diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzznwam" "b/data_all_eng_slimpj/shuffled/split2/finalzznwam" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzznwam" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nInterferometric observations of Cepheid binary dynamics are being used to place more stringent constraints on masses of individual Cepheids \\citep[see, e.g.,][]{2018ApJ...863..187E}. Cepheid masses have also been derived using envelope pulsation models \\citep[see, e.g.,][]{2005ApJ...629.1021C}. When these Cepheids are plotted on the Hertzsprung-Russell (H-R) diagram, their observed luminosities are higher than those of standard stellar evolution model tracks of the same mass. In other words, the evolution models need to be 10-30\\% more massive to reach the luminosities of the observed Cepheids, with the magnitude of the difference increasing toward lower mass\/shorter period Cepheids. The evolution model luminosities and the extent of 'blue loops', where stars evolve into the Cepheid pulsation instability region during their core helium-burning phase, are affected by many factors, including rotation \\citep{2014A&A...564A.100A,2018A&A...616A.112S}, mass loss \\citep{2006MmSAI..77..207B, 2011A&A...529L...9N}, convective overshooting \\citep{2011A&A...529L...9N}, and nuclear reaction rates \\citep{2010A&A...520A..41M}. To date, no single modification of physical inputs has resolved definitively this Cepheid mass discrepancy.\n\nMotivated by this mass discrepancy, we explore Cepheid evolution and pulsation models using the Modules for Experiments in Astrophysics (MESA) open-source stellar evolution code\\footnote{mesa.sourceforge.net. See `getting started' tutorial and example ``star\/test\\_suite\/5M\\_cepheid\\_blue\\_loop'' for Cepheid evolution, and ``star\/test\\_suite\/rsp\\_Cepheid'' for RSP model example.} \\citep{2019ApJS..243...10P}, version r12115. In particular, we consider increases in the triple-$\\alpha$ and $^{12}$C($\\alpha$,$\\gamma$)$^{16}$O reaction rates, and opacity enhancements that have been proposed to explain the pulsation frequencies of main-sequence hybrid $\\beta$ Cepheid\/Slowly-Pulsating B (SPB) stars that will evolve to become Cepheids. We also use the new radial stellar pulsation (RSP) capability in MESA to model the nonlinear radial envelope pulsations of three Galactic Cepheids: $\\delta$ Cep, Polaris, and V1334 Cyg. \n\n\\section{Effects of reaction rate and opacity increases on Cepheid evolution}\n\\label{evolution}\n\nFigure \\ref{reaction_rates} compares evolution models with nominal MESA nuclear reaction rates, and with triple-$\\alpha$ and $^{12}$C($\\alpha$,$\\gamma$)$^{16}$O reaction rates, important during helium burning, multiplied by a factor of three. These models use AGSS09 \\citep{2009ARA&A..47..481A} abundance mixture, helium mass fraction Y=0.28, metallicity Z=0.02, OPAL \\citep{1996ApJ...464..943I} opacities, a standard MESA convective overshoot treatment, and do not include rotation or mass loss. While these models with higher reaction rates do show an increase in luminosity ($\\sim$0.2 dex) and extent of the blue loops, the effect is not large enough by itself to solve the mass discrepancy; for a 4.5 M$_{\\odot}$ model, a luminosity increase of $\\sim$0.5 dex is needed. Furthermore, reaction rates are assessed to be uncertain by less than a factor of 1.5 \\citep{2018ApJS..234...19F}.\n\n\\articlefigure[width=.75\\textwidth]{reaction_rates.eps}{reaction_rates}{Post-main sequence MESA evolution tracks for massive stars with unmodified reaction rates (blue), and with triple-$\\alpha$ and $^{12}$C($\\alpha$,$\\gamma$)$^{16}$O multiplied by a factor of three (orange). The red and blue vertical lines mark the boundaries of the Cepheid instability strip for Z=0.02 from \\cite{2000ApJ...529..293B}.}\n\nOpacity increases have been proposed to explain the pulsation frequency content of main-sequence hybrid $\\beta$ Cep\/SPB pulsators, e.g., $\\nu$ Eri \\citep{2017MNRAS.466.2284D}. We applied the same temperature-dependent multipliers for OPAL opacities used for $\\nu$ Eri to Cepheid evolution models. These multipliers are a pair of Gaussians centered around log $T$ = 5.3 and 5.46, with amplitudes of 1.5 and 2.5, respectively, and mainly affect the `Z-bump' region that causes the $\\kappa$-effect driving of main-sequence B-type star pulsations. These opacity enhancements turned out to affect Cepheid evolution tracks negligibly, only slightly decreasing the extent of the blue loop tip. \n\n\\section{MESA Radial Stellar Pulsation (RSP) Models}\nWe next applied the MESA RSP capability \\citep[see also][]{2008AcA....58..193S}, including a time-dependent convection treatment, to model Cepheid envelope pulsations. The RSP code does not allow (as yet) the import of an envelope structure and composition profile directly from an evolution model; however, envelope composition is changed insignificantly during evolution before core helium exhaustion, especially without mass loss. The RSP model input requires mass (M), luminosity (L), effective temperature (T$_{\\rm eff}$), Y and Z. Using the nominal settings, a 150-zone envelope model is built from the stellar surface down to a temperature of 2 million K. The linear periods and growth rates for fundamental (F), 1st overtone (OT), and 2nd OT modes are calculated including the time-dependent convection treatment. \n \n\nWe started with inputs based on observed parameters for Galactic Cepheids from the literature, and then varied the input parameters, mainly T$_{\\rm eff}$, but also L or M if needed to identify a model calculated to have the observed pulsation period and positive linear growth rate in the desired pulsation mode. The model's pulsation is then initialized in this mode with radial velocity amplitude 0.1 km\/s. It may be necessary to run several thousands of pulsation periods, or tens of millions of timesteps, until the model converges to a limiting radial velocity amplitude. We applied this procedure to models of $\\delta$ Cep, with estimated mass 5-5.3 M$_{\\odot}$ \\citep{2015ApJ...804..144A}, as well as to models of Polaris and V1334 Cyg, with dynamical masses constrained using the orbit of a binary companion of 3.45 $\\pm$ 0.75 M$_{\\odot}$ and 4.228 $\\pm$ 0.133 M$_{\\odot}$, respectively \\citep{2018ApJ...867..121G}. These models use Y=0.27, and Z=0.015 (assumed near solar) for V1334 Cyg and $\\delta$ Cep, and Z=0.016, slightly above solar, for Polaris \\citep{2001A&A...373..159C}. We also calculated models for each Cepheid using this same procedure including the temperature-dependent opacity multiplier discussed in Section \\ref{evolution}.\n\nTable \\ref{table} summarizes the RSP model results. Note that the Polaris model mass needed to be increased to 5.93 M$_{\\odot}$ to find a model with a pulsation period in agreement with the observed value. Figure \\ref{delta_Cep} shows results for the $\\delta$ Cep model without the opacity multiplier. For the default choices of viscous dissipation in the nonlinear hydrodynamics simulation, the model reaches a limiting radial velocity amplitude 22 km\/s, not far from the observed amplitude of 25 km\/s \\citep{2005ApJS..156..227B}. The observed radial velocity amplitudes of Polaris and V1334 Cyg are a few km\/s \\citep{2008AJ....135.2240L} and about 5 km\/s \\citep{2018ApJ...867..121G}, respectively. In general, the radial velocity amplitudes of the models without the opacity multiplier are in good agreement with the observed values, while the opacity multiplier causes a significant decrease in limiting amplitude. It is interesting that the V1334 Cyg model without the opacity multiplier has positive growth rates for both the F and 1st OT modes; even though the observed 1st OT mode has the highest linear growth rate and is initialized in this mode, the model switches modes after many periods to pulsate in the fundamental mode. Figure \\ref{HRD_3models} shows the observed and model L and T$_{\\rm eff}$ on the H-R diagram. Note that the V1334 Cyg and Polaris models and observations lie well above the MESA 5 M$_{\\odot}$ evolution track (without rotation or mass loss), illustrating the mass discrepancy for these two stars that have dynamical mass determinations.\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=8.5cm]{RV.eps}\n\\includegraphics[width=8.5cm]{KEGrowth.eps}\n\\includegraphics[width=8.5cm]{Period.eps}\n\\end{center}\n\\caption{Photospheric radial velocity in km\/sec (top), kinetic energy growth rate per period (center), and pulsation period in days (bottom) vs. stellar age in years after the start of the hydrodynamic simulation for $\\delta$ Cep RSP model without opacity enhancement.}\\label{delta_Cep}\n\\end{figure}\n\n\\clearpage\n\n\\articlefigure[width=.58\\textwidth]{HRD_3models.eps}{HRD_3models}{H-R diagram showing locations of observed Cepheids, and of MESA RSP models that match the observed pulsation periods. The green line shows a 5 M$_{\\odot}$ MESA evolution track with (Y, Z) = (0.27, 0.015), without mass loss or rotation.}\n\n\\begin{table}[!ht]\n\\caption{Properties of MESA RSP models with standard opacities and with opacity enhancement (+$\\kappa$)}\n\\label{table}\n\\begin{center}\n{\\small\n\\begin{tabular}{lccccccc} \n\\tableline\n\\noalign{\\smallskip}\n& $\\delta$ Cep & & Polaris & & V1334 Cyg &\\\\\n\\noalign{\\smallskip}\n\\tableline\n\\noalign{\\smallskip}\nProperty & Normal & +$\\kappa$ & Normal & +$\\kappa$ & Normal & +$\\kappa$ \\\\\n\\tableline\nMass (M$_{\\odot}$) & 5.3 & 5.3 & 5.93& 5.93 & 4.288 & 4.288 \\\\\nLuminosity (L$_{\\odot}$) & 2089 & 1800 & 2818 & 2818 & 1800 & 1800 \\\\\nT$_{\\rm eff}$ (K) & 5861 & 5761 & 6048 & 6064 & 6072 & 6092 \\\\\nP$_{\\rm o}$ (days) & 5.3676 & 5.3676 & 3.9716 & 3.9720 & 3.3322 & 3.3316 \\\\\nPulsation mode & F & F & 1st OT & 1st OT & 1st OT & 1st OT \\\\\n\\noalign{\\smallskip}\nLinear Growth \\\\\nRate per Period & 0.00819 & 0.00119 & 0.00105 & 0.00268 & 0.00488 (F) & 0.00532 \\\\\n & & & & & 0.00688 (OT) & \\\\\nLimiting \\\\\nAmplitude (km\/s) & 22 & 7.3 & 3.6 & 0.61 & 18 (F) & 11 (OT) \\\\\n\\tableline\\\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\n\\vspace*{-1.0cm}\n\n\\section{Summary and Conclusions}\nThe MESA code and new RSP capabilities are useful tools to model Cepheid evolution and pulsation. Increasing helium-burning reaction rates increases the luminosity and extent of Cepheid blue loops, but implausibly high increases are needed to resolve the mass discrepancy. Including opacity enhancements around 200,000-400,000 K in the envelope, as proposed for SPB\/$\\beta$ Cep variables, does not significantly affect the evolution tracks or the extent of blue loops. Radial velocity amplitudes of MESA RSP envelope models of $\\delta$ Cep, Polaris, and V1334 Cyg agree well with observed amplitudes. However, including an opacity enhancement as proposed for SPB\/$\\beta$ Cep stars results in lower radial velocity amplitudes, although these amplitudes could be increased by decreasing the viscous dissipation settings in the RSP models. The Polaris and V1334 Cyg RSP models with periods matching observations have locations in the H-R diagram above that of a 5 M$_{\\odot}$ evolution track, neglecting mass loss and rotation, inconsistent with their lower dynamical masses.\n\n\n\n\\acknowledgements This research was supported in part by the National Science Foundation under Grant No.~NSF ACI-1663688. We are grateful to KITP for the opportunity to participate in the 2019 MESA summer school. We thank Bill Paxton, Frank Timmes, Josiah Schwab, the 2019 MESA summer school students, and LANL undergraduate summer student Stephanie Flynn.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{The next steps in extragalatic astrophysics}\n\\label{sec:Intro}\n\\vskip+0.5truecm\n\\begin{multicols}{2}\nThe era of extragalactic astrophysics began in earnest around the\ntime of the Great Debate between Shapley and Curtis in\n1920 \\citep{Trimble:95}. The debate focused on the nature of a number\nof intriguing ``nebulae'', at a time when the consensus rested on a Universe\nin which the Milky Way was its main constituent,\na scenario that harks back to the model of our Galaxy laid out by\n\\citet{Herschel:1785}. The discoveries\nduring the 1930s, pioneered by Slipher, Hubble, and\nHumason resulted in the concept of island Universes,\nwhere each ``stellar system'', a galaxy, constitutes a\nfundamental building block tracing the largest scales in the\nCosmos. It has been nearly a century since this Great Debate, and our\nunderstanding of extragalactic astrophysics has come a long\nway.\n\\bigskip\n\nDevelopments in telescopes, instrumentation and analysis techniques\nhave allowed us to decipher the intricacies of galaxy formation. At\npresent, the established paradigm rests on a dark matter dominated\ncosmic web within which a comparatively small mass fraction consists\nof ordinary matter (``baryons''), mostly in the form of stars, gas and\ndust. The first stage of galaxy formation is driven by the \n(linear) growth of the dark matter density fluctuations imprinted\nduring the earliest phases of cosmic evolution. \nStable dark matter structures, termed halos, collapse and virialise,\nconstituting the basic units in this scenario. At the same time, \ngas accumulates in the central regions of these\nhalos, leading to cooling and star formation. The general aspects of\nthis complex process can be explained within the current framework\n\\citep[see, e.g.][]{SM:12}, resulting in an overall very successful\ntheory that matches the observations. \nHowever, many of the key processes are\nonly roughly understood, most notably the ``baryon physics'' that\ntransforms the smooth distribution of gas at early times into the\ngalaxies we see today. This complex problem requires large, targeted \ndata sets probing the most important phases of galaxy\nformation and evolution. This proposal addresses the next steps that\nthe astrophysics community will follow in the near future to\nunderstand structure formation. High-quality spectroscopic\nobservations of galaxies are required to probe these important\nphases. {\\sl We motivate below the need for a large, space-based, ultra-deep survey of\ngalaxy spectra in the near-infrared, and present the technological\nchallenges that must be addressed.}\n\nThe extremely weak fluxes of the targets, combined with the need to work\nat near infrared wavelengths imply such a task must be pursued from space,\nfree of the noise from atmospheric emission and absorption. Moreover, the\nneed to simultaneously observe many sources spectroscopically, from an\nunmanned, unserviceable mission, defines arguably one of the toughest challenges\nin space science. Such a task is optimally suited for the 2035-2050 period\nenvisioned by ESA within the Voyage 2050 call. We emphasize that this science case\ncomplements the succesful track record of ESA in this field,\nwith missions such as Herschel (tracing the evolution of dust in galaxies),\nGaia (tracing the gravitational potential of our Galaxy), as well as \nthe cosmology-orientated missions, {\\sl Planck} and {\\sl Euclid}. We will show below how\nthe fundamental science case of galaxy formation and evolution requires\na future space-based observatory, beyond the capabilities of the\nupcoming {\\sl JWST} or large 30-40m ground-based telescopes such as ESO's {\\sl ELT}.\n\n\\end{multicols}\n\n\\section{The evolution of galaxies at the peak of activity}\n\\label{sec:D4000}\n\\begin{multicols}{2}\n\n\n\\subsection{Star formation across cosmic time}\n\nThe observational evidence reveals that the overall level of star\nformation in nearby galaxies is comparatively low with respect to\nearlier epochs. Fig.~\\ref{fig:CSFH}, derived from various\nobservational traces of star formation, shows a characteristic peak in\nthe cosmic star formation activity in galaxies between redshifts 1 and\n3, roughly corresponding to a cosmic time between 2 and 6\\,Gyr after\nthe Big Bang (or a lookback time between 8 and 12\\,Gyr ago). Such a\ntrend can be expected as the gas from the initial stages is gradually\nlocked into stars and, subsequently, remnants. This trend is highly\npacked with complex information\n\n\\begingroup\n\\centering\n\\includegraphics[width=85mm]{CH_f1.pdf}\n\\captionof{figure}{Cosmic star formation history: This diagram shows\n the redshift evolution of the\n star formation rate density. Note that detailed\n spectroscopic {\\sl optical} galaxy surveys exist only out to\n z$\\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$}$1, whereas the epochs of maximum star formation \n (z$\\sim$1--3), and the first stages of formation (z$\\lower.5ex\\hbox{$\\; \\buildrel > \\over \\sim \\;$}$7)\n are poorly understood (from \\citealt{CSFH_HB06}; see also \\citealt{CSFH_MD14})}\n\\label{fig:CSFH}\n\\vskip+0.3truecm\n\\endgroup\n\n\\noindent\nregarding the efficiency of star\nformation, the mechanisms of gas infall and outflows, the ejection\nof gas from evolved phases of stellar evolution and the bottom-up\nhierarchy of structure formation.\n\nIn addition, the z$\\sim$1--3\nredshift window corresponds to the peak of AGN activity \\citep{Richards:06},\nand merger rate \\citep{Ryan:08}. \nMoreover, it is the epoch\nwhen the dark matter halos hosting massive galaxies allow for\ncold accretion via cosmic streams (see \\S\\S\\ref{SS:ColdAcc}).\nDecoding this complex puzzle\nrequires a detailed study of the different phases of evolution. At\npresent we only have complete galaxy samples amounting to $\\sim$1 million\nhigh quality spectra at low redshift (z$\\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$} 0.2$,\ne.g. SDSS, \\citealt{SDSS}), along with samples of spectra at\nintermediate redshift (z$\\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$}$1.5), e.g. VIPERS \\citep{vipers},\nVVDS \\citep{vvds}, zCOSMOS \\citep{zcosmos}, GAMA \\citep{gama}, BOSS\n\\citep{boss} or LEGA-C \\citep{LegaC}. Future spectroscopic surveys\nwill also probe similar redshift ranges within the optical spectral\nwindow -- e.g., WAVES \\citep{WAVES}; WEAVE \\citep{WEAVE}; DESI \\citep{DESI},\nMSE \\citep{MSE}. In the NIR, ESO's VLT\/MOONS \\citep{MOONS} will constitute\nthe state-of-the art ground based survey, but the expected S\/N will not\nbe high enough for studies comparable to those perfomed on SDSS spectra\nat z$\\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$}$0.2. We note that many of the spectroscopic surveys (past and future) \nare mostly designed as a ``redshift machine'' (i.e. optimised for cosmology, using\ngalaxies simply as ``test particles''), and the S\/N of the data in the\ncontinuum is too low for any of the science presented here to be\nsuccessfully delivered. {\\sl None of the current and future observing\n facilities, both ground- and space-based, will be capable of\n creating the equivalent of the spectroscopic SDSS catalogue at these\n redshifts.}\n\n\n\\subsection{Bimodality and galaxy assembly}\n\nOn a stellar mass vs colour (or age) diagram, galaxies populate two well\ndefined regions: the red sequence and the blue cloud \\citep[see,\n e.g.,][]{Kauff:03,Taylor:15}. Galaxies on the red sequence are mostly massive,\npassively-evolving systems with little or no ongoing star\nformation. Although the red sequence extends over a wide range in\nstellar mass, the most massive galaxies tend to be on the red\nsequence, with a preferential early-type morphology. In contrast, blue\ncloud galaxies have substantial ongoing star formation, and extend\ntowards the low-mass end. A third component is also defined, the\ngreen valley \\citep{M07}, between these two. However, the identification of this\nregion as a transition stage between the blue cloud and the red\nsequence is far from trivial \\citep{Schaw:14,Ang:19}. There are many studies\ntracing the redshift evolution of galaxies in these regions\n\\citep[e.g.][]{Bell:04,Ilbert:10,Muzz:13}, revealing a downsizing\ntrend, so that the bulk of star formation (i.e. the ``weight'' of the\nblue cloud) shifts from the most massive galaxies at high redshift, to\nlower mass systems in the present epoch. This simple diagram allows\nus to present a simplified version of star formation in galaxies,\nincluding the usual bottom-up hierarchy that begins with small star\nforming systems, leading to more massive galaxies through in situ star\nformation and mergers, both with (``wet'') and without (``dry'')\nadditional star formation.\n\nModels such as those proposed by\n\\citet[][see Fig.~\\ref{fig:BiMod}]{Faber:07} allow us to express\ngraphically the complex processes involved. However, the problem with\nthese analyses is how to properly characterize the formation stage of\na galaxy by a simple descriptor such as galaxy colour. More detailed\nanalyses have been presented of the colour-mass diagram, combining\nphotometry and spectroscopy in relatively nearby samples\n\\citep[e.g.][]{Schaw:07} showing interesting processes that relate the\nvarious sources of feedback (see \\S\\S\\ref{SS:feedback}\nbelow).\n\n\\begingroup \\centering\n\\includegraphics[width=60mm]{CH_f2.pdf}\n\\captionof{figure}{Schematics of galaxy evolution from the blue cloud\n to the red sequence. Three different scenarios are considered,\n as labelled, with the black arrows representing evolution \n through wet mergers and quenching, and white arrows symbolising\n stellar mass growth through dry mergers \\citep[adapted from][]{Faber:07}}\n\\label{fig:BiMod}\n\\vskip+0.3truecm\n\\endgroup\n\nHowever, such studies are complicated by the fact that the\nunderlying stellar populations span a wide range of ages and chemical\ncomposition, and the star formation processes do not involve a\nsubstantial fraction of the baryonic mass of the galaxy. Therefore,\nit is necessary to extend these studies, including high quality\nspectroscopic data, to explore the evolution on the colour-stellar mass\ndiagram with galaxies targeted during the peak of galaxy formation.\nAt these redshifts (z$\\sim$1-3), we will be dealing with the most\nimportant stages of formation.\n\n\\subsection{The role of star formation and AGN}\n\\label{SS:feedback}\n\nThe bimodality plot (Fig.~\\ref{fig:BiMod}) illustrates the key \nprocesses underlying galaxy evolution. Most importantly, the presence\nof a large population of passive galaxies on the red sequence, without\nan equivalent counterpart of massive galaxies on the blue cloud requires\nphysical mechanisms by which star formation is quenched. As the fuel\nfor star formation is cold gas, quenching of any type must resort to\nreducing this component, either by heating, photoionisation or mechanical\nremoval of the cold phase.\n\nVarious theoretical models have been explored over the\npast decades, most notably based on the expulsion of\ngas from supernovae-driven winds \\citep[stellar feedback, e.g.][]{DekelSilk:86}\nor from a central supermassive black\nhole \\citep[AGN feedback, e.g.][]{SilkRees:98}.\n\n\\begingroup\n\\centering\n\\includegraphics[width=80mm]{CH_f3.pdf}\n\\captionof{figure}{Correlation between the stellar-to-halo mass ratio and halo\n mass. Even at the peak of the curve ($\\sim$3\\%) the stellar mass\n is significantly lower that the cosmic baryon to dark matter ratio,\n revealing an inefficient process of star formation. \n Furthermore, the decrease of this fraction towards both the high-\n and low-mass end reveals the complexity of feedback mechanisms\n \\citep[from][]{Behroozi:10}.}\n\\label{fig:AM}\n\\vskip+0.3truecm\n\\endgroup\n\nA comparison of the observed stellar mass function of galaxies and\nN-body simulations of dark matter halos (see Fig.~\\ref{fig:AM})\nsuggests at least two distinct mechanisms to expel gas from galaxies,\none dominant at the low-mass end, and the other one controlling the\nhigh-mass end. Since the efficiency of stellar winds is expected to\nincrease in weaker gravitational potentials, one would assume stellar\nfeedback is responsible for the low-mass trend. Similarly, the increasing\nefficiency of AGN feedback with black hole mass would produce \n the trend at the high-mass end. Furthermore, the strong\ncorrelation between bulge mass (or velocity dispersion) and the mass\nof the central supermassive black hole\n\\citep[Fig.~\\ref{fig:MBHvSig}, see, e.g.][]{KorHo:13,Saglia:16} gives\nfurther support to the role of AGN activity in shaping galaxy\nformation. However, this picture is too simplistic,\n\nrequiring a\nbetter understanding of the physics.\nDetailed analyses of winds driven by nearby starbursting galaxies\npresent a complex scenario that is not properly described by the\nlatest numerical codes of galaxy formation \\citep{HB:15}. The\nprevalence of outflows increases towards the younger phases of galaxy\nformation. Therefore, detailed studies over complete samples during\nthe critical phases of galaxy evolution are needed to understand \nfeedback in detail.\n\n\\begingroup\n\\centering\n\\includegraphics[width=85mm]{CH_f4.pdf}\n\\captionof{figure}{Correlation between black hole mass and velocity\n dispersion in local galaxies, from {\\sl direct} measurements of the\n SMBH mass \\citep[from][]{HB:14}.}\n\\label{fig:MBHvSig}\n\\vskip+0.2truecm\n\\endgroup\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=120mm]{CH_f5.pdf}\n\\captionof{figure}{Redshift evolution of the merger rate, ${\\cal R}(z)$,\n as measured by \\citet{Mundy:17}. The\n observational constraints, shown as points with different\n symbols, are in stark contrast with respect to state-of-the-art\n predictions from the Illustris numerical simulations of galaxy\n formation \\citep[dashed lines, labelled at different stellar masses;][]{Illustris}.}\n\\label{fig:Merg}\n\\vskip-0.3truecm\n\\end{figure*}\n\n\\subsection{Galaxy growth through mergers}\n\nOne of the main methods by which galaxies form is through the merger\nprocess, whereby separate galaxies combine together to form a new\nsystem. Merging is a significant channel of galaxy formation, and\nneeds to be measured with high precision if we are to understand how\ngalaxy formation proceeds. Closer to home, the complex structure of the\nstellar populations found around the Milky Way, its vicinity and the\nnearby Andromeda galaxy reflects the contribution of mergers to galaxy\ngrowth \\citep[e.g.][]{Ferguson:02,Ivezic:12}. Whilst mergers are\narguably not the way in which galaxies obtain the majority of their mass, this\nprocess is still likely the main one for triggering AGN and black hole\nformation and accounts for 25-50\\% of the formation of massive galaxies\nsince z=3 \\citep{Owns:14}. Thus, a detailed quantitative assessment\nof galaxy merger rates is a critical step that has not yet been fully\ncarried out, due to the lack of complete spectroscopic samples.\nFurthermore, there are inconsistencies with the results\nobtained so far and a disagreement with theory, showing that more\nwork, and better data are needed in this area.\n\nFirstly, the exact role of mergers in galaxy formation is not clear,\nwith conflicting results, particularly at higher redshifts (z$>$1). The\nmerger fraction at z$\\lower.5ex\\hbox{$\\; \\buildrel > \\over \\sim \\;$}$1 is likely high, with a merger rate of\n$\\sim$0.5--1 mergers Gyr$^{-1}$ \\citep[e.g.][]{Bluck:12,Tasca:15}.\nMany merger rates at high redshift z$>$1 are measured with galaxy\nstructures, or based on samples of galaxies in kinematic or photometric redshift\npairs. However, our best estimates of the merger rate differ from\ntheory by up to an order of magnitude (see Fig.~\\ref{fig:Merg}\ncontrasting observational results with the latest, state-of-the-art\nsimulations by the Illustris collaboration). Moreover, we do not\nhave robust estimates about the role of minor mergers\nin galaxy formation -- recovering\nthese will require very deep spectroscopic observations.\n\n\nThe best way to measure the merger rate at high redshift is through\nspectroscopic pairs which requires both position and accurate radial\nvelocity information \\citep[e.g.][]{CLS:12}. However, the most up to\ndate studies have only used 12 pairs at z$>$2 to measure this\nimportant quantity \\citep{Tasca:14} with a merger fraction with\nrather large errors ($19.4^{+9}_{-6}$\\%) due to small number\nstatistics.