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+{"text":"\\section{Introduction} \\label{sec:intro}\n\nMillisecond pulsars (MSPs) are rapidly spinning and highly magnetized neutron stars (NSs), formed through mass and angular momentum accretion from an evolving companion star to a slowly rotating NS \\citep{alpar82,stairs04,tauris11}. At the end of the accretion stages, which are commonly observed in low-mass X-ray binaries \\citep[e.g.][]{tauris06,papitto13,ferraro15}, the re-accelerated NS is reactivated as a MSP in the radio bands and the companion star is expected to be an exhausted and deeply peeled star, typically a white dwarf (WD) with a He core \\citep[e.g.][]{stairs04}, although deviations from this scenario exist \\citep[e.g.][]{ferraro01a,ransom05,lynch12,pallanca12,pallanca13b,cadelano15a}.\nGlobular clusters (GCs) are the ideal factory for the formation of MSPs. In fact, the large stellar densities in the cores of GCs favor dynamical interactions, such as exchange interactions and tidal captures, which can lead to the formation of a large variety of binary systems whose evolution generates stellar exotica like blue stragglers \\citep[e.g.][]{ferraro09,ferraro12,ferraro18a}, low-mass X-ray binaries \\citep[e.g.][]{pooley03}, cataclysmic variables \\citep[e.g.][]{paresce92,ivanova06} and MSPs as well \\citep[e.g.][]{ransom05,hessels07,cadelano18}. As a consequence, the number of MSPs per unit mass in the global GC system of the Milky Way turns out to be $\\sim 10^3$ times larger than in the Galactic field. \n\n{To date, 150 MSPs are known in 28 different GCs. Among these clusters, the case of Terzan 5 and 47 Tucanae are particularly worthy of attention: the former is a stellar system known to host $\\sim25\\%$ of the entire MSP population in GCs \\citep{ransom05,cadelano18,andersen18} and proposed to be the remnant of a pristine fragment of the Galactic bulge \\citep{ferraro09,lanzoni10,massari14,ferraro16,prager17}; the latter is the second cluster in terms of MSP abundance, hosting 25 systems that allowed studies ranging from binary evolution to the cluster structure and gas content \\citep[see e.g.][]{freire01,cadelano15b, riverasandoval15,ridolfi16,freire17,abbate18}}. Therefore the study of MSPs deserves special attention since it allows to study the physical properties of the host stellar system \\citep[e.g.][]{freire17,prager17,abbate18} and also to probe binary  and stellar evolution under extreme conditions. This can be done, for example, through the identification of the optical counterparts, whose emission is dominated by the companion stars \\citep{ferraro01a,ferraro03a,mucciarelli13,pallanca14,cadelano15b,cadelano17}. In the case of degenerate companions, like WDs or NSs, precise mass measurements of both the components of the binary can be performed directly through timing analysis if relativistic effects are observed. Such values, not only are highly valuable to study in detail the formation and evolution of these systems, but are also of extreme importance to put constraints on the equation of state of ultra-dense matter and to test general relativity \\citep[e.g.][]{freire08,demorest10,antoniadis13}. If relativistic effects are not observed, mass measurements can be obtained through the optical identification of the WD companion. In fact, by comparing the observed magnitudes with appropriate WD cooling sequences, the mass of the companion star can be evaluated and such a value, combined with the mass function obtained through radio timing, provides the determination of the NS mass. { Furthermore, if the companion star is bright enough, the radial velocity curve, and thus the mass ratio,  can be determined through spectroscopic observations, together with more precise measurements of the companion mass} \\citep[see e.g.][]{ferraro03b,antoniadis12,antoniadis13}. \n\nM3 (NGC 5272) is a bright and well studied GC located at about 10 kpc from the Sun \\citep[see][]{ferraro93,buonanno94,ferraro97a,ferraro97b,ferraro97c,carretta98,rood99,ferraro18b}. Its stellar population is characterized by an intermediate metallicity ($[Fe\/H]=-1.5$), a very small reddening $E(B-V)=0.01$ and a large number of variable stars \\citep[see][for more information about the cluster properties]{harris10}. \\citet{hessels07} reported on the secure identification of 3 MSPs in this cluster plus a fourth candidate. Among these, only two MSPs, namely J1342$+$2822B and J1342$+$2822D (hereafter M3B and M3D) have been clearly detected  in a number of observations large enough to provide the astrometric and orbital properties of the binaries, although the position of M3D is still affected by a large uncertainty in both right ascension and declination. The properties of these two systems, as reported by \\citet{hessels07}, are summarized in Table~\\ref{tab:MSP}. M3B is a 2.39 ms MSP, located in binary with a $\\sim 1.4$ days circular orbit. Its general radio properties suggest that it is a canonical system and therefore it should be in a binary system with a WD companion. On the other hand, M3D has a spin period of 5.45 ms and it is in a very long and slightly eccentric orbit of about 129 days. The vast majority of binary MSPs in GCs have orbital periods shorter than few days, while just a dozen have orbital periods longer than 10 days\\footnote{see \\url{http:\/\/www.naic.edu\/~pfreire\/GCpsr.html}}. Therefore M3D likely underwent an anomalous evolution; indeed, binaries with orbital periods longer then 10 days are believed to be formed through an exchange interaction between an isolated NS and a primordial cluster binary \\citep{hut92,sigurdsson93}. Its eccentricity, which is likely the result of fly-by encounters with other stars \\citep{phinney92,phinney93,bagchi09}, is unexpected for systems formed through the canonical low-mass X-ray binary channel and confirms a non-standard evolution for this system.\n\nHere we report on the secure identification of the companion star to M3B and on a candidate companion to M3D. These companions have been discovered through high resolution near-ultraviolet (near-UV) and optical observations. In Section~\\ref{sec:data} we present the data-set we used in this work and the data reduction procedures. In Section~\\ref{sec:comM3B} we present the identification and characterization of the companion star to M3B, while in Section~\\ref{sec:comM3D} we discuss a possible candidate companion to M3D. Finally, in Section~\\ref{sec:conc} we draw our conclusions.\n\n\n\\begin{deluxetable*}{lcc}\n\\tablecolumns{3}\n\\tablewidth{0pt}\n\\tablecaption{Main radio timing\nparameters for M3B and M3D\\tablenotemark{a}, from \\citet{hessels07}.\\label{tab:MSP}}\n\\tablehead{\\colhead{Parameter} & \\colhead{M3B} & \\colhead{M3D}}\n\\startdata\nRight ascension, $\\alpha$ (J2000)\\dotfill  & 13$^{\\rm h}\\,42^{\\rm m}\\,11\\fs0871(1)$  & 13$^{\\rm h}\\,42^{\\rm m}\\,10\\fs2(6)$    \\\\\nDeclination, $\\delta$ (J2000)  \\dotfill  & $28^\\circ\\,22'\\,40\\farcs141(2)$   & $28^\\circ\\,22'\\,36(14)\\arcsec$  \\\\\nAngular offset from cluster center, $\\theta_{\\perp}$ (\\arcsec) \\dotfill & 8.4 & 14(7) \\\\\nSpin period, $P$ (ms)  \\dotfill  & 2.389420757786(1) &   5.44297516(6) \\\\\nOrbital period, $P_b$ (days) \\dotfill & 1.417352298(2) & 128.752(5) \\\\\nTime of ascending node passage, $T_{asc}$ (MJD) \\dotfill & 52485.9679712(6) & 52655.38(4) \\\\\nProjected semi-major axis, $x$ (s) \\dotfill & 1.875655(2) & 38.524(4)\\\\\nEccentricity, $e$ \\dotfill & ... & 0.0753(5) \\\\\nMSP mass function, $f \\ (M_{\\odot})$ \\dotfill  & 0.003526842(6) & 0.0037031(8) \\\\\n\\hline\n\\enddata\n\\tablenotetext{a}{Numbers in parentheses are uncertainties in the last digits quoted.}\n\\end{deluxetable*} \n\n\n\\section{Observations and DAta analysis} \\label{sec:data}\n\nThis work is based on a archival data-set of deep, high resolution near-UV and optical observations obtained through the UVIS channel of the Wide Field Camera 3 mounted on the Hubble Space Telescope (GO 12605, PI: Piotto). The data-set is composed of 6 images in the F275W {(near-ultraviolet)} filter with exposure time of 415 s, 4 images in the F336W (U) filter with exposure time of 350 s and 4 images in the F438W (B) filter with exposure time of 42 s. All the images {have been obtained on May 15th 2012 during $\\sim5$ hours of continual observations and thus they only cover $\\sim15\\%$ and $\\sim0.16\\%$ of the orbits of M3B and M3D, respectively}. The images are dithered by few arcseconds each, in order to avoid spurious effects due to bad pixels and to fill the inter-chip gap. The photometric analysis has been performed with {\\rm DAOPHOT IV} \\citep{stetson87} adopting the so-called ``UV-route'' as described in \\citealt{raso17} (see also \\citealt{ferraro97b,ferraro01b,ferraro03c,dalessandro18a,dalessandro18b}). As a first step, about 200 stars have been selected in each image in order to model the point spread function, which has been then applied to all the sources detected at more than $5\\sigma$ from the background level. Then, we built a master catalog with stars detected in at least half the near-UV (F275W) images. At the corresponding position of these stars, the photometric fit was forced in all the other frames  by using {\\rm DAOPHOT\/ALLFRAME} \\citep{stetson94}. Using such a near-UV master list, the crowding effect due to the presence of giants and turn-off stars is mitigated and several blue stars like blue stragglers and white dwarfs are recovered. Finally, for each star we homogenized the magnitudes estimated in different images, and their weighted mean and standard deviation have been adopted as the star magnitude and its related photometric error. The instrumental magnitudes have been reported to the VEGAMAG system by cross-correlation with the catalog of the ``{\\it The Hubble Space Telescope UV Legacy Survey of Galactic globular clusters}'' \\citep{piotto15}, obtained from the same data-set used in this work and publicly available.\n\nThe star detector positions have been corrected from geometric distortion following the procedure described by \\citet{bellini11}. Then, these positions have been reported to the absolute coordinate system ($\\alpha$,$\\delta$) using the stars in common with the Gaia DR2 catalog \\citep{gaia18}. The coordinate system of this catalog is based on the International Celestial Reference System, which allows an appropriate comparison with the MSP positions derived from timing using solar system ephemerids, being the latter referenced to the same celestial system. The resulting $1\\sigma$ astrometric uncertainty is of $\\sim 0.1 \\arcsec$ in both $\\alpha$ and $\\delta$.\n\n\n\\section{IDENTIFICATION OF THE COMPANION STAR to M3B} \\label{sec:comM3B}\n\n\\begin{figure}[t] \n\\centering\n\\includegraphics[scale=0.72]{chartB.png}\n\\caption{$3\\arcsec \\times 3 \\arcsec$ finding chart of the region around the radio position of M3B. The left, middle and right panels are extracted from a F275W, F336W and F438W image, respectively. The red cross is centered on the MSP position and the red circle has a radius $r=0.15\\arcsec$. The only star located within this circle is the identified companion to M3B.}\n\\label{fig:chartB}\n\\end{figure}\n\n\\begin{figure}[h!] \n\\centering\n\\includegraphics[scale=0.49]{cmd_M3B.pdf}\n\\caption{($m_{F275W}$, $m_{F275W}-m_{F336W}$) CMD of M3 obtained from the data-set used in this work. The positions of the companion star to M3B is marked with a red square.}\n\\label{fig:cmd}\n\\end{figure}\n\nIn order to search for the companion star to M3B, we investigated all the stellar sources detected in a $3\\arcsec \\times 3 \\arcsec$ region surrounding the MSP timing position reported in Table~\\ref{tab:MSP}. The stellar source closest to the radio position turned out to be a very blue object, located at $\\alpha=13^{\\rm h}\\,42^{\\rm m}\\,11\\fs0881$ and $\\delta=28^\\circ\\,22'\\,40\\farcs141$, at only $0.01 \\arcsec$ from the radio MSP position. The finding chart of this object in the three different filters is reported in Figure~\\ref{fig:chartB}. In the color-magnitude diagram (CMD) this star is located along the red side of the WD cooling sequence, in a region where WDs with a He core are expected to be found (see Figure~\\ref{fig:cmd}). The combination of the excellent agreement between the radio and the optical position and the peculiar location in the CMD allows us to safely conclude that the identified WD is the companion star to M3B. Its magnitudes in the three filters are the following: $m_{F275W}=22.45\\pm0.05$, $m_{F336W}=22.75\\pm0.05$ and $m_{F438W}=23.4\\pm0.2$. \nAlthough multiple epoch images are available for each filter, no evidence of photometric variability has been observed for the companion star. While this could be due to the poor and sparse orbital period coverage provided by the available data-set, {we stress that the photometric variability is only rarely observed for degenerate companion stars and is not due to the re-heating of the star by the MSP emitted energy, which is generally negligible, but most likely to pulsations (global stellar oscillations) of the WD itself \\citep[e.g.][]{maxted13,kilic15}.} \n\n\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[scale=0.4]{cmdwdB.pdf}\n\\caption{Same as in Figure~\\ref{fig:cmd} but zoomed into the WD region. As reported in the top legend, the blue curve is a cooling track for $0.55 \\, M_{\\odot}$ C-O WDs from the {\\it BaSTI} database, while the cyan curves are He WD cooling tracks with masses, from left to right, of $0.24 \\, M_{\\odot}$, $0.23 \\, M_{\\odot}$, $0.20 \\, M_{\\odot}$, $0.19 \\, M_{\\odot}$ and $0.18 \\, M_{\\odot}$, calculated at the cluster metallicity $(Z\\sim0.0005)$ following \\citet{istrate14,istrate16}. Points at different cooling ages are highlighted with gray symbols as reported in the secondary legend.}\n\\label{fig:cmdwd}\n\\end{figure}\n\nWe can now compare the measured magnitudes with those predicted by He WD cooling sequences in order to derive the main physical properties of the companion star. As a first step, we checked the validity of the photometric calibration by comparing the observed sequence of standard WDs with a theoretical cooling model for C-O WDs with masses $M=0.55 \\ M_{\\odot}$ taken from the {\\it BaSTI} database \\citep{salaris10}. Absolute magnitudes have been reported to the observed frame by adopting a color-excess $E(B-V)=0.01$, a distance modulus $(m-M)_V=15.07$ \\citep{harris10}, and appropriate extinction coefficients for each filter: $A_{F275W}\/A_V =1.94$, $A_{F336W}\/A_V =1.66$ and $A_{F438W}\/A_V =1.33$ \\citep{cardelli89,odonnell94}. The corresponding curve is shown in Figure~\\ref{fig:cmdwd} and its agreement with the observed WD sequence confirms the accuracy of the adopted photometric calibration and cluster parameters.\n\nIn the case of the He WDs, we built a set of theoretical tracks using the new models for extremely low-mass He WD by \\citet{istrate14,istrate16}. These models are particularly appropriate for the present case study, since they simulate the entire evolution of systems within the low-mass X-ray binary channel, also including the effects of rotational mixing and element diffusion {and adopting the appropriate cluster metallicity for the secondary stars}. A selection of the obtained cooling tracks is shown in Figure~\\ref{fig:cmdwd} (cyan lines).\n\nIt is now possible to use the theoretical tracks to constrain the main physical properties of the detected WD such as its mass, surface temperature and cooling age\\footnote{The WD cooling age is defined, according to \\citet{istrate16}, as the time passed since the proto-WD reached the maximum surface temperature along the evolutionary track.}. To do this, we used an approach similar to that implemented by \\citealt{dai17} (see also \\citealt{pallanca13a}, \\citealt{testa15}). We interpolated all the available models to create a fine grid in the mass range $0.16 \\, M_{\\odot}$ - $0.35 \\, M_{\\odot}$, temperature range 9000~K~-~21000~K and cooling age range 0.1~Gyr~-~3~Gyr. Assuming Gaussian photometric errors,  we thus estimated the likelihood of each point of the grid (i.e. of each possible combination of the three parameters) as: \n$$\nL = \\prod_{j} \\frac{exp\\left[\\frac{-\\left(m_j - m_j^{mod}\\right)^2}{2\\delta_j^2}\\right]}{\\sqrt{2\\pi \\delta_j^2}} \n$$\nwhere the index $j$ runs through the three photometric filters F275W, F336W and F438W, $m_j$ and $\\delta_j$ are the observed magnitudes and related errors and $m_j^{mod}$ is the magnitude predicted by the models at that point of the grid (see equation 5 in \\citealt{dai17}).\n\nThe marginalized 1D and 2D likelihood distributions are presented in the corner plot\\footnote{see \\url{https:\/\/corner.readthedocs.io\/en\/latest\/} \\citep{foreman16}} of Figure~\\ref{fig:cornerM3B}. For each parameter, the best-fit value, the lower and the upper uncertainties have been estimated as the $50^{th}$, $16^{th}$ and $84^{th}$ percentile of its likelihood distribution, respectively. We therefore found that the He WD has a mass $M_{COM}~=~0.19~\\pm~0.02~ \\ M_{\\odot}$, a cooling age of $1.0^{+0.2}_{-0.3}$ Gyr and an effective temperature of $T=12\\pm 1\\cdot10^3$ K. The derived cooling age is consistent with the MSP spin-down age ($>1.1$~Gyr) estimated by \\citet{hessels07}.