stau\nmass-squares (see Eq.~\\ref{rgequation}) become smaller also, which\nwould increase the stop\/stau masses at the weak scale. Together\nwith the reduction of $A_{H_uQ_3U_3}$, this effect on the RG\nevolution eventually makes the physical lighter stop mass\n$m_{\\tilde t_1}$ larger compared to the case with $a_i=c_i=1$. On\nthe other hand, stau masses are more affected by the change of the\nboundary values, thus their weak scale values become lighter\ncompared to the case of $a_i=c_i=1$.\n Smaller $X_t$ leads to\nalso a smaller $|m_{H_u}^2|$ at the weak scale, resulting the\nreduction of the Higgsino mass $\\mu$ and the pseudoscalar Higgs\nboson mass $m_A$. Figs.~\\ref{fig:density-nzero}.a and\n\\ref{fig:density-nzero}.b show all of these features. Again, as\n$\\alpha$ increases, the neutralino LSP changes from Bino-like to\nHiggsino-like. Comparing to Fig.~\\ref{fig:density-tb35}, the\nlighter stop becomes heavier, while the lighter stau and the\npseudoscalar Higgs become lighter. As a consequence, the stop LSP\nregion disappears, but the stau LSP region at large $\\tan\\beta$\nbecomes larger. The magenta regions of\nFigs.~\\ref{fig:density-nzero}.c and \\ref{fig:density-nzero}.d\ncorrespond to the parameter region giving the WMAP DM density\nunder the assumption of pure thermal production. They clearly show\nthe Higgsino-like LSP at $\\alpha>1$ and also the pseudoscalar\nHiggs resonance effect for the Bino-like LSP at smaller $\\alpha$.\n\n\\vskip 0.8cm\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=7cm,width=7cm]{nzmw3-tb10-mass.eps}\\hskip 1cm\n\\includegraphics[height=7cm,width=7cm]{nzmw3-tb35-mass.eps} \\vskip\n1cm\n\\includegraphics[height=7cm,width=7cm]{tb10-nq05-nh1.eps}\\hskip 1cm\n\\includegraphics[height=7cm,width=7cm]{tb35-nq05-nh1.eps}\n\\end{center}\n\\vskip -0.3cm \\caption{The results for the case in which $a_{\\rm\nH}=c_{\\rm H}=0$ and $a_{\\rm M}=c_{\\rm M}=1\/2$.}\n\\label{fig:density-nq05-nh1}\n\\end{figure}\n\n\n\nFig.~\\ref{fig:density-nzero2} shows the results for the case in\nwhich $a_{\\rm H}=c_{\\rm H}=0$ for the Higgs multiplets, while\n$a_{\\rm M}=c_{\\rm M}=1$ for the quark\/lepton matter multiplets. A\ncharacteristic feature of this case is that the lightest\nneutralino is the LSP over the entire region of parameter space\nallowing the electroweak symmetry breaking. Compared to the case\nin which $a_i=c_i=1$ for both the Higgs and matter multiplets,\n$X_I$ ($I=t,b,\\tau$) for the RG evolution (\\ref{rgequation}) have\nsmaller values, while the boundary values of stop\/stau\nmass-squares remain the same. This results in heavier stop and\nstau at the weak scale. Except for the absence of stop\/stau LSP\nregion, other features are somewhat similar to other cases.\nThe brown regions are excluded by the $b \\to s \\gamma$ constraint. \nOn the brown region with small $M_0$ in Fig. 11.c, the chargino loop\ncontribution to $b\\to s\\gamma$ dominates, which results in Br$(b\n\\to s \\gamma)$ smaller than the experimentally allowed range. \nOn the other hand, the charged Higgs boson loop becomes significant\nin the large $M_0$ region, making the predicted Br$(b \\to s\n\\gamma)$ exceed the experimental bound. The region between those\ntwo brown regions is allowed due to the cancellation between the\nchargino and charged Higgs boson loop contributions.\n\nFinally, Fig.~\\ref{fig:density-nq05-nh1} is for the case with\n$a_H=c_H=0$ and $a_{\\rm M}=c_{\\rm M}=1\/2$. The results are quite\nsimilar to the case in which $a_i=c_i=1\/2$ for both the Higgs and\nmatter multiplets. The $\\alpha=2$ region of this case corresponds\nto the TeV scale mirage mediation model proposed in\n\\cite{tevmirage} as a model to minimize the fine tuning for the\nelectroweak symmetry breaking in the MSSM.\n\n\n\n\n\n\n\\subsection{Dark matter detections} \\vspace{0.5cm}\n\n\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=6cm,width=6cm]{tb10-si.eps}\\hskip 1.0cm\n\\includegraphics[height=6cm,width=6cm]{tb35-si.eps}\\vskip 0.5cm\n\\end{center}\n\\vskip -0.5cm \\caption{Spin-independent neutralino and proton\nscattering cross section in case with $a_i=c_i=1$.}\n\\label{fig:sigsip-tb35}\n\\end{figure}\n\n\nTo see the prospect of direct DM detection, spin-independent cross\nsection of the neutralino-proton scattering is presented in\nFig.~\\ref{fig:sigsip-tb35} for the case with $a_i=c_i=1$. Here, we\nimposed the experimental bounds on sparticle and Higgs masses, and\n$b\\rightarrow s\\gamma$ branching ratio. In the figures, the red\npoints give the WMAP DM density: $0.085 < \\Omega_\\chi h^2 <\n0.119$, the cyan corresponds to the region giving $\\Omega_\\chi\nh^2< 0.085$, and the rest gives $\\Omega_\\chi h^2 > 0.119$, under\nthe assumption of pure thermal production of neutralino DM.\nOne can notice that there are two distinct branches of the WMAP\npoints which correspond to the Bino branch and the Higgsino branch, respectively.\nIn our scan, Higgsino-like LSP gives a larger $\\sigma_{SI}$ for a given\n$m_\\chi$. The dominant contribution to $\\sigma_{SI}$ usually comes\nfrom the Higgs exchange process which becomes\n significant if the LSP neutralino is a mixed Bino-Higgsino state.\nOn the other hand, for $a_i=c_i=1$, the LSP neutralino is either\nBino-like or Higgsino-like since the mixed Bino-Higgsino region\ngives a stop or stau LSP. Therefore, it is expected that the\ncross section is rather small for the case with $a_i=c_i=1$.\nIndeed, Fig.~\\ref{fig:sigsip-tb35} shows that the predicted values\nare all less than the current and near future experimental\nsensitivity.\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=6cm,width=6cm]{nzmw-tb10-si.eps}\\hskip 1cm\n\\includegraphics[height=6cm,width=6cm]{nzmw-tb35-si.eps}\\vskip\n0.5cm\n\\includegraphics[height=6cm,width=6cm]{nzmw2-tb10-si.eps}\\hskip 1cm\n\\includegraphics[height=6cm,width=6cm]{nzmw2-tb35-si.eps}\\vskip 0.5cm\n\\includegraphics[height=6cm,width=6cm]{nzmw3-tb10-si.eps}\\hskip 1cm\n\\includegraphics[height=6cm,width=6cm]{nzmw3-tb35-si.eps}\\vskip 0.5cm\n\\end{center}\n\\vskip -0.5cm \\caption{Spin-independent neutralino and proton\nscattering cross section for other values of $(a_i,c_i)$ giving a\nmixed Bino-Higgsino LSP over a significant fraction of the\nparameter space.} \\label{fig:sigsip-tb36}\n\\end{figure}\n\n\nHowever, the prospect of direct DM detection is dramatically\nchanged if one considers other choices of $a_i$ and $c_i$.\nFig.~\\ref{fig:sigsip-tb36} shows the predictions for\nspin-independent cross section of the neutralino-proton scattering\nfor the three other choices of $(a_i,c_i)$ giving a mixed\nBino-Higgsino LSP over a significant fraction of the parameter\nspace and also a reduced value of the pseudoscalar Higgs mass.\nThese values of $a_i$ and $c_i$ give heavier stop, thereby the\n$b\\rightarrow s\\gamma$ constraint becomes less significant\ncompared to the case with $a_i=c_i=1$. Again, the red points\nrepresent the parameter values giving the WMAP DM density $ 0.085\n< \\Omega_\\chi h^2 < 0.119$, the cyan points give $\\Omega_\\chi\nh^2 < 0.085$, and the rest stands for $\\Omega_\\chi h^2 > 0.119$,\nunder the assumption of thermal production of neutralino LSP.\nAs expected, the scattering cross sections are largely enhanced\ncompared to the case with $a_i=c_i=1$. Now, much of the WMAP\npoints give $\\sigma_{SI}$ exceeding the sensitivity limit of the\nplanned SuperCDMS experiment. If one includes the cyan points,\nthe cross section can be much bigger, reaching even at the current\nCDMS sensitivity limit.\n\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=6cm,width=6cm]{tb10-gamma.eps}\\hskip 0.5cm\n\\includegraphics[height=6cm,width=6cm]{tb35-gamma.eps}\\vskip 1.0cm\n\\includegraphics[height=6cm,width=6cm]{tb10-mono.eps}\\hskip 0.5cm\n\\includegraphics[height=6cm,width=6cm]{tb35-mono.eps}\n\\end{center}\n\\vskip -0.5cm \\caption{Continuum (a and b) and monochromatic (c\nand d) gamma ray flux from the Galactic Center vs. $m_\\chi$ for\nthe case with $a_i=c_i=1$ and $\\rm{tan}\\beta=10$ or $35$.}\n\\label{fig:fluxgam-tb35}\n\\end{figure}\n\n\nGamma rays induced by neutralino annihilation in Galactic Center\nmight provide an indirect detection of neutralino DM.\nFig.~\\ref{fig:fluxgam-tb35} shows the predicted continuum (a and\nb) and monochromatic (c and d) gamma ray fluxes from the Galactic\nCenter as a function of the LSP neutralino mass for the case with\n$a_i=c_i=1$. Here we chose the same halo density profile as the\nprevious section, giving $\\bar J(\\Delta\\Omega=10^{-3} {\\rm sr})\n\\sim 30$. The red points in the figures give the WMAP DM density,\nwhile the cyan and the rest give $\\Omega_\\chi h^2<0.085$ and\n$\\Omega_\\chi h^2>0.119$, respectively, under the assumption of\npure thermal production. Again the WMAP points have two distinct\nbranches, the Bino-branch and the Higgsino-branch. \nIncluding the cyan points\ngiving smaller thermal relic DM density, the case with $a_i=c_i$\ncan give a continuum\ngamma ray flux up to $2\\times 10^{-11} {\\rm cm^{-2} s^{-1}}$ and\n$10^{-10} {\\rm cm^{-2} s^{-1}}$ for $\\tan\\beta = 10$ and 35,\nrespectively.\n This maximum flux of the continuum gamma rays barely touch\nthe expected reach of GLAST. However, the real gamma ray flux can\nbe much bigger than these predictions if the actual halo density\nprofile is denser than the assumed profile. For Higgsino LSP,\nunsuppressed annihilation into W or Z boson pair is the major\nsource of continuum gamma rays. As can be noticed from\nFig.~\\ref{fig:fluxgam-tb35}, for some parameter values, the gamma\nray flux from Bino LSP is largely enhanced by the pseudoscalar\nHiggs resonance effect.\n\n\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=6cm,width=6cm]{nzmw3-tb10-gamma.eps}\\hskip 0.5cm\n\\includegraphics[height=6cm,width=6cm]{nzmw3-tb35-gamma.eps}\\vskip 1.0cm\n\\includegraphics[height=6cm,width=6cm]{nzmw3-tb10-mono.eps}\\hskip 0.5cm\n\\includegraphics[height=6cm,width=6cm]{nzmw3-tb35-mono.eps}\n\\end{center}\n\\vskip -0.4cm \\caption{Continuum (a and b) and monochromatic (c\nand d) gamma ray flux from the Galactic Center vs. $m_\\chi$ for\nthe case in which $a_{\\rm H}=c_{\\rm H}=0$, $a_{\\rm M}=c_{\\rm\nM}=1\/2$, and $\\rm{tan}\\beta=10$ or 35.} \\label{fig:fluxgam-nzmw3}\n\\end{figure}\n\n\n\n\n\nIn fact, it is quite nontrivial to discriminate the continuum\ngamma rays produced by neutralino annihilation from the diffuse\ngalactic gamma ray backgrounds. On the other hand, the\nmonochromatic gamma ray line from $\\chi\\chi \\rightarrow\n\\gamma\\gamma$ or $\\gamma Z$ can be considered as a 'smoking gun'\nsignal of WIMP dark matter. Figs.~\\ref{fig:fluxgam-tb35}.c and\n\\ref{fig:fluxgam-tb35}.d show the gamma ray line flux produced by\nneutralino pair annihilation in Galactic Center for the case with\n$a_i=c_i$. One can notice that there is a clear distinction\nbetween the Bino and Higgsino LSP regions. The gamma ray line flux\nranges from $10^{-19}$ to $10^{-16}\\, {\\rm cm^{-2} s^{-1}}$ for\nthe Bino LSP branch of WMAP points, while it ranges from\n$10^{-16}$ to $10^{-15}\\, {\\rm cm^{-2} s^{-1}}$ for the Higgsino\nLSP branch. For Higgsino-like LSP, the gamma ray line flux comes\ndominantly from the $W^\\pm \\chi_1^\\mp$ loop diagrams resulting in\na large cross section for $\\chi\\chi \\rightarrow \\gamma\\gamma$ or\n$\\gamma Z$ \\cite{higgsinoDM}. While GLAST will probe the photon\nenergies only up to 300 GeV with a low energy threshold,\nAtmospheric Cherenkov Telescopes (ACT) such as H.E.S.S. will be\nable to cover higher photon energy ranges and probe the gamma ray\nflux down to $10^{-14} {\\rm cm^{-2} s^{-1}}$. The predicted\nmonochromatic fluxes in Figs.~\\ref{fig:fluxgam-tb35}.c and\n\\ref{fig:fluxgam-tb35}.d are still below this sensitivity limit.\nHowever, as we have stressed, these results are based on a rather\nconservative halo density profile. In view of that the predicted\nflux can increase even by a factor of $10^4$ if one uses an\nextreme halo model like the spiked profile, the monochromatic\ngamma ray signal for the Higgsino dark matter might be measurable\nin case of a cuspy halo density profile.\n\n\n\n\nAs we have anticipated, other values of $(a_i,c_i)$ specified in\n(\\ref{4choices}) allow a mixed Bino-Higgsino LSP over a\nsignificant fraction of parameter space. It is thus expected that\nthose other cases can give a larger gamma ray flux compared to the\ncase with $a_i=c_i=1$. In Fig.~\\ref{fig:fluxgam-nzmw3}, we\ndepicted the results for the case with $a_{\\rm H}=c_{\\rm H}=0$ and\n$a_{\\rm M}=c_{\\rm M}=1\/2$. Indeed, this case gives a larger flux,\nalthough not dramatically different. The red WMAP points can give\na continuum gamma ray flux up to $3 \\times 10^{-11} {\\rm cm^{-2}\ns^{-1}}$, while the cyan points giving smaller thermal relic DM\ndensity can reach up to $2\\times 10^{-10} {\\rm cm^{-2} s^{-1}}$.\nThe maximal flux of monochromatic gamma ray is about $10^{-15}\n{\\rm cm^{-2} s^{-1}}$ for the red WMAP points and about $7 \\times\n10^{-15} {\\rm cm^{-2} s^{-1}}$ for the cyan points. Again, these\nresults are obtained for the conservative halo density profile\ngiving $\\bar J(\\Delta\\Omega=10^{-3} {\\rm sr}) \\sim 30$. The real\ngamma ray flux can be significantly bigger than these predictions\nif the actual halo density profile is denser than the assumed\nprofile.\n\n\n\n\n\n\\section{Conclusions}\n\nIn this paper, we have examined the prospect of neutralino dark\nmatter in mirage mediation scenario of SUSY breaking in which soft\nmasses receive comparable contributions from modulus mediation and\nanomaly mediation. Depending upon the model parameters, especially\nthe anomaly to modulus mediation ratio, the nature of the lightest\nneutralino changes from Bino-like to Higgsino-like via\nBino-Higgsino mixing region. For Bino-like LSP, the conventional\nthermal production mechanism can give a right amount of relic DM\ndensity, i.e. the WMAP observation $0.085 < \\Omega_{DM}h^2 <\n0.119$, through the stop\/stau-neutralino coannihilation process or\nthe pseudo-scalar Higgs resonance effect. In overall, compared to\nthe mSUGRA scenario, a significantly larger fraction of the\nparameter space can give the WMAP DM density under the assumption\nof thermal production, while satisfying all known phenomenological\nconstraints. This is partly because the lightest neutralino is a\nmixed Bino-Higgsino over a sizable fraction of the parameter\nspace.\n\n\nWe also studied the detection possibilities of neutralino dark\nmatter in mirage mediation. For the parameter region giving the\nWMAP density of Bino-like or Higgsino-like LSP, direct detection\nvia elastic scattering between neutralino DM and nuclear target\nturns out to be mostly under the sensitivity of near future\nexperiments. However the other parameter region giving the WMAP\ndensity of mixed Bino-Higgsino LSP predicts typically a cross\nsection above the expected sensitivity limit of SuperCDMS. The\ncontinuum and monochromatic gamma ray fluxes from neutralino\nannihilation in Galactic Center have been analyzed also.\nGenerically, Higgsino-like LSP gives a larger gamma ray flux than\nBino-like LSP, however the continuum gamma ray flux from Bino LSP\ncan be significantly enhanced for some particular parameter values\ndue to the pseudo-scalar Higgs resonance effect. Although the\ngamma ray fluxes predicted within a conservative halo model are\nbelow the sensitivity of ongoing and planned experiments, it might\nbe detectable if the actual halo density is denser than the\nconservative profile used in our analysis.\n\n\n\\bigskip\n\n\n\n\\acknowledgments We thank Kwang-Sik Jeong for helpful discussions\nand also for clarifying various conventions for soft terms. This\nwork is supported by the KRF Grant KRF-2005-201-C00006 funded by\nthe Korean Government (K.C. and Y.S.), the KOSEF Grant\nR01-2005-000-10404-0 (K.C. and Y.S.), the Center for High Energy\nPhysics of Kyungpook National University (K.C.), the BK21 program\nof Ministry of Education (K.Y.L.), and the Astrophysical Research\nCenter for the Structure and Evolution of the Cosmos funded by the\nKOSEF (Y.G.K.). K.O. has been supported by the grant-in-aid for\nscientific research on priority areas (No. 441): \"Progress in\nelementary particle physics of the 21 century through discoveries\nof Higgs boson and supersymmetry\" (No. 16081209) from the Ministry\nof Education, Culture, Sports, Science and Technology of Japan.\nK.O. and Y.S. thank Yukawa Institute in Kyoto University for the\nuse of Altix3700 BX2 by which much of the numerical calculation\nhas been made. Y.S. also thanks the Particle Theory and Cosmology\nGroup at Tohoku University for the use of the computer facility.\n\n\n\n\n\n\n\\section*{Appendix A.}\n\n\\vskip 0.5cm In this appendix, we summarize the notations and\nconventions used in this paper. The quantum effective action in\n$N=1$ superspace is given by \\begin{eqnarray} && \\int d^4\\theta\n\\left[-3CC^*e^{-K\/3} +\\frac{1}{16}\\left(\nG_aW^{a\\alpha}\\frac{D^2}{\\partial^2}W^a_\\alpha+{\\rm\nh.c.}\\right)\\right] +\\left(\\,\\int d^2\\theta\\, C^3W+{\\rm h.c.}\\, \\right) \\nonumber \\\\\n&=& \\int d^4\\theta\n\\left[\\,-3CC^*e^{-K_0\/3}+CC^*e^{-K_0\/3}Z_i\\Phi^*_ie^{2V_aT_a}\\Phi_i\n +\\frac{1}{16}\\left(\\,\n G_aW^{a\\alpha}\\frac{D^2}{\\partial^2}W^a_\\alpha+{\\rm\n h.c.}\\,\\right)\\,\\right]\\nonumber \\\\\n &&+\\,\\left(\\,\\int d^2\\theta\\,\n C^3\\left[\\,W_0+\\frac{1}{6}\\lambda_{ijk}\\Phi_i\\Phi_j\\Phi_k\\,\\right]+{\\rm h.c.}\n\\,\\right)+..., \\end{eqnarray} where the gauge kinetic terms are written as a\n$D$-term operator to accommodate the radiative corrections to\ngauge couplings, and the ellipsis stands for the irrelevant higher\ndimensional operators. The K\\\"ahler potential $K$ is expanded as\n\\begin{eqnarray} K=K_0(T_A,T_A^*)+Z_i(T_A,T_A^*)\\Phi^*_ie^{2V_aT_a}\\Phi_i+...,\n\\end{eqnarray} where $V_a$ and $\\Phi_i$ denote the visible gauge and matter\nsuperfields given by \\begin{eqnarray}\n\\Phi^i&=&\\phi^i+\\sqrt{2}\\,\\theta\\psi^i+\\theta^2F^i,\\nonumber \\\\\nV^a &=& -\\theta\\sigma^\\mu\\bar\\theta A^a_\\mu\n-i\\bar\\theta^2\\theta\\lambda^a + i\\theta^2\\bar\\theta\\bar\\lambda^a +\n\\frac{1}{2}\\theta^2\\bar\\theta^2 D^a, \\end{eqnarray} and $T_A=(C,T)$ are the\nSUSY breaking messengers including the conformal compensator\nsuperfield $C=C_0+\\theta^2F^C$ and the modulus superfield\n$T=T_0+\\sqrt{2}\\theta\\tilde{T}+\\theta^2F^T$. The radiative\ncorrections due to renormalizable gauge and Yukawa interactions\ncan be encoded in the matter K\\\"ahler metric $Z_i$ and the gauge\ncoupling superfield $G_a$ which is given by\n \\begin{eqnarray} G_a\\,=\\,{\\rm Re}(f_a)+\\Delta G_a, \\end{eqnarray} where\n$f_a$ is the holomorphic gauge kinetic function and $\\Delta G_a$\nincludes the $T_A$-dependent radiative correction to gauge\ncoupling. The superpotential is expanded as \\begin{eqnarray}\nW=W_0(T)+\\frac{1}{6}\\lambda_{ijk}(T)\\Phi_i\\Phi_j\\Phi_k+..., \\end{eqnarray}\nwhere $W_0(T)$ is the modulus superpotential stabilizing $T$. Here\nwe do not specify the mechanism to generate the MSSM Higgs\nparameters $\\mu$ and $B$, and treat them as free parameters\nconstrained only by the electroweak symmetry breaking condition.\nFor a discussion of $\\mu$ and $B$ in mirage mediation, see\nRef.~\\cite{Choi:2005uz}.\n\n\nFor the canonically normalized component fields, the above\nsuperspace action gives the following form of the running gauge\nand Yukawa couplings, the supersymmetric gaugino-matter fermion\ncoupling ${\\cal L}_{\\lambda\\psi}$, and the soft SUSY breaking\nterms: \\begin{eqnarray} \\frac{1}{g_a^2}&=& {\\rm Re}(G_a),\\quad y_{ijk}\\, =\\,\n\\frac{\\lambda_{ijk}}{\\sqrt{e^{-K_0}Z_iZ_jZ_k}},\\nonumber \\\\\n {\\cal\nL}_{\\lambda\\psi}&=&i\\sqrt{2}\\left(\\phi_i^* T^a\\psi_i\\lambda^a\n-\\bar\\lambda^aT^a\\phi_i\\bar\\psi_i \\right), \\nonumber\n\\\\\n{\\cal L}_{\\rm soft}&=&-m^2_i\\phi^i\\phi^{i*}\n-\\left(\\,\\frac{1}{2}M_a\\lambda^a\\lambda^a\n+\\frac{1}{6}A_{ijk}y_{ijk}\\phi^i\\phi^j\\phi^k +{\\rm h.c.}\\right),\n\\end{eqnarray} where \\begin{eqnarray} M_a &=& F^A\\partial_A\\ln ({\\rm Re}(G_a)),\n\\nonumber \\\\\nA_{ijk} &=& -F^A\\partial_A \\ln\\left(\n\\frac{\\lambda_{ijk}}{e^{-K_0}Z_iZ_jZ_k}\\right),\n\\nonumber \\\\\nm^2_i &=& -F^AF^{B*}\\partial_A\\partial_{\\bar B} \\ln\\left(\ne^{-K_0\/3}Z_i\\right) \\end{eqnarray} for \\begin{eqnarray} F^T&=&\n-e^{K_0\/2}(\\partial_T\\partial_{T^*})^{-1}(D_TW_0)^*,\\nonumber \\\\\nF^C&=&m_{3\/2}^*+\\frac{1}{3}\\partial_TK_0F^T \\quad(m_{3\/2}=\ne^{K_0\/2}W_0). \\end{eqnarray} In the approximation ignoring the off-diagonal\ncomponents of $w_{ij}=\\sum_{pq}y_{ipq}y^*_{jpq}$, the 1-loop RG\nevolution of soft parameters is determined by \\begin{eqnarray}\n{16\\pi^2}\\frac{dM_a}{d\\ln\\mu}&=& 2 \\left[-3\\,{\\rm\ntr}\\Big(T_a^{2}({\\rm Adj})\\Big) +\\sum_i {\\rm\ntr}\\Big(T_a^{2}(\\phi^i)\\Big) \\right] g^2_aM_a,\n\\nonumber \\\\\n{16\\pi^2}\\frac{dA_{ijk}}{d\\ln\\mu} &=& \\left[\n\\sum_{p,q}|y_{ipq}|^2A_{ipq} - 4 \\sum_a g^2_aC_2^a(\\phi^i) M_a\n\\right] + \\Big[i \\leftrightarrow j\\Big] + \\Big[i \\leftrightarrow\nk\\Big],\n\\nonumber \\\\\n{16\\pi^2}\\frac{d m^2_i}{d\\ln\\mu} &=&\n\\sum_{j,k}|y_{ijk}|^2\\left(m^2_i+m^2_j+m^2_k+|A_{ijk}|^2\\right)\n\\nonumber \\\\ &-& 8 \\sum_a g^2_aC_2^a(\\phi^i)|M_a|^2 +2g_1^2q_i\n\\sum_j q_j m^2_j,\n\\end{eqnarray} where the quadratic Casimir $C^a_2(\\phi_i)=(N^2-1)\/2N$ for a\nfundamental representation $\\phi_i$ of the gauge group $SU(N)$,\n$C_2^a(\\phi_i)=q_i^2$ for the $U(1)$ charge $q_i$ of $\\phi_i$.\n\n\n\nIn mirage mediation, soft terms at $M_{GUT}$ are determined by the\nmodulus mediation of ${\\cal O}(F^T\/T)$ and the anomaly mediation\nof ${\\cal O}(F^C\/8\\pi^2 C_0)$ which are comparable to each other.\nIn the presence of the axionic shift symmetry \\begin{eqnarray} U(1)_T: \\quad\n{\\rm Im}(T)+ \\mbox{real constant}\\end{eqnarray} which is broken by the\nnon-perturbative term in the modulus superpotential \\begin{eqnarray}\nW_0=w-Ae^{-aT},\\end{eqnarray} one can always make that $m_{3\/2}$ and $F^T$\nare simultaneously real. Also since $F^T\/T\\sim m_{3\/2}\/4\\pi^2$, we\nhave \\begin{eqnarray} \\frac{F^C}{C_0}= m_{3\/2}\\left(\\,1+{\\cal\nO}\\left(\\frac{1}{8\\pi^2}\\right)\\,\\right). \\end{eqnarray} Then, upon ignoring\nthe parts of ${\\cal O}(F^T\/8\\pi^2 T)$, the resulting soft\nparameters at $M_{GUT}$ are given by\n\\begin{eqnarray}\n M_a&=& M_0 +\\frac{m_{3\/2}}{16\\pi^2}\\,b_ag_a^2,\n\\nonumber \\\\\nA_{ijk}&=&\\tilde{A}_{ijk}-\n\\frac{m_{3\/2}}{16\\pi^2}\\,(\\gamma_i+\\gamma_j+\\gamma_k),\n\\nonumber\\\\\nm_i^2&=& \\tilde{m}_i^2-\\frac{m_{3\/2}}{16\\pi^2}M_0\\,\\theta_i\n-\\left(\\frac{m_{3\/2}}{16\\pi^2}\\right)^2\\dot{\\gamma}_i,\n\\end{eqnarray} where \\begin{eqnarray} M_0&=&F^T\\partial_T\\ln{\\rm Re}(f_a),\n\\nonumber\n\\\\\n\\tilde{A}_{ijk}&\\equiv& (a_i+a_j+a_k)M_0\n\\,=\\,F^T\\partial_T\\ln(e^{-K_0}Z_iZ_jZ_k),\\nonumber \\\\\n\\tilde{m}_i^2&\\equiv& c_iM_0^2\\,=\\,\n-|F^T|^2\\partial_T\\partial_{\\bar{T}} \\ln(e^{-K_0\/3}Z_i),\\end{eqnarray} and\n \\begin{eqnarray} b_a&=&-3{\\rm tr}\\left(T_a^2({\\rm\nAdj})\\right)+\\sum_i {\\rm tr}\\left(T^2_a(\\phi_i)\\right),\n\\nonumber \\\\\n\\gamma_i&=&2\\sum_a\ng_a^2C^a_2(\\phi_i)-\\frac{1}{2}\\sum_{jk}|y_{ijk}|^2, \\nonumber\n\\\\\n\\theta_i&=& 4\\sum_a g_a^2 C^a_2(\\phi_i)-\\sum_{jk}|y_{ijk}|^2\n(a_i+a_j+a_k), \\nonumber\n\\\\\n\\dot{\\gamma}_i&=&8\\pi^2\\frac{d\\gamma_i}{d\\ln\\mu},\\end{eqnarray} where\n$\\omega_{ij}=\\sum_{kl}y_{ikl}y^*_{jkl}$ is assumed to be diagonal.\nHere we have used that $\\lambda_{ijk}$ are $T$-independent\nconstant as ensured by the axionic shift symmetry $U(1)_T$.\n\n\nLet us now summarize our conventions for the MSSM. The\nsuperpotential of canonically normalized matter superfields is\ngiven by \\begin{eqnarray} W &=& y_DH_d\\cdot QD^c+y_LH_d\\cdot LE^c-y_UH_u\\cdot\nQU^c - \\mu H_d\\cdot H_u, \\end{eqnarray} where the $SU(2)_L$ product is\n$H\\cdot Q=\\epsilon_{ab}H^aQ^b$ with\n$\\epsilon_{12}=-\\epsilon_{21}=1$, and color indices are\nsuppressed. Then the chargino and neutralino mass matrices are\ngiven by \\begin{eqnarray} -\\frac{1}{2}\\,\\tilde\\psi^{-T}{\\cal M}_C\\tilde\\psi^+\n-\\frac{1}{2}\\,\\tilde\\psi^{0T}{\\cal M}_N\\tilde\\psi^0 + {\\rm h.c.},\n\\end{eqnarray} where \\begin{eqnarray} {\\cal M}_C &=& \\left(\n\\begin{array}{cc}\n- M_2\\,\\,\n& g_2 \\langle H^0_u \\rangle \\\\\ng_2 \\langle H^0_d \\rangle & \\mu\n\\end{array}\n\\right),\\nonumber \\\\\n{\\cal M}_N &=& \\left(\n\\begin{array}{cccc}\n-M_1 & 0 & -\\frac{1}{\\sqrt 2}\\,g_Y \\langle H^0_d \\rangle\n& \\frac{1}{\\sqrt 2}\\,g_Y \\langle H^0_u \\rangle \\\\\n0 & -M_2 & \\frac{1}{\\sqrt 2}\\,g_2 \\langle H^0_d \\rangle\n& -\\frac{1}{\\sqrt 2}\\,g_2 \\langle H^0_u \\rangle \\\\\n -\\frac{1}{\\sqrt 2}\\,g_Y \\langle H^0_d \\rangle\n& \\frac{1}{\\sqrt 2}\\,g_2 \\langle H^0_d \\rangle\n& 0 & -\\mu \\\\\n \\frac{1}{\\sqrt 2}\\,g_Y \\langle H^0_u \\rangle\n& -\\frac{1}{\\sqrt 2}\\,g_2 \\langle H^0_u \\rangle & -\\mu & 0\n\\end{array}\n\\right), \\end{eqnarray} in the field basis \\begin{eqnarray} \\tilde\\psi^{+T} &=&\n-i\\left(\\tilde W^+,\\, i\\tilde H^+_u \\right), \\quad \\tilde\\psi^{-T}\n\\,=\\, -i\\left(\\tilde W^-,\\, i\\tilde H^-_d \\right),\n\\nonumber \\\\\n\\tilde\\psi^{0T} &=& -i\\left( \\tilde B,\\,\\tilde W^3,\\, i\\tilde\nH^0_d,\\,i\\tilde H^0_u \\right), \\end{eqnarray} for $\\tilde W^{\\pm}=(\\tilde\nW^1\\mp i \\tilde W^2)\/\\sqrt 2.$\n\n\nThe one-loop beta function coefficients $b_a$ and anomalous\ndimension $\\gamma_i$ in the MSSM are given by \\begin{eqnarray} b_3&=&-3, \\qquad\nb_2=1,\\qquad b_1=\\frac{33}{5},\n\\nonumber \\\\\n\\gamma_{H_u} &=& \\frac{3}{2}g_2^2+\\frac{1}{2}g_Y^2 -3y_t^2,\n\\nonumber \\\\\n\\gamma_{H_d} &=& \\frac{3}{2}g_2^2+\\frac{1}{2}g_Y^2 - 3 y_b^2 - y_\\tau^2\n\\nonumber \\\\\n\\gamma_{Q_a} &=& \\frac{8}{3} g_3^2 + \\frac{3}{2} g_2^2\n +\\frac{1}{18} g_Y^2 - (y_t^2 + y_b^2) \\delta_{3a},\n\\nonumber \\\\\n\\gamma_{U_a} &=& \\frac{8}{3} g_3^2 + \\frac{8}{9} g_Y^2\n - 2 y_t^2 \\delta_{3a},\n\\nonumber \\\\\n\\gamma_{D_a} &=& \\frac{8}{3} g_3^2 + \\frac{2}{9} g_Y^2\n - 2 y_b^2 \\delta_{3a},\n\\nonumber \\\\\n\\gamma_{L_a} &=& \\frac{3}{2} g_2^2 + \\frac{1}{2} g_Y^2\n - y_\\tau^2 \\delta_{3a},\n\\nonumber \\\\\n\\gamma_{E_a} &=& 2 g_Y^2 - 2 y_\\tau^2 \\delta_{3a}, \\end{eqnarray} where\n$g_2$ and $g_Y=\\sqrt{3\/5}g_1$ denote the $SU(2)_L$ and $U(1)_Y$\ngauge couplings.