\n\nA near infrared spectroscopic survey of distant galaxies at z$>$1 will\ngive us the information we need to address this issue in detail. A\nsurvey with a high completeness level over the z=1--3 range will give\nus a surface density over 10 times higher than previous surveys at\n1$<$z$<$3 such as DEEP2, VVDS, and UDSz. To address this type of\nscience, the survey strategy needs to incorporate the option of\nincluding such targets in the mask layout (if the method is to proceed\nwith a reconfigurable focal plane, see~\\S\\S\\ref{SS:ReconfFP}). Given\nthe density of targets at the redshifts of interest, the merger\nfraction will be measured to an accuracy an order of magnitude better\nthan what is currently known at these redshifts.\n\nThis is necessary to\nultimately pin down the amount of mass assembled through merging, as\nwell as to determine the role of merging on the triggering and\nquenching of star formation, and on central AGN activity. For\nreference, in the most massive systems with M$_* > 10^{10}$M$_\\odot$\nit will be possible to measure merger fraction ratios of up to 1:30\ndown to a stellar mass limit of M$_* =10^{9.5}$M$_\\odot$, such that we\ncan study, for the first time, the role of minor mergers in these\nprocesses.\n\n\n\\subsection{The role of cold accretion}\n\\label{SS:ColdAcc}\n\nThe evolution of the gaseous component -- and its subsequent\ntransformation into stars -- is arguably one of the most complicated\nproblems in extragalactic astrophysics.\nHydrodynamical processes driving the gas flows, and feedback from star\nformation, AGN activity or dynamical evolution of the baryon-dominated\ncentral regions of halos lead to a significant mismatch between\nthe mass assembly history of dark matter halos, and the star formation\nhistories of galaxies embedded in these halos. In fact,\nFig.~\\ref{fig:AM} illustrates this mismatch.\n\nOne key observable of the difference between dark matter growth and\ngalaxy growth is the presence of massive galaxies at early \ntimes \\citep[e.g.][]{CI:04,McCarthy:04,Fontana:06,PPG:08}. A naive\nmapping of dark matter growth into stellar mass growth leads \nto late star formation in massive galaxies, as found in\nthe first, pioneering computer simulations of galaxy formation\n\\citep[e.g.,][]{Kauff:96}. The presence of massive galaxies (stellar\nmass $\\lower.5ex\\hbox{$\\; \\buildrel > \\over \\sim \\;$} 10^{11}$M$_\\odot$) with quiescent populations at\nredshifts z$\\sim$2-3 \\citep[e.g.][]{FW4871} requires a mechanism by\nwhich the commonly adopted process of star formation through\nshock-heating of gas after the virialization of the halo, followed by\ncooling \\citep{RO:77} cannot be the main growth channel in these\nsystems.\n\n\\begingroup\n\\centering\n\\includegraphics[width=80mm]{CH_f6.pdf}\n\\captionof{figure}{Mechanism proposed by \\citet{Barro:13}\nto explain the size evolution of massive galaxies.\nThe grey contour shows the galaxy distribution at low redshift. Two main\ngrowth channels are proposed, involving a mixture of processes such as\nmerging, star formation quenching or secular processes. Large,\nhigh quality spectroscopic data at these redshifts will allow us to\ntest in detail these proposals.}\n\\label{fig:SizeEv}\n\\vskip+0.2truecm\n\\endgroup\n\n\nWe find ourselves in a similar quandary with strong AGN\nactivity at very high redshift, z$\\lower.5ex\\hbox{$\\; \\buildrel > \\over \\sim \\;$}$6 -- reflecting the presence\nof very massive black holes within the first billion years of cosmic\ntime \\citep{Fan:06}. In addition to the traditional hot-mode growth,\ncold gas can also flow towards the centres of halos, following the\nfilamentary structure of the dark matter distribution, efficiently\nfeeding the central sites of star formation at early times\n\\citep{Dekel:09}. Moreover, this process is found to operate in the\nmost massive systems at early times \\citep{DB:06}. Numerical\nsimulations suggest that clump migration and angular momentum transfer\nprovides an additional mechanism leading to the creation of massive\nstellar cores at early times \\citep{Ceve:10}.\n\nHowever, observational constraints of the role of cold accretion are\nfew, and no conclusive evidence has been found to date. A large\nspectroscopic galaxy survey probing the peak of evolution would allow\nus to study the hot- and cold-mode growth channels of star formation\nand black hole growth, and the connection with redshift and\nenvironment. A detailed analysis of the shape of targeted spectral\nlines will allow us to detect and quantify gas inflows, but a large\nvolume of data is necessary given the small covering factor\nof accretion flows \\citep{FG:11}. The high S\/N\nof this survey will make studies of individual galaxies (not stacked\nspectra) available. As of today, state-of-the-art samples comprise\n$\\sim$100 spectra with just enough S\/N to study bright emission lines\n(see, e.g. \\citealt{Genzel:14} with VLT\/KMOS; or \\citealt{Kacprzak:16}\nwith Keck\/MOSFIRE). These studies give promising results about the\npresence of this important process of galaxy growth. Note studies\nin the Ly-$\\alpha$ region (i.e. concerning the {\\sl cosmic dawn} survey,\n\\S\\ref{sec:Lyman}) can also be used to obtain constraints\non gas inflows \\citep{Yajima:15}.\n\n\n\\subsection{Size evolution}\n\nAn additional conundrum raised by the study of massive galaxies at\nhigh redshift is the issue of size evolution. The comoving number\ndensity of massive ($\\lower.5ex\\hbox{$\\; \\buildrel > \\over \\sim \\;$} 10^{11}$M$_\\odot$) galaxies has been found\nnot to decrease very strongly with redshift (z$\\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$} 2$), with respect\nto the predictions from simple models of galaxy formation that mostly\nlink galaxies to the evolution of the dark matter halos\n\\citep[e.g.][]{CC:07,IF:09}.\n\nThis would reflect an early formation of these type of galaxies,\nwhereby the bulk of the stellar mass is in place by redshift\nz$\\sim$2--3. However, the sizes of these galaxies at z$\\lower.5ex\\hbox{$\\; \\buildrel > \\over \\sim \\;$}$1--2 are\nsignificantly smaller than their low-redshift counterparts\n\\citep[e.g.][]{Daddi:05,Trujillo:06}. A large volume of publications\nhas been devoted to propose mechanisms that could explain this puzzle,\nincluding gas outflows as a mechanism to alter the gravitational\npotential, ``puffing-up'' the dense central region.\n\n\nHowever, the (old) stellar populations typically found in massive\ngalaxies do not allow for significant quantities of recent star\nformation, or cold gas flows to explain this size evolution\n\\citep{I3}, suggesting instead a growth process through gas-free (dry)\nmerging. This merging can proceed dramatically -- through a small\nnumber of major mergers \\citep{KS:09}, where the merging progenitors\nhave similar mass -- or through a more extended and smooth process of\nminor merging \\citep{Naab:09}. In addition, one should consider\nwhether these evolved compact cores end up as massive (and extended)\nearly-type galaxies in high density regions \\citep{Pogg:13}, or as\nmassive bulges of disk galaxies \\citep{IGDR:16}.\n\nFig.~\\ref{fig:SizeEv} shows a diagram of how this may work, from an analysis of\nmassive galaxies in CANDELS \\citep{Barro:13}, with an interesting\nevolution from massive compact systems with a strong star\nformation rate, towards the quiescent galaxies we see today, involving\nboth secular processes, galaxy mergers and star formation\nquenching. Establishing such connections requires a large volume\nof galaxy spectra at the peak of galaxy formation activity.\nAll these studies are based on relatively small samples\n($\\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$} 10^3$) with mostly high-quality photometry (from HST) but comparatively poor\nspectroscopic data. {\\sl Accurate characterization of the stellar\npopulation content of these galaxies will enable us to robustly\nconstrain the processes by which galaxies grow.}\n\n\n\\subsection{Reaching out: the role of environment}\nThe environment where galaxies reside plays a significant\nrole in shaping their observed properties and thus their evolution. It\nessentially deprives them of their hot and cold gas reservoirs, thus\nquenching their star formation activity, and also can literally\ndisrupt them by removing their stars \\citep{AP:15}. The observed\nproperties of galaxies in the local Universe have provided us with a\nwealth of evidence towards environmental processes, whose time scales\nand amplitudes are unfortunately known only at a qualitative level. A\nrobust quantitative estimate of the dependence of such parameters on\nenvironment and redshift largely remains an open problem.\n\nThe Sloan Digital Sky Survey (SDSS) has been the very first survey to\nperform an unprecedented and statistically significant census of the\nphotometric and spectroscopic properties of z$\\simeq$0 galaxies at optical\nwavelengths. It has permitted us to detail star formation activity in\ngalaxies across several orders of magnitude, with respect to galaxy\nstellar mass, environment and infall time at z$\\simeq$0. We are now aware that the\nnumber of quenched galaxies -- not forming new stars any longer --\nrises with their stellar mass at a fixed kind of environment, and with\nenvironment magnitude (from small galaxy groups to large clusters) at\nfixed stellar mass \\citep{Wein:06,VdB:08,AP:09,Wetz:12}.\n\n\\begingroup\n\\vskip+0.2truecm\n\\centering\n\\includegraphics[width=75mm]{CH_f7.pdf}\n\\captionof{figure}{The stellar age of galaxies less massive\n than $10^{10}$M$_\\odot h^{-2}$ is shown as a function of infall\n time. Galaxies are colour-coded regarding halo mass: red circles,\n orange squares and yellow triangles identify galaxies in clusters,\n rich groups and low-mass groups, respectively. The grey stripe\n indicates the stellar age of equally-massive galaxies in the\n field \\citep[from][]{AP:19}.}\n\\label{fig:Env}\n\\vskip+0.2truecm\n\\endgroup\n\nWe have also learnt from SDSS that the age of the bulk of stars in a\ngalaxy grows progressively older i) as their infall time increases\n(i.e. galaxies accreted onto their present-day host environment early\non are now older than those accreted more recently); 2) as their\nenvironment, at fixed infall time, becomes more massive, indicating\nthat the environment mass enhances the efficiency of those physical\nprocesses able to quench star formation in galaxies\n\\citep{AP:10, AP:19, Smith:19}. In addition, recently accreted cluster galaxies\nappear to be older than equally-massive field galaxies, an\nobservational result that has been attributed to group-preprocessing:\nthe star-formation quenching of these recent infallers started already\nwhile they were still living in smaller groups, that later merged with\nclusters \\citep[see Fig.~\\ref{fig:Env},][]{AP:19}. Such group-preprocessing has extensively been\nadvocated by semi-analytic models of galaxy formation and evolution in\norder to explain the large number of quenched galaxies observed in\nclusters \\citep{DeLuc:12,Wetz:13}\n\nThe observational evidence described above highlights the importance\nof knowing the accretion epoch of a galaxy if we want to understand\nthe role of environment. Unfortunately, we can not use observations of\nz$\\simeq$0 galaxies to accurately derive their infall epochs when they\nbecame exposed to environmental effects for the first time. To\ndetermine such an important moment in the evolution of galaxies we\nneed to quantify and study environment at different redshifts; this is\nwhat a deep-wide NIR spectroscopic galaxy survey will enable us to do,\nby tracing the assembly history of environments with cosmic time,\nproviding us with a direct measurement of the redshift of infall of\ngalaxies as a function of their stellar mass. Moreover, the lensing\nand X-ray information from {\\sl Euclid} and {\\sl eRosita},\nrespectively, combined with the accurate spectroscopic information\nproduced by {\\tt Chronos} will probe the dependence of the star\nformation histories on the dark matter halos. While the data from {\\sl\n Euclid} and {\\sl eRosita} will mainly target the assembly of massive\nenvironments, thus introducing a significant bias towards star-forming\ngalaxies, {\\tt Chronos} will broaden the study to smaller environments\nand consequently will avoid the selection bias of the {\\sl Euclid}\nsample.\n\nWhen and in which environments did the quenching of the star formation\nactivity of galaxies start? How fast did it proceed? The quantitative\nand direct replies to these inquiries are provided by our measurements\nof star formation rates, star formation histories and chemical\nenrichment of galaxies of different stellar mass, in different\nenvironments at different epochs, from z$\\sim$1--3 to z=0. Only these\nobservables allow us to directly estimate the typical time scales of\nstar formation in galaxies, and to achieve a model-independent value\nof the time scales over which galaxy groups and clusters switched\ngalaxy star formation off, and produced the observed present-day\ngalaxy populations.\n\nWith increasing redshift these measurements shift to infrared\nwavelengths and become challenging even for modern ground-based\ntelescopes. Ground-based measurements allow for only a partial\ncharacterization of the properties of galaxies at z$>$0.5, for which we can\nmostly measure emission lines (thus star formation rates) since their\nabsorption lines (used as age and metallicity indicators) become less\nand less accessible. The data gathered so far on galaxies at $0.3<$z$<0.8$\nindicate that the fraction of quenched galaxies is larger in\ngalaxy groups than in the field, but definitively lower than the\nfraction of quenched galaxies in groups at z$\\simeq$0 \\citep{Wil:05,McGee:11}.\nAt intermediate redshifts, the fraction of star forming galaxies\ndiminishes from 70-100\\% in the field to 20-10\\% in the more massive\ngalaxy clusters \\citep{Pogg:06}. However, the star formation rates of\ngroup galaxies do not significantly differ from those in the field;\nonly star forming galaxies in clusters show star formation rates a\nfactor of 2 lower than in the field at fixed stellar mass\n\\citep{Pogg:06,Vul:10,McGee:11}.\n\nAt the highest redshifts probed for environment,\n$0.8<$z$<1$, the more massive galaxy groups and clusters are mostly populated\nby quenched galaxies and both exhibit a 30\\% fraction of\npost-starburst galaxies \\citep[i.e. with a recently truncated star\n formation activity,][]{Bal:11}. In particular, the fraction of\npost-starburst galaxies in clusters exceeds that in the field by a\nfactor of 3. Cluster and field galaxies still able to form new stars\nshare instead similar star formation rates. On the basis of these\nresults, \\citet{Muzz:12} have argued that, at z$\\sim$1, either\nthe quenching of star formation due to the secular evolution of\ngalaxies is faster and more efficient than the quenching induced by\ngalaxy environment, or both mechanisms occur together with the same\ntime scale. Which mechanism prevails and over which time scale?\nAt present, we do not know.\n\nTo further progress on this issue, we require a facility such as\n{\\tt Chronos} to observe a complete stellar-mass limited sample of\nenvironments at z$\\geq$1--3, and to derive the star formation histories\nof their galaxies with an unprecedented accuracy. {\\tt Chronos}\nobservations will thus deliver the fading time scales of star\nformation of galaxies of different stellar mass residing in groups and\nclusters. This is not simply an incremental step in our knowledge of\nenvironment-driven galaxy evolution. This is the {\\sl still missing,\nfundamental quantitative change} from the simple head-count of\nquenched or star-forming galaxies to the measurement of physical\nproperties of galaxies in environments at {\\sl cosmic noon}.\n\\end{multicols}\n\n\n\\section{First galaxies and the epoch of reionization}\n\\label{sec:Lyman}\n\\begin{multicols}{2}\n\n\n\\subsection{Leaving the dark ages}\n\nCosmic reionization is a landmark event in the history of the\nUniverse. It marks the end of the ``Dark Ages'', when the first stars\nand galaxies formed, and when the intergalactic gas was heated to tens\nof thousands of Kelvin from much colder temperatures. This\nglobal transition, during the first billion years of cosmic history,\nhad far-reaching effects on the formation of early cosmological\nstructures and left deep impressions on subsequent galaxy and star\nformation, some of which persist to the present day.\n\nThe study of this epoch is thus a key frontier in completing our\nunderstanding of cosmic history, and is currently at the forefront of\nastrophysical research \\citep[e.g.][]{Robertson:15}. Nevertheless,\ndespite the considerable recent progress in both observations and\ntheory (e.g. see recent reviews by \\citealt{Dunlop:13} and \\citealt{Loeb:13})\nall that is really established about this crucial era is that Hydrogen\nreionization was completed by redshift z$\\sim$6 (as evidenced by\nhigh-redshift quasar spectra; \\citealt{Fan:06}) and probably commenced\naround z$\\sim$12 (as suggested by the {\\sl Planck} \npolarisation measurements, which favour a `mean' redshift of\nz$_{\\rm re} = 8.8^{+1.7}_{-1.4}$\\,; \\citealt{Planck:15}). However, within\nthese bounds the reionization history is essentially unknown. New\ndata are required to construct a consistent picture of reionization\nand early galaxy formation\/growth (see Fig.~\\ref{fig:reioniz}).\n\nUnderstanding reionization is therefore a key science goal for a\nnumber of current and near-future large observational projects. In\nparticular, it is a key science driver for the new generation of major\nlow-frequency radio projects (e.g. LOFAR, MWA and SKA) which aim to\nmap out the cosmic evolution of the neutral atomic Hydrogen via 21-cm\nemission and absorption. However, such radio surveys cannot tell us\nabout the sources of the ionizing flux, and in any case radio\nobservations at these high redshifts are overwhelmingly difficult, due\nto the faintness of the emission and the very strong foregrounds. It\nis thus essential that radio surveys of the neutral gas are\ncomplemented by near-infrared surveys which can both map out the\ngrowth of ionized regions, and provide a complete census of the\nionizing sources.\n\n\\begingroup\n\\vskip+0.2truecm\n\\centering\n\\includegraphics[width=75mm]{CH_f8.jpg}\n\\captionof{figure}{Measures of the neutrality\n $1-{{Q}_{{{{\\rm H}}_{{\\rm II}}}}}$ of the intergalactic medium as a\n function of redshift. Shown are the observational constraints, along\n with model predictions of the evolving IGM neutral fraction (in\n red). The bottom panel shows the IGM neutral fraction near the end\n of the reionization epoch, where the model fails to capture the\n complexity of the reionization process.\n \\citep[from][]{Robertson:15}. }\n\\label{fig:reioniz}\n\\vskip+0.2truecm\n\\endgroup\n\nA genuine multi-wavelength approach is required,\nand cross-correlations between different types of observations will be\nnecessary both to ascertain that the detected signals are genuine\nsignatures of reionization, and to obtain a more complete\nunderstanding of the reionization process.\nIt has thus become increasingly clear that a wide-area, sensitive,\nspectroscopic near-infrared survey of the z=6--12 Universe is required\nto obtain a proper understanding of the reionization process and early\ngalaxy and black-hole formation. Such a survey cannot be undertaken\nfrom the ground (due to Earth's atmosphere), nor with {\\sl JWST} (inadequate\nfield-of-view), nor {\\sl Euclid} or {\\sl WFIRST} (inadequate\nsensitivity with slitless spectra). Only a mission such as\n{\\tt Chronos} can undertake such a survey and simultaneously address the\nthree, key, interelated science goals which we summarize\nbelow. Moreover, detailed studies of z$>$6 galaxies in the Ly-$\\alpha$\nregion will complement the information provided at longer wavelengths\nby ALMA \\citep[e.g.][]{Capak:15}.\n\n\n\\subsection{The clustering of Ly-$\\alpha$ emitters as a probe of reionization}\n\nCosmological simulations of reionization predict that the\nhighly-clustered, high-redshift sources of Lyman-continuum photons\nwill lead to an inhomogeneous distribution of ionized regions. The\nreionization process is expected to proceed inside-out, starting from\nthe high-density peaks where the galaxies form. Thus, as demonstrated\nby the state-of-the-art simulations shown in Fig.~\\ref{fig:reioniz2},\nreionization is predicted to be highly patchy in nature. This\nprediction is already gaining observational support from the latest\nlarge-area surveys for Ly-$\\alpha$ emitters at z$\\sim$6.5, where it has been\nfound that, depending on luminosity, their number density varies by a\nfactor of 2--10 between different $\\frac{1}{4}$\\,deg$^2$ fields\n\\citep{Ouchi:10,Nakamura:11}. It is thus clear that surveys over many\nsquare degrees are required to gain a representative view of the\nUniverse at z$>$6. Crucially, with such a survey, the differential\nevolution and clustering of Lyman-break galaxies and Ly-$\\alpha$\nemitting galaxies can be properly measured for the first time,\noffering a key signature of the reionization process.\n\nHigh-redshift galaxies can be selected on the basis of\neither their redshifted Lyman break (the sudden drop in emission from\nan otherwise blue galaxy, due to inter-galactic absorption at\nwavelengths $\\lambda_{\\rm rest} < 1216$\\AA), or their\nredshifted Ly-$\\alpha$\nemission. The former class of objects are termed Lyman-Break Galaxies\n(LBGs) while the latter are termed Ly-$\\alpha$ Emitters (LAEs).\nIn principle, LAEs are simply the subset of LBGs with detectable\nLy-$\\alpha$ emission, but the current sensitivity limitations of\nbroad-band near-infrared imaging over large areas has meant that\nnarrow-band imaging has been successfully used to yield samples of\nlower-mass galaxies which are not usually identified as LBGs\n\\citep[e.g.][]{Ono:10}. Nevertheless, as demonstrated by\nspectroscopic follow-up of complete samples of bright LBGs\n\\citep[e.g.][]{Stark:10, Vanzella:11,Schenker:12}, the fraction of\nLBGs which are LAEs as a function of redshift, mass, and environment\nis a potentially very powerful diagnostic of both the nature of the\nfirst galaxies, and the physical process of reionization.\n\n\\begingroup\n\\vskip+0.2truecm\n\\centering\n\\includegraphics[width=75mm]{CH_f9.png}\n\\captionof{figure}{The geometry of the epoch of reionization, as\n illustrated by a slice through a (165\\,Mpc)$^3$ simulation volume at\n z=9. Shown are the density (green\/yellow), ionized fraction\n (red\/orange), and ionizing sources (dark dots) \\citep{Iliev:12}.\n The necessity of a deep, near-infrared spectroscopic survey\n covering many square degrees is clear.}\n\\label{fig:reioniz2}\n\\vskip+0.2truecm\n\\endgroup\n\n\nWith the unique combination of deep, wide-area near-infrared imaging,\nprovided by surveys such as {\\sl Euclid} and {\\sl WFIRST}, and deep, complete\nfollow-up near-infrared spectroscopy, made possible with {\\tt Chronos}, we\npropose to fully exploit the enormous potential of this\napproach.\n\nThe essential idea of using {\\tt Chronos} to constrain\nreionization is as follows: while the Ly-$\\alpha$ luminosity of LAEs is\naffected both by the intrinsic galaxy properties, and by the \\HI\ncontent (and hence reionization), the luminosity of LBGs (which is\nmeasured in the continuum) depends only on the intrinsic galaxy\nproperties. Thus, a deep, wide-area, complete survey for LBGs at\nz$\\sim$6--12 with accurate redshifts secured by {\\tt Chronos} will deliver a\ndefinitive measurement of the evolving luminosity function and\nclustering of the emerging young galaxy population, while the analysis\nof the follow-up spectroscopy will enable us to determine which LBGs\nreside in sufficiently large ionized bubbles for them to also be\nobserved as LAEs. In order to prevent strong damping wing absorption\nof Ly-$\\alpha$ photons, a galaxy must carve out a bubble of radius R$_I$ of\n500--1000 physical kpc at z$\\sim$8. According to the most recent\nreionization history predictions from cosmological simulations,\nconsistent with the various reionization constraints, the \\HI fraction\nat this redshift is around $\\chi \\sim 0.5$--$0.7$. R$_I$ for a typical\ngalaxy with a star-formation rate of $\\dot{\\rm M}_* = 1$\\,M$_\\odot$\\,yr$^{-1}$\nis expected to be considerably smaller (though it depends on poorly\nestablished values of the ionizing photon escape fraction; cf.\n\\citealt{RM:03}). Thus, such galaxies will be only marginally\ndetectable in the Ly-$\\alpha$ line if they are isolated. In practice, some of\nthese galaxies will be highly clustered and therefore will help each\nother in building a \\hbox{H$\\scriptstyle\\rm II\\ $} region which is large enough to clear the\nsurrounding \\HI and make it transparent to Ly-$\\alpha$ photons.\n\nThis argument emphasizes the importance of clustering studies of LAEs,\nfor which the proposed survey is optimally designed. A key aim is to\ncompute in great detail the two-point correlation function of LAEs and\nits redshift evolution. For the reasons outlined above, reionization\nis expected to increase the measured clustering of emitters and the\nangular features of the enhancement would be essentially impossible to\nattribute to anything other than reionization. \n\nIn fact, under some scenarios, the apparent clustering of LAEs can be\nwell in excess of the intrinsic clustering of halos in the concordance\ncosmology. Observing such enhanced clustering would confirm the\nprediction that the \\hbox{H$\\scriptstyle\\rm II\\ $} regions during reionization are large\n\\citep{McQuinn:07}. As required to meet our primary science goals,\nthe {\\tt Chronos} surveys will result in by far the largest and most\nrepresentative catalogues of LBGs and LAEs ever assembled at\nz$>$6. Detailed predictions for the number of LBGs as extrapolated\nfrom existing ground-based and HST imaging surveys are deferred to the\nnext subsection. However, here we note that the line sensitivity of\nthe 100\\,deg$^2$ spectroscopic survey will enable the identification\nof LAEs with a Ly-$\\alpha$ luminosity $\\ge 10^{42.4}$\\,erg\\,s$^{-1}$,\nwhile over the smaller ultra-deep 10\\,deg$^2$ survey this\nline-luminosity limit will extend to $\\ge 10^{41.6}$\\,erg\\,s$^{-1}$.\nCrucially this will extend the Ly-$\\alpha$ detectability of LBG\ngalaxies at z$\\sim$8, with brightness J$\\sim$27AB, down to ``typical''\nequivalent widths of $\\sim$15\\AA\\ \\citep{Stark:10,Vanzella:11,CL:12,\n Schenker:12}.\n\n\nThe total number of LAEs in the combined surveys (100 + 10\\,deg$^2$)\nwill obviously depend on some of the key unknowns that {\\tt Chronos} is\ndesigned to measure, in particular the fraction of LBGs which display\ndetectable Ly-$\\alpha$ emission as a function of redshift, mass and\nenvironment. However, if the observed LAE fraction of bright LBGs at\nz$\\sim$7 is taken as a guide, the proposed surveys will uncover $\\sim$10,000\nLAEs at z$>$6.5.\n\n\n\\subsection{The emerging galaxy population at z$>$7, and the supply of reionizing photons}\n\nThe proposed survey will provide a detailed spectroscopic\ncharacterization of an unprecedently large sample of LBGs and\nLAEs. Crucially, as well as being assembled over representative\ncosmological volumes of the Universe at z$\\sim$6--12, these samples\nwill provide excellent sampling of the brighter end of the galaxy UV\nluminosity function at early epochs. As demonstrated by the most\nrecent work on the galaxy luminosity function at z$\\sim$7--9\n\\citep{McLure:13}, an accurate determination of the faint-end slope of\nthe luminosity function (crucial for understanding reionization) is in\nfact currently limited by uncertainty in L$_*$ and\n$\\Phi_*$. Consequently, a large, robust, spectroscopically-confirmed\nsample of brighter LBGs over this crucial epoch is required to yield\ndefinitive measurements of the evolving luminosity functions of LBGs\nand LAEs.