\n\\begin{figure}[b] \n\\centering\n\\includegraphics[scale=0.45]{cornerM3B.pdf}\n\\caption{Constraints on the mass, cooling age and surface temperature of the companion star to M3B. The 1D histograms show the likelihood marginalized distributions for each of the three parameters and the blue solid and black dashed lines are, respectively, the $50^{th}$, $16^{th}$ and $84^{th}$ percentiles of each distribution, that have been used as estimates of the best-fit value of each parameter and their related uncertainty. The contours in the 2D histograms correspond to $1\\sigma$, $2\\sigma$ and $3\\sigma$ levels and the best values for each parameter are marked with the blue point and lines. The text at the top  reports the derived mass, cooling age and temperature values.}\n\\label{fig:cornerM3B}\n\\end{figure}\nFrom the best-fit model we have also inferred that the proto-WD phase\\footnote{{ It is the phase following the mass-transfer stage, when the He core contracts at an almost constant luminosity before starting its cooling phase \\citep[see][]{istrate14,istrate16}}.} lasted for $1.0^{+0.1}_{-0.5}$ Gyr and that this star is composed of a core with a mass of about $0.18 \\ M_{\\odot}$ and a thin envelope with mass of about $0.01 \\ M_{\\odot}$. The total He mass is around $0.187 \\ M_{\\odot}$, while the H mass is only around $0.003 \\ M_{\\odot}$. Finally, its central density and  temperature are around $1.3 \\cdot 10^{5}$~g\/cm$^{-3}$ and $2\\cdot 10^{7}$~K, respectively.\n\n{{The companion star parameters just derived can be used to investigate the physical properties of its progenitor star.} We used the {\\rm PARSEC} evolutionary tracks \\citep{bressan12,bressan13} to study the evolution of the He core of isolated stars with different masses and ages at the cluster metallicity. Assuming a cluster stellar population age of $12.5\\pm0.5$ Gyr \\citep{dotter10} and using the derived cooling age and proto-WD phase duration, we can estimate that the mass-transfer stopped (i.e. the Roche-Lobe detachment occurred) when the cluster had an age around $10.5\\pm0.8$ Gyr. {Within this age range, only stars in the mass range $0.82 \\ M_{\\odot} - 0.85 \\ M_{\\odot}$ have grown a He core with a mass comparable with that derived for the companion to M3B (see left panel of Figure~\\ref{fig:massacore}). On the other hand, at an age $t=10.5\\pm0.8$ Gyr and at the cluster metallicity, the mass of a star at the main sequence turn-off is $0.81^{+0.01}_{-0.02} \\ M_{\\odot}$ and stars as massive as $0.82 \\ M_{\\odot} - 0.85 \\ M_{\\odot}$ are already evolved toward the red giant branch. Indeed, as shown in the evolutionary tracks plotted in right panel of Figure~\\ref{fig:massacore}, stars with mass in the range $0.82 \\ M_{\\odot} - 0.85 \\ M_{\\odot}$ have grown a $0.19\\pm0.02 \\ M_{\\odot}$ He core at the base of the red giant branch. By assuming that the mass-transfer phase lasted approximately $\\sim1$ Gyr, we can infer that the it started just after the star left the main sequence stage and therefore the bulk of the mass-transfer essentially occurred during the sub-giant branch phase, where the star is expected to expand as a consequence of its canonical evolution and so is able to fill its Roche-Lobe. Therefore the observational properties of the M3B companion measured here appear to be fully consistent with a suitable scenario for the formation of the MSP.}}\n\n\\begin{figure}[b!]\n\\centering\n\\includegraphics[scale=0.32]{masscore2.png}\n\\includegraphics[scale=0.32]{hr.png}\n\\caption{{{\\it Left Panel:} He core mass as a function of the stellar mass  as predicted by \\citet{bressan12,bressan13} evolutionary tracks. The black solid curve and the light gray shaded area represent the values for an age of $10.5\\pm0.8$ Gyr. The horizontal line (and dark gray band) marks the mass of the He WD companion to M3B (and its related uncertainty) as derived in Section~\\ref{sec:comM3B}. {\\it Right Panel:} evolutionary tracks for different stellar masses as reported in the top-left legend. The red region of the tracks highlights the phase where the stars have grown a He core with a mass comparable with the one measured for the companion to M3B at an age of $10.5\\pm0.8$ Gyr (corresponding to the mass-transfer end, i.e. the Roche-Lobe detachment phase). The blue shaded area marks the region of the tracks 1 Gyr before the Roche-Lobe detachment, when the mass-transfer probably started.}}\n\\label{fig:massacore}\n\\end{figure}\n\n\nThe He WD mass here obtained ($M_{COM}$), combined with the MSP mass function ($f$) derived from radio timing (see Table~\\ref{tab:MSP}) can be use to constrain the orbital inclination angle $i$ and, most importantly, the mass of the NS ($M_{NS}$). Indeed these quantities are related by the following equation:\n\\begin{equation}\nf(M_{NS}, M_{COM}, i) = \\frac{(M_{COM}\\sin{i})^3}{(M_{NS}+M_{COM})^2}\n\\end{equation}\nWe used the affine-invariant Markov Chain Monte Carlo (MCMC) ensemble sampler {\\rm emcee\\footnote{\\url{https:\/\/emcee.readthedocs.io\/en\/stable\/}}} \\citep{foreman13} to constrain $M_{NS}$ and $i$. We set a Gaussian prior on $M_{NS}$, centered at $1.4 \\ M_{\\odot}$, which is the mass typically measured for NSs in binary MSPs \\citep{ozel16}, and with a standard deviation of $0.5 \\ M_{\\odot}$, {large enough to include the observed NS mass distribution \\citep{antoniadis16}}. $M_{NS}$ is sampled in the range $0.5 \\ M_{\\odot}$ - $2.5 \\ M_{\\odot}$. On the other hand, we set an uniform prior on the distribution of $\\cos{i}$ in the range 0 - 1. The results are presented in Figure~\\ref{fig:mpsrB}. The best value for the NS mass is $M_{NS}=1.1\\pm 0.3 \\ M_{\\odot}$, {thus suggesting that this system hosts a low-mass NS, although the value is still compatible with the typical NS masses measured for these kind of systems. Results do not change if wider NS mass distributions are used}. The probability distribution of the inclination angle is shaped as a truncated Gaussian and clearly shows that this system is observed almost edge-on. We therefore assumed that the best value for the inclination angle corresponds to the maximum in the probability distribution and its lower uncertainty to the $16^{th}$ percentile of an identical but symmetric probability distribution. Doing so we have estimated the inclination angle to be $i=89^{+1}_{-26}$ degrees. { All the physical properties of the binary here derived are summarized in Table~\\ref{tab:comMSP}.}\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=0.6]{corner_mpsrB.pdf}\n\\caption{Constraints on the mass of the NS and on the orbital inclination angle of M3B. The 1D histograms are the marginalized probability distributions of the two parameters, where the solid blue and black dashed lines are the best values and their related uncertainties (see text). The bottom left panel is the joint 2D posterior probability distribution and the contours corresponds to $1\\sigma$, $2\\sigma$ and $3\\sigma$ confidence levels.}\n\\label{fig:mpsrB}\n\\end{figure}\n\n\n\n\n\\section{A He WD ORBITING M3D TOO?}\\label{sec:comM3D}\n\n\\begin{figure}[h] \n\\centering\n\\includegraphics[scale=0.16]{chartD.png}\n\\caption{Left panel: $15\\arcsec \\times 18\\arcsec$ chart of the region surrounding M3D, obtained from a F275W image. The black box is centered on the MSP position and corresponds to the error box quoted in \\citet{hessels07}. The red circle indicates the position of the candidate companion. Right panel: same as in the left panel, but zoomed into the position of the candidate companion.}\n\\label{fig:chartD}\n\\end{figure}\n\n\\begin{figure}[h!] \n\\centering\n\\includegraphics[scale=0.35]{cmd_M3D.pdf}\n\\includegraphics[scale=0.35]{cmdwdD.pdf}\n\\caption{{{\\it Left panel:} same as in Figure~\\ref{fig:cmd}. The position of the candidate companion star to M3D is marked with a large red square, while the positions of all the other stars within the investigated area is marked with smaller red squares. {\\it Right panel:} same as in Figure~\\ref{fig:cmdwd} but for the candidate companion to M3D.}}\n\\label{fig:cmdD}\n\\end{figure}\n\nThe timing solution of M3D reported by \\citet{hessels07} provides the celestial position of this binary affected by a modest uncertainty ($0.6\\arcsec$) in $\\alpha$ and a large uncertainty ($14\\arcsec$) in $\\delta$. Therefore the search for the optical counterpart to this system is challenging. We carefully investigated all the stellar sources within and nearby the position error box: {this region of interest is centered on the best MSP position (see Table~\\ref{tab:MSP}) and is as large as the $3\\sigma$ uncertainty in $\\alpha$ and $2\\sigma$ uncertainty in $\\delta$. We only found a possible candidate in a position marginally compatible with the radio one (see Figure~\\ref{fig:chartD} and Figure~\\ref{fig:cmdD}). In fact, all the stars detected within the investigated area likely belong to the canonical evolutionary sequences but one source located again on the red side of the WD cooling sequence, in a position compatible with that of the He WDs}. This candidate counterpart is located at $\\alpha=13^{\\rm h}\\,42^{\\rm m}\\,10\\fs3041$ and $\\delta=28^\\circ\\,22'\\,44\\farcs786$. The right ascension is shifted by $1.5\\arcsec$ from the radio value, which is larger than the combined radio uncertainty and optical astrometric precision.  Its observed magnitudes are: $m_{F275W}=23.06\\pm0.08$, $m_{F336W}=23.35\\pm0.08$ and $m_{F438W}=24.1\\pm0.3$.\nFollowing the same method used for M3B and presented in Section~\\ref{sec:comM3B}, we found that this star has a mass $M_{COM}=0.22 \\pm 0.02 \\ M_{\\odot}$, a cooling age of $1.1^{+0.7}_{-0.6}$ Gyr and an effective temperature of $13\\pm 2 \\cdot 10^{3}$ K. Using the MSP mass function (Table~\\ref{tab:MSP}) we estimated the NS mass to be $M_{NS}=1.3 \\pm 0.3 \\ M_{\\odot}$ and the orbital inclination angle to be $i=72\\pm16$ degrees (see Table~\\ref{tab:comMSP}). Again, the $M_{NS}$ value would be in agreement with the typical values observed for binary MSPs.\nWe tried to establish phase-connection of the MSP timing solution using the available radio data and fixing the MSP position to that of the candidate counterpart. Unfortunately, this did not lead to an improvement of the timing solution obtained by \\citet{hessels07}. \nGiven all this, we cannot solidly confirm that the detected He WD is the optical counterpart to M3D, {which could be still under the detection threshold. Furthermore, the peculiar evolution that this system likely underwent implies that the companion star might not be a classical He WD. However it is worth noting that a He WD has been identified orbiting B1620$-$26 in M4, the only similar MSP with an optical counterpart to date \\citep{sigurdsson03}.} Future radio observations and timing analysis are needed to improve the position measurement of this object.\n\n\n\\begin{deluxetable*}{lcc}\n\\tablecolumns{3}\n\\tablewidth{0pt}\n\\tablecaption{Physical properties of the binaries M3B and M3D as derived in Section~\\ref{sec:comM3B} and ~\\ref{sec:comM3D}. \\label{tab:comMSP}}\n\\tablehead{\\colhead{Parameter} & \\colhead{M3B} & \\colhead{M3D?}}\n\\startdata\nCompanion mass, $M_{COM} \\ (M_{\\odot})$\\dotfill  & $0.19\\pm0.02$ & $0.22\\pm0.02$    \\\\\nEffective temperature, $T$ ($10^3$ K) \\dotfill & $12\\pm1$ & $13\\pm2$ \\\\\nCooling age, Age (Gyr) \\dotfill & $1.0^{+0.2}_{-0.3}$ & $1.1^{+0.7}_{-0.6}$ \\\\\nNeutron star mass, $M_{NS} \\ (M_{\\odot})$\\dotfill  & $1.1\\pm0.3$ & $1.3\\pm0.3$    \\\\\nInclination angle, $i$ (deg) \\dotfill & $89^{+1}_{-26}$ & $72\\pm16$ \\\\\n\\hline\n\\enddata\n\\end{deluxetable*} \n\n\\section{CONCLUSIONS}\\label{sec:conc}\n\nWe used deep and high resolution images obtained at near-UV and optical wavelengths to search for the companion stars to the binary MSPs in the GC M3. By exploiting the ``UV route'' to dilute the crowding issues in the cluster center and increase the sensitivity to blue\/hot stars, we have been able to firmly identify the companion star to the canonical MSP M3B and find a candidate counterpart to the anomalous (long period and mild eccentricity) system M3D.\nThe companion star to M3B turned out to be a WD with a He core, as expected from the canonical formation scenario. Interestingly, despite the fact that this cluster hosts a stellar population with an intermediate metallicity, the companion is an extremely low-mass object, with a mass of about $0.19 \\ M_{\\odot}$. Indeed, the lower is the metal content of the secondary stars, the larger is expected to be the minimum mass of the WD remnants. This is due to the fact that low metallicity stars have shorter evolutionary timescales and smaller radii and therefore their Roche Lobe is filled (i.e. the mass-transfer starts) in a more advanced stage of their evolution, when the He core is grown more massive than it would have for a more metal-rich system \\citep{istrate16}. Our mass measurement is indeed close to the minimum possible mass produced by the adopted models at the cluster metallicity. This stresses the importance of the study of extremely low-mass WD systems, with special care to the physical processes occurring during the evolution. In fact, the models we used take into account effects like rotational mixing and element diffusion. Models not including these effects are characterized by bluer cooling tracks that would have not been able to reproduce the observed magnitudes of the companion or, in the case they do, the resulting companion mass would have been so small that the corresponding NS mass becomes unreasonable (i.e. $<1 \\ M_{\\odot}$). {We have also shown that the progenitor of this WD was likely a $\\sim0.83 \\ M_{\\odot}$ star which filled its Roche-Lobe after leaving the main sequence, thus implying that the bulk of the mass-transfer activity occurred during the sub-giant branch phase. All the derived physical properties of the companion star, combined with the information obtained through radio timing, allowed to infer that this binary is observed almost edge-on and probably hosts a low-mass NS with a mass around $\\sim 1.1 \\ M_{\\odot}$.}\n\nIn the case of the candidate companion star to M3D, we have identified a He WD at a position marginally compatible with the highly uncertain radio one. Therefore the association between this degenerate object and the MSP cannot be confirmed yet. This WD is again a low-mass object with a mass around $0.22 \\ M_{\\odot}$ and a cooling age of $\\sim1$~Gyr\n\n{\\citet{tauris99} discussed the correlation between the masses of the WD companion stars and the orbital periods of the binary MSPs (the so-called ``TS99'' relation; see also \\citealt{istrate14} for a more recent investigation on this). According to this relation, the $\\sim1.4$~days orbit of M3B implies a companion star with a mass of $\\sim0.21 \\ M_{\\odot}$, fully compatible, within the uncertainties, with the mass value derived in this work. On the other hand, the long $\\sim 129$ days orbit of M3D implies a companion star with a mass around $\\sim 0.35 \\ M_{\\odot}$, a value significantly larger than the one here derived for its candidate companion star. This could suggests that the proposed optical counterpart is not the real companion star, even though its mass is compatible with the measured MSP mass function if a typical NS mass is assumed. If this WD is truly the companion star to M3D, the discrepancy between the measured mass and the one predicted by the TS99 relation could be explained by the formation of this system through an exchange interaction occurred after the NS recycling phase. Such an exchange, however, is expected to produce highly eccentric binaries \\citep[see][]{prince91,freire04,lynch12}, while only a mild eccentricity is measured for this system.}\n\n\\acknowledgments\n{We thank the anonymous referee for the careful reading of the manuscript}. This research is part of the scientific project Cosmic-Lab at the Department of Physics and Astronomy at Bologna University. \n\n\\vspace{5mm}\n\\facilities{HST(WFC3)}\n\\software{DAOPHOT \\citep{stetson87}, {\\rm emcee} \\citep{foreman13}, {\\rm corner.py} \\citep{foreman16}}\n\n\n\n\n\\section{} \n\n\\textit{Research Notes of the \\href{https:\/\/aas.org}{American Astronomical Society}}\n(\\href{http:\/\/rnaas.aas.org}{RNAAS}) is a publication in the AAS portfolio\n(alongside ApJ, AJ, ApJ Supplements, and ApJ Letters) through which authors can \npromptly and briefly share materials of interest with the astronomical community\nin a form that will be searchable via ADS and permanently archived.\n\nThe astronomical community has long faced a challenge in disseminating\ninformation that may not meet the criteria for a traditional journal article.\nThere have generally been few options available for sharing works in progress,\ncomments and clarifications, null results, and timely reports of observations\n(such as the spectrum of a supernova), as well as results that wouldn't\ntraditionally merit a full paper (such as the discovery of a single exoplanet\nor contributions to the monitoring of variable sources). \n\nLaunched in 2017, RNAAS was developed as a supported and long-term\ncommunication channel for results such as these that would otherwise be\ndifficult to broadly disseminate to the professional community and persistently\narchive for future reference.\n\nSubmissions to RNAAS should be brief communications - 1,000 words or fewer\n\\footnote{An easy way to count the number of words in a Research Note is to use\nthe \\texttt{texcount} utility installed with most \\latex\\ installations. 