\n The\n$\\theta_i$ and $\\dot{\\gamma}_i$ which determine the soft scalar\nmasses at $M_{GUT}$ are given by \\begin{eqnarray} \\theta_{H_u} &=&\n3g_2^2+g_Y^2 -6y_t^2(a_{H_u}+a_{Q_3}+a_{U_3}), \\nonumber \\\\\n\\theta_{H_d} &=& 3g_2^2+g_Y^2 - 6y_b^2(a_{H_d}+a_{Q_3}+a_{D_3}) -\n2y_\\tau^2(a_{H_d}+a_{L_3}+a_{E_3})\n\\nonumber \\\\\n\\theta_{Q_a} &=& \\frac{16}{3} g_3^2 + 3 g_2^2\n +\\frac{1}{9} g_Y^2 - 2\\Big(y_t^2(a_{H_u}+a_{Q_3}+a_{U_3}) + y_b^2(a_{H_d}+a_{Q_3}+a_{D_3})\\Big) \\delta_{3a},\n\\nonumber \\\\\n\\theta_{U_a} &=& \\frac{16}{3} g_3^2 + \\frac{16}{9} g_Y^2\n - 4y_t^2(a_{H_u}+a_{Q_3}+a_{U_3}) \\delta_{3a},\n\\nonumber \\\\\n\\theta_{D_a} &=& \\frac{16}{3} g_3^2 + \\frac{4}{9} g_Y^2\n - 4y_b^2(a_{H_d}+a_{Q_3}+a_{D_3}) \\delta_{3a},\n\\nonumber \\\\\n\\theta_{L_a} &=& 3 g_2^2 + g_Y^2\n - 2y_\\tau^2 (a_{H_d}+a_{L_3}+a_{E_3})\\delta_{3a},\n\\nonumber \\\\\n\\theta_{E_a} &=& 4 g_Y^2 - 4 y_\\tau^2(a_{H_d}+a_{L_3}+a_{E_3})\n\\delta_{3a}, \\end{eqnarray} and \\begin{eqnarray} \\dot\\gamma_{H_u} &=& \\frac{3}{2} g_2^4\n+ \\frac{11}{2} g_Y^4\n - 3 y_t^2 b_{y_t},\n\\nonumber \\\\\n\\dot\\gamma_{H_d} &=& \\frac{3}{2} g_2^4 + \\frac{11}{2} g_Y^4\n - 3 y_b^2 b_{y_b} - y_\\tau^2 b_{y_\\tau},\n\\nonumber \\\\\n\\dot \\gamma_{Q_a} &=& -8 g_3^4 + \\frac{3}{2} g_2^4 + \\frac{11}{18} g_Y^4\n -(y_t^2 b_{y_t} + y_b^2 b_{y_b}) \\delta_{3a},\n\\nonumber \\\\\n\\dot\\gamma_{U_a} &=& - 8 g_3^4 + \\frac{88}{9} g_Y^4\n - 2 y_t^2 b_{y_t} \\delta_{3a},\n\\nonumber \\\\\n\\dot\\gamma_{D_a} &=& - 8 g_3^4 + \\frac{22}{9} g_Y^4\n - 2 y_b^2 b_{y_b} \\delta_{3a},\n\\nonumber \\\\\n\\dot\\gamma_{L_a} &=& \\frac{3}{2}g_2^4 + \\frac{11}{2} g_Y^4\n - y_\\tau^2 b_{y_\\tau} \\delta_{3a},\n\\nonumber \\\\\n\\dot\\gamma_{E_a} &=& 22 g_Y^4 - 2 y_\\tau^2 b_{y_\\tau} \\delta_{3a},\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nb_{y_t} &=& - \\frac{16}{3} g_3^2 - 3 g_2^2 - \\frac{13}{9} g_Y^2\n + 6 y_t^2 + y_b^2,\n\\nonumber \\\\\nb_{y_b} &=& - \\frac{16}{3} g_3^2 - 3 g_2^2 - \\frac{7}{9} g_Y^2\n + y_t^2 + 6 y_b^2 + y_\\tau^2,\n\\nonumber \\\\\nb_{y_\\tau} &=& - 3 g_2^2 - 3 g_Y^2 + 3 y_b^2 + 4 y_\\tau^2. \\end{eqnarray}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nTwo classfiles namely \\file{cas-sc.cls} and \\file{cas-dc.cls} were\nwritten for typesetting articles submitted in journals of Elsevier's\nComplex Article Service (CAS) workflow.\n\n\\subsection{Usage}\n\\begin{enumerate}\n\\item \\file{cas-sc.cls} for single column journals. \n\n\\begin{vquote}\n \\documentclass[]{cas-sc}\n\\end{vquote}\n\\item \\file{cas-dc.cls} for single column journals. \n\n\\begin{vquote}\n \\documentclass[]{cas-dc}\n\\end{vquote}\n\\end{enumerate}\nand have an option longmktitle to handle long front matter. \n\n\\section{Front matter}\n\n\\begin{vquote}\n\\title [mode = title]{This is a specimen $a_b$ title} \n\\tnotemark[1,2]\n\n\\tnotetext[1]{This document is the results of the research\n project funded by the National Science Foundation.}\n\n\\tnotetext[2]{The second title footnote which is a longer text \n matter to fill through the whole text width and overflow into\n another line in the footnotes area of the first page.}\n\n\\author[1,3]{CV Radhakrishnan}[type=editor,\n auid=000,bioid=1,\n prefix=Sir,\n role=Researcher,\n orcid=0000-0001-7511-2910]\n\\cormark[1]\n\\fnmark[1]\n\\ead{cvr_1@tug.org.in}\n\\ead[url]{www.cvr.cc, cvr@sayahna.org}\n\\end{vquote}\n\n\\begin{vquote}\n\n\\credit{Conceptualization of this study, Methodology, \n Software}\n\n\\address[1]{Elsevier B.V., Radarweg 29, 1043 NX Amsterdam, \n The Netherlands}\n\n\\author[2,4]{Han Theh Thanh}[style=chinese]\n\n\\author[2,3]{CV Rajagopal}[%\n role=Co-ordinator,\n suffix=Jr,\n ]\n\\fnmark[2]\n\\ead{cvr3@sayahna.org}\n\\ead[URL]{www.sayahna.org}\n\n\\credit{Data curation, Writing - Original draft preparation}\n\n\\address[2]{Sayahna Foundation, Jagathy, Trivandrum 695014, \n India}\n\n\\author[1,3]{Rishi T.}\n\\cormark[2]\n\\fnmark[1,3]\n\\ead{rishi@stmdocs.in}\n\\ead[URL]{www.stmdocs.in}\n\n\\address[3]{STM Document Engineering Pvt Ltd., Mepukada,\n Malayinkil, Trivandrum 695571, India}\n\n\\cortext[cor1]{Corresponding author}\n\\cortext[cor2]{Principal corresponding author}\n\\fntext[fn1]{This is the first author footnote. but is common \n to third author as well.}\n\\fntext[fn2]{Another author footnote, this is a very long \n footnote and it should be a really long footnote. But this \n footnote is not yet sufficiently long enough to make two lines \n of footnote text.}\n\\end{vquote}\n\n\\begin{vquote}\n\\nonumnote{This note has no numbers. In this work we \n demonstrate $a_b$ the formation Y\\_1 of a new type of \n polariton on the interface between a cuprous oxide slab \n and a polystyrene micro-sphere placed on the slab.\n }\n\n\\begin{abstract}[S U M M A R Y]\nThis template helps you to create a properly formatted \n \\LaTeX\\ manuscript.\n\n\\noindent\\texttt{\\textbackslash begin{abstract}} \\dots \n\\texttt{\\textbackslash end{abstract}} and\n\\verb+\\begin{keyword}+ \\verb+...+ \\verb+\\end{keyword}+ \nwhich contain the abstract and keywords respectively. \nEach keyword shall be separated by a \\verb+\\sep+ command.\n\\end{abstract}\n\n\\begin{keywords}\nquadrupole exciton \\sep polariton \\sep \\WGM \\sep \\BEC\n\\end{keywords}\n\n\\maketitle\n\\end{vquote}\n\n\\begin{figure}\n\\includegraphics[width=\\textwidth]{sc-sample.pdf}\n\\caption{Single column output (classfile: cas-sc.cls).}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\textwidth]{dc-sample.pdf}\n\\caption{Double column output (classfile: cas-dc.cls).}\n\\end{figure}\n\n\\subsection{Title}\n\n\\verb+\\title+ command have the below options:\n\\begin{enumerate}\n\\item \\verb+title:+ Document title\n\\item \\verb+alt:+ Alternate title\n\\item \\verb+sub:+ Sub title\n\\item \\verb+trans:+ Translated title\n\\item \\verb+transsub:+ Translated sub title\n\\end{enumerate}\n\n\\begin{vquote}\n \\title[mode=title]{This is a title}\n \\title[mode=alt]{This is a alternate title}\n \\title[mode=sub]{This is a sub title}\n \\title[mode=trans]{This is a translated title}\n \\title[mode=transsub]{This is a translated sub title}\n\\end{vquote}\n\n\n\\subsection{Author}\n\\verb+\\author+ command have the below options: \n\n\\begin{enumerate}\n\\item \\verb+auid:+ Author id\n\\item \\verb+bioid:+ Biography id\n\\item \\verb+alt:+ Alternate author\n\\item \\verb+style:+ Style of author name chinese\n\\item \\verb+prefix:+ Prefix Sir\n\\item \\verb+suffix:+ Suffix\n\\item \\verb+degree:+ Degree\n\\item \\verb+role:+ Role\n\\item \\verb+orcid:+ ORCID\n\\item \\verb+collab:+ Collaboration\n\\item \\verb+anon:+ Anonymous author\n\\item \\verb+deceased:+ Deceased author\n\\item \\verb+twitter:+ Twitter account\n\\item \\verb+facebook:+ Facebook account\n\\item \\verb+linkedin:+ LinkedIn account\n\\item \\verb+plus:+ Google plus account\n\\item \\verb+gplus:+ Google plus account\n\\end{enumerate}\n\n\\begin{vquote}\n\\author[1,3]{Author Name}[type=editor,\n auid=000,bioid=1,\n prefix=Sir,\n role=Researcher,\n orcid=0000-0001-7511-2910,\n facebook=,\n twitter=,\n linkedin=,\n gplus=]\n\\end{vquote}\n\n\\subsection{Various Marks in the Front Matter}\n\nThe front matter becomes complicated due to various kinds\nof notes and marks to the title and author names. Marks in\nthe title will be denoted by a star ($\\star$) mark;\nfootnotes are denoted by super scripted Arabic numerals,\ncorresponding author by of an Conformal asterisk (*) mark.\n\n\\subsubsection{Title marks}\n\nTitle mark can be entered by the command, \\verb+\\tnotemark[]+\nand the corresponding text can be entered with the command\n\\verb+\\tnotetext[]+ \\verb+{}+. An example will be:\n\n\\begin{vquote}\n\\title[mode=title]{Leveraging social media news to predict\n stock index movement using RNN-boost}\n\n\\tnotemark[1,2]\n\n\\tnotetext[1]{This document is the results of the research\n project funded by the National Science Foundation.}\n\n\\tnotetext[2]{The second title footnote which is a longer \n text matter to fill through the whole text width and \n overflow into another line in the footnotes area of \n the first page.}\n\\end{vquote}\n\n\\verb+\\tnotetext+ and \\verb+\\tnotemark+ can be anywhere in\nthe front matter, but shall be before \\verb+\\maketitle+ command.\n\n\\subsubsection{Author marks}\n\nAuthor names can have many kinds of marks and notes:\n\n\\begin{vquote}\n footnote mark : \\fnmark[]\n footnote text : \\fntext[]{}\n affiliation mark : \\author[]\n email : \\ead{}\n url : \\ead[url]{}\n corresponding author mark : \\cormark[]\n corresponding author text : \\cortext[]{}\n\\end{vquote}\n\n\\subsubsection{Other marks}\n\nAt times, authors want footnotes which leave no marks in\nthe author names. The note text shall be listed as part of\nthe front matter notes. Class files provides\n\\verb+\\nonumnote+ for this purpose. The usage\n\n\\begin{vquote}\n\\nonumnote{}\n\\end{vquote}\n\n\\noindent and should be entered anywhere before the \\verb+\\maketitle+\ncommand for this to take effect. \n\n\\subsection{Abstract and Keywords}\n\nAbstract shall be entered in an environment that starts\nwith \\verb+\\begin{abstract}+ and ends with\n\\verb+\\end{abstract}+. Longer abstracts spanning more than\none page is also possible in Class file even in double\ncolumn mode. We need to invoke longmktitle option in the\nclass loading line for this to happen smoothly.\n\nThe key words are enclosed in a \\verb+{keyword}+\nenvironment.\n\n\\begin{vquote}\n\\begin{abstract}\n This is a abstract. \\lipsum[3]\n\\end{abstract}\n\n\\begin{keywords}\n First keyword \\sep Second keyword \\sep Third \n keyword \\sep Fourth keyword\n\\end{keywords}\n\\end{vquote}\n\n\\section{Main Matter}\n\\subsection{Tables}\n\\subsubsection{Normal tables}\n\n\\begin{vquote}\n\\begin{table}\n \\caption{This is a test caption.}\n \\begin{tabular*}{\\tblwidth}{@{} LLLL@{} }\n \\toprule\n Col 1 & Col 2\\\\\n \\midrule\n 12345 & 12345\\\\\n 12345 & 12345\\\\\n 12345 & 12345\\\\\n \\bottomrule\n \\end{tabular*}\n\\end{table}\n\\end{vquote}\n\n\\subsubsection{Span tables}\n\n\\begin{vquote}\n\\begin{table*}[width=.9\\textwidth,cols=4,pos=h]\n \\caption{This is a test caption.}\n \\begin{tabular*}{\\tblwidth}{@{} LLLLLL@{} }\n \\toprule\n Col 1 & Col 2 & Col 3 & Col4 & Col5 & Col6 & Col7\\\\\n \\midrule\n 12345 & 12345 & 123 & 12345 & 123 & 12345 & 123 \\\\\n 12345 & 12345 & 123 & 12345 & 123 & 12345 & 123 \\\\\n 12345 & 12345 & 123 & 12345 & 123 & 12345 & 123 \\\\\n \\bottomrule\n \\end{tabular*}\n\\end{table*}\n\\end{vquote}\n\n\\subsection{Figures}\n\\subsubsection{Normal figures}\n\\begin{vquote}\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[scale=.75]{Fig1.pdf}\n\t\\caption{The evanescent light - $1S$ quadrupole coupling\n\t($g_{1,l}$) scaled to the bulk exciton-photon coupling\n\t($g_{1,2}$). The size parameter $kr_{0}$ is denoted as $x$ and\n\tthe \\PMS is placed directly on the cuprous oxide sample ($\\delta\n\tr=0$, See also Fig. \\protect\\ref{FIG:2}).}\n\t\\label{FIG:1}\n\\end{figure}\n\\end{vquote}\n\n\\subsubsection{Span figures}\n\n\\begin{vquote}\n\\begin{figure*}\n\t\\centering\n\t \\includegraphics[width=\\textwidth,height=2in]{Fig2.pdf}\n\t\\caption{Schematic of formation of the evanescent polariton on\n\tlinear chain of \\PMS. The actual dispersion is determined by \n the ratio of two coupling parameters such as exciton-\\WGM \n coupling and \\WGM-\\WGM coupling between the microspheres.}\n \\label{FIG:2}\n\\end{figure*}\\end{vquote}\n\n\\subsection{Theorem and theorem like environments}\n\nCAS class file provides a few hooks to format theorems and\ntheorem like environments with ease. All commands the\noptions that are used with \\verb+\\newtheorem+ command will work\nexactly in the same manner. Class file provides three\ncommands to format theorem or theorem like environments:\n\n\\begin{enumerate}\n\\item \\verb+\\newtheorem+ command formats a theorem in\n\\LaTeX's default style with italicized font for theorem\nstatement, bold weight for theorem heading and theorem\nnumber typeset at the right of theorem heading. It also\noptionally accepts an argument which will be printed as an\nextra heading in parentheses. Here is an example coding and\noutput:\n\n\\begin{vquote}\n\\newtheorem{theorem}{Theorem}\n\\begin{theorem}\\label{thm}\n The \\WGM evanescent field penetration depth into the \n cuprous oxide adjacent crystal is much larger than the \n \\QE radius: \n \\begin{equation*}\n \\lambda_{1S}\/2 \\pi \\left({\\epsilon_{Cu2O}-1}\n \\right)^{1\/2} = 414 \\mbox{ \\AA} \\gg a_B = 4.6 \n \\mbox{ \\AA} \n \\end{equation*}\n\\end{theorem}\n\\end{vquote}\n\n\\item \\verb+\\newdefinition+ command does exactly the same\nthing as with except that the body font is up-shape instead\nof italic. See the example below:\n\n\\begin{vquote}\n\\newdefinition{definition}{Definition}\n\\begin{definition}\n The bulk and evanescent polaritons in cuprous oxide\n are formed through the quadrupole part of the light-matter\n interaction:\n \\begin{equation*}\n H_{int} = \\frac{i e }{m \\omega_{1S}} {\\bf E}_{i,s} \n \\cdot {\\bf p}\n \\end{equation*}\n\\end{definition}\n\\end{vquote}\n\n\\item \\verb+\\newproof+ command helps to define proof and\ncustom proof environments without counters as provided in\nthe example code. Given below is an example of proof of\ntheorem kind.\n\n\\begin{vquote}\n\\newproof{pot}{Proof of Theorem \\ref{thm}}\n\\begin{pot}\n The photon part of the polariton trapped inside the \\PMS\n moves as it would move in a micro-cavity of the effective\n modal volume $V \\ll 4 \\pi r_{0}^{3} \/3$. Consequently, it\n can escape through the evanescent field. This evanescent\n field essentially has a quantum origin and is due to\n tunneling through the potential caused by dielectric\n mismatch on the \\PMS surface. Therefore, we define the\n \\emph{evanescent} polariton (\\EP) as an evanescent light -\n \\QE coherent superposition.\n\\end{pot}\n\\end{vquote}\n\n\\end{enumerate}\n\n\\subsection{Enumerated and Itemized Lists}\n\nCAS class files provides an extended list processing macros\nwhich makes the usage a bit more user friendly than the\ndefault LaTeX list macros. With an optional argument to the\n\\verb+\\begin{enumerate}+ command, you can change the list\ncounter type and its attributes. You can see the coding and\ntypeset copy. \n\n\\begin{vquote}\n\\begin{enumerate}[1.]\n \\item The enumerate environment starts with an optional\n argument `1.' so that the item counter will be suffixed\n by a period as in the optional argument.\n \\item If you provide a closing parenthesis to the number in the\n optional argument, the output will have closing \n parenthesis for all the item counters.\n \\item You can use `(a)' for alphabetical counter and `(i)' for\n roman counter.\n \\begin{enumerate}[a)]\n \\item Another level of list with alphabetical counter.\n \\item One more item before we start another.\n \\begin{enumerate}[(i)]\n \\item This item has roman numeral counter.\n\\end{vquote}\n\n\\begin{vquote}\n \\item Another one before we close the third level.\n \\end{enumerate}\n \\item Third item in second level.\n \\end{enumerate}\n \\item All list items conclude with this step.\n\\end{enumerate}\n\n\\section{Biography}\n\n\\verb+\\bio+ command have the below options:\n\\begin{enumerate}\n \\item \\verb+width:+ Width of the author photo (default is 1in).\n \\item \\verb+pos:+ Position of author photo.\n\\end{enumerate}\n\n\\begin{vquote}\n\\bio[width=10mm,pos=l]{tuglogo.jpg}\n \\textbf{Another Biography:}\n Recent experimental \\cite{HARA:2005} and theoretical\n \\cite{DEYCH:2006} studies have shown that the \\WGM can travel\n along the chain as \"heavy photons\". Therefore the \\WGM \n acquires the spatial dispersion, and the evanescent \n quadrupole polariton has the form (See Fig.\\ref{FIG:3}):\n\\endbio\n\\end{vquote}\n\n\\section[CRediT...]{CRediT authorship contribution statement}\n\nGive the authorship contribution after each author as \n\n\\begin{vquote}\n \\credit{Conceptualization of this study, Methodology, \n Software}\n\\end{vquote}\n\nTo print the details use \\verb+\\printcredits+ \n\n\\begin{vquote}\n \\author[1,3]{V. {{\\=A}}nand Rawat}[auid=000,\n bioid=1,\n prefix=Sir,\n role=Researcher,\n orcid=0000-0001-7511-2910]\n\\end{vquote}\n\n\\begin{vquote}\n \\cormark[1]\n \\fnmark[1]\n \\ead{cvr_1@tug.org.in}\n \\ead[url]{www.cvr.cc, www.tug.org.in}\n\n \\credit{Conceptualization of this study, Methodology, \n Software}\n\n \\address[1]{Indian \\TeX{} Users Group, Trivandrum 695014, \n India}\n\n \\author[2,4]{Han Theh Thanh}[style=chinese]\n\n \\author[2,3]{T. Rishi Nair}[role=Co-ordinator,\n suffix=Jr]\n \\fnmark[2]\n \\ead{rishi@sayahna.org}\n \\ead[URL]{www.sayahna.org}\n\n \\credit{Data curation, Writing - Original draft preparation}\n\n . . .\n . . .\n . . .\n \\printcredits\n\\end{vquote}\n\n\\section{Bibliography}\n\nFor CAS categories, two reference models are recommended.\nThey are \\file{model1-num-names.bst} and \\file{model2-names.bst}.\nFormer will format the reference list and their citations according to\nnumbered scheme whereas the latter will format according name-date or\nauthor-year style. Authors are requested to choose any one of these\naccording to the journal style. You may download these from \n\nThe above bsts are available in the following location for you to\ndownload:\n\n\\url{https:\/\/support.stmdocs.in\/wiki\/index.php?title=Model-wise_bibliographic_style_files} \n\\hfill $\\Box$\n\n\\end{document}\n\n\n\\section{Introduction}\n\nThe inherent property of wetting refers to the preferential affinity of a fluid that is immersed in another immiscible fluid to coat a solid material \\cite{de1985wetting,bonn2009wetting,de2013capillarity}. Due to the ubiquitous existence of colloidal and interfacial phenomena in nature and applications, understanding the role of wetting is a key principal of interest. This intriguing interest in wetting behavior is motivated by numerous advanced technologies in nanotechnology, biological engineering, material science and geosciences \\cite{powell2011electric,xu2014proteins,blossey2003self,bartels2017oil}. For instance, the design of bio-inspired fluidics, directional fluid transportation, composite functional materials, nano\/micro-fluidics and energy storage systems require a precise and effective way to describe the wetting state therein. \n\nSince 1805, Young's equation, based on thermodynamic laws, is firmly established to infer the wettability of a flat and chemically homogeneous solid surface at equilibrium \\cite{young1805iii}, $\\cos \\theta_Y = \\frac{\\sigma_{ls} - \\sigma_{vs}}{\\sigma_{lv}}$, where the subscripts denote the immiscible fluids (liquid $l$ and vapor $v$) and solid ($s$) for the associated surface free energies. $\\theta_Y$ is the contact angle formed at the microscopic contact point on the surface, as depicted in Fig. \\ref{fig1}. Wetting hysteresis due to the contribution of surface heterogeneity and contact line dynamics has been studied in detail over the last century \\cite{wenzel1936resistance,cassie1944wettability,decker1997contact}. However, despite theoretical and experimental investigations through the last 70 years \\cite{de1985wetting,bonn2009wetting,yu2015wetting,turmine2000thermodynamic,blunt2019thermodynamically,mahani2015kinetics}, several fundamental challenges in characterizing wetting state of complex multiphase systems remain unsolved and are currently pending. \n\nFor disordered and complex porous geometries ranging from catalysts in fuel cells to lungs in the respiratory system to porous glass filters to subsurface rocks, there are multiple length scales, various surface free energies, and surface heterogeneity involved. The convoluted interplay of physicochemical properties with multi-scale complexities therein presents significant spatial variability of contact angles along the three-phase contact line \\cite{sun2020probing,holtzman2015wettability}, which leads to the resulting wetting hysteresis behavior and pinning effects. Accordingly, these features trigger a formidable challenge to characterizing wetting behavior, which are far from being solved. It is therefore controversial whether Young's law is still applicable for these disordered and complex solids \\cite{garfi2019fluid}. \n\nIn the past, contact angle (and curvature) measurements were mainly based on two-dimensional (2D) projections, and in configurations that are optically transparent such as a typical sessile drop setup \\cite{kwok1997contact}. However, three-dimensional (3D) imaging provides the possibility to determine contact angles within opaque porous media \\cite{scanziani2017automatic,alratrout2017automatic,ibekwe2020automated,dalton2018methods}. Possible 3D imaging technologies are 3D X-ray microcomputed tomography (micro-CT) \\cite{wildenschild2013x} and confocal microscopy \\cite{sundberg2007contact}. Given the rapid developments in 3D imaging technologies, there is an opportunity for interface science to make use of this methodology where the wetting of various porous domains would be of interest. The first community that started adopting this new technology is the porous media community, out of pure necessity to measure the wettability of immiscible fluids in geological materials, which are naturally opaque \\cite{andrew2014pore,alratrout2018wettability,tudek2017situ}. These efforts rely on micro-CT images of porous rocks saturated with immiscible phases \\cite{wildenschild2013x}. The image voxels of a 3D system are segmented into respective phases \\cite{schluter2014image} followed by the identification of the three-phase contact line to measure spatially distributed apparent contact angles, $\\theta_{app}$, for each microscopic three-phase contact point along the contact line. As shown in the right side of Fig. \\ref{fig1}, the method measures the angle between the local tangential plane of liquid\/vapor interface and the solid surface in the vicinity of three-phase contact points. Thus, the approach represents \\textit{in situ} $\\theta_{app}$ directly along the contact line, which refers to microscopic wetting. However, the simplicity of $\\theta_{app}$ computed by the local method is susceptible to errors due to image quality and pixelation-related segmentation errors \\cite{armstrong2012linking,klise2016automated}. In particular, the identification of the three-phase contact line and measurement of an angle over a few voxels is inherently error-prone, as will be investigated herein. Ultimately, these systematic errors complicate the process of quantifying the wetting state of a porous multiphase system. \n\nAnother issue arises by using for the purpose of wetting characterization the concept of contact angle, which is subject to hysteresis. Due to hysteresis that is emphasized by complex geometries, surface heterogeneity, and interface dynamics, it is very questionable whether contact angle is a representative measure for characterizing wetting in such systems \\cite{wenzel1936resistance,cassie1944wettability,johnson1964contact,morrow1975effects,priest2007asymmetric}. In addition, the line tension and disjoining\/cojoining pressure occur along the contact line. Contact line tension, which is the excess free energy per unit length, is a dominant parameter in microscopic wetting \\cite{indekeu1994line}. It contributes to intermolecular force balance, resulting in contact line pinning and the associated hysteresis loop of contact angles \\cite{de1985wetting,bonn2009wetting,de2013capillarity}. Therefore, the variations on contact line curvature and the associated $\\theta_{app}$ vary from a microscopic point of view due to the effects of surface heterogeneity and flow dynamics, which yields unexpected wetting behavior. Despite these parameters influencing microscopic wetting and hysteresis behavior, the question arises as to whether $\\theta_{app}$ can capture enough information to represent the wetting state of the system and provide sufficient guidance for the design of functional surfaces?\n\n\\begin{figure}\n\\hspace*{-0.2cm}\\includegraphics[width=0.51\\textwidth]{FIG1.png}\n\\caption{The schematic illustration of topological principle based on integral geometry \\cite{sun2020probing} and Young's equation based on thermodynamics at a microscopic contact point when a sessile droplet is deposited on a flat solid substrate exposed to vapor phase. The schematic illustration of apparent contact angle $\\theta_{app}$ extraction by local measurement along the contact line.}\n\\label{fig1}\n\\end{figure}\n\nPrevious studies demonstrate that $\\theta_{app}$ alone provides an incomplete description of wetting where the contact line is asymmetric \\cite{rabbani2018pore}; especially for characterizing the macroscopic wetting behavior of multiphase systems, which has remained unexplored from a theoretical perspective. The wetting behavior synergistic affects on the phase topology and contact area with the solid surface, which must also be considered. It can be representative of the macroscopic wetting of the system and captures the complete microscopic information related to the thermodynamics. To this end, in our previous works \\cite{sun2020linking,sun2020probing}, we developed a theory based on topological principles that can effectively describe wetting behavior, which links across various length scales pertinent to wetting phenomena. Herein, we aim to quantitatively and qualitatively unravel the effective and robust nature of the proposed theory by providing in-depth analysis and comparison to other recent methods for characterizing the wettability of multiphase systems. \n\n\\section{Theoretical Concepts}\n\\subsection{Deficit Curvature from Gaussian Curvature}\nTopological principle is applied to characterize the wetting state of a given multiphase system by the link of the Gauss-Bonnet theorem. For a fluid droplet $D$ that has a closed interface in the three-phase system, where liquid ($l$), vapor ($v$) and solid ($s$) are present, the total curvature of the fluid surface and its global topology are related by the Gauss-Bonnet theorem \\cite{chern1944simple,sun2020linking}. As a consequence, the Euler characteristic $\\chi$ and its total curvature for the surface $I$ of the droplet $D$ obey the following expression,\n\n\\begin{equation} \\label{eq:1}\n 2\\pi \\chi(D)=\\int_{I} \\kappa_G dA + \\int_{\\partial I} \\kappa_g dC,\n\n\n\n\\end{equation}\nwhere $dA$ and $dC$ are the droplet interfacial area element along the surface $I$ and the line element along the contact line $\\partial I$, respectively. $\\kappa_G$ and $\\kappa_g$ are Gaussian curvature along the droplet interface and geodesic curvature along the contact line $\\partial I$, respectively. \n\\begin{figure*}\n\\centering\\includegraphics[width=0.85\\textwidth]{FIG2}\n\\caption{Schematic diagram of 3D droplets on contact with a complex solid (transparent) with labeled definitions where grey color denotes the liquid\/vapor interface ($I_{lv}$), and blue color denotes the liquid\/solid interface ($I_{ls}$). The contact line loops are formed by three-phase contact points (yellow circles). From left to right, there is an increasing complexity of droplet morphology.