\n\nLeaving aside the uncertainties in the numbers of LAEs discussed\nabove, we can establish a reasonable expectation of the number of\nphotometrically-selected LBGs which will be available within the\ntimescales expected for such a mission. For example, scaling from\nexisting HST and ground-based studies, the ``Deep'' component of the\n{\\sl Euclid} survey (reaching J$\\sim$26AB at 5$\\sigma$ over $\\sim$40\\,deg$^2$),\nis expected to yield $\\sim$6000\\,LBGs in the redshift range 6.5$<$z$<$7.5\nwith J$<$26AB (selected as ``z-drops''), $\\sim$1200 at 7.5$<$z$<$8.5\n(``Y-drops''), and several hundred at z$>$8.5 (``J-drops'')\n\\citep{Bouwens:10,Bowler:12,McLure:13}.\n\nTherefore, the planned spectroscopic follow-up over 10\\,deg$^2$, will be\nable to target (at least) $\\sim$1500\\,LBGs in the redshift range 6.5$<$z$<$7.5,\n$\\sim$300 in the redshift bin 7.5$<$z$<$8.5, and an as yet to be\ndetermined number of candidate LBGs at 8.5$<$z$<$9.5. The proposed\ndepth and density of the {\\tt Chronos} near-infrared spectroscopy will allow\ndetection of Ly-$\\alpha$ line emission from these galaxies down to a 5$\\sigma$\nflux limit $10^{-18}$\\,erg\\,cm$^{-2}$\\,s$^{-1}$, enabling rejection of any low-redshift\ninterlopers, determination of the LAE fraction down to equivalent widths of $\\sim$10\\AA,\nand accurate spectroscopic redshifts for the LAE subset.\n\n\\subsection{The contribution of AGN to reionization \\& the early growth of black holes}\n\nSDSS has revolutionised studies of quasars at the highest redshifts,\nand provided the first evidence that the epoch of reionization was\ncoming to an end around z$>$6 \\citep{Becker:01}. As with the\nstudies of galaxies discussed above, pushing to higher redshifts is\nimpossible with optical surveys, regardless of depth, due to the fact\nthat the Gunn-Peterson trough occupies all optical bands at z$>$6.5.\nTherefore, to push these studies further in redshift needs deep\nwide-field surveys in the near-infrared.\n\nThe wide-area, ground-based VISTA near-infrared public surveys such as\nVIKING and the VISTA hemisphere survey are slowly beginning to uncover\na few bright quasars at z$\\sim$7 \\citep[e.g.][]{Mortlock:11}. Recent\nevidence combining X-ray and near-IR data suggests that faint quasars\nat z$\\sim$6 may be commoner than previously thought, and might contribute\nto reionization significantly \\citep{Giallongo:15,MH:15}.\nIt is expected that {\\sl Euclid} and {\\sl WFIRST} will be able to provide\na good determination of the bright end of the QSO luminosity function\nat z$>$6. However, the shape of the QSO luminosity function at these\nredshifts can only be studied with detailed near-infrared spectroscopy\nover a significant survey area. This is the only direct way to\nproperly determine the contribution of accreting black holes to the\nreionization of the Universe and constrain the density of black-holes\nwithin the first Gyr after the Big Bang; the combination of depth and\narea proposed in this NIR survey provides the ideal way in which to\nmeasure the evolving luminosity function of quasars at 6.5$<$z$<$10.\n\\end{multicols}\n\n\\eject\n\n\\section{Precision Cosmology}\n\\label{sec:cosmo}\n\n\n\\begin{multicols}{2}\n\nThe {\\sl Euclid} mission will revolutionize cosmology, however the ultimate\nprecision of {\\sl Euclid} will be limited by our understanding of galaxy\nevolution on small-scales ($\\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$}$1\\,Mpc) due to baryonic feedback\nmechanisms. For example \\citet{vDaal:11} predicted that\nAGN feedback should have a sizeable 20\\% effect on the amplitude of\nthe matter power spectrum, amongst many other studies. Without\ncalibration data on small-scales from large complete spectroscopic\nsamples, {\\sl Euclid} will be required to either marginalize over such\neffects, remove them from the analyses using filter techniques, or\nmodel them using a phenomenological ansatz such as the halo model.\n\nUnderstanding galaxy evolution will therefore enable precision\ncosmology to be extended beyond the {\\sl Euclid} baseline to smaller scales, \nallowing for an increased sensitivity of modified\ngravity models, and up to a ten fold improvement on dark energy\nconstraints than from {\\sl Euclid} alone. As example of beyond-{\\sl Euclid}\ncosmology enabled by small-scale information, we list the following:\n\n\\begin{itemize}\n\\item Neutrino Physics. Massive neutrinos impact the matter\n power spectrum on both linear and non-linear scales. In particular, \n information on the neutrino hierarchy is amplified on small-scales\n \\citep{Jimenez:10}.\n\n\\item Warm Dark Matter. The temperature, and particle mass,\n of dark matter is still unknown. In fact models in which dark matter\n has a small temperature are still allowed by the data. If dark\n matter is warm then any signature of its effects will be seen on\n small-scales, e.g. in the stellar mass function.\n\n\\item Modified Gravity. The accelerated expansion could be\n a symptom of our gravity model, general relativity, being\n incorrect. Models that change general relativity can have a\n scale-dependence, and chameleon mechanisms can act on relatively\n small scales \\citep{Amen:13}.\n\\end{itemize}\n \nFigure~\\ref{fig:Cosmo} shows the sensitivity of three beyond-{\\sl Euclid}\ncosmological models to small-scale information. The deep redshift\nrange would also constrain early-dark energy models, complementing the\n{\\sl Euclid} cosmology objectives using techniques such as those used by\n\\citet{Mandel:12} in SDSS.\n\n\\begingroup\n\\centering\n\\includegraphics[width=80mm]{CH_f10.png}\n\\captionof{figure}{A sample of new physical effects that can be tested using\n small-scale information. Shown is the ratio of the unaffected power\n spectrum compared to that with the new physical effects, as a\n function of scale, at redshift zero. The solid line\n (from \\citealt{Markovic:11}) shows the impact of a 1.25\\,keV warm dark matter\n particle, The dot-dashed line shows the impact of a massive\n neutrinos with total mass of 0.4\\,eV from \\citet{Zhao:13}, and the\n dashed line shows a Hu-Sawicki modified gravity model with an\n amplitude deviation in the Lagrangian of $10^{-6}$ \\citep{Baldi:14}.}\n\\label{fig:Cosmo}\n\\endgroup\n\n \n\\end{multicols}\n\n\\section{Scientific requirements}\n\\label{sec:scireq}\n\\begin{multicols}{2}\n\n\\subsection{Introduction}\nThe study of galaxy formation and evolution involves a large range of\nmeasurement concepts. A deep spectroscopic galaxy survey -- combined\nwith high resolution NIR imaging from \n{\\sl Euclid} and {\\sl WFIRST} -- provides the optimal dataset. Note, however, the\ninherently more complex task of gathering high-quality\nspectroscopic data with respect to imaging. A spectral resolution\n$R\\equiv\\lambda\/\\Delta\\lambda\\sim 1500-3000$ is needed both for \naccurate velocity dispersion measurements, \nand to beat the degeneracies present in spectral features. This limit is\nmainly set by the typical stellar velocity dispersions found in\ngalaxies (50--300\\,km\\,s$^{-1}$), and by the need to adequately\nresolve targeted emission lines and absorption features. Fig.~\\ref{fig:Hlimit} quantifies the\nmagnitude limit within the targeted redshift range. Ideally, a H=26AB\nlimit, {\\sl in the continuum}, would provide complete samples down to\na stellar mass of M$_*\\lower.5ex\\hbox{$\\; \\buildrel > \\over \\sim \\;$} 10^9$M$_\\odot$ across the peak of\ngalaxy formation activity (z$\\sim$1--3). Note that at\nhigher redshifts, the analysis will rely on emission lines, although\nit will be possible to work in the continuum of the most massive\ngalaxies (M$_*\\lower.5ex\\hbox{$\\; \\buildrel > \\over \\sim \\;$} 10^{9.5}$M$_\\odot$ at z$\\sim6$).\n\nRegarding the issue of target selection for spectroscopy, H=26AB is\nthe sensitivity limit expected for the deep fields with {\\sl\n Euclid}\/NISP \\citep{Euclid}, and {\\sl WFIRST}\/WFI will provide\nphotometry slightly deeper than this \\citep{WFIRST}. \n\n\\begingroup\n\\vskip+0.3truecm\n\\centering\n\\includegraphics[width=85mm]{CH_f11.pdf}\n\\captionof{figure}{Stellar mass of a range of stellar populations with respect\n to redshift, with apparent magnitude H=24AB and H=26AB\n \\citep[from the synthetic models of ][for a Chabrier IMF, at solar metallicity]{BC:03}.\n The shaded regions extend from old stellar populations (formed at $z_{\\rm FOR}=10$) to\n a younger galaxy (age 50\\,Myr). Real galaxies will mostly sit within the shaded\n regions. \n}\n\\label{fig:Hlimit}\n\\vskip+0.3truecm\n\\endgroup\n\n\nNote the highly challenging measurements: at the faint \nend, a H=26AB distant galaxy produces a flux of $\\sim$3\nphotons per second in a perfect, unobstructed 3\\,m diameter telescope\nthrough the WFC3\/F160W passband. Furthermore, the same collecting area\nyields $\\sim$30 photons per {\\sl hour}, per spectral resolution\nelement, in the continuum of a spectrum at R=2000. The\nsky brightness at the best ground-based sites reach\n$\\mu_{\\rm H,AB}^{\\rm Sky}\\sim$19.5\\,mag\\,arcsec$^{-2}$ \\citep{SkyH}, and the\nzodiacal background can be as high as $\\mu_{\\rm H,AB}^{\\rm Zodi}\\sim$21.5\\,mag\\,arcsec$^{-2}$\nin the same spectral region\\footnote{Wide Field Camera 3 Instrument Handbook\n for cycle 24 (STScI, v8.0, Jan 2016)}.\nAt these limiting magnitudes, any successful project must be \nbased in space, and requires very long integration\ntimes, pointing towards the darkest regions away from the galactic plane\nand the ecliptic. For reference, the best spectroscopic samples of galaxies at\nz$\\sim$2--3 with state-of-the-art, ground-based facilities\n(e.g. VLT\/X-SHOOTER) reach K$\\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$}$21.5{\\sc AB}, and have noisy\ncontinua \\citep[e.g.][]{Marsan:16}.\n\nIn a presentation for the future\nESA L2\/L3 science cases \\citep{Chronos}, we argued that {\\sl any}\nground-based facility, including future telescopes such as {\\sl ELT}\nor {\\sl TMT}, will not be capable of providing a clean spectrum over\na wide spectral window, needed to trace in\ndetail the continuum associated to the stellar populations of galaxies\nat the peak of formation.\n\n\n\\begin{center}\n{\\sl \\vskip-2truemm Tentative mission concept}\n\\end{center}\n\\vskip-3truemm\n\nThe proposed science case will require a\nlarge aperture survey telescope in space (between 3 and 6\\,m\ndiameter), ideally at L2, although bolder options in the future may\nconsider a lunar platform (allowing for service missions,\nand providing added value to a future manned programme to the moon).\nThe survey will entail \nlong total integration times per field, over the 100\\,ks mark -- requiring fine\npointing accuracy. {\\sl Such a survey would be, by\nfar, the deepest ever taken.} The baseline concept proposed in\n\\citet{Chronos} was equivalent to taking one Hubble Ultra Deep Field\nevery fortnight for five years. \nSuch characteristics places {\\sl Chronos} as an L-type mission, ideally\nincluding cross-collaborations with international space agencies outside\nthe ESA domain. A smaller, M-type, mission could be envisioned for technology\ndevelopment, targetting the most luminous galaxies in the two cosmic\nintervals under study.\n\n\\subsection{Why target one million spectra?}\n\\label{SS:WhySoMany}\nThe aim of the survey is to provide a legacy database of high quality\ngalaxy spectra, sampling both the peak (z$\\sim$1--3) as well as the first\nphases (z$\\lower.5ex\\hbox{$\\; \\buildrel > \\over \\sim \\;$}$6) of galaxy formation. In contrast with\ncosmology-driven missions -- such as {\\sl Euclid} or {\\sl Planck} -- \nthat have a unique figure of merit for the constraint of a reduced set\nof cosmological parameters, {\\tt Chronos} will be a\n``general-purpose'' survey. Regarding sample size, we use as\nreference, the best spectroscopic dataset of galaxy spectra at\nz$\\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$}$0.2, namely the ``classic'' Sloan Digital Sky Survey (SDSS), comprising\napproximately 1 million optical spectra of galaxies brighter than\n$r$$\\sim$18\\,AB \\citep[e.g.,][]{SDSS}. The problems facing galaxy formation studies are not as\n``clean'' as, for instance, finding $w$ in a dark energy-dominated cosmology,\nor water vapour in an \nexoplanet. Galaxy formation is a highly complex field, involving a\nlarge set of physical mechanisms. Such complex questions need large\ndatasets to be able to probe in detail variations of the observables\nwith respect to properties such as the stellar mass, size or\nmorphology of the galaxy under consideration, the mass of its host\nhalo, the potential nuclear activity (ongoing or recent), the presence\nof infall\/outflows, or tidal interactions from nearby\ninterlopers. Therefore, {\\sl it is necessary to probe this\nmulti-parameter space in sufficient depth to understand in detail\nthe role of the mechanisms driving galaxy formation}. This is where a\nlarge multiplex mission such as {\\tt Chronos} exceeds the specifications\nof future large facilities such as {\\sl JWST} or {\\sl ELT}. Although\nJWST\/NIRSpec will obviously have the capability of observing deep NIR\nspectra of distant galaxies, its small field of view, lower multiplex\nand oversubscription -- across a wide range of disciplines -- will\nallow such a powerful telescope to gather, at most, $\\sim$1,000 galaxy\nspectra at similar spectral resolution, within the remit of this science\ncase \\citep{Rieke:19}. Doubtlessly, it will\nhelp tackle the science drivers listed above. However, such a small\nsample will always lead to the question of whether the observed\nsources are representative. Furthermore, if one wants to explore the\neffect of one of the parameters\/observables listed above, it will be\nnecessary to divide the sample accordingly. As an example, studies of\nenvironment-related processes done at lower redshift with SDSS or\nGAMA, work with samples between 10 and 100 times larger than the\npotential output of JWST. {\\sl{\\tt Chronos} should be considered a\n successor to JWST and ELT-class facilities in galaxy formation\n studies.}\n\n\\subsection{Why the proposed spectral mode?}\n\\label{SS:WhySpec}\n\nThe table in Fig.~\\ref{fig:DMD} (right) shows the overal properties of\nthe proposed survey. Choosing a wider wavelength coverage would enable\nus to target additional spectral features. In principle, it would be\npossible to extend the coverage to $K$ band. Note that the targeted\nspectral coverage is suitable for the analysis of the feature-rich\nregion around the 4000\\AA\\ break in the ``cosmic-noon'' sample. Those\nfeatures will be good enough to determine accurate kinematics, stellar\npopulation properties (age and chemical composition) and gas\nparameters. Extending the data, e.g. to 3\\,$\\mu$m would add H$\\alpha$\nat the highest redshifts of the ``cosmic noon'' sample (z$\\sim$4),\nwhich will obviously increase the science return, as the H$\\alpha$\nregion will allow us to improve on estimates of star formation rates,\nor characterize in more detail the ionization state of the gas. In the\n``cosmic dawn'' sample, the key region is the Ly$\\alpha$ interval,\nfully covered at the required redshifts by the proposed wavelength\nrange, so a limited extended spectral coverage in this sample is not\nso beneficial.\n\nHowever, we emphasize that a significant increase in wavelength\ncoverage at fixed spectral resolution could make the spectrograph\nprohibitively expensive in a high multiplex system such as {\\tt\n Chronos}. The main science drivers cannot be fulfilled at lower\nspectral resolution, and a lower multiplex will make the proposed\nsurvey size unfeasible within a 5 year mission concept, so a wider\nspectral coverage should not be the major direction to improve on this\nconcept. Note also that {\\tt Chronos} will operate with ultra-faint\nsources. At longer wavelengths, the thermal background of the\ntelescope will impose additional costly solutions to keep the\nbackground at acceptable levels.\n\n\n\\subsection{Comparison with current and future projects}\n\\label{SS:Sci_Comp}\n\n{\\tt Chronos} will play a unique role in the landscape of future\nnear-infrared spectroscopic surveys. The MOONS multi-fiber\nspectrograph at the ESO VLT will have a multiplex of about 1000 fibers\nover a field of view (FoV) of 500\\,arcmin$^2$, and will cover\n0.6--1.8\\,$\\mu$m at R$\\sim$5000 \\citep{MOONS}. The Subaru Prime Focus\nSpectrograph (PFS) will use up to 2400 fibers over 1.3\\,deg$^2$, and\nwill cover 0.38-1.26\\,$\\mu$m at R$\\sim$4300 in the near-IR ($\\lambda\n>$0.94\\,$\\mu$m) \\citep{PFS}. The Maunakea Spectroscopic Explorer (MSE)\nis planned as a dedicated 10m-class telescope with a high multiplex\n(2,000--3,000) spectrograph that will work at low and high spectral\nresolution \\citep{MSE}.\nHowever, the data taken with these\npromising facilities will be inevitably affected by the atmosphere\n(opaque spectral windows, telluric absorption lines, OH emission line\nforest, high sky background) which will severely limit the\nsensitivity, the quality of the spectra and the capability to observe\nthe continuum of faint objects. Having both continuum and absorption\ninformation in galaxy spectra over a wide spectral window allows us\nto break the degeneracies that entangle the properties of the stellar\npopulations. In the case of space-based facilities,\n{\\sl JWST} will have a very small survey efficiency due to its small\nFoV and is therefore expected to play a complementary role in the\ndetailed study of small samples of objects. {\\sl Euclid} and {\\sl\n WFIRST} will survey very wide sky areas (15,000 and 5000\\,deg$^2$,\nrespectively) in the near-infrared ($\\sim$1--2\\,$\\mu$m), but the\nspectroscopy will be slitless and with low resolution (R$<$500). This\nmakes {\\sl Euclid} and {\\sl WFIRST} powerful missions for redshift\nsurveys based on fairly bright emission lines, but less suitable for\ndetailed spectroscopic studies. For these reasons, {\\tt Chronos} will\nplay a unique and unprecedented role thanks to its uninterrupted and\nwide near-IR spectral range (rest-frame optical for z$>$1 objects),\nextremely high sensitivity due to the low background, capability to\ndetect the continuum down to H=24-26AB, high S\/N ratio suitable to\nperform astrophysical and evolutionary studies, very high\nmultiplexing, wide sky coverage and large (SDSS-like) samples of\nobjects.\n\n\\subsection{Star formation history of galaxies}\n\\label{SS:Sci_SFH}\n\nThe stellar component in a galaxy is made up of a complex mixture of\nages and chemical composition, reflecting its past formation\nhistory. For instance, galaxies that underwent recent episodes of star\nformation will include a young stellar component, characterised\nby strong Balmer absorption \\citep[e.g.][]{Wild:09}; an efficient process\nof gas and metal outflows will be reflected in the chemical\ncomposition, targeted through metallicity-sensitive spectral indices\nsuch as Mgb and $\\langle$Fe$\\rangle$ \\citep[e.g.][]{SCT:00}; abundance variations between different\nchemical elements, such as [Mg\/Fe], map the efficiency of star\nformation \\citep[e.g.][]{Thomas:05};\nvariations in the stellar initial mass function (a fundamental component\nof any galaxy formation model) can be constrained through the\nanalysis of gravity-sensitive indices \\citep[e.g.][]{Ferreras:13,AH:18}.\n\n\\begingroup\n\\centering\n\\includegraphics[width=85mm]{CH_f12.pdf}\n\\captionof{figure}{The red lines are model predictions from\n \\citep{BC:03} for two age-sensitive ({\\sl left}) and two metallicity\n sensitive ({\\sl right}) line strengths for a galaxy with velocity\n dispersion $\\sigma=$200\\,km\\,s$^{-1}$, as a function of age and\n metallicity, respectively (the bottom axes show the age and\n metallicity ranges). The orange lines are the estimated\n measurements, along with a 1\\,$\\sigma$ error bar, given as a\n function of S\/N (shown in the top axes). The simulated data\n correspond to a population at solar metallicity and age 3\\,Gyr,\n marked with vertical dashed blue lines, along with a $\\pm$0.1\\,dex\n interval in grey.}\n\\label{fig:SNR}\n\\vskip+0.2truecm\n\\endgroup\n\nThe stellar component of a galaxy encodes a\nfossil record of its evolution. In contrast, the gaseous component\ngives a snapshot of the ``ongoing'' processes. The analysis of the\nunresolved stellar populations in distant galaxies is tackled\nthrough targeted line strengths and spectral fitting, by comparing\nhigh-quality spectroscopic data with the latest stellar population\nsynthesis models \\citep[e.g.][]{Vazdekis:12,Vazdekis:15}. Such\nmethods have been very successful at understanding the formation\nhistory of low redshift galaxies by use of spectra from the Sloan\nDigital Sky Survey \\citep[e.g.][]{Gallazzi:05}. Similar type of\nstudies at high redshift are fraught with the difficulties of dealing\nwith very faint sources, in an observer frame (NIR) where the complex\nand highly variable airglow and telluric absorption makes ground-based\nobservations tremendously challenging. Figure~\\ref{fig:SNR} shows a\ntest with synthetic spectra of the S\/N level required to constrain\nstellar population parameters from a set of line strengths. For a\n0.1\\,dex (statistical) accuracy in log(Age) or log(Z\/Z$_\\odot$),\ntypical values of S\/N of $\\sim$10--20 per resolution element are required\n{\\sl in the continuum}. This is a challenging target for galaxies at\nz$\\sim 2-3$, given the faint flux levels in the continuum shown in\nFig.~\\ref{fig:Hlimit}.\n\n\\subsection{The role of AGN}\n\nStudies of the past star formation histories of galaxies\n(\\S\\S\\ref{SS:Sci_SFH}) need to be compared with diagnostics of AGN\nactivity, to understand the connection between galaxy growth and that\nof the central SMBH. Such studies are based on emission line diagrams\n\\citep[e.g.][]{BPT} that trace the ionisation state of the\ninterstellar medium. The requirements with regards to the S\/N and\nspectral resolution are similar to the limits imposed by the analysis\nof stellar populations, although we note that emission line\nconstraints will be less stringent, in general, to those in the\ncontinuum. At high enough S\/N, it may be possible to separate the\ncentral component (dominated by the AGN) from the bulk of the\ngalaxy. As reference, a 0.1\\,arcsec resolution element maps into a\nprojected physical distance of 0.8--1\\,kpc at z$\\sim$1--3.\n\n\n\\subsection{Environment and Merger history of galaxies}\n\\label{SS:Sci_Env}\nLarge spectroscopic redshift surveys are needed to characterize the\nenvironment of galaxies in detail\n\\citep[e.g.][]{Yang:07,Robotham:11}. A mass-limited complete survey\nwill allow us to probe the merging history of galaxies, either from\nthe study of dynamically close pairs \\citep[e.g.][]{CLS:12,SH4} or\nthrough morphological studies \\citep[e.g.][]{Lofthouse:16}.\n \nAlthough deep NIR imaging surveys will be available at the time of a\npotential L4 mission, there will not be a comprehensive counterpart of\nspectroscopic observations, except for reduced sets of galaxies\n($\\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$}$1,000) observed by {\\sl JWST}, or {\\sl E-ELT}-like telescopes\nfrom the ground. In order to beat cosmic variance it is necessary to\nobtain spectroscopic redshifts covering large enough volumes. As a\nrough estimate, we use the state-of-the-art Sloan Digital Sky Survey\nas reference. The original low-redshift dataset, limited to\nr$<17.7${\\sc AB}, can be considered ``complete'' out to redshift\nz$\\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$}$0.2, covering a comoving volume of $5.5\\times\n10^{-5}$\\,Gpc$^3$ per square degree. At the peak of galaxy formation\nactivity, z$\\sim 1-3$, the equivalent volume is $0.02$\\,Gpc$^3$ per\nsquare degree. Since the SDSS footprint extends over $\\sim\n10^4$\\,deg$^2$ on the sky, a similar comoving volume will be probed by\n{\\tt Chronos} if covering 30\\,deg$^2$. Although a detailed analysis is\nbeyond the scope of this proposal, it may be advisable to opt for a\ntiered survey, from shallower samples (H$<$24{\\sc AB}) over\n100\\,deg$^2$ to deeper regions, covering $\\sim 5-10$\\,deg$^2$ at\nH$<$26{\\sc AB}.\n\n\n\\subsection{Gas and stellar kinematics and chemistry}\n\nThe emission and absorption line positions and shapes are a valuable\ntool to study the kinematics and chemical composition of the stellar\nand gaseous components. Through high volumes of high S\/N data with\nhigh enough spectral resolution, it will be possible to trace stellar\nkinematics and the mechanisms of gas outflows and stellar\nfeedback. Moreover, information such as the velocity dispersion\nor the spin parameter can be used to constrain the\nproperties of the dark matter halos hosting galaxies at z$\\sim$1-3\n\\citep[e.g.][]{Burkert:16,Wuyts:16}. More detailed analyses can be\ngathered by integral field units, where the spectra of different\nregions of the galaxy are extracted separately. Such instruments have\nfacilitated detailed analyses of the stellar and gaseous components in\nnearby \\citep[e.g. ATLAS$^{\\rm 3D}$][]{Cap:11} and distant\n\\citep[e.g. KMOS$^{\\rm 3D}$][]{Wisnioski:15} galaxies. Due to the\nfaintness of the sources and the need for a high multiplex system\ncovering a wide field of view, we would, in principle, decide against\nan IFU-based instrument, although this issue would be an important one\nto tackle during the definition phase (see \\S\\ref{sec:Measure}). Also\nnote that at the redshifts probed, the (spatial) resolving power is\nrather limited, expecting a resolution -- measured as a physical\nprojected distance -- around 1\\,kpc at z$\\sim$1--3. The high S\/N\nrequirements of the previous cases align with this one, but on the\nissue of kinematics, a slightly higher spectral resolution may be\ndesired. Note also that at high resolution, high S\/N spectra may be\nused to disentangle different components \\citep[such as bulge and\n disk,][]{Ocvirk:06}.\n\n\\subsection{Observations of galaxies in the high redshift Universe}\n\nDeep NIR spectroscopy from space is the only way to confirm the\ncontinuum break at 1216\\AA\\ in the high redshift Universe\n(z$\\lower.5ex\\hbox{$\\; \\buildrel > \\over \\sim \\;$}$6). Ground-based instruments only detect these objects when\nthey have strong Ly-$\\alpha$ lines in clean regions of the night sky\nspectrum. This line can be scattered by neutral intergalactic gas, and\nis likely to be weaker at such redshifts. The goal of achieving enough\nS\/N in the continuum is important to properly characterize the\nproperties of the underlying stellar populations, something that could\nbe done with individual galaxies in the deep survey, and with stacked\nsubsamples in the wide survey.\n\n\\end{multicols}\n\n\\section{Measurement Concept}\n\\label{sec:Measure}\n\n\\begin{multicols}{2}\n\n\n\\subsection{Reconfigurable Focal Planes for Space Applications}\n\\label{SS:ReconfFP}\n\n{\\sl JWST} will be the first astronomy mission to have a true multi-object\nspectroscopic capability via the micro-shutter arrays in the NIRSpec\ninstrument, which can observe up to ~100 sources simultaneously over a\nfield-of-view around $3^\\prime\\times 3^\\prime$ \\citep{Li:07}. Scaling\nthis technology to the field sizes and multiplex advantage required\nfor the next generation of space-based spectroscopic survey\ninstruments is not straightforward however, and will likely require a\nnew approach. There are currently three technologies which show\npromise in this area.\n\n\\begin{figure*}\n\\centering\n\\raisebox{-0.55\\height}{\\includegraphics[width=70mm]{CH_f13.jpg}}\n\\hspace{0.5truecm}\n\\centering \n{\\parbox{105mm}{\\begin{tcolorbox}[tab1,tabularx={X|C{21mm}|C{21mm}},title=Summary of {\\tt Chronos} survey specifications,boxrule=0.5pt]\nSpectral range & \\multicolumn{2}{c}{0.8--2$\\mu$m}\\\\ \\hline\nSpectral resolution & \\multicolumn{2}{c}{1500-3000}\\\\ \\hline\nTarget Multiplex & \\multicolumn{2}{c}{$\\lower.5ex\\hbox{$\\; \\buildrel > \\over \\sim \\;$} 5000$}\\\\ \\hline\nField of View (deg$^2$) & \\multicolumn{2}{c}{$\\lower.5ex\\hbox{$\\; \\buildrel > \\over \\sim \\;$}$0.2}\\\\ \\hline\\hline\nSURVEY & Wide & Deep\\\\ \\hline\nSensitivity (@S\/N=20) & H=24{\\sc AB} & H=26{\\sc AB}\\\\ \\hline\nLine Sensitivity (@5$\\sigma$) & $5\\times 10^{-19}$cgs & $8\\times 10^{-20}$cgs\\\\ \\hline\nGalaxy density (z=1--3, deg$^{-2}$) & $4.8\\times 10^4$ & $1.2\\times 10^5$\\\\ \\hline\nCoverage (deg$^2$) & 100 & 10\\\\ \\hline\n\\end{tcolorbox}\n}}\n\\captionof{figure}{{\\sl Left:} Photomicrograph of tilted DMD\n micromirrors. The neighbouring mirrors have been removed to reveal\n the substructure (courtesy ASME\/Texas Instruments). {\\sl Right:}\n General specifications of the proposed survey.