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Each RNAAS article is\nissued a DOI and indexed by ADS \\citep{2000A&AS..143...41K} to create a\nlong-term, citable record of work.\n\nArticles can be submitted in \\latex\\ (preferably with the new \"RNAAS\"\nstyle option in AASTeX v6.2), MS\/Word, or via the direct submission in the\n\\href{http:\/\/www.authorea.com}{Authorea} or\n\\href{http:\/\/www.overleaf.com}{Overleaf} online collaborative editors.\n\nAuthors are expected to follow the AAS's ethics \\citep{2006ApJ...652..847K},\nincluding guidance on plagiarism \\citep{2012AAS...21920404V}.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.85,angle=0]{aas.pdf}\n\\caption{Top page of the AAS Journals' website, \\url{http:\/\/journals.aas.org},\non October 15, 2017.  Each RNAAS manuscript is only allowed one figure or\ntable (but not both). Including the\n\\href{http:\/\/journals.aas.org\/\/authors\/data.html\\#DbF}{data behind the figure}\nin a Note is encouraged, and the data will be provided as a link in the\npublished Note.\\label{fig:1}}\n\\end{center}\n\\end{figure}\n\n\n\\acknowledgments\n\nAcknowledge people, facilities, and software here but remember that this counts\nagainst your 1000 word limit.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nThe state of the inhomogeneous cosmic matter distribution (and the time-dependent\ndark energy density) can be measured by\nthe gravitational lensing shearing and magnification of background luminous\nsources. Using galaxies as the back lights for gravitational lensing (``cosmic shear'')\nallows tomographic reconstruction of the matter distribution along the line of sight.\nBut, inference of galaxy lensing shears is clouded by the unknown distribution\n(and possibly correlation) of galaxy intrinsic morphologies as well as image systematics\nthat induce spurious ellipticity correlations.\n\nCommon inference algorithms for gravitational lensing shear involve cross-correlating\nthe ellipticities of galaxies in a two-point function estimator under the assumption\nof calibrated intrinsic ellipticity distributions and alignments~\\citep[e.g.,][]{jee2015}.\nThis is a lossy procedure because the angular phase information in the lensing\nshear and magnification fields on the sky is discarded. Traditional algorithms\nare also necessarily biased because of the need to calibrate the unknown galaxy\nellipticity distributions. Said another way, the two-point function summary statistics of\ncosmological large-scale structure do not capture the full statistical information in the multivariate\ndistribution of the lensing observable\n\\citep[e.g.,][]{pan2005,hamilton2006,takada2013,carron2014,2016arXiv160501100P}\n\nIn this paper we revisit the problem of inferring the lensing convergence posterior distribution\nfrom a catalog of galaxy ellipticities to potentially capture more cosmological information from\ncosmic shear than is available in two-point function estimators of the shear. We pursue a\nprobabilistic approach to lensing convergence inference as a means to propagate the large\nuncertainties that can come in the presence of galaxy shape noise, finite survey areas, masking,\nand sample selection. Moreover, we aim to build an inference framework that can fit within our previous\nwork on probabilistic cosmic shear~\\citep{mbi-theory}.\n\n\\citet{1990ApJ...349L...1T} first presented a method to reconstruct the projected mass distribution\nfrom weak gravitational lensing shears of source galaxy images.\n\\citet{1993ApJ...404..441K} derived the lensing convergence in terms\nof the shear (a nonlocal relation) and applied this theory to estimators of the convergence given a\nmeasured galaxy ellipticity catalog. The method of \\citet{1993ApJ...404..441K}, while theoretically sound,\nrequires unbroken sky coverage and low-pass filtering to yield finite noise in the lensing\nconvergence estimator.\nThe \\citet{1993ApJ...404..441K} method remains useful, however, for visualization\npurposes~\\citep{2015PhRvL.115e1301C,2015PhRvD..92b2006V}.\nSubsequent papers have extended, applied, and explored the limitations of filtering algorithms for\ncluster mass mapping~\\citep{1995A&A...303..643B,1995A&A...294..411S,1998A&A...335....1L,2002ApJ...568..141G,2005A&A...440..453D,2007Natur.445..286M,2012A&A...540A..34D,2012MNRAS.424..553A,2013MNRAS.433.3373V}. Remaining data analysis challenges\nusing such methods include finite survey boundaries and masks,\nseparation of E and B modes in the shear\nfield, noise or significance characterization for shear or convergence extrema, and the requirement\nin many algorithms to smooth or average the ellipticities of galaxies before the convergence inference\nprocess.\n\n\nMaximum likelihood (ML) estimators for the lensing\nconvergence~\\citep[e.g.,][]{1996ApJ...464L.115B,1998A&A...337..325S,2008MNRAS.385.1431H,2009ApJ...702..980K}\ncan help mitigate biases arising from survey masks and admit mathematically consistent noise\ncharacterization~\\citep{2000MNRAS.313..524V}. However such estimators are biased. And many\napproaches often still require a preliminary smoothing of the observed galaxy ellipticity field.\n\n\\citet{bridle1998} and \\citet{bridle2000} introduced a `maximum entropy'\nBayesian prior for the lensing convergence from information theoretic and Bayesian analysis perspectives to\nderive an estimator for the projected mass distribution of galaxy clusters with desirable noise\nproperties in a finite field.\n\\citet{Marshall:531310} and \\citet{2001A&A...368..730S} refined the maximum entropy method to\nspecify the optimal smoothing\nlength scale for galaxy cluster mass inference via the Bayesian evidence of the observed\nellipticities. \\citet{2013ascl.soft08004M} released a code\\footnote{{\\sc LensEnt2}\\xspace, \\url{http:\/\/www.slac.stanford.edu\/~pjm\/lensent\/version2\/index.html}}\nimplementing the algorithm from \\citet{Marshall:531310}.\nThe algorithm of \\citet{Marshall:531310} is close to meeting all the requirements for the current\nanalysis, except the choice of smoothing scale and application to field rather than cluster lensing\nis not demonstrated in the literature. We will show further benefits of the algorithm developed\nin this paper below.\n\\citet{2006A&A...451.1139S} applied a modification of the maximum entropy method to a cosmic\nshear analysis using $N$-body cosmological simulations to create mock observations.\nThe work of \\citet{2006A&A...451.1139S} and also \\citet{jiao2011} include maximum entropy algorithm\nmodifications to better handle masking, shape noise, and degrees of smoothing.\n\nFor constraining cosmological parameters, several groups have considered the abundance of peaks\nin lensing convergence maps~\\citep{2010PhRvD..81d3519K,2011ApJ...735..119S,2014MNRAS.442.2534S,2015PhRvD..91f3507L,2016MNRAS.456..641R,2016PhRvD..94d3533L,PhysRevLett.117.051101,2016arXiv160501100P}. These studies have a common\napproach in direct calculation or estimation of lensing peaks without an attempt to infer\nassociated peaks in the 3D cosmological mass density. The measurement process is thus direct as\nlensing by multiple structures or voids along any given line of sight can easily confuse the\n2D to 3D mass inference. Just as the abundance of galaxy clusters provides tight constraints on the\ncosmological model, so too can the abundance of lensing peaks.\nHowever, the estimators for lensing peaks often involve averages or line integrals over contiguous\nsky areas, which are confounded by survey masks~\\citep{0004-637X-784-1-31}.\nThe bias introduced by masking may be overcome with forward simulations of the nonlinear lensing\nconvergence field, which has been achieved with $N$-body cosmological simulations of large-scale\nstructure combined with ray-tracing predictions of the lensing statistics~\\citep[e.g.,][]{2016ApJ...819..158B,2016arXiv160501100P}.\n\nIn a complimentary approach to cosmological parameter inference, other groups have considered a \nlinear theory approximation to (suitably smoothed) lensing mass maps. \nSpecifically, if (i) the shear field can be approximated as Gaussian distributed, (ii) the noise \ndistribution in the galaxy ellipticity measurements (and intrinsic shape distribution) can also \nbe approximated as Gaussian distributed, and (iii) the weak shear approximation is valid, then \nmass maps can be obtained using the Wiener Filter (WF)~\\citep{wiener1949extrapolation}. \nThe WF is a well-studied signal-inference algorithm that has been applied to cosmic microwave \nbackground (CMB) analysis with great success in a Bayesian context~\\citep{2004PhRvD..70h3511W}.\n\\citet{Alsing2015} and \\citet{2016arXiv160700008A} have recently extended the WF technology from\nthe CMB literature to the problem of weak lensing shear field and power spectrum inference. However,\nthese WF approaches are limited not only by the WF assumptions above but also by requirements\nto smooth the measured galaxy ellipticity field to a uniform grid on the sky and to work solely\nwithin a linear theory approximation for the distribution of cosmological mass density perturbations.\nWorking within the stated assumptions, the primary challenge for WF approaches is the computation\nof large matrix solve operations. Novel and effective algorithms for sampling from the WF\ndistribution have been demonstrated for the CMB~\\citep{2013A&A...549A.111E,2016ApJ...820...31R}\nthat are also effective for smoothed, weak-shear, and linear theory cosmic shear\ninferences~\\citep{2016arXiv160700008A}.\n\n\nThis paper is structured as follows. In \\autoref{sec:method} we describe the statistical\nframework for lensing convergence and shear inference given galaxy images or a galaxy\nellipticity catalog. In \\autoref{sec:framework} we describe the Gaussian Process (GP)\nprior for the lensing potential and how this informs the correlated inferences of shear\nand convergence. We give a prescription for GP parameter optimization in\n\\autoref{sub:optimizing_interpolation_parameters}.\nWe apply our method to infer lensing convergence maps of simulated ellipticity catalogs\nin \\autoref{sub:simulation_study} and of an observed galaxy cluster in \\autoref{sub:abell}.\nWe describe our main conclusions in \\autoref{sec:conclusions}.\nWe provide details of the GP covariance derivation for lensing shear and convergence\nfields in \\autoref{sec:Gaussian process covariances} and provide\nlensing map inferences under an analogous covariance\nfor a cosmological model with a linear theory approximation in\n\\autoref{sec:cosmology_dependent_covariance_model}.\n\n\\section{Method}\n\\label{sec:method}\n\nWe describe a joint probabilistic model for the lensing shear and convergence given galaxy imaging\ndata. We previously presented the complete statistical framework for cosmic shear\ninference~\\citep{mbi-theory} and here focus on the specifics of lensing convergence inference\nas a function of sky coordinates in both linear and nonlinear regimes of the cosmological\nmass density perturbations. In \\autoref{fig:pgm} (left panel) we show the relationships in\nthe probabilistic model for CCD galaxy imaging data for multiple observation epochs, multiple\ngalaxies, and multiple galaxy samples (e.g., samples selected in different photometric redshift bins).\n\\begin{figure*}[!htb]\n  \\centerline{\n    \\includegraphics[width=0.48\\textwidth]{shear_general_pgm.png}\n    \\includegraphics[width=0.4\\textwidth]{shear_gp_pgm.png}\n  }\n  \\caption{Two successive levels of approximation for our statistical model for sampling\n  probabilistic lensing shear and convergence $\\Upsilon$ fields.\n  The unshaded circles indicate sampling parameters.\n  Shaded circles indicate observed parameters while dots indicate parameters with fixed values\n  rather than being sampled.\n  %\n  Left panel: The model for galaxy image pixel data ${\\mathbf{d}}_{n,i}$ for each galaxy $n$ and observation\n  epoch $i$ requires specification of the pixel noise $\\sigma_{n,i}$, the intrinsic (i.e., unlensed)\n  galaxy ellipticity $e_n$, the lensing fields $\\Upsilon$, and the PSF at each galaxy location in\n  each epoch ${\\Pi}_{n,i}$. The distribution of galaxy intrinsic ellipticities is described by the\n  parameters $\\alpha_n$, which can specify distinct distributions for each galaxy $n$.\n  In \\citet{mbi-theory} we infer marginal constraints on $\\alpha_n$ under a Dirichlet Process (DP)\n  prior. For this initial study, we assert a Gaussian Process (GP) prior on the lens fields\n  $\\Upsilon$ in this paper, with the assumption that the posterior inferences of $\\Upsilon$\n  will be related to a cosmological model in a separate analysis pipeline as we describe in the\n  text. Here we assert a known PSF ${\\Pi}_{n,i}$ at every galaxy location, again deferring the\n  inference and marginalization of PSFs to a separate paper. The inference of the lens fields also\n  depends on our assertion of the survey window function $W$.\n  %\n  Right panel: The approximate statistical model when the galaxy imaging pixel data is summarized as\n  a galaxy ellipticity catalog ($e_{n}, \\sigma_{e,n}$ for $n=1,\\dots,n_{\\rm gal}$) requires specification of\n  the intrinsic ellipticity distribution parameters $\\alpha$ (now assumed the same for all galaxies),\n  the ellipticity measurement uncertainties $\\sigma_{\\rm ms,n}$ per galaxy, and the lens fields\n  $\\Upsilon$.\n  We now also assert the variance, $\\alpha$ of the assumed zero-mean\n  intrinsic ellipticity distribution.\n  }\n  \\label{fig:pgm}\n\\end{figure*}\n\n\n\\begin{table*\n\\begin{center}\n\\caption{Parameters for the statistical model. }\n\\label{tab:sampling_parameters}\n\\begin{tabular}{cl}\n\\hline\nParameter & Description \\\\\n\\hline\n$i$ & index over epochs, or different exposures of a galaxy\\\\\n$n$ & index over galaxies\\\\\n${\\Pi}_{n,i}$ & Point Spread Function (PSF) for galaxy $n_s$ in epoch $i$ \\\\\n$e_{n}$ & Intrinsic (pre-lensing) ellipticity of a galaxy\\\\\n$\\Upsilon_{s}$ & lensing shear and convergence that modify a galaxy image\\\\\n$\\psi_{s}$ & lensing potential \\\\\n$\\theta$ & cosmological parameters\\\\\n$W$ & survey window function \\\\\n$\\sigma_{n,i}$ & noise properties in a galaxy image\\\\\n$\\alpha_{n}$ & parameters specifying the intrinsic ellipticity distribution for a galaxy\\\\\n$a_{\\rm DP}$ & parameters specifying the distribution over $\\alpha_{n_s}$\\\\\n${\\bf d_{n}}$ & data vector (measured $e_{1,2}$ for each galaxy $n$)\\\\\n$\\sigma_{{\\rm ms}; n}$ & ellipticity measurement error for galaxy $n$ \\\\\n$\\mathbf{x}$ & vector of 2D spatial locations of galaxies \\\\\n$\\epsilon^\\mathrm{int}_{n}$ & intrinsic galaxy ellipticity for galaxy $n$ \\\\\n$\\alpha\\equiv\\sigma_{e}^2$ & parameters of the distribution of galaxy parameters \\\\\nW & window function for the survey footprint \\\\\n$a_{\\rm GP}$ & parameters of the GP kernel\\\\\n$\\lambda_{\\rm GP}$ & precision of the GP kernel (element of $a_{\\rm GP}$) \\\\\n$\\ell^{2}_{\\rm GP}$ & squared GP correlation length (element of $a_{\\rm GP}$) \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\nThe model for observed galaxy images requires (at least) specification of the point-spread\nfunction (PSF), ${\\Pi}_{n_s,i}$, at all $n_s$ galaxy locations in all epochs $i$, the intrinsic\nor pre-lensing shape or morphology of the galaxy,\n$e_{n_s}$, the applied lensing shear and magnification, $\\Upsilon_{s}$, and the noise properties\nof the pixelated image, $\\sigma_{n_s,i}$ (e.g., Gaussian with an asserted r.m.s.).\n\nTo simplify the discussion for this paper focused on shear inference, we assume the PSF ${\\Pi}_{n_s,i}$ is\nknown at the locations of all galaxies in all epochs. We do not further consider errors in the\nPSF inference in this work. Following \\citet{mbi-theory} we allow for the intrinsic ellipticity\n(and potentially size, fluxes, etc.) of each galaxy to be drawn from galaxy-specific distributions\nwith parameters $\\alpha_{n_s}$. The parameters $\\alpha_{n_s}$ in turn are hierarchically \ndistributed under a distribution with with parameters $a_{\\rm DP}$, allowing \ninference of the effective `shape noise' in the \nshear inference. We use the label $\\Upsilon$ for \nthe model for the lensing shear $\\gamma$ and convergence $\\kappa$, which uniquely specify the \ncomponents of the trace-free linear distortion matrix  under a weak lensing ($\\kappa\\ll1$) \napproximation of the lens equation~\\citep[e.g.,][]{bartelmann01}.\n\nAs illustrated by the `plates' in the left panel of \\autoref{fig:pgm}, the PSF is unique to each\nobservation epoch\nbut the intrinsic properties of a galaxy image are common across epochs (for each unique galaxy).\nA key feature in the graphical model of the left panel in \\autoref{fig:pgm} is that the lens fields\n$\\Upsilon_{s}$ are common across all galaxy images in all epochs, indicating that the we require\na spatially correlated model for the coherent lensing shear and magnification patterns on the sky.\nSpecifying an appropriate correlated model for the lens fields is the main focus of this paper.\nWe allow the lens field models to be distinct for different galaxy samples $s$ in\n\\autoref{fig:pgm} because, e.g., galaxies in different photometric redshift bins will be lensed\nby partially different foreground mass distributions. Only at the top level of the graphical\nmodel do the lens field inferences from different galaxy samples become connected under a\ncosmological model with parameters $\\theta$.