}\n\\label{fig2}\n\\end{figure*}\n\nFor a 3D droplet, we can arrive at a generalized form of the expression by subdividing the fluid surface into liquid\/vapor ($lv$) and liquid\/solid ($ls$) interfaces,\n\n\\begin{equation} \\label{eq:2}\n\\begin{split}\n 4\\pi\\chi(D) &=2 \\pi \\chi(I_{lv}) + 2 \\pi \\chi (I_{ls}) \\\\\n &=\\int_{I_{lv}} \\kappa_{G}dA+\\int_{I_{ls}} \\kappa_{G}dA +\\int_{\\partial I}(\\kappa_{g_{lv}}+\\kappa_{g_{ls}})dC. \n\\end{split}\n\\end{equation}\n\n\\noindent It indicates that the Euler characteristic of the droplet always remains constant and the geodesic curvature source term, $\\int_{\\partial I}(\\kappa_{g_{lv}}+\\kappa_{g_{ls}})dC$, will change accordingly based on the contribution of total surface curvature due to the wetting behavior of the system. The deficit curvature, $\\Theta$, is defined as the summation of geodesic curvatures along the contact line relative to the tangential plane for each interface and corresponds to a total angle of change. By considering a droplet deposited on a flat, smooth and homogeneous surface as shown in the top of Fig.\\ref{fig1}, $\\Theta$ along the contact line can be expressed by the labeled notations as,\n\n\\begin{equation} \\label{eq:3}\n\\begin{split}\n\\Theta &= \\int_{\\partial I}(\\kappa_{g_{lv}}+\\kappa_{g_{ls}})dC \\\\\n&=\\int_{\\partial I} \\kappa_{lvs}\\mathbf{n}_{lvs}\\cdot \\big[\\mathbf{n}_s \\sin \\theta_Y + \\mathbf{n}_{ls} (1-\\cos \\theta_Y) \\big]dC,\n\\end{split}\n\\end{equation}\n\n\\noindent where $\\mathbf{n}_{lvs}$ is the normal vector to the contact line, which points in the direction of the curvature for the contact line. $\\mathbf{n}_{lv}$ is the outward normal vector along the contact line relative to the droplet interface. $\\mathbf{n}_{ls}$ is the outward normal vector along the contact line relative to the liquid\/solid interface. From Eq. (\\ref{eq:3}), it is evident that $\\Theta$ obtained by applying the topological principle is an explicit average of intrinsic contact angle, i.e. Young's angle ($\\theta_Y$) for this situation. Therefore, $\\Theta$ can be interpreted in terms of fluid morphology in a way that is not affected by the contact angle hysteresis due to surface heterogeneity, geometry complexity, and interface dynamics \\cite{sun2020probing}.\n\nFor solid surfaces that contain disordered and complex geometries or even with confined domains as shown in Fig. \\ref{fig2}, we can revisit Eq. (\\ref{eq:2}) to obtain $\\Theta$. Macroscopic contact angle $\\theta_{macro}$ can be obtained by using a normalizing factor $\\lambda$ to scale the desired contact angle interval in $\\theta_{macro}$ $\\in [0,\\pi]$,\n\n\\begin{equation} \\label{eq:4}\n\\begin{split}\n\\theta_{macro} &= \\lambda \\Theta\\\\\n&= \\lambda \\Big[4\\pi\\chi(D) - \\Big(\\int_{I_{lv}} \\kappa_{G}dA+\\int_{I_{ls}} \\kappa_{G}dA \\Big)\\Big].\n\\end{split}\n\\end{equation}\nThe nature of this formulation resolves a few pressing issues. Firstly, the angle along the contact line is not directly computed as an average from the sequence of local geometric contact angle point measurements. $\\theta_{macro}$ is inferred from interfacial curvature and area measurements along with a topological measurement. While these measures are still susceptible to pixelization effects, it is expected that these effects would be less than that resulting from local measurements along the three-phase contact line (as tested herein). Secondly, $\\theta_{macro}$ accounts for the complete geodesic curvature of the contact line, which is the curvature of the three-phase contact line relative to both the solid surface and liquid\/vapor interface. Therefore, the formulation captures wetting effects as evident by line tension and $\\theta_{app}$, which are both known to influence wetting behavior. \n\n\\subsection{Derivation of Young's Equation}\nIn this section, we show the direct link between the topological and thermodynamic concepts for determining the wetting state of the system by applying surface energy minimization and variational principles \\cite{seo2015re}. The Gauss-Bonnet theorem allows us to express the total curvature of a droplet $D$ in terms of deficit curvature, corresponding average curvature and surface area, \n\n\\begin{align}\n4 \\pi \\chi(D) = \\Theta + \\kappa_{lv} A_{lv} + \\kappa_{ls} A_{ls} \\;.\n\\label{eq:5}\n\\end{align}\n\nHere, we consider a variational principle of the internal energy as applied in thermodynamic approaches. The variation of the internal energy is given by Euler's homogeneous function theorem,\n\n\\begin{align}\n\\delta U = T \\delta S - p_v \\delta V_v - p_l\\delta V_l + \\sigma_{lv} \\delta A_{lv}\n\\nonumber\\\\ + \\sigma_{ls}\\delta A_{ls} + \\sigma_{vs} \\delta A_{vs} \\;.\n\\label{eq:6}\n\\end{align}\nBased on this, we consider a closed system with $\\delta U=0$. Droplet rearrangements can occur provided that the volume is not changed, meaning that $\\delta V_l = \\delta V_v = 0$. Therefore, entropy production in the system is strictly linked to the minimization of the surface energy,\n\n\\begin{align}\n\\delta S = - \\frac 1 T \\Big[ \\sigma_{lv} \\delta A_{lv}\n+ \\sigma_{ls}\\delta A_{ls} + \\sigma_{vs} \\delta A_{vs} \\Big] \\ge 0 \\;.\n\\label{eq:7}\n\\end{align}\nThe total surface area of the solid substrate is constant, then \n\n\\begin{align}\n\\delta A_s = \\delta A_{vs} + \\delta A_{ls} = 0 \\;,\n\\label{eq:8}\n\\end{align}\nwhich can be used to eliminate one of the surface areas from Eq. (\\ref{eq:7}). The topological constraint from Eq. (\\ref{eq:5}) determines the condition that should be imposed to ensure that geometric variation occurs without changing the topology,\n\n\\begin{equation} \\label{eq:9}\n\\begin{split}\n\\delta \\big[ 4 \\pi \\chi(D)\\big] & = \\delta \\Theta + \n\\kappa_{lv} \\delta A_{lv} + A_{lv} \\delta \\kappa_{lv} \\\\ &\\quad\n\\kappa_{ls} \\delta A_{ls} + A_{ls} \\delta \\kappa_{ls} \\\\ & = 0\n \\;.\n\\end{split}\n\\end{equation}\n\nWe shall impose Eq. (\\ref{eq:9}) as a constraint on Eq. (\\ref{eq:7}) using the method of Lagrange multipliers, also using Eq. (\\ref{eq:8}) to eliminate $A_{vs}$. Now, we can express the entropy production as\n\n\\begin{equation}\n\\begin{split}\n\\delta S &= - \\frac 1 T \\Big[\n\\underbrace{\\Big( \\sigma_{ls} - \\sigma_{vs} + \\frac{\\sigma_{lv}\\kappa_{ls}}{\\kappa_{lv}} \\Big)\n}_{\\mbox{surface area variation}}\\delta A_{ls} \\\\\n &\\quad + \\frac{\\sigma_{lv}}{\\kappa_{lv}} \n\\underbrace{\\big( \\delta \\Theta + \n A_{lv} \\delta \\kappa_{lv} + A_{ls} \\delta \\kappa_{ls} \\big)}_{\\mbox{total curvature variation}} \\Big]\n \\;. \n\\end{split}\n \\label{eq:10}\n\\end{equation}\nThis expression separates terms associated with the variation of the surface area from terms that redistribute curvature along the cluster boundary. Put another way, the second term corresponds to a redistribution of the total curvature that occurs at constant surface area due to the deformation of the droplet. One of the ways to redistribute the total curvature is by altering the deficit curvature, which changes the contact angle. Since $T > 0$, the inequality can be re-expressed in the form,\n\n\\begin{equation}\n\\Big( \\frac{\\sigma_{ls} - \\sigma_{vs}}{\\sigma_{lv}} \\kappa_{lv} + \\kappa_{ls} \\Big) \\delta A_{ls} \n+ \\delta \\Theta + A_{lv} \\delta \\kappa_{lv} + A_{ls} \\delta \\kappa_{ls} \\le 0\\;,\n \\ \n\\label{eq:11}\n\\end{equation}\nwhich is equivalent to stating that the surface energy of the system must decrease.\n\n\\noindent Since the solid surface is flat, which means $\\kappa_{ls} = 0$ and $\\delta \\kappa_{ls} = 0$. The variations can be computed directly based on expressions for a spherical cap with droplet radius $R$ and droplet height $h$ to the solid surface, where\n\n\\begin{equation}\n\\begin{split}\n\\Theta &= 2 \\pi (1-\\cos \\theta) \\\\\n\\kappa_{lv} &= \\frac{1}{R^2} \\\\\nA_{lv} &= 2 \\pi R h \\\\\nA_{ls} &= \\pi h (2R-h)\n\\end{split}\n\\label{eq:12} \n\\end{equation}\n\nand the associated variations\n\n\\begin{equation} \\label{eq:13}\n\\begin{split}\n\\delta \\Theta &= \\delta \\big [ 2 \\pi (1-\\cos \\theta) \\big ] \n = 2 \\pi (h R^{-2}\\delta R - R^{-1} \\delta h ) \\\\\n \\delta \\kappa_{lv} &= \\delta \\big [ R^{-2} \\big] = -2 R^{-3} \\delta R \\\\\n\\delta A_{ls} &= \\delta \\big [\\pi h (2R-h) \\big] =\n \\pi \\big( 2 h \\delta R + 2 R \\delta h - 2 h \\delta h \\big) \n\\end{split}\n\\end{equation}\nThe volume of a spherical cap is $V = \\frac 1 3 \\pi h^2 \\big( 3 R-h \\big)$, and setting $\\delta V=0$ imposes the relationship $\\delta R = \\Big( 1 - \\frac{2R}{h} \\Big) \\delta h$. Inserting this into expressions above, we obtain\n\n\\begin{equation}\n\\begin{split}\n\\delta \\Theta &= 2 \\pi \\Big(\\frac{h}{R^2} - \\frac{3}{R} \\Big) \\delta h \n\\\\\n\\delta \\kappa_{lv} &= \\Big(\\frac{4}{R^2h} - \\frac{2}{R^3} \\Big) \\delta h \\\\\n\\delta A_{ls} &= -2 \\pi R \\delta h\n\\end{split}\n\\label{eq:14}\n\\end{equation}\nInserting Eqs. (\\ref{eq:12}) and (\\ref{eq:14}) into Eq. (\\ref{eq:11}) and rearranging terms gives\n\n\\begin{eqnarray}\n \\frac{2\\pi \\delta h }{R} \\Bigg\\{\n1 - \\frac{h}{R} -\\frac{\\sigma_{cs} - \\sigma_{as}}{\\sigma_{ca}} \n\\Bigg\\}\\le 0\\;.\n\\end{eqnarray}\nNoting that $cos \\theta = 1 - h\/R$, we observe that for a general variation $\\delta h$\nand $R>0$, the entropy maximum will be obtained based on the condition\n\n\\begin{align}\n \\cos \\theta -\\frac{\\sigma_{ls} - \\sigma_{vs}}{\\sigma_{lv}} = 0\\;,\n\\end{align}\nwhich is Young's equation for the contact angle. The classical result is thereby obtained when the shape of the contact line is radially symmetric. This outcome provides evidence to demonstrate that the proposed topological concept is a general explicit geometric statement and also has a direct link to classical thermodynamics. \n\n\\section{Computation of Macroscopic Contact Angle}\n\\begin{algorithm}\n \\caption{Implementation in computation of macroscopic contact angles for droplets in multiphase system.}\n \\label{codel}\n \\begin{algorithmic}\n \\For{each droplet in the system}\n \\State 2D surface manifold generation by marching cubes algorithm\n \\State Gaussian smooth for the manifold:\n \\State $v_i' = v_i +\\alpha \\sum w_{ij}(v_j - v_i)$\n \\For{each triangle on the manifold}\n \\State Compute triangle area:\n \\State $A_i = \\frac{1}{2}|v_{12} \\times v_{13}|$\n \\State Compute principal curvatures $\\kappa_1$ and $\\kappa_2$\n \\State Compute Gaussian curvature:\n \\State $\\kappa_{Gi} = \\kappa_1 \\kappa_2$\n \\State Compute mean curvature:\n \\State $\\kappa_{Mi} = \\frac{\\kappa_1 + \\kappa_2}{2}$\n \\If {$\\kappa_M < 0$}\n \\State $\\kappa_{Gi} A_i = - \\kappa_{Gi} A_i$\n \\EndIf\n \\EndFor\n \\State Compute $\\theta_{macro}$ by normalizing deficit curvature with the number of contact line loop $N$:\n \\State $\\theta_{macro} = \\lambda \\Theta = \\frac{4 \\pi - \\sum \\kappa_{Gi} A_i}{4N}$\n \\EndFor\n \\end{algorithmic}\n\\end{algorithm}\n\\begin{figure}\n\\hspace*{-0.5cm}\\includegraphics[width=0.5\\textwidth]{FIG3}\n\\caption{The distribution of (a) - (c) Gaussian curvatures and (d) - (f) mean curvatures on the droplet interface for various degrees of $n_{Layer}$, which determines the number of triangles considered to be neighbours of a given point. Consequently, it determines the quadratic form equation of the surface patch. As the value of $n_{Layer}$ increases, the number of the surrounding triangles to be considered as neighbours increases, which leads to more accurate results.}\n\\label{fig3}\n\\end{figure}\n\nTo explain the implementation of the proposed topological principle to characterize the wetting state of the system, we first chose a droplet in a complex and confined domain from the segmented image for illustration. By modeling the whole droplet surface manifold using a generalized marching cubes algorithm \\cite{hege1997generalized}, we performed a triangular approximation of the interface for each region (e.g., liquid\/vapor and liquid\/solid) to preserve the droplet topology. To obtain accurate results and remove extreme outliers of curvatures, a simplification of the triangles by applying an edge collapse algorithm for the droplet surface was required to omit small non-smooth regions. For instance, the small concave spots in the convex regions will be degenerated during the triangle contraction process. The area of each triangle $A_i$ can be computed by the cross product of the two adjacent vectors ($v_{12}$, $v_{13}$) formed by the vertices ($v_1$, $v_2$ and $v_3$), which is $A_i = \\frac{1}{2}|v_{12} \\times v_{13}|$. Further, we smoothed the surface manifold by shifting its vertices to minimize the voxelization effects and segmentation errors. Each new vertex position ($v_i'$) was shifted towards the average position of its neighbours ($v_j$) by the weights ($w_{ij}$). Therefore, the vector average for the vertex can be obtained,\n\n\\begin{equation}\n \\Delta v_i=\\sum w_{ij}(v_j - v_i),\n\\end{equation}\nand the updated vertex position will be\n\n\\begin{equation}\n v_i' = v_i +\\alpha \\Delta v_i,\n\\end{equation}\nwhere $\\alpha$ represents a scale factor ranging from $0$ to $1$. We then computed the magnitude and direction of the principal curvatures ($\\kappa_1$ and $\\kappa_2$) by fitting a quadratic form equation on the desired surface patch and obtaining its corresponding eigenvalues and eigenvectors \\cite{armstrong2012linking}. The Gaussian curvature ($\\kappa_{Gi}$) and mean curvature ($\\kappa_{Mi}$) for each triangle can thereby be obtained in terms of the principal curvatures,\n\n\\begin{equation}\n\\begin{split}\n \\kappa_{Gi} &= \\kappa_1 \\kappa_2,\\\\\n \\kappa_{Mi} &= \\frac{\\kappa_1 + \\kappa_2}{2}.\n\\end{split}\n\\end{equation}\n\\begin{figure}\n\\hspace*{-0.5cm}\\includegraphics[width=0.53\\textwidth]{FIG4}\n\\caption{Measured contact angles for different simulated droplets with increasing window size. It shows that an optimal window size of $60$ pixels is sufficient to accurately measure the apparent contact angle by the 2D local method.}\n\\label{fig4}\n\\end{figure}\nThe values of mean curvature provide the information of droplet interface, where the positive values of mean curvature indicate the convex interface and negative values indicate the concave interface as shown in Fig. \\ref{fig3}d-f. In addition, there is a negative sign assigned to $\\kappa_{Gi}A_i$ for concave interfaces and a value of $0$ assigned for flat interfaces. Finally, we can obtain the macroscopic contact angle, $\\theta_{macro}$, for the droplet by normalizing the deficit curvature $\\Theta$ as illustrated in Eq. (\\ref{eq:4}),\n\n\\begin{equation}\n \\theta_{macro} = \\lambda \\Theta = \\frac{4 \\pi - \\sum \\kappa_{Gi} A_i}{4N}\n\\end{equation}\nwhere $N$ is the count of contact line loops. For more information, please see \\cite{sun2020probing}. For multiphase systems where a large number of droplets are present, such as porous media and subsurface rocks, the proposed implementation procedure as described in Algorithm \\ref{codel} enables the computation of macroscopic contact angles on a droplet-by-droplet basis.\n\n\\section{Results and Discussion}\n\\subsection{Wetting in Flat and Ideally-smooth Surfaces}\n\nTo unravel the effective and robust nature of the developed theory, we first compare the macroscopic contact angle $\\theta_{macro}$ and the apparent microscopic contact angles $\\theta_{app}$ on a flat and smooth surface. We performed a quasi-steady-state simulation of 3D sessile oil droplets immersed in immiscible ambient water on various flat surfaces that have different intrinsic contact angles defined by Young's equation. It is a more ideal solution for performing simulation than experiments where the droplet after deposition achieves a different contact angle, i.e. wetting hysteresis, when compared to the intrinsic contact angle. This is due to the difficulties in attaining experimental equilibrium condition and collecting ideally-flat surface \\cite{arjmandi2017kinetics}. The equilibrium topologies of the oil droplets on the surface were simulated by minimizing the overall system energy and were subjected to constraints such as surface tension and gravitational energy. In the system, the overall energy is the sum of its interfacial potential energy and is governed by the Young-Laplace equation. For the interfacial potential energy of the droplet ($E$), it can be expressed as the sum of the respective interfacial energies,\n\n\\begin{equation}\n E=\\int\\int_{A_{ls}}\\sigma_{ls}dA+\\int\\int_{A_{lv}}\\sigma_{lv}dA+\\int\\int_{A_{vs}}\\sigma_{vs}dA,\n\\label{eq:21}\n\\end{equation}\nwhere $\\sigma$ and $A$ represent interfacial tension and interfacial area, respectively. By applying Young's equation, Eq. (\\ref{eq:21}) becomes\n\n\\begin{equation}\n \\frac{E}{\\sigma_{lv}}=A_{lv}-\\int\\int_{A_{ls}}\\cos \\theta_Y dA.\n\\end{equation}\n\n\\begin{figure*}\n\\centering\\includegraphics[width=0.8\\textwidth]{FIG5}\n\\caption{The comparison diagram of $\\theta_{macro}$ and the contact angle distributions of the 3D local measurement by \\cite{alratrout2017automatic} and modified 2D local measurement by \\cite{scanziani2017automatic} for the simulated droplets with different intrinsic contact angles.}\n\\label{fig5}\n\\end{figure*}\n\nAs shown in Fig. \\ref{fig5}, the equilibrium topologies of oil droplets were simulated on a surface that gradually change from oleophobic to oleophilic. To obtain the microscopic apparent contact angles along the contact line, we implemented the 2D local method similar to \\cite{scanziani2017automatic} and the 3D local method developed by \\cite{alratrout2017automatic}. For the 2D local method, the contact points were smoothed using a moving average to determine the position and direction of the contact line. A 2D image was then extracted normal to the direction of the contact line for each contact point. In Fig. \\ref{fig4}, we computed the contact angle measurement with different window (slice) sizes for each simulated droplet. The results demonstrate that a window (slice) size of $60$ pixels is sufficient to provide an accurate contact angle. In contrast to \\cite{scanziani2017automatic}, both circular and linear regressions were applied to best fit the solid surface. The best fit of either the circular or linear regression was determined by the lower of the two root-mean-square deviations. If the circle approximation was selected as the best fit, a line tangent to the circle at the contact point was calculated to represent the slope of the surface at that point. In addition, a constant curvature of the liquid\/vapor interface was assumed under the assumption that the system is at equilibrium. Consequently, the apparent contact angle was measured as the angle between the solid surface and liquid\/vapor interface tangent lines \\cite{meisenheimer2020optimizing}. For the 3D local method, Gaussian smoothing was applied to the droplet surface. Then, the two vectors that have a direction perpendicular to liquid\/vapor and liquid\/solid interface were identified for each contact point along the contact line. The apparent contact angle was thereby computed from the dot product of these vectors. \n\n\\begin{figure*}\n\\centering\\includegraphics[width=0.9\\textwidth]{FIG6}\n\\caption{Top: The comparison of droplet topology as down-sampling the image resolution with measured local contact angles for each three-phase contact points. Bottom: The corresponding macroscopic contact angle ($\\theta_{macro}$) and 3D local measurements ($\\theta_{app}$) associated with their change by comparing with the original resolution. As deficit curvature is less sensitive to resolution effects, it is particularly attractive as a way to capture wetting on rough surfaces where sub-resolution heterogeneity can have a demonstrated influence on the measured macroscopic contact angle.}\n\\label{fig6}\n\\end{figure*}\n\nIn Fig. \\ref{fig5}, we compare the macroscopic contact angle $\\theta_{macro}$ distributions from the developed topological principle to the apparent contact angle distributions by the two local methods for each intrinsic contact angle. It is shown that the 2D local measurement has a higher standard deviation compared with the 3D local measurement. However, the mean values of the contact angle distribution for the 2D local measurement are closer to the intrinsic contact angle $\\theta_{in}$. Furthermore, the contact angle distribution calculated by 3D local method tends to have higher contact angles for more oleophobic surfaces and lower contact angles for more oleophilic surfaces, which provides a systematic bias (error) towards intermediate-wet conditions. Conversely, $\\theta_{macro}$ is slightly less than the intrinsic contact angle for oleophobic surfaces and slightly greater than the intrinsic contact angle for oleophilic surfaces. The trend observed for $\\theta_{macro}$ has a theoretical explanation since the total deficit curvature accounts for both local contact angles and the deformation of the contact line along the solid surface (line tension). The theoretical development for this particular aspect is explained in detail by Sun et al. \\cite{sun2020probing}. Overall, contact angles measured for flat and homogeneous surfaces are comparable for all three tested methods, which supports the theoretical developments in Section 2.2 since the proposed topological approach under these ideal conditions can be simplified to Young's equation. \n\n\\subsection{Wetting in Complex Geometries}\n\\begin{figure}\n\\centering\\includegraphics[width=0.5\\textwidth]{FIG7}\n\\caption{The contact angle distributions of different image resolution for sintered glass by applying (b) 3D local method for each contact point and (c) the topological approach on a droplet-by-droplet basis. The plot of (d) mean and (e) standard deviation of the distribution for each image resolution.}\n\\label{fig7}\n\\end{figure}\n\nTo further investigate the wetting state on heterogeneous surfaces where surface chemistry and roughness are present, we performed two quasi-static capillary forces dominated flow experiments for sintered glass at the Swiss Light Source and Bentheimer sandstone at the Australian National University \\cite{dataset}. The details of the experimental setup and image processing can be found in \\cite{schluter2016pore} and \\cite{sun2020probing}. The sintered glass has a relatively lower degree of surface geometry with smooth surfaces and a lower degree of surface chemistry. While for Bentheimer sandstone, the surface chemical heterogeneity, surface geometry and roughness are relatively higher. In Table \\ref{tab}, a summary of experimental setup and image processing is given for both sintered glass and Bentheimer sandstone. The 3D fluid configurations and spatial arrangements for both samples are displayed in Figs. \\ref{fig7}a and \\ref{fig8}a.\n\n\\begin{table*}\n \\centering\n \\caption{Summary of experimental setup and image processing.}\n \\begin{tabular}{L{2.7cm}C{5cm}C{5cm}}\n \\toprule \n & Sintered Glass & Bentheimer Sandstone\\\\\n \\midrule\n Porosity & $31.8 \\%$ & $24.1 \\%$\\\\\n Permeability & $21.5 \\pm 2$ $\\rm D$ & $4.3$ $\\rm D$\\\\\n Sample diameter & $4$ $\\rm mm$ & $4.9$ $\\rm mm$\\\\\n Sample length & $10$ $\\rm mm$ & $10$ $\\rm mm$\\\\\n Non-wetting phase & n-decane & ambient air\\\\\n Wetting phase & brine & brine\\\\\n Injection rate & $0.1$ $\\rm \\mu L\/min$ & $0.3$ $\\rm \\mu L\/min$\\\\\n Brine saturation & $78 \\%$ & $93 \\%$\\\\\n Imaging source & fast synchrotron-based tomography & bench-top helical micro-tomography\\\\\n Image resolution & $4.2$ $\\rm \\mu m$ & $4.95$ $\\rm \\mu m$\\\\\n Time step & $30$ $\\rm s$ & $1.35$ $\\rm hr$\\\\\n \\bottomrule\n \\end{tabular}\n \\label{tab}\n\\end{table*}\n\n\\begin{figure}\n\\centering\\includegraphics[width=0.5\\textwidth]{FIG8}\n\\caption{The contact angle distributions of different image resolution for Bentheimer sandstone by applying (b) 3D local method for each contact point and (c) the topological approach on a droplet-by-droplet basis. The plot of (d) mean and (e) standard deviation of the distribution for each image resolution.}\n\\label{fig8}\n\\end{figure}\n\nTo investigate the resolution effects, we first performed a resolution study by picking a well-resolved droplet from the dataset of Bentheimer sandstone. The original dimension of image for the droplet is $47 \\times 51 \\times 48$ voxels. We then down-sample the image of the droplet by halving the resolution twice as shown in Fig. \\ref{fig6}a-c. It is intuitive in Fig. \\ref{fig6} how the down-sampling impacts the droplet interfacial curvature and contact line loops. The results demonstrate that the macroscopic contact angle based on the deficit curvature is less sensitive to resolution effects than local contact angle measurements, since the change in the mean value of $\\theta_{app}$ is larger than the change in $\\theta_{macro}$. Local contact angle measurements cannot be performed without resolving the contact line region. It is clear that the resolution necessary to measure the angles along the contact line is lost before the topological structure of the contact line loops is destroyed. In Fig. \\ref{fig6}a-c, we investigate that the local contact angles vary significantly as the resolution decreases. For the topological approach based on the Gauss-Bonnet theorem, there remains a basis to obtain deficit curvature when the topological structure of the contact line loops is captured, even if only a single voxel represents the contact line loop as shown on the bottom right of Fig. \\ref{fig6}c.