}\n\\label{fig:DMD}\n\\end{figure*}\n\n\\begin{enumerate}\n\\item\\underline{Digital Micromirror Arrays (DMDs):} Digital micromirror\ntechnology was developed in the 1990s by Texas Instruments for use in\nlight projection systems (see Fig.~\\ref{fig:DMD}). The current\nstate-of-the-art is 2kx1k devices with 13\\,$\\mu$m pixels but larger\nformat devices (up to 16 million pixels) are under development\n(c.f. 62,000 micro-shutters in {\\sl JWST} NIRSpec). These devices are\nalso well matched in pixel size to the focal planes of small to medium\nsize telescopes. DMDs were first proposed for the ESA M-class SPACE\nmission concept \\citep{CI:09} which later evolved into the {\\sl\n Euclid} mission. The primary technological challenges in exploiting\nDMDs in space are: (i) developing radiation-hard electronics to drive\nthe DMDs (the MEMS technology used in the mirrors themselves are not\nsusceptible to damage except by extreme micrometeorite events), (ii)\ndemonstrating reliable operation at cryogenic temperatures as required\nfor observations at near-infrared wavelengths, (iii) modifying the\nvisible-light windows on commercial devices to allow extended\noperation into the near-infrared, (iv) improving the\ncontrast\/scattered light for bright objects. Preliminary work has been\nundertaken during studies for SPACE\/{\\sl Euclid} and elsewhere\n\\citep{ZA:11,ZA:17} but further work is required to raise the TRL.\n\n\\item\\underline{Reconfigurable Slits:} A near-infrared multi-object\nspectroscopy target selection system which has been successfully\ndeployed on the ground-based MOSFIRE instrument at Keck is the\nConfigurable Slit Unit (CSU) \\citep{Span07}. This is a form of\nmicro-mechanical system which employs voice-coil actuated\n``inch-worm'' motors to position up to 46 slitlets (each 5\\,mm long)\nin a $4^\\prime\\times 4^\\prime$ field. This technology has been proven\nto be reliable in cryogenic operation (at 120K) over several\nyears. Whilst the 1-D motion of the slits in the focal plane reduces\ntarget acquisition efficiency somewhat, the contiguous slits allow for\nimproved sky-subtraction compared to devices (like the DMD) where the\nlocal sky is obtained via separate apertures. The primary technology\nchallenges of adopting this technology for space applications would be\nthe substantial miniaturization required and increasing the multiplex\ngain by 1 or 2 orders of magnitude, possibly using a piezo-electric\ndrive system.\n\n\\item\\underline{Liquid Crystal Masks:} Liquid crystal (LC) masks are\nwidely used as spatial light modulators in a number of laboratory\napplications. Whilst fundamentally relying on the ability of\npolarizing crystals to transmit or block linearly polarized light,\nthey can be made to work more efficiently on unpolarized light using\npolymer dispersed liquid crystals (PDLCs). By combining an optically\nactive material with an appropriate electrode structure,\nreconfigurable masks can be obtained which only transmit light in a\nspecific spatial pattern \\citep{WS92}. Devices in formats up to\n1024x768 are commercially available with up to 36\\,$\\mu$m pixels. The\nprimary technological challenges would be: (i) operation efficiency\n(switching times) at low temperatures due to the properties of the LC\nmedium, (ii) contrast limits (many commercial devices have contrast\nratios <100:1), (iii) limitations on bandwidth due to the chromatic\nproperties of LCs, (iv) limitations due to non-orthogonal\nillumination.\n\\end{enumerate}\n\nIn addition to the above reconfigurable focal plane solutions, which\nsegment the focal plane spatially according to preselected target\npositions, an alternative approach is to select targets from a fixed\ngrid of sub-areas across the focal plane (one target per sub-area)\nusing a ``beam-steering'' approach. Many of these rely on similar\nunderlying technologies to those discussed above (i.e. MEMS and\/or\nvariable prisms) and should be explored in the context of specific\nmission requirements. They are particularly suitable to selecting\ntargets for spatially-resolved (``integral field'') spectroscopic\nstudies.\n\n\\subsection{Large format Integral Field Units}\n\nA complementary approach to massively multiplexed spectroscopy when\nthe target densities are high enough, is to use some form of integral\nfield spectroscopy (IFS) which delivers a full spectral datacube for a\ncontiguous region of sky. The IFS approach also opens up a large\nserendipity space since no imaging surveys are required to pre-select\ntargets.\n\nIntegral field units have been widely used on ground-based telescopes\n\\citep[e.g.][]{AllS:06} and a small-format device ($30\\times 30$\nspatial pixels) will be launched on {\\sl JWST} as one of the observing modes\nfor the NIRSpec spectrograph \\citep{Birkman:14}. Integral field\nsystems can be realized using a number of techniques but the favoured\napproach for space infrared systems is the diamond-machined image\nslicer \\citep{Lobb:08} which can take advantage of monolithic\nmanufacturing methods and a robust thermal design approach. Technology\ndevelopments would be required to develop wide-field integral field\nsystems for space applications, but the generic approach using a\n``field-splitter'' front-end optic to feed multiple sub-systems is\nwell-understood from ground-based instruments\n\\citep[e.g.][]{Pares:12}. Mass, power and data rate budgets remain to\nbe explored but will be common to all wide-field spectroscopic\nfacilities.\n\n\n\\subsection{NIR detector technology}\n\nA successful outcome of a survey such as {\\tt Chronos} also rests on highly\nefficient NIR detector technology, with minimal noise and\nwell-understood systematics. The survey operates at a very low-photon\nregime, where it is essential to control the noise sources, and to\nunderstand in detail the response of the detector. For instance,\ncross-talk and persistence are substantial problems that can hinder\nthe observations, and need to be characterised in exquisite detail.\n\nThe best available technology for this science case involves\nHgCdTe-based detectors, where the spectral range can be optimised by\nthe choice of the ratio of Hg to Cd, that modifies the band gap\nbetween 0.1 and 1.5\\,eV. As of today, US-based companies can provide\n4k$\\times$4k HgCdTe arrays with high enough TRL for a space mission\n\\citep[e.g. Teledyne,][]{Teledyne}. However, given the long\ntimescales expected for a potential mission, and aligned with ESA's\ninvestment in NIR detector technology development \\citep{Nelms:16}, it\nwould be desirable to involve European groups (such as CEA-LETI, Selex\nES or Caeleste) in the development of ultra-sensitive NIR detectors\nand the associated electronics. New HgCdTe-based technology with\navalanche photodiodes, developed by Selex appears quite promising for\nastrophysics applications \\citep{Saphira}.\n\n\\subsection{Photonics-based approach}\n\nAn alternative approach to the traditional spectrograph design is to\nadopt a photonics-based instrument, creating the equivalent version of\nan integrated circuit in electronics. Astrophotonics has produced\nseveral revolutionary technologies that are changing the way we think\nof conventional astronomical instrumentation. In particular, the\ninvention of the photonic lantern \\citep{LBB:05} allows us to reformat\nthe input to any instrument into a diffraction-limited output. As was\nfirst described in \\citet{JBH:06} and \\citet{JBH:10}, this means that,\nin principle, {\\sl any spectrograph operating at any resolving power\n can be designed to fit within a shoebox}. These authors refer to\nthis as the photonic integrated multimode microspectrograph (PIMMS)\nconcept and it has been demonstrated at the telescope and in space\n(see Fig.~\\ref{fig:pimms}). Suitable optical designs are presented in\n\\citet{RB:12}. Presently, the main limitation is that the ideal\ndetector has yet to be realized, although discussions are ongoing with\ndetector companies. This technology is ideally suited to optical and\ninfrared spectroscopy, and may overcome the technological challenges faced\nby conventional spectrograph designs within ESAs Voyage 2035-2050\nlong-term plan.\n\n\\begingroup\n\\vskip+0.2truecm\n\\centering\n\\includegraphics[width=80mm]{CH_f14.png}\n\\captionof{figure}{As seen above, the PIMMS concept has been\n demonstrated at the telescope \\citep{Cvet:12}, on a balloon\n (2012) and onboard the Inspire cubesat (2017-8) flown by the\n University of Sydney \\citep{Cairns:19}.}\n\\label{fig:pimms}\n\\vskip+0.2truecm\n\\endgroup\n\n\n\\end{multicols}\n\n\\clearpage\n\n\\begin{multicols}{2}\n\n \n\n\n\n\n\\setlength{\\bibsep}{2.0pt}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe diversity of the electronic property of condensed matter wholly originates from the ionic configuration. One of its most striking consequences is anomalous concentration of the electronic one-particle states characterized by the peaks in the spectrum of the density of states (DOS) $D(E)$. The concentration of the DOS near the Fermi level indicates that many electrons contribute to the low-energy phenomena, as well as implies that effects of the electron-electron and electron-ion interactions become significant. \n\nHow to make the DOS concentrate in tiny energy ranges has therefore been of continuous interest in the field of the band theory. An extreme example is the flat band,~\\cite{Sutherland1986, Lieb1989,Mielke1-1991,Mielke2-1991,Tasaki1992, Mielke-Tasaki1993, Tasaki-review, Bergholtz-review} where the electronic one-particle energy eigenvalues $\\varepsilon({\\bf k})$ with ${\\bf k}$ being the Bloch wave number is constant in the entire Brillouin zone. Theoretically, the flat bands emerge from tight binding models with specially designed features such as decoration of the unit cell and carefully tuned model hopping parameters between the orbitals. A recent interesting realization of almost flat band is seen in the twisted bilayer graphene.~\\cite{TBG1,TBG2,MacDonald2012} Band structures with flat dispersions within partial regions of the ${\\bf k}$ space and the resulting divergence (or peak) in $D(E)$ are often reported; we list a few which have been understood as a consequence of the special configurations of the hopping parameters.~\\cite{Ochi-RCo5-2015, Jelitto1969, Akashi-interfere-PRB2017}\n\nA theory from a contrasted viewpoint, where we only assume differentiability of $\\varepsilon({\\bf k})$ in the ${\\bf k}$ space, has been established by van Hove.~\\cite{vHS} He has pointed out that, the critical points defined by $|\\nabla \\varepsilon({\\bf k})|=0$ always yield singular points in the DOS (van Hove singularity; vHS) and the minimum number of critical points is nonzero due to the topology of the ${\\bf k}$ space. Remarkably, in the one and two dimensional cases, divergent singularities in $D(E)$ related to the isolated critical points always emerge. The saddle point in (quasi-) two dimensional systems has therefore been of recurrent interest in different contexts; in cuprates,~\\cite{PhysRevLett.73.3302, Markiewicz-review} topological surface states,~\\cite{PhysRevB.97.075125,Ghosh-Pt2HgSe3-arxiv} and the monolayer and multilayer graphene,~\\cite{PhysRevB.77.113410, PhysRevLett.104.136803, Chubukov-graphene-SC, PhysRevB.95.035137, Yuan-TBLG-arxiv} etc. However, such divergence is not generally assured to be present in the three-dimensional case. The isolated critical point with the quadratic {\\bf k} dependence induces the divergence of the derivative of the DOS, but not of the DOS value. The peaks in the DOS are nevertheless observable in various three dimensional systems with the model and first principle electronic structure calculations, even if dimensional reduction is not apparent from the crystal structure of the systems. This fact motivates us to seek for commonplace deformations to $\\varepsilon({\\bf k})$ that can yield divergent DOS singularities in three dimensions.\n\nIn the present article, we characterize a general but unrecognized mechanism of forming divergent singularities in $D(E)$ in three dimensions. Energetically degenerate continuous extension of the critical points generally enhances the degree of singularity by reducing the effective dimension.~\\cite{Markiewicz-review} We point out that, by ``pushing\" the isolated critical point on the hypersurface $\\varepsilon({\\bf k})$, nearly degenerate loop or shell structure of new critical points are formed around it, giving rise to the DOS peaks. We discuss the archetype of this mechanism and how it appears in reality with a simple model. \nAlso, we show that our theory successfully describes how the DOS peak forms in the recently discovered pressure induced high-$T_{\\rm c}$ superconductor H$_{3}$S,~\\cite{Eremets} which boosts its $T_{\\rm c}$ up to 200~K by incorporating a large number of electrons into the pair condensation. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Theory}\n\n\\begin{figure}[h]\n \\begin{center}\n \\includegraphics[scale=0.30]{vHS3D_2D_1D_all_190625.pdf}\n \\caption{Asymptotic forms of the DOS near the vHSs for (a) three, (b) two, and (c) one dimensions. $A (A')$ and $B (B')$ represent the vHSs corresponding to the extrema and saddle points, respectively. See the main text for more specific definitions of the vHSs.}\n \\label{fig:vHSclass}\n \\end{center}\n\\end{figure}\n\n\\subsection{Critical points and the van Hove singularities}\nLet us first start from the review on the relation between the critical points and vHSs in the electronic DOS. The critical point is defined as the points ${\\bf k}$ which satisfy $|\\nabla \\varepsilon({\\bf k})|=0$. The band dispersion near the critical point is therefore generally approximated to the following bilinear form\n\\begin{eqnarray}\n\\varepsilon_{\\rm 3D}({\\bf k})\n&\\simeq &\n\\frac{1}{2}\n\\left(\n\\begin{array}{ccc}\nk_{x} &\\ k_{y} &\\ k_{z}\n\\end{array}\n\\right)\n\\left(\n\\begin{array}{ccc}\na_{xx} &\\ a_{xy} &\\ a_{xz} \\\\\na_{xy} &\\ a_{yy} &\\ a_{yz} \\\\\na_{xz} &\\ a_{yz} &\\ a_{zz}\n\\end{array}\n\\right)\n\\left(\n\\begin{array}{c}\nk_{x} \\\\\nk_{y} \\\\\nk_{z}\n\\end{array}\n\\right)\n\\nonumber \\\\\n&=&\n\\frac{k^{2}_{1}}{2m_{1}}\n+\n\\frac{k^{2}_{2}}{2m_{2}}\n+\n\\frac{k^{2}_{3}}{2m_{3}}\n,\n\\label{eq:bilinear3D}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\varepsilon_{\\rm 2D}({\\bf k})\n&\\simeq &\n\\frac{1}{2}\n\\left(\n\\begin{array}{cc}\nk_{x} &\\ k_{y}\n\\end{array}\n\\right)\n\\left(\n\\begin{array}{cc}\na_{xx} &\\ a_{xy} \\\\\na_{xy} &\\ a_{yy} \n\\end{array}\n\\right)\n\\left(\n\\begin{array}{c}\nk_{x} \\\\\nk_{y}\n\\end{array}\n\\right)\n\\nonumber \\\\\n&=&\n\\frac{k^{2}_{1}}{2m_{1}}\n+\n\\frac{k^{2}_{2}}{2m_{2}}\n,\n\\\\\n\\varepsilon_{\\rm 1D}({\\bf k})\n&\\simeq &\n\\frac{k^{2}_{1}}{2m_{1}}\n,\n\\end{eqnarray}\n\nwhere $(k_{1}, k_{2}, k_{3})^{T} \\equiv M_{\\rm 3D} (k_{x}, k_{y}, k_{z})^{T}$ [$(k_{1}, k_{2})^{T} \\equiv M_{\\rm 2D} (k_{x}, k_{y})^{T}$] with $M_{\\rm 3D}$ ($M_{\\rm 2D}$) being an orthogonal matrix. The structure in the DOS corresponding to the vHS is classified by the relations between the values of the effective mass $m_{i}$. Let us define the types of the critical points for the three dimensional case as \n\\begin{eqnarray}\n&&A_{\\rm 3D}: m_{1} > 0,\\ m_{2} > 0,\\ m_{3} > 0,\n\\\\\n&&B_{\\rm 3D}: m_{1} > 0,\\ m_{2} > 0,\\ m_{3} < 0,\n\\\\\n&&B'_{\\rm 3D}: m_{1} > 0,\\ m_{2} < 0,\\ m_{3} < 0,\n\\\\\n&&A'_{\\rm 3D}: m_{1} < 0,\\ m_{2} < 0,\\ m_{3} < 0,\n\\end{eqnarray}\nfor the two dimensional case as\n\\begin{eqnarray}\n&&A_{\\rm 2D}: m_{1} > 0,\\ m_{2} > 0,\n\\\\\n&&B_{\\rm 2D}: m_{1} > 0,\\ m_{2} < 0,\n\\\\\n&&A'_{\\rm 2D}: m_{1} < 0,\\ m_{2} < 0,\n\\end{eqnarray}\nand for the one dimensional case as\n\\begin{eqnarray}\n&&A_{\\rm 1D}: m_{1} > 0,\n\\\\\n&&A'_{\\rm 1D}: m_{1} < 0,\n\\end{eqnarray}\n\n\nFigure 1 represents the typical behavior of the DOS near the vHS related to the critical points in the respective dimensions.~\\cite{Grosso-Parravicini} In the three dimensional case, the critical points are classified into two: (i) all the effective mass have the same sign (extremum; $A_{\\rm 3D}, A'_{\\rm 3D}$), or (ii) either of the three has the different sign (saddle point; $B_{\\rm 3D}, B'_{\\rm 3D}$). Near the vHSs related to those critical points (commonly represented by $E_{\\rm vHS}$ below), the DOS behaves\n\\begin{eqnarray}\nD(E)\n&\\propto&\n\\sqrt{E-E_{\\rm vHS}} \\ (A_{\\rm 3D})\n\\\\\nD(E) &\\propto& \n\\left\\{\n\\begin{array}{c}\n-\\sqrt{E_{\\rm vHS}-E}+{\\rm const.} \\ (E < E_{\\rm vHS})\\\\\n{\\rm const.} \\hspace{65pt} (E > E_{\\rm vHS})\n\\end{array}\n\\right.\n (B_{\\rm 3D})\n .\n\\end{eqnarray}\nThe classification for two dimensional case is as follows: (i) all the effective mass have the same sign (extremum; $A_{\\rm 2D}, A'_{\\rm 2D}$), or (ii) the signs of the values of the effective mass are different (saddle point; $B_{\\rm 2D}$). \n \\begin{eqnarray}\nD(E)\n&\\propto&\n\\theta(E-E_{\\rm vHS}) \\ (A_{\\rm 2D})\n\\\\\nD(E) &\\propto&\n{\\rm log}\\frac{1}{|E_{\\rm vHS}-E|} \n\\ (B_{\\rm 2D})\n ,\n\\end{eqnarray}\nwhere $\\theta$ denotes the theta function $\\theta(x)=0 \\ (x<0); 1 (x>0)$. The dependence related to $A'_{\\rm 2D}$ is inverse of $A_{\\rm 2D}$ as well. In the one dimensional case, the one effective mass is either positive ($A_{\\rm 1D}$) or negative ($A'_{\\rm 1D}$)\n\\begin{eqnarray}\nD(E)\n&\\propto&\n\\frac{1}{\\sqrt{E-E_{\\rm vHS}}} \\ (A_{\\rm 1D})\n\\end{eqnarray}\nThe dependences related to $A'_{\\rm 3D}$, $B'_{\\rm 3D}$, $A'_{\\rm 2D}$ and $A'_{\\rm 1D}$ are inverse of those without primes, respectively. In later discussions we often focus only on the critical points with (without) prime, as parallel arguments are obviously applicable to those without (with) prime by the energy inversion.\n\n\\begin{figure}[h]\n \\begin{center}\n \\includegraphics[scale=0.35]{Saddle_caldera_all_190611.pdf}\n \\caption{Schematic picture of the concept of perturbing the saddle point $P$ in three dimensions. (a) Two dimensional band structure near the original saddle point and (b) the corresponding form of the DOS near its energy level $E_{\\rm s}$. In the third dimension (not shown) the band is assumed parabolic. (c) The band structure with the circular symmetric perturbation turning $P$ from the maximum to minimum in the two dimensions. The dashed line indicates the saddle loop. (d) Form of the DOS, where the divergence occurs at the energy level of the saddle loop. (e) The band structure with anisotropic deformation to (c), where we indicate the ridge as a remnant of the saddle loop as dotted line. The emergent saddle points are denoted by $E_{\\rm s}$ and $E'_{\\rm s}$. (f) The corresponding schematic form of the DOS, where shoulders are formed at $E_{\\rm s}$ and $E'_{\\rm s}$, respectively.}\n \\label{fig:caldera}\n \\end{center}\n\\end{figure}\n\n\\begin{figure}[h]\n \\begin{center}\n \\includegraphics[scale=0.35]{Extremum_sphere_all_190625.pdf}\n \\caption{Schematic picture of the concept of perturbing the extremum $P$ in three dimensions. (a) Cross sections of the three dimensional band structure near the original maximum. Here, the value of $\\varepsilon({\\bf k})$ is represented by the color scheme. (b) The corresponding form of the DOS near its energy level $E_{\\rm e}$. (c) The band structure with the spherically symmetric perturbation turning $P$ from the maximum to minimum. The dotted line indicates the extremum shell. (d) The form of the DOS, where the divergence occurs at the energy level of the extremum shell $E_{\\rm e}$. (e) The schematic form of the DOS for the case with anisotropic deformation to (c), where termination point corresponding to the maximum $E_{\\rm e}$ and the shoulder corresponding to the saddle point $E_{\\rm s}$ are formed, respectively (The corresponding three dimensional band structure is not shown). (f) The variant of (e) with additional saddle point $E'_{\\rm s}$ on the shell, where the corresponding shoulder is formed.}\n \\label{fig:sphere}\n \\end{center}\n\\end{figure}\n\nThe above discussion clarifies that, for possible divergence of DOS in three dimensions, it is necessary that the bilinear expansion should be broken down at certain band critical points. In other words, the rank of the coefficient matrix in Eq.~(\\ref{eq:bilinear3D}) must be lower than 3 (Ref.~\\onlinecite{Yuan-TBLG-arxiv}). This means that the expansion series in certain coordinates begin with the higher order terms, or, as a limiting case, the critical point extends over a continuous one or higher dimensional region in the {\\bf k} space. For simplicity we ignore the higher order terms and concentrate on the latter possibility in this paper; the band dispersion is either parabolic or flat. From this view, by taking the limits where any of the effective masses becomes infinity, the form of the DOS around the vHS converges to any low-dimensional counterpart. For example, for the $B_{\\rm 3D}$ case, in the limit $m_{2}\\rightarrow \\infty$ the DOS form converges to that of $B_{\\rm 2D}$, whereas the limit $m_{3}\\rightarrow \\infty$ corresponds to $A_{\\rm 2D}$. These correspondences reflects that the effective dimension of the band structure is reduced due to the extension of the critical point. The higher order components of $\\varepsilon({\\bf k})$ can make subtle changes to the degree of singularity, though we ignore them. What we do in this work is consider a simple modification of general three dimensional band structures which yields the extended critical points for possible utilization of the diverging nature of $D(E)$ in lower dimensions.\n\n\n\\subsection{Saddle loop and extremum shell}\nIn this subsection, we characterize a general mechanism for emergence of the peak in the DOS in three dimensional system. Let us consider a saddle point $P$ of class $B'_{\\rm 3D}$ for example (Fig.~\\ref{fig:caldera}(a), left), where in the $k_{1}$ and $k_{2}$ directions the band dispersion is concave ($m_1 = m_2 <0$). We assume that the dispersion is kept parabolic in the $k_{3}$ direction with positive definite effective mass around $P$ in the equal-$k_{3}$ plane ($m_{3}>0$) in the later discussion in this paragraph. In this case, the resulting DOS exhibit the shoulder like vHS [Fig.~\\ref{fig:caldera}(b)] corresponding to the critical point $B'_{\\rm 3D}$ in Fig.~\\ref{fig:vHSclass}. Suppose any perturbation is exerted on the system, which significantly reduces the energy eigenvalues in the close vicinity of $P$ (push the $P$ point). With sufficiently large amount of the perturbation, $P$ turns into the energy minimum and caldera-like structure emerges. If the change of the energy eigenvalues is ideally circular symmetric with respect to the $k_{1}$--$k_{2}$ plane, a closed loop of the saddle points, or saddle loop, appears as depicted in Fig.~\\ref{fig:caldera}(c). At all the points on this line, the band dispersion is concave in the direction perpendicular to the line, flat along the line, and convex in the $k_{3}$ direction as assumed above. As a result, a divergent singularity of the DOS corresponding to the case $B_{\\rm 2D}$ is formed [Fig.~\\ref{fig:caldera}(d)]. Interestingly, the original saddle point $P$ is then modified into the local minimum $A_{\\rm 3D}$ and it does not appear in the DOS spectrum since the contribution around the point, being proportional to $\\sqrt{E-E_{P}}$, is overwhelmed by the contribution from the vicinity of the saddle loop. \n\nOne can also conceive a similar situation for the maximum point $P$ corresponding to type $A'_{\\rm 3D}$. For simplicity let us assume the band isotropy around this point as well ($m_{1}=m_{2}=m_{3}<0$). The DOS form near the singularity trivially corresponds to $\\propto \\sqrt{E_{P}-E}$ [Fig.~\\ref{fig:sphere} (a)(b)]. By introducing a perturbation with spherical symmetry that appreciably reduces the energy eigenvalues around $P$, the maxima of the resulting spectrum form a closed shell, or extremum shell, around $P$[Fig.~\\ref{fig:sphere}(c)]. At any points on this shell, the band dispersion is zero and parabolic in the two tangential directions and one perpendicular direction, respectively. The divergent singularity of the $A'_{1D}$ type then appears at the energy value on the shell [Fig.~\\ref{fig:sphere}(d)]. \n\n\n\\begin{figure}[h]\n \\begin{center}\n \\includegraphics[scale=0.21]{Higher_order_LTs_190819.pdf}\n \\caption{Higher order Lifshitz transitions. (a) The edge pair switching transition in the equal energy surface, where the surfaces at the energy levels slightly above and below $E_{\\rm s}$ for the band structure of Fig.~\\ref{fig:caldera}(c). (b) The surface pair formation transition, where the surfaces at the energy levels slightly above and below $E_{\\rm e}$ for the band structure of Fig.~\\ref{fig:sphere}(c). The close up view of the cross sections is also displayed, where the white lines indicate the cutting planes.}\n \\label{fig:LTs}\n \\end{center}\n\\end{figure}\n\n\n\\subsubsection{Higher order Lifshitz transitions}\nWe here show the character of the saddle loop and extremum shell from the perspective of the equal energy surface. The equal energy surface is defined by the manifold in the ${\\bf k}$ space where $\\varepsilon({\\bf k})$ is equal to a certain energy value. The surface at the Fermi energy--Fermi surface--is of particular interest as it displays the electronic states dominating low-energy phenomena. The change of the topology of the equal energy surface by varying the energy value is called Lifshitz transition.~\\cite{Lifshitz} In three dimensions, generally two types of the transition occur: pocket appearance and the neck disruption. These transitions are observed when the energy value varies through the levels of critical points of types $A_{\\rm 3D}$ and $B_{\\rm 3D}$, respectively. When we change the energy value through the vHS due to the saddle loop or extremum shell, topologically distinct transitions occur in the equal energy surface: so to speak, (i) the edge pair switching transition~[Fig.~\\ref{fig:LTs}(a)]--two adjacent surfaces are attached along the saddle loop and a pocket is detached with the switching of the edge connections--and (ii) surface pair formation transition~[Fig.~\\ref{fig:LTs}(b)]--two parallel surfaces emerge along the extremum shell from the void. Those transitions are actually variants of the known neck disruption transition, in that they are understood as the simultaneous occurrence of the neck disruption (or formation) on a one and two dimensional manifolds, respectively. For this reason we later refer to those transitions as higher-order Lifshitz transitions. We propose to give them specific names, since those transitions are directly related to the source of the divergence of DOS; namely, the linear (planar) region in the ${\\bf k}$ space where ``edge pair switching (surface pair formation)\" occurs is responsible for the divergent contribution. Note that the neck disruption at a point indicate the divergence of the derivative of DOS, but not of the DOS itself.\n\n\\subsubsection{Effect of anisotropy}\nIn realistic three dimensional systems, we cannot generally expect the ideal saddle loop and extremum shell since they require that the gradient of the band is exactly zero along them. Still, we can state that the DOS peaks resulting from those, though not divergent, persist against subtle deformations of the band structure, as long as the the ${\\bf k}$ dependence of $\\varepsilon({\\bf k})$ is not drastic. In the above discussions of the saddle loop, we have assumed circular symmetry for the perturbation around the original saddle point $P$. When some anisotropy is introduced, the saddle loop is deformed into a (not ideally circular) loop of ridge line encircling $P$ [Fig.~\\ref{fig:caldera}(e)], along which the band dispersion is not exactly zero. In the middle of this line, there must emerge one or more minima and maxima, which corresponds to the critical points of type $B_{\\rm 3D}$ and $B'_{\\rm 3D}$, respectively. The divergent DOS singularity then evolves into two (or more) neighboring shoulders, whose energy distance corresponds to the band dispersion along the ridge loop. The whole structure appears as a ``peak\" in the calculated DOS. Note that the presence of the two types of critical points is due to the periodicity of $\\varepsilon({\\bf k})$ along the loop, and they are not assured for $\\varepsilon({\\bf k})$ on an open linear region.