\n\nWe therefore explore an interim probabilistic model for the lens fields inferred from distinct\ngalaxy samples $s$ such that inferences of $\\Upsilon_s$ are statistically independent for\ndifferent $s$. This will allow us to separate computationally expensive components of a cosmic\nshear inference pipeline, explore multiple cosmological and systematics models for the data,\nand eventually perform rigorous uncertainty propagation and marginalization of image and intrinsic\ngalaxy nuisance parameters as outlined in \\citet{mbi-theory}.\n\nIn measuring lensing shear of galaxy images, we must marginalize over the\nintrinsic ellipticities, $\\epsilon^\\mathrm{int}$, of the images. This is often done by averaging the ellipticities\nof galaxies in neighboring regions of the sky and redshift, where the weights can include the\nmeasurement and shape noise models. But, here we perform a\nmarginalization over an explicit intrinsic ellipticity distribution,\n\\begin{multline}\\label{eq:marg_like}\n  \\prf{{\\mathbf{d}} | \\Upsilon(\\mathbf{x}), \\alpha} = \\prod_{n=1}^{n_{\\rm gal}}\n  \\int d^{m}\\epsilon^\\mathrm{int}_n\\,\n  \\prf{{\\mathbf{d}}_n | \\epsilon^\\mathrm{int}_n, \\Upsilon(\\mathbf{x}_n)}\n  \\\\ \\times\n  \\prf{\\epsilon^\\mathrm{int}_n|\\alpha},\n\\end{multline}\nwhere we assume the likelihood functions for each galaxy image $n=1,\\dots,n_{\\rm gal}$ are\nstatistically independent~\\citep[see][for more discussion of this assumption]{mbi-theory}.\nThe data vector ${\\mathbf{d}}_{n}$ is composed of either the pixel values contributing to the image of\ngalaxy $n$ or a summary statistic of those pixel values.\n\nAs a pedagogical step in the development of our probabilistic lens field model,\nwe will consider an approximate likelihood function for summary statistics of the pixel data;\nnamely estimators for the\nellipticity of each galaxy image $e\\equiv e_1 + i e_2$ along with an associated measurement\nerror per ellipticity component $\\sigma_{\\rm ms}$. See the right panel of \\autoref{fig:pgm}.\nWe will develop the approximate model for ellipticity measurements as a data vector\nin this paper but advocate for the more complete algorithm of \\citet{mbi-theory}\nfor any data analysis because of the known large biases in using ellipticity estimators for\ncosmic shear~\\citep[e.g.,][]{Refregier++2012,Kacprzak++2012}.\n\nAssuming Gaussian distributed ellipticity measurement errors the likelihood\nfunction is then,\n\\begin{equation}\\label{eq:likelihood}\n  \\prf{{\\mathbf{d}}_n|e_n, \\Upsilon(\\mathbf{x}_n)} = \\mathcal{N}_{\\hat{e}_n}\n  \\left(e_n, \\sigma^2_{{\\rm ms}; n}\\mathbb{1}} % identity matrix: \\usepackage{bbold_{2}\\right),\n\\end{equation}\nwhere we explicitly label the data ${\\mathbf{d}}_n\\equiv\\hat{e}_{n}$ as ellipticity estimators,\nand the distribution is bivariate given the two ellipticity components.\n\nA general likelihood function depends on the observable galaxy properties\nsuch as ellipticity, size, and flux, which are modified by lensing from the intrinsic\nproperties described by $\\epsilon^\\mathrm{int}$.\nSo, to evaluate \\autoref{eq:likelihood}, we define the lensed galaxy parameters,\n\\begin{equation}\\label{eq:lens_transform}\n  e_n(\\Upsilon(\\mathbf{x}_n)) \\equiv f(\\epsilon^\\mathrm{int}_n, \\Upsilon(\\mathbf{x}_n)),\n\\end{equation}\nwhere $f(\\cdot)$ denotes the function that transforms the intrinsic, unlensed, galaxy\nellipticities $\\epsilon^\\mathrm{int}_n$ to those in the lensed model $e_n$\nunder the action of the lensing convergence and shear specified by $\\Upsilon(\\mathbf{x}_n)$\nat the galaxy sky location $\\mathbf{x}_n$.\nIn the weak shear limit defined by $\\kappa \\ll 1$, \\autoref{eq:lens_transform} reduces to\n\\begin{equation}\\label{eq:weak_shear}\n  \\tilde{e}_{n}^{\\rm weak-shear} = \\epsilon^\\mathrm{int}_{n} + g(\\mathbf{x}_n),\n\\end{equation}\nwhere $g\\equiv\\gamma\/(1-\\kappa)\\approx\\gamma$ is the reduced shear.\n\nFor our pedagogy, we specify a Gaussian distribution for the \\emph{unlensed}\ngalaxy properties $\\epsilon^\\mathrm{int}$ to use in evaluating \\autoref{eq:marg_like},\n\\begin{equation}\n  \\prf{\\epsilon^\\mathrm{int}_{n}|\\alpha,\\Upsilon(\\mathbf{x}_n)} =\n  \\mathcal{N}_{\\epsilon^\\mathrm{int}_n} \\left(0, \\sigma_{e}^2\\mathbb{1}} % identity matrix: \\usepackage{bbold_{2}\\right).\n\\end{equation}\n\nUsing the weak shear approximation in \\autoref{eq:weak_shear}, we can perform\nthe marginalization integral in \\autoref{eq:marg_like} analytically,\n\\begin{align}\n  \\prf{{\\mathbf{d}} | \\Upsilon(\\mathbf{x}), \\alpha} &= \\prod_{n=1}^{n_{\\rm gal}}\n    \\int d^{2}\\epsilon^\\mathrm{int}_n\\,\n    \\mathcal{N}_{\\epsilon^\\mathrm{int}}\\left(\\hat{e}_n - g(\\mathbf{x}_n), \\sigma_{{\\rm ms};n}^2\\mathbb{1}} % identity matrix: \\usepackage{bbold_2\\right)\n    \\notag\\\\\n    &\\quad\\times\n    \\mathcal{N}_{\\epsilon^\\mathrm{int}}\\left(0, \\sigma_{e}^{2}\\mathbb{1}} % identity matrix: \\usepackage{bbold_{2}\\right)\n    \\\\\n    &= \\prod_{n=1}^{n_{\\rm gal}}\n    \\mathcal{N}_{g(\\mathbf{x}_n)}\\left(\\hat{e}_n, \\left(\\sigma_{{\\rm ms};n}^2 + \\sigma_{e}^2\\right)\\mathbb{1}} % identity matrix: \\usepackage{bbold_{2}\\right)\n    \\\\\n    &= \\mathcal{N}_{\\Matrix{g}}\\left(\\hat{\\Matrix{e}}, \\mathsf{N}\\right),\n    \\label{eq:marg_like_analytic}\n\\end{align}\nwhere in the final line we defined the $2n_{\\rm gal}$-length vector of observed ellipticities $\\hat{\\Matrix{e}}$,\nthe same-length vector of reduced shear components at each galaxy locations $\\Matrix{g}$, and\nthe diagonal $2n_{\\rm gal}\\times 2n_{\\rm gal}$ dimensional covariance matrix $\\mathsf{N}$ with the $n$th diagonal\nentry equal to $\\sigma^2_{{\\rm ms};n} + \\sigma^2_{e}$.\nNote the weak shear likelihood does not depend on the lensing convergence $\\kappa$ (because the\nreduced shear is approximated as equal to the non-reduced shear and we ignore lensing magnification\neffects on the galaxy sizes and fluxes).\n\n\n\\subsection{A maximum entropy prior for lensing fields}\n\\label{sec:framework}\n\nWe want an interim prior on the lensing convergence and shear that is not only\nindependent of cosmology but also broadly encompassing of the possible\ncosmological interpretations and spatially varying systematics contributions.\nThat is, by choosing a functional form of a prior\nfor interpolating shear over the sky and marginalizing statistical uncertainties,\nwe should not restrict the class of physical models that might explain the data.\nA mathematical version of this sentiment is that we want a\nmaximum entropy prior~\\citep{PhysRev.106.620,shore1980} on the lensing fields.\n\nFor an assumed mean and (co-)variance, the Gaussian distribution is the maximum entropy\ndistribution~\\citep{10.2307\/2984828}. Therefore, because it is principally the second moment of the\nlensing convergence that we use to constrain cosmological models, we choose a\nGaussian Process (GP) interim prior for sampling convergence (and shear) fields on\nthe sky given measurements of galaxy image moments.\n\nGravitational lensing shear is a spin-2 field and is non-locally\nrelated to the lensing convergence, which both present modeling challenges.\nHowever, both the lensing convergence and shear can be derived as second\nderivatives of a scalar valued lensing potential $\\psi$. So we impose the\nGP prior on the potential $\\psi$ and derive the related priors on the\nconvergence $\\kappa$ and shear $\\gamma$ from this starting assertion.\n\nEach of the lensed observables, $\\Upsilon(\\mathbf{x}) \\equiv \\left[\\kappa, \\gamma_1, \\gamma_2\\right]$\nis related to the lensing potential $\\psi$ via derivatives in the form of:\n\\begin{align}\n  \\kappa &=\\frac{1}{2}\\left(\\frac{\\partial^2 \\psi}{\\partial x_1^2} +\n    \\frac{\\partial^2 \\psi}{\\partial x_2^2 }\\right) = \\frac{1}{2}(\\psi_{,11} +\n    \\psi_{,22})\n  \\label{eq:kappa_from_psi}\n  \\\\\n  \\gamma_1 &=\\frac{1}{2}\\left(\\frac{\\partial^2 \\psi}{\\partial x_1^2} -\n    \\frac{\\partial^2 \\psi}{\\partial x_2^2}\\right) = \\frac{1}{2}(\\psi_{,11} -\n    \\psi_{,22})\n  \\label{eq:gamma1_from_psi}\n  \\\\\n  \\gamma_2 &=\\frac{1}{2}\\left(\\frac{\\partial^2 \\psi}{\\partial x_1 \\partial\n    x_2} +\\frac{\\partial^2 \\psi}{\\partial x_2 \\partial x_1}\\right) =\n    \\frac{1}{2}(\\psi_{,12} + \\psi_{,21}).\n  \\label{eq:gamma2_from_psi}\n\\end{align}\nIt is straightforward to derive via integration by parts on the moments of the field \nthat if $\\psi(\\mathbf{x})$ is Gaussian\ndistributed for given $\\mathbf{x}$, then so too is $\\Upsilon(\\mathbf{x})$, but with a\nmodified covariance.\nBy specifying a GP prior on the lens potential, we therefore can derive\na GP prior on the combination of lensing convergence and shear fields that preserves the\nphysical correlations between these fields. We will show that we can infer the lensing \nconvergence and shear via correlated draws from a GP distribution given a galaxy \nellipticity catalog. We do not then compute lensing convergence and shear from \nspatial derivatives of a GP-distributed lens potential. The latter operation is only \nprecisely defined when we have (in principle) knowledge of the lens potential at all \nsky locations. However, galaxies provide a non-contiguous background for measuring the lens \npotential. The relations in \\autoref{eq:kappa_from_psi}, \\autoref{eq:gamma1_from_psi}, and \n\\autoref{eq:gamma2_from_psi} are then imposed only by the GP covariance structure during \nsampling and marginalization.\n\nThe derivations of using a GP to represent the lensing convergence and shear fields were\nfirst presented in \\citet{ng2016}. We refer readers to \\citet{ng2016} for an introduction to\nthe basics of a GP. We show the derivations from \\citet{ng2016} in\n\\autoref{sec:Gaussian process covariances} for the convenience of the reader.\nIn particular, the GP prior for $\\Upsilon$\nderived from that for $\\psi$ should not mix E and B modes in the two-point function\nbecause we only allow for GP realizations that preserve the combinations of fields that satisfy\nEquations~\\ref{eq:kappa_from_psi}--\\ref{eq:gamma2_from_psi} in the two-point correlations.\nIt is also straightforward\nto extend the scalar valued potential $\\psi$ to a complex-valued potential\n$\\psi\\rightarrow\\psi_{E} + i\\psi_{B}$\nto model or infer both E and B mode contributions to a measured shear signal.\n\nWe choose a squared exponential kernel for the GP model of the lensing potential $\\psi$,\n\\begin{align}\\label{eq:gp_cov}\n  \\mathsf{S}_{\\psi}(\\mathbf{x}, \\mathbf{y}; \\lambda_{\\rm GP}, \\ell^{2}_{\\rm GP}) =\n \n  \\lambda_{\\rm GP}^{-1} \\exp\\left(-\\frac{1}{2} \\frac{s^2(\\mathbf{x},\\mathbf{y})}{\\ell^{2}_{\\rm GP}}\\right),\n\\end{align}\nwhere $\\lambda_{\\rm GP}$ is a precision parameter that sets the amplitude of fluctuations of the GP,\n$\\ell^{2}_{\\rm GP}$ is a squared distance defining the correlation length of the GP kernel,\nand the squared distance $s^2$ between pairs of galaxy locations is,\n\\begin{equation}\\label{eq:euclid_dist}\n  s^2 \\equiv (\\mathbf{x} - \\mathbf{y})^T (\\mathbf{x} - \\mathbf{y}).\n\\end{equation}\nThe squared exponential kernel is useful in interpolating smooth response functions between \nthe observed galaxy locations, as we expect for low resolution or low signal-to-noise ratio (SNR)\nweak lensing mass reconstructions. Exploration of other kernel choices for different mass \nreconstruction resolutions and SNRs is an interesting question that we leave for later work.\nA practical motivation for our kernel choice is that the two parameters of the squared exponential \nkernel can be optimized for different data sets without particular numerical or computational \nchallenges. A further justification for our smooth kernel choice comes from our intention to use the \nGP as merely an interim prior for sampling lens fields. We will later describe how these interim lens \nfield realizations may be re-weighted under cosmologically informed priors.\nWe list the derivatives of \\autoref{eq:gp_cov} in Appendix \\ref{sec:Gaussian process covariances} \nthat are required to build the joint covariance for the lensing shear and convergence.\nWe will denote the covariance constructed from second derivatives of \\autoref{eq:gp_cov} as\n$\\mathsf{S}_{\\rm GP}$.\n\nWe now return to the pedagogical derivation of the likelihood function for galaxy ellipticity\nestimators adding the GP distribution for the shear.\nBecause both the marginal likelihood in \\autoref{eq:marg_like_analytic} and the shear prior are\nGaussian distributions in the shear, we can specify the shear (and convergence) joint posterior\nfor all galaxy and grid locations as a multivariate Gaussian distribution (otherwise known\nas the Wiener filter),\n\\begin{multline}\\label{eq:shear_posterior}\n  \\prf{\\Upsilon(\\mathbf{x},\\mathbf{x}') | {\\mathbf{d}}, \\alpha, a_{\\rm GP}}\n  =\n  \\mathcal{N}_{\\Upsilon(\\mathbf{x},\\mathbf{x}')}\n  \\left({\\bm \\mu}_{\\Upsilon}, \\mathsf{S}_{\\Upsilon} \\right)\n  \\\\ \\times\n  \\mathcal{N}_{\\hat{\\Matrix{e}}}\\left(0, \\mathsf{N} + \\mathsf{S}_{\\rm GP}\\right),\n\\end{multline}\nwhere,\n\\begin{align}\n  \\mathsf{S}_{\\Upsilon} &\\equiv \\mathsf{S}_{\\rm GP}\\left(\\mathsf{S}_{\\rm GP} + \\mathsf{N}\\right)^{-1}\\mathsf{N}\n  \\label{eq:posterior_cov}\n  \\\\\n  {\\bm \\mu}_{\\Upsilon} &\\equiv \\mathsf{S}_{\\rm GP} \\left(\\mathsf{S}_{\\rm GP} + \\mathsf{N}\\right)^{-1} \\hat{\\Matrix{e}}.\n  \\label{eq:posterior_mean}\n\\end{align}\nCalculation of ${\\bm \\mu}_{\\Upsilon}$ in \\autoref{eq:posterior_mean} yields a posterior mean\nestimate of the lensing shear and convergence at every galaxy location. However, this estimator\nrequires inversion of an $2n_{\\rm gal}\\times2n_{\\rm gal}$ covariance matrix, which can be computationally expensive.\n\nOther works have attempted to reduce the dimensionality of the covariances\nin \\autoref{eq:shear_posterior} by interpolating and averaging\nthe measured galaxy ellipticities onto a grid of coarser resolution than that\nsampled by the galaxy angular distribution~\\citep[e.g.,][]{Alsing2015}.\nHowever, such interpolations not only restrict the measured dynamic range,\nbut also can be expected to introduce artefacts in the inferred shear field on the grid based on\nthe shape of the smoothing kernel, to propagate\nshape measurement systematics to a broad range of angular scales,\nand to ignore error propagation from individual galaxy shape measurements to the shear inference.\nOur method defines an explicit interpolation from galaxy to other sky locations, with error\npropagation included.\n\nWe can marginalize the lensing fields at the galaxy locations to obtain the marginal\nposterior for the lensing fields at just the smaller number of grid locations $\\mathbf{x}'$,\n\\begin{multline}\\label{eq:lens_posterior}\n  \\prf{\\Upsilon(\\mathbf{x}') | {\\mathbf{d}}, \\alpha, a_{\\rm GP}} \\propto\n  \\int d\\Upsilon(\\mathbf{x})\\,\n  \\prf{{\\mathbf{d}} | \\Upsilon(\\mathbf{x}), \\alpha}\n  \\\\ \\times\n  \\prf{\\Upsilon(\\mathbf{x}') | \\Upsilon(\\mathbf{x}), a_{\\rm GP}}\n  \\prf{\\Upsilon(\\mathbf{x}) | a_{\\rm GP}}.\n\\end{multline}\nWhen all distributions are Gaussians, \\autoref{eq:lens_posterior} reduces to evaluating\n\\autoref{eq:shear_posterior} at only the locations of the grid $\\mathbf{x}'$ given the input\nellipticities $\\hat{\\Matrix{e}}$. Note however that we still require in \\autoref{eq:posterior_mean}\nthe evaluation of the shear model at every galaxy location, which is then interpolated by the\nWF to arbitrary sky locations. Our approach is therefore different from algorithms that require\nan initial averaging of galaxy ellipticity components over local sky regions.\n\\autoref{eq:lens_posterior} thus specifies the interim distribution of lensing shear and convergence\nthat can be later propagated to cosmological model analyses.\n\n\n\n\\subsubsection{Optimizing interpolation parameters}\n\\label{sub:optimizing_interpolation_parameters}\n\nWe can further marginalize the lens fields $\\Upsilon$ at all locations $\\mathbf{x}$ and $\\mathbf{x}'$ to obtain\nthe marginal likelihood for the GP parameters,\n\\begin{equation}\\label{eq:gp_marg_like}\n  \\prf{{\\mathbf{d}} | \\alpha, a_{\\rm GP}} = \\mathcal{N}_{{\\mathbf{d}}}\n  \\left(0, \\mathsf{S}_{\\rm GP} + \\mathsf{N}\\right).\n\\end{equation}\nWe maximize the density in \\autoref{eq:gp_marg_like} to determine suitable values of the\nGP parameters for interpolation of the lens fields for a given ellipticity\ncatalog~\\citep[see a similar approach in][]{Marshall:531310}.\n\n\\autoref{eq:gp_marg_like} is informative on the GP parameters if the data vector ${\\mathbf{d}}$ includes\nsufficient numbers of galaxies to beat down the shape noise. In cases with smaller\nsignal to noise ratios we can also add a cosmologically informed prior to help constrain\nthe GP parameters. In this case we can replace the data vector in \\autoref{eq:gp_marg_like}\nwith a simulation of the data vector derived from a cosmological model. Then, by marginalizing\nover realizations of the simulated data vector we get a marginal prior for the GP parameters,\n\\begin{align}\\label{eq:marg_dist_cosmo}\n  \\prf{a_{\\rm GP}| \\theta, \\alpha} &=\n  \\int d{\\mathbf{d}}^{\\rm sim}\\,\n  \\prf{a_{\\rm GP} | {\\mathbf{d}}^{\\rm sim}, \\alpha} \\prf{{\\mathbf{d}}^{\\rm sim} | \\theta},\n  \\notag\\\\\n  %\n  &=\n  \\prf{a_{\\rm GP}}\n  \\\\\n  &\\times\n  \\int d{\\mathbf{d}}^{\\rm sim}\\, \\frac{\\prf{{\\mathbf{d}}^{\\rm sim} | a_{\\rm GP}, \\alpha}}{\\prf{{\\mathbf{d}}^{\\rm sim} | \\alpha}}\n  \\prf{{\\mathbf{d}}^{\\rm sim} | \\theta}.