\n\nFurther, we performed the contact angle distribution measurements for both samples on a collection of droplets by applying the topological approach for each droplet and 3D local method for each contact point. By gradually decreasing the image resolution as shown in Figs. \\ref{fig7} and \\ref{fig8}, the changes in the mean and standard deviation of the contact angle distributions for the topological approach are notably less than that of the local measurements. For sintered glass, the mean of the contact angle distribution decreases by an additional $7 \\%$ while there is an additional $31 \\%$ reduction in the standard deviation for the topological approach. Similar results for the topological method are investigated in the Bentheimer sandstone with a decrease in mean contact angle and standard deviation of $12 \\%$ and $27 \\%$, respectively. Overall, at sufficiently high resolution, the topological approach and local measurements are comparable. While at lower resolutions, the topological approach deviates less from the high resolution results than the local contact angle measurements. \n\nLastly, we performed contact angle measurements for both samples by applying the topological approach, 2D and 3D local methods with the highest image resolution obtained as shown in Fig. \\ref{fig9}. It reveals that the 2D local measurement provides a lower mean and a higher standard deviation of the contact angle distribution. However, the mean value of the 3D local measurement is the highest, which can be explained in the simulation of droplets results for the flat and homogeneous surfaces. For more oleophobic (water-wet) surfaces, contact angle measurements computed by the 3D local method is larger than the intrinsic contact angles. The $\\theta_{macro}$ distribution computed by the topological approach falls between the two local methods. Overall, the topological and 3D local methods are comparable at sufficiently high resolution, suggesting that microscopic wetting information is captured. \n\n\\begin{figure}\n\\hspace*{-0.5cm}\\includegraphics[width=0.5\\textwidth]{FIG9}\n\\caption{The contact angle distributions for (a) sintered glass and (b) Bentheimer sandstone by applying topological approach, 2D and 3D local methods with the highest resolution obtained together with the associated mean and standard deviation.}\n\\label{fig9}\n\\end{figure}\n\n\\section{Conclusions}\n\nRecent methods for characterizing wetting behavior rely on restrictive thermodynamic laws such as Young's equation for flat homogeneous surfaces, and Wenzel and Cassie-Baxter models for heterogeneous surfaces \\cite{young1805iii,wenzel1936resistance,cassie1944wettability}. In this work, we developed a new theory based on topological principles to determine the wetting state of multiphase systems. Specifically, the concept of thermodynamic equilibrium is not necessary since the formulation is a geometrical statement between the total surface curvature and global topology of a fluid object. In this regard, the proposed methodology does not require visualization of the three-phase contact line. It thus would be advantageous for applications where a large fluid body is resolved while the contact line pinning occurs at a smaller length scale. This could be particularly informative for problems where the droplet is connected to a thin film, such as a droplet slipping along a declining plane and\/or the corner flow mechanism observed in porous media flows. We also highlight that the proposed theory is consistent with Young's equation for flat and homogeneous surfaces by performing a variational analysis.\n\nBy comparing with traditional local contact angle measurements \\cite{alratrout2017automatic,scanziani2017automatic,klise2016automated}, we assess the sensitivity and robustness of the proposed topological approach. It is found that the proposed theory has less sensitivity to resolution effects than the tested local methods. While at sufficiently high image resolution, the results are comparable. As observed in Fig. \\ref{fig6}, the resolution necessary to measure the contact line is lost before the topological structure of the contact line loops are destroyed. Contact angles cannot be measured without resolving the contact line region. However, the contact line region must always form a loop even if it is captured by only one voxel. Therefore, there remains a basis to measure $\\theta_{macro}$ even at low image resolution since topology and interfacial curvatures of fluid surface are still captured. Another way to think about why $\\theta_{macro}$ works well at low resolution is because fluid blobs are physically larger than the contact lines, and we can use information embedded in the fluid interface structure to infer the wetting state.\n\nOverall, the results demonstrate that the proposed theory provides an accurate macroscopic wetting description based on microscopic wetting information that is less susceptible to resolution-based errors due to its robust nature of measuring interfacial area and curvature rather than angles along the contact line. The theoretical links of the proposed theory explored in other recent publications \\cite{sun2020linking,sun2020probing} are further highlighted by the variational analysis presented herein. The particular example was provided for a simple sessile droplet system. However, more complex systems with morphologically and mineralogically heterogeneous geometries could be considered leading to a more general condition for the constrained entropy inequality. This could be particularly important for the development of multiphase flow models that explicitly account for wettability and will be the focus of further work. \n\n\\section*{Acknowledgments}\nA part of this work was performed at the Swiss Light Source, Paul Scherrer Institute, Villigen, Switzerland. C. S. acknowledges an Australian Government Research Training Program Scholarship. A. H. acknowledges ARC DE180100082 and the ANU\/UNSW Digicore Research Consortium. J. M. acknowledges an award of computer time provided by the Department of Energy Early Science Program. This work was supported by the U.S. National Science Foundation, Hydrologic Sciences Program under award No.1344877. This research also used resources of the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC05-00OR22725.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe Galactic interstellar medium (ISM) has a multiphase structure \nwith neutral hydrogen being distributed between the \ncold neutral (CNM), warm neutral (WNM) and warm \nionized (WIM) media. A large fraction of the gas is also found in \ndiffuse, translucent and dense molecular clouds. Newly formed stars\nare associated with these dense molecular clouds and \nstrongly influence the physical state of the rest of the gas in different\nforms through radiative and mechanical inputs. \nThe physical conditions in the multiphase ISM depend on the UV background \nradiation field, metallicities, dust content and the\ndensity of cosmic rays \n\\citep[see Figs.~5, 6 and 7 in][]{Wolfire95}.\nIn addition, \nthe filling factor of the different phases depends sensitively\non the supernova rate \\citep{deavillez2004}.\nTherefore, detecting and studying the multiphase ISM\nin external galaxies has great importance for our understanding of \ngalaxy evolution.\n\nDamped Lyman-$\\alpha$ systems (DLAs) are the highest H~{\\sc i} column\ndensity absorbers seen in QSO spectra, with \n$N$(H~{\\sc i})$\\ge 2\\times 10^{20}$~cm$^{-2}$.\nThese absorbers trace the bulk of the neutral hydrogen at $2\\le z\\le 3$ \n\\citep [][]{Prochaska05,Noterdaeme09dla} and have long been identified as \nrevealing the interstellar medium of the high-redshift precursors\nof present day galaxies \\citep[for a review see,][]{wolfe05}.\n\nThe typical dust-to-gas ratio of DLAs, is less than one tenth of that observed \nin the local ISM, and only a small fraction ($<\u223c10$\\%) of DLAs show detectable\namounts of molecular hydrogen \\citep{Petitjean00,Ledoux03,Noterdaeme08} with\nthe detection rate being correlated to the dust content of the gas \n\\citep{Petitjean06}. \nThe estimated temperature and molecular fraction in these \nsystems are consistent with them \noriginating from the CNM \\citep{Srianand05}.\nIt has been shown recently that strong C~{\\sc i} absorbers detected in \nlow-resolution Sloan Digital Sky Survey (SDSS) \nspectra are good candidates for H$_2$ bearing systems.\nIndeed these absorbers\nhave yielded the first detections of CO molecules in high-$z$ DLAs\n\\citep{Srianand08,Noterdaeme09co,Noterdaeme10co,Noterdaeme11}.\nThe properties of these absorbers are similar to those of translucent \nmolecular clouds. \nThe fact that no DLA is found to be associated with a dense molecular \ncloud, a fundamental ingredient \nof star-formation, is most certainly related to \nthe large extinction that these clouds are expected to produce and\/or\nthe small size of such regions \\citep{Zwaan06} making \ndetections difficult.\n\nThus, most DLAs detected in optical spectroscopic surveys seem to \nprobe the diffuse H~{\\sc i} \ngas \\citep{Petitjean00}. However, \nabout 50\\% of the DLAs show detectable C~{\\sc ii}$^*$ absorption\n\\citep{Wolfe08}, and \\citet{Wolfe03b} argued that a \nconsiderable fraction of the\nC~{\\sc ii}$^*$ absorption in DLAs originates from CNM \ngas \\citep[see however][]{Srianand05}. \n{ Detection of 21-cm absorption is the best way to estimate the CNM\nfraction of DLAs as it is sensitive to both $N$(H~{\\sc i}) and\nthermal state of the gas \\citep{Kulkarni88}.}\n\\begin{figure*}\n\\centerline{\n\\vbox{\n\\hbox{\n\\includegraphics[trim= 25.0mm 50.0mm 90.0 40.0mm, clip, scale=0.33,angle=90.0]{J0733+2721c.ps}\n\\includegraphics[trim= 25.0mm 50.0mm 90.0 50.0mm, clip, scale=0.33,angle=90.0]{J0801+4725c.ps}\n\\includegraphics[trim= 25.0mm 50.0mm 90.0 50.0mm, clip, scale=0.33,angle=90.0]{J0816+4823c.ps}\n}\n\\hbox{\n\\includegraphics[trim= 25.0mm 50.0mm 70.0 40.0mm, clip, scale=0.33,angle=90.0]{J0839+2002c.ps}\n\\includegraphics[trim= 25.0mm 50.0mm 70.0 50.0mm, clip, scale=0.33,angle=90.0]{J0852+2431c.ps}\n\\includegraphics[trim= 25.0mm 50.0mm 70.0 50.0mm, clip, scale=0.33,angle=90.0]{J1017+6116c.ps}\n}\n\\hbox{\n\\includegraphics[trim= 25.0mm 50.0mm 70.0 40.0mm, clip, scale=0.33,angle=90.0]{J1223+5037c.ps}\n\\includegraphics[trim= 25.0mm 50.0mm 70.0 50.0mm, clip, scale=0.33,angle=90.0]{J1237+4708c.ps}\n\\includegraphics[trim= 25.0mm 50.0mm 70.0 50.0mm, clip, scale=0.33,angle=90.0]{J1242+3720c.ps}\n}\n\\hbox{\n\\includegraphics[trim= 25.0mm 50.0mm 70.0 40.0mm, clip, scale=0.33,angle=90.0]{J1406+3433c.ps}\n\\includegraphics[trim= 25.0mm 50.0mm 90.0 50.0mm, clip, scale=0.33,angle=90.0]{J1413+4505c.ps}\n\\includegraphics[trim= 25.0mm 50.0mm 90.0 50.0mm, clip, scale=0.33,angle=90.0]{J1435+5434c.ps}\n}\n}}\n\\caption[]{\nSDSS spectra showing the \\lya lines for 12 DLAs in our sample. \nThe best fitted Voigt profiles (solid curves) together with the \nassociated 1$\\sigma$ errors (shaded regions) are over-plotted. The dotted\ncurve gives the best fitted continuum (in some cases the continuum fit\nincludes the emission lines also). We have used VLT UVES spectra\nto get $N$(H~{\\sc i}) in the case of \\zabs = 3.1745 system towards J1337+3152\n\\citep[see][]{Srianand10}\nand \\zabs = 2.595 and 2.622 systems towards J0407$-$4410 \\citep[CTS 247, see][]{Ledoux03}.\n}\n\\label{dlafits}\n\\end{figure*}\nThis is why it is important to search for 21-cm absorption in DLAs \nover a wide redshift range. While a good fraction of DLAs\/sub-DLAs \npreselected through\nMg~{\\sc ii} absorption seems to show 21-cm absorption at $z\\sim 1.3$\n\\citep[see for example][]{Gupta09, Kanekar09mg2},\nsearches for 21-cm absorption in DLAs at $z_{\\rm abs}\\ge2$ have\nmostly resulted in null detections \\citep[see][]{Kanekar03,Curran10}\nwith only four detections reported till now \\citep[see][]{Wolfe85,Kanekar06,Kanekar07,York07}. \n{\nThe low detection rate of 21-cm absorption in high-$z$ DLAs can be\nrelated to either the gas being warm (i.e high spin temperature, $T_{\\rm S}$, \nas suggested by \\citet{Kanekar03}) \nand\/or the low value of covering factor ($f_c$)\nthrough high-z geometric effects\n\\citep{Curran06}. \n\nThe best way to address the covering factor issue is to perform \nmilliarcsecond scale spectroscopy in the redshifted 21-cm line \nusing very long baseline interferometry (VLBI) to measure the extent \nof absorbing gas \\citep{Lane00}. Unfortunately due to limited \nfrequency coverage and sensitivity of the receivers available with \nVLBI such studies cannot be extended to high redshift DLAs. \nAlternatively, the core fraction measured in the milliarcsecond scale\nimages can be used to get an estimate of the covering factor\n\\citep[see][] {Briggs89, Kanekar09vlba}. Here one assumes that the \nabsorbing gas completely covers at least the emission from the milliarcsec\nscale core.\nTherefore, to address this issue, \none needs, not only to increase the number of systems searched for \n21-cm absorption but also to perform milliarcsecond scale imaging of\nthe background radio sources.\n\nWe report here the results of a search for 21-cm absorption in\n10 DLAs at $z>2$ we have carried out using GBT and GMRT, \ncomplemented by L-band VLBA images of the background QSOs. \n\\begin{table}\n\\caption{Log of GBT and GMRT observations to search for 21-cm absorption}\n\\begin{center}\n\\begin{tabular}{lccccc}\n\\hline\n\\hline\n\\multicolumn{1}{c}{Source}& Tele- & Date & Time & BW & Ch. \\\\\n\\multicolumn{1}{c}{name} & scope & & & & \\\\\n & & yy-mm-dd & (hr) & (MHz) & \\\\\n\\multicolumn{1}{c}{(1)} & (2) & (3) & (4) & (5) & (6) \\\\\n\\hline\n\nJ0407$-$4410 & GBT & 2006-10-20 & 4.7 & 0.625 & 512 \\\\ \n(CTS247) & & 2007-01-05 & & & \\\\ \n & & 2007-01-06 & & & \\\\ \n & & 2007-01-08 & & & \\\\ \n J0733+2721 & GBT & 2007-12-05 & 10.7 & 1.25 & 512 \\\\ \n & & 2007-12-06 & & & \\\\ \n J0801+4725 & GMRT & 2006-12-22 & 10.8 & 1 & 128 \\\\ \n & & 2006-11-23 & & & \\\\\n J0852+2431 & GBT & 2009-08-07 & 5.6 & 1.25 & 1024 \\\\ \n & & 2009-08-08 & & & \\\\ \n & & 2009-09-06 & & & \\\\ \n J1017+6116 & GBT & 2008-10-15 & 4.5 & 1.25 & 1024 \\\\ \n & & 2008-10-16 & & & \\\\ \n & & 2008-10-18 & & & \\\\\n J1242+3720 & GMRT & 2007-06-08 & 6.1 & 0.5 & 128 \\\\ \n & & 2007-06-10 & & & \\\\ \n J1337+3152 & GMRT & 2009-01-13 & 6.2 & 1 & 128 \\\\ \n & & 2009-03-17 & 7.8 & 0.25 & 128 \\\\ \n J1406+3433 & GBT & 2009-03-05 & 8.0 & 1.25 & 1024 \\\\ \n & & 2009-03-06 & & & \\\\ \n & & 2009-03-07 & & & \\\\ \n & & 2009-03-08 & & & \\\\ \n & & 2009-04-07 & & & \\\\ \n J1435+5435 & GMRT & 2007-06-10 & 6.7 & 1 & 128 \\\\ \n & & & & & \\\\ \n\\hline\n\\end{tabular}\n\\end{center}\n\\begin{flushleft}\nColumn 1: Source name. \nColumn 2: Telescope used for 21-cm absorption search. \nColumn 3: Date of observations. \nColumn 4: Total time on source i.e. after excluding telescope set-up time and calibration overheads. \nColumns 5 and 6: Spectral setup for the observations i.e. bandwidth (BW) and number of \nspectral channels respectively. \n\\end{flushleft}\n\\label{obslog}\n\\end{table}\nThis survey has resulted in the detection of 21-cm absorption in \nthe \\zabs = 3.1745\nDLA towards J1337+3152. A detailed analysis of this system and\ntwo sub-DLAs close to this system are presented in \\citet{Srianand10}.\nSection~\\ref{samp} presents the details of our sample. \nIn Section 3 we present the details of GBT and GMRT spectroscopic \nobservations, VLBA continuum observations, and data reduction. \nThe detection rate of 21-cm absorption in DLAs is discussed in \nSection~\\ref{detect}. In Section~\\ref{gencor} we study the\ncorrelations between the parameters derived from 21-cm observations, \n$N$(H~{\\sc i}), metallicity\nand redshift. In Section~\\ref{mole} we study the relation between\n21-cm and \\h2 absorption. The results are summarized in \nSection~\\ref{results}. In this work we assume a flat Universe with\n$H_0$~=~71~\\kms~Mpc$^{-1}$, $\\Omega_{\\rm m}$~=~0.27 and $\\Omega_\\Lambda$~=~0.73.\n\n\\section{The sample of DLAs}\n\\label{samp}\n\n\\begin{table*}\n\\caption{GMRT low-frequency flux density measurements for the DLAs observed with the GBT \n}\n\\begin{center}\n\\begin{tabular}{lcccc}\n\\hline\n\\hline\n\\multicolumn{1}{c}{Source name} & $S_{\\rm 610MHz}$ & Date & $S_{\\rm 325MHz}$ & Date \\\\\n & & & & \\\\\n & (mJy) & yy-mm-dd & (mJy) & yy-mm-dd \\\\\n\\multicolumn{1}{c}{(1)} & (2) & (3) & (4) & (5) \\\\\n\\hline \nJ0407$-$4410 & 124 & 2006-05-23 & 52 & 2007-11-24 \\\\\nJ0733$+$2721 & 248 & 2007-11-04 & 549 & '' \\\\\nJ0852$+$2431 & 198 & 2008-12-26 & 237 & 2009-03-17 \\\\\nJ1017$+$6116 & 274 & '' & 266 & ,, \\\\\nJ1406$+$3433 & 165 & '' & 185$^\\dag$ & (WENSS) \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\begin{flushleft}\nColumn 1: Source name. \nColumns 2 and 4: GMRT flux density measurements at 610 and 325\\,MHz respectively. \nColumns 3 and 5: Dates of the 610 and 325\\,MHz observations respectively. \\\\\n$^\\dag$ From the WENSS catalog.\\\\ \n\\end{flushleft}\n\\label{flux}\n\\end{table*}\n\\begin{table*}\n\\caption{Results from the GBT and GMRT observations}\n\\begin{center}\n\\begin{tabular}{lccccccccc}\n\\hline\n\\hline\nSource name & \\zem & \\zabs & log\\,$N$(H~{\\sc i}) &S$_{1.4\\,GHz}$& $\\delta $ &Spectral rms & S$_{\\nu_{abs}}$ & $\\int\\tau$dv & {${ T}_{\\rm s}\\over f_{\\rm c}$} \\\\\n & & & (cm$^{-2}$) & (mJy) & (km\\,s$^{-1}$) &(mJy\\,b$^{-1}$\\,ch$^{-1}$) & (mJy)& (km\\,s$^{-1}$)&(K) \\\\\n~~~~~~~~(1) & (2) & (3) & (4) & (5) & (6) & (7) & ~~(8) & (9) & (10) \\\\\n\\hline\n\\\\\n J040718$-$441013 & 3.020 &2.595 & 21.05$\\pm$0.10 & - & 3.7 & 5.9 & 67 & $<$1.61 & $>$382 \\\\ \n J040718$-$441013 & '' &2.622 & 20.45$\\pm$0.10 & - & '' & 7.1 & 67 & $<$1.93 & $>$81 \\\\ \n J073320.49+272103.5 & 2.938 &2.7263 & 20.25$\\pm$0.20 & 240 & 3.8 & 3.4 & 451 & $<$0.14 & $>$692 \\\\ \n J080137.68+472528.2 & 3.276 &3.2235 & 20.80$\\pm$0.15 & 78 & 7.0 & 1.5 & 164 & $<$0.22 & $>$1563 \\\\ \n J085257.12+243103.2 & 3.617 &2.7902 & 20.70$\\pm$0.20 & 160 & 3.9 & 3.9 & 228 & $<$0.32 & $>$850 \\\\ \n J101725.89+611627.5 & 2.805 &2.7681 & 20.60$\\pm$0.15 & 477 & 3.9 & 4.2 & 268 & $<$0.29 & $>$758 \\\\ \n J124209.81+372005.7 & 3.839 &3.4135 & 20.50$\\pm$0.30 & 662 & 3.6 & 3.6 & 615 & $<$0.11 & $>$1567 \\\\ \n J133724.69+315254.5 & 3.174 &3.1745 & 21.36$\\pm$0.10 & 83 & 6.9 & 1.3 & 69 &2.08$\\pm$0.17 & 600$^{+220}_{-160}$\\\\ \n J140653.84+343337.4 & 2.566 &2.4989 & 20.20$\\pm$0.20 & 167 & 3.6 & 3.0 & 178 & $<$0.31 & $>$356 \\\\ \n J143533.78+543559.4 & 3.811 &3.3032 & 20.30$\\pm$0.20 & 96 & 7.1 & 1.5 & 145 & $<$0.26 & $>$418 \\\\ \n & & & & & & & & & \\\\ \n\\hline\n\\end{tabular}\n\\end{center}\n\\begin{flushleft}\nColumn 1: Source name. \nColumn 2: QSO emission redshift. \nColumn 3: Absorption redshift of DLAs as determined from the metal absorption lines. \nColumn 4: H~{\\sc i} column density. \nColumn 5: Flux density at 1.4\\,GHz from the FIRST catalog.\nColumns 6 and 7: Spectral resolution and rms for the survey spectrum. \nColumn 8: Continuum flux density of source at the redshifted 21-cm absorption frequency. \nColumn 9: Integrated 21-cm optical depth or 3-sigma upper limit to $\\int \\tau dv$ for the equivalent spectral resolution of \n10\\,km\\,s$^{-1}$. \nColumn 10: Ratio of spin temperature and covering factor of absorbing gas. \n\\\\\n\\end{flushleft}\n\\label{dlasamp}\n\\end{table*}\nTo construct our sample, we cross-correlated the overall sample of \nDLA-bearing QSO sightlines from SDSS-DR7 \n\\citep[][including systems that are not part\nof the published statistical sample used to measure $\\Omega_{\\rm HI}$]{Noterdaeme09dla} with the VLA FIRST catalog to\nidentify DLAs in front of radio sources brighter than 50~mJy at 1.4 GHz. \nWe excluded radio sources with the DLA 21-cm absorption frequencies redshifted \ninto GBT and GMRT frequency ranges known to be affected by \nstrong radio frequency interference (RFI). There are 13 DLAs from the SDSS-DLA catalog that satisfy these conditions. \nIn addition, there are 4 DLAs along the sight line towards J0407$-$4410\n(also known as CTS\\,247) at \\zabs~=~1.913, 2.550, 2.595 \\& 2.622. Two of these,\nat \\zabs~=~2.595 and 2.622, have redshifted 21-cm absorption frequency in the \nrelatively RFI free frequency range of GBT. Including these two DLAs towards \nCTS\\,247, we have a sample of 15 DLAs for which a search\nfor 21-cm absorption was carried-out using either GMRT or GBT. We observed 14 DLAs, (the exception is the \\zabs = 3.079 system towards \nJ1413+4505), but obtained useful spectra for only 10 DLAs.\nIn addition, we have obtained milliarcsecond scale \nimages at 1.4\\,GHz for all the QSOs except CTS\\,247 to understand the role of \nradio structure in detectability of 21-cm absorption in DLAs. \nThe details of the GBT, GMRT and VLBA observations are given below. \n\nThe Lyman-$\\alpha$ profiles for 12 DLAs selected from the SDSS-DLA catalog \nare shown in the Fig.~\\ref{dlafits}. The H~{\\sc i} column density for \neach of these DLAs has been estimated using Voigt profile fits to the \nLyman-$\\alpha$ absorption line. The QSO continuum was approximated by a \nlower order spline using absorption free regions on both sides of \nthe H~{\\sc i} trough (dotted curves in each panel). In addition, \nspecial care was taken to fit the emission line profiles whenever \nthe \\lya absorption is close to QSO emission lines. \nFor the remaining three DLAs in our sample, \\zabs=3.1745 DLA towards \nJ1337+3152 and the two DLAs towards CTS\\,247, we use the column \ndensities measured by \n\\citet{Srianand10} and \\citet{Ledoux03} respectively from high \nresolution VLT UVES spectra.\n\n\n\n\\section{Details of Observations and data reduction}\n\\subsection{The GBT and GMRT observations}\n\nWe observed our sample of 14 DLAs using the GBT prime focus receivers \nPF1-340\\,MHz and PF1-450\\,MHz, and the GMRT P-band receiver. Although \nwe selected DLAs such that the redshifted 21-cm absorption frequencies were \nnot affected by strong RFI, no useful data could be obtained for \n4 absorption systems either due to \nRFI or other technical reasons. The observing log for the remaining \n10 DLAs and the spectral set-up used for these observations are \nprovided in Table~\\ref{obslog}. \nGBT observations were performed in the standard position-switching mode with \ntypically 5\\,min spent on-source and 5\\,min spent off-source. \nThe data were acquired in the orthogonal polarization \nchannels XX and YY. We used the \nGBT spectral processor as the backend for these observations. \nThe two DLAs towards \nCTS\\,247 were observed simultaneously using two bands of 0.625\\,MHz split \ninto 512 channels.\nFor the GMRT observations, typically a \nbandwidth of 0.5 or 1\\,MHz \nsplit into 128 frequency channels was used. The data were acquired in the two \northogonal polarization channels RR and LL. \nFor the flux density\/bandpass calibration of GMRT \ndata, standard flux density calibrators were observed for 10-15\\,min every \ntwo hours. A phase calibrator was also observed for 10\\,min every \n$\\sim$45\\,min to get reliable phase solutions. \n\nWe used NRAO's GBTIDL package to develop a pipeline to automatically \nanalyse the GBT spectral-line data sets. After excluding time ranges for \nwhich no useful data were obtained, the data were processed through this \npipeline. The pipeline calibrates each data record individually and flags \nthe spectral channels with deviations larger than 5$\\sigma$ as \naffected by RFI. After subtracting a second order baseline these data \nare averaged to produce baseline (i.e. continuum) subtracted spectra for \nXX and YY. The baseline fit and statistics \nfor the flagging are determined using the spectral region that excludes the\ncentral 25\\% and last \n10\\% channels at both ends of the spectrum. If necessary, a first-order cubic spline was \nfitted to the averaged XX and YY spectra obtained from the pipeline, which were then combined \nto produce the Stokes-I spectrum. The spectrum was then shifted to the heliocentric frame. \nThe multi-epoch spectra for a source were then resampled onto the same frequency scale and \ncombined to produce the final spectrum.\n\n\nThe GMRT data were reduced using the NRAO AIPS package \nfollowing the standard procedures described in \\citet{Gupta06}. \nSpecial care was taken to exclude the baselines and time stamps affected \nby RFI. \nThe spectra at the quasar positions were extracted from the RR and LL \nspectral cubes and compared for consistency. If necessary, a first-order \ncubic-spline was fitted to remove the residual continuum from the spectra. \nThe two polarization channels were then combined to get the stokes I spectrum \nwhich was then shifted to the heliocentric frame.\n\n\n\\begin{figure*}\n\\centerline{\\hbox{\n\\psfig{figure=\"dlaspec.ps\",height=16.0cm,width=17.0cm,angle=0}\n}}\n\\caption[]{GBT and GMRT spectra of DLAs in our sample. \nShaded regions mark features that are due to RFI. The arrows in the case of non-detections indicate the expected\npositions of the 21-cm absorption. In the case of \\zabs = 3.1745 system towards\nJ1337+3152 we show the high resolution 21-cm absorption spectrum only. Two arrows in the case of\nCTS247a indicate the expected position of 21-cm absorption from the two H$_2$ components.\n}\n\\label{21cm}\n\\end{figure*}\nThe FWHM of the GBT beam at 400\\,MHz is 30$^\\prime$ and the rms confusion is \n500\\,mJy. This is comparable \nto the flux densities of the background radio sources observed with the GBT. \n{Therefore, to \ncorrect for the effect of other confusing sources in the GBT beam and determine \nthe QSO flux densities at the redshifted 21-cm frequency, we observed these with the GMRT at 610 and 325\\,MHz. For these observations we \nhave used 32\\,MHz bandwidth. Details of these GMRT observations \nand the measured flux densities are provided in Table~\\ref{flux}.} For J1406+3433, the 325\\,MHz \nflux density is taken from the Westerbork Northern Sky Survey (WENSS). We interpolate these flux density measurements to determine \nthe flux densities at redshifted 21-cm frequencies for the quasars observed with the GBT. \nSince, flux densities for these 5 QSOs are not measured at the same epoch as the \nGBT spectroscopic observations, in principle, radio flux density \nvariability can affect \nour estimates of 21-cm optical depths for the corresponding DLAs. However, \nthis effect is much smaller than the error caused by confusion from other sources in the beam and should not \naffect the statistical results derived later in the paper. \n\nThe GBT and GMRT observations of our DLA sample have resulted in useful \n21-cm absorption spectra for 10 DLAs. These spectra are presented \nin Fig.