\n\nA parallel discussion applies to the case of the extremum shell. With the deformation of the band structure from the above ideal case, the shell is somehow deformed and at least two critical points respectively of $A'_{\\rm 3D}$ and $B_{\\rm 3D}$ are assured to appear anywhere on the shell, and can $B'_{\\rm 3D}$ appear occasionally. The divergent DOS singularity then evolves into adjacent termination point and shoulder(s), between which the DOS rapidly varies [Fig.~\\ref{fig:sphere} (e)(f)]. Note that the apparent width of the DOS peak does not exactly corresponds to the energy distance between the $A'_{\\rm 3D}$ and $B_{\\rm 3D}$ points [$E_{\\rm e}-E_{\\rm s}$ in panel (f)]. \n\nThe anisotropy also changes the appearance of the higher order Lifshitz transitions depicted above. When the energy level is tuned through the peak positions, the transition initiates from the isolated vHSs anywhere on the loop or shell. The starting and end points of the transitions would appear as the well-known neck disruption or formation. By comparing the equal energy surfaces below the lower vHS and above the higher vHS, we see the edge pair switching or surface pair formation process. A statement therefore remains valid that, even with the anisotropy, the higher order Lifshitz transitions occurring between a certain (nonzero) energy range indicate the peaked concentration of the DOS within that range, as well as displays the ${\\bf k}$ point region responsible for the peak.\n\nIn the literature, one sometimes finds discussions relating the peak of the DOS and any vHS appearing in the band structure along specific linear paths. For the above two cases, the DOS peaks are better understood as remnant of the ideal saddle loop or extremum shell. The total DOS peak is formed by the states near the loop or shell, not only by those in the vicinity of the isolated vHSs. The vHSs would correspond to any detailed feature of the peak such as shoulders, but what dominates the peak width is the degree of the band dispersion over the loop or shell, where specific positions of the vHS could change depending on subtle perturbations. We believe that many of the DOS peaks found so far in the model and first-principles band structure calculations of the three dimensional systems should be of the present classes. Later, we demonstrate a simple model system that exhibit the DOS peaks due to the saddle loop and extremum shell and analyze the first-principle band structure of a realistic system, superconducting H$_3$S under extreme compression,~\\cite{Eremets} from the viewpoint of the present theory.\n\n\n\n\n\n\\section{Tight binding model}\nIn the previous section, we have discussed the possible DOS peaks in three dimensions due to formation of the saddle loops and extremum shell. This mechanism is apparently so general that one would expect it is ubiquitous, though the requirements for its occurrence seems difficult to realize in terms of the Hamiltonian. The essential factor is that the perturbation acts strongly only at the close vicinity of the original saddle point or extremum in the ${\\bf k}$ space. Another factor in the case of the saddle loop is that the band dispersion in the other direction must remain parabolic with definite sign of the effective mass. Here we exemplify a simple model system where those requirements are satisfied.\n\n\\begin{figure}[t]\n \\begin{center}\n \\includegraphics[scale=0.30]{cos1k_plus_cos2k_all_190616.pdf}\n \\caption{Modification of the maximum (at $k=\\pi$) to minimum by a rapidly varying Fourier component. $A$ denotes a parameter.}\n \\label{fig:cos-cos2}\n \\end{center}\n\\end{figure}\n\n\n\\subsection{Promising perturbations}\nBefore going to the specific model, we argue possible perturbations in terms of the tight-binding model that can yield the saddle loops and extremum shells. The tight-binding model on a periodic lattice is generally written as\n\\begin{eqnarray}\n\\mathcal{H}\n=\\sum_{{\\bf R}, {\\bf R}'}\\sum_{ij} t_{ij}({\\bf R}-{\\bf R}')c_{i}^{\\dagger}({\\bf R})c_{j}({\\bf R}')\n.\n\\end{eqnarray}\nHere, ${\\bf R}$ and ${\\bf R}'$ denote the lattice vectors and $i$ and $j$ are the index of orbitals. $c^{\\dagger}_{i}({\\bf R})$ ($c_{i}({\\bf R})$) is the creation (annihilation) operator of electrons of state $i$ at site ${\\bf R}$. $t_{ij}({\\bf R})$ is the hopping between states $i$ and $j$ across the unit cells connected by vector ${\\bf R}$. The hopping with ${\\bf R}$ yields the ${\\bf k}$ dependence of the energy eigenvalues $\\varepsilon_{n}({\\bf k})$ ($n:$ band index) with the form $\\sim {\\rm cos}({\\bf k}\\cdot {\\bf R})$; the near-site hopping (=small ${\\bf R}$) thus gives slowly varying component in the ${\\bf k}$ space. If the band structure near the saddle or maximum point is dominated by such slowly varying components, it is modified to the minimum by adding more rapidly oscillating contributions which originates from the hopping with larger ${\\bf R}$. In Fig.~\\ref{fig:cos-cos2}, we depict an example: In one direction, the maximum of the dispersion $\\sim {\\rm cos}k$ at $k=\\pi$ is modified to the minimum by adding the component with the ${\\rm cos}2k$ form. Note that in this case the relative amount of the ${\\rm cos}2k$ component represented by parameter $A$ in Fig.~\\ref{fig:cos-cos2} must be larger than some threshold value in order to change the maximum at $k=\\pi$ to local minimum. \n\nIn multiband systems, there can also be a possibility that the composition of the states labeled by $n{\\bf k}$ varies rapidly with respect to ${\\bf k}$, as observed in the topological insulator.~\\cite{RevModPhys.83.1057} In such a situation, one could render the target saddle or maximum to minimum point by exerting any orbital-selective perturbation.\n\nThe farther neighbor hopping is especially promising as the candidate perturbation in that it is generally expected to become appreciable in systems under strong compression. Below, we first demonstrate the formation of the DOS peak in a tight-binding model due to the farther neighbor hopping. We later discuss the origin of the DOS peak at the Fermi level in the compressed H$_{3}$S, where we find that the perturbation responsible for the peak formation is the farther neighbor orbital selective hopping.\n\n\\begin{figure}[h]\n \\begin{center}\n \\includegraphics[scale=0.50]{SC_tb_model_tp_bands_dos_sum_ed_191201.pdf}\n \\caption{(1) Simple cubic tight binding model with farther neighbor hopping. (b) The simple cubic Brillouin zone, where the ${\\bf k}$-point path for the band structure calculation is depicted. The point $X'$ is a symmetrically equivalent to $X$. (c) The calculated band structure and (b) the DOS of the model of the spectrum Eq.(\\ref{eq:ek-sc-tp}). Here and hereafter $t$ is fixed to 1~eV.}\n \\label{fig:sc-band}\n \\end{center}\n\\end{figure}\n\n\\begin{figure*}[t!]\n \\begin{center}\n \\includegraphics[scale=0.32]{SC_tb_model_tp_bands_dos_sum_close_190829.pdf}\n \\caption{Close up views of the band structure and DOS around the DOS peaks for (a) $t'=0.2t$, (b) $0.3t$ and (c) $0.4t$. The dashed lines are guides to the eye. The small difference of the DOS values from those in Fig.~\\ref{fig:sc-band} is due to the smearing scheme used for the DOS calculation.}\n \\label{fig:sc-band-close}\n \\end{center}\n\\end{figure*}\n\n\\begin{figure*}[t!]\n \\begin{center}\n \\includegraphics[scale=0.15]{SC_Lifshitz_transitions_tp04_190829.pdf}\n \\caption{Snapshots of the higher order Lifshitz transitions for the equal energy surfaces with $t'=0.4t$. The simple cubic Brillouin zones are depicted as white frames. Labels (1)--(8) indicates the energy levels at which the equal energy surfaces were calculated. (upper) The surface pair formation transition around the extremum shell indicated by dashed lines. (lower) The edge-pair switching transition around the saddle loop indicated by dashed lines.}\n \\label{fig:sc-LTs}\n \\end{center}\n\\end{figure*}\n\n\\subsection{Simple cubic lattice model}\nWe consider the single orbital nearest neighbor tight-binding model on the simple cubic lattice [Fig.~\\ref{fig:sc-band}(a)]\n\\begin{eqnarray}\n\\mathcal{H}\n=-t\\sum_{\\langle {\\bf R}, {\\bf R}' \\rangle_{\\rm sc}} c^{\\dagger}({\\bf R})c({\\bf R}')\n,\n\\end{eqnarray}\nwhere $\\langle {\\bf R}, {\\bf R}' \\rangle_{\\rm sc}$ denotes that ${\\bf R}$ and ${\\bf R}'$ are the nearest neighbor sites in the simple cubic lattice. The resulting energy eigenvalue of the total Hamiltonian reads\n\\begin{eqnarray}\n\\varepsilon({\\bf k})\n&=&\n-2t({\\rm cos}k_{x}+{\\rm cos}k_{y}+{\\rm cos}k_{z})\n. \n\\end{eqnarray}\nIn the simple cubic Brillouin zone [Fig.~\\ref{fig:sc-band}(b)], the isolated vHSs of types $A_{\\rm 3D}$, $B_{\\rm 3D}$, $B'_{\\rm 3D}$ and $A'_{\\rm 3D}$ are located at the $\\Gamma$, $X$, $M$ and $R$ points, respectively. Therefore, the DOS does not exhibit divergence [solid lines in Fig.~\\ref{fig:sc-band}(c)(d)].\n\nIn order to induce the band deformation around the critical points at the $M$ and $R$ points, we here introduce the farther neighbor hopping $t({\\bf R}\\!\\!-\\!\\!{\\bf R}' \\!\\!=\\!\\!2{\\bf a}_{i})=-t'$ with ${\\bf a}_{i} (i=x,y,z)$ being the primitive lattice vectors. The energy eigenvalue is then modified to\n\\begin{eqnarray}\n\\varepsilon({\\bf k})\n&=&\n-2t({\\rm cos}k_{x}+{\\rm cos}k_{y}+{\\rm cos}k_{z})\n\\nonumber \\\\\n&&-2t'({\\rm cos}2k_{x}+{\\rm cos}2k_{y}+{\\rm cos}2k_{z})\n. \n\\label{eq:ek-sc-tp}\n\\end{eqnarray}\n\nThe necessary condition for turning the saddle point at $M$ to minimum is that the maxima in the band dispersions along the $X$-$M$ and $\\Gamma$-$M$ are located in the middle of the paths, which corresponds to $t' > t\/4$. This is also the necessary condition for turning the maximum at $R$ to minimum. We therefore examined the band structure and DOS with $t'$ changed through $t\/4$. In Fig.~\\ref{fig:sc-band}(c), we indeed find maxima with $t' =0.3t$ and $0.4t$ at fractional points near $M$ and $R$, and concomitantly the peaks in the DOS are formed. In the present case, the critical points responsible for the shoulders of the DOS peaks are located exactly on the regular Brillouin zone paths (Fig.~\\ref{fig:sc-band-close}). It must be, however, noted that those points are just cross sections of the high dimensional saddle loops and extremum shell, and all the states in the vicinity of those high dimensional structures contribute to the peaks. We also note that the critical points at the $M$ and $R$ points are, as displayed in Fig.~\\ref{fig:sc-band-close} (b) and (c), no longer responsible for any appreciable structures in the DOS peaks.\n\nTo verify the saddle loop and extremum shell, we also examined the evolution of the equal energy surfaces through the DOS peaks with $t'=0.4t$. The surface pair formation at the extremum shell is ideally observed when the energy level is reduced through the corresponding DOS singularity. The upper panels of Fig.~\\ref{fig:sc-LTs} show the change of the equal energy surfaces. The pocket formation is first observed at the maximum between the $\\Gamma$-$R$ path, the surfaces gradually evolves and finally bifurcates into two nested surfaces sandwiching the deformed extremum shell (indicated by bold dashed line) when the energy level is below the lower DOS shoulder. Comparing the end points [(1) to (4)], the whole change is interpreted as the surface pair formation. Similarly, the edge pair switching transition should be observed when the energy level passes down the energy position of the ideal saddle loop. According to the lower panels, the transition starts with the neck formation at a point on the $\\Gamma$-$M$ line and the resulting hole gradually evolves into the void running along the deformed saddle loop (indicated by bold dashed line), and thus the edge pair switching transition is completed. The higher-order Lifshitz transitions occurring on the single band within the nonzero energy range thus indicates the saddle loop and extremum shell.\n\n\n\n\n\n\n\\section{DOS peak in the cubic H$_{3}$S superconductor}\nIn this section, we study the electronic structure of the recently discovered high-temperature superconductor H$_{3}$S~(Refs.~\\onlinecite{Eremets} and \\onlinecite{Shimizu}). After its theoretical and experimental discoveries, first principle calculations have revealed its band structure.\\cite{Duan2014, Bernstein-Mazin-PRB2015,Papacon-Pickett-ele-str-PRB2015,Flores-Livas2016,Quan-Pickett-vHs-PRB2016, Fan-H3AB-JPCM2016} Its particularly interesting feature is the peak in the DOS at the Fermi level. According to the Bardeen-Cooper-Schrieffer theory of the phonon-mediated conventional superconductors,\\cite{BCS1, BCS2} an approximate formula of the superconducting transition temperature $T_{\\rm c}$ is written as\n\\begin{eqnarray}\nT_{\\rm c}\n\\propto\n\\omega\n{\\rm exp}[-\\frac{1}{\\lambda}]\n,\n\\end{eqnarray}\nwhere $\\omega$ denotes the frequency of the phonon mode mediating the electron pairing, and $\\lambda$ is the dimensionless parameter representing the total pairing strength. Because $\\lambda$ is proportional to the DOS at the Fermi level, the peaked DOS is thought to be a crucial factor for the high $T_{\\rm c}$. In fact, several groups have reported that the experimentally observed $T_{\\rm c}$ is accurately reproduced with the Eliashberg theory~\\cite{Duan2014, Errea-PRL2015, Sano-vHS-PRB2016, Errea-Nature2016} and density functional theory for superconductors,\\cite{Flores-Livas2016, Akashi-PRB2015, Akashi-Magneli-PRL2016} where its $T_{\\rm c}$ is boosted by the large DOS. Although the conventional phonon mechanism thus explains the high $T_{\\rm c}$, potential impacts of the DOS shoulder as the vHS and small electronic energy scale indicated by the DOS peak width have also attracted attention for possible unconventional mechanism.\\cite{Bianconi-NSM, Bianconi-EPL, Bianconi-scirep} In either direction, the presence of the DOS peak at the Fermi level is a pivotal basis of the theoretical discussions. Nevertheless, its existence has yet been verified with direct experimental observations because of the difficulty of the high pressure experiments. It is hence important to see if there is any general mechanism behind the formation of the DOS peak that makes it persist against subtle changes of the structure and calculation conditions, or it is a consequence of the accidentally fine-tuned crystal structure.\n\nThe ``minimal\" modeling of the electronic states in H$_{3}$S has been first proposed by Bernstein and coworkers.\\cite{Bernstein-Mazin-PRB2015} They have pointed out that the inversion of the on-site energies between the hydrogen-1$s$ and sulfur 3$p$ is important and proposed a simple tight binding model with the sulfur 3$s$, 3$p$ and hydrogen 1$s$ orbitals with only the nearest neighbor hopping parameters. Later, Quan and Pickett~\\cite{Quan-Pickett-vHs-PRB2016} have published a set of tight binding parameters with farther neighbor hopping obtained by the construction of the Wannier functions.\\cite{MLWF1, MLWF2} Ortenzi and coworkers, on the other hand, have argued that important hopping parameters are missing in their models and proposed an alternative model~\\cite{Ortenzi-TBmodel-PRB2016} constructed by the recipe of Slater and Coster.\\cite{Slater-Coster} They focused on the critical point in the middle of the $N$-$X$ path in the base centered cubic (BCC) Brillouin zone and imposed a criterion that the energy level of this point, as well as the topology of the Fermi surfaces, are reproduced, but their resulting DOS peak is rather broadened compared with the first-principle calculation. Souza and Marsiglio~\\cite{Souza-Marsiglio-IJMPB,Souza-Marsiglio-PRB} have took an opposite approach; they studied the features of the band structure intrinsic to the Bravais lattices. They recalled the finding by Jelitto~\\cite{Jelitto1969} that the single $s$ orbital nearest neighbor tight binding model on the BCC lattice yield the divergence of the DOS, which is due to the completely flat band dispersion on planar manifolds (boundaries of the reduced simple cubic Brillouin zone) induced by the interference of the Bloch phase of the wave function.\\cite{Akashi-interfere-PRB2017} However, its relation to the DOS peak of H$_{3}$S is not apparent as such flat dispersion is yet found.\n\nFor the modeling of the system, it is important to determine which of the features of the first-principle band structure are essential. We concentrate on its DOS peak; we first examine the mechanism how the DOS peak is formed and later explore the simple model that at least retains the DOS peak formed by the common mechanism. Through the previous close analysis, the DOS peak has been found to have two adjacent shoulder structures and the locations of the corresponding critical points in the ${\\bf k}$ space have also been specified.~\\cite{Quan-Pickett-vHs-PRB2016} We first see that the actual feature responsible for the whole peak is the above mentioned saddle loop and the previously found critical points are just its cross sections.\n\n\\begin{figure}[t]\n \\begin{center}\n \\includegraphics[scale=0.50]{H3S_struct_band_dos_sum_190831.pdf}\n \\caption{(a) Crystal structure of the cubic H$_{3}$S, where large and small balls represent sulfur and hydrogen atoms, respectively. The primitive and conventional unit cells are indicated by thin and bold lines, respectively. (b) The Brillouin zones corresponding to the respective definitions of the unit cell. (c) The electronic (left) band structures and (right) DOS spectra, where dashed line indicates the Fermi level. Thin and bold lines are the results of the first principle calculation and effective tight binding model derived from the Wannier functions, respectively. The dotted line is the upper bound of the frozen window for the construction of the Wannier functions (see main text).}\n \\label{fig:H3S-str-band}\n \\end{center}\n\\end{figure}\n\n\\begin{figure*}[t]\n \\begin{center}\n \\includegraphics[scale=0.30]{H3S_cubic_band_special_paths_190920.pdf}\n \\caption{Band dispersions along unconventional paths. (a) The ${\\bf k}$-point paths passing the target point $P$ indicated by dark star and (b) the corresponding band dispersions. (c) The ${\\bf k}$-point paths for location of the saddle loop and (d) (left) the corresponding band dispersions. The light stars indicate the minimum and maximum along the line of saddle points, whereas squares represent the midpoints of the saddle line. (right) Total DOS $D(E)$ from first principles, from the Wannier model and partial DOS $D(E)-D_{\\rm SL}(E)$ from the Wannier model. The region of integration for $D_{\\rm SL}(E)$ ($\\Omega_{\\rm SL}$) [shown by shaded volume in panels (a) and (c)] is defined by $\\{{\\bf k} | 0.20 < \\sqrt{(k_{x}-0.5)^2 + (k_{y}-0.5)^2} < 0.35; |k_{z}| < 0.15\\}$ with ${\\bf k}$ in the unit of $2\\pi\/a$. The horizontal dot-dashed line indicates the upper bound of the frozen window same as in Fig.~\\ref{fig:H3S-str-band}(c). (e) Schematic picture of the saddle loops, where the anomaly is concentrated.}\n \\label{fig:H3S-special-path}\n \\end{center}\n\\end{figure*}\n\n\n\\subsection{First principle band structure}\nWe calculated the first principle electronic band structure of the cubic H$_{3}$S with the plane wave pseudopotential method as implemented in {\\sc quantum espresso}.\\cite{QE} The Perdew-Burke-Ernzerhof generalized gradient approximation~\\cite{GGAPBE} was adopted for the exchange correlation functional. The pseudopotentials were made with the Troullier-Martins scheme.\\cite{TM, FHI} The plane-wave cutoff for the wave function was set to 80~Ry. The cubic lattice parameter was set to 5.6367~bohr, which is the optimized value under 200GPa. We derived the Wannier model using {\\sc wannier90}.\\cite{wannier90} The sulfur-$s$, $p$ and hydrogen $s$ orbitals were adopted as the initial guess. We imposed the frozen window constraint,\\cite{MLWF2} within which the Hilbert space is assured to be spanned by the resulting Wannier orbitals. Very wide energy window up to Fermi level plus $\\gtrsim$ 40 eV, within which the states are used for constructing the Wannier orbitals, was set for accurate reproduction of the first principle band structure. We did not minimize the gauge-dependent part of the Wannier spreads to get the projected Wannier orbitals rather than the maximally localized ones so that subtle possible shift of the Wannier centers from the atomic sites does not occur.~\\cite{PhysRevB.74.195118}\n\nWe show the calculated band structure in Fig.~\\ref{fig:H3S-str-band} (c). Here we took the non-primitive simple cubic cell~\\cite{Bianconi-scirep} [panel (a)] for convenience in the later discussions, by which the Brillouin zone is folded from the BCC one as shown in panel (b). Our calculated band structure (thin line) reproduces the features shown in the previous studies, especially the sharp peak in the DOS at the Fermi level.\\cite{Quan-Pickett-vHs-PRB2016} The apparent local maxima seen in the middle of the $X$--$M$ and $M$--$\\Gamma$ paths have been related to the DOS peak in previous discussions.\\cite{Bianconi-scirep} We name the maximum along the $X$--$M$ path $P$ for later analysis. We further scrutinize the relation between the details of the band structure and DOS peak. \n\n\n\\begin{figure}[t]\n \\begin{center}\n \\includegraphics[scale=0.50]{H3S_cubic_die_cut_190903.pdf}\n \\caption{Edge switching transition in the equal energy surfaces with variable energy levels through the DOS peak in H$_{3}$S. The bold and thin dashed lines indicate the transition lines which are and are not responsible for the DOS peaks, respectively.}\n \\label{fig:H3S-edge-switch}\n \\end{center}\n\\end{figure}\n\n\nWe find that the band dispersion along $k_{z}$ through $P$ is convex as shown in Fig.~\\ref{fig:H3S-special-path} (b). To see the band dispersion in the direction normal to the BZ boundary, we also calculated the band dispersions along the paths depicted in Fig.~\\ref{fig:H3S-special-path} (b). Departing from the boundary, the degenerate bands split, and one of the maxima are aligned in a curly line indicated by stars. Those points, around which the band is concave in the path directions and convex in the $k_{z}$ direction, form the saddle loop encircling the $M$ point. Note that there are in total three symmetrically equivalent saddle loops interchanged by the three fold rotation in the (1 1 1) axis. We find that the small dispersion along the loop corresponds to the width of the DOS peak [panel (d)], which suggests that the DOS peak is dominated by the saddle loops. Furthermore, we calculated the local DOS originating from the saddle loops by\n\\begin{eqnarray}\nD_{\\rm SL}(E)=N_{\\rm sym}\\sum_{n{\\bf k}\\in \\Omega_{\\rm SL}}\\delta(E-\\varepsilon_{n{\\bf k}})\n,\n\\label{eq:D-SL}\n\\end{eqnarray}\nwhere the volume of integration $\\Omega_{\\rm SL}$ is limited to a tiny shaded region depicted in Fig.~\\ref{fig:H3S-special-path}(a)(c) and $N_{\\rm sym}$(=12) is the symmetry factor. By subtracting $D_{\\rm SL}(E)$ from the total DOS $D(E)$, the peak structure completely vanished [Fig.~\\ref{fig:H3S-special-path}(d)]. This result directly evidences that the DOS peak structure is wholly due to the saddle loop. \n\nFinally, we also calculated the equal energy surfaces with different energy levels through the DOS peak position. As indicated by dashed lines in Fig.~\\ref{fig:H3S-edge-switch}, we observed two edge pair switching transitions on the paths around the $M$ point. According to the above analysis, the region where the inner transition occurs is responsible for the DOS peak. The position of the higher order Lifshitz transition through the energy range of the target DOS peak thus indicates the ``hot spot\" corresponding to the peak.\n\nWe thus establish the simplistic view on the complicated electronic structure of this system as depicted in Fig.~\\ref{fig:H3S-special-path} (e). Its anomalous aspects, peaked concentration of the DOS and possible competition of the electronic and phononic energy scales, originates from tiny regions in the ${\\bf k}$ space around the $M$ points where the saddle loops are located. This fact should encourage further scrutiny on the electronic states of the saddle loop regions and their interplay with phonons, not only the multiple hole pockets around the $\\Gamma$ point,~\\cite{PhysRevB.92.205125,Ghosh-H3S-ph-break} though it is out of the scope of the present work.\n \n\\begin{figure}[t]\n \\begin{center}\n \\includegraphics[scale=0.50]{H3S_bipartition_model_191201.pdf}\n \\caption{(a) Partitioning of the crystal structure of H$_{3}$S into two identical simple cubic lattices. (b) The simple cubic (ReO$_{3}$) H$_{3}$S structure. (c) The tight binding hopping parameters of the minimal model. The symmetrically equivalent parameters are not shown. Note that the actual form of the Wannier orbitals are depicted in Fig.~\\ref{fig:Wannier}.}\n \\label{fig:bipartite}\n \\end{center}\n\\end{figure}\n\n\\subsection{Wannier model and its analysis}\nThe present result motivates us to seek for a simple model that reproduces the saddle loop as the minimal model for the DOS peaking in H$_{3}$S. Starting from the seven orbital Wannier model (sulfur-$s$, -$p$ and hydrogen $s\\times 3$) that perfectly reproduces the {\\it ab initio} band structure [Fig.~\\ref{fig:H3S-str-band}(c); see Table~\\ref{tab:tbparam-app} for the original hopping parameters], we conducted a thorough examination on how the band structure and DOS change by omitting any hopping parameters. The crystal structure of the cubic H$_{3}$S is bipartite, in that it is formed by identical simple cubic sublattices shifted from each other by $(a\/2, a\/2, a\/2)$ [Fig.~\\ref{fig:bipartite}(a)]. In preceding analyses,\\cite{Heil-Boeri2015, Quan-Pickett-vHs-PRB2016} it has been pointed out that the electronic density contributed to by the states near the DOS peak is spatially concentrated along the individual simple cubic frames. Inspired by this fact, we calculated the electronic structure of a ``bi-partitioned\" simple cubic lattice (known as the ReO$_{3}$ structure) by neglecting all the hopping parameters across the sublattices. Interestingly, we find that the saddle loop structure around the $M$ point, as well as the DOS peak emerging from this, are retained with this simple cubic lattice model as shown in Fig. ~\\ref{fig:Spectr}(c). The local maxima along the $X$-$M$ and $M$-$\\Gamma$ paths indicated by arrows are the cross sections of the saddle loop around the $M$ point. \n\n\\begin{figure*}[t]\n \\begin{center}\n \\includegraphics[scale=0.45]{H3S_Wannier_spectra_Ss_Hzs_190903.jpg}\n \\caption{Electronic band structure of the Wannier models with different levels of approximation. (left) Band structures and DOS spectra derived from (a) the Wannier model that shows the perfect agreement with the first principle calculation for the low energy states [See Fig.~\\ref{fig:H3S-str-band}(c)], (c) the Wannier model where the hopping between the two bipartite lattices are ignored, (e) the bipartite simple cubic Wannier model with only the nearest neighbor hopping parameters [$t^{(1)}_{{\\rm S}s-{\\rm H}s}$ and $t^{(1)}_{{\\rm S}p\\sigma-{\\rm H}s}$ in Fig.~\\ref{fig:bipartite}], and (d) the ``minimum\" model where the farther neighbor hopping parameters [$t^{(4)}_{{\\rm S}s-{\\rm S}s}$ and $t^{(4)}_{{\\rm H}s-({\\rm H})-{\\rm H}s}$ in Fig.~\\ref{fig:bipartite}] are added on top of (c). (b) (d) (f) (h) The corresponding spectral functions. }\n \\label{fig:Spectr}\n \\end{center}\n\\end{figure*}\n\nThe above result suggests an interesting possibility: the saddle loop structure is persistent against the doubling of the simple cubic cell, and then the seven-orbital simple cubic lattice model captures the mechanism of the DOS peak formation. The former point can be supported theoretically, by considering the interference effect of the Bloch functions.