\n  \\notag\n\\end{align}\nIn \\autoref{eq:marg_dist_cosmo} we specify a model for drawing simulated data realizations\n${\\mathbf{d}}^{\\rm sim}$ given cosmological parameters $\\theta$. A standard approach to such a model\nis to first draw a realization of the 3D mass density perturbations from a Gaussian distribution\nwith a cosmologically determined initial power spectrum and then to evolve the initial\nmass density perturbations under a numerical (e.g., $N$-body) simulation of gravitational\nevolution.\n\nModeling only linear density perturbation evolution, the late-time model for the\nprojected lensing mass density, and hence the shear, is also a Gaussian distribution,\n\\begin{equation}\\label{eq:linear_cosmo_model}\n  \\prf{{\\mathbf{d}}^{\\rm sim}|\\theta}=\\mathcal{N}_{{\\mathbf{d}}^{\\rm sim}}(0, \\mathsf{\\Sigma}(\\theta)),\n\\end{equation}\nfor some cosmological covariance $\\mathsf{\\Sigma}(\\theta)$.\nTo evaluate \\autoref{eq:marg_dist_cosmo}  we also must determine the evidence\n\\begin{equation}\\label{eq:evidence}\n  \\prf{{\\mathbf{d}}^{\\rm sim}|\\alpha, I} \\equiv\n  \\int da_{\\rm GP}\\, \\prf{{\\mathbf{d}}^{\\rm sim} | \\alpha, a_{\\rm GP}}\n  \\prf{a_{\\rm GP} | I},\n\\end{equation}\nwhere $I$ denotes prior information on the GP parameters.\nEven under a linear density perturbation approximation the evidence\ncannot be calculated analytically. \n\nFor illustration of the GP parameter prior informed by cosmology, we use \\autoref{eq:linear_cosmo_model} \nto evaluate \\autoref{eq:marg_dist_cosmo} via Monte Carlo integration. We draw $N$ samples \nof ${\\mathbf{d}}^{\\rm sim}_{i}$ from $\\prf{{\\mathbf{d}}^{\\rm sim}|\\theta}$ and evaluate \n$\\prf{{\\mathbf{d}}^{\\rm sim}_{i}|a_{\\rm GP}, \\alpha}$ for each $i=1,\\dots,N$. We approximate \nthe evidence in \\autoref{eq:evidence} via 2D numerical integration and then calculate \n\\begin{equation}\n  \\prf{a_{\\rm GP}| \\theta, \\alpha} \\approx \\prf{a_{\\rm GP}}\n  \\frac{1}{N} \\sum_{i=1}^{N} \\frac{\\prf{{\\mathbf{d}}^{\\rm sim}_{i}|a_{\\rm GP}, \\alpha}}\n  {\\prf{{\\mathbf{d}}^{\\rm sim}_{i}|\\alpha, I}}.\n\\end{equation}\nWe show the resulting posterior constraints on the GP parameters in \\autoref{fig:gp_cosmo_prior}.\n\\begin{figure}[!htb]\n    \\centerline{\n        \\includegraphics[width=0.4\\textwidth]{gp_cosmo_prior.png}\n    }\n    \\caption{Posterior constraints on the GP parameters after marginalizing realizations of \n    simulated lens fields under a cosmological model. The correlations and variance in the simulated \n    data serve to communicate the cosmological covariance structure into constraints on the \n    GP parameters. The tight degeneracy shows the tradeoff between low correlations and large \n    GP precision (equivalent to white noise as the GP precision becomes large) and larger correlations \n    and smaller precision as the simulated maps are fit with smoother models. The lines show the 3-$\\sigma$\n    confidence interval.}\n    \\label{fig:gp_cosmo_prior}\n\\end{figure}\nThe cosmological simulation realizations impose a tight constraint in a linear combination \nof the logarithm of the two GP parameters. The constraints in \\autoref{fig:gp_cosmo_prior} \nindicate the cosmological lens field simulations can be fit either with a large GP precision \nand small correlation length (i.e., essentially as white noise), or with a smaller precision \nand larger correlation lengths (i.e., as a smooth correlated map). \nWe thus obtain tight constraints on the GP parameters if we can include some prior knowledge \nof the correlation length scale via $\\prf{a_{\\rm GP}}$.\n\nSee \\autoref{sec:cosmology_dependent_covariance_model} for\na description of our linear theory cosmological model for the lensing convergence.\n\n\n\n\n\\subsection{Cosmological parameter inference}\n\\label{sub:cosmological_parameter_inference}\n\nUnder a standard cosmological model the lensing potential $\\psi$ is related to the\n3D cosmological late-time gravitational potential $\\Psi^{\\rm LT}$ by a projection along\nthe line of sight weighted by the lensing efficiency $K(a;A_s)$,\n\\begin{equation}\\label{eq:lenspot_proj}\n  \\bar{\\psi}_{s}(\\mathbf{x}) \\equiv\n \n  W(\\mathbf{x})\n  \\int dz\\,\n  \\Psi^{\\rm LT}(\\mathbf{x},z)\n  K(z ; Z_s),\n\\end{equation}\nwhere $W(\\mathbf{x})$ is the survey window function, $z$ is the cosmological redshift,\nand $Z_s$ defines the redshift distribution of source galaxies for a galaxy sample $s$.\nThe late-time gravitational potential is in turn related to the potential of the\ncosmological initial conditions $\\Psi^{\\rm IC}$ via a deterministic function $G$\ndefining gravitational evolution~\\citep{2013MNRAS.432..894J},\n\\begin{equation}\\label{eq:gravpot_ic}\n  \\Psi^{\\rm LT}(\\mathbf{x}, z) = G\\left(\\Psi^{\\rm IC}, \\theta, z\\right).\n\\end{equation}\nWe assume the initial conditions $\\Psi^{\\rm IC}$ are Gaussian distributed with mean\nzero and covariance $\\mathsf{\\Sigma}^{\\rm IC}(\\theta)$, which is entirely determined by the\npotential power spectrum from inflation.\n\nA common method to simulate $\\Psi^{\\rm LT}$ according to \\autoref{eq:gravpot_ic} is\nto run a cosmological $N$-body simulation with initial conditions $\\Psi^{\\rm IC}$\ndrawn from a Gaussian distribution with a specified power spectrum.\nThe numerical $N$-body solver allows evaluation of the function $G$, which does not\nhave a known analytic form beyond low-order perturbation theory.\nThe standard approach to evaluate \\autoref{eq:lenspot_proj} for the lens potential\nin a numerical simulation is to trace light rays through the observer's light cone\ngiven the simulated $\\Psi^{\\rm LT}(\\mathbf{x}, a)$. The bundle of light rays is\nevaluated at a discrete set of sky locations to predict the lensing shear and\nconvergence. The predicted lens fields must then be interpolated over the sky to\ngalaxy locations to complete the numerical cosmological model prediction.\n\nWe have already defined a probabilistic interpolation of lens fields over the sky\nvia the GP. We therefore define the conditional distribution of the lens potential\nat galaxy locations $\\mathbf{x}$ given the simulations evaluated at positions $\\mathbf{x}'$ via the\nmodel of \\autoref{eq:gravpot_ic} and \\autoref{eq:lenspot_proj}\nfollowed by GP interpolation,\n\\begin{equation}\\label{eq:cond_model_comparison}\n  \\prf{\\psi(\\mathbf{x}) | \\bar{\\psi}(\\mathbf{x}', \\theta),a_{\\rm GP}}\n  = \\mathcal{N}_{\\psi} \\left(\n    \\mu_{\\psi}, \\mathsf{\\Sigma}_{\\psi}\n  \\right),\n\\end{equation}\nwhere,\n\\begin{align}\n  \\mu_{\\psi} &\\equiv \\mathsf{S}(\\mathbf{x},\\mathbf{x}')\\mathsf{S}^{-1}(\\mathbf{x}',\\mathbf{x}')\n  \\bar{\\psi}_{s}(\\mathbf{x}'; \\Psi^{\\rm IC}, \\theta, A_s, W)\n  \\label{eq:mu_psi}\n  \\\\\n  \\mathsf{\\Sigma}_{\\psi} &\\equiv\n  \\mathsf{S}(\\mathbf{x}, \\mathbf{x}) - \\mathsf{S}(\\mathbf{x},\\mathbf{x}')\\mathsf{S}^{-1}(\\mathbf{x}',\\mathbf{x}')\\mathsf{S}(\\mathbf{x}',\\mathbf{x}),\n  \\label{eq:sigma_psi}\n\\end{align}\ndefine the mean and covariance for the conditional multivariate Gaussian.\nWith \\autoref{eq:cond_model_comparison} we have thus derived a conditional probability\ndistribution to compare theoretical predictions of the lens potential with interim\nsamples of the potential drawn under the GP prior.\nSaid another way, our comparison of conditional posterior\nsamples of the lens fields with the cosmological models is mediated by the interpolation\nover the sky using the GP.\n\nBy marginalizing the initial conditions realizations and the lens fields realizations\nwe can now derive the posterior distribution for the cosmological parameters,\n\\begin{multline}\\label{eq:marg_post_cosmo}\n  \\prf{\\theta | {\\mathbf{d}}, a_{\\rm GP}, \\alpha} \\propto\n  \\prf{\\theta}\n  \\\\ \\times\n  \\int d\\Upsilon\\, \\int d\\Upsilon'\\, \\int d\\Psi^{\\rm IC}\\,\n  \\delta_{D}\\left(\\Upsilon' - h(\\Psi^{\\rm IC})(\\mathbf{x}')\\right)\n  \\\\ \\times\n  \\frac{\\prf{\\Psi^{\\rm IC}(\\mathbf{x}') | \\mathsf{\\Sigma}^{\\rm IC}(\\theta)}}\n    {\\prf{\\Upsilon' | a_{\\rm GP}}}\n  \\\\ \\times\n  \\prf{{\\mathbf{d}} | \\Upsilon, \\alpha}\n  \\prf{\\Upsilon | \\Upsilon', a_{\\rm GP}} \\prf{\\Upsilon' | a_{\\rm GP}},\n\\end{multline}\nwhere $h(\\Psi^{\\rm IC})$ indicates the deterministic gravitational evolution of the\ninitial conditions potential to late times where the lensing is observed.\n\nThe final line of \\autoref{eq:marg_post_cosmo} is the interim sampling distribution\nwe defined in the previous section. We can thus perform the integrals in\n\\autoref{eq:marg_post_cosmo} via Monte Carlo with the same interim lens field\nsamples that are generated in the map making algorithm.\n\\begin{figure}[!htb]\n  \\centerline{\n    \\includegraphics[width=0.5\\textwidth]{shear_gp_pgm_cosmo_split.png}\n  }\n  \\caption{\n   \n   \n   \n   \n   \n    In our approximate cosmological parameter inference pipeline we assume\n    the lens fields at the galaxy locations $\\Upsilon$ are entirely determined\n    by GP interpolation from the lens fields at a regular set of sky locations\n    $\\Upsilon'$ at which we compute theory predictions.\n    The lens fields on the regular set are informed by the cosmological\n    model with parameters $\\theta$.\n    \n  }\n  \\label{fig:pgm_cosmo}\n\\end{figure}\n\n\nWe propose two sampling algorithms depending on whether or not we can directly evaluate the\nprobability density for the lens fields under the cosmological model.\n\\begin{description}\n  \\item[Sampling algorithm 1] This is an algorithm to use when it is possible to analytically\n  evaluate the cosmological probability density function.\n  With $K$ samples of $\\Upsilon(\\mathbf{x}')$ from the interim sampling distribution,\n  we use importance sampling to approximate the marginalizations in\n  \\autoref{eq:marg_post_cosmo},\n  \\begin{multline}\\label{eq:marglikecosmo}\n    \\prf{\\theta | {\\mathbf{d}}, a_{\\rm GP}, \\alpha} \\approx\n    \\prf{\\theta}\n   \n    \\frac{1}{K}\\sum_{k=1}^{K}\n    \\frac{\\prf{\\Upsilon_{k}(\\mathbf{x}') | \\theta}}\n      {\\prf{\\Upsilon_{k}(\\mathbf{x}') | a_{\\rm GP}}},\n  \\end{multline}\n  where from \\autoref{eq:marg_post_cosmo},\n  \\begin{multline}\n    \\prf{\\Upsilon(\\mathbf{x}') | \\theta} \\equiv\n    \\int d\\Psi^{\\rm IC}\\,\n    \\prf{\\Psi^{\\rm IC}(\\mathbf{x}') | \\mathsf{\\Sigma}^{\\rm IC}(\\theta)}\n    \\\\ \\times\n    \\delta_{D}\\left(\\Upsilon' - h(\\Psi^{\\rm IC}(\\mathbf{x}'))\\right).\n  \\end{multline}\n\n  \\item[Sampling algorithm 2] This is an algorithm to use when cosmological modeling of the lens\n  fields is only possible via forward simulation.\n  For most cosmological models of interest, we can only predict the model for the\n  lens fields via forward simulation of the cosmological mass density perturbation evolution.\n  In this case, we have no direct mechanism to evaluate the density\n  $\\prf{\\Upsilon(\\mathbf{x}')|\\theta}$.\n  Instead, we follow the steps,\n  \\begin{enumerate}\n    \\item Draw $\\Psi^{\\rm IC}(\\mathbf{x}')$ from $\\prf{\\Psi^{\\rm IC}(\\mathbf{x}') | \\mathsf{\\Sigma}^{\\rm IC}(\\theta)}$.\n    \\item Compute the predicted lens fields given the drawn initial conditions\n    $h(\\Psi^{\\rm IC})$ (e.g., via $N$-body simulation and ray-tracing prediction of the lensing\n    shear and convergence).\n    \\item Select $K$ samples of $\\Upsilon(\\mathbf{x})$ from the set of interim samples under the\n    GP prior.\n    \\item Evaluate the density, $\\prf{\\Upsilon|\\Upsilon', a_{\\rm GP}}$, as in\n    \\autoref{eq:cond_model_comparison} for each sample $K$ and compute,\n    \\begin{equation}\n      \\prf{\\theta | {\\mathbf{d}}, a_{\\rm GP}, \\alpha} \\approx \\prf{\\theta}\n      \\frac{1}{K} \\sum_{k=1}^{K}\n      \\prf{\\Upsilon_{k}(\\mathbf{x})|\\Upsilon(\\mathbf{x}',\\theta), a_{\\rm GP}}.\n    \\end{equation}\n    Note that $\\Upsilon'$ here is that from the cosmological forward model, not that from the\n    interim sampling.\n  \\end{enumerate}\n\\end{description}\n\n\nFor Sampling Algorithm 1, if we assume linear cosmological perturbations we can approximate,\n  \\begin{equation}\n    \\prf{\\Upsilon(\\mathbf{x}')| \\theta} \\approx\n    \\mathcal{N}_{\\Upsilon(\\mathbf{x}')}\n    \\left(\\mathbf{0}, \\mathsf{\\Sigma}(P_{\\Upsilon}(\\theta))\\right).\n  \\end{equation}\n  Under this approximation we can combine the terms from \\autoref{eq:marg_post_cosmo},\n  \\begin{equation}\\label{eq:lens_field_cond_factor}\n    \\prf{\\Upsilon | \\Upsilon',a_{\\rm GP}} \\prf{\\Upsilon'|\\theta} =\n    \\mathcal{N}_{\\Upsilon(\\mathbf{x},\\mathbf{x}')} \\left(\n      0, \\mathsf{M}(\\mathbf{x},\\mathbf{x}'; a_{\\rm GP}, \\theta)\n    \\right),\n  \\end{equation}\n  where $\\mathsf{M}(\\mathbf{x},\\mathbf{x}'; a_{\\rm GP}, \\theta)$ is defined in \\autoref{eq:signal_cov_cosmo}.\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n  If we further assume, as above, that the likelihood function is a Gaussian distribution in the\n  galaxy ellipticities with a linear weak shear applied, then \\autoref{eq:marg_post_cosmo} becomes,\n  \\begin{equation}\\label{eq:marg_post_cosmo_linear}\n    \\prf{\\theta | {\\mathbf{d}}, a_{\\rm GP}, \\alpha} \\propto \\prf{\\theta}\n    \\mathcal{N}_{{\\mathbf{d}}}\\left(\n      0, \\mathsf{M}(a_{\\rm GP}, \\theta) + \\mathsf{N}\n    \\right),\n  \\end{equation}\n  with the signal covariance as defined in \\autoref{eq:signal_cov_cosmo}.\n\n  To summarize, the approximations required for \\autoref{eq:marg_post_cosmo_linear} are,\n  \\begin{enumerate}\n    \\item a likelihood function that is Gaussian in the galaxy ellipticities,\n    \\item galaxy ellipticity measurements that are unbiased estimators of the reduced shear,\n      \\item weak shear ($\\kappa \\ll 1$),\n    \\item linear cosmological perturbations.\n  \\end{enumerate}\n  We can drop assumptions 1, 2, and 3 and still use Sampling Algorithm 1. But to avoid assumption\n  4 we must resort to Sampling Algorithm 2.\n\n\n\n\\section{Results}\n\\label{sec:results}\n\nWe evaluate the mean and covariance of the marginal posterior for the lensing shear and convergence\nfrom \\autoref{eq:shear_posterior} for a simulated data set, where we can compare with a known\ntruth (\\autoref{sub:simulation_study}). We also analyze\nthe ellipticity catalog~\\citep{jee2013} in the\nDeep Lens Survey (DLS)\\footnote{\\url{http:\/\/dls.physics.ucdavis.edu}}~\\citep{2002SPIE.4836...73W}\nin the vicinity of a massive galaxy cluster where the\namplitudes of the lensing fields are large (\\autoref{sub:abell}).\n\n\\subsection{Simulation study}\n\\label{sub:simulation_study}\n\nTo validate that \\autoref{eq:posterior_mean} can recover the correct shear and convergence fields,\nwe create simulated galaxy ellipticity catalogs with artificially small shape noise so we can\nmeasure the lensing shear and convergence to a tuneable precision.\n\nOur procedure for simulating galaxy ellipticity catalogs is,\n\\begin{enumerate}\n  \\item Calculate a lensing convergence angular power spectrum using the cosmology theory\n  code {\\sc CHOMP}\\xspace\\footnote{\\url{https:\/\/github.com\/karenyyng\/chomp}}~\\citep{2013JCAP...11..009M}.\n    We assume a standard\n  $\\Lambda$CDM cosmology with $\\Omega_m=0.3$, $\\sigma_8=0.8$ and a source redshift distribution\n  with a narrow peak at $z=1$.\n  \\item Simulate Gaussian-distributed lensing shear and convergence maps on a grid using the\n  code {\\sc GalSim}\\xspace\\footnote{\\url{https:\/\/github.com\/GalSim-developers\/GalSim}}~\\citep{galsim}.\n  \\item Place one galaxy in each grid cell of the simulated lensing shear maps. These galaxies are simply\n  sources of illumination for measuring the lensing fields, not cosmologically clustered galaxies.\n  \\item For each galaxy, draw intrinsic ellipticity components from a 2D Gaussian distribution\n  with mean 0 and a specified variance, $\\sigma_e^2$.\n  \\item Calculate lensed ellipticities for each galaxy by adding the lensing shear to the intrinsic\n  ellipticities, assuming a weak shear approximation.\n  \\item Save the galaxy angular sky locations and ellipticity components to a catalog file.\n\\end{enumerate}\nGiven a simulated ellipticity catalog, we find the GP parameters\n$\\lambda_{\\rm GP}, \\ell^{2}_{\\rm GP}$ that maximize \\autoref{eq:gp_marg_like}.\nWe then evaluate \\autoref{eq:posterior_mean} and \\autoref{eq:posterior_cov} using the optimized\nGP parameters to obtain the\nmarginal posterior distribution of the lensing fields $\\kappa, \\gamma$ at all galaxy locations\nas well as on a regular grid of locations.\nWe expect that in practical applications to large data sets the shear will only need to be\nevaluated at the galaxy locations and the converngence (or lens potential) will only need to be\nevaluated at a smaller number of sky grid locations, thus reducing the overall dimensionality of\nthe linear system to be solved.\n\n\nWe show an example of the output of this procedure in\n\\autoref{fig:mass_map_comparison_galsim}\ncompared to the input shear fields used to generate the mock ellipticity catalog. In the\nleft column in \\autoref{fig:mass_map_comparison_galsim} we show the posterior mean fields from\n\\autoref{eq:posterior_mean}.\nIn the adjacent panel we show the input shear fields.