~\\ref{21cm}. GBT spectra, typically acquired at a spectral \nresolution, of $\\sim$1\\,\\kms \nhave been smoothed to $\\sim$4\\,\\kms for presentation. \nThe 21-cm absorption is detected only for one DLA (i.e. \\zabs~=~3.1745 DLA towards J1337+3152)\nand a detailed analysis of this system is presented in \\citet{Srianand10}.\nNone of the other ``absorption-like features'', marked as shaded regions, \nare reproduced in spectra from different polarizations and epochs, \nbut are due to RFI. For CTS247b (i.e for \\zabs = 2.622 DLA towards\nCTS247) these \nfeatures are present only in one polarization at certain times. For J0852+2431 \nand J1017+6116, using a combination of high spectral resolution ($\\sim$1\\,\\kms) and\/or \nmulti-epoch observations we rule out the possibility of these features \nbeing real 21-cm absorption. \nDetails of the optical depth measurements and other observational results for all the \n10 DLAs are summarized in Table~\\ref{dlasamp}. \n\n\n\n\\begin{table}\n\\caption{Details of phase-referencing calibrators used for the VLBA observations }\n\\begin{center}\n\\begin{tabular}{lcc}\n\\hline\n\\hline\nSource & Calibrator & Separation \\\\\n & &(degrees)\\\\\n~~~~~~~~(1) & (2) & (3) \\\\\n\\hline\n J0733+2721 & J0732+2548 & 1.5 \\\\ \n J0801+4725 & J0754+4823 & 1.5 \\\\ \n J0816+4823 & J0808+4950 & 1.9 \\\\\n J0839+2002 & J0842+1835 & 1.6 \\\\\n J0852+2431 & J0856+2111 & 3.5 \\\\ \n J1017+6116 & J1031+6020 & 2.0 \\\\ \n J1223+5037 & J1227+4932 & 1.3 \\\\\n J1237+4708 & J1234+4753 & 0.9 \\\\\n J1242+3720 & J1242+3751 & 0.5 \\\\ \n J1337+3152 & J1329+3154 & 1.6 \\\\ \n J1406+3433 & J1416+3444 & 1.9 \\\\ \n J1413+4505 & J1417+4607 & 1.2 \\\\\n J1435+5435 & J1429+5406 & 1.0 \\\\ \n\\hline\n\\end{tabular}\n\\end{center}\n\\begin{flushleft}\nColumn 1: Source name. \nColumn 2: Phase-referencing calibrator. \nColumn 3: Separation between the radio source and phase-referencing calibrator. \n\\end{flushleft}\n\\label{vlbalog}\n\\end{table}\n\n\\subsection{Continuum observations with VLBA}\nThe sample of quasars presented here was observed as part of a \nlarger VLBA survey to obtain milliarcsecond scale images for QSOs \nwith foreground DLAs and Mg~{\\sc ii} systems, \nand understand the relationship between radio structure \nand detectability of 21-cm \nabsorption. We have observed using VLBA 21-cm receiver for 11~hrs and 18~hrs\non 21\/02\/2010 and 10\/06\/2010 respectively.\nWe used eight 8\\,MHz baseband channels, i.e. the total bandwidth of 64\\,MHz. \nEach baseband channel was split into 32 spectral points. Both the right \nand left-hand circular polarization channels were recorded. Two bit \nsampling and a post-correlation time resolution of 2 seconds were used. \n\nThe observations were done using nodding-style phase-referencing with a cycle \ntime of $\\sim$5\\,min, i.e. 3\\,min on the source and $\\sim$1.5\\,min on the \nphase-referencing calibrator. The phase-referencing calibrators were selected \nfrom the VLBA calibrator survey (VCS) at 2.3 and 8.6\\,GHz \n(Table~\\ref{vlbalog}). \nIn order to improve the uv-coverage, the total observing time was split \ninto snapshots over a number of different hour angles. Each source, except \nCTS247 which was excluded due to observational constraints, was typically \nobserved for a total of $\\sim$30\\,min. During both observing runs, strong \nfringe finders\/bandpass calibrators such as J0555+3948, J0927+3902, \nJ1800+3848 and J2253+1608 were also observed every $\\sim$3\\,hr for 4-5\\,min. \n\\begin{figure*}\n\\centerline{\n\\vbox{\n\\hbox{\n\\psfig{figure=\"J0733MAP_NOLABELS.PS\",height=5.0cm,width=5.5cm,angle=-90}\n\\psfig{figure=\"J0801MAP_NOLABELS.PS\",height=5.0cm,width=5.5cm,angle=-90}\n\\psfig{figure=\"J0816MAP_NOLABELS.PS\",height=5.0cm,width=5.5cm,angle=-90}\n}\n\\hbox{\n\\psfig{figure=\"J0839MAP_NOLABELS.PS\",height=5.0cm,width=5.5cm,angle=-90}\n\\psfig{figure=\"J0852MAP_NOLABELS.PS\",height=5.0cm,width=5.5cm,angle=-90}\n\\psfig{figure=\"J1017MAP_NOLABELS.PS\",height=5.0cm,width=5.5cm,angle=-90}\n}\n\\hbox{\n\\psfig{figure=\"J1223MAP_NOLABELS.PS\",height=5.0cm,width=5.5cm,angle=-90}\n\\psfig{figure=\"J1237MAP_NOLABELS.PS\",height=5.0cm,width=5.5cm,angle=-90}\n\\psfig{figure=\"J1242MAP_NOLABELS.PS\",height=5.0cm,width=5.5cm,angle=-90}\n}\n\\hbox{\n\\psfig{figure=\"J1337MAP_NOLABELS.PS\",height=5.0cm,width=5.5cm,angle=-90}\n\\psfig{figure=\"J1406MAP_NOLABELS.PS\",height=5.0cm,width=5.5cm,angle=-90}\n\\psfig{figure=\"J1413MAP_NOLABELS.PS\",height=5.0cm,width=5.5cm,angle=-90}\n}\n}\n}\n\\caption[]{Contour plots of VLBA images at 1.4\\,GHz. The rms in the images are listed \nin Table~\\ref{vlbares}. \n{At the bottom of each image the restoring beam is shown as an ellipse, \nand the first contour level (CL) in mJy\\,beam$^{-1}$ and FWHM are noted}. \nThe contour levels are plotted as CL$\\times$($-$1, 1, 2, 4, 8,...)\\,mJy\\,beam$^{-1}$. {Depending upon the detailed structure of the radio sources,\nthe emission could be more extended at the redshifted\n21-cm frequencies.}\n}\n\\vskip -22.3cm\n\\begin{picture}(400,400)(0,0)\n\\put(-019,373){\\small J0733$+$2721}\n\\put( 084,272){\\tiny CL = 1.5}\n\\put( 043,265){\\tiny B:0.011$^{\\prime\\prime}$$\\times$0.004$^{\\prime\\prime}$,$+$13$^\\circ$ }\n\\put( 144,373){\\small J0801$+$4725}\n\\put( 245,272){\\tiny CL = 0.8}\n\\put( 205,265){\\tiny B:0.012$^{\\prime\\prime}$$\\times$0.004$^{\\prime\\prime}$,$+$21$^\\circ$ }\n\\put( 304,373){\\small J0816$+$4823}\n\\put( 405,272){\\tiny CL = 0.6}\n\\put( 364,265){\\tiny B:0.010$^{\\prime\\prime}$$\\times$0.004$^{\\prime\\prime}$,$+$ 1$^\\circ$ }\n\\put(-017,230){\\small J0839$+$2002}\n\\put( 084,129){\\tiny CL = 0.7}\n\\put( 043,122){\\tiny B:0.010$^{\\prime\\prime}$$\\times$0.004$^{\\prime\\prime}$,$-$ 1$^\\circ$ }\n\\put( 144,230){\\small J0852$+$2431}\n\\put( 244,129){\\tiny CL = 0.6}\n\\put( 204,122){\\tiny B:0.010$^{\\prime\\prime}$$\\times$0.005$^{\\prime\\prime}$,$-$ 2$^\\circ$ }\n\\put( 304,230){\\normalsize J1017$+$6116}\n\\put( 404,129){\\tiny CL = 2.5}\n\\put( 363,122){\\tiny B:0.014$^{\\prime\\prime}$$\\times$0.004$^{\\prime\\prime}$,$+$46$^\\circ$ }\n\\put(-017,087){\\normalsize J1223+5037}\n\\put( 084,-13){\\tiny CL = 2.5}\n\\put( 043,-20){\\tiny B:0.008$^{\\prime\\prime}$$\\times$0.005$^{\\prime\\prime}$,$-$12$^\\circ$ }\n\\put( 144,087){\\normalsize J1237$+$4708}\n\\put( 244,-13){\\tiny CL = 0.7}\n\\put( 204,-20){\\tiny B:0.008$^{\\prime\\prime}$$\\times$0.005$^{\\prime\\prime}$,$+$ 7$^\\circ$ }\n\\put( 304,087){\\normalsize J1242$+$3720}\n\\put( 404,-13){\\tiny CL = 4.0}\n\\put( 363,-20){\\tiny B:0.008$^{\\prime\\prime}$$\\times$0.005$^{\\prime\\prime}$,$+$ 4$^\\circ$ }\n\n\\put(-017,-56){\\normalsize J1337$+$3152}\n\\put( 084,-156){\\tiny CL = 0.5}\n\\put( 043,-163){\\tiny B:0.009$^{\\prime\\prime}$$\\times$0.005$^{\\prime\\prime}$,$-$11$^\\circ$ }\n\\put( 144,-56){\\normalsize J1406$+$3433}\n\\put( 244,-156){\\tiny CL = 1.5}\n\\put( 204,-163){\\tiny B:0.010$^{\\prime\\prime}$$\\times$0.004$^{\\prime\\prime}$,$+$ 8$^\\circ$ }\n\\put( 304,-56){\\normalsize J1413$+$4505}\n\\put( 404,-156){\\tiny CL = 0.7}\n\\put( 363,-163){\\tiny B:0.010$^{\\prime\\prime}$$\\times$0.004$^{\\prime\\prime}$,$-$10$^\\circ$ }\n\\end{picture}\n\\vskip +9.5cm\n\\label{vlbamap}\n\\end{figure*}\n\\begin{figure}\n\\addtocounter{figure}{-1}\n\\centerline{\n\\vbox{\n\\hbox{\n\\psfig{figure=\"J1435MAP_NOLABELS.PS\",height=6.0cm,width=6.5cm,angle=-90}\n}\n}\n}\n\\caption[]{ {\\sl Continued}. \n}\n\\vskip -7.1cm\n\\begin{picture}(400,400)(0,0)\n\\put( 050,373){\\normalsize J1435$+$5435}\n\\put( 172,252){\\tiny CL = 0.8}\n\\put( 131,245){\\tiny B:0.013$^{\\prime\\prime}$$\\times$0.005$^{\\prime\\prime}$,$-$19$^\\circ$ }\n\\end{picture}\n\\vskip -7.1cm\n\\label{vlbamap}\n\\end{figure}\n\n\\begin{sidewaystable*}[]\n\\vskip -7 in\n\\vskip 0.3in\n\\centering\n\\begin{tabular}{lcccccccccccccc}\n\\multicolumn{15}{l}{{\\bf Table 5}: ~Results from the VLBA data. }\n\\\\\n\\hline\n\\hline\nSource &$z_{\\rm abs}$& Right & Declination & rms & Comp. & S & r &$\\theta$& a & b\/a &$\\phi$& $S_{\\rm T}$ & $f_{\\rm VLBA}$ &LLS \\\\\nname & & ascension & & & & & & & & & & & & \\\\\n & & (J2000) & (J2000)&(mJy\\,beam$^{-1}$)& &(mJy)&(mas)& ($^\\circ$)&(mas)& &($^\\circ$)& (mJy) & &(pc)\\\\\n (1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9) &(10)&(11) &(12) & (13) & (14) &(15)\\\\\n\\hline\nJ0733+2721 & 2.7263& 07 33 20.4830 & +27 21 03.430 & 0.3 & 1 & 97 & 0 & - &4.52 &0.05 &-88 & 240 & 0.62 &348 \\\\%\n & & & & & 2 & 3 & 9.1 &-129 &13.24&0.00 &-18 & & & \\\\%\n & & & & & 3 & 19 &32.7 &-140 &5.98 &0.26 & 32 & & & \\\\%\n & & & & & 4 & 29 &10.8 & 51 &8.37 &0.44 &-17 & & & \\\\%\nJ0801+4725 & 3.2235& 08 01 37.6930 & +47 25 28.082 & 0.2 & 1 & 28 & 0 & - &6.77 &0.00 & 72 & 78 & 0.67 &342 \\\\%\n & & & & & 2 & 8 &9.34 &-112 &7.09 &0.00 & 72 & & & \\\\%\n & & & & & 3 & 16 &44.7 &-112 &8.63 &0.22 & 64 & & & \\\\%\nJ0816+4823 & & 08 16 19.0044 & +48 23 28.490 & 0.2 & 1 & 42 & 0 & - &4.59 &0.24 & 87 & 69 & 0.75 &68 \\\\%\n & & & & & 2 & 10 & 9.0 & 132 &8.75 &0.43 &-63 & & & \\\\%\nJ0839+2002 & & 08 39 10.8970 & +20 02 07.391 & 0.2 & 1 & 113 & 0 & - &4.91 &0.24 & 84 & 130 & 0.87 &$\\le$39\\\\%\nJ0852+2431 & 2.7902& 08 52 57.1211 & +24 31 03.271 & 0.2 & 1 & 78 & 0 & - &4.92 &0.31 & 71 & 160 & 0.55 & 175\\\\%\n & & & & & 2 & 10 & 21.9& 42 &15.2 &0.16 & 53 & & & \\\\%\nJ1017+6116 &2.7681 & 10 17 25.8865 & +61 16 27.414 & 0.5 & 1 &388 & 0 & - &1.50 &0.68 & 55 & 477 & 0.86 &38\\\\%\n & & & & & 2 & 24 & 4.7 & 145 &2.45 &0.00 & 70 & & & \\\\%\nJ1223+5037 & & 12 23 43.1740 & +50 37 53.344 & 0.5 & 1 & 96 & 0 & - &2.50 &0.00 & 78 & 229 & 0.60 &554 \\\\%\n & & & & & 2 & 16 &13.6 & 80 &3.01 &0.83 &-19 & & & \\\\%\n & & & & & 3 & 25 &71.4 & 78 &8.61 &0.00 & 89 & & & \\\\%\nJ1237+4708 & & 12 37 17.4413 & +47 08 06.964 & 0.2 & 1 & 64 & 0 & - &3.17 &0.20 &-74 & 80 & 0.80 & $\\le27$\\\\%\nJ1242+3720 &3.4135 & 12 42 09.8121 & +37 20 05.692 & 0.6 & 1 & 848 & 0 & - &1.93 &0.76 & 22 & 662 & 1.00 & $\\le14$ \\\\%\nJ1337+3152 &3.1747 & 13 37 24.6931 & +31 52 54.642 & 0.2 & 1 & 83 & 0 & - &3.85 &0.38 & 74 & 83 & 1.00 &$\\le30$ \\\\%\nJ1406+3433 &2.4989 & 14 06 53.8532 & +34 33 37.339 & 0.4 & 1 &127 & 0 & - &3.24 &0.22 &-23 & 167 & 0.87 & 153 \\\\%\n & & & & & 2 & 18 & 18.7& -30 &23.79&0.23 &-25 & & & \\\\%\nJ1413+4505 & & 14 13 18.8652 & +45 05 22.990 & 0.2 & 1 & 105 & 0 & - &2.19 &0.42 &-67 & 140 & 0.88 & 216 \\\\%\n & & & & & 2 & 12 & 3.4 & -77 &5.37 &0.00 &-86 & & & \\\\%\n & & & & & 3 & 6 &27.9 & -72 &6.28 &0.03 &-67 & & & \\\\%\nJ1435+5435 &3.3032 & 14 35 33.7812 & +54 35 59.312 & 0.2 & 1 & 31 & 0 & - &2.66 &0.17 &-29 & 96 & 0.55 &155 \\\\%\n & & & & & 2 & 17 &20.4 & 155 &4.00 &0.47 &-36 & & & \\\\%\n & & & & & 3 & 5 & 7.6 & 153 &5.52 &0.00 &-7 & & & \\\\%\n & & & & & & & & & & & & & & \\\\%\n\\hline\n\\end{tabular}\n\\begin{flushleft}\nColumn 1:~Source name. Column 2:~absorption redshift. \nColumns 3 and 4:~right ascension and declination of component-1 (see column 6)\nfrom the multiple Gaussian fit to the source, respectively. \nColumn 5:~rms in the map in mJy\\,beam$^{-1}$. \nColumn 6:~component id. \nColumn 7:~flux density of the component in mJy.\nColumns 8 and 9:~radius and position angle of the component with respect to component-1, respectively.\n{Columns 10, 11 and 12: major axis, axial ratio and position angle \nof the deconvolved Gaussian component, respectively.}\nColumn 13:~flux density in mJy from FIRST\/NVSS. \nColumn 14:~$c_f$ is the ratio of 1.4 GHz flux density in \nVLBA image to that in the FIRST image, \nColumn 15:~largest projected linear size in pc. \\\\\n\\end{flushleft}\n\\label{vlbares}\n\\end{sidewaystable*}\n\n\nData were calibrated and imaged using AIPS and DIFMAP in a standard way. \nGlobal fringe fitting was performed on the phase-referencing calibrators. \nThe delays, rates and phases estimated from these were transferred to the \nsources which were then self-calibrated until the final images were obtained. \nRadio sources were characterised by fitting Gaussian models to the self-calibrated \nvisibilities. \nVLBA maps of the 13 QSOs are shown in Fig.~\\ref{vlbamap} \nand the results of model fitting are listed in columns\\,\\#\\,6-12 of Table~\\ref{vlbares}. \n\nNon-detection of 21-cm absorption in a DLA could be due to the small covering \nfactor of the absorbing gas. The typical spatial resolution achieved in our \nVLBA observations is $\\sim$8\\,mas. \nIf the extent of absorbing gas is of the order of the scales probed by our \nVLBA observations (i.e. $>$20pc) then we expect the detectability of \n21-cm absorption to depend on the fraction and spatial extent of radio \nflux density detected in these images. In column\\,\\#\\,14 of Table~\\ref{vlbares} \nwe give the ratio of total flux densities detected \nin the VLBA and FIRST images at 20cm, i.e. $f_{\\rm VLBA}$. \\\nThe last column of this table \ngives the largest linear size (LLS), i.e the separation between the farthest radio components, of the \nradio source at the redshift of the DLA. \nOut of the 13 QSOs presented in Fig.~\\ref{vlbamap} that have DLAs along \ntheir line of sight, we have 21-cm absorption \nspectra for only 8 DLAs. For the DLA towards \nJ0816+4823, we use the 21-cm absorption measurement from \n\\citet{Curran10}. Thus we have a sample of 9 DLAs with both 21-cm \nabsorption measurements and VLBA 21-cm maps for the background QSOs. \nThe $f_{\\rm VLBA}$ for this sample ranges from 0.6 to 1, and LLS from \n$<$15\\,pc to 340\\,pc. \n\n\n\n\nIn the absence of VLBI spectroscopy at the redshifted 21-cm line\nfrequency, the ratio of VLBA core flux density to the total flux density measured in the arcsecond scale\nimages (called core fraction $c_f$) has been used as an\nindicator of the covering factor $f_c$ \\citep[see][]\n{Briggs89, Kanekar09vlba}. \nHere we use the term `core' to refer to the flat spectrum\nunresolved radio component coincident with\nthe optical QSO in the VLBA image.\n\nFor J1242+3720 and J1337+3152 the radio source is modelled \nas a single unresolved component and within the uncertainties all \nthe flux density in the arcsecond scale \nFIRST images is recovered in our VLBA images (Fig.~\\ref{vlbamap}). \nBoth of these sources, have \nflat spectra\n{suggesting the radio emission even at lower frequencies\noriginates predominantly from the compact core.\nTherefore we take $f_c$=1 for this case.}\n\nThe radio source J1017+6116 has an inverted radio spectrum ($\\alpha$=$-$0.4) \n and 86\\% of the \nflux density in the FIRST image is recovered in the VLBA image, 94\\% of which is contained in the main unresolved\ncomponent (Fig.~\\ref{vlbamap}). \nFor another 3 flat-spectrum sources with 21-cm absorption measurements, i.e. \nJ0816+4823, J0852+2431, and J1406+3433, more than 80\\% of the VLBA flux \ndensity is present in a \nsingle unresolved component. \nFor these 4 sources, based on the flat spectral index, the dominant\ncomponent in the VLBA image can be identified with radio core\/optical\nQSO. \nTherefore for the 6 sources mentioned above, we have estimated \nthe core fraction, \n$c_f$, and used it as the covering factor, $f_c$, of the gas}. \nThe remaining three sources, J0733+2721, J0801+4725 and J1435+5435, exhibit multiple components in their \nVLBA 20-cm images. The identification of the component coincident with the \nQSO, \nand the estimation of their $c_f$ from VLBA images for these three\nsources are highly uncertain. \n\n\n\n\\section{Detectability of 21-cm absorption}\n\\label{detect}\n\\begin{figure}\n\\centerline{\n\\includegraphics[scale=0.4,angle=0.0]{fig6.ps}\n}\n\\caption{ The allowed range of fraction of DLAs having harmonic mean\nspin temperature $T_{\\rm S}$ greater than a limiting value $T_{\\rm S}^l$\nas a function of $T_{\\rm S}^l$. We use only those DLAs for which\n$c_f$ measurements are available. The lower envelope of the shaded \nregion is obtained considering all the lower limits \non $T_{\\rm S}$ as measurements. The upper envelope is obtained \nassuming all the lower limits as measurements with \n$T_{\\rm S}\\ge T_{\\rm S}^l$.\n}\n\\label{fwnm}\n\\end{figure}\n\nIn this Section we investigate the detectability of 21-cm absorption in DLAs \nand the implication of non-detections for the physical state of the H~{\\sc i} \ngas.\nIt is clear from the last column of Table~\\ref{dlasamp}, \nthat for most of the DLAs, our data has good sensitivity to detect \n$T_{\\rm S}\/f_{\\rm c}$$\\sim$100~K gas.\n\nThe H~{\\sc i} 21-cm absorption is detected only in the \\zabs~=~3.1745 system\ntowards SDSS~J1337+3152. {This is one of the weakest radio sources in\nour sample (with a 3$\\sigma$ $\\int\\tau dv$ limit of 0.4 \\kms). However, \nthanks to high $N$(H~{\\sc i}) our spectrum is sensitive enough to detect\nany gas with $T_{\\rm S}\\le 3100$ K.\n}\n This source is unresolved in our\nVLBA observations (see Fig~\\ref{vlbamap}). The L-band flux density\nmeasured in our VLBA image is consistent with that\nmeasured by the FIRST survey. Therefore, the core fraction is,\n$c_{\\rm f} \\sim 1$, and the size of the VLBA beam is less than 30~pc at \nthe redshift of the absorber.\nThe spin-temperature, measured from the ratio of 21-cm optical depth and\nthe $N$(H~{\\sc i}) column density derived from the Lyman-$\\alpha$ trough,\nis 600$^{+220}_{-160}$~K which is consistent with the upper limit on $T_{\\rm S}$ \nobtained from the width of the single component Gaussian fit to the \n21-cm absorption \\citep{Srianand10}. \n\n\\begin{table*}\n\\addtocounter{table}{1}\n\\caption{Summary of 21-cm searches in $z\\ge2$ DLAs. Column 1: QSO. \nColumn 2: absorption redshitft.\nColumn 3: log $N$(H~{\\sc i}). Column 4: Integrated optical depth. \nColumn 5: Reference for $\\int\\tau dv$\ngiven in column 4. Column 6: the core fraction $c_f$. Column 9: the fraction\nof CNM (see the text for its definition). Column 10: the H$_2$ fraction\nand Column 11: References for $N$(H~{\\sc i}) and\/or f(H$_2$) measurements.\n}\n\\begin{tabular}{lcccccccccc}\n\\hline\n\\multicolumn{1}{c}{QSO} & \\zabs & log$N$(H~{\\sc i}) &$\\int \\tau dv$ & Refs&$c_f$ & $T_{\\rm S}\/f_{\\rm c}$ & $T_{\\rm S}$ & $f({\\rm CNM})$& log f(\\h2) & Refs\\\\\n\\multicolumn{3}{c}{} & (\\kms) &\\multicolumn{2}{c}{} & (K) & (K) \\\\ \n\\multicolumn{1}{c}{(1)} & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9) &(10)&(11) \\\\\n\\hline\n\\hline\n\\multicolumn{10}{c}{\\bf 21-cm detections}\\\\\nJ020346.6+113445& 3.38714& 21.26$\\pm$ 0.08 & 0.71$\\pm$0.02 & 1 &0.76 &1397 &1062&$\\sim$0.19 &$-6.2,-4.6$&a\\\\ \nJ031443.6+431405& 2.28977& 20.30$\\pm$ 0.11 & 0.82$\\pm$0.09 & 2 &.... &133 &....&$\\sim$1.00 &.... &d\\\\\nJ044017.2$-$433309& 2.34747& 20.78$\\pm$ 0.10 & 0.22$\\pm$0.03 & 3 &0.59 &1493 & 881&$\\sim$0.23 &.... &e\\\\ \nJ050112.8$-$015914& 2.03955& 21.70$\\pm$ 0.10 & 7.02$\\pm$0.16 & 4 &.... &390 & ...&.... &$\\le -6.40$ &b\\\\ \nJ133724.6+315254& 3.17447& 21.36$\\pm$ 0.10 & 2.08$\\pm$0.17 & 5 &1.00 &600& 600&$\\sim$0.33&$-7.00$ &c\\\\ \n\\multicolumn{10}{c}{\\bf 21-cm non-detections having metallicity measurements} \\\\\nJ033755.7$-$120412& 3.1799 & 20.65$\\pm$ 0.10&$\\le$0.06 & 6 & 0.62&$\\ge$4057 &$\\ge$2515 &$\\le 0.08$ &$\\le -5.10$ &b\\\\ \nJ033901.0$-$013318& 3.0619 & 21.10$\\pm$ 0.10&$\\le$0.06 & 6 & 0.68&$\\ge$11435&$\\ge$7775 &$\\le 0.03$ &$\\le -6.90$ &b\\\\ \nJ040733.9$-$330346& 2.569 & 20.60$\\pm$ 0.10&$\\le$0.12 & 7 & 0.44&$\\ge$1807 &$\\ge$795 &$\\le 0.25$ & ....& e\\\\ \nJ040718.0$-$441013& 2.59475& 21.05$\\pm$ 0.10&$\\le$1.61 & 8 & ....&$\\ge$380 &.... & .... &$-2.61^{+0.17}_{-0.20}$&b\\\\ \nJ040718.0$-$441013& 2.62140& 20.45$\\pm$ 0.10&$\\le$1.93 & 8 & ....&$\\ge$80 &.... & .... &$\\le -6.20$&b\\\\ \nJ043404.3$-$435550& 2.30197& 20.95$\\pm$ 0.10&$\\le$0.33 & 7 & ....&$\\ge$1471 &.... & .... &$\\le -5.15$&b\\\\ \nJ053007.9$-$250330& 2.81115& 21.35$\\pm$ 0.07&$\\le$0.58 & 9$^\\dag$ & 0.94&$\\ge$2103 &$\\ge$1977&$\\le 0.10$ &$-2.83^{+0.18}_{-0.19}$&b\\\\ \nJ091551.7+000713& 2.7434 & 20.74$\\pm$ 0.10&$\\le$0.37 & 7 & ....&$\\ge$809 &.... & .... &.... &e\\\\ \nJ135646.8$-$110129& 2.96680& 20.80$\\pm$ 0.10&$\\le$0.33 & 6 & ....&$\\ge$1042 &.... & .... &$\\le -6.75$&b\\\\ \nJ135706.1$-$174402& 2.77990& 20.30$\\pm$ 0.15&$\\le$0.14 & 7 & ....&$\\ge$777 &.... & .... &$\\le -5.99$ &a\\\\ \nJ142107.7$-$064356& 3.44828& 20.50$\\pm$ 0.10&$\\le$0.14 & 7 & 0.69&$\\ge$1231 &$\\ge$849 &$\\le 0.24$ &$\\le -5.69$&b\\\\ \nJ234451.2+343348& 2.90910& 21.11$\\pm$ 0.10&$\\le$0.21 & 6 & 0.71&$\\ge$3343 &$\\ge$2373 &$\\le 0.08$ &$\\le -6.19$ &a\\\\ \n\\multicolumn{10}{c}{\\bf 21-cm non-detections having no metallicity measurements} \\\\ \nJ053954.3$-$283956& 2.9742 & 20.30$\\pm$ 0.11&$\\le$0.06 & 6 & 0.47 & $\\ge$1812&$\\ge851$ &$\\le 0.23$ & ....&e\\\\ \nJ081618.9+482328& 3.4358 & 20.80$\\pm$ 0.20&$\\le$1.43 & 9 & 0.60 & $\\ge$240 &$\\ge144$ &$\\le 1.00$ & ....&a\\\\ \nJ073320.4+272103& 2.7263 & 20.25$\\pm$ 0.20&$\\le$0.14 & 8 & .... & $\\ge$692 &.... &.... & ....&a\\\\ \nJ080137.6+472528& 3.2235 & 20.80$\\pm$ 0.15&$\\le$0.22 & 8 & .... & $\\ge$1563&.... &.... & ....&a\\\\ \nJ085257.1+243103& 2.7902 & 20.70$\\pm$ 0.20&$\\le$0.32 & 8 & 0.49 & $\\ge$854 &$\\ge418$ &$\\le 0.48$ & ....&a\\\\\nJ101725.8+611627& 2.7681 & 20.60$\\pm$ 0.15&$\\le$0.29 & 8 & 0.81 & $\\ge$748 &$\\ge606$ &$\\le 0.33$ & ....&a\\\\ \nJ124209.8+372005& 3.4135 & 20.50$\\pm$ 0.30&$\\le$0.11 & 8 & 1.00 & $\\ge$1566&$\\ge1566$&$\\le 0.13$ & ....&a\\\\\nJ140501.1+041536& 2.708 & 21.07$\\pm$ 0.24&$\\le$0.19 & 9 & .... & $\\ge$3369& .... &.... & ....&f\\\\ \nJ140501.1+041536& 2.485 & 20.20$\\pm$ 0.20&$\\le$0.08 & 9 & .... & $\\ge$1080 & .... &.... & ....&f\\\\ \nJ140653.8+343337& 2.4989 & 20.20$\\pm$ 0.20&$\\le$0.31 & 8 & 0.76 & $\\ge$279 &$\\ge212$ &$\\le 0.94$ & ....&a\\\\ \nJ143533.7+543559& 3.3032 & 20.30$\\pm$ 0.20&$\\le$0.26 & 8 & .... & $\\ge$418 & .... &.... & ....&a\\\\ \n\\hline\n\\end{tabular}\n\\begin{flushleft}\n{\nReferences in column \\#5: 1) \\citet{Kanekar07}, 2) \\citet{York07}, 3) \\citet{Kanekar06}, 4) \\citet{Briggs89}, 5) \\citet{Srianand10}, \n6) \\citet{Kanekar03}, 7) \\citet{Kanekar09ts}, 8) This paper, and 9) \\citet{Curran10}. $^\\dag$ Archival data from GBT08A\\_003 (PI: Curran) was processed \nthrough our pipeline. See text for details. \n\\\\}\n{References for $N$(H~{\\sc i}) and\/or f(H$_2$) measurements (column \\# 11): a) This paper, b) \\citet{Noterdaeme08}, c) \\citet{Srianand10}, d) \\citet{Ellison08}, e) \\citet{Akerman05}\nf) \\citet{Curran10}.}\n\\end{flushleft}\n\\label{tablesum}\n\\end{table*} \n\nIn Table~\\ref{tablesum} we provide various details of our measurements \ntogether with the previous measurements at $z\\ge 2$ from the literature. \nWe present the results dividing the sample into three groups. These are systems\n with 21-cm detections (five systems), systems with 21-cm absorption upper limits with (twelve systems) and without (eleven systems) high-resolution optical \nspectra from which to derive accurate metallicities. The first\ntwo groups are used to investigate the connection between UV measurements\nand 21-cm optical depth. \nIn all cases the 3$\\sigma$ upper limits on the integrated\n21-cm optical depth are computed assuming a line width of 10~\\kms. \n\nThe 21-cm detection rate from our sample, \nwithout putting any sensitivity limit, \nis 10\\%. This is 13\\% when we restrict to $\\int \\tau dv$ limit\nof 0.4 \\kms ( the limit achieved in the case of J1337+3152\nwhere we have 21-cm detection). Taken at face value, \nthe extended sample listed in Table~\\ref{tablesum}\ngives a 21\\% detection rate for $\\int \\tau dv$ limit of 0.4 \\kms.\nFor a $\\int \\tau dv$ limit of 0.2 \\kms we get the detection rate of\n28\\%.\nHowever, these may not be representative values as the list of \nsystems compiled from the literature may be biased towards detections \nas some authors may not have reported their non-detections systematically\n\nSince we know $N$(H~{\\sc i}) from the damped Lyman-$\\alpha$ line, the \ndetection limit on the integrated optical depth implies a lower limit on the \nratio $T_{\\rm S}\/f_{\\rm c}$. \nThe $T_{\\rm S}\/f_{\\rm c}$ measurements are reported in column 7 \nof Table~\\ref{tablesum}. \nIn column 6 of this table, we give the core fraction $c_{ f}$. \nAs mentioned above, $c_f$ is basically the ratio of flux density in the unresolved core seen in\nVLBA images to the total flux density measured in the arcsecond scale FIRST images. \nFor the objects from the literature we use the $c_f$ values given in\n\\citet{Kanekar09vlba}. These measurements were made at 327~MHz, close\nto the redshifted 21-cm frequencies. \nFollowing \\citet{Kanekar09vlba} we use core fraction ($c_f$)\nas the estimate of the covering factor ($f_c$).\nThe $T_{\\rm S}$ measurements given in column 8 of Table~\\ref{tablesum}\nare obtained by assuming $f_c = c_f$.\n\nIn Fig.~\\ref{fwnm} we plot the percentage of DLAs having $T_{\\rm S}$ \ngreater than a limiting value $T_{\\rm S}^l$ as a function of \n$T_{\\rm S}^l$ for systems with $T_{\\rm S}$ measurements given in column\n8 of Table~\\ref{tablesum}. The lower envelope of the shaded region is \nobtained considering all the lower limits on $T_{\\rm S}$ as \nmeasurements. The upper envelope is obtained assuming\nall the lower limits as measurements with $T_{\\rm S}\\ge T_{\\rm S}^l$.\nIt is clear from the figure that\nmore than 50\\% of the DLAs have $T_{\\rm S}\\ge 700 K$. Remember that the $T_{\\rm S}$ measured in an individual DLA\nis the harmonic mean temperature of different phases that contribute to the observed\n$N$(H~{\\sc i}).\nAssuming that the gas is simply a two phase medium with similar \ncovering factors the\nfraction of H~{\\sc i} in the CNM (called $f{\\rm (CNM)}$) can be written as,\n\\begin{equation}\nf{\\rm (CNM)} = {1 \\over T_{\\rm S}^W} \\bigg{[} {T_{\\rm S}^{\\rm C}T_{\\rm S}^W \\over T_{\\rm S}} - T_{\\rm S}^{\\rm C}\\bigg{]} \n\\end{equation}\nwhere, ${T_{\\rm S}^{\\rm C}}$ and ${T_{\\rm S}^{\\rm W}}$ are the spin-temperature of the CNM and WNM\nrespectively.