\\cite{Akashi-interfere-PRB2017} Consider any Bloch state composed of orbitals on a single sublattice $|\\Phi_{s; n{\\bf k}}\\rangle$, where $s(=1,2)$ denotes the index of the sublattice. The Wannier Hamiltonian $\\mathcal{H}$ is block diagonalized for each ${\\bf k}$. Due to the commensurate shift between the sublattices and mirror symmetries, the following formula holds\n\\begin{eqnarray}\n\\langle \\Phi_{1; n{\\bf k}}|\\mathcal{H}|\\Phi_{2; n{\\bf k}}\\rangle=0\n\\ \\ ({\\bf k}\\in {\\rm boundary\\ of\\ the\\ SC\\ BZ}).\n\\nonumber \\\\\n\\label{eq:sub-decouple}\n\\end{eqnarray}\nThis implies that the band structure at the BZ boundary is less affected by the hopping across the sublattices, compared with the ${\\bf k}$-point regions far from the boundary. With the saddle points along the $X$-$M$ and $X'$-$M$ paths [Fig.~\\ref{fig:sc-band}(b)] retained against the intersublattice coupling, from the physical requirement that the variation of the band dispersion is not drastic in the ${\\bf k}$ space, the saddle loop connecting those points must be persistent as well. We hence assert that the decoupled simple cubic lattice model is the minimal one.\n\nTo seek the origin of the saddle loop, we next calculated the electronic structure of the single sublattice model with only the nearest neighbor hopping parameters retained [Fig.~\\ref{fig:Spectr}(e)]. The saddle loop structure then converged to the saddle {\\it point} at the $M$ point as indicated by an arrow. The corresponding DOS peak was smeared. We found that the saddle loop is recovered by considering farther neighbor hopping parameters $t^{(4)}_{{\\rm S}s-{\\rm S}s}$ and $t^{(4)}_{{\\rm H}s-({\\rm H})-{\\rm H}s}$ [Fig.~\\ref{fig:bipartite}(c)]. The superscript $(4)$ implies that the hopping connects the fourth nearest neighbor sites in the original crystal structure. Notably, those parameters are positive and yield contributions to the energy spectrum in the $+[{\\rm cos}({\\bf a}_{1}\\cdot {\\bf k}) +{\\rm cos}({\\bf a}_{2}\\cdot {\\bf k})]$ form, with which the energy eigenvalue at $M$ is reduced (pushed). The orbital resolved spectral function (Ref.~\\onlinecite{Fetter-Walecka}; see Appendix~\\ref{app:spectr} for its formal definition) displayed in Fig.~\\ref{fig:Spectr}(f) shows that the electronic state at the saddle point $M$ is mainly composed of the sulfur $s$ and hydrogen H$_{z}$ $s$ orbitals [Fig.~\\ref{fig:Spectr}(f)]. Introduction of $t^{(4)}_{{\\rm S}s-{\\rm S}s}$ and $t^{(4)}_{{\\rm H}s-({\\rm H})-{\\rm H}s}$ therefore substantially reduces the energy eigenvalue at $M$, forming the saddle loop around it.\n\nThe origin of the large farther neighbor hopping with positive sign can be suggested from the Wannier orbitals displayed in Fig.~\\ref{fig:Wannier}. Usually, the hopping between two orbitals without phase shift is negative. In the present case, however, both the sulfur $s$-like and hydrogen $s$-like orbitals show the sign inversion of the wave functions and have tails in a length scale comparable to half of the lattice parameter. These features cause the relatively large farther neighbor hopping with the sign inversion. \n\n\n\\begin{figure}[t]\n \\begin{center}\n \\includegraphics[scale=0.50]{Wannier_funcs_190831.pdf}\n \\caption{Isosurfaces of the calculated Wannier functions, where signs of the functions are distinguished by colors; (a) Sulfur $s$-like orbital, (b) sulfur $p$-like orbital and (c) hydrogen $s$-like orbital.}\n \\label{fig:Wannier}\n \\end{center}\n\\end{figure}\n\n\\begin{table*}[t!]\n\\caption{Tight binding parameters (eV) of the minimal model that explains the saddle loop around the $M$ point and related DOS peak. The onsite energies are calculated from the level of sulfur $s$-like orbital. The sign $\\pm$ indicates the arbitrariness in the definition of the $s$-$p\\sigma$ type hopping. See Fig.~\\ref{fig:bipartite} (c) for the definitions of the hopping parameters.}\n\\begin{tabular}{|c|cccc|cc|}\n \\hline\n Pressure (GPa) &120 &160& 200&240&\\multicolumn{2}{c|}{120} \\\\ \\hline\n Distortion &\\multicolumn{4}{c|}{No}& Molecule & Lattice \\\\ \\hline \n Lattice parameter (a.u.) &5.8990& 5.7542&5.6367&5.5375 &\\multicolumn{2}{c|}{(see text)}\\\\ \\hline\n Onsite &&&& &&\\\\\n Sulfur $s$ like &---&---& --- & ---&---& ---\\\\\n Sulfur $p$ like &8.16&7.82& 7.49 & 7.20 &7.98&8.32\\\\\n Hydrogen $s$ like &6.42&5.77& 5.18 & 4.62 &6.27&6.50\\\\ \\hline\n Hopping & &&&&&\\\\\n $t^{(1)}_{{\\rm S}s-{\\rm H}s}$ &-3.84&-4.09& -4.30 & -4.48 &-4.69, -2.98&-3.84\\\\\n $t^{(1)}_{{\\rm S}p\\sigma-{\\rm H}s}$ &$\\pm$5.04&$\\pm$5.35& $\\pm$5.62 & $\\pm$5.85 &$\\pm$5.69, $\\pm$4.31&$\\pm$5.07\\\\\n $t^{(4)}_{{\\rm S}s-{\\rm S}s}$ &+0.81&+0.91& +0.99 & +1.07 &+0.78&+0.81 \\\\\n $t^{(4)}_{{\\rm H}s-({\\rm H})-{\\rm H}s}$ &+0.43&+0.46& +0.49 & +0.51 &+0.45&+0.46\\\\ \\hline\n\\end{tabular}\n\\label{tab:tbparam}\n\\end{table*}\n\n\\begin{figure*}[t]\n \\begin{center}\n \\includegraphics[scale=0.50]{H3S_cubic_120GPa_to_240GPa_SC_bands_dos_zoom_ed_191129.pdf}\n \\caption{Pressure dependence of the first principles (a) band structure and (b) DOS for the cubic H$_{3}$S. The scale of the {\\bf k}-point axis, which actually depends on the lattice parameter, is fixed for comparison.}\n \\label{fig:pressure-band-dos}\n \\end{center}\n\\end{figure*}\n\n\n\\subsection{Summary: what makes the DOS peaked?}\nWe have thus reached the minimal model of H$_{3}$S with seven (or six, if we regard the onsite energy of the Sulfur-$s$ like orbital arbitrary) parameters (Table~\\ref{tab:tbparam}) that demonstrates the DOS peak formation, The basis set is composed of the sulfur $s$, $p$ and hydrogen $s$ orbitals with structural deformation in the cubic ReO$_{3}$ configuration. The nearest neighbor hopping model gives rise to a band with small dispersion, on which a saddle point is located at $M$. The further nearest neighbor hopping modifies this saddle point into the local minimum and the saddle loop is formed around it. The doubling of the lattice substantially modifies the whole electronic structure, but its important features---the maxima along the $X$-$M$ and $X'$-$M$ paths---are protected due to the interference of the Bloch states, which forces the saddle loop through those points to persist. The relevance of the present model to the actual H$_{3}$S is validated by analysis of the spectral functions [Figs.~\\ref{fig:Spectr} and \\ref{fig:Spectr-app}]. The {\\bf k} point dependence of the orbital character around the saddle loop is surprisingly consistent among the four models; the original model that perfectly reproduces the {\\it ab initio} band structure, bipartite one, bipartite nearest neighbor one and the final minimal one. \n\nIn the above description we ignored the hopping across the different sublattices in order to highlight the mechanism that ``pushes\" the saddle critical point. We also note a quantitative effect of some intersublattice hopping parameters. We found that concurrent inclusion of $t^{(1)}_{{\\rm H}s-{\\rm H}s}$ and $t^{(2)}_{{\\rm S}p\\sigma-{\\rm H}s}$ (Table~\\ref{tab:tbparam-app}) in combination with the farther neighbor intrasublattice hopping parameters $t^{(4)}_{{\\rm S}s-{\\rm S}s}$ and $t^{(4)}_{{\\rm H}s-({\\rm H})-{\\rm H}s}$ have also effect of enhancing the ``M\"-shape dispersion of the relevant band along the $X-M-\\Gamma$ path in Fig.~\\ref{fig:Spectr}(c), though its mechanism is not obvious. The original first principles band structure is then reproduced better. Note that those parameters do not solely yield the saddle loop, whereby we conclude that their effect is secondary compared with the farther neighbor hopping. The detail on this point is summarized in Appendix~\\ref{app:intersub}.\n\n\n\\begin{figure*}[t]\n \\begin{center}\n \\includegraphics[scale=0.40]{H3S_cubic_120GPa_distort_bands_dos_zoom_ed_191129.pdf}\n \\caption{Distortion dependence of the band structure and DOS at 120GPa. Although the reciprocal vectors vary by the distortions, we plot the bands in the same scale by defining the special points in terms of the reciprocal vector coordinates. (a)(b) Views of the distortions, where the (1 1 1) axis is indicated by solid line. The degrees of the distortions are exaggerated. (c) and (d) are the first principles band structures and DOS with and without the distortions, respectively. (e) The saddle loop contribution $D_{\\rm SL}(E)$ defined in Eq.~(\\ref{eq:D-SL}) and (f) the remnant $D(E)-D_{\\rm SL}(E)$, calculated with the Wannier models.}\n \\label{fig:distort-band-dos}\n \\end{center}\n\\end{figure*}\n\n\nFinally, we try to reconcile the previous representative proposals of the tight binding modeling of this system [Refs.~\\onlinecite{Bernstein-Mazin-PRB2015,Quan-Pickett-vHs-PRB2016,Ortenzi-TBmodel-PRB2016}] with our ``minimal\" explanation in view of the reproduction of the saddle loops and DOS peaks. Among them, only Ref.~\\onlinecite{Ortenzi-TBmodel-PRB2016} employed a different method for calculating the parameters; prepare the model first and fit the parameters so that the band energies at selected {\\bf k} points agree with the first-principles values. In Table~\\ref{tab:tbparam-app}, we summarize the first, second, third and fourth neighbor hoppings connecting the sites departed by distances $a\/2$, $a\/\\sqrt{2}$, $\\sqrt{3}a\/2$ and $a$, respectively. The subscripts ${\\rm S}s$, ${\\rm S}p$, and ${\\rm H}s$ denote the Wannier orbitals displayed in Fig.~\\ref{fig:Wannier}. To our observation, the fifth neighbor hopping parameters (across the sites departed by $\\sqrt{5}a\/2$) also have appreciable effects on the whole band structure, which we do not append in the text, though. The full set of the parameters is available online in the output format of {\\sc Wannier90}.\\cite{Wannier-github}\n\n\n\n The inversion of the onsite energy levels of the sulfur $p$(-like) and hydrogen $s$(-like) states are wholly consistent among all the models. Also, the {\\it published} Wannier model parameters in Refs.\\onlinecite{Bernstein-Mazin-PRB2015,Quan-Pickett-vHs-PRB2016} are in fair agreement with ours, where the subtle differences are thought to originate from the gauge degree of freedom of the Wannier orbitals or setting of the energy window parameters. Bernstein and coworkers~\\cite{Bernstein-Mazin-PRB2015} published only the nearest neighbor hopping parameters; we found that those can yield DOS peak related to isolated saddle points at $M$ similarly to Fig.~\\ref{fig:Spectr} (e) but not the saddle loop. The list published by Quan and Pickett~\\cite{Quan-Pickett-vHs-PRB2016} includes the positive farther neighbor hopping parameters ($t^{(4)}_{{\\rm S}s-{\\rm S}s}$ and $t^{(4)}_{{\\rm H}s-({\\rm H})-{\\rm H}s}$ in our expression), from which we expect that the DOS peak as well as the saddle loops are sufficiently reproduced. They also reported that the DOS calculated with the single sublattice model with the nearest neighbor hopping is ``nothing like the original one\". This result probably corresponds to our Fig.~\\ref{fig:Spectr} (e); incorporating $t^{(4)}_{{\\rm S}s-{\\rm S}s}$ and $t^{(4)}_{{\\rm H}s-({\\rm H})-{\\rm H}s}$ into their sublattice model would reproduce the DOS peak and saddle loops. The hopping parameter between sulfur $s$-like and $p$-like states departed by $(a\/2,a\/2,a\/2)$ ($W_{sp}$ in Ref.~\\onlinecite{Ortenzi-TBmodel-PRB2016}; $t^{(3)}_{{\\rm S}s-{\\rm S}p\\sigma}$ in our expression) was, in our calculation, small, which is probably why Quan and Pickett did not mention this parameter. More on this parameter, we found that it has an effect of {\\it raising} the energy level at the midpoint of the $M$--$\\Gamma$ path, which corresponds to the maximum in the $N$--$X$ path of the BCC Brillouin zone. Ortenzi and coworkers~\\cite{Ortenzi-TBmodel-PRB2016} referred to this point, not the loop, for their parameter fitting and did not introduce the farther neighbor hoppings; therefore, the isolated vHS was reproduced, but the whole saddle loop structures was not, which is likely the reason of their DOS peak smearing. \n\n\n\\begin{table*}[t!]\n\\caption{Tight binding parameters of the present work and Refs.~\\onlinecite{Bernstein-Mazin-PRB2015,Quan-Pickett-vHs-PRB2016,Ortenzi-TBmodel-PRB2016}. The sign $\\pm$ indicates the arbitrariness in the definition of the $s$-$p\\sigma$ type hopping. Dagger ($\\dagger$) indicates the parameters included in our minimal model (Table~\\ref{tab:tbparam}).}\n\\begin{tabular}{c|ccc|c}\n\\hline\n Parameter (eV)& Present & Ref.~\\onlinecite{Bernstein-Mazin-PRB2015} & Ref.~\\onlinecite{Quan-Pickett-vHs-PRB2016} & Ref.~\\onlinecite{Ortenzi-TBmodel-PRB2016} \\\\ \\hline\n Lattice parameter (a.u.) &5.6367 & 5.6409& 5.6& 5.64 \\\\ \\hline\n Method &\\multicolumn{3}{c|}{Wannier orbital construction}& Analytical fit \\\\ \\hline\n Onsite& & & & \\\\\n Sulfur $s$ like & --- & --- & --- &---\\\\\n Sulfur $p$ like & 7.49 & 7.3 & 7.95& 11.38\\\\\n Hydrogen $s$ like & 5.18 & 3.6& 2.52 &10.29\\\\ \\hline\n 1st NN & & & &\\\\ \n $(\\dagger)$ $t^{(1)}_{{\\rm S}s-{\\rm H}s}$ & -4.30 & -4.2 &-4.37 &+2.81\\\\\n $(\\dagger)$ $t^{(1)}_{{\\rm S}p\\sigma-{\\rm H}s}$ & $\\pm$5.62 & -5.2&-5.42 &+4.65\\\\\n $t^{(1)}_{{\\rm H}s-{\\rm H}s}$ & -2.26 & -2.7 & -2.80 &-2.73\\\\ \\hline \n 2nd NN & & & &\\\\\n $t^{(2)}_{{\\rm S}s-{\\rm H}s}$ & -0.10 & (N. A.) & (N. A.)& --- \\\\\n $t^{(2)}_{{\\rm S}p\\sigma-{\\rm H}s}$ & $\\pm$1.00 &(N. A.) & +0.93& --- \\\\ \n $t^{(2)}_{{\\rm H}s-{\\rm H}s}$ & +0.08 & (N. A.) & (N. A.) & --- \\\\ \\hline\n 3rd NN & & & &\\\\\n $t^{(3)}_{{\\rm S}s-{\\rm S}s}$ & +0.06 & (N. A.)& +0.30&+2.31\\\\\n $t^{(3)}_{{\\rm S}s-{\\rm S}p \\sigma}$ & $\\pm$0.30 & (N. A.) & (N. A.) &+3.33 \\\\ \n $t^{(3)}_{{\\rm S}p-{\\rm S}p \\sigma}$ & +1.38 & (N. A.) &(N. A.) &+1.69\\\\ \n $t^{(3)}_{{\\rm S}p-{\\rm S}p \\pi}$ & +0.29 & (N. A.) &(N. A.) &-0.07\\\\ \n $\\frac{1}{4}t^{(3)}_{{\\rm S}p-{\\rm S}p \\sigma}+\\frac{3}{4}t^{(3)}_{{\\rm S}p-{\\rm S}p \\pi}$ & +0.56 & (N. A.) &+0.60 &+0.37\\\\ \n $t^{(3)}_{{\\rm H}s-{\\rm H}s}$ & +0.01 & (N. A.) &(N. A.) & ---\\\\ \\hline \n 4th NN & & & &\\\\\n $(\\dagger)$ $t^{(4)}_{{\\rm S}s-{\\rm S}s}$ & +0.99 & (N. A.) & +0.94&--- \\\\\n $t^{(4)}_{{\\rm S}s-{\\rm S}p \\sigma}$ & $\\pm$1.11 & (N. A.) & 1.29 &--- \\\\ \n $t^{(4)}_{{\\rm S}p-{\\rm S}p \\sigma}$ & -1.78 & (N. A.) & -1.83& ---\\\\ \n $t^{(4)}_{{\\rm S}p-{\\rm S}p \\pi}$ & -0.05 & (N. A.) & (N. A.)& ---\\\\ \n $(\\dagger)$ $t^{(4)}_{{\\rm H}s-({\\rm H})-{\\rm H}s}$ & +0.49 & (N. A.) & +0.55& ---\\\\ \n $t^{(4)}_{{\\rm H}s-({\\rm S})-{\\rm H}s}$ & -1.28 & (N. A.)& -1.14& ---\\\\ \\hline \n\\end{tabular}\n\\label{tab:tbparam-app}\n\\end{table*}\n\n\n\n\\subsection{Pressure dependence}\nThe two pivotal statements established through this section are that (i) the DOS peak at the Fermi level originates from the saddle loops around the $M$ and symmetrically related points and (ii) the single sublattice model explains how they are formed. We examine if those points are also valid against the change of the external pressure. We recalculated the electronic structures with the cubic lattice paremeters optimized at 120, 160, and 240 GPa and derived the Wannier model parameters with the same procedure. As a whole, the band structure changed only slightly. We show only the behavior in the energy range $\\pm 2$eV in Fig.~\\ref{fig:pressure-band-dos}. The peak height of the DOS were almost the same, whereas the Fermi level shifted a little toward the middle of the peak by increasing the pressure. This result is consistent with a previous calculation.~\\cite{Bianconi-NSM} We confirmed that the DOS peak commonly originates from the saddle loop region $\\Omega_{\\rm SL}$ defined in Fig.~\\ref{fig:H3S-special-path} (see Appendix~\\ref{app:D-SL-pressure} for specific data). The Wannier model parameters for different pressures are summarized in Table ~\\ref{tab:tbparam}. We observe that the hopping parameters monotonically increases in absolute value by pressure, which is simply due to the compression of the system. Importantly, the farther neighbor hopping parameters $t^{(4)}_{{\\rm S}s-{\\rm S}s}$ and $t^{(4)}_{{\\rm H}s-({\\rm H})-{\\rm H}s}$ remained positive and sizable regardless of the pressures. One is reminded that the modification of the saddle point into the loop generally requires that the perturbation is stronger than any threshold value (See Fig.~\\ref{fig:sc-band}). We confirmed that the single sublattice nearest neighbor hopping model yields only the saddle points and they are modified into the saddle loops by those farther hopping parameters. These results support that the validity of the above statements (i) and (ii) across the experimental pressure range.\n\n\n\nThe above calculations were performed with the $Im\\bar{3}m$ cubic structure, but in the experiments, the crystal structure is thought to suffer from distortions especially at relatively low pressures. For more relations to the experiments, we also calculated the band structure with two types of distortions at 120 GPa. One is the molecular ($R3m$) distortion;~\\cite{Duan2014} the three hydrogens are displaced from the center of the bonds so that the local H$_{3}$S molecules are formed [Fig.~\\ref{fig:distort-band-dos}(a)]. This structure is more stable than the high symmetry cubic structure in terms of the Born-Oppenheimer energy surface, whereas the high-symmetry structure may be more stable if the quantum nature of the hydrogen positions are considered.~\\cite{Errea-Nature2016} The other is the lattice distortion; In a previous experiment,~\\cite{Goncharov-PRB2017} a uniform stretch of the crystal was observed in the cubic $(1 1 1)$ direction [Fig.~\\ref{fig:distort-band-dos}(b)], though its origin is yet unclear.~\\cite{Bianco-PRB2018} To obtain the input crystal structures with these distortions, we performed the variable cell structure optimization for the molecular $R3m$ phase and simply changed the angles between the lattice vectors with the hydrogen positions kept in the middle of the bonds, respectively. In the former case, the lattice parameter was 5.9233 and the cosine of the angle between the lattice vectors, which is exactly zero for the ideal cubic structure, was $\\simeq$0.001. The hydrogen positions shifted from the bond centers by about 0.21 bohr in the bonding directions. In the latter, we set the angle cosine to $\\simeq$ 0.030 so that the degree of the distortion defined by Ref.~\\onlinecite{Goncharov-PRB2017} is 3\\%.\n\nWe show the first principles band structures and DOS in Fig.~\\ref{fig:distort-band-dos} (c) and (d). Although the whole band structure is again consistent, appreciable changes in the band structure and DOS were seen in the vicinity of the Fermi level. The molecular distortion has large effect in particular around the $\\Gamma$ point, with which some hole Fermi surfaces disappears. This change is largely responsible for the apparent reduction of the total DOS peak height from the cubic case seen in panel (d), which is also implied by the partial DOS from the region other than the saddle loop one in panel (f). On the other hand, the effect of the lattice distortion is significant on the saddle loop, as indicated at the local band maxima in the $X-M$ and $M-\\Gamma$ paths. It lifts the degeneracy at the former maximum and the resulting DOS peak splits into two [panel (d)]. The partial DOS analysis for the saddle loop and other regions [(e) and (f)] shows that the contribution from outside the saddle loop region $D(E)-D_{\\rm SL}(E)$ is almost invariant, indicating that the impact of this distortion is local in the ${\\bf k}$ space. The origin of this is thought to be the intersublattice coupling [Eq.~(\\ref{eq:sub-decouple})] switched on by the reduced symmetry. The DOS peak separation could be observable as a ``pseudogap\" structure through any probes that detects the electronic transition processes between the split bands, though its actual magnitude is expected to depend on the degree of the distortion. We note that, apart from the subtle changes of the DOS peak structure, the concentrating trend of the DOS near the Fermi level around the saddle loops is persistent against those distortions. The positive sizable farther neighbor hopping parameters were also obtained as summarized in Table~\\ref{tab:tbparam} as well, suggesting the robustness of the mechanism extracted by the sublattice model. These results again validate the statements (i) and (ii) above, as well as clarify the importance of the saddle loops in the experiments.\n\n\n\\section{Conclusions}\nIn this paper, we have characterized an archetype of the mechanisms that yield the DOS peak in three dimensional crystals. Modifying the second order saddle point (maximum) into local minimum by ``pushing\" the band hypersurface, the saddle loop (extremum shell) structure certainly appears and it induces concentration of the DOS at a tiny energy range. Being the high dimensional structure, the saddle loop and extremum shell are difficult to recognize from the standard scheme of visualizing the band structures along linear paths in the Brillouin zone. The existence of the critical points {\\it anywhere} (not necessarily at the special points) on these structures are assured by their closed nature, which appear in the DOS spectra as adjacent shoulders. The width of the DOS peak is determined by the band dispersion over the entire loop and shell. We have pointed out that the higher order Lifshitz transition through the energy levels across the DOS peak is a useful indicator to locate those structures in the ${\\bf k}$ space. Our theory gives us a deep insight into an important feature, the DOS peaks, of the electronic structure, as well as provides a simple guiding principle for design of electronic materials.\n\nWe have demonstrated how the electronic structure in the superconducting H$_{3}$S under pressure is understood with the present theory. The DOS peak in this system, which is thought to be the source of the high temperature superconductivity, is accompanied by several puzzling features such as critical points at apparently fractional low symmetry points. Through the close analysis, we have extracted the saddle loop structure and successfully derived a minimal model for the DOS peak formation. The cubic ReO$_{3}$ structure with sizable positive farther neighbor hopping results in the saddle loops around the $M$ and symmetrically equivalent points, which remains relevant against the lattice doubling. Although we do not go so far as to state that this is the minimal appropriate modeling for the superconductivity of this system, it is confirmed that one of its ingredients---formation of the DOS peak---is more than accidental and the same mechanism can be applicable to various isomorphous systems.\n\n\n\\begin{acknowledgments}\nThe author thanks to Peter Maksym for fruitful discussions. This work was supported by MEXT Element Strategy Initiative to Form Core Research Center in Japan. Some of the calculations were performed at the Supercomputer Center at the Institute for Solid State Physics in the University of Tokyo. {\\sc Vesta}~\\cite{Vesta} and {\\sc Fermisurfer}~\\cite{Fermisurfer} were used for visualization of the data\n\\end{acknowledgments}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{intro}\n\\renewcommand{\\baselinestretch}{1.3}\n\\normalsize\nThe term \\emph{integrated information} (denoted $\\phi$, for short) has been introduced by Giulio Tononi~\\cite{Tono-01-AIB, ToSp-03-BMC, Tono-04-BMC} to characterize the capacity of a system to integrate information acquired by its parts. Informally speaking, the integrated information owned by a system in a given state can be described as the information (in the Theory of Information sense) generated by a system in the transition from one given state to the next one as a consequence of the causal interaction of its parts above and beyond the sum of information generated independently by each of its parts.\n\nSuch a theory was first introduced as a linear model~\\cite{ToSE-94-PNAS, ToSE-96-PNAS, ToES-98-TCS, ToSp-03-BMC, Tono-04-BMC}, then reformulated as a discrete one~\\cite{BaTo-08-PCB, Tono-08-BB, BaTo-09-PCB} and was aimed at trying to formally capture what is consciousness in living beings~\\cite{ToEd-98-Sci, EdTo-00, Tono-08-BB}. Its description is not always clear from a mathematical point of view, and to best of our knowledge this is the first formal description where all steps of the model are presented in detail using the framework of probabilistic boolean networks.\n\nIn our presentation we also provides a more general formulation of the model, which can be used for analyzing the system at a generic time instant, and which does not require the assumption of uniformity of the probability distribution at the initial time instant.\n\nWe also formally prove here, for the first time in the literature to the best of our knowledge, that integrated information is null for a disconnected system, that is a system made up by independent components.\n\n\\smallskip\\noindent\nThe characterization of integrated information is based on another concept, always defined by Tononi and coauthors, named \\emph{effective information} and modeling how much information is gained by an external observer on the previous state of a system from checking which is its current state, with respect to what can \"a priori\" be deduced on the previous state from the known dynamics of the system itself. Given this emphasis on the experimental side of the knowledge acquisition process, we suggest here to use the terms \"experimental information\" or \"Galileian information\" as synonyms for \"effective information\".\n\nEffective information is zero for static systems or uniformly random systems, which is consistent with everyday scientist's experience. And, similarly, integrated information is also zero for disconnected systems, independently from their kind.\n\n\n\n\\section{Probabilistic Boolean Networks}\n\\label{probabilistic-boolean-networks}\nLet $X=(V,E)$ be a directed graph with $n$ boolean nodes, i.e. taking values in $\\{0,1\\}$. The value taken by a node is called also its \\emph{state}. Edge $(u,v) \\in E$ models the fact that node $v$ gets in input the state of $u$. We assume time runs in discrete steps or instants, and nodes may change their value with the flow of time depending on (the value of) the states of their input nodes.\n\nTemporal evolution of state of node $i$ is given by a law $f_i:\\{0,1\\}^{n_i} \\rightarrow \\{0,1\\}$ computing state of $i$ at the next time instant as a function only of the current state of its $n_i\\leq n$ input nodes. Self loops are admitted. Nodes can all have the same law $f$ or each node can have its specific law. In any case laws are constant with time.\n\nWe call $X$ as defined above a \\emph{Deterministic Boolean Network}. To put things into context, \\emph{Random Boolean Networks} have been defined in the literature since many years, differing from the deterministic version only in the fact that each $f_i$ is randomly chosen when building the network. Random boolean networks have been widely studied as model for gene expression in biological systems.\n\nVarious probabilistic versions of Boolean Networks have also been defined, different from ours, for example~\\cite{SDKZ-02-BI}, where each node at each time instant randomly chooses, according to a given probability distribution, the law to be used from a finite domain of admissible laws.\n\n\\smallskip\\noindent\nOur version of \\emph{Probabilistic Boolean Network} (PBN, for short) assumes the probabilistic law $r_i:\\{0,1\\}^{n_i} \\rightarrow [0,1]$ associated to node $i$ provides for each configuration of the states of the $n_i$ input nodes the probability $r_i$ that at the next time instant node $i$ has (equivalently, is in) state $1$ (being then $1-r_i$ the probability $i$ is in state $0$). It can be shown that this model can describe every network defined according to the model introduced in~\\cite{SDKZ-02-BI}. In the following we use interchangeably the terms system and network.\n\n\\smallskip\\noindent\nAt each time instant $t$ a PBN can be in any of its $2^n$ states, we assume are provided of some arbitrary enumeration $\\{x_i\\}$. State of network $X$ at time $t$ is denoted $X_t$. A PBN can also be considered as a \\emph{Markov chain with a finite space state}.