\nWe show the `true' convergence that we calculated at the same time as the input shear, but we\ndo not use the convergence at any point in our calculation. The ``estimated convergence'' comes\nfrom interpolating the measured galaxy ellipticities with the GP kernel.\nThe right panel in \\autoref{fig:mass_map_comparison_galsim} shows the\nsignal-to-noise ratio (SNR) of the lens field maps, defined as,\n\\begin{equation}\\label{eq:snr_def}\n  {\\rm SNR} \\equiv \\Upsilon \/ \\sqrt{{\\rm diag}\\left(\\mathsf{S}_{\\Upsilon}\\right)},\n\\end{equation}\nwith $\\mathsf{S}_{\\Upsilon}$ defined in \\autoref{eq:posterior_cov}.\nIn the limit that the variance of the lens fields dominates the intrinsic shape\nand ellipticity measurement variances, \\autoref{eq:posterior_cov} reduces to\n$\\mathsf{N}$. Then \\autoref{eq:snr_def} becomes\n${\\rm SNR} \\rightarrow \\Upsilon \/ \\sqrt{\\sigma_e^2 + \\sigma_{\\rm ms}^{2}}$,\nwhich is similar to SNR definitions in other weak lensing mass mapping\nanalyses~\\citep{2000MNRAS.313..524V,2014MNRAS.442.2534S}.\n\nThe simulation in\n\\autoref{fig:mass_map_comparison_galsim} has an artificially low intrinsic ellipticity r.m.s.\\\nof $\\sigma_e=0.0026$ compared to $\\sigma_e=0.26$ in the Deep Lens Survey that\nwe analyze in \\autoref{sub:abell}.\nWe choose a small shape noise r.m.s.\\ to\nallow us to validate our convergence inference with a small number of only 1600 galaxies.\nWe place the simulated galaxies on a $40\\times40$ grid, so that we do not have to interpolate\nthe simulated shear fields to build the mock ellipticity catalog.\n\nBecause the Gaussian model for the shape noise r.m.s.\\ scales with the number of galaxies\n$n_{\\rm gal}$ as $n_{\\rm gal}^{-1\/2}$, the shape noise r.m.s.\\ in\nour simulation is equivalent to a DLS-like galaxy sample with 16 million galaxies.\nThis is about 64 times the number of galaxies we have in a single four square degree field\nof the DLS. We show the effect of increasing the shape noise r.m.s.\\ by a factor\nof $\\sqrt{64}$ later in this section.\n\\begin{figure*}[!htb]\n  \\centerline{\n    \\includegraphics[width=0.54\\textwidth]{mass_map_comparison_galsim.png}\n    \\includegraphics[width=0.35\\textwidth]{mass_map_snr_galsim.png}\n  }\n  \\caption{Left: Comparison of the convergence and shear maps in our simulation study between that\n  used to generate our mock galaxy ellipticity catalog (right column) and the output of our\n  GP interpolation (left column). The rows show the maps for the two shear components $\\gamma_{1,2}$\n  and the convergence $\\kappa$. These maps cover a $2\\times2$ square degree field. The simulated\n  intrinsic ellipticity r.m.s.\\ is set to an artificially small value of $\\sigma_e=0.0026$,\n  which is 100 times smaller than that observed for the complete Deep Lens Survey catalog.\n  Right: signal-to-noise ratio (SNR) maps for the same simulations. We calculate SNR as the\n  ratio of the map to the square root of the diagonal of the covariance\n  in \\autoref{eq:posterior_cov}.\n  These simulations use 1600 galaxies and grid sizes for the GP interpolated shear and convergence\n  with 24 grid cells per dimension.\n  }\n  \\label{fig:mass_map_comparison_galsim}\n\\end{figure*}\n\nThe reconstructed shear and convergence maps in \\autoref{fig:mass_map_comparison_galsim} are a\ngood match to the simulation inputs, but are noticeably smoothed. This is for two reasons;\n(1) we evaluate the interpolated lens fields on a smaller $24\\times24$ grid than the\n$40\\times40$ grid on which the inputs are evaluated,\n(2) we use a value of $\\ell^{2}_{\\rm GP}=0.0123$ that is larger than the Nyquist scale in the maps, even\nfor the $24\\times24$ interpolated grid because of the way the GP parameters are optimized.\nWe will discuss the GP parameter optimization below.\n\nIn \\autoref{fig:power_spectra_galsim} we show the E and B mode power spectrum estimators obtained\nfrom the posterior mean shear fields shown in \\autoref{fig:mass_map_comparison_galsim}.\nWe compare the E-mode power spectrum estimator from the mean posterior shear maps with that\nusing the higher-resolution shear maps that were used to generate the mock ellipticity\ncatalog (which we label `simulation truth'). We also show in \\autoref{fig:power_spectra_galsim}\nthe `theory' power spectrum that we used to generate the `simulation truth' shear maps.\nThe `theory' and `simulation truth' spectra agree on scales below the Nyquist frequency,\nshown in \\autoref{fig:power_spectra_galsim} as the vertical blue line. The mean posterior\npower spectrum agrees with the `simulation truth' spectrum on scales below the effective\nsmoothing frequency derived from the value of $\\ell^{2}_{\\rm GP}$ and shown by the vertical\ndot-dashed line in \\autoref{fig:power_spectra_galsim}.\n\\begin{figure}[!htb]\n  \\centerline{\n    \\includegraphics[width=0.5\\textwidth]{power_galsim_lownoise.png}\n  }\n  \\caption{Comparison of power spectrum estimators in our simulation study to the `truth' input\n  spectrum.\n  The vertical solid lines show the Nyquist frequencies of the grids.\n  The vertical dot-dashed line shows the multipole corresponding to the length scale\n  $\\sqrt{\\ell_{\\rm GP}^{2}}$ set by the GP kernel parameter $\\ell_{\\rm GP}^{2}=0.0123$ for\n  field coordinates are normalized to the unit square.\n  }\n  \\label{fig:power_spectra_galsim}\n\\end{figure}\n\nThe B-mode power spectrum estimated\nfrom the mean posterior shear maps is shown by the dashed red line in \\autoref{fig:power_spectra_galsim}.\nWe expect the B-mode power to be consistent with zero because we simulated only E-mode power.\nThe nonzero B-mode power spectrum in \\autoref{fig:power_spectra_galsim} is explained by\nexamining \\autoref{fig:EB_maps}, which shows the E and B mode mean posterior maps from which the\nE and B mode mean posterior power spectra were derived. The B-mode map in \\autoref{fig:EB_maps}\nis near zero throughout the field except near the boundaries. These edge effects in the B-mode map\ncan be explained by mathematical ambiguities in the definition of E and B mode separation in a\nfinite field~\\citep{bunn2003}. The small value of the B-mode map in \\autoref{fig:EB_maps} away from\nthe field boundaries indicates our GP interpolation method does not create spurious B-modes.\n\\begin{figure*}\n  \\centerline{\n    \\includegraphics[width=0.6\\textwidth]{EB_maps_galsim.png}\n  }\n  \\caption{E and B mode maps derived from the posterior mean shear maps shown in the left column\n  of \\autoref{fig:mass_map_comparison_galsim}.\n  The E-mode map closely matches the interpolated convergence map in\n  \\autoref{fig:mass_map_comparison_galsim} as expected.\n  The B-mode map is near zero throughout the center of the field, but shows non-zero\n  values around the field edge because of mathematical ambiguities in the E\/B mode separation\n  at the field boundaries~\\citep{bunn2003}. Note the value of the plotted B-mode map\n  has been multiplied by 10 for easier visualization.\n  }\n  \\label{fig:EB_maps}\n\\end{figure*}\n\nTo illustrate the GP parameter optimization procedure we show in \\autoref{fig:sigma_e_maps} the\nmean posterior convergence, the associated SNR maps, and the marginal likelihood surface for\nthe GP parameters (left to right columns) for increasing intrinsic ellipticity variance\n$\\sigma_e^2$ (top to bottom rows). The top row of panels in \\autoref{fig:sigma_e_maps} show a repeat\nof the $\\sigma_e^2$ and $a_{\\rm GP}$ values used in \\autoref{fig:mass_map_comparison_galsim}.\nFor the very small shape noise in this case, there is a narrow peak in the log-likelihood surface\nfor the two GP kernel parameters. We thus select the maximum likelihood (ML) values for $a_{\\rm GP}$\nand obtain the convergence and SNR maps that closely resemble the `true' convergence\nas shown in \\autoref{fig:mass_map_comparison_galsim}. However, the ML value of $\\ell^{2}_{\\rm GP}$ is somewhat\nlarger than the Nyquist frequency of the grid to which we interpolate as shown in\n\\autoref{fig:power_spectra_galsim} (compare dot-dashed to red solid vertical lines). Because the\nnoise is sub-dominant in this example, we would expect a value of $\\ell^{2}_{\\rm GP}$ matching the grid\nNyquist frequency to yield a more accurate convergence map reconstruction. We are likely to\nobtain more accurate results, therefore, if we impose a prior on $a_{\\rm GP}$ that encodes this\nexpectation.\n\\begin{figure*}\n  \\centerline{\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\sigma_{e}=0.0026$}}{kappa_galsim_sigmae1.png}\n    \\includegraphics[width=0.3\\textwidth]{snr_galsim_sigmae1.png}\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\lambda_{\\rm GP}=2.2\\times 10^{8}, \\ell^2=0.012$}}{lnp_galsim_sigmae1.png}\n  }\n  \\centerline{\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\sigma_{e}=0.0077$}}{kappa_galsim_sigmae2.png}\n    \\includegraphics[width=0.3\\textwidth]{snr_galsim_sigmae2.png}\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\lambda_{\\rm GP}=2.3\\times 10^{7}, \\ell^2=0.061$}}{lnp_galsim_sigmae2.png}\n  }\n  \\centerline{\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\sigma_{e}=0.013$}}{kappa_galsim_sigmae3.png}\n    \\includegraphics[width=0.3\\textwidth]{snr_galsim_sigmae3.png}\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\lambda_{\\rm GP}=2.3\\times 10^{7}, \\ell^2=0.061$}}{lnp_galsim_sigmae3.png}\n  }\n  \\centerline{\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\sigma_{e}=0.023$}}{kappa_galsim_sigmae5.png}\n    \\includegraphics[width=0.3\\textwidth]{snr_galsim_sigmae5.png}\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\lambda_{\\rm GP}=2.3\\times 10^{6}, \\ell^2=0.30$}}{lnp_galsim_sigmae5.png}\n  }\n  \\caption{Posterior mean convergence maps (left), signal-to-noise ratio (SNR) maps (middle),\n  and marginal likelihood contours (right) for simulated galaxy catalogs with varying intrinsic\n  ellipticity r.m.s.\\ $\\sigma_e$.\n  From top to bottom, $\\sigma_e = \\left(0.00258, 0.00774, 0.0129, 0.0232\\right)$ (these values are\n  also annotated in the bottom left corner of each convergence map).\n  The top row matches the simulation shown in \\autoref{fig:mass_map_comparison_galsim}.\n  Following the right column from top to bottom shows how the maximum-likelihood estimates for the\n  GP parameters shifts to longer correlation lengths and smaller precision parameters with increasing\n  shape noise. That is, as the data becomes more noise dominated, the marginal GP parameter likelihood\n  changes shape to prefer smoothing, or effectively averaging, more galaxies to retain a more\n  significant shear and convergence signal. The ML values for the GP parameters are listed\n  in each panel showing the log-likelihood contours.\n  }\n  \\label{fig:sigma_e_maps}\n\\end{figure*}\n\nAs the shape noise increases (for the same input signal and measurement uncertainties), the peak\nin the marginal log-likelihood surface for $a_{\\rm GP}$ becomes broader and eventually disappears\nas shown by the rows of panels from top to bottom in \\autoref{fig:sigma_e_maps}. The ML value\nfor $\\ell^{2}_{\\rm GP}$, while less well defined, continues to yield maps that are more smoothed as\n$\\sigma_e^2$ increases. This helps to preserve large amplitudes of the peaks in the SNR maps (see\nthe color bar scales in the SNR maps of \\autoref{fig:sigma_e_maps}), but\nin compensation erases structures at all but the lowest spatial frequencies in the maps. This\nprocedure, with flat priors in the log of $a_{\\rm GP}$, appears useful for visualizing the posterior\nconvergence maps, but is undesirable for subsequent cosmological analyses. We see again that\nwe would prefer a prior favoring smaller $\\ell^{2}_{\\rm GP}$ even as the shape noise becomes large.\n\nWe assert such a prior in \\autoref{fig:sigma_e_maps_with_prior}, by imposing Gaussian priors\nseparately in $\\ln(\\lambda_{\\rm GP})$ and $\\ln(\\ell^{2}_{\\rm GP})$ with parameters given in\n\\autoref{tab:prior_params_galsim}. Our Gaussian prior is informed by the cosmological simulation study \nshown in \\autoref{fig:gp_cosmo_prior} combined with our prior that the GP correlation length be large enough \nso that white noise does not dominate the fits to the lens fields. \nFor each value of $\\sigma_e$ in \\autoref{fig:sigma_e_maps_with_prior}\nwe see that the signal-to-noise ratio (SNR) is comparable to that in \\autoref{fig:sigma_e_maps}\nbut the convergence maps include higher spatial frequency structures.\n\\begin{table}[!htb]\n  \\begin{center}\n    \\caption{\\label{tab:prior_params_galsim}Parameters for the Gaussian prior on the\n      GP parameters for our simulation study.}\n    \\begin{tabular}{lcc}\n      \\hline\\hline\n      Parameter & mean & std. dev. \\\\\n      \\hline\n      $\\ln\\left(\\lambda_{\\rm GP}\\right)$ & $18$ & 0.43 \\\\\n      $\\ln\\left(\\ell^{2}_{\\rm GP}\\right)$    & $-4.0$ & 0.1\\\\\n      \\hline\\hline\n    \\end{tabular}\n  \\end{center}\n\\end{table}\n\n\\begin{figure*}\n  \\centerline{\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\sigma_{e}=0.0026$}}{kappa_galsim_prior_sigmae1.png}\n    \\includegraphics[width=0.3\\textwidth]{snr_galsim_prior_sigmae1.png}\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\lambda_{\\rm GP}=2.2\\times 10^{8}, \\ell^2=0.012$}}{lnp_galsim_prior_sigmae1.png}\n  }\n  \\centerline{\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\sigma_{e}=0.0077$}}{kappa_galsim_prior_sigmae2.png}\n    \\includegraphics[width=0.3\\textwidth]{snr_galsim_prior_sigmae2.png}\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\lambda_{\\rm GP}=2.3\\times 10^{7}, \\ell^2=0.027$}}{lnp_galsim_prior_sigmae2.png}\n  }\n  \\centerline{\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\sigma_{e}=0.013$}}{kappa_galsim_prior_sigmae3.png}\n    \\includegraphics[width=0.3\\textwidth]{snr_galsim_prior_sigmae3.png}\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\lambda_{\\rm GP}=2.2\\times 10^{8}, \\ell^2=0.027$}}{lnp_galsim_prior_sigmae3.png}\n  }\n  \\centerline{\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\sigma_{e}=0.023$}}{kappa_galsim_prior_sigmae5.png}\n    \\includegraphics[width=0.3\\textwidth]{snr_galsim_prior_sigmae5.png}\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\lambda_{\\rm GP}=2.2\\times 10^{8}, \\ell^2=0.027$}}{lnp_galsim_prior_sigmae5.png}\n  }\n  \\caption{Same as \\autoref{fig:sigma_e_maps} except with a prior asserted for the GP\n  parameters $a_{\\rm GP}$ that limits the degree of smoothing in the posterior mean maps.\n  }\n  \\label{fig:sigma_e_maps_with_prior}\n\\end{figure*}\n\n\n\\subsubsection{Cosmological parameter constraints}\n\\label{sub:cosmo_params}\n\nAlthough we assert a non-cosmological GP prior on the lens fields to infer regular convergence\nand shear maps, we now demonstrate how we can recover cosmological information in a manner\nsimilar to common algorithms in the literature. That is, we compute an angular power spectrum\nestimator for the lensing convergence from the lens field posterior distribution.\n\nThe posterior distribution for the lens fields given the GP parameters is a multivariate\nGaussian distribution characterized by a mean field and covariance as given in\n\\autoref{eq:posterior_mean} and \\autoref{eq:posterior_cov}. The posterior mean is thus a\nconvenient and useful summary statistic (as we have shown above).\nAlso, because an isotropic Gaussian random field is fully described by the angular power spectrum,\nit is common in the literature to reduce cosmological large-scale structure statistics\nto two-point function estimators for cosmological parameter estimation.\n\nWe showed the angle-averaged (E-mode) convergence power spectrum estimator for our low-noise\nGaussian simulated maps in \\autoref{fig:power_spectra_galsim}. We use this power spectrum\nestimator as a summary statistic derived from the observed ellipticity catalog. We further\nassert a multivariate Gaussian likelihood function for the power spectrum estimator\nwith covariance~\\citep[e.g.,][]{2001ApJ...554...56C},\n\\begin{equation}\n  \\mathrm{Cov}\\left(P_{\\kappa}(\\ell)\\right) = \\frac{2}{N_{\\ell}}P_{\\kappa}^{2}(\\ell),\n\\end{equation}\nwhere $N_{\\ell}$ is the number of modes contributing to the band power estimator for a multipole\nbin centered at $\\ell$.\n\nIn \\autoref{fig:cosmo_param_contours} we show 68\\% and 95\\% contours of the 2D posterior\ndistribution for the cosmological mass density $\\Omega_m$ and density fluctuations r.m.s.\\\n$\\sigma_8$ given the angular power spectrum likelihood just described and flat priors.\nThe values used to generate the mock data are $\\Omega_m=0.3, \\sigma_8=0.8$.\nWe limit the multipole range of the power spectrum in the likelihood to $100 < \\ell < 1300$,\nwhere the upper bound is set by the effective smoothing length imposed by the asserted GP correlation\nparameter $\\ell^{2}_{\\rm GP}$.\n\\begin{figure}\n  \\centerline{\n    \\includegraphics[width=0.47\\textwidth]{cosmo_param_contours.png}\n  }\n  \\caption{Marginal constraints on the cosmological parameters using the E-mode power spectrum\n  estimator derived from the mean posterior convergence field.}\n  \\label{fig:cosmo_param_contours}\n\\end{figure}\n\nWe see from \\autoref{fig:cosmo_param_contours} that the GP interim prior used to derive the\nconvergence map from the galaxy ellipticity catalog has not biased the cosmological\nparameter constraints (within the uncertainties) obtained from a reduced summary statistic\nof the lens field posterior. We defer to later work more complete demonstrations of the\ncosmological parameter inference algorithms in \\autoref{sub:cosmological_parameter_inference}\nbased on marginalizing the lens field realizations.