\n\\citet{Srianand05} have noticed that the H~{\\sc i} phase traced by the H$_2$ \nabsorption has temperature typically in the range 100-200~K. Thus we\nconsider the CNM temperature to be 200 K (instead of 70 K as\nseen in CNM of the\nGalaxy) so that the $f$(CNM) we get will be\na conservative upper limit. Assuming ${T_{\\rm S}^{\\rm C}}\\sim200$~K and ${T_{\\rm S}^{\\rm W}}\\sim10^4$~K,\n$T_S = 700$ K can be obtained for a combination of $f{\\rm (CNM)}$ = 0.27 and \n$f{\\rm (WNM)}$ = 0.73. Therefore $f{\\rm (CNM)}$ is less than 0.27 in \nat least 50\\% of the DLAs. Note that choosing ${T_{\\rm S}^{\\rm W}}\\sim8000$~K\n(as suggested for the Galactic ISM) instead of the 10$^4$ K used here, \ndoes not change the results appreciably. \n\nWe estimate $f{\\rm (CNM)}$ for the four 21-cm detections\n(excluding J0501-0159 (B0458-020) for which we do not have the covering factor value). \nApart from J0314+4314 (3C082) which seems to be a special case \\citep{York07}, \nthe CNM seems to represent roughly 20 to \n30\\% of the total $N$(H~{\\sc i}) measured in these DLAs. \nFor individual non-detections, we can calculate conservative upper limits of the \nfraction of $N$(H~{\\sc i}) in the CNM phase assuming $T_{\\rm S}^{\\rm C}$~=~200~K.\nThe values of $f$(CNM) are given in column \\#9 of Table~\\ref{tablesum} for systems with \n$f_{\\rm c}$ measurements. The upper limits vary between 0.10 and 1.0 with a\nmedian value of 0.23. \nThus the analysis presented here, under the assumption that $f_c = c_f$, \nsuggests that most of the neutral hydrogen in \nhigh-$z$ DLAs is warm. This is very much consistent with the conclusion of \\citet{Petitjean00} \nbased on the lack of \\h2 detections in most high-$z$ DLAs.\n\\begin{figure}\n\\centerline{\n\\includegraphics[scale=0.4,angle=0.0]{fig_metvsts.ps}\n}\n\\caption{Metallicity vs $T_S\/f_c$. The vertical dashed line marks the\nmedian metallicity measured in our sample. }\n\\label{tsvmet}\n\\end{figure}\n\n\\section{Results of Correlation analysis}\n\\label{gencor}\n\\begin{figure*}\n\\centerline{\n\\vbox{\n\\hbox{\n\\includegraphics[scale=0.4,angle=0.0]{fig1.ps}\n\\includegraphics[scale=0.4,angle=0.0]{fig2.ps}}\n\\hbox{\n\\includegraphics[scale=0.4,angle=0.0]{fig3.ps}\n\\includegraphics[scale=0.4,angle=0.0]{fig4.ps}\n}\n}}\n\\caption{Properties of 21-cm absorptions vs redshift and $N$(H~{\\sc i}).\nThe vertical dashed lines give the median value of the quantity plotted\nin the x-axis.}\n\\label{figcor}\n\\end{figure*}\n\nIn this Section we explore correlations between the 21-cm\noptical depth and other observable parameters. As we have only\na few 21-cm detections and mostly \nupper limits we use survival analysis and in particular the generalised rank \ncorrelation test \\citep{Isobe86}. For this purpose we use\nthe Astronomical SURVival analysis (ASURV) package.\n\n\\subsection{Metallicity vs T$_s$\/$f_c$}\nFirstly we study the importance of the metallicity of the gas.\nOnly 3 DLAs in our sample have measurements of metallicity from\nhigh resolution optical spectroscopy. In the extended sample\nat $z\\ge2$ (see Table~\\ref{tablesum}) there are 17 systems with metallicity measurements\nand 21-cm spectra. In Fig.~\\ref{tsvmet}, we plot $T_{\\rm S}\/f_{\\rm c}$ versus\nmetallicity. The vertical long-dashed line marks the median metallicity of the points \nplotted in the figure. The only detection found in the low metallicity half is for \n\\zabs~=~3.1745 towards J1337+3152 reported from\nour survey. The other four detections are from the high metallicity half. \nThe non-parametric generalized Kendall rank correlation test \nsuggests only a weak correlation between Z and $T_{\\rm S}\/f_{\\rm c}$ (at the 1.42$\\sigma$ level) with the probability that \nit can arise due to chance being 0.15. The significance is even lower (i.e 0.9$\\sigma$ with\na chance probability of 0.37) when we use $T_{\\rm S}$ (instead\nof $T_{\\rm S}\/f_{\\rm c}$) for cases where we have estimated \n$f_{\\rm c}$ measurements. \nWe wish to point out that a correlation between $T_s\/f_c$ and metallicity\nis reported in the literature \\citep[][]{Curran07,Kanekar09ts,Curran10}.\nThe lack of correlation in our sample (with systems in a restricted\nredshift range) could either reflect redshift evolution of the relationship\nor small range in metallicity covered by the sample.\nMetallicity measurements for the remaining 11 systems in Table~\\ref{tablesum} would \nallow us to address this issue in a statistically significant manner.\n\nWe also looked at the possible correlation between the 21-cm\noptical depth and the velocity width ($\\Delta v$) of the low \nionization lines. This information is available for 14 sources.\nAgain we find no statistically significant correlation \nbetween the two. This is inconsistent with the 2.2 to 2.8 $\\sigma$ level\nweak correlation between $W$(Mg~{\\sc ii}) and $\\Delta v$\nreported by \\citet{Curran07}.\nHowever, this is consistent with the finding of\n\\citet{Gupta09}, that\n$W$(Mg~{\\sc ii}) \nand 21-cm optical depth are not correlated for a sample of 33 strong Mg~{\\sc ii} systems at \n1.10$\\le z \\le 1.45$ \\citep[see also][]{Kanekar09mg2}. \n\n\\subsection{Redshift dependence}\n\nIn the left hand side panels of Fig.~\\ref{figcor} we plot \n$T_s\/f_c$ and integrated 21-cm optical depth vs redshift.\nNo clear correlation is evident in this figure. The non-parametric\nKendall test finds no significant correlation between $\\int \\tau dv$ \n(or T$_S$) and $z$. Note our sample probes only a restricted \nredshift range in terms of time interval probed.\nHowever, the lack of correlation found here is consistent\nwith the near constancy of T$_s$\/$f_c$ as a function of redshift\nfound by \\citet{Curran10}. Understanding the redshift dependence of\n$T_S$ is very important in particular to address whether there is \nany evolution in $T_s$ \\citep[][]{Kanekar03} or\ngeometric effects \\citep[][]{Curran06}. \nTo make an unbiased comparison we need to have 21-cm measurements \nat low $z$ for a well defined sample of DLAs detected \nbased on \\lya\\ absorption.\n \n\n\\subsection{Dependence on $N$(H~{\\sc i})}\n\nRecently \\citet{Curran10} have found a 3$\\sigma$ level\ncorrelation between $N$(H~{\\sc i}) and $T_s\/f_c$. To check whether\nthis correlation holds at $z>2$, we plot, in the top panels \nof Fig.~\\ref{figcor}, the integrated\n21-cm optical depth as a function of redshift and $N$(H~{\\sc i}).\nWe note\nthat there is a tendency for more 21-cm detections in DLAs\nwith higher $N$(H~{\\sc i}). \nHowever, the non-parametric\nKendall test finds no significant correlation between\n$\\int \\tau dv$ and $N$(H~{\\sc i}).\nIn the bottom right panel we plot $T_{\\rm S}\/f_{\\rm c}$ against log~$N$(H~{\\sc i}). \nThe Kendall test does not show any significant relation between the two \nquantities (1.28$\\sigma$ with a probability of 0.2 for this to be due to chance). \nThus we do not find any evidence for the 21-cm optical depth to depend on\n$N$(H~{\\sc i}) in our sample.\n\n\\section{21-cm absorption and H$_2$}\n\\label{mole}\nAs 21-cm absorption and H$_2$ molecules can give complementary information\non the physical state of the gas. In this Section, we study the relationship\nbetween these two indicators. \nThere are 13 DLAs in our extended sample for which\nthe expected optical wavelength range of redshifted \\h2 \nabsorptions has been observed at high spectral resolution. \nNine of these sources are part of the UVES sample of \\citet{Noterdaeme08}. \n\\citet{Srianand10} have reported the detection of \\h2 in J1337+3152 \nand here we report the search \nfor \\h2 in the remaining three DLAs (\\zabs = 3.3871 towards \nJ0203+1134, \\zabs = 2.7799 towards J1357$-$1744\nand \\zabs = 2.9091 towards J2344+3433). In the 10th column of \nTable~\\ref{tablesum}, we summarize \nthe molecular fraction \n$f$(\\h2)~=~2$N$(\\h2)\/(2$N$(\\h2)+$N$(H~{\\sc i}))] derived for these\n13 systems.\n\nIn 8 systems, neither 21-cm absorption nor H$_2$ molecules are detected\nwith typical upper limits of the order of 10$^{-6}$ for $f$(\\h2).\nApart from the system at \\zabs~=~2.6214 towards J0407$-$4410, the lower\nlimits on $T_{\\rm S}\/f_{\\rm c}$ for the remaining 7 systems are higher \nthan 700~K. \nThere are 4 cases where $f_{\\rm c}$ measurements are available. In three\ncases (\\zabs~=~3.1799 towards J0337$-$1204, \\zabs~=~3.0619\ntowards J0339$-$0133 and \\zabs~=~2.9019 towards J2344+3433), the \nlower limit on $T_{\\rm S}$ is more than 2000~K. These are in line\nwith the suggestion by \\citet{Petitjean00} that the absence of\n\\h2 in most of the DLAs is due to the low density and high\ntemperature of the gas.\n\nIn two cases (\\zabs~=~2.5947 towards J0407$-$4410 (CTS 247) and \\zabs~=~2.8112 \ntowards J0530$-$2503 (PKS 0528-250)), strong H$_2$ absorption is detected with \nrotational excitations consistent with the \\h2-bearing gas being a CNM.\nHowever, 21-cm absorption is not detected in either case. We\ndiscuss these two systems in detail below. \n\nAmong the five 21-cm absorbers, high resolution UVES spectra covering \nthe expected wavelength range of \\h2 absorption are available for four \nsystems. The exception is the \\zabs~=~2.28977 \nsystem towards J0314+4314 (B0311+430). For the \\zabs~=~2.3474 system towards \nJ0440$-$4333 (B0438$-$436) the continuum flux in the expected wavelength range is removed \nby high ionization lines from an associated system, as well as by a high-$z$ Lyman \nlimit system present along the line of sight.\nBelow we discuss the five systems where simultaneous\nanalysis of \\h2 and 21-cm absorption is possible.\n\n\\begin{figure*}\n\\centerline{\n\\includegraphics[scale=0.6,angle=270]{b0201_h2fit.ps}\n}\n\\caption{Voigt profile fits to H$_2$ Lyman and Werner band\nabsorption lines in the \\zabs~=~3.3868 DLA system towards J0203+1134 (PKS~0201+113).\nThe zero of the velocity scale is defined at $z = 3.38716$. The two\nvertical lines at $v = 0$ and $-$68~\\kms~ show the locations of\ntwo 21-cm absorption components reported by \\citep{Kanekar07}. \nThe vertical dotted line indicates the location of \\h2 absorption.\n In each panel we also show the error spectrum with dotted curves.\n}\n\\label{h2b0201}\n\\end{figure*}\n\n\n\\subsection{\\zabs = 3.3868 DLA towards J0203+1134 (PKS 0201+113)}\n\nSearches for 21-cm absorption in this system have yielded\nconflicting results \\citep{DeBruyn96, Briggs97}.\nBased on GMRT spectra taken at three different epochs, \n\\citet{Kanekar07} reported the detection of 21-cm absorption \nin two components at \\zabs~=~3.387144(17) and 3.386141(45). \nUsing $N$(\\ion{H}{i})$\\sim$ (1.8$\\pm$0.3)$\\times 10^{21}$~cm$^{-2}$ they \nobtained $T_{\\rm S} = [955\\pm160](f_{\\rm c}\/0.69)$~K. \nUsing high resolution optical spectrum,\n\\citet{Ellison01Q0201} have\nfound a gas phase metallicity of 1\/20 of solar with\nvery little dust depletion.\nThe gas cooling rate,\nlog~$l_{\\rm c} = -26.67\\pm0.10$~erg s$^{-1}$ Hz$^{-1}$,\nderived using C~{\\sc ii*} absorption is consistent with\nthis DLA being part of high-cool population defined by\n \\citet{Wolfe03a}. \nFrom \\citet{Kanekar07} we \ncan see that the strongest 21-cm absorption does not correspond to the strongest velocity\ncomponent in either C~{\\sc ii$^*$} or Fe~{\\sc ii}. \n\nHere we report the detection of \nH$_2$ absorption from J=0 and J=1 levels originating \nfrom both Lyman and Werner bands \n(see Fig.~\\ref{h2b0201}). A single component Voigt profile fit reproduces the data well. As the \nLyman-$\\alpha$ forest is dense and the spectral signal-to-noise ratio is not very high due to the\nfaintness of the QSO, we considered a range of $b$ values (i.e between 1 and 5~\\kms) to get the best\nfit values of log[$N$(H$_2$, J=0)] in the range 16.10$-$14.48 and log[$N$(H$_2$, J=1)] = 16.03$-$14.57. \nWe estimated the kinetic temperature using the ortho-to-para ratio (i.e $T_{01}$) \nand found it to be in the range 48$-$108~K for the range of $b$ parameters\nconsidered above. We note that for $b$ parameters greater than 2~\\kms, the H$_2$ lines are \nmainly in the linear portion of the curve of growth and the column density estimate is insensitive to the \nassumed value. \nThe average molecular fraction, $log~f({\\rm H_2})$, in the range, \n$-4.6\\le log~f({\\rm H_2})\\le -6.2$. \n\nDespite the gas being cold, there is no 21-cm absorption\ndetected at the position of the H$_2$ component (at z = 3.38679) \nwhich is well separated from the 21-cm absorption components \nIf we use $f_c=0.76$, as found by \\citet{Kanekar09vlba} using 326 MHz\nobservations, we find\nlog~$N$(H~{\\sc i}) $\\le19.12$. This is less than 1\\% of the total H~{\\sc i} column density measured in this system.\n\nUnlike most of the strong \\h2 systems, this system does not show detectable C~{\\sc i} absorption. This means we do not have,\nunfortunately, an independent estimate of the density \nfrom fine-structure excitation.\n\n\\subsection{\\zabs = 2.5948 towards J0407-4410 (CTS 247)}\n\n As the radio source is faint, our GBT spectrum only gives \na weak limit on the spin temperature, $T_{\\rm S} \\ge 380$~ K when we use a \nline width of 10 \\kms.\n\\citet{Srianand05} have reported log~$N$(C~{\\sc ii$^*$})~=~13.66$\\pm$0.13. This, together with log~$N$(H~{\\sc i})~=~21.05$\\pm$0.10, \ngives a gas cooling rate of log~$l_{\\rm c} = 26.92\\pm0.16$. This is very close to the value $l_{\\rm c}^{\\rm crit}$\nthat seems to demarcate between the high and low cool systems defined by \\citet{Wolfe08}. \n\n\\citet{Ledoux03} reported the detection of H$_2$ from this system.\nThe H$_2$ absorption is well fitted with two components at \\zabs~=~2.59471 and 2.49486\nwith log~$N$(H$_2$)~=~18.14 and 15.51 respectively \\citep{Srianand05}. These components have $T_{01}$~=~121$\\pm$10 \nand 91$\\pm$6~K respectively. The average molecular fraction, log~$f$(H$_2$), is found to be $-2.42^{+0.07}_{-0.12}$ with \nan average metallicity of $-1.02\\pm0.12$ and moderate dust depletion \\citep{Ledoux03}. \n\nThe absence of 21-cm absorption from this system is intriguing\nas H$_2$ components have T$\\sim$100 K.\nWith the same $b$ parameters as used to fit the \\h2 lines and \nthe rms from the GBT spectrum, we get a 3$\\sigma$ upper limit\nof $\\int \\tau dv = 0.88$ \\kms. This translates to a constraint, $f_{\\rm c}\\times$$N$(H~{\\sc i}) $\\le$ 2$\\times 10^{20}$~cm$^{-2}$\nin the H$_2$ components where we have assumed $T_{\\rm S}$~=~$T_{01}$.\nUnfortunately we do not have a VLBA image of this source\nand it is difficult to constrain the covering factor of the gas. \nIf we assume $f_{\\rm c}\\sim 1$ then the upper limit on $N$(H~{\\sc i}) implies \nthat the H$_2$ component is a sub-DLA with\nlog~$f$(\\h2) $>-1.85$.\n\n From the column densities of the C~{\\sc i} fine-structure lines,\n\\citet{Srianand05} have constrained the particle density in the gas to be in the range \n$4.52$ using GMRT and GBT. \nWe detect 21-cm absorption in only one of them. From our sample we \nfind the 21-cm detection rate is 13\\% \nfor a $\\int \\tau dv$ limit of 0.4 km\/s (the detection limit reached in \nthe case of J1337+3152).\nWe also obtained 1420~MHz VLBI images for the sources in our sample.\n\n\nThe 21-cm detection at $z\\ge 2$ seems to favour systems with high \nmetallicity and\/or high $N$(H~{\\sc i}) \\citep[see also][]{Kanekar09ts, Curran10}.\nThis basically means that the probability of detecting cold components that can \nproduce detectable 21-cm absorption is higher in systems with high values \nof $N$(H~{\\sc i}) and Z. However, we do not\nfind any correlation between the integrated optical depth \n(or T$_{\\rm S}$\/$f_{\\rm c}$) and $N$(H~{\\sc i}) or metallicity. \n\n\nIt is important to address the covering factor issue before drawing\nany conclusions on $T_S$. Ideally one should do high spatial resolution\nVLBA spectroscopy for this purpose \\citep[see for example][]{Lane00}.\nHowever, this is not possible at present specially for $z\\ge2$ absorbers.\nTherefore, we proceed by assuming that the core fraction found in the VLBA \nimages\nas the covering factor of the absorbing gas \\citep[as in the case of][]{Kanekar09vlba}.\nWe find that more than 50\\% of DLAs have weighted mean spin temperature \n($T_{\\rm S}$) in excess of 700 K. For the assumed temperature\nof the CNM gas $T_{\\rm S}^C = 200$ K (as seen in \\h2 components in high-z DLAs) \nwe find that more than 73\\% of H~{\\sc i} in such systems is\noriginating from WNM. The median value CNM fraction (i.e $f$(CNM)) \nobtained for the detections and the\nmedian value of upper limits in the case of non-detections are in the range 0.2 to 0.25. \n\nWe study the connection between 21-cm and \\h2 absorption in a sub-sample of 13 DLAs where both these species can be searched for.\nWe report the detection and detailed analysis of \\h2 molecules in the \\zabs=3.3871 DLA system towards J0203+1134 where 21-cm \nabsorption is also detected. For a $b$ parameter in the range 1-5~\\kms\\ we find 14.57$\\le$log~$N$(\\h2)$\\le$16.03. The inferred kinetic\ntemperature is in the range 48-108~K based on $T_{01}$ of H$_2$. However no 21-cm absorption is detected at the very position of \nthis \\h2\\ component. This suggests that the H~{\\sc i} column density associated with this component is $\\le$ 10$^{19}$~cm$^{-2}$. \nHowever, the lack of proper coincidence between 21-cm and any of the strong \nUV absorption components may also mean that \nthe radio and optical sight lines probe different volumes of the gas.\n\nIn the case of 8 DLAs, neither 21-cm nor H$_2$ are detected. Typical upper \nlimits on the molecular fraction ($f_{\\rm H_2}$) in these systems are \n$\\le 10^{-6}$. \nThe lack of \\h2 in DLAs can be explained if the H~{\\sc i} gas originates\nfrom low density regions photoionized by the metagalactic UV\n\\citep[see for example,][]{Petitjean92,Petitjean00,Hirashita05}.\nThis also indicates that the volume filling factor of \\h2 in DLAs \nis small \\citep{Zwaan06}. Typical limits obtained for $T_{\\rm S}$\nin these systems are consistent with only a small fraction of\nthe H~{\\sc i} gas originating from the CNM phase as suggested by the\nlack of \\h2 absorption.\n\nIn two cases strong \\h2 absorption is detected and kinetic temperatures are in the range 100-200~K, but 21-cm absorption is not \ndetected. Even in two cases where both the species are detected they do not originate from the same velocity component. \nThe lack of 21-cm absorption directly associated with \\h2\\ indicates that only a small fraction (typically $\\le$ 10\\%)\nof the neutral hydrogen seen in the DLA \nis associated with the \\h2 components \\citep[see also][]{Noterdaeme10co}. This implies that the molecular fractions $f$(\\h2)\nreported from the \\h2 surveys should be considered as conservative lower limits for the \\h2 components.\n\nFor two of the \\h2-bearing DLAs with density measurements based on \nC~{\\sc i} fine-structure excitation we derive an upper limit on the line of \nsight thickness of $\\le 15$~pc. { This is consistent with the size estimate for the H$_2$-bearing gas in \\zabs = 2.2377 DLA\ntowards Q1232+082 based on partial coverage \\citep{Balashev11}.}\n\nIn principle, the presence of \\h2 and 21-cm absorptions in a single component provides \na unique combination to simultaneously constrain the variation\nof the fine-structure constant ($\\alpha$), the electron-to-proton mass ratio ($\\mu$) and the proton G-factor. \nAs shown here, DLAs with 21-cm \nand \\h2 detections are rare. Even in these cases the presence of multiphase structure at parsec scale is evident, \nintroducing velocity shifts\nbetween the different absorption components that will affect the constraints on the variation of constants.\n\n\\section{acknowledgements}\nWe thank GBT, GMRT, VLBA and VLT staff for their support\nduring the observations and the anonymous referee for some\nuseful comments. We acknowledge the use of SDSS\nspectra from the archive (http:\/\/www.sdss.org\/). \nThe National Radio Astronomy Observatory is a facility of the\nNational Science Foundation operated under cooperative agreement by\nAssociated Universities, Inc. \nVLBA data\nwere correlated using NRAO's implementation of the DiFX\nsoftware correlator that was developed as part of the Australian\nMajor National Research Facilities Programme and operated under\nlicence.\nRS and PPJ gratefully acknowledge support from the Indo-French\nCentre for the Promotion of Advanced Research (Centre Franco-Indien pour\nla promotion de la recherche avanc\\'ee) under Project N.4304-2.\n\n\\def\\aj{AJ\n\\def\\actaa{Acta Astron.\n\\def\\araa{ARA\\&A\n\\def\\apj{ApJ\n\\def\\apjl{ApJ\n\\def\\apjs{ApJS\n\\def\\ao{Appl.~Opt.\n\\def\\apss{Ap\\&SS\n\\def\\aap{A\\&A\n\\def\\aapr{A\\&A~Rev.\n\\def\\aaps{A\\&AS\n\\def\\azh{AZh\n\\def\\baas{BAAS\n\\def\\bac{Bull. astr. Inst. Czechosl.\n\\def\\caa{Chinese Astron. Astrophys.\n\\def\\cjaa{Chinese J. Astron. Astrophys.\n\\def\\icarus{Icarus\n\\def\\jcap{J. Cosmology Astropart. Phys.\n\\def\\jrasc{JRASC\n\\def\\mnras{MNRAS\n\\def\\memras{MmRAS\n\\def\\na{New A\n\\def\\nar{New A Rev.\n\\def\\pasa{PASA\n\\def\\pra{Phys.~Rev.~A\n\\def\\prb{Phys.~Rev.~B\n\\def\\prc{Phys.~Rev.~C\n\\def\\prd{Phys.~Rev.~D\n\\def\\pre{Phys.~Rev.~E\n\\def\\prl{Phys.~Rev.~Lett.\n\\def\\pasp{PASP\n\\def\\pasj{PASJ\n\\def\\qjras{QJRAS\n\\def\\rmxaa{Rev. Mexicana Astron. Astrofis.\n\\def\\skytel{S\\&T\n\\def\\solphys{Sol.~Phys.\n\\def\\sovast{Soviet~Ast.\n\\def\\ssr{Space~Sci.~Rev.\n\\def\\zap{ZAp\n\\def\\nat{Nature\n\\def\\iaucirc{IAU~Circ.\n\\def\\aplett{Astrophys.~Lett.\n\\def\\apspr{Astrophys.~Space~Phys.~Res.\n\\def\\bain{Bull.~Astron.~Inst.~Netherlands\n\\def\\fcp{Fund.~Cosmic~Phys.\n\\def\\gca{Geochim.~Cosmochim.~Acta\n\\def\\grl{Geophys.~Res.~Lett.\n\\def\\jcp{J.~Chem.~Phys.\n\\def\\jgr{J.~Geophys.~Res.\n\\def\\jqsrt{J.~Quant.~Spec.~Radiat.~Transf.\n\\def\\memsai{Mem.~Soc.~Astron.~Italiana\n\\def\\nphysa{Nucl.~Phys.~A\n\\def\\physrep{Phys.~Rep.\n\\def\\physscr{Phys.~Scr\n\\def\\planss{Planet.~Space~Sci.\n\\def\\procspie{Proc.~SPIE\n\\let\\astap=\\aap\n\\let\\apjlett=\\apjl\n\\let\\apjsupp=\\apjs\n\\let\\applopt=\\ao\n\\bibliographystyle{mn2e}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\\begin{figure*}[htbp]\n\t\n\t\\includegraphics[width=\\textwidth]{fig\/motivation}\n\t\\caption{Illustration of a toy sequential recommendation example, where the blue node is the target user, the green nodes are the neighbors of the target user in the social network, and the pink nodes denote the items. The black dashed lines denote the anticipated recommended results between items and target user $u_1$. (a)-(b) Existing methods, which rarely consider the temporally dependent dynamics of the heterogeneous graph, lead to inaccurate prediction results. (c) On the contrary, the proposed \\textbf{TEA} method simultaneously aggregates the historical user behavior sequence and the dynamic heterogeneous graph, thus resulting in more accurate predictions than existing methods.}\n\t\\label{fig:motivation}\n\\end{figure*}\n\nThe sequential recommendation system is achieving more and more attention because of its practicality and effectiveness \\cite{eskandanian2019modeling, he2017translation, tang2013exploiting, tang2016recommendation}. \\textcolor{black}{In a sequential recommendation system,\nthe users access different items at different time stamps and frequently interact with each other.} The difficulties of sequential recommendation mainly come from two aspects:\n\\textcolor{black}{the temporal dependency of historical behaviors and the nonstationarity of users} \\textcolor{black}{The temporal dependency of historical behaviors means that the decision of a user is influenced by the historical behaviors.\nAnd the nonstationarity of users means that the decision of a user is influenced by the social relationship with the neighbors and the user-item interactions of the neighbors.}\nTherefore, one important challenge is how to effectively leverage the historical behaviors and the social relationship between users.\n\nFocusing on the above challenge, numerous sequential recommendation algorithms have been proposed in recent years, which leverage the user behavior sequence and employ the Markov Chains to model the transition of items. Eskandanian et al. \\cite{eskandanian2019modeling} mine the user preference and identify change-points\nin the sequence of user interactions by using Hidden Markov Model (as shown in Figure \\ref{fig:motivation} (a)). \nHe et al. \\cite{he2017translation} model the personalized sequential behavior by using the personalized translation vectors and the previous item embedding to predict the next item. \\textcolor{black}{These transition-based methods assume that the users are independent of each other, which ignores the influence between the users. Considering that the behavior of a user is easily affected by the neighbors, ignoring the dependence between users will suffer from limited performance in sequential recommendation.} \n\nAnother kind of recommendation algorithm \\cite{fan2019graph, ma2011recommender, tang2016recommendation,jamali2010matrix} focuses on analyzing the social relationships between users and user-item interactions in a static user-item graph.\nThe typical methods include the traditional Collaborative Filtering (CF) methods \\cite{hu2008collaborative, koren2008factorization, rendle2009bpr}, the deep learning enhanced approaches \\cite{liu2020deep, wang2019multi, deng2019deepcf, covington2016deep}, the recently developed graph neural networks based methods \\cite{battaglia2018relational}, as well as the social-network-based methods \\cite{fan2019graph,song2019session,yu2020enhance}. These methods reveal that both the interactions among users in social networks and the user-item bipartite graphs are beneficial to the performance of recommendation system. \nHowever, almost all the aforementioned methods assume the heterogeneous graphs of the users and the items are static, which ignores the dynamical influence \\textcolor{black}{of the temporal interaction between items and social networks}, and further results in the suboptimal performance of recommendation systems. Take Figure \\ref{fig:motivation} (b) for a toy example. The existing methods without considering the dynamic user-item heterogeneous graph might recommend the $v_1$ in preference to $v_2$ since more friends of user $U_1$ choose $v_1$. \n\n\n\nThus, it is essential to devise a unified framework to take advantage of both the historical behaviors of a user and dynamic interactions between the neighbors and items. Figure \\ref{fig:motivation} (c) illustrates our main idea that models the temporally user-item heterogeneous graphs and generates a more accurate prediction.