\n\n\\smallskip\\noindent\nA PBN is completely described by its \\emph{state transition matrix} $S$, whose elements $s_{i j}$ are:\n\\[\n s_{i j} \\circeq p(X_{t+1}=x_j \\: | \\: X_t=x_i) \\nonumber\n\\]\nthat is, element $s_{i j}$ is the probability that at time $t+1$ the network is in state $x_j$ {\\em conditioned} to the fact that at time $t$ the network was in state $x_i$. Note that since the probabilistic law associated to each node is time constant, state transition matrix $S$ is also time constant, hence we can speak of an \\emph{homogeneous Markov chain}. A square matrix of real numbers is a state transition matrix if $0\\leq s_{i j}\\leq 1$ e $\\sum_{i=1}^{n} s_{i j} = 1$.\n\n\\smallskip\\noindent\nValues of $s_{i j}$ can be easily computed by means of the $r_k$ values for each node $k$ as it follows. Let $i=\\sigma_n \\sigma_{n-1} \\ldots \\sigma_1$ be the bit string representing the network state at instant $t$, where $\\sigma_k$ represent state of node $k$ at instant $t$. The network state at the next instant $t+1$ is $j=\\sigma'_n \\sigma'_{n-1} \\ldots \\sigma'_1$ where $\\sigma'_k$ is the state of node $k$ computed by law $r_k$ for instant $t+1$. It is $\\sigma'_k=1$ with probability $r_k(\\sigma_n \\sigma_{n-1} \\ldots \\sigma_1)$ and $\\sigma'_k=0$ with probability $1-r_k(\\sigma_n \\sigma_{n-1} \\ldots \\sigma_1)$. Then\n\\begin{equation}\n\\label{s-from-r}\ns_{i j}=\\prod_{k=1}^{n} \\rho_k \\nonumber\n\\end{equation}\nwhere $\\rho_k=r_k(\\sigma_n \\sigma_{n-1} \\ldots \\sigma_1)$ if $\\sigma'_k=1$ and $\\rho_k=1-r_k(\\sigma_n \\sigma_{n-1} \\ldots \\sigma_1)$ if $\\sigma'_k=0$.\n\n\\smallskip\\noindent\nLet us denote with $\\vect{p}_t(i)=p(X_t=x_i)$ probability that network is in state $x_i$ at instant $t$. State distribution probability at $t+1$ is given by:\n\\[\n \\vect{p}_{t+1}(x_i) = \\sum_{j=1}^{2^n} {\\vect{p}_t(x_j) s_{j i} }\n\\]\nNote that, even if $S$ is time constant (i.e., stationary), state probability distribution is not necessarily so. Let $\\vect{p}_t$ be the row vector with elements $\\vect{p}_t(i)$. Previous formula can be written in a matrix form as\n\\[\n\\vect{p}_{t+1} = \\vect{p}_t \\cdot S\n\\]\nand, denoting with $S^i$ the $i$-th column of $S$, it is\n\\[\n\\vect{p}_{t+1}(i) = \\vect{p}_t \\cdot S^i\n\\]\nIf for some $t$ it is $\\vect{p}_{t+1}(\\cdot)= \\vect{p}_t(\\cdot)$ then we say the network is in the \\emph{stationary regime}. It is then\n\\[\n\\vect{p} = \\vect{p} \\cdot S\n\\]\nthat is $\\vect{p}$ is an eigenvector of $S$ with eigenvalue $1$. Note that not every eigenvector of $S$ can be a stationary probability distribution, since it has to fulfill probability distribution constraints. For example, the null eigenvector is never a stationary probability distribution.\n\nRow $S_i$ of the state transition matrix provides the conditional probability distribution $p(X_{t+1}\\,|\\,X_{t}=x_i)$ describing network state at the instant \\emph{next} to the one the network is in state $x_i$.\n\n\\smallskip\\noindent\nNetwork dynamics can also be analyzed backwards in time. Let us assume that we have observed or measured that network at instant $t$ is in a given state. We can then compute state distribution probability for instant $t-1$, that is we can compute the law by which states at instant $t-1$ might have caused the state actually observed or measured at instant $t$. This is provided by defining a \\emph{state backward-transition matrix} $B$, describing probabilities obtained inverting through Bayes rule the relations between events. Its elements $b_{i j}$ are:\n\\[\n b_{i j}(t) \\circeq p(X_{t-1}=x_j \\, | \\, X_{t}=x_i)\n\\]\nthat can be written as\n\\[\n b_{i j}(t) = \\frac{p(X_{t-1}=x_j , \\, X_{t}=x_i)}{p(X_t = x_i)}\n\\]\nand applying again Bayes rule we have\n\\[\n b_{i j}(t) =\n \\frac{ p(X_{t} =x_i \\, | \\, X_{t-1}=x_j) p(X_{t-1} = x_j) }{ p(X_{t} = x_i) } =\n \\frac{ s_{j i} p(X_{t-1} = x_j) }{ p(X_{t} = x_i)} =\n \\frac{\\vect{p}_{t-1}(j) s_{j i}}{\\vect{p}_t(i)} = \\frac{\\vect{p}_{t-1}(j) s_{j i}}{\\vect{p}_{t-1} \\cdot S^i}\n\\]\nIf at instant $t-1$ state probability distribution is uniform then last formula becomes\n\n\\begin{equation}\n\\label{bij-uniforme}\nb_{i j}(t) = \\frac{s_{j i}}{\\sum_k s_{k i}}\n\\end{equation}\nNote that if state probability distribution is uniform then state backward-transition matrix $B$ is a kind of transpose of the state transition matrix $S$. Note also that while $S$ is time constant, $B$ is not so, in general.\n\nRow $B_i(t)$ of the state backward-transition matrix $B$ provides the conditional probability distribution $p(X_{t-1}\\,|\\,X_{t}=x_i)$ describing network state at the instant \\emph{previous} to the one the network is in state $x_i$.\n\n\n\n\\section{Effective Information}\n\\subsection{Introduction}\nEffective information can be informally described as \\emph{the quantity of information on possible predecessors of current states} acquired \\emph{additionally} from actually measuring the current network state \\emph{with respect to what can be acquired from the knowledge of state transition matrix only}.\nWe propose calling it \\emph{experimental information} or \\emph{Galileian information}, given the emphasis it gives to experimentally acquired knowledge with respect to purely theoretical knowledge. Here quantity of information is intended in the standard sense of the Shannon's Information Theory.\n\n\\smallskip\\noindent\nThe main question effective informations answers to is: if network observation finds that its current state is $x_i$, which is the additional knowledge provided by this measure with respect to what can be known on the network by its state transition matrix only, i.e. without knowing which is the current state of the network?\n\nStill remaining at the informal level this additional knowledge can be described as the reduction in uncertainty provided by the actual measurement with respect to the uncertainty existing on the basis of the state transition matrix only.\n\nOn one side there are those systems whose regime trajectory in the space state is a deterministic cycle. For such systems the observation provides an effective information of $\\log_2 k$ bits\\footnote{from now on all logarithms are to the base $2$} (where $k$ is the number of the nodes on the cycle, i.e. its length). Since a deterministic closed trajectory of length $k$ in the state space corresponds to a suitable subset of $k$ rows of the state transition matrix each containing exactly one value $1$, and since before measuring the system the uncertainty is maximum -- given that the system can be in any of these $k$ states -- while after measuring the systems it is univocally known the predecessor of the current state, the information acquired through observation is maximum and equal, according to the standard way of measuring information, to $\\log k$ bits.\n\nOn the other side there are those systems whose behavior in the state space is uniformly random, that is those systems where each state can be, with equal probability, the predecessor of the current state. Measuring the actual current state in these systems provides an effective information of $0$ bits since no reduction in uncertainty is provided through the observation (complete uncertainty both before and after the measurement). Also for completely static systems, that is systems whose state is constant while time runs there is no reduction in uncertainty provided through the observation (no uncertainty either before or after the measurement).\n\n\n\\subsection{Formal definition}\nWe define the effective information obtained by observing that system $X$ is in state $x_i$ at instant $t$ as\n\\begin{equation}\n\\label{ei}\n ei(t,x_i) \\circeq D_{KL}(B_i(t) \\; || X_{t-1})\n\\end{equation}\nwhere $D_{KL}$ is the Kullback-Leibler divergence\\footnote{The Kullback-Leibler divergence (or distance) of probability distribution $q(x)$ from probability distribution $p(x)$ is defined as $ D_{KL}(p||q) \\circeq \\sum_{x \\in \\Omega_x}p(x)\\log{ \\frac{p(x)}{q(x)}} = \\avg{\\log{\\frac{p(x)}{q(x)}}}_p $ and note it is asymmetric.}. Then\n\\begin{eqnarray}\nei(t,x_i) & = &\\sum_j{b_{i j}(t) \\log{\\frac{b_{i j}(t)}{p(X_{t-1}=x_j)}}} \\nonumber \\\\\n & = & -H(B_i(t)) -\\sum_j{b_{i j}(t)\\log{p(X_{t-1}=x_j)}} \\nonumber \\label{ei-2}\n\\end{eqnarray}\nOur definition is a generalization of the one provided by Tononi and coauthors (cfr. equations 1A and 1B of~\\cite{BaTo-08-PCB}). Ours in fact allows to study system behavior for each time instant and for each probability distribution $X_0$, while in~\\cite{BaTo-08-PCB} the time instant under investigation is always $t=1$ and it is always assumed probability distribution $X_0$ is the uniform one. Our formulation hence allows to model both the transient and the stationary regime of a system.\n\n\\smallskip\\noindent\nFor the case when the state probability distribution $X_{t-1}$ is uniform the formula above becomes:\n\\begin{eqnarray}\nei(t,x_i) & = & -H(B_i(t)) -\\sum_j{b_{i j}(t)\\log{\\frac{1}{2^n}}} \\nonumber \\\\\n & = & -H(B_i(t)) +n \\sum_j{b_{i j}(t)} \\nonumber \\\\\n & = & n - H(B_i(t)) \\nonumber \\label{ei-unif}\n\\end{eqnarray}\n\n\n\\smallskip\\noindent\nEffective information in the regime phase of a system is provided by considering equation~\\eqref{ei} in the limit for the instant $t$ tending to infinity\n\\[\n ei(x_i) \\circeq D_{KL}(B_i \\; || X_\\infty)\n\\]\nwhere $B_i$ ed $X_\\infty$ are the stationary probability distributions defined by the limits, if they exist, of the probability distributions for instant $t$, which describe the regime phase of the system. That is:\n\\[\np(X_\\infty = x_i) \\circeq \\lim_{t \\to \\infty}{p(X_t = x_i)} \\circeq p_i\n\\]\nand\n\\begin{equation}\n\\nonumber \\label{b-infinity}\np(B_i = x_j) \\circeq p(X_{\\infty} = x_j | X_{\\infty} = x_i) = \\frac{s_{ji}p_j}{p_i}\n\\end{equation}\nhence\n\\begin{equation}\n\\nonumber \\label{ei-infinity}\n ei(x_i) = \\sum_j{b_{i j} \\log{\\frac{b_{i j}}{p(X_\\infty=x_j)}}}\n = -H(B_i) -\\sum_j{b_{i j}\\log{p_j}}\n\\end{equation}\nA system which has a uniformly random behavior in the regime phase has $H(B_i)=n$, since state probability distribution $p(X_{t-1} | X_t)$ is $p(x_j) = \\frac{1}{2^n}$, hence\n\\[\n ei(x_i) = -n -\\sum_j{\\frac{1}{2^n}\\log{\\frac{1}{2^n}}} = \\sum_j{\\frac{n}{2^n}} -n = n-n = 0\n\\]\nA system completely static in the regime phase, i.e. which remains fixed in a single attraction state $x_i$, has $H(B_i)=0$ since the unique possible predecessor is $x_i$ itself and $p(x_j) = 0$ if $i \\neq j$ from which we have\n\\[\n ei(x_i) = \\log{1} = 0\n\\]\nNote that sum is computed only on observable states (i.e. where $p(x_j) \\neq 0$), to avoid the undeterminate form $0\\log{\\frac{0}{0}}$.\n\nA system having in the regime phase a single cyclic attractor containing all states, i.e. a deterministic closed trajectory in the space state walking through all states, has $H(B_j)=0$ since each state has exactly one predecessor while $p(x_j) = \\frac{1}{2^n}$ and hence\n\\[\nei(x_i) = 0-\\log{\\frac{1}{2^n}} = n\n\\]\nThe same holds, assuming the stationary state space distribution is uniform, when the system has more cyclic attractors partitioning all the space state.\n\nIf the system has a single cyclic attractor with $k<2^n$ states (or more cyclic attractors partitioning a subset of size $k<2^n$ of all states, still assuming a uniform stationary state space distribution) then it is $ei(x_i) = \\log{k}$.\n\n\\smallskip\\noindent\nThe analysis in~\\cite{BaTo-08-PCB} assumes the maximum uncertainty and uniformity on the initial systems conditions and is focused on computing effective information in the instant right after the initial state. The formulation of effective information in~\\cite{BaTo-08-PCB} is therefore the following particular case of ours:\n\\[\n ei_1(x_i) = D_{KL}(B_i(1) \\, || \\, X_0)\n\\]\nNote also that since for this particular case the assumptions used for the derivation of~\\eqref{bij-uniforme} hold, it can be written\n\\[\nb_{ij}(1) = \\frac{s_{ji}}{\\sum_k{s_{ki}}}\n\\]\n\n\n\\subsection{Effective information of subsets}\n\\label{effective-information-subsets}\nFor the definition of integrated information it is required to define how to measure effective information for subsets of a given network $X$. Let $A \\subseteq X$. When $X$ is in state $x_i$ we denote with $\\pi_A(x_i) = {}^A\\,\\!x_i$ the state of $A$.\nLet $A_t$ be the random variable representing state of $A$ at instant $t$. We can define for $A$ state transition matrix ${}^A\\!S$ and state backward-transition matrix ${}^A\\!B$ in analogy with the general case as\n\\[\n {}^A\\!s_{ij} \\circeq p(A_{t+1}=a_j \\, | \\, A_{t}=a_i)\n\\]\nand\n\\[\n {}^A\\,\\!b_{i j}(t) \\circeq p(A_{t-1}=a_j \\, | \\, A_{t}=a_i)\n\\]\nBoth can be obtained from $S$ e $p(\\cdot)$ after some long but straightforward computations. Intuitively and informally speaking, the computation is based on summing transition probabilities over all states of $X$ which are equivalent with respect to subset $A$, averaged with their state probabilities.\n\n\\smallskip\\noindent\nNow, all definitions introduced for a network $X$ can be applied to any of its subset of nodes $A$ by substituting in the previous formulas $S$, $B$, and $X$ respectively with ${}^A\\!S$, ${}^A\\!B$, and $A$. We then obtain\n\\begin{equation}\n\\label{ei-t-A-ah}\nei(t,A,a_h) \\circeq D_{KL}({}^A\\!B_h(t)||A_{t-1})\n\\end{equation}\n\n\n\n\\section{Integrated Information}\nWe are now ready to formally define integrated information, that is the quantity of information generated in a system transitioning from one state to the next by the causal interaction of its parts, above and beyond the quantity of information generated independently by each of its parts.\n\nGiven a system $X$ let $V \\subseteq X$ and $\\{M_k\\}$ a partition of $V$ in $m$ subsets. Let $M_k(t)$ be the random variables describing the state of the $k$-th component of the partition at instant $t$.\nLet $X$ be in state $x_i$ at instant $t$. Then $V$ at the same instant is in state ${}^V\\!x_i$ and the $k$-th component is in state ${}^{M_k}x_i$. In the following we use $v_h$ and $\\mu_k$ as a shorthand for ${}^V\\!x_i$ and ${}^{M_k}x_i$, respectively.\n\nPartition-dependent integrated information is first defined for a subset $V$ as a function of partition $\\{M_k\\}$, time instant $t$, and current state $v_h$ as\n\\begin{equation}\n\\label{phi-V-M}\n\\phi(t,V,\\{M_k\\},v_h) \\circeq ei(t,V,v_h) - \\sum_{k=1}^m{ei(t,M_k,\\mu_k)}\n\\end{equation}\nValue computed by this formula clearly depends on the considered partition. Tipically, an unbalanced partition produces a lower value of $\\phi$ (see~\\cite{BaTo-08-PCB}). Hence the following normalization function is introduced\n\\begin{equation}\n\\label{N} \\nonumber\nN(t,V,\\{M_k\\},v_h) \\circeq (m-1)\\min_k\\{H(M_k(t))\\}\n\\end{equation}\nThen, the \\emph{Minimum Information Partition} (\\emph{MIP}) is defined as the partition providing the minimum value for the integrated information after the normalization process, that is\n\\begin{equation}\n\\label{P-MIP} \\nonumber\nP_{\\mbox{\\scriptsize \\emph{MIP}}}(t,V,v_h) \\circeq \\argmin_P\\Big \\{\\frac{\\phi(t,V,P,v_h)}{N(t,V,P,v_h)}\\Big \\}\n\\end{equation}\nThe above formula has been defined by Tononi for generic partitions, but in all of its papers and here it is only discussed the case of bi-partitions, i.e. partitions in two subsets.\n\n\\smallskip\\noindent\nIntegrated information $\\phi$ for subset $V$, in state $v_h$ at instant $t$, is now formally defined as the value of the partition-dependent integrated information computed on \\emph{MIP}, that is\n\\begin{equation}\n\\label{phi-V} \\nonumber\n\\phi(t,V,v_h) \\circeq \\phi(t,V,P_{\\mbox{\\scriptsize \\emph{MIP}}}(t,V,v_h),v_h)\n\\end{equation}\n\n\\smallskip\\noindent\nAnd it is now possible to formally define the value of integrated information for the whole system $X$.\n\\label{complex}\nA subset $V \\subseteq X$ having $\\phi > 0$ is called \\emph{complex}. If it is not a proper subset of another subset with a larger $\\phi$ it is called \\emph{main complex}. The value of integrated information of $X$, in state $x_i$ at instant $t$, is defined as the value of integrated information of its main complex of maximum value.\n\\begin{equation}\n\\label{phi} \\nonumber\n\\phi(t,x_i) \\circeq \\max_{V \\subset X} \\, {\\phi(t,V,P_{\\mbox{\\scriptsize \\emph{MIP}}}(t,V,v_h),v_h)}\n\\end{equation}\nThe value of integrated information averaged over all states of the system is provided through the state distribution probability $p_t(\\cdot)$, that is\n\\begin{equation}\n\\label{avg_phi} \\nonumber\n\\phi(t) \\circeq \\sum_{x_i \\in X} {\\phi(t,x_i)\\,p_t(i)}\n\\end{equation}\n\n\n\n\n\n\n\\section{Integrated information in disconnected systems}\nIntuitively, any system having a partition in two independent subsets, i.e. that can be partitioned in two subsets such that no node in a subset affects the state value of nodes in the other subset, should have zero as value of its integrated information.\n\nWe now give a formal proof of this property, to the best of our knowledge never appeared in the literature. We consider the value of integrated information assuming at instant $t-1$ the system has a uniform state probability distribution, consistently with discussion in~\\cite{BaTo-08-PCB}. Remember that for a subset $V$ of the system $X$ in state $x_h$ we use $v_h$ as a shorthand for ${}^V\\!x_h$, the restriction of $x_h$ to nodes in $V$.\n\n\\begin{theorem}[Integrated information in a disconnected network]\nLet $A'$ and $A''$ be two disjoint subsets of a network $X$, $A' \\cup A'' = V \\subseteq X$. Let us denote with $v_h$ the current state of $V$, and with $a'_h$ e $a''_h$ the current states of subsets $A'$ and $A''$, respectively.\n\nFor each state $v_h$ and time instant $t$ it is\n\\[\n\\phi(t,V,\\{A',A''\\},v_h) = 0\n\\]\n\\end{theorem}\n\\begin{proof}\nFrom the definition~\\eqref{phi-V-M} of partition-dependent integrated information and the definition~\\eqref{ei-t-A-ah} of the effective information for a subset it is\n\\begin{eqnarray}\n\\phi(t,V,\\{A',A''\\},v_h) & = & ei(t,V,v_h) -ei(t,A',a'_h) -ei(t,A'',a''_h) \\nonumber \\\\\n & = & D_{KL}({}^V\\!B_h(t) \\, || \\, V_{t-1}) -D_{KL}({}^{A'}\\!B_{i}(t) \\, || \\, A'_{t-1}) -D_{KL}({}^{A''}\\!B_{j}(t) \\, || \\, A''_{t-1}) \\label{all-DKL}\n\\end{eqnarray}\nFrom the definition of the Kullback-Leibler divergence it is\n\\[\nD_{KL}({}^V\\!B_h(t) \\, || \\, V_{t-1}) = -H({}^V\\!B_h(t)) -\\sum_j{{}^V\\!b_{h j}(t)\\log{p(V_{t-1}=v_j)}}\n\\]\nRemember that ${}^V\\!B_h(t)$ is a conditional probability distribution for the state preceding the current one\n\\begin{eqnarray}\np({}^V\\!B_h(t) = v_j) & = & p(V_{t-1} = v_j \\,|\\, V_{t} = v_h) \\nonumber \\\\\n & = & p(A'_{t-1} = a'_j \\wedge A''_{t-1} = a''_j \\,|\\, V_{t} = v_h) \\nonumber\n\\end{eqnarray}\nApplying the chain rule of entropy it is\n\\[\nH({}^V\\!B_h(t)) = H(A'_{t-1} \\,|\\, V_{t} = v_h) + H\\Big( (A''_{t-1}\n\\,|\\, V_{t} = v_h) \\big | A'_{t-1} \\Big)\n\\]\nand given the independence between $A''$ and $A'$ it follows that\n\\begin{eqnarray}\nH({}^V\\!B_h(t)) & = & H(A'_{t-1} \\,|\\, V_{t} = v_h) + H(A''_{t-1} \\,|\\, V_{t} = v_h) \\nonumber \\\\\n & = & H(A'_{t-1} \\,|\\, A'_{t} = a_{h'}) + H(A''_{t-1} \\,|\\, A''_{t} = a_{h''}) \\nonumber \\\\\n & = & H({}^{A'}\\!B_{h'}(t)) + H({}^{A''}\\!B_{h''}(t)) \\nonumber\n\\end{eqnarray}\nFrom the assumption of uniform state probability distribution at $t-1$ it is\n\\begin{eqnarray}\nD_{KL}({}^V\\!B_h(t) \\, || \\, V_{t-1}) & = & -H({}^V\\!B_h(t)) + |V| \\nonumber \\\\\nD_{KL}({}^{A'}\\!B_h(t) \\, || \\, A'_{t-1}) & = & |A'|-H({}^{A'}\\!B_h(t)) \\nonumber \\\\\nD_{KL}({}^{A''}\\!B_h(t) \\, || \\, A''_{t-1}) & = & |A''|-H({}^{A''}\\!B_h(t)) \\nonumber\n\\end{eqnarray}\nand substituting the above right members for the left ones in equation~\\eqref{all-DKL} and considering that $|V| = |A'|+|A''|$ we obtain\n\\[\n\\phi(t,V,\\{A',A''\\},v_h) = |V| -|A'| -|A''| -H({}^V\\!B_h(t)) + H({}^{A'}\\!B_{h'}(t)) +\nH({}^{A''}\\!B_{h''}(t)) = 0\n\\]\n\\end{proof}\n\n\n\n\\section{Conclusions}\n\\label{conclusions}\nIn this paper we have given a thorough presentation of a model proposed by Giulio Tononi~\\cite{Tono-01-AIB, ToSp-03-BMC, Tono-04-BMC} for modeling \\emph{integrated information}, i.e. how much information is generated in a system by causal interaction of its parts and \\emph{above and beyond} the information given by the sum of its parts. The model was aimed at trying to formally capture what is consciousness in living beings~\\cite{ToEd-98-Sci, EdTo-00, Tono-08-BB} and the reader is referred to Tononi's papers for detailed motivations of the model.\n\n\\smallskip\\noindent\nWe have considered the discrete version of the model~\\cite{BaTo-08-PCB, Tono-08-BB, BaTo-09-PCB}. The original papers describing the model are not always fully clear in their mathematical formulation and here we have given the first formal description of such a model where all steps are detailed presented.\n\nIn doing so we have provided a more general formulation of such a model, which is independent from the time chosen for the analysis and from the uniformity of the probability distribution at the initial time instant.\n\nFinally, we have also given here the first formal proof that a system made up by independent parts has a value of integrated information equal to zero.\n\n\n\\paragraph{Acknowledgments.} We would like to thank Luciano Gual\\`{a} and Guido Proietti for useful and interesting discussions related to the work here described.\n\n\\renewcommand{\\baselinestretch}{1.0}\n\\small\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\subsection{Motivation}\nTop physics is an important probe of theories of new physics at the TeV scale, as many of these theories posit TeV-scale partners to the top quark in order to resolve the Higgs hierarchy problem. These theories in general have chiral structure, thus a measurement of the top polarization from the decays of top partners can establish useful constraints on them.\n\nIn the case of supersymmetry with $R$-parity, the composition of the scalar top partner ``stop'' ${\\tilde t}$ in terms of the weak eigenstates ${\\tilde t}_R$ and ${\\tilde t}_L$ can be constrained by observing the polarization of tops in the decay ${\\tilde t}\\rightarrow t {\\tilde \\chi}_1^0$. The fermionic top partners in extra-dimensions theories with KK parity and little Higgs theories with T-parity have the analogous decays $t^{(1)}\\rightarrow t B^{(1)}$ and $T'\\rightarrow t B_H$. As discrete parities are desirable because they limit dangerous contributions to electroweak precision variables~\\cite{takeuchi} and admit WIMP dark matter candidates (such as ${\\tilde \\chi}_1^0$, $B^{(1)}$ and $B_H$), the $t+\\met$ collider signature provides a useful handle on a broad class of well-motivated TeV-scale theories.\n\nThere are also numerous theories with and without top partners containing extra massive gauge bosons $Z'$ with decays such as $Z'\\rightarrow t\\bar t$. Top polarization measurements can constrain the chiral structure of their couplings to the top quark. In general, $t\\bar t$ may be produced in new heavy resonances.\n\nTo date there have been a number of studies on measuring top polarization at the Large Hadron Collider (LHC). Ref.~\\cite{hisano} performed a signal-only Monte Carlo-level analysis in the context of gluino decay, and Ref.~\\cite{agashe} performed a Monte Carlo-level analysis for KK gluons. For the $t+\\met$ class, Ref.~\\cite{shelton} performed a signal-only parton-level formal calculation, followed by Refs.~\\cite{perelstein} and \\cite{berger} performing a Monte Carlo-level analysis with backgrounds, acceptance cuts and smearing effects. For heavy resonances, Ref.~\\cite{rehermann} considered the $W+$jets background and smearing effects, later Ref.~\\cite{godbole} performed a signal-only Monte Carlo-level analysis, and Ref.~\\cite{sakurai} studied various measurables in a fast detector simulation without backgrounds. Ref.~\\cite{krohn} elucidated the benefit of jet substructure for measuring the polarization of boosted hadronic tops for both classes of theories, although without backgrounds and at Monte Carlo-level. \n\nIn this paper we study top polarization for the $t+\\met$ class of theories at {\\em detector level} including all contamination sources (e.g., ISR\/FSR and MPI), relevant detector effects (e.g., magnetic field) and backgrounds. To this end we focus on pair production of 600 GeV and 800 GeV ${\\tilde t}_1$ at the 14~TeV LHC under the simplified model in which they decay entirely to $t\\chi_1^0$, where $\\chi_1^0\\simeq {\\tilde B}$ and $m_{\\chi_1^0}=100$~GeV. We choose to focus on supersymmetry also because it may be the most well-motivated of this class of theories, solving the hierarchy problem up to Planck scale as well as enhancing gauge coupling unification at high scale.\n\nFirst we will briefly review the kinematics of top polarization and the phenomenology of ${\\tilde t}_L$-${\\tilde t}_R$ mixing. Then we will describe our simulation and analysis methodology, and present our results for the expected sensitivity to the stop mixing angle. Finally, we will look ahead to possibilities for improving and ramifying our methodology.\n\n\\subsection{Top polarization} \nMeasurement of top polarization is possible because top quarks undergo weak decay prior to hadronization, so the top decay products carry information on the polarization of the parent quark undisturbed by the hadronization process. The kinematics of top polarization is presented in Refs.~\\cite{hisano,shelton,perelstein,godbole,kane}. The decay products of the top quark have the angular distributions\n\\begin{equation}\n\\frac{1}{\\Gamma}\\frac{d\\Gamma}{d(\\cos\\theta_{tf})}\\propto 1+{\\cal P}_t k_f \\cos\\theta_{tf}\n\\end{equation}\nwhere $\\cos\\theta_{tf}$ is the angle between the daughter momentum and the top spin axis in the top rest frame; we can take the latter as the direction of top momentum in the lab frame. ${\\cal P}_t=\\pm 1$ is the polarization of the top quark, and $k_f$ is the ``spin analyzing power'' of the daughter flavor. For the $b$-quark,\n\\begin{equation}k_b = -\\frac{m_t^2-m_W^2}{m_t^2+m_W^2} \\simeq -0.4\\end{equation}\nwhereas for the lepton daughter of a leptonic top decay one finds $k_l = 1$. Consequently we can measure ${\\cal P}_t$ by observing the distribution of $\\cos\\theta_{tf}$. For the case of hadronic tops, this can be done directly by reconstructing the top and resolving the $b$-quark daughter. However, for leptonic tops, one cannot fully reconstruct the top momentum due to the neutrino from the $W$ decay. It has been proposed to define $\\cos\\theta_{tl}$ in an ``approximate rest frame'' in semileptonic events with reconstruction of the accompanying hadronic top~\\cite{perelstein}, or require the leptonic top in a semileptonic event to be highly boosted such that one can use alternative measurables which are insensitive to top momentum in the limit $\\beta\n\\rightarrow 1$~\\cite{shelton,berger}. Yet another proposal is to use judicious cuts to preserve polarization information in the lab frame despite the boost of the leptonic top~\\cite{godbole}.\n\nIn our analysis we measure the polarization of hadronic tops, which has not only the advantage of being more simple than leptonic methods, but also that of greater statistics, as 89\\% of top pairs have one hadronic top whereas only 44\\% of top pairs are semileptonic~\\cite{pdg}. Moreover, leptonic top analysis entails all the difficulties of identifying isolated leptons in a real hadron collider environment. Standard cone-based lepton isolation lose efficiency with increasing boost, hampering polarization measurements for heavy parent states. However, this may be ameliorated by narrowing the isolation cone size as lepton $p_T$ increases~\\cite{rehermann,todt}.\n\n\n\n\\subsection{Stop mixing}\nAs written in the review~\\cite{martin}, the stop mass matrix in the weak basis under the minimal supersymmetric standard model (MSSM) is\n\\begin{equation}\n{\\cal L}_{m_{\\tilde t}}=-\\left(\\begin{array}{cc}\\tilde t^*_L & \\tilde t^*_R\\end{array}\\right){\\bf m_{\\tilde t}^2 }\\left(\\begin{array}{c} \\tilde t_L \\\\ \\tilde t_R\\end{array}\\right)\n\\end{equation}\n\\begin{equation}\n{\\bf m_{\\tilde t}^2} = \\left(\\begin{array}{cc}m_{Q_3}^2 + m_t^2+\\Delta_{\\tilde u_L} & v(a_t^* \\sin\\beta - \\mu y_t\\cos\\beta)\\\\v(a_t\\sin\\beta-\\mu^*y_t\\cos\\beta) & m_{\\bar u_3}^2+m_t^2+\\Delta_{\\tilde u_R}\\end{array}\\right)\n\\end{equation}\n\\begin{eqnarray}\n\\Delta_{{\\tilde u}_L}&=&\\left(\\frac{1}{2}-\\frac{2}{3}\\sin^2\\theta_W\\right)\\cos(2\\beta)m_Z^2\\\\\n\\Delta_{{\\tilde u}_R}&=&\\frac{2}{3}\\sin^2\\theta_W\\cos(2\\beta)m_Z^2\\quad .