\n\n\n\\subsection{Abell cluster in the Deep Lens Survey}\n\\label{sub:abell}\n\nWe apply our lens field inference algorithm to the galaxy ellipticity catalog derived from the\nDLS\\footnote{\\url{http:\/\/dls.physics.ucdavis.edu}}~\\citep{2002SPIE.4836...73W}.\nThe DLS is a 20 square degree optical imaging survey optimized for cosmic shear measurements.\nWe analyze a $\\sim1$~square~degree field centered on the Abell cluster 781, which was\npreviously analyzed using DLS lensing shear measurements in\n\\citet{2006ApJ...643..128W,2009ApJ...702..980K,wittman2014}.\nAbell 781 consists of four massive galaxy clusters, which is a useful case study for our\nalgorithm because it provides a large lensing signal with a modest number of galaxies\nand because the distribution of mass density perturbations is decidedly not Gaussian\ndistributed.\n\n\\citet{jee2013} presented galaxy ellipticity measurements with the DLS $R$-band imaging calibrated\nto produce shear estimates with biases well below the statistical uncertainties for two-point\ncosmic shear correlation function estimators. The DLS shear pipeline in \\citet{jee2013} includes\ncorrelated PSF size and ellipticity corrections in each DLS exposure, calibration of additive\nand multiplicative shear biases with image simulations, and a set of null tests validating the PSF\nshear calibration corrections. The shear estimation pipeline used in \\citet{jee2013}, \\textsf{sFIT},\nwas further validated as the winning algorithm in the blinded community shear measurement\nchallenge \\textsf{GREAT3}~\\citep{great3-paper1}.\nThe DLS was performed with four optical pass-bands ($BVRz$) that allow photometric\nredshift (photo-$z$) estimates for all lensing source galaxies~\\citep{schmidt2013}.\n\\citet{jee2015} extended the DLS shear analysis to a tomographic cosmic shear measurement using\nthe photo-$z$ estimates.\n\nThe DLS galaxy ellipticity catalog produced for \\citet{jee2013} includes a catalog-level selection\nbased on measured galaxy magnitudes, sizes, ellipticity measurement error, photo-$z$ estimates, and\nproximity to masks as listed in \\citet{jee2013} Table 2.\nWe perform a further set of selections on this catalog as listed in\n\\autoref{tab:dls_selection}.\n\\begin{table}[!htb]\n  \\begin{center}\n    \\caption{\\label{tab:dls_selection}Selection criteria applied to the DLS galaxy catalog.}\n    \\begin{tabular}{ccc}\n    \\hline\\hline\n    Parameter        & min         & max \\\\\n    \\hline\n    $z_b$            & 0.45        & --  \\\\\n    $de$             & --          & 0.1 \\\\\n    $\\sqrt{a^2+b^2}$ & $0.8^{''}$  & --  \\\\\n    $R$              & 22          & 23  \\\\\n    \\hline\\hline\n    \\end{tabular}\n  \\end{center}\n\\end{table}\nWe choose the lower bound on (maximum posterior) photo-$z$, $z_b$, to select source galaxies\nthat are likely to be at redshifts larger than that of the highest redshift sub-cluster in the field\nat $z\\approx0.43$~\\citep{wittman2014}. We further select only those galaxies in the ellipticity\ncatalog with ellipticity measurement errors, $de$, less than 0.1 and sizes greater than 0.8~arcseconds to\nobtain galaxies likely to have more precisely measured shapes for informing the lensing shear.\nWe exclude galaxies with sizes, as determined from the geometric mean of the semi-major and semi-minor axes $a,b$,\nless than 0.8 arcseconds because the ellipticities tend to be less\nwell measured when the galaxy size is similar to that of the PSF.\nWe select the brighter galaxies based on $R$-band magnitude that are still likely to be faint\nenough to avoid significant contamination from cluster members. After all the selections listed\nin \\autoref{tab:dls_selection} we measure an ellipticity r.m.s.\\ of $\\sigma_e=0.21$. This measurement\nincludes the ellipticities with lensing effects included, but because lensing is sub-dominant\nto the intrinsic ellipticity dispersion we assert $\\sqrt{\\alpha} = \\sigma_e=0.21$ for the posterior\ninference for A781.\n\nWe show our mean posterior inference of the lensing convergence of A781 in\n\\autoref{fig:mass_map_Abell} using 6000 galaxies randomly selected from the cut sample described\nin \\autoref{tab:dls_selection}. We select only 6000 galaxies to limit the size of the lens field\njoint covariance that we must invert.\nThe left column of panels shows the convergence while the right\ncolumn shows the signal-to-noise ratio for the mean posterior. Our calculation is in the\ntop row of panels in \\autoref{fig:mass_map_Abell}, which we compare with the aperture mass\nalgorithm of \\citet{2012ApJ...747L..42D} in the bottom row of panels.\nNote we use the same galaxy sample as input to each mass mapping algorithm.\nThe algorithm of \\citet{2012ApJ...747L..42D},\ncalled `aperture densomitry', provides a mass estimator that is more localized on the sky. The shear\nin apertures is averaged with weighting functions that account for both angular selections and\nthe expected line-of-sight lensing kernel with the aide of the photometric redshift information\nin the DLS catalog. However, to make a more direct comparison with the algorithm in this paper,\nwe recomputed the aperture densomitry weights without using any photometric redshift information\nfor the source galaxies.\n\n\n\\autoref{fig:mass_map_Abell} shows that we obtain consistent results for the two main A781\nsub-clusters using our mean posterior map and the method used in a previous analysis. The white\ncrosses in \\autoref{fig:mass_map_Abell} indicate the locations of all sub-clusters detected\nin \\citet{wittman2014}. We do not detect all the same sub-clusters, which is likely\nbecause we use a significantly\nsmaller number of galaxies (6000 versus $\\sim 50000$) while we test the performance and scaling\nof our codes. However one sub-cluster denotedin \\autoref{fig:mass_map_Abell} is only detected\nin the literature in x-ray emission~\\citep{sehgal2008}.\n\\citet{wittman2014} also weight the source galaxies according to the expected\nlensing kernel and the photo-$z$ estimates. We make no use of photo-$z$ information other than\nin the sample selection.\n\\begin{figure*}[!htb]\n\t\\centerline{\n\t\t\\includegraphics[width=0.47\\textwidth]{kappa_Abell781.png}\n    \\includegraphics[width=0.47\\textwidth]{snr_Abell781.png}\n\t}\n  \\centerline{\n    \\includegraphics[width=0.47\\textwidth]{kappa_Abell781_WD.png}\n    \\includegraphics[width=0.47\\textwidth]{snr_Abell781_WD.png}\n  }\n\n\t\\caption{Posterior mean lens field maps (left) and SNR maps (right) for a\n  field centered on the galaxy cluster Abell 781. We use 6000 galaxies selected\n  to have photometric redshifts larger than the known redshifts of the two primary\n  clusters in this field of view ($z=0.296$ for Abell 781 and $z=0.43$ for a chance\n  alignment of a second cluster)\n  The top row of panels show the posterior mean map and SNR using our GP\n  prior. The bottom row of panels show the algorithm of \\citet{2012ApJ...747L..42D}\n  applied to the same galaxies with the same uniform per-galaxy weighting\n  as in the top row, but evaluated on a finer grid. The normalization of the\n  convergence in the lower left panel is arbitrary~\\citep[see][]{2012ApJ...747L..42D}.\n  The white crosses indicate sub-clusters identified by~\\citet{sehgal2008} with\n  x-ray detections. There is no associated mass for the white cross second\n  from the right in any published analyses of this system.\n\t}\n\t\\label{fig:mass_map_Abell}\n\\end{figure*}\n\n\\begin{table}[!htb]\n  \\begin{center}\n    \\caption{\\label{tab:prior_params_abell}Parameters for the Gaussian prior on the\n      GP parameters for A781.}\n    \\begin{tabular}{lcc}\n      \\hline\\hline\n      Parameter & mean & std. dev. \\\\\n      \\hline\n      $\\ln\\left(\\lambda_{\\rm GP}\\right)$ & $\\ln\\left(10^{6}\\right)$ & 0.1 \\\\\n      $\\ln\\left(\\ell^2\\right)$           & $\\ln\\left(10^{-4}\\right)$ & 0.5\\\\\n      \\hline\\hline\n    \\end{tabular}\n  \\end{center}\n\\end{table}\nIn \\autoref{fig:abell_lnp} we show the marginal log-likelihood for the A781 ellipticities in\nthe plane of the two GP parameters. Unlike in \\autoref{fig:sigma_e_maps} the noise covariance\nis now significant in defining the contours in \\autoref{fig:abell_lnp} such that a large\nGP precision and small GP correlation length is favored (indicating a sub-dominant\nsignal covariance). We therefore impose a Gaussian prior in the logarithm of the GP parameters\nwith parameters listed in \\autoref{tab:prior_params_abell}. We infer maximum posterior values\nof $\\lambda_{\\rm GP}=2.7\\times10^{6}$ and $\\ell^2=0.012$, which we use in the convergence inference\nin \\autoref{fig:mass_map_Abell}. However, as shown in \\autoref{fig:abell_lnp}, the marginal\nposterior for the GP parameters is only weakly peaked for this data set, and a range of GP\nparameters would be acceptable for the mass map inference.\n\\begin{figure}[!htb]\n    \\centerline{\n        \\includegraphics[width=0.45\\textwidth]{lnp_Abell781.png}\n    }\n    \\caption{The marginal posterior distribution for the GP parameters with\n    priors imposed to favor correlation lengths smaller than the expected cluster sizes.}\n    \\label{fig:abell_lnp}\n\\end{figure}\n\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nWe have demonstrated a probabilistic model inference of lensing shear and convergence\nusing a Gaussian Process (GP) prior. Our method is an extension of previous Bayesian maximum entropy\nmass mapping methods that is applicable in both cosmic shear (i.e., `field') and cluster\nlensing regimes. We validated our algorithm using simulated Gaussian distributed\nshear maps with comparisons to the input `truth' and by reconstructing the mass distribution\nin a cluster field in the Deep Lens Survey where we compare with the results of a previously\npublished `aperture densometry' method.\n\nBecause our GP kernel for the lensing shear and convergence is derived from a consistent lens\npotential, we find excellent separation of E and B modes in the posterior lens field maps. \nRecently \\citet{2016arXiv161003345B} presented an algorithm to project Wiener Filter maps into \n`pure' E and B modes, removing ambiguity in the E\/B decomposition from survey masks. Because we compute \nWiener Filter solutions for the lensing convergence from galaxy ellipticities, the method of \n\\citet{2016arXiv161003345B} is a simple extension of the algorithm we present in this paper. \n\nWe have also described an algorithm for optimization of the GP kernel parameters given a\nmeasured galaxy ellipticity catalog or a cosmologically-informed prior based on simulation\nof lensing fields. An interesting future extension of this work could include both galaxy \npositions and ellipticities in the reconstruction of the gravitational potential. \n\nOur algorithm is computationally challenging in the solution of linear systems of dimension equal\nto the number of galaxies. Next generation surveys will have $10^{7}$--$10^{10}$ galaxies, requiring\nboth parallelization of the linear solver routines and optimizations such as sparse matrix\napproximations via tapering~\\citep{Kaufman2008} or Fast Fourier Transform (FFT)-based matrix\nmultiplications~\\citep[exploiting the special structure of the isotropic cosmological covariances][]{padmanabhan++2003}.\nAs a next step in the validation and scaling of\nour code we plan to apply the algorithms in this paper to the variable shear `branches' of the\n{\\sc GREAT3}\\xspace challenge\\footnote{\\url{http:\/\/great3challenge.info\/}}~\\citep{Mandelbaum++2013}, which\ncontain $2\\times 10^{5}$ galaxies per simulated field. Coincidentally, each of the five\nDLS four square degree fields contains a comparable number of $\\sim2\\times10^5$ lensing source\ngalaxies.\n\nFor the algorithm we validated in this paper we assumed an approximate data vector of galaxy\nellipticity measurements rather than the more informative imaging pixel data for each galaxy, as we\ndescribed previously in \\citet{mbi-theory}.\nIf we use galaxy samples rather than an ellipticity catalog then we have to go back to\n\\autoref{eq:marg_like} and replace the integral over the intrinsic ellipticities with a numerical\nimportance sampling formula from \\citet{mbi-theory}. In this case we do not need to assume\nconjugacy of the intrinisc ellipticity prior and the likelihood function, which is good because\nthe pixel-level likelihood will not be conjugate. The algorithms we presented in\n\\autoref{sub:cosmo_params} for cosmological parameter estimation are applicable, however, when using\neither an approximate ellipticity data vector or the interim sampling of galaxy image model\nparameters. \n\nWe can still derive a Wiener Filter from the product of the Gaussian intrinsic ellipticity\ndistribution (or DP base distribution) and the interim variable shear GP prior assuming a weak\nshear approximation. But then we need to evaluate \\autoref{eq:posterior_cov} and\n\\autoref{eq:posterior_mean} for every interim sample.\nThe parameters of $\\mathsf{S}+\\mathsf{N}$ include the GP parameters in $\\mathsf{S}$ and the $\\alpha$ intrinsic\nellipticity distribution parameter in $\\mathsf{N}$. If we update either parameter we need to perform a\nnew matrix factorization and solve operation at every step (but these operations can be done once\nfor all interim samples of galaxy image model parameters in a given step). Exploring optimized\nlinear algebra approaches and effective sampling strategies for this more complete framework will\nbe a focus of future work.\n\nWe have not included any redshift information about the source galaxies in our model. However, the \nuse of such information (via photometric redshifts) is a critical component of the weak lensing \nanalyses for cosmic shear surveys as well as cluster mass reconstructions with optimized \nsignal-to-noise ratios. A simple extension of our work to include redshifts could be to impose \nseparable GP priors on the lens fields for galaxy samples binned in redshift (or redshift estimator).\nThe physical correlations between the lens fields inferred in each source bin can be modeled in \nthe hierarchical inference stage when marginalizing lens fields to infer cosmological parameters\n\\citep[as outlined in][]{mbi-theory}. A more thorough approach would be to include probabilistic \nredshifts for each source galaxy and marginalize over each source redshift distribution as \npart of the lens field inference. \n\n\\section*{Acknowledgments}\nWe thank Chris Paciorek for discussion about the\nstatistical framework for performing shear inference and the use of Gaussian Processes.\nWe thank Alex Malz for critical reviews of draft versions of this work.\nThanks to M. James Jee for providing the Deep Lens Survey shear catalog.\nPart of this work performed under the auspices of the U.S. Department of Energy\nby Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.\nFunding for this work was provided by LLNL Laboratory Directed Research and Development\ngrant 16-ERD-013.\nThis work was also supported by the Director, Office of Science, Office of Advanced\nScientific Computing Research, Applied Mathematics program of the U.S. Department of Energy under\nContract No.DE-AC02-05CH11231.\nThis research used resources of the National Energy Research Scientific\nComputing Center (NERSC), a DOE Office of Science User Facility supported by\nthe Office of Science of the U.S. Department of Energy under Contract No.\nDE-AC02-05CH11231.\nThis work uses a modified version of the public code \\texttt{George} available at\n\\url{https:\/\/github.com\/karenyyng\/george}, which was forked from\n\\url{https:\/\/github.com\/dfm\/george}. We also made use of the \\texttt{daft} package \n(developed by Dan Foreman-Mackey, David W. Hogg, and contributors, and available at \n\\url{http:\/\/daft-pgm.org\/}) for plotting probabilistic graphical models.\n\n\n\\bibliographystyle{apj3auth}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThe probably presence of an unknown form of energy in the universe, called dark energy, and confirmed by a large number of observations, starting by the data of Supernovae IA in 1998 \\cite{riess}, have leaded to explore the possible theoretical origin for this fluid. Since it is the direct responsible of the present accelerating expansion of the universe, a negative pressure is required which leads to a negative equation of state parameter. The most popular candidate, the cosmological constant, which posses a constant equation of state (EoS), $p_{\\Lambda}=-\\rho_{\\Lambda}$, can explain quite well the cosmological evolution. However,  the open possibility that the EoS is not completely constant but evolutes dynamically (even crossing the phantom barrier more), and the quite large difference between the observed dark energy density and the vacuum energy density predicted by quantum field theories, have leaded to explore other possibilities, as the existence of scalar fields, vector fields or modifications of General Relativity (GR), among others (for a review on dark energy candidates, see \\cite{copeland}). \n\\par\nIn the context of modified gravities, a wide range of possibilities have been explored, being $f(R)$ gravity probably the most popular one due to its simplicity since it generalizes the Hilbert-Einstein action to a more complex function of the Ricci scalar (for a review on $f(R)$ gravity, see \\cite{reviews,Bamba:2012cp}). Nevertheless, other kind of theories have been suggested, where other curvature invariants are included as the Gauss-Bonnet gravity. In this paper, we study the so-called $f(T)$ gravity, which in analogy to f(R) gravity, consists in a generalization of the action of Teleparallel gravity, a theory that assumes Weitzenbock connection instead of the Levi-Civita connection, which yields to a null curvature but a non-vanishing torsion (for a review see \\cite{pereira1}). In this gravitational theory, the main field is represented by the so-called tetrads instead of the metric as in GR. This kind of theories has become very popular recently as can also explain the accelerated expansion of the universe with no need of dark energy, and even the inflationary epoch (see \\cite{ferraro1}-\\cite{ratbay}). Then, a wide number of aspects have been studied in the context of $f(T)$ gravity, as  its local Lorentz invariance \\cite{barrow}, static solutions \\cite{daouda1}, non-diagonal tetrads \\cite{daouda3}, or the presence of wormholes \\cite{boehmer}, as well as other aspects \\cite{x,cemsinan}. Also a large effort has been done to study cosmological solutions for this class of theories, as well as possible cosmological predictions (see Refs.