\nIn the figure, the decision of whether a user $u_i$ will choose a given item $v_{t+1}$ is controlled by two important factors: (1) the historical interactions between him or her and items; (2) the temporal dynamic heterogeneous graph, including the interactions between the neighbors and items. Hence, the goal of the proposed method is to estimate the conditional probability of $v_{t+1}$ given a user $u_i$, the historical accessed item sequence $v_{1:t}$, as well as the heterogeneous graph sequence $H_{1:t+1}$, which can be formulated as $P(v_{t+1}|u_i, \\mathcal{H}_{1:t+1}; v_{1:t})$.\n\nBased on the above idea, we propose the \\textbf{T}emporally \\textbf{E}volving \\textbf{A}ggregation (\\textbf{TEA} in short) framework for sequential recommendation by aggregating the user behavior sequence as well as the dynamic user-item heterogeneous graph. Inspired by the sequence labeling in natural language processing \\cite{panchendrarajan2018bidirectional, hao2021semi} to model the joint probability distribution), we adopt CRF to model the item decision sequence and estimate $P(v_{t+1}|u_i, \\mathcal{H}_{1:t+1}; v_{1:t})$. In order to alleviate the issue of the large item space, we use the pseudo likelihood method to approximate the aforementioned conditional probability. \n\\textcolor{black}{By doing this, the training procedure can be performed by estimating the unary score and transition score in CRF, which are implemented by our designed modules.}\n{Technically, we design a \\textit{Time-Restricted User Behavior Sequence Aggregation Module} to estimate the transition score of \\text{CRF}, and a \\textit{Temporal Dynamic Heterogeneous Graphs Aggregation Module }to estimate the Unary Scores of \\text{CRF}.}\nWe further provide two different practical implementations based on the proposed framework. Extensive experimental studies demonstrate that our \\textbf{TEA} framework outperforms the state-of-the-art recommendation methods on two published datasets and one real-world WeChat official accounts dataset.\n\n\nThe remainder of this paper is organized as follows. In Section \\ref{sec:related}, we review related researches into recommendation systems, including social recommendation and sequential recommendation. In Section \\ref{sec:model}, we define the problem of sequential recommendation under the dynamical heterogeneous graph and further derive the objective function based on the conditional random field. In Section \\ref{implementation}, we provide the implementation details of the proposed \\textbf{TEA} model. We further analyze the connection to existing methods in Section \\ref{connection}. And then, we present our experimental results based on two standard benchmarks and one real-world dataset in Section \\ref{sec:exp}. Finally, we give our conclusion of the proposed method. \n\n\n\\section{Related Works} \\label{sec:related}\nIn this section, we mainly discuss the existing techniques on social recommendation and sequential recommendation.\n\nIn order to effectively mine the deep demands of users, researchers set their sights on social relations, hence social recommendation has received more and more attention. One of the most important methods is Matrix Factorization (MF) \\cite{mnih2007probabilistic,baltrunas2011matrix,he2016fast}. Based on the traditional matrix factorization methods, Hao et al. \\cite{ma2008sorec} proposed a co-factorization method, which shares a common latent user-feature matrix factorized by both ratings and social relations. With the development of deep learning methods, He et al. \\cite{he2017neural} propose NeuMF by replacing the inner product with a neural architecture that can learn an arbitrary function from data. \nFan et al. \\cite{fan2018deep} propose a deep neural network-based model to learn non-linear features of each user from social relations and to integrate them into probabilistic matrix factorization for the social recommendation.\nDeng et al. \\cite{deng2016deep} propose a two-phase recommendation process to utilize deep learning to calculate the impact of community effect from the interests of users' trusted friends for recommendations.\n\nRecently, graph neural networks (GNNs) \\cite{battaglia2018relational,kipf2016semi} are widely used to aggregate node information and topological structure from social networks, hence GNNs are employed to address the social recommendation problem. In order to well aggregate the heterogeneous information, Fan et al. \\cite{fan2019graph} propose the GraphRec for the social recommendation. Fu et al. \\cite{fu2020magnn} leverage the metapaths \\cite{shi2018heterogeneous} to obtain the heterogeneous graph embedding. Considering that the influences in the social network may be context-dependent, Song et al. \\cite{song2019session} address the session-based social recommendation by using a dynamic-graph-attention neural network architecture. \nHowever, the aforementioned methods rarely consider the fact that different friends in social networks choose different items. In this work, considering the fact that social influence and user behaviors are time-dependent, the proposed \\textbf{TEA} method focuses on aggregating the temporally evolving social influence and the user behavior sequence.\n\nSince users usually access the items in chronological order, the users are likely to choose the items that are closely relevant to those they just accessed. Many works on sequential recommendation follow this assumption. Aiming to model the item-item transition probabilities, some traditional works borrow the idea of the Markov chain. Rendle et al. \\cite{rendle2010factorizing} bridge the Matrix Factorization (MF) and Markov Chains (MC). He et al. \\cite{he2017translation} propose TransRec to model such third-order relationships \\textcolor{black}{(e.g. the relationships among a user, the previously accessed item and the next item)} for large-scale sequential prediction. Motivated by the advantages of sequence learning in natural language processing, many neural network-based methods are proposed to learn the sequential dynamics. Tang et al. \\cite{tang2018personalized} leverage convolutional neural networks to encode the sequences into the embeddings. Other works \\cite{hidasi2018recurrent, quadrana2017personalizing} leverage recurrent neural networks and their variants to model the sequences of items. Kang et al. \\cite{kang2018self} further leverage attention-mechanism and propose the SASRec to balance the goal of MC-based methods and RNNs based methods. Moreover, Sun et al. \\cite{sun2019bert4rec} argue that such left-to-right unidirectional models are sub-optimal. So they propose BERT4Rec, which employs deep bidirectional self-attention to model user behavior sequences.\n\\textcolor{black}{In this paper, the proposed \\textbf{TEA} leverage the Conditional Random Field (CRF) to model the translation of items, which calculates the transition score and the unary score by respectively aggregating the user behavior sequence information as well as the dynamic user-item heterogeneous graph.}\n\n\n\n\\section{Model}\\label{sec:model}\n\\newcommand{\\tabincell}[2]{\\begin{tabular}{@{}#1@{}}#2\\end{tabular}} \n\\begin{table}\n\\caption{Notation and Descriptions.}\n\t\\centering\n\t\\begin{tabular}{c|c}\n\t\t\\hline\n\t\t\\small{Notations} & Descriptions \\\\\n\t\t\\hline\n\t\t$U,V$ & User and item set. \\\\\n\t\t\\hline\n\t\t$m,n$ & The size of user set and item set. \\\\\n\t\t\\hline\n\t\t$\\mathcal{G}^b, \\mathcal{G}^b_t$ & \\tabincell{c}{The bipartite graph only includes the user-item \\\\ interaction and that at the $t$-th timestamp.} \\\\\n\t\t\\hline\n\t\t$\\mathcal{E}^b, \\mathcal{E}^b_t$ & \\tabincell{c}{The edges set of bipartite graph \\\\ and that at the $t$-th time-step.} \\\\\n\t\t\\hline\n\t\t$\\mathcal{G}^s$ & The social networks. \\\\\n\t\t\\hline\n\t\t$\\mathcal{E}^s$ & The edges among users in social network\\\\\n\t\t\\hline\n\t\t$\\mathbf{H}_t$ & \\tabincell{c}{The heterogeneous graph that includes the \\\\ social network $\\mathcal{G}^s$ and the bipartite graph $\\mathcal{G}^b_t$ \\\\ at $t$-th time-step.} \\\\ \n\t\t\\hline\n\t\t$\\mathbf{p}_i$ & The embedding of user $u_i$. \\\\\n\t\t\\hline\n\t\t$\\mathbf{q}_j$ & The embedding of item $v_j$. \\\\\n\t\t\\hline\n\t\t$\\mathbf{k}_j$ & The embedding of the $j$-th position in item sequences.\\\\\n\t\t\\hline\n\t\t$\\mathbf{W}, \\mathbf{b}$ & Weights and biases in neural networks. \\\\\n\t\t\\hline\n\t\t$d$ & The dimension number of representation. \\\\\n\t\t\\hline\n\t\t$\\Theta_f$ & The parameters of unary scores function. \\\\\n\t\t\\hline\n\t\t$\\Theta_g$ & The parameters of transition scores function. \\\\\n\t\t\\hline\n\t\t$\\oplus$ & The concatenation operator of any two vectors. \\\\\n\t\t\\hline\n\t\t$\\bm{x}$ & The observed item sequence.\\\\\n\t\t\\hline\n\t\t$\\bm{y}$ & The label item sequence of $\\bm{x}$ \\\\\n\t\t\\hline\n\t\t$\\mathcal{N}(u_i)$ & The 1st-order neighbourhood of user $u_i$ \\\\\n\t\t\\hline\n\t\t$\\mathcal{I}_t({u_i})$ & The accessed items of user $u_i$ at $t$-th time-step. \\\\\n\t\t\\hline\n\t\t$\\tau$ & \\tabincell{c}{Time window for selecting the walks in the duration \\\\ of [t-$\\tau$, t+$\\tau$].} \\\\\n\t\t\\hline\n\t\t$d$ & The dimension of user and item embedding. \\\\\n\t\t\\hline\n\t\\end{tabular}\n\t\\label{tab:notation}\n\\end{table}\nIn this section, we begin with the problem definition of sequential recommendation. Then we derive the unified objective function under conditional probability $P(v_{t+1}|u_i, \\mathcal{H}_{1:t+1}; v_{1:t})$.\n\n\\subsection{Problem Definition}\nLet $U=\\{u_1, u_2, \\cdots, u_n\\}$ and $V=\\{v_1, v_2, \\cdots, v_m\\}$ denote the sets of users and items respectively, in which $n$ is the number of users and $m$ is the number of items. For user-item interactions, we let $\\mathcal{G}^b=\\{U \\cup V, \\mathcal{E}^b\\}$ be the user-item bipartite graph with edges $(u_i, v_j) \\in \\mathcal{E}^b$. As for user-user relations, we let $\\mathcal{G}^s=\\{U, \\mathcal{E}^s\\}$ be the social graph with edges $(u_i, u_j) \\in \\mathcal{E}^s$. If we combine the bipartite graph and the social graph, we obtain the following heterogeneous graph $\\mathbf{H}=\\{U \\cup V, \\mathcal{E}^b \\cup \\mathcal{E}^s\\}$. Let $v_{1:t}$ be the user behaviors sequence for $u_i$. Since we consider the temporal evolving social influence, we let $\\mathcal{H}_{t+1}=\\{\\mathbf{H}_1, \\mathbf{H}_2, \\cdots, \\mathbf{H}_{t+1}\\}$ be the heterogeneous graph sequence, where $\\mathbf{H}_t=\\{U \\cup V, \\mathcal{E}^b_t \\cup \\mathcal{E}^s\\}$ and $\\mathcal{E}^b_t$ is the user-item interactions in $t$-th time-step. For user $u_i$, given the behavior sequence $v_{1:t}$ and the heterogeneous graph sequence $\\mathcal{H}_{t+1}$ as well as the item $v_{t+1}$, our goal is to estimate the conditional probability of $P(v_{t+1}|v_{1:t},u_i,\\mathcal{H}_{t+1})$. The mathematical notation and the corresponding descriptions are summarized in Table \\ref{tab:notation}.\n\n\n\n\\subsection{Methodology}\nWe begin with the traditional Conditional Random Field (CRF), which is a probabilistic graphical model widely used in sequence labeling \\cite{panchendrarajan2018bidirectional}. CRF has shown to be very effective since it can jointly model the label decision by capturing the dependencies across adjacent labels. Considering the general definition of CRF, let $\\bm{x}=\\{x_1, \\cdots,x_t,\\cdots, x_T\\}$ and $\\bm{y}=\\{y_1, \\cdots, y_t, \\cdots, y_T\\}$ denote the observed sequence and its corresponding labels respectively. Formally, the conditional distribution $p(\\bm{y}|\\bm{x})$ of Linear Chain CRF\\cite{ma2016end} is given by:\n\\begin{equation}\n\\label{equ:crf}\n\\begin{split}\np(\\bm{y}|\\bm{x}) &= \\frac{1}{Z(\\bm{x})}\\exp(\\sum_{t=1}^{T}f(x_t,y_t;\\Theta_f) + \\sum_{t=1}^{T-1}g(y_t, y_{t-1};\\Theta_g)), \\\\\nZ(\\bm{x}) &= \\sum_{\\bm{y'}}\\exp(\\sum_{t=1}^{T}f(x_t,y'_t;\\Theta_f) + \\sum_{t=1}^{T-1}g(y'_t, y'_{t-1};\\Theta_g)),\n\\end{split}\n\\end{equation}\nin which $\\Theta_f$ and $\\Theta_g$ are the trainable parameters.\n\nThere are three important components in the above CRF model: the partition function $Z(\\bm{x})$, the unary scores function $f(x_t,y_t)$ and the transition scores function $g(y_t, y_{t-1})$. The partition function $Z(\\bm{x})$ is a normalization factor in order to obtain a probability. The unary scores function $f(x_t,y_t)$ is used to estimate the probability of $y_t$ given the observed $x_t$. And the transition scores function $g(y_t, y_{t-1})$ is used to estimate the probability of $y_t$ given $t_{t-1}$.\n\nThe three components framework provides us a unified solution to aggregate both the historical behaviors of users and the dynamic social influence from the social networks. Following the formulation of CRF, the purpose of our model is to estimate the conditional distribution as follow:\n\\begin{equation}\n\\label{equ:social_crf}\n\\begin{split}\nP(v_{1:t+1}|u_i, \\mathcal{H}_{t+1}) &= \\\\\n\\frac{1}{Z(\\mathcal{H}_{t+1}, u_i)}\\exp(&\\sum_{t=1}^{T}f(\\mathbf{H}_{t+1}, u_i, v_{t+1};\\Theta_f)\\\\&+\\sum_{t=1}^{T-1}g(v_{t+1}, v_{t};\\Theta_g)), \\\\\nZ(\\mathcal{H},u_i)=\\sum_{S'^{u_i}_{t+1}}\\exp(&\\sum_{t=1}^{T}f(\\mathbf{H}_{t+1}, u_i, v_{t+1};\\Theta_f)\\\\&+\\sum_{t=1}^{T-1}g(v_{t+1}, v_{t};\\Theta_g)),\n\\end{split}\t\n\\end{equation}\nin which $f(\\mathbf{H}_{t+1}, u_i, v_{t+1};\\Theta_f)$ denotes the aggregation of temporal evolving social influence, $g(v_{t+1}, v_{t};\\Theta_g)$ denotes the aggregation of user behaviors. In specific, $f(\\mathbf{H}_{t+1}, u_i, v_{t+1};\\Theta_f)$ describes the relationship between the the dynamic heterogeneous graph $\\mathbf{H}_{t+1}$ and the available item $v_{t+1}$ and $g(v_{t+1}, v_{t};\\Theta_g)$ models the dependency between the available item $v_{t+1}$ and the user behavior sequence.\n\nHowever, it is almost impossible to calculate $Z(\\mathcal{H}, u_i)$ since the sequence length is too large. In order to address this issue, we employ the pseudo likelihood method as an effective approximation method \\cite{besag1975statistical,ma2018cgnf}, and further derive the following estimation of the conditional probability:\n\\begin{equation}\n\\label{equ:social_crf_app}\n\\begin{split}\nP(v_{1:t+1}|u_i,\\mathcal{H}_{t+1}) \\approx PL(v_{1:t+1}|u_i,\\mathcal{H}_{t+1}) =\\\\ \\prod \\limits_{t} P(v_{t+1}|v_{1:t},u_i,\\mathcal{H}_{t+1}).\n\\end{split}\n\\end{equation}\n\nCombining Equation (\\ref{equ:social_crf}) and Equation (\\ref{equ:social_crf_app}), we further derive the following estimation of the conditional probability $P(v_{t+1}|v_{1:t},u_i,\\mathcal{H}_{t+1})$:\n\\begin{equation}\n\\begin{split}\nP(v_{t+1}|v_{1:t}&,u_i,\\mathcal{H}_{t+1})=\\\\\n&\\frac{\\exp(f(\\mathbf{H}_{t+1}, u_i, v_{t+1};\\Theta_f)+g(v_{t+1}, v_{1:t};\\Theta_g))}\n{\\sum_{v_j\\in V}\\exp(f(\\mathbf{H}_{t+1}, u_i, v_j;\\Theta_f)+g(v_j, v_{1:t};\\Theta_g))}.\n\\end{split}\n\\end{equation}\n\nFinally, we can obtain the objective function of our proposed model as follows:\n\\begin{equation}\n\\mathcal{L}_{crf} = \\frac{1}{n}\\sum_{i=1}^{n} \\sum_{t=1}^{T}\\log P(v_{t+1}|v_{1:t},u_i,\\mathcal{H}_{t+1}).\n\\end{equation}\n\nThe aforementioned objective function is usually impractical because the size of the item set is very large and the computation cost is unaffordable. \nInspired by \\cite{mikolov2013distributed}, we employ the negative sampling strategy to obtain the tractable unified objective function of sequential recommendation as follows:\n\\begin{equation}\n\\label{equ:final_loss}\n\\begin{split}\n\\mathcal{L}_{crf} =&\\frac{1}{n}\\sum_{i=1}^{n}\\sum_{t=1}^{T-1} \\log\\sigma(f(\\mathbf{H}_{t+1}, u_i, v_{t+1};\\Theta_f) \\\\&+ g(v_{t+1}, v_{1:t};\\Theta_g)) +\\\\\n& \\sum_{k=1}^{n_s}[\\log\\sigma(-f(\\mathbf{H}_{t+1}, u_i, v_{k};\\Theta_f) \\\\&- g(v_{k}, v_{1:t};\\Theta_g))],\n\\end{split}\n\\end{equation}\nwhere $\\sigma$ is the sigmoid activation function and $v_k$ is the negative item uniformly sampled from the whole item set $V$.\n\nThe objective function enjoys an \\textcolor{black}{appealing} physical meaning.\nIt provides the insights of how to design the model for sequential recommendation: $f(\\mathbf{H}_{t+1}, u_i, v_{t+1};\\Theta_f)$ models the information of temporal evolving heterogeneous graph in the forms of the unary energy function; meanwhile $g(v_{t+1}, v_{1:t};\\Theta_g)$ not only models the alternative item $v_t$ but also the user behavior sequence in the form of the pairwise energy function. \n\n\\section{Implementation of Temporally Evolving Aggregation Framework}\\label{implementation}\n\n\\begin{figure*}[htbp]\n\t\\includegraphics[width=\\textwidth]{fig\/model}\n\t\\caption{The framework of the temporally evolving aggregation model for the sequential recommendation. (a) The overview of the proposed model, the temporally dependent heterogeneous graphs aggregated representation $h_{t+1}^u$, the user behavior aggregated representation $h_{t+1}^v$ and the item embedding $\\mathbf{q}_{t+1}$ are fed into the CRF layer and $P(v_{t+1}|u_i, \\mathcal{H}_{t+1}; v_{1:t})$ is estimated. (b) The time-restricted user behavior sequence aggregation block is based on the user behavior sequence aggregation and the time-restricted aggregation. Note that the GRU used in this module is different from that in (a). (c) The dynamic temporally heterogeneous graph aggregation block, which is based on the bipartite graph aggregation and social network aggregation, takes $\\mathcal{H}_t$ as input, the arrows denote the message passing direction.}\n\t\\label{fig:model}\n\\end{figure*}\nIn this section, we provide the implementation details of the proposed temporally evolving aggregation model. As illustrated in Figure \\ref{fig:model}(a), our implementation takes both the aggregation of user behavior sequences and the aggregation of temporally dependent heterogeneous graphs into consideration and employs the GRU cells \\cite{cho2014learning} and CRF layers to predict the final results. The details of the two aggregation modules are presented in Figure \\ref{fig:model} (b) and Figure \\ref{fig:model} (c) respectively. We will give detailed descriptions of these two aggregation modules in the following subsections. \n\n\\subsection{Time-Restricted User Behavior Sequence Aggregation for the Transition Scores}\nIn this subsection, we will introduce the technical details of $g(v_{t+1}, v_{1:t};\\Theta_g)$. Given user $u_i$ and the corresponding behavior sequence $v_{1:t}$, we aim to calculate the user-specific item transition score. \n\n\\subsubsection{User Behavior Sequence Aggregation}\nConsidering that the future behavior of a user is not only influenced by the latest accessed items but also the items that the user has accessed before, the user behavior sequence aggregation block should consider both the transition between items and the long-term dependency of items. Inspired by the great success of the self-attention mechanism \\cite{vaswani2017attention} in various tasks like machine translation, we propose an extension of the self-attention mechanism to model the personalized item transition and long-term dependency by simultaneously leveraging the item information and the position information. Formally, given the $j$-th candidate item, we calculate the weights of each historical item as follows:\n\\begin{equation}\n\\begin{split}\n a_{\\tau j} = \\text{softmax}(\\frac{\\mathbf{W}_{Q}(\\mathbf{q}_j+\\mathbf{k}_j) \\left(\\mathbf{W}_{K} (\\mathbf{q}_\\tau + \\mathbf{k}_\\tau) \\right)^{\\mathsf{T}}}{\\sqrt{d}}), &\\tau < j ,\n\\end{split}\n\\end{equation}\nwhere $\\mathbf{q}_j$ is the embedding of item $v_j$, $\\mathbf{k}_j$ is the position embedding at $j$-th position of the input sequence, $\\mathbf{W}_Q, \\mathbf{W}_K$ are trainable projection parameters and $\\sqrt{d}$ is the scaling factor, and $d$ is the dimension of the embedding. As a result, we can calculate the historical item aggregated representation as follows:\n\\begin{equation}\n\\mathbf{z}_j = \\sum_{\\tau=1}^{\\tau=j-1} a_{\\tau j}\\mathbf{W}_V\\left(\\mathbf{q}_\\tau + \\mathbf{k}_\\tau\\right),\n\\end{equation}\nin which $\\mathbf{W}_V$ are trainable projection parameters. \n\n\\subsubsection{Time-Restricted Aggregation}\nSince the temporal interactions between users and items are very sparse, for the users that contain limited social relationships and items interactions, it is hard to obtain a ideal user embedding for the sparse social substructure, and it is also difficult to obtain a debiased item embedding. Therefore, it is a challenging task to well aggregate the information from the users to the items and vice verse. Fortunately, we find that the users that select the same items usually share the same interests and intent. Inspired by this intuition, we further proposed the time-restricted aggregation module.\n\nFirst, we selected the walk with three nodes (e.g., USER-ITEM-USER) with the restriction of time window $\\tau$. In detail, given the interaction $(u_i, v_t)$, we find the other users that select the same item in the time window of $[t-\\tau, t+\\tau]$, where $\\tau$ is the window size. In our experimental implementation, we choose $\\tau=60$ days. Therefore, we can collect the $\\tau-$restricted walks for example $u_i-v_t-u'$. Sequentially, we employ another GRU to aggregate the information from the dense substructures to the sparse substructures, which can be formalized as follow:\n\\begin{equation}\n \\mathbf{h}_{u_i}, \\mathbf{h}_{v_t}, \\mathbf{h}_{u'} = \\text{GRU}(\\mathbf{p}_i,\\mathbf{q}_t,\\mathbf{p}';\\mathcal{W}_R),\n\\end{equation}\nin which we take the walk $u_i-v_t-u'$ as input and $\\mathbf{h}_{u_i}, \\mathbf{h}_{v_t}, \\mathbf{h}_{u'}$ are the output of GRU of each timestamp; $\\mathcal{W}_R$ are the trainable parameters.\n\n\\subsubsection{Calculate the Transition Scores}\nIn order to well perform the personalized user behavior sequence aggregation, we further add the user embedding $\\mathbf{p}_i$ into the transformed item representation. Formally, we can calculate the transition score $s_t$ as follows:\n\\begin{equation}\n\\begin{split}\ns_t &= \\left(\\mathbf{W}_g^{(3)}\\left[\\mathbf{h}_t^{v_j}\\oplus \\bm{h}_{u_i} \\oplus \\bm{h}_{v_t}\\oplus \\mathbf{p}_i \\right]\\right)^\\mathsf{T} \\bm{q}_j, \\\\\n\\mathbf{h}_t^{v_j} &= \\mathbf{p}_i + \\mathbf{W}_g^{(2)}\\left(\\text{ReLU}(\\mathbf{W}_g^{(1)}\\mathbf{z}_j + \\mathbf{b}_g^{(1)}) \\right) + \\mathbf{b}_g^{(2)},\n\\end{split}\n\\end{equation}\nin which $\\mathbf{W}_g^{(1)}, \\mathbf{W}_g^{(2)}, \\mathbf{b}_g^{(1)}, \\mathbf{b}_g^{(2)}$ are the trainable parameters. For convenience, we let $\\Theta_g=\\{\\mathbf{W}_Q, \\mathbf{W}_K, \\mathbf{W}_V, \\mathbf{W}_g^{(1)}, \\mathbf{W}_g^{(2)}, \\mathbf{W}_g^{(3)},\\mathbf{b}_g^{(1)}, \\mathbf{b}_g^{(2)}, \\mathbf{p}, \\mathbf{q}, \\mathbf{k},\\bm{\\omega}_R\\}$ be the trainable parameters of $g(v_{t+1}, v_{1:t};\\Theta_g)$.\n\n\\subsection{Dynamic Temporal Heterogeneous Graphs Aggregation for the Unary Scores}\nIn this part, we will introduce the details of the dynamic temporally heterogeneous graphs aggregation $f(\\mathbf{H}_{t+1}, u_i, v_{t+1};\\Theta_f)$, which is used to calculate the unary scores. The dynamic temporally heterogeneous graphs aggregation contains the bipartite graph aggregation and the social network aggregation. \n\n\\subsubsection{Bipartite Graph Aggregation} In this part, we aim to obtain the aggregated of the bipartite graph at $t$-th time-step. Given user $u_i$ and the heterogeneous graph sequence $\\mathcal{H}_{t+1}$, we first obtain the user-specific representation $\\hat{\\mathbf{h}_{t}}$ of $\\mathcal{H}_t$. Specifically, we employ two different aggregated strategies and raise two variants of the proposed method: the GraphSAGE \\cite{hamilton2017inductive} based method (named TEA-S) and the graph attention networks \\cite{velivckovic2017graph} based method (named TEA-A). More experimental details will be introduced in the next section.\n\nAs for the TEA-S variation, we can obtain the user-specific representation $\\hat{\\mathbf{h}_{t}}$ as follows: \n\\begin{equation}\n\\hat{\\mathbf{h}_{t}} = \\text{ReLU} \\left({\\mathbf{W}_{A}} \\operatorname{MEAN}\\left(\\mathbf{q}_{k}, \\forall k \\in \\mathcal{I}_t(\\mathcal{N}(u_i))\\right)\\right) ,\n\\end{equation}\nwhere $\\mathbf{W}_{A}$ are the trainable parameters and $\\mathcal{I}_t(\\mathcal{N}(u_i))$ denotes the items interacted by $u_i$'s neighbors at between $t$-th and $t+1$-th time-step; and $\\operatorname{MEAN}$ denotes the average pooling operation.\n\nAs for the TEA-A variation, we aggregate the item information to the user with the help of the graph attention mechanism, which can be formulated as: \n\\begin{equation}\n\\begin{split}\n&\\qquad\\qquad\\hat{\\mathbf{h}_{t}} = \\text{ReLU} \\left(\\sum_{j \\in \\mathcal{I}_t(\\mathcal{N}(u_i)) } \\alpha_{i j}\\mathbf{q}_{j}\\right),\\\\\n\\end{split}\n\\end{equation}\n{where $\\alpha_{ij}$ is the weight of user $u_i$ and item $v_j$ and is defined as}\n\\begin{equation}\n\\small\n\\begin{split}\n\\alpha_{i j} = &\\frac{\\exp \\left(\\operatorname{LeakyReLU}\\left({\\mathbf{w}_{A}}^{\\mathsf{T}}\\left[ {\\mathbf{W}_{A}\\mathbf{q}_{t}} \\oplus \\mathbf{W}_{A}\\mathbf{q}_{j}\\right]\\right)\\right)}{\\sum_{k \\in \\mathcal{I}_t(\\mathcal{N}(u_i)) } \\exp \\left(\\operatorname{LeakyReLU}\\left({\\mathbf{w}_{A}}^{\\mathsf{T}}\\left[ \\mathbf{W}_{A}\\mathbf{q}_{t} \\oplus \\mathbf{W}_{A}\\mathbf{q}_{k}\\right]\\right)\\right)}\n,\n\\end{split}\n\\end{equation}\nin which $\\mathbf{q}_t$ is the embedding of the item interacted by $u_i$ at $t$-th time-step and $\\oplus$ is the concatenation operation. $\\mathbf{w}_{A}$ and $\\mathbf{W}_{A}$ are trainable parameters. And $LeakyReLU$ is the leaky version of a rectified linear unit.