\n\\end{eqnarray}\nIf $a_t$ and $\\mu$ are real, the mixing can be represented by the rotation\n\\begin{equation}\n\\left(\\begin{array}{c}{\\tilde t}_1 \\\\ {\\tilde t}_2\\end{array}\\right)= \\left(\\begin{array}{cc}\\cos\\theta & -\\sin\\theta \\\\ \\sin\\theta & \\cos\\theta\\end{array}\\right)\\left(\\begin{array}{c}{\\tilde t}_L \\\\ {\\tilde t}_R\\end{array}\\right)\n\\end{equation}\nwhere\n\\begin{equation}\n\\tan 2\\theta = \\frac{2m_t(A_t-\\mu\/\\tan\\beta)}{m_{Q_3}^2- m_{\\bar u_3}^2+\\Delta_{\\tilde u_L}-\\Delta_{\\tilde u_R}}\n\\end{equation}\nand in which $m_t=y_t v \\sin\\beta$ and $A_t = a_t\/y_t$ have been substituted.\n\nLet us consider the case that $\\tan\\beta\\gg 1$ and $\\mu$, $m_{Q_3}$ and $m_{\\bar u_3}$ are of the same order at low scale; we also assume $m_{Q_3}>m_{\\bar u_3}$ at low scale due to renormalization.\nThen if $A_t = 0$ we see that ${\\tilde t}_1\\simeq {\\tilde t}_R$ ($\\theta\\simeq\\pi\/2$); conversely, if $A_t\\mathrel{\\rlap{\\lower4pt\\hbox{\\hskip1pt$\\sim$} \\mu$ then ${\\tilde t}_1$ will have a significant ${\\tilde t}_L$ component ($\\theta\\sim\\pi\/4$). If indeed the mixing angle $\\theta$ can be measured, then under these assumptions $A_t$ may be strongly constrained.\n\nThese assumptions can be eased by incorporating other measurables. For example, knowledge of the ${\\tilde t}_1$ mass (e.g., from production rates or kinematic constraints) and the Higgs mass, which takes large corrections at one-loop that depend on $A_t$, $\\mu$, $\\tan\\beta$, $m_{{\\tilde t}_1}$ and $m_{{\\tilde t}_2}$~\\cite{mihoko}, would leave only the relationship between $m_{Q_3}$ and $m_{\\bar u_3}$ as a model-dependent quantity. The recent discovery of a Higgs-like boson~\\cite{higgsdiscovery_atlas,higgsdiscovery_cms} with a mass suggesting a large value of $A_t$ for TeV-scale supersymmetric scalars~\\cite{higgsmass} may already be hinting at such a correction. \n\nIt has also been proposed to consider the ratios of the branching fractions of stops decaying to neutralinos and charginos in order to constrain stop mixing parameters. However, it may require input from both a hadron collider and a linear collider to have sufficient information~\\cite{rolbiecki}. In either case, this may be another set of observables which may be useful in tandem with direct polarization measurements.\n\nTo connect the stop mixing angle with top polarization, we consider the interaction term for the decay in our simplified model\\footnote{For a discussion with completely general ${\\tilde\\chi}_1^0$, see e.g. Ref.~\\cite{perelstein}.} in which $\\chi_1^0\\simeq \\tilde B$,\n\\begin{eqnarray}\n\\notag\\Delta {\\cal L} &=& g'\\left[\\left(\\frac{1}{6}\\right){\\tilde t}_L^*{\\tilde \\chi}_1^0 t_L +\\left(-\\frac{2}{3}\\right) {\\tilde t}_R^* {\\tilde \\chi}_1^0t_R\\right]\\\\\n&=& g'{\\tilde t}_1^*{\\tilde\\chi}_1^0\\left[\\cos\\theta\\left(\\frac{1}{6}\\right) t_L + \\sin\\theta \\left(\\frac{2}{3}\\right)t_R\\right]\\\\\\notag &-&g'{\\tilde t}_2^*{\\tilde\\chi}_1^0\\left[\\sin\\theta\\left(\\frac{1}{6}\\right) t_L + \\cos\\theta \\left(\\frac{2}{3}\\right)t_R\\right]\n\\end{eqnarray}\nwhere the quantities in the parentheses are the hypercharges. Then the observed ``effective'' mixing angle from the decay ${\\tilde t}_1\\rightarrow t {\\tilde\\chi}_1^0$ is\n\\begin{equation}\n\\tan\\theta_{\\rm obs} = 4\\tan\\theta\\quad .\n\\end{equation}\nThus for or simplified model the stop mixing angle is amplified in the top polarization mixing angle, increasing the sensitivity to ${\\tilde t}_R$ and reducing the sensitivity to small ${\\tilde t}_L$ admixtures. Under the assumptions given before, this therefore reduces the sensitivity to small values of $A_t$.\n\nPutting together the Higgs mass correction and top polarization arising from stop mixing, we can see the theoretical sensitivity of $A_t$ to the measured top polarization in Figure~\\ref{fig:susy}. Here, the top polarization is defined as $(c_L^2-c_R^2)\/(c_L^2+c_R^2)$, where $c_L$ and $c_R$ are the coupling strengths $(1\/6)\\cos\\theta$ and $(2\/3)\\sin\\theta$ to the left-handed and right-handed tops, respectively. The Higgs mass correction is computed using {\\tt FeynHiggs}~\\cite{feynhiggs}. The Higgs mass window is chosen to be the intersection of the $\\pm 1\\sigma$ ATLAS and CMS regions, $125.2 < m_h < 126.2$. Indeed, sensitivity improves as the tops become more left-handed.\\footnote{For this parameter scan, $m_t=173$~GeV, $\\tan\\beta=10$, $\\mu=1692$~GeV, $m_A=1791$~GeV, $m_0=2000$~GeV, $A_b=A_\\tau=1009$~GeV, $M_1=393$~GeV, $M_2=720$~GeV and $M_3=1966$~GeV, consistent with the assumptions in our discussion.}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{susy.pdf}\n\\caption{Constraint on trilinear coupling $A_t$ vs. top polarization measurement from the MSSM Higgs mass correction for 800~GeV light stop and $125.2 < m_h < 126.2$. Blue region is for $A_t>0$, and the red region for $A_t < 0$.}\n\\label{fig:susy}\n\\end{figure}\n\n\n\\section{Simulation and analysis}\nThe goal of our analysis is to show that top polarization information can be obtained by reconstructing boosted tops in a realistic hadron collider environment. First we describe the Monte Carlo generation of our data, then the detector simulation, and finally the reconstruction of top jets using the physics objects from the simulation.\n\\subsection{Event generation and detector simulation}\nUsing {\\tt Herwig++ 2.5.0}~\\cite{herwig} with all physics effects (hadronization, ISR\/FSR, MPI) included, we generated left ($\\sin\\theta = 0$), mixed ($\\tan\\theta = 0.25$) and right ($\\sin\\theta = 1$) ${\\tilde t}_1{\\tilde t}^*_1$ samples with masses 600 GeV and 800 GeV for the 14~TeV LHC under our simplified model. We calculated NLO production cross sections using {\\tt Prospino2.1}~\\cite{prospino}, shown in Table~\\ref{table:crosssections}.\n\nWe consider the backgrounds $t\\bar t$+jets, $Z$+jets, $W$+jets and $t\\bar t+Z$, which we generated using {\\tt MadEvent\/MadGraph 5.1.3}~\\cite{madgraph,matching} + {\\tt PYTHIA 6.4.25}~\\cite{pythia} (also with all physics effects), taking their leading order cross sections which are also shown in Table~\\ref{table:crosssections}. All extra jets are five-flavor ($g$ + $u$, $d$, $c$, $s$, $b$).\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\nProcess & Generator-level cut & Cross section \\\\\n\\hline\n${\\tilde t}_1 {\\tilde t}_1^*$, $m=600$~GeV & --- & 218~fb\\\\\n${\\tilde t}_1 {\\tilde t}_1^*$, $m=800$~GeV & --- & 36.8~fb\\\\\n\\hline\n$t\\bar t+\\leq$ 2 jets & $p_{T,j1}>300$ GeV & 40.6 pb\\\\\n$(Z\\rightarrow \\nu \\bar\\nu)+\\leq$ 3 jets & $p_{T,j1}\\,,\\,\\met>250$ GeV & 7.8 pb\\\\\n$(W\\rightarrow [l,\\tau]\\nu)+\\leq$ 3 jets & $p_{T,j1}\\,,\\,\\met>300$ GeV & 4.57 pb\\\\\n$t\\bar t+(Z\\rightarrow \\nu \\bar\\nu)$ & --- & 0.11 pb\\\\\n\\hline\n\\end{tabular}\n\\caption{Signal and background cross-sections at 14~TeV LHC.}\n\\label{table:crosssections}\n\\end{table}\n\nFor our detector simulation we used {\\tt Delphes 2.0.2}~\\cite{delphes}. We modified the {\\tt Delphes} codebase to use {\\tt FastJet 3.0.3}~\\cite{fastjet} instead of the bundled version, as the newer version has an interface to manipulate subjets at specific clustering scales or steps. This allows us to ``prune'' the clustering tree to remove contamination, then store the resulting subjets, all within the {\\tt Delphes} analysis pipeline. \n\nThe {\\tt Delphes} detector settings are tuned to ATLAS, with the hadronic calorimeter grid set to match that in the ATLAS TDR~\\cite{atlastdr}. The magnetic field is turned on in the simulation.\n\n\\subsection{Cuts}\n\\label{sub:cuts}\nWe implement the following event cuts, designed to increase the significance $S\/\\sqrt{S+B}$ for our characteristic $t+\\met$ signature:\n\\begin{enumerate}\n\\item {\\em $\\met >$300~GeV.}\n\\item {\\em Leading fat jet $p_T >$400~GeV.}\n\\item {\\em If there are no leptons w\/ $p_T>5$~GeV, require subleading fat jet $p_T>$100~GeV.} This cut suppresses processes like $W\/Z$+jets in which the second fat jet is from QCD, as these jets are likely to be soft. Since the majority of signal events have leptons due to $W\\rightarrow l\\nu$ and the decay of $b$-flavored mesons, we require that there are no leptons for this cut. Thus, this cut is most effective against $Z$+jets.\n\\item {\\em Lepton is not collimated with $\\met$}. For every lepton with $p_T>5$~GeV, require\n\\begin{equation}\n\\frac{\\cos(\\phi_\\met - \\phi_l)}{(\\met+p_{T,l})\/(350\\;{\\rm GeV})} < 0.4\\quad .\n\\end{equation}\nThis selects against high $\\met$ arising from boosted leptonic $W$ decays in $t\\bar t$+jets and $W$+jets, as the opening angle between the lepton and $\\vec\\met$ is likely to be much smaller in these processes than from a top partner decay.\n\\item {\\em Hard subjet is not collimated with $\\met$}. For every subjet with $p_T>50$~GeV, require the same as above. This works against hadronic $\\tau$ from $W$ decays in $t\\bar t$+jets and $W$+jets, as well as highly collimated $b$-subjets from top decays in $t{\\bar t}+$jets.\n\\item {\\em Require at least one top tagged jet}, using the procedure described in the next section.\n\\item {\\em $225< {M_T}_2 < 650$ for 600~GeV stops, and $325 < {M_T}_2 < 850$ for 800~GeV stops.} ${M_T}_2$ is calculated for the leading and subleading jet, with $m_\\chi=0$. For the leading jet we used the reconstructed top jet if top tagged, otherwise we used the trimmed jet; similarly for the subleading jet. We employed the ${M_T}_2$ code of Ref.~\\cite{ucdavismt2}. \n\\end{enumerate}\nThe resulting cut flow for signal events is shown in Table~\\ref{table:cutflowsignal}, and for background events in Table~\\ref{table:cutflowbackground} for 14~TeV LHC @ 100~fb$^{-1}$. The cuts reveal some preference for right stops, which is noted in Ref.~\\cite{perelstein}.\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\n\\multicolumn{2}{|c|}{\\multirow{2}{*}{Cut}} & \\multicolumn{3}{|c|}{Stop 600 GeV} & \\multicolumn{3}{|c|}{Stop 800 GeV}\\\\\n\\cline{3-8}\n\\multicolumn{2}{|c|}{} & Left & Mixed & Right & Left & Mixed & Right\\\\\n\\hline\n\\# & Pre-cut & 21900 & 21900 & 21900 & 3680 & 3680 & 3680\\\\\n\\hline\\hline\n1 & $\\met > 300$~GeV & 9513 & 9739 & 9857 & 2394 & 2411 & 2433\\\\\n2 & $p_{T,j1} > 400$~GeV & 5415 & 5496 & 5472 & \t1816 & 1835 & 1825\\\\\n3 & If $n_l=0$, $p_{T,j2}>100$~GeV & 5150 & 5220 & \t5192 & 1756 & 1773 & 1764\\\\\n4 & lepton\/$\\met$ collimation & 4155 & 4315 & 4394 &\t1515 & 1555 & 1559\\\\\n5 & subjet\/$\\met$ collimation & 2914 & 3046 & 3084 & 1171 & 1209 & 1208\\\\\n\\hline\n6 & \\# top tag $\\geq 1$ & 1014 & 1065 & 1082 & 450 & 456 & 463\\\\\n7a & $225< {M_T}_2 < 650$ & 908 & 954 & 969 & --- & \t--- & ---\\\\\n7b & $325 < {M_T}_2 < 850$ & --- & --- & --- & 364 & \t369 & 374\\\\\n\\hline\n\\end{tabular}\n\\caption{Cut flow for signal events at 14~TeV LHC @ 100 fb$^{-1}$.}\n\\label{table:cutflowsignal}\n\\end{table}\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline\n\\multicolumn{2}{|c|}{Cut} & $t{\\bar t}+$jets & $Z+$jets & $W+$jets & $t{\\bar t}+Z$ \\\\\n\\hline\n\\# & Generator-level & $4.06\\times 10^6$ & $7.8\\times 10^5$ & $4.57\\times 10^5$ & 11000\\\\\n\\hline\\hline\n1 & $\\met > 300$~GeV & $1.30\\times 10^5$ & $4.96\\times 10^5$ & $4.36\\times 10^5$ & 815\\\\\n2 & $p_{T,j1} > 400$~GeV & 90503 & $2.28\\times 10^5$ & $3.20\\times 10^5$ & 351\\\\\n3 & If $n_l=0$, $p_{T,j2}>100$~GeV & 88133 & $1.01\\times 10^5$ & $2.68\\times 10^5$ & 326\\\\\n4 & lepton\/$\\met$ collimation & 21518 & 98441 & 69865 & 237\\\\\n5 & subjet\/$\\met$ collimation & 4412 & 60860 & 37852 &\t149 \\\\\n\\hline\n6 & \\# top tag $\\geq 1$ & 1140 & 305 & 99 & 52\\\\\n7a & $225< M_{T2} < 650$ & 554 & 222 & 56 & 43\\\\\n7b & $325 < M_{T2} < 850$ & 275 & 141 & 45 & 30\\\\\n\\hline\n\\end{tabular}\n\\caption{Cut flow for background events at 14~TeV LHC @ 100 fb$^{-1}$.}\n\\label{table:cutflowbackground}\n\\end{table}\n\n\n\\subsection{Top jet reconstruction}\nMany aspects of this analysis are well-reviewed in Ref.~\\cite{tilmantop}.\n\n\\subsubsection{Jet clustering and grooming}\nHadrons are clustered as fat jets using the Cambridge\/Aachen algorithm~\\cite{cajets} with cone size $R=1.2$ and subjet cone size $\\Delta R = 0.2$. These numbers are chosen such that the distribution in the number of subjets per fat jet peaks at $\\sim 3$ in our signal samples after the following grooming procedures:\n\\begin{enumerate}\n\\item The jet clustering trees are ``pruned''~\\cite{pruning} using the mass-drop condition~\\cite{massdrop}\n\\begin{equation}\nm_{j_{n-1}} < 0.8 \\times m_{j_n}\n\\end{equation}\nwhere $m_{j_{n-1}}$ is the invariant mass of the hardest parent jet, and $m_{j_n}$ is the invariant mass of the child jet at clustering step $n$. We also require the subjet separation condition\n\\begin{equation}\nd_{k_T}(j_{n-1,1}, j_{n-1,2}) > (\\Delta R)^2\\cdot m_{j_n}^2\n\\end{equation}\nwhere $d_{k_T}$ is the $k_T$ distance between the two parents jets. This removes contamination from MPI and ISR, improving top reconstruction quality. We take the fat jet (and its subjets) at the clustering step where both conditions are satisfied.\n\\item This jet is then ``trimmed''~\\cite{trimming}, removing subjets with $p_T<10$~GeV. This further reduces contamination.\n\\end{enumerate}\n\nThe trimmed jet is then fed to our top reconstruction algorithm.\n\\subsubsection{Reconstruction and tagging}\nThe following algorithm is attempted for every fat jet:\n\\begin{enumerate}\n\\item Require $p_T>400$~GeV for the {\\em untrimmed} jet.\n\\item Require at least three subjets.\n\\item Require that one $b$-subjet $j_b$ is reconstructed in the jet.\n\\item Find the subjet combination $j_1 j_2 j_b$, of which no pair of subjets are within $\\Delta R$ of each other, and which gives the closest invariant mass to $m_t$. Require also that this invariant mass be in the window (150, 200)~GeV.\n\\item If there is no successful tag, require at least four subjets and retry the step above with four-subjet combinations $j_1 j_2 j_3 j_b$.\n\\item Optionally require one two-subjet combination to have an invariant mass in the loose $W$ mass window (50, 110)~GeV. We present our main results both with and without this requirement.\n\\end{enumerate}\nThis algorithm differs from the Johns Hopkins top tagger~\\cite{hopkinstagger} by declustering more than two steps in the pruning stage if necessary, using a mass drop condition rather than a $p_T$ drop condition, having an absolute rather than fractional trimming threshold, requiring a $b$-tag instead of imposing a $W$ mass condition, and not imposing a top helicity angle condition (as this is our observable). The algorithm also differs from the CMS tagger~\\cite{cmstagger} by not requiring a minimum two-subjet invariant mass.\n\nOur algorithm differs also from the HEPTopTagger~\\cite{heptoptagger} by not imposing the various two-subjet mass requirements, having a mandatory $b$-tag, and also by implementing four-subjet reconstruction.\n\nWe make these choices to enhance top tagging efficiency, presuming that the top parent particle has already been discovered. Nonetheless, top mistagging does not overwhelm the signal as will be apparent in our results.\n\n\\subsubsection{$b$-tagging subjets}\n\\label{sec:btagging}\nWe require $b$-tagging in our top reconstruction algorithm to reconstruct the observable $\\cos\\theta_{tb}$ with high fidelity, but also to suppress top mistagging from background processes. Utilizing recent advances in $b$-tagging by the LHC detector collaborations, we employ the $b$-tagging efficiencies (shown in Table~\\ref{table:btag}) recently validated at 7~TeV LHC by CMS for their CSVM tagger~\\cite{cmsbtagging}. We impose the upper limit $p_T = 1000$~GeV to be conservative, though it is not indicated by the CMS study. We choose to use the CMS efficiencies since they are validated up to $p_T=670$~GeV, though a recent ATLAS study~\\cite{atlasbtagging} shows similar efficiencies up to 200~GeV.\n\n\\begin{table}\n\\centering\n\\begin{tabular}{|c|c|}\n\\hline\nKinematic region & Efficiency\\\\\n\\hline\n$p_T < 30$ GeV & 0\\%\\\\\n$30\\;{\\rm GeV} < p_T < 60\\;{\\rm GeV}$ & 60\\%\\\\\n$60\\;{\\rm GeV} < p_T < 450\\;{\\rm GeV}$ & 70\\%\\\\\n$450\\;{\\rm GeV} < p_T < 1000\\;{\\rm GeV}$ & 60\\%\\\\\n$1000\\;{\\rm GeV}400$~GeV cut (blue), requiring successful pruning (yellow), trimming (gray), requiring three or more subjets (cyan), $b$-tagging (magenta), and finally top tagging (green).}\n\\label{fig:jet_invm}\n\\end{figure}\n\nThe quality of reconstruction is also apparent in the signal-only $\\cos\\theta_{tb}$ distributions shown in Figure~\\ref{fig:ctb_stops}. One sees that the parton-level and reconstructed distributions coincide up to statistical fluctuations. Near $\\cos\\theta_{tb}=-1$ there is depletion due to poor $b$-tagging efficiency, whereas near $\\cos\\theta_{tb}=+1$ there is depletion due to the $W$ subjets being too soft to pass the trimming threshold. One sees for this reason that there is less contrast between left and mixed stops with mass 600~GeV than with mass 800~GeV. The distributions at pre-cut\/tag parton-level match Ref.~\\cite{shelton}, except they have a downward left-to-right tilt due to the harder $b$-partons losing energy to FSR.\n\\begin{figure}[h]\n\\includegraphics[width=\\columnwidth]{ctb_stops.pdf}\n\\caption{ $\\cos\\theta_{tb}$ distributions for stops only, normalized to one. Left to right are left, mixed, and right stop samples; upper row is for 600~GeV stops, the lower for 800~GeV stops. Shown in each panel are the reconstructed distribution (black), parton-level distribution (blue), and parton-level distribution before cuts and tagging (green).}\n\\label{fig:ctb_stops}\n\\end{figure}\n\nFinally, we show sample ${M_T}_2$ distributions for 600~GeV and 800~GeV mixed stops in Figure~\\ref{fig:mt2} at reconstruction level, signal only. One can clearly see the expected inflection points at ${M_T}_2=600$~GeV and ${M_T}_2=800$~GeV. As described in Subsection~\\ref{sub:cuts}, we consider only top-tagged events, calculating ${M_T}_2$ with the two leading fat jets; for each we use the reconstructed jet if it is top-tagged, otherwise we use the trimmed jet.\n\\begin{figure}[h]\n\\includegraphics[width=\\columnwidth]{mt2.pdf}\n\\caption{Sample ${M_T}_2$ distributions for mixed stops at reconstruction level, signal only. Left panel is for stop mass 600~GeV, and the right panel is for mass 800~GeV.}\n\\label{fig:mt2}\n\\end{figure}\n\n\n\\subsection{$t\\bar t$ + jets control region}\nAs experimenters are likely to rely on data rather than Monte Carlo for the $t\\bar t$+jets background, we demonstrate a control region by inverting cuts \\#4 and \\#5, {\\em requiring} that there be at least one lepton or hard subjet highly collimated with $\\vec\\met$. An array of $\\cos\\theta_{tb}$ distributions for this region is shown in Figure~\\ref{fig:ttcontrol}. They evince little contamination from signal and other backgrounds.\n\nIt may be of concern that the collimation cuts \\#4 and \\#5 might introduce distortions to the $\\cos\\theta_{tb}$ distributions obtained from top tagging. In Figure~\\ref{fig:ttcutcheck} we show that the polarization distribution for $t\\bar t$+jets is invariant under these cuts up to statistical fluctuations. We show also in Figure~\\ref{fig:tthilo} that the ${M_T}_2$ cut does not distort the polarization distribution of $t\\bar t$+jets in the control region. Thus, it is safe to use a data-driven $t\\bar t$+jets background from this region.\n\n\\begin{figure}[h]\n\\includegraphics[width=\\columnwidth]{ttcontrol.pdf}\n\\caption{Event distributions of $\\cos\\theta_{tb}$ for $t\\bar t+$jets in the control region at 14~TeV LHC @ 100~fb$^{-1}$. Left-to-right are left, mixed and right stop samples; upper row is for 600~GeV stops, the lower row for 800~GeV stops. Shown in each panel top to bottom is signal (red), ${t\\bar t}+$jets (blue), $Z+$jets (green), $W$+jets (yellow), and $t\\bar t+Z$ (gray; hardly visible).}\n\\label{fig:ttcontrol}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{combo_ttcutcheck.pdf}\n\\caption{Distributions of $\\cos\\theta_{tb}$ for ${t\\bar t}$+jets, normalized to one. Shown are the distribution without collimation cuts (black), with the lepton collimation cut only (blue), with the subjet collimation cut only (red), and with both collimation cuts applied (cyan).}\n\\label{fig:ttcutcheck}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{combo_tthilo.pdf}\n\\caption{Distributions of $\\cos\\theta_{tb}$ for the ${t\\bar t}$+jets control sample, normalized to one. Shown are the distribution for ${M_T}_2< 225$~GeV (black), and for ${M_T}_2 > 225$~GeV.}\n\\label{fig:tthilo}\n\\end{figure}\n\n\n\\subsection{Sensitivity to stop mixing}\nTo calculate sensitivity, we sum the $\\cos\\theta_{tb}$ distributions from signal and background processes. For the $t\\bar t+$jets contribution, we use the control region distribution normalized to the total of the signal region distribution. This sum is shown in Figure~\\ref{fig:combostack}, where we have rebinned the data to use only two bins in order to minimize the trials penalty. The corresponding $p$-values for distinguishing stop mixing hypotheses are shown in Table~\\ref{table:pvalues}. For 600~GeV stops at 14~TeV LHC @ 100~fb$^{-1}$, left and right mixtures can be distinguished to better than $4\\sigma$, and left\/right can be distinguished from the mixed state to better than $1.5\\sigma$. For 800~GeV stops, left and right can be distinguished to nearly $3\\sigma$.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{combo_1Dstack.pdf}\n\\caption{Sum of $\\cos\\theta_{tb}$ distributions from different processes for 14~TeV LHC @ 100~fb$^{-1}$, {\\em without} a $W$ mass condition. Left to right are left, mixed and right stop samples; upper row is for 600~GeV stops, and the lower is for 800~GeV stops. Shown are the contributions from signal (red), $t\\bar t$+jets (blue), $Z+$jets (green), $W+$jets (yellow), $t\\bar t + Z$ (gray).}\n\\label{fig:combostack}\n\\end{figure}\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{|l||c|c|c||c|c|c|}\n\\hline\n\\multirow{2}{*}{{\\em Truth}} & \\multicolumn{3}{|c||}{{\\em Hypothesis ($m_{{\\tilde t}_1}=$600 GeV)}} & \\multicolumn{3}{|c|}{{\\em Hypothesis ($m_{{\\tilde t}_1}=$800 GeV)}}\\\\\n\\cline{2-7}\n & left & mixed & right & left & mixed & right\\\\\n\\hline\nleft & 1 & 0.047 & $2.6\\times10^{-6}$ & 1 & 0.16 & 0.0014\\\\\nmixed & 0.058 & 1 & 0.030 & 0.17 & 1 & 0.24\\\\\nright & $8.1\\times 10^{-6}$ & 0.032 & 1 & 0.0018 & 0.25 & 1\\\\\n\\hline\n\\end{tabular}\n\\caption{$p$-values for distinguishing stop mixing hypotheses at 14~TeV LHC @ 100~fb$^{-1}$, with no $W$ mass condition.}\n\\label{table:pvalues}\n\\end{table}\n\nExcept for $t\\bar t +Z$, most of the tops from the background processes surviving the cuts and reconstruction are due to extra QCD jets carrying $b$-flavor. The results therefore are improved somewhat by imposing a loose $W$ mass condition in the top reconstruction algorithm. If we require the mass of some two-subjet combination (not including the $b$-subjet) to be in the window $(50, 110)$~GeV, we obtain the sum plot shown in Figure~\\ref{fig:combostack_mW} and the corresponding $p$-values in Table~\\ref{table:pvalues_mW}. For 600~GeV stops, left and right can now be distinguished to better than $4.5\\sigma$, and left\/right can be distinguished from mixed to roughly $2\\sigma$. For 800~GeV stops, left and right can now be distinguished to better than $3\\sigma$. However, imposing this $W$ mass condition may increase systematic errors from the $W\/Z+$jets backgrounds due to the greater variation between the negative and positive bins. This occurs because the subjets of the fake $W$s have lower $p_T$ at larger $\\cos\\theta_{tb}$, and are less likely to satisfy the $W$ mass condition.\n\nWe conclude that stop mixing hypotheses can be distinguished at the 14~TeV LHC with $\\sim100$~fb$^{-1}$ of data.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{combo_1Dstack_mW.pdf}\n\\caption{Sum of $\\cos\\theta_{tb}$ distributions from different processes for 14~TeV LHC @ 100~fb$^{-1}$ {\\em with} a $W$ mass condition. Left to right are left, mixed and right stop samples; upper row is for 600~GeV stops, and the lower is for 800~GeV stops. Shown are contributions from signal (red), $t\\bar t$+jets (blue), $Z+$jets (green), $W+$jets (yellow), $t\\bar t + Z$ (gray).}\n\\label{fig:combostack_mW}\n\\end{figure}\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{|l||c|c|c||c|c|c|}\n\\hline\n\\multirow{2}{*}{{\\em Truth}} & \\multicolumn{3}{|c||}{{\\em Hypothesis ($m_{{\\tilde t}_1}=$600 GeV)}} & \\multicolumn{3}{|c|}{{\\em Hypothesis ($m_{{\\tilde t}_1}=$800 GeV)}}\\\\\n\\cline{2-7}\n & left & mixed & right & left & mixed & right\\\\\n\\hline\nleft & 1 & 0.018 & $1.7\\times10^{-6}$ & 1 & 0.17 & $5.6\\times 10^{-4}$\\\\\nmixed & 0.025 & 1 & 0.017 & 0.17 & 1 & 0.14\\\\\nright & $9.0\\times 10^{-7}$ & 0.018 & 1 & $7\\times 10^{-4}$ & 0.15 & 1\\\\\n\\hline\n\\end{tabular}\n\\caption{$p$-values for distinguishing stop mixing hypotheses at 14~TeV LHC @ 100~fb$^{-1}$, with a $W$ mass condition.}\n\\label{table:pvalues_mW}\n\\end{table}\n\n\\section{Conclusion and outlook}\nIn summary, after reviewing the motivation for top polarization measurement, the kinematics of top polarization, and the phenomenology of stop mixing, we described a simulation and analysis methodology for the $t+\\met$ collider signature that can distinguish stop mixing hypotheses at 14~TeV LHC with $\\sim 100$~fb$^{-1}$ of data.\n\nThere are several possible improvements to the methodology. The first and foremost is to include polarization information from leptonic decays, perhaps using the techniques of~\\cite{rehermann,todt}. Also if, in consultation with experimenters, the trimming threshold of $p_T =10$~GeV or the subjet cone size $\\Delta R = 0.2$ can be reduced, this would enhance the performance of the top reconstruction algorithm. Finally, it may be useful to implement the top reconstruction technique of Ref.~\\cite{krohn} to supplement $b$-tagging, especially at large boosts.\n\nOther possible improvements are to:\n\\begin{itemize}\n\\item Implement charm mistagging in the $b$-tagging algorithm, though this may be non-trivial as discussed in Section~\\ref{sec:btagging}.\n\\item Use spin-correlated backgrounds, e.g. from {\\tt ALPGEN}~\\cite{alpgen}. We did produce a small sample of $t\\bar t+$jets in {\\tt ALPGEN} but found no difference in the $\\cos\\theta_{tb}$ distribution from the {\\tt MadGraph\/MadEvent}+{\\tt PYTHIA} events used in this analysis.\n\\item Calculate backgrounds at NLO using, e.g., {\\tt MC@NLO}~\\cite{mcatnlo} to increase accuracy. However, this is likely beyond our computational capabilities unless it is used only to normalize the leading-order backgrounds. We estimate that the $K$-factor for the $t\\bar t$ background to be less than 1.2 given our cuts $p_{T,j1}>400$ and $\\met > 300$~GeV~\\cite{melnikov}, though there is evidence that with an additional jet one may actually have $K < 1$~\\cite{dittmaier}.\n\\end{itemize}\n\nIn conclusion, the LHC running at 14~TeV may provide new discoveries in the top sector, in which case top polarization will be an important tool for constraining this new physics. The methodology in this paper may be useful for this purpose.\n\n\\section*{Acknowledgments}\nThis work is supported by the World Premier International Research Center Initiative (WPI Initiative) as well as Grant-in-Aid for Scientific Research Nos. 22540300, 23104005 from the Ministry of Education, Science, Sports, and Culture (MEXT), Japan. The authors would like to thank T.~T.~Yanagida and S.~Matsumoto for preliminary discussions on this topic. We would also like to thank Z.~Heng, K.~Sakurai and C.~Wymant for useful comments.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}