\n\nIn order to model temporally dependent heterogeneous graphs propagation, we feed $\\hat{\\mathbf{h}_t}$ into the Gated Recurrent Unit \\cite{cho2014learning}. The GRU cell operation at the $t$-th time-step can be formulated as: \n\\begin{equation}\n\\mathbf{h}_t = \\text{GRU}(\\hat{\\mathbf{h}_t}, \\mathbf{h}_{t-1}; \\mathcal{\\bm{W}}_G),\n\\end{equation}\nin which $\\mathcal{\\bm{W}}_G$ denotes all trainable parameters of the GRU cell. \\\\ \n\\subsubsection{Social Network Aggregation}\nTo propagate the information of neighbors' interests, we further aggregate the information from the social network. For simplicity, we only formulate the GraphSAGE aggregation as follows: \n\\begin{equation}\n\\mathbf{h}_{s} = \\text{ReLU}\\left(\\mathbf{W}_{S} \\operatorname{MEAN}\\left( \\mathbf{p}_{k}, \\forall k \\in \\mathcal{N}(u_i) \\right)\\right),\n\\end{equation}\nwhere $\\mathbf{W}_{S}$ is the trainable parameters. \n\n\n\\subsubsection{Calculate the Uunary Scores}\nBased on the aforementioned aggregation, we fuse the time-dependent representation $\\mathbf{h}_t$ and time-independent representation $\\mathbf{h}_s$ into one vector and calculate the social influence score $s_f$, i.e., the output of unary scores function $f(\\cdot)$. It is formulated as: \n\\begin{equation}\n\\begin{split}\ns_f &= {\\mathbf{h}_t^{u_i}}^\\mathsf{T} \\mathbf{q}_j,\\\\\n\\mathbf{h}_t^{u_i} = \\mathbf{W}_f^{(2)}\\text{ReLU}&(\\mathbf{W}_f^{(1)} [\\mathbf{h}_{t} \\oplus \\mathbf{h}_{s} ] + \\mathbf{b}_f^{(1)}) + \\mathbf{b}_f^{(2)},\n\\end{split}\n\\end{equation}in which $\\mathbf{W}_f^{(1)}, \\mathbf{W}_f^{(2)}, \\mathbf{b}_f^{(1)}, \\mathbf{b}_f^{(2)}$ are trainable parameters. In summary, we let $\\Theta_f=\\{ \\mathbf{W}_{A}, \\mathbf{W}_{S}, \\bm{\\omega}_{G},\\mathbf{W}_f^{(1)}, \\mathbf{W}_f^{(2)}, \\mathbf{b}_f^{(1)}, \\mathbf{b}_f^{(2)}, \\mathbf{p}, \\mathbf{q}\\}$ be the trainable parameters of $f(\\mathbf{H}_{t+1}, u_i, v_{t+1};\\Theta_f)$.\n\n\\subsection{Model Summarization}\n\n\nThe total loss of our proposed model is summarized as follow:\n\\begin{equation}\n\\mathcal{L} = \\mathcal{L}_{crf} + \\gamma \\mathcal{L}_{reg},\n\\end{equation}\nwhere $\\mathcal{L}_{reg}$ is the L2 normalization on all parameters and $\\gamma$ is a trade-off hyper-parameter. \n\nBased on this objective function, our model is trained by the following procedure:\n\\begin{equation}\n(\\hat{\\Theta_g},\\hat{\\Theta_f}) = \\underset{\\Theta_g, \\Theta_f}{\\arg \\min }\\mathcal{L}.\n\\end{equation}\nAll parameters are jointly optimized using the Adam\\cite{kingma2014adam} algorithm. \n\nIn the testing, we estimate the probability of $P(v_{t+1}|v_{1:t},u_i,\\mathcal{H}_{t+1})$ as follows:\n\\begin{equation}\n\\begin{split}\nP(&v_{t+1}|v_{1:t},u_i,\\mathcal{H}_{t+1}) =\\\\&\\sigma(f(\\mathbf{H}_{t+1}, u_i, v_{t+1};\\hat{\\Theta_f})+g(v_{t+1}, v_{1:t};\\hat{\\Theta_g})).\n\\end{split}\n\\end{equation}\n\n\n\\section{Connections to Existing Models}\\label{connection}\nWe will discuss the connections to the existing transition-based sequential recommendation methods. Most of the existing works of transition-based sequential recommendation methods \\cite{he2017translation,rendle2010factorizing} are based on Markov Chains. These methods mainly consider two important factors: (1) the interactions between users and items to capture the inherent intent of users, (2) the sequential dynamics between items to capture the relationships between items. Thus, we find that our method is more general and some of the existing works can be taken as special cases of ours. The detailed discussions for each work are as follows. \n\nRegarding the work FPMC \\cite{rendle2010factorizing}, it simplifies the huge state space problem by introducing the basket of items and consequently ignores the sequence information of historical items in each basket. In the contrast, our method utilizes the historical item sequence by using the self-attention mechanism with position embedding and is more general than FPMC.\n\nRegarding the work TransRec\n\\cite{he2017translation}, it models the personalized sequential behavior by using the personalized translation vectors and the previous item embedding to predict the next items but ignores the long-term dependencies since it only considers the relationships between any two items. Moreover, TransRec addresses the problem of the huge state space of items by introducing the subspace, while our method utilizes the negative sampling strategy. Thus, our method is more feasible and efficient to capture the dynamic social influence of the target users.\n\n\n\n\n\\section{Experiment}\\label{sec:exp}\n\nIn this section, we experimentally evaluate the performance of our method on three datasets against the state-of-the-art compared methods. The preprocessed scripts and the source code can be found at \\footnote{{https:\/\/github.com\/DMIRLAB-Group\/TEA}}.\n\\begin{table}[t]\n\t\\caption{Statistics of the datasets.}\n\t\\label{tab:stat}\n\t\\centering\n\t\\scalebox{1.0}{\n\t\t\\begin{tabular}{p{2.0cm}p{1.5cm}p{1.5cm}p{1.5cm}}\n\t\t\t\\toprule\n\t\t\tDataset & \\textbf{Epinions} & \\textbf{Yelp} & \\textbf{Wechat}\\\\\n\t\t\t\\midrule\n\t\t\t\\# users & 22,167 & 270,770 & 568,100 \\\\\n\t\t\t\\# items & 296,278 & 184,134 & 242,702 \\\\\n\t\t\t\\# interactions & 798,620 & 3,602,495 & 9,422,722 \\\\\n\t\t\t\\# social links & 355,813 & 5,974,526 & 5,667,864 \\\\\n\t\t\tdensity & 0.0121\\% & 0.0072\\% & 0.0068\\% \\\\\n\t\t\tsocial density & 0.0724\\%\t& 0.0081\\% & 0.0018\\% \\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t}\n\\end{table}\n\\subsection{Datasets}\nWe evaluate our proposed TEA framework on two public datasets (Epinions and Yelp) and a large-scale private dataset (WeChat Official Accounts Dataset). \nThe statistics of datasets are summarized in Table \\ref{tab:stat}. The brief information of the datasets is as follows: \n\\begin{itemize}\n\t\\item {Epinions}\\footnote{{http:\/\/www.trustlet.org\/extended\\_epinions.html}}: A benchmark dataset for the recommendation. In Epinions, a user can rate and give comments on items. Besides, a user can also select other users as their trusters, and we use the trust graphs \\textcolor{black}{(which are composed of the trust relationships)} as the network information. \n\t\\item {Yelp}\\footnote{{https:\/\/www.kaggle.com\/yelp-dataset\/yelp-dataset}}: An online review platform where users review local businesses (e.g., restaurants and shops). The user-item interactions and the social networks are extracted in the same way as Epinions. \n\t\\item {WeChat Official Accounts Dataset}: WeChat is a Chinese multi-purpose messaging, social media, and mobile payment application developed by Tencent. And WeChat official accounts dataset is one of the functions. On the WeChat Official Account platform, users can read and share articles. This dataset is constructed by user-article clicking records and user-user social networks on this platform. \n\\end{itemize}\n\nWe preprocess the datasets following the approach in \\cite{he2017translation}. \\textcolor{black}{Specifically, for all these datasets, we follow the previous works \\cite{kang2018self,sun2019bert4rec} and treat a rating or review as implicit feedback.} We further use the timestamps to determine the sequence order of actions. We discard users and items with fewer than 5 associated actions. In cases where star ratings are available, we take the item with a rating higher than 3 as users' positive feedback. \n\nFor data splitting, we employ the widely used leave-one-out evaluation \\cite{rendle2009bpr, he2017neural}. We hold out the latest interaction of each user as the test set and select the second latest interaction as the validation set. The remaining data are used for training. \n\n\\begin{table*}[htb]\n\t\\caption{The performance evaluation of the compared methods on Epinions dataset. The value presented are averaged over 5 replicated with different random seeds.}\n\t\\label{tab:epin}\n\t\\centering\n\t\\scalebox{1.1}{\n\t\\begin{tabular}{|l|lcccccc|}\n\t\t\\hline\n\t\tModel Class & \\multicolumn{1}{l|}{Models} & HR@5 & NDCG@5 & HR@10 & NDCG@10 & HR@20 & {NDCG@20} \\\\ \\hline\n\t\t& \\multicolumn{1}{l|}{BPRMF \\cite{rendle2009bpr}} & 38.72$\\pm$0.10 & 29.66$\\pm$0.12 & 47.53$\\pm$0.10 & 32.50$\\pm$0.07 & 57.21$\\pm$0.22 & 34.95$\\pm$0.13 \\\\\n\t\t& \\multicolumn{1}{l|}{NeuMF \\cite{he2017neural}} & 41.35$\\pm$0.59 & 31.13$\\pm$0.69 & 51.15$\\pm$0.43 & 34.31$\\pm$0.64 & 60.93$\\pm$0.34 & 36.78$\\pm$0.59 \\\\\n\t\t& \\multicolumn{1}{l|}{SocialMF \\cite{jamali2010matrix}} & 41.78$\\pm$0.16 & 32.57$\\pm$0.29 & 50.01$\\pm$0.18 & 35.23$\\pm$0.29 & 58.23$\\pm$0.14 & 37.31$\\pm$0.25 \\\\\n\t\t\\multirow{-4}{*}{\\begin{tabular}[c]{@{}l@{}}Matrix \\\\ Factorization \\\\ based\\end{tabular}}\n\t\t& \\multicolumn{1}{l|}{SoRec \\cite{ma2008sorec}} & 40.81$\\pm$0.33 & 31.14$\\pm$0.30 & 49.61$\\pm$0.16 & 33.99$\\pm$0.24 & 58.42$\\pm$0.19 & 36.22$\\pm$0.25 \\\\ \\hline\n\t\t& \\multicolumn{1}{l|}{GraphRec \\cite{fan2019graph}} & 39.50$\\pm$0.35 & 30.16$\\pm$0.27 & 48.94$\\pm$0.42 & 33.21$\\pm$0.21 & 58.87$\\pm$0.29 & 35.72$\\pm$0.20 \\\\\n\t\t& \\multicolumn{1}{l|}{LightGCN \\cite{he2020lightgcn}} & 42.59$\\pm$0.07 & 32.20$\\pm$0.09 & 51.92$\\pm$0.08 & 35.22$\\pm$0.07 & 60.54$\\pm$0.09 & 37.41$\\pm$0.08 \\\\\n\t\t\\multirow{-3}{*}{\\begin{tabular}[c]{@{}l@{}}Graph Neural \\\\ Network based\\end{tabular}}\n\t\t& \\multicolumn{1}{l|}{DGRec \\cite{song2019session}} & 40.36$\\pm$0.25 & 30.52$\\pm$0.16 & 49.67$\\pm$0.14 & 33.53$\\pm$0.15 & 59.26$\\pm$0.19 & 35.95$\\pm$0.15 \\\\ \\hline\n\t\t& \\multicolumn{1}{l|}{DMAN \\cite{Tan_Zhang_Liu_Huang_Yang_Zhou_Hu_2021}} & 35.15$\\pm$0.27 & 27.06$\\pm$0.33 & 45.01$\\pm$0.06 & 30.23$\\pm$0.24 & 55.85$\\pm$0.27 & 32.98$\\pm$0.30 \\\\\n\t\t& \\multicolumn{1}{l|}{TransRec \\cite{he2017translation}} & 44.79$\\pm$0.12 & 36.09$\\pm$0.21 & 52.51$\\pm$0.11 & 38.58$\\pm$0.17 & 60.98$\\pm$0.11 & 40.72$\\pm$0.07 \\\\\n\t\t& \\multicolumn{1}{l|}{SASRec \\cite{kang2018self}} & 43.32$\\pm$0.20 & 33.97$\\pm$0.20 & 51.88$\\pm$0.20 & 36.74$\\pm$0.20 & 60.31$\\pm$0.20 & 38.87$\\pm$0.18 \\\\\n\t\t\\multirow{-3}{*}{\\begin{tabular}[c]{@{}l@{}}Sequence \\\\ based\\end{tabular}}\n\t\t& \\multicolumn{1}{l|}{ASAS \\cite{manotumruksa2020sequential}} & 44.97$\\pm$0.34 & 35.59$\\pm$0.29 & 53.44$\\pm$0.29 & 38.33$\\pm$0.27 & 61.41$\\pm$0.29 & 40.35$\\pm$0.28 \\\\ \n\t\t\\hline\n\t\tOurs \n\t\t& \\multicolumn{1}{l|}{TEA-A} & 47.84$\\pm$0.04 & 38.40$\\pm$0.41 & 55.99$\\pm$0.04 & 41.04$\\pm$0.41 & 63.51$\\pm$0.29 & 42.95$\\pm$0.38 \\\\\n\t\t& \\multicolumn{1}{l|}{TEA-S} & \\textbf{48.13}$\\pm$0.25 & \\textbf{38.65}$\\pm$0.18 & \\textbf{56.10}$\\pm$0.17 & \\textbf{41.24}$\\pm$0.15 & \\textbf{63.58}$\\pm$0.08 & \\textbf{43.13}$\\pm$0.11 \\\\\n\t\t\\hline \n\t\\end{tabular}}\n\\end{table*}\n\n\n\n\\subsection{Implementation Details}\nWe use PyTorch to implement our model and deploy it on RTX 2080 GPU. Hyper-parameter settings for all three datasets are as follows: embedding dimension $d=64$, batch size $B=1024$, dropout rate $p_\\text{drop}=0.5$, L2 regularization weight $\\gamma$=5e-4, negative sampling size $n_s=50$, sequence truncation length $L_s=50$, neighbor truncation length $L_n=20$, and learning rate $\\eta=0.01$. We train all the methods with five different random seeds and report the means and standard deviations. \n\n\\subsection{Evaluation Metrics}\nWe evaluate all the models with two widely used Top-N metrics: Hit Rate@$K$ (HR@$K$) and Normalized Discounted Cumulative Gain@$K$ (NDCG@$K$).\nHR measures the percentage that recommended items contain at least one correct item interacted by the user, while NDCG considers the positions of correct recommended items. \nIn the context of sequential recommendation, since we only test on the latest item in a user behavior sequence, HR is identical to recall and proportional to precision \\cite{kang2018self}. \n\nSince it is time-consuming to rank all items for each user during the evaluation, we followed the strategy in \\cite{kang2018self}. Specifically, for each user, we randomly sample 100 negative items and rank these items with the ground-truth item. HR and NDCG are estimated based on the ranking results. We report the experiment results for $K=5\/10\/20$. \n\n\n\\subsection{Compared Methods}\nWe compare our proposed models (TEA-S and TEA-A) based on TEA framework with three kinds of baselines: the matrix factorization based models, the graph neural networks based models, and the sequence recommendation methods. \\\\\n\\textbf{Matrix Factorization based Methods}:\n\\begin{itemize}\n\t\\item BPRMF\\cite{rendle2009bpr}: A general learning framework for personalized ranking recommendation uses implicit feedback. \n\t\\item NeuMF\\cite{he2017neural}: It replaces the inner product with a multilayer perception (MLP) to learn the user-item interaction function.\n\t\\item SocialMF \\cite{jamali2010matrix}: It considers the social information and propagation of social information into the matrix factorization model.\n\t\\item SoRec\\cite{ma2008sorec}: It performs co-factorization on the user-item rating matrix and user-user social relations matrix.\n\\end{itemize}\n\\textbf{Graph Neural Network based Methods}:\n\\begin{itemize}\n\t\\item GraphRec\\cite{fan2019graph}: It uses the graph neural network to combine user behavior information and social network information into the recommendation task. For fairness, we discard the opinion\/rate embedding in our implementation. \n\n\t\\item LightGCN \\cite{he2020lightgcn}: A state-of-the-art graph-based collaborative filtering method. It explicitly integrates a bipartite graph structure into the embedding learning process to model the high-order connectivity in the user-item interaction graph\n\t\\item DGRec \\cite{song2019session}: A session-based recommendation method that combines the user action-temporal information and the social information via recurrent neural networks and dynamic graph attention networks.\n\\end{itemize}\n\\textbf{Sequential Recommendation Methods:}\n\\begin{itemize}\n\t\\item TransRec\\cite{he2017translation}: A sequential recommendation method that models each user as a translation vector to capture the transition from the current item to the next.\n\t\\item SASRec \\cite{kang2018self}: It leverages the Transformer\\cite{vaswani2017attention} to capture users' sequential behaviors.\n\t\\item ASASRec \\cite{manotumruksa2020sequential}: An improved version of SASRec with an adversarial training strategy. \n\t\\item DMAN \\cite{Tan_Zhang_Liu_Huang_Yang_Zhou_Hu_2021}: It effectively utilizes the sequential data by segmenting the overall behavior sequence and maintaining the long-term interests of users. \n\\end{itemize}\n\\textbf{Model Variants:}\n\\begin{itemize}\n\t\\item TEA-S: We use the GraphSAGE based aggregation method in the bipartite graph aggregation. \n\t\\item TEA-A: We use the Graph Attention mechanism based aggregation method in the bipartite graph aggregation.\n\t\\item TEA-RS: We remove the time-restricted aggregation and use the GraphSAGE based aggregation method in the bipartite graph aggregation. \n\t\\item TEA-RA: We remove the time-restricted aggregation and use the Graph Attention mechanism based aggregation method in the bipartite graph aggregation. \n\\end{itemize}\n\n\\subsection{Results}\n\n\n\n\\begin{table*}[htb]\n\t\\caption{The performance evaluation of the compared methods on Yelp dataset. The value presented are averaged over 5 replicated with different random seeds.}\n\t\\label{tab:yelp}\n\t\\centering\n\t\\scalebox{1.1}{\n\t\\begin{tabular}{|l|lcccccc|}\n\t\t\\hline\n\t\tModel Class & \\multicolumn{1}{l|}{Models} & HR@5 & NDCG@5 & HR@10 & NDCG@10 & HR@20 & {NDCG@20} \\\\ \\hline\n\t\t& \\multicolumn{1}{l|}{BPRMF \\cite{rendle2009bpr}} & 66.33$\\pm$0.27 & 52.46$\\pm$0.16 & 76.51$\\pm$0.26 & 55.77$\\pm$0.16 & 84.59$\\pm$0.22 & 57.82$\\pm$0.15 \\\\\n\t\t& \\multicolumn{1}{l|}{NeuMF \\cite{he2017neural}} & 70.38$\\pm$0.26 & 56.14$\\pm$0.28 & 79.35$\\pm$0.12 & 59.06$\\pm$0.24 & 86.14$\\pm$0.12 & 60.79$\\pm$0.22 \\\\\n\t\t& \\multicolumn{1}{l|}{SocialMF \\cite{jamali2010matrix}} & 64.82$\\pm$0.24 & 49.69$\\pm$0.24 & 76.27$\\pm$0.28 & 53.42$\\pm$0.21 & 84.99$\\pm$0.28 & 55.63$\\pm$0.19 \\\\\n\t\t\\multirow{-4}{*}{\\begin{tabular}[c]{@{}l@{}}Matrix \\\\ Factorization \\\\ based\\end{tabular}}\n\t\t& \\multicolumn{1}{l|}{SoRec \\cite{ma2008sorec}} & 70.41$\\pm$0.10 & 54.55$\\pm$0.10 & 81.45$\\pm$0.04 & 58.15$\\pm$0.07 & 89.03$\\pm$0.04 & 60.08$\\pm$0.06 \\\\ \\hline\n\t\t& \\multicolumn{1}{l|}{GraphRec \\cite{fan2019graph}} & 68.37$\\pm$0.23 & 51.44$\\pm$0.27 & 81.55$\\pm$0.17 & 55.74$\\pm$0.18 & 90.61$\\pm$0.17 & 58.05$\\pm$0.16 \\\\\n\t\t& \\multicolumn{1}{l|}{LightGCN \\cite{he2020lightgcn}} & 73.04$\\pm$0.21 & 57.10$\\pm$0.21 & 84.39$\\pm$0.07 & 60.80$\\pm$0.19 & 92.08$\\pm$0.07 & 62.76$\\pm$0.17 \\\\\n\t\t\\multirow{-3}{*}{\\begin{tabular}[c]{@{}l@{}}Graph Neural \\\\ Network based\\end{tabular}}\n\t\t& \\multicolumn{1}{l|}{DGRec \\cite{song2019session}} & 76.22$\\pm$0.24 & 60.18$\\pm$0.28 & 86.57$\\pm$0.18 & 63.55$\\pm$0.26 & 92.93$\\pm$0.08 & 65.18$\\pm$0.16 \\\\ \\hline\n\t\t& \\multicolumn{1}{l|}{DMAN \\cite{Tan_Zhang_Liu_Huang_Yang_Zhou_Hu_2021}} & 72.93$\\pm$0.33 & 57.45$\\pm$0.16 & 83.64$\\pm$0.34 & 60.94$\\pm$0.29 & 91.03$\\pm$0.25 & 62.82$\\pm$0.26 \\\\\n\t\t& \\multicolumn{1}{l|}{TransRec \\cite{he2017translation}} & 75.81$\\pm$0.15 & 60.63$\\pm$0.16 & 80.19$\\pm$0.20 & 64.00$\\pm$0.15 & 93.13$\\pm$0.12 & 65.78$\\pm$0.15 \\\\\n\t\t& \\multicolumn{1}{l|}{SASRec \\cite{kang2018self}} & 69.28$\\pm$0.39 & 53.18$\\pm$0.43 & 81.66$\\pm$0.08 & 57.21$\\pm$0.37 & 90.36$\\pm$0.08 & 59.43$\\pm$0.34 \\\\\n\t\t\\multirow{-3}{*}{\\begin{tabular}[c]{@{}l@{}}Sequence \\\\ based\\end{tabular}}\n\t\t& \\multicolumn{1}{l|}{ASASRec \\cite{manotumruksa2020sequential}} & 72.97$\\pm$0.13 & 56.76$\\pm$0.10 & 84.53$\\pm$0.04 & 60.53$\\pm$0.09 & 92.18$\\pm$0.04 & 62.48$\\pm$0.07 \\\\ \n\\hline\n\t\tOurs\n\t\t& \\multicolumn{1}{l|}{TEA-A} & 80.38$\\pm$0.25 & 65.42$\\pm$0.36 & \\textbf{88.99}$\\pm$0.14 & 68.23$\\pm$0.33 & \\textbf{94.11}$\\pm$0.10 & 69.54$\\pm$0.21 \\\\\n\t\t& \\multicolumn{1}{l|}{TEA-S} & \\textbf{80.43}$\\pm$0.18 & \\textbf{65.59}$\\pm$0.26 & 88.97$\\pm$0.08 & \\textbf{68.37}$\\pm$0.23 & 94.09$\\pm$0.07 & \\textbf{69.68}$\\pm$0.21 \\\\ \n\t\t\\hline \n\t\\end{tabular}}\n\\end{table*}\n\n\n\\begin{table*}[htb]\n\t\\caption{The performance evaluation of the compared methods on WeChat dataset. The value presented are averaged over 5 replicated with different random seeds.}\n\t\\label{tab:Wechat}\n\t\\centering\n\t\\scalebox{1.1}{\n\t\\begin{tabular}{|l|ccccccc|}\n\t\t\\hline\n\t\tModel Class & \\multicolumn{1}{l|}{Models} & HR@5 & NDCG@5 & HR@10 & NDCG@10 & HR@20 & {NDCG@20} \\\\ \\hline\n\t\t& \\multicolumn{1}{l|}{BPRMF \\cite{rendle2009bpr}} & 62.33$\\pm$0.12 & 56.38$\\pm$0.12 & 68.55$\\pm$0.18 & 58.38$\\pm$0.14 & 75.60$\\pm$0.10 & 60.16$\\pm$0.14 \\\\\n\t\t& \\multicolumn{1}{l|}{NeuMF \\cite{he2017neural}} & 68.58$\\pm$0.16 & 61.66$\\pm$0.16 & 75.26$\\pm$0.18 & 63.82$\\pm$0.16 & 82.53$\\pm$0.14 & 65.65$\\pm$0.15 \\\\\n\t\t& \\multicolumn{1}{l|}{SocialMF \\cite{jamali2010matrix}} & 68.25$\\pm$0.13 & 61.30$\\pm$0.42 & 74.62$\\pm$0.07 & 63.36$\\pm$0.39 & 81.44$\\pm$0.04 & 65.09$\\pm$0.37 \\\\\n\t\t\\multirow{-4}{*}{\\begin{tabular}[c]{@{}l@{}}Matrix \\\\ Factorization \\\\ based\\end{tabular}}\n\t\t& \\multicolumn{1}{l|}{SoRec \\cite{ma2008sorec}} & 73.66$\\pm$0.04 & 66.43$\\pm$0.11 & 79.60$\\pm$0.02 & 68.36$\\pm$0.10 & 85.56$\\pm$0.03 & 69.86$\\pm$0.09 \\\\ \\hline\n\t\t\n\t\t& \\multicolumn{1}{l|}{GraphRec \\cite{fan2019graph}} & 66.04$\\pm$0.33 & 52.17$\\pm$0.29 & 76.80$\\pm$0.25 & 55.66$\\pm$0.26 & 85.72$\\pm$0.18 & 57.93$\\pm$0.24 \\\\\n\t\t& \\multicolumn{1}{l|}{LightGCN \\cite{he2020lightgcn}} & - & - & - & - & - & - \\\\\n\t\t\\multirow{-2}{*}{\\begin{tabular}[c]{@{}l@{}}Graph Neural \\\\ Network based\\end{tabular}}\n\t\t& \\multicolumn{1}{l|}{DGRec \\cite{song2019session}} & 74.99$\\pm$0.22 & 63.94$\\pm$0.24 & 82.29$\\pm$0.14 & 66.31$\\pm$0.24 & 88.52$\\pm$0.09 & 67.89$\\pm$0.20 \\\\ \\hline\n\t\t& \\multicolumn{1}{l|}{DMAN \\cite{Tan_Zhang_Liu_Huang_Yang_Zhou_Hu_2021}} & - & - & - & - & - & - \\\\\n\t\t& \\multicolumn{1}{l|}{TransRec \\cite{he2017translation}} & 73.54$\\pm$0.17 & 64.87$\\pm$0.16 & 80.06$\\pm$0.14 & 66.99$\\pm$0.15 & 86.03$\\pm$0.14 & 68.50$\\pm$0.14 \\\\\n\t\t& \\multicolumn{1}{l|}{SASRec \\cite{kang2018self}} & 76.37$\\pm$0.35 & 65.97$\\pm$0.35 & 83.76$\\pm$0.21 & 68.37$\\pm$0.29 & 89.94$\\pm$0.11 & 69.94$\\pm$0.23 \\\\\n\t\t\\multirow{-3}{*}{\\begin{tabular}[c]{@{}l@{}}Sequence \\\\ based\\end{tabular}}\n\t\t& \\multicolumn{1}{l|}{ASASRec \\cite{manotumruksa2020sequential}} & 78.28$\\pm$0.23 & 68.13$\\pm$0.25 & 85.19$\\pm$0.21 & 70.38$\\pm$0.19 & 90.81$\\pm$0.14 & 71.80$\\pm$0.13 \\\\ \n \\hline\n\t\tOurs\n\t\t& \\multicolumn{1}{l|}{TEA-A} & 82.06$\\pm$0.25 & 73.47$\\pm$0.26 & 87.61$\\pm$0.16 & 75.28$\\pm$0.21 & 92.18$\\pm$0.17 & 76.44$\\pm$0.13 \\\\\n\t\t& \\multicolumn{1}{l|}{TEA-S} & \\textbf{83.23}$\\pm$0.19 & \\textbf{76.12}$\\pm$0.21 & \\textbf{88.12}$\\pm$0.13 & \\textbf{77.70}$\\pm$0.18 & \\textbf{92.42}$\\pm$0.17 & \\textbf{78.79}$\\pm$0.16 \\\\\n\t\t\\hline \n\t\\end{tabular}}\n\\end{table*}\n\n\n\n\n{Tables \\ref{tab:epin}, \\ref{tab:yelp} and \\ref{tab:Wechat} present the recommendation performance of all the methods on the three datasets, respectively. We do not report the performance of LightGCN, DMAN on WeChat Official Accounts dataset because of the limitation of memory.}\n\nFirst, by modeling social influence, the performances of social-aware methods (SocialMF, SoRec, GraphRec, and DGRec) are improved compared with that of BPRMF in most cases, which is consistent with previous works. This {observation} indicates that social information reflects users' interests effectively. \nSecond, the sequence based methods (DGRec, TransRec, SAS, and ASAS) also perform comparably well. These improvements reflect the importance of temporal information on recommendation tasks. \nThird, DGRec and our proposed methods (including TEA-S and TEA-A) that combine social information and temporal information achieve much better performance, especially on large datasets. \nAt last, our proposed TEA-S and TEA-A consistently outperform all the {compared methods} on both public and real-world datasets with an average improvement of 3.15\\% on HR@10 and 8.38\\% on NDCG@10 against the best {competitor}. The significant improvements validate the effectiveness of aggregating the user behavior sequence and the the influence between the users. We also observe that performance of TEA-A is slightly {lower} than that of TEA-S, indicating that the graph attention mechanism is difficult to handle the high sparsity of temporally evolving heterogeneous graphs.\n\n\\begin{figure}[t]\n\t\\label{fig:hypm_emb}\n\t\\centering\n\t\\scalebox{0.48}\n\t{\\includegraphics[width=\\textwidth]{fig\/hypm_emb}}\n\t\\caption{The sensitivity of the embedding dimension $d$.}\n\\end{figure}\n\\subsection{The Sensitivity of Hyper-parameters}\n\n\\begin{figure}[htbp]\n\\centering\n\\begin{minipage}[htbp]{0.47\\textwidth}\n\\centering\n\\includegraphics[width=\\textwidth]{fig\/ns_line_hr10}\n\\caption{The sensitivity of the negative sampling size $n_s$.}\n\\end{minipage}\n\\begin{minipage}[htbp]{0.47\\textwidth}\n\\centering\n\\includegraphics[width=\\textwidth]{fig\/ns_line_nd10}\n\\caption{The sensitivity of the negative sampling size $n_s$.}\n\\end{minipage}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\centering\n\\begin{minipage}[t]{0.24\\textwidth}\n\\centering\n\\label{fig:ablation_yelp}\n\\includegraphics[width=\\textwidth]{fig\/yelp_albation_hr20}\n\\caption{The Experiment results of ablation studies on Yelp dataset.}\n\\end{minipage}\n\\begin{minipage}[t]{0.24\\textwidth}\n\\centering\n\\label{fig:ablation_epinion}\n\\includegraphics[width=\\textwidth]{fig\/epinion_albation_hr20}\n\\caption{The Experiment results of ablation studies on Epinion dataset.}\n\\end{minipage}\n\\end{figure}\n\n\\subsubsection{Embedding Dimension.}\n\n\n\n\nIn {Figure} 3 we analyze the sensitivity of the embedding dimension $d$ by showing HR@10 and NDCG@10 of our proposed {TEA-S} with $d$ varying from 8 to 64. We can observe that our model significantly benefits from a larger dimension when the dataset is large. {A small embedding dimension ($d=16$) is enough for {TEA-S} to achieve the best performance on Epinions.} \n\n\n\n\n\n\\subsubsection{Sensitivity of the Number of Negative Samples.}\n{Figures} 4 and 5 shows the sensitivity of the number of negative samples $n_s$ in Equation (\\ref{equ:final_loss}) by showing HR@10 and NDCG@10 of our proposed {TEA-S} with $n_s$ varying from 1 to 100. The variant with $n_s=5$ performs comparably well, though using $n_s \\geq 10$ still boosts performance especially on the large-scale dataset, which means that using more negative samples is helpful to estimate the item transition probability. \nThe variant with $n_s=100$ achieves similar performance to the default setting $n_s=50$, which indicats that our model is stable with $n_s$. \n\n\\subsection{Ablation Study}\nIn order to evaluate the effectiveness of the time-restricted aggregation, we remove the aforementioned aggregation module and obtain the variants \\textbf{TEA-RS} and \\textbf{TEA-RA}. {experimental results are} shown in {Figures} 6 and 7. {From these results}, we can find that the models with time-restricted aggregation achieve a better performance, especially the results on the Yelp dataset. We also find that the promotion in Epinion dataset is not so remarkable, this is {since} the social networks in Epinion are much denser than that of Yelp. To some extent, the experiment results reflect that the proposed time-restricted aggregation can mitigate the drawbacks of sparse social networks and user-item interactions.\n\n\n\n\\section{Conclusion}\\label{sec:conclu}\nThis paper presents a temporally evolving aggregations framework for the sequential recommendation. Beginning from the original conditional random field, we derive the unified objective function for the sequential recommendation, \\textcolor{black}{which leverages the social influence between users and the dynamic user-item heterogeneous graph.} The proposed framework provides the insights and principles of designing the sequential recommendation model. We further provide two different implementations of the proposed framework. Experimental results on three real-world datasets show that the \\textbf{TEA} framework outperforms state-of-the-art methods.\n\n\\section{Acknowledgments}\n\nWe would like to thank Lingling Yi and Li Li from WeChat for their help and supports on this work.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\ifCLASSOPTIONcaptionsoff\n \\newpage\n\\fi\n\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}