~\\cite{houndjo}-\\cite{li}).\n\\par\nAt the present work, we are interested to study some particular cosmological solutions in $f(T)$ gravity, where the appropriate action is reconstructed for each case. Specifically, the Bianchi type-I, Kantowski-Sachs (KS) and Bianchi type-III models are considered, and particularly some important solutions, such as power law and de Sitter (dS) expansion, or more complex ones as exponential functions for the scale factor in each direction of the space. Since power law and dS solutions can provide a good description for some specific phases of the universe evolution, their reconstruction in $f(T)$ gravity becomes a crucial point in order to consider this class of theories as serious candidates for explaining the whole cosmological history. In addition, here we assume more general cosmological metrics than Friedmann-Lema\\^itre-Robertson-Walker (FLRW) metrics,  in particular anisotropic universes described by the Bianchi type-I, Kantowski-Sachs (KS) and Bianchi type-III metrics, in order to provide the most general description of the cosmological evolution in the context of $f(T)$ gravity.  Moreover, exponential solutions are also considered, this kind of expansions has become very popular recently as they may conduct the universe to a non-singular state, where some bounded systems may be broken. Such state suggested in Ref.~\\cite{LittleRip}, and called {\\it Little Rip},  has already been studied in $f(R)$ gravity (see Ref.~\\cite{Nojiri:2011kd}), as well as in $f(T)$ theories \\cite{bambamyr}. Even more,  the possible occurrence of a   {\\it Little Rip} has been also explored in the context of the so-called viable modified gravities (see Ref.~\\cite{SaezGomez:2012ek}). Note that anisotropic\ncosmological metrics have been already studied in the context of GR with the presence of isotropic and anisotropic fluids, as well as the stability of the solutions \\cite{barrow3,stability}.\n\\par\nFurthermore, the use of an auxiliary scalar field, in analogy to the equivalence of Brans-Dicke theories for $f(R)$ gravity (see for instance Ref.~\\cite{STFR}), is also implemented, from which may result a useful tool to reconstruct the appropriate action as well as for studying the properties of $f(T)$ gravity. \n\\par\nThe main motivations of assuming  the assumption of a model with anisotropic geometry are based on several physical aspects as: the famous problem of the CMB quadrupole can be solved by considering a universe with planar symmetry \\cite{campanelli1} where eccentricity in decoupling, generated by a uniform cosmic magnetic field whose current strength,  $ B(t_0) \\sim 10^{-9}$ Gauss, should be close to $e_{dec}\\sim 10^{-2}$; the Bianchi type models in Loop Quantum Cosmology \\cite{lqc}, $^4He$ abundance \\cite{campanelli2}, cosmic parallax \\cite{campanelli3,fontanini}, small anisotropic pressures \\cite{barrow2}, cosmological solutions of the low energy string effective action \\cite{massimo}, anisotropic inflationary universe \\cite{inflation} and some other \\cite{x-1}. In the $f(R)$ theory, we already have some good results \\cite{can,sharif2}, therefore, we propose to establish the equations here and get the first results in $f(T)$ gravity, for the Bianchi type-I, type-III and KS models.\n\\par\nThen, the paper is organized as follows: in section \\ref{sec2}, the basic concepts of $f(T)$ gravity are introduced. In section \\ref{BK} , the equations for general  Bianchi type-I, type-III and Kantowski-Sachs (KS) models are deduced in a particular coordinate system and diagonal tetrads. Section \\ref{sec2a} deals with the reconstruction of the $f(T)$ action for some relevant solutions, and where several techniques are considered, including a kind of scalar-tensor theory for torsion gravity. Finally, section \\ref{conclusions} is devoted to the conclusions and discussions on the results found in the paper.\n\\section{\\large Preliminary definitions and equations of motion}\\label{sec2}\n\n\nAs previously mentioned, the $f(T)$ theory of gravity is defined in the Weitzenbock's space time in which the line element is described by \n\\begin{equation}\\label{el}\ndS^{2}=g_{\\mu\\nu}dx^{\\mu}dx^{\\nu}\\; ,\n\\end{equation} \nwhere $g_{\\mu\\nu}$ are the components of the metric which is symmetric and possesses $10$ degrees of freedom. One can describe the theory in the spacetime or in the tangent space, which allows us to rewrite the line element (\\ref{el}) as follows \n\\begin{eqnarray}\ndS^{2} &=&g_{\\mu\\nu}dx^{\\mu}dx^{\\nu}=\\eta_{ij}\\theta^{i}\\theta^{j}\\label{1}\\; ,\\\\\ndx^{\\mu}& =&e_{i}^{\\;\\;\\mu}\\theta^{i}\\; , \\; \\theta^{i}=e^{i}_{\\;\\;\\mu}dx^{\\mu}\\label{2}\\; ,\n\\end{eqnarray} \nwhere $\\eta_{ij}=diag[1,-1,-1,-1]$ and $e_{i}^{\\;\\;\\mu}e^{i}_{\\;\\;\\nu}=\\delta^{\\mu}_{\\nu}$ or  $e_{i}^{\\;\\;\\mu}e^{j}_{\\;\\;\\mu}=\\delta^{j}_{i}$. The square root of the metric determinant is given by  $\\sqrt{-g}=\\det{\\left[e^{i}_{\\;\\;\\mu}\\right]}=e$ and the matrix $e^{a}_{\\;\\;\\mu}$ are called tetrads and represent the dynamic fields of the theory.\n\\par\nBy using theses fields, one can define the Weitzenbock's connection as \n\\begin{eqnarray}\n\\Gamma^{\\alpha}_{\\mu\\nu}=e_{i}^{\\;\\;\\alpha}\\partial_{\\nu}e^{i}_{\\;\\;\\mu}=-e^{i}_{\\;\\;\\mu}\\partial_{\\nu}e_{i}^{\\;\\;\\alpha}\\label{co}\\; .\n\\end{eqnarray}\nThe main geometrical objects of the spacetime are constructed from this connection. The components of the tensor torsion are defined by the antisymmetric part of this connection\n\\begin{eqnarray}\nT^{\\alpha}_{\\;\\;\\mu\\nu}&=&\\Gamma^{\\alpha}_{\\nu\\mu}-\\Gamma^{\\alpha}_{\\mu\\nu}=e_{i}^{\\;\\;\\alpha}\\left(\\partial_{\\mu} e^{i}_{\\;\\;\\nu}-\\partial_{\\nu} e^{i}_{\\;\\;\\mu}\\right)\\label{tor}\\;.\n\\end{eqnarray}\nThe components of the contorsion are defined as \n\\begin{eqnarray}\nK^{\\mu\\nu}_{\\;\\;\\;\\;\\alpha}&=&-\\frac{1}{2}\\left(T^{\\mu\\nu}_{\\;\\;\\;\\;\\alpha}-T^{\\nu\\mu}_{\\;\\;\\;\\;\\alpha}-T_{\\alpha}^{\\;\\;\\mu\\nu}\\right)\\label{contor}\\; .\n\\end{eqnarray}\nIn order to make more clear the definition of the scalar equivalent to the curvature scalar of RG, we first define a new tensor $S_{\\alpha}^{\\;\\;\\mu\\nu}$, constructed from the components of the tensors torsion and contorsion as\n\\begin{eqnarray}\nS_{\\alpha}^{\\;\\;\\mu\\nu}&=&\\frac{1}{2}\\left( K_{\\;\\;\\;\\;\\alpha}^{\\mu\\nu}+\\delta^{\\mu}_{\\alpha}T^{\\beta\\nu}_{\\;\\;\\;\\;\\beta}-\\delta^{\\nu}_{\\alpha}T^{\\beta\\mu}_{\\;\\;\\;\\;\\beta}\\right)\\label{s}\\;.\n\\end{eqnarray}\nWe can now define the torsion scalar by the following contraction\n\\begin{eqnarray}\nT=T^{\\alpha}_{\\;\\;\\mu\\nu}S^{\\;\\;\\mu\\nu}_{\\alpha}\\label{te}\\; .\n\\end{eqnarray}\n\nThe action of the theory is defined by generalizing  the Teleparallel theory, as \n\\begin{eqnarray}\\label{action}\nS=\\int e\\left[f(T)+\\mathcal{L}_{Matter}\\right]d^4x\\;,\n\\end{eqnarray}\nwhere $f(T)$ is an algebraic function of the torsion scalar $T$. Making the functional variation of the action  (\\ref{action}) with respect to the tetrads, we get the following equations of motion \\cite{barrow,daouda1,daouda2}\n\\begin{eqnarray}\nS^{\\;\\;\\nu\\rho}_{\\mu}\\partial_{\\rho}Tf_{TT}+\\left[e^{-1}e^{i}_{\\mu}\\partial_{\\rho}\\left(ee^{\\;\\;\\alpha}_{i}S^{\\;\\;\\nu\\rho}_{\\alpha}\\right)+T^{\\alpha}_{\\;\\;\\lambda\\mu}S^{\\;\\;\\nu\\lambda}_{\\alpha}\\right]f_{T}+\\frac{1}{4}\\delta^{\\nu}_{\\mu}f=4\\pi\\mathcal{T}^{\\nu}_{\\mu}\\label{em}\\; ,\n\\end{eqnarray}\nwhere $\\mathcal{T}^{\\nu}_{\\mu}$ is the energy momentum tensor, $f_{T}=d f(T)\/d T$ and $f_{TT}=d^{2} f(T)\/dT^{2}$. By setting $f(T)=a_1T+a_0$,  the equations of motion (\\ref{em}) are the same as that of the Teleparallel theory with a cosmological constant, and this is dynamically equivalent to the GR.  These equations clearly depend on the choice made for the set of tetrads \\cite{cemsinan}. \n\\par\nThe contribution of the interaction with the matter fields is given by the energy momentum tensor which, is this case, is defined as \n\\begin{eqnarray}\n\\mathcal{T}^{\\,\\nu}_{\\mu}=  diag\\left(1,-\\omega_x,-\\omega_y,-\\omega_z\\right)\\rho                         \\label{tem}\\; ,\n\\end{eqnarray}\nwhere the $\\omega_i$ ($i=x,y,z$) are the parameters of equations of state related to the pressures $p_x$, $p_y$ and $p_z$.\n\n\\section{Field equations for Bianchi type-I, type-III and Kantowski-Sachs models} \\label{BK}\n\n\nLet us first establish the equations of motion of a set of diagonal tetrads using the Cartesian coordinate metric, for  describing models of Bianchi type-I, type-III and Kantowski-Sachs (KS). We propose to start with the Bianchi type-III case, from which Bianchi type-I and KS can be recovered. For the Bianchi type-III case, the metric reads \n\\begin{equation}\ndS^2=dt^2-A^2(t)dx^2-e^{-2\\alpha x}B^2(t)dy^2-C^2(t)dz^2\\,\\,\\,,\\label{metrictype3}\n\\end{equation}\nwhere $\\alpha$ is a constant parameter. Note that the Bianchi type-I is recovered by setting $\\alpha=0$ from the Bianchi type-III, while KS is recovered when one takes $\\alpha=0$ and $B(t)=C(t)$. Let us choose the following set of diagonal tetrads related to the metric (\\ref{metrictype3})\n\\begin{eqnarray}\n\\left[e^{a}_{\\;\\;\\mu}\\right]=diag\\left[1,A,e^{-\\alpha x}B,C\\right]\\;. \\label{matrixtype3}\n\\end{eqnarray}\nThe determinant of the matrix (\\ref{matrixtype3}) is $e=e^{-\\alpha x}ABC$. The components of the tensor torsion (\\ref{tor}) for the tetrads (\\ref{matrixtype3}) are given by\n\\begin{eqnarray}\nT^{1}_{\\;\\;01}=\\frac{\\dot{A}}{A}\\,,\\,T^{2}_{\\;\\;02}=\\frac{\\dot{B}}{B}\\,,\\, T^{2}_{\\;\\;21}=\\alpha\\,,\\,T^{3}_{\\;\\;03}=\\frac{\\dot{C}}{C}\\;,\\label{torsiontype3}\n\\end{eqnarray}\nand the components of the corresponding tensor contorsion are \n\\begin{eqnarray}\nK^{01}_{\\;\\;\\;\\;1}=\\frac{\\dot{A}}{A}\\,,\\,K^{02}_{\\;\\;\\;\\;2}=\\frac{\\dot{B}}{B}\\,,\\,K^{12}_{\\;\\;\\;\\;2}=\\frac{\\alpha}{A^2}\\,,\\,K^{03}_{\\;\\;\\;\\;3}=\\frac{\\dot{C}}{C}\\;.\\label{contorsiontype3}\n\\end{eqnarray} \nThe components of the tensor $S_{\\alpha}^{\\;\\;\\mu\\nu}$, in (\\ref{s}),  are given by\n\\begin{eqnarray}\nS_{0}^{\\;\\;01}=S_{3}^{\\;\\;31}=\\frac{\\alpha}{2A^2}\\,,\\,S_{1}^{\\;\\;10}=\\frac{1}{2}\\left(\\frac{\\dot{B}}{B}+\\frac{\\dot{C}}{C}\\right)\\,,\\,S_{2}^{\\;\\;20}=\\frac{1}{2}\\left(\\frac{\\dot{A}}{A}+\\frac{\\dot{C}}{C}\\right)\\,,\\,S_{3}^{\\;\\;30}=\\frac{1}{2}\\left(\\frac{\\dot{A}}{A}+\\frac{\\dot{B}}{B}\\right)\\;.\\label{tensortype3}\n\\end{eqnarray}\nBy using the components (\\ref{torsiontype3}) and (\\ref{tensortype3}),  the torsion scalar (\\ref{te}) is given by\n\\begin{eqnarray}\nT=-2\\left(\\frac{\\dot{A}\\dot{B}}{AB}+\\frac{\\dot{A}\\dot{C}}{AC}+\\frac{\\dot{B}\\dot{C}}{BC}\\right)\\; \\label{torsionScalar1}.\n\\end{eqnarray}\nThe equations of motion are given by \n\\begin{eqnarray}\n16\\pi \\rho &=& f+4f_T\\Big[\\frac{\\dot{A}\\dot{B}}{AB}+\\frac{\\dot{A}\\dot{C}}{AC}+\\frac{\\dot{B}\\dot{C}}{BC}-\\frac{\\alpha^2}{2A^2}\\Big]\\,\\,,\\label{densitytype3}\\\\\n-16\\pi p_x &=& f+2f_T\\left[\\frac{\\ddot{B}}{B}+\\frac{\\ddot{C}}{C}+\\frac{\\dot{A}\\dot{B}}{AB}+\\frac{\\dot{A}\\dot{C}}{AC}+2\\frac{\\dot{B}\\dot{C}}{BC}\\right]\\nonumber\\\\&+&2\\left(\\frac{\\dot{B}}{B}+\\frac{\\dot{C}}{C}\\right)\\dot{T}f_{TT}\\;,\\label{radialpressuretype3}\\\\\n-16\\pi p_y&=&f+2f_T\\left[\\frac{\\ddot{A}}{A}+\\frac{\\ddot{C}}{C}+\\frac{\\dot{A}\\dot{B}}{AB}+2\\frac{\\dot{A}\\dot{C}}{AC}+\\frac{\\dot{B}\\dot{C}}{BC}\\right]\\nonumber\\\\&+&2\\left(\\frac{\\dot{A}}{A}+\\frac{\\dot{C}}{C}\\right)\\dot{T}f_{TT}\\;,\\label{tangentialpressure1type3}\\\\\n-16\\pi p_z=f&+&2f_T\\left[\\frac{\\ddot{A}}{A}+\\frac{\\ddot{B}}{B}+2\\frac{\\dot{A}\\dot{B}}{AB}+\\frac{\\dot{A}\\dot{C}}{AC}+\\frac{\\dot{B}\\dot{C}}{BC}-\\frac{\\alpha^2}{A^2}\\right]\\nonumber\\\\&+&2\\left(\\frac{\\dot{A}}{A}+\\frac{\\dot{B}}{B}\\right)\\dot{T}f_{TT}\\;,\\label{tangentialpressure2type3}\\\\\n\\frac{\\alpha}{2A^2}\\left[\\left(\\frac{\\dot{A}}{A}-\\frac{\\dot{B}}{B}\\right)f_T-\\dot{T}f_{TT}\\right]&=&0 \\,\\,\\,, \\label{constraint1} \\\\\n\\alpha\\left(\\frac{\\dot{A}}{A}-\\frac{\\dot{B}}{B}\\right)f_T&=&0\\;.\\label{constraint2}\n\\end{eqnarray}\nIn the particular case where $f(T)=T-2\\Lambda$, the equations (\\ref{densitytype3})-(\\ref{constraint2}) are identical to that of the GR \\cite{akarsu}. The equation of constraint (\\ref{constraint2}) appears in both the GR as in $f(R)$ gravity \\cite{sharif2}. But here we have a second equation of constraint (\\ref{constraint1}), which appears as a generalization of the previous one, because here we have a contribution of a term of second derivative of the function $f(T)$ with respect to $T$.\n\\par\n\nBy setting $\\alpha=0$, the Bianchi type-I case is recovered and the equations of motions read\n\\begin{eqnarray}\n16\\pi \\rho &=& f+4f_T\\Big[\\frac{\\dot{A}\\dot{B}}{AB}+\\frac{\\dot{A}\\dot{C}}{AC}+\\frac{\\dot{B}\\dot{C}}{BC}\\Big]\\,\\,\\,,\\label{densitytype1}\\\\\n-16\\pi p_x &=& f+2f_T\\left[\\frac{\\ddot{B}}{B}+\\frac{\\ddot{C}}{C}+\\frac{\\dot{A}\\dot{B}}{AB}+\\frac{\\dot{A}\\dot{C}}{AC}+2\\frac{\\dot{B}\\dot{C}}{BC}\\right]+2\\left(\\frac{\\dot{B}}{B}+\\frac{\\dot{C}}{C}\\right)\\dot{T}f_{TT}\\;,\\label{radialpressuretype1}\\\\\n-16\\pi p_y&=&f+2f_T\\left[\\frac{\\ddot{A}}{A}+\\frac{\\ddot{C}}{C}+\\frac{\\dot{A}\\dot{B}}{AB}+2\\frac{\\dot{A}\\dot{C}}{AC}+\\frac{\\dot{B}\\dot{C}}{BC}\\right]+2\\left(\\frac{\\dot{A}}{A}+\\frac{\\dot{C}}{C}\\right)\\dot{T}f_{TT}\\;,\\label{tangentialpressure1type1}\\\\\n-16\\pi p_z&=&f+2f_T\\left[\\frac{\\ddot{A}}{A}+\\frac{\\ddot{B}}{B}+2\\frac{\\dot{A}\\dot{B}}{AB}+\\frac{\\dot{A}\\dot{C}}{AC}+\\frac{\\dot{B}\\dot{C}}{BC}\\right]+2\\left(\\frac{\\dot{A}}{A}+\\frac{\\dot{B}}{B}\\right)\\dot{T}f_{TT}\\;\\label{tangentialpressure2type1}\\;.\n\\end{eqnarray}\nThe equations of motion corresponding to KS model are obtained by setting $\\alpha=0$ and $B=C$, yielding\n\\begin{eqnarray}\n16\\pi\\rho&=&f+4f_T\\left[\\left(\\frac{\\dot{B}}{B}\\right)^2+2\\frac{\\dot{A}\\dot{B}}{AB}\\right]\\;,\\label{densityks}\\\\\n-16\\pi p_x&=&f+4f_T\\left[\\frac{\\ddot{B}}{B}+\\left(\\frac{\\dot{B}}{B}\\right)^2+\\frac{\\dot{A}\\dot{B}}{AB}\\right]+4\\frac{\\dot{B}}{B}\\dot{T}f_{TT}\\;,\\label{radialpressureks}\n\\end{eqnarray}\n\\begin{eqnarray}\n-16\\pi p_y&=&f+2f_T\\left[\\frac{\\ddot{A}}{A}+\\frac{\\ddot{B}}{B}+\\left(\\frac{\\dot{B}}{B}\\right)^2+3\\frac{\\dot{A}\\dot{B}}{AB}\\right]+2\\left(\\frac{\\dot{A}}{A}+\\frac{\\dot{B}}{B}\\right)\\dot{T}f_{TT}\\;,\\label{tangentialpressureks}\\\\\np_y&=&p_z\\nonumber\\,\\,\\,.\n\\end{eqnarray}\nIn the next section we will perform the reconstruction scheme of the action of the system for some particular cases.\n\n\\section{ Reconstructing $f(T)$ gravity in inhomogeneous universes}\\label{sec2a}\n\n\nLet us now consider the reconstruction of the $f(T)$ action for some particular solutions of the class of metrics explored in the previous section. Specifically, we consider solutions of the type of de Sitter, power law evolutions and exponential solutions. Note that de Sitter and power law solutions have been widely explored in other contexts of modified gravity, as $f(R)$ and Gauss-Bonnet gravities (see Ref.~\\cite{Nojiri:2009kx}), since they can provide a well description of  the cosmological evolution along its particular phases.\\\\\nLet's start by considering for simplicity Bianchi type-I and Kantowski-Sachs $(\\alpha=0)$ metrics. Then, the conservation equation for the energy momentum tensor (\\ref{tem}) can be easily obtained,\n\\begin{equation}\n\\dot{\\rho}+\\left(H_x+H_y+H_z\\right)\\rho+H_x p_x+H_y p_y +H_z p_z=0\\ ,\n\\label{D1}\n\\end{equation}\nwhere we have defined $H_x=\\frac{\\dot{A}}{A}\\ \\ H_y=\\frac{\\dot{B}}{B}\\ \\ H_z=\\frac{\\dot{C}}{C}$. We can now analyse de Sitter, power law solutions and exponential expansion in  Bianchi type-I metric by one side, and Kantowski-Sachs metric by the other, where $B=C$ that implies $p_y=p_z$.\n\n\\subsection{De Sitter solutions}\n\nDe Sitter solutions are well known in the context of cosmology since the current epoch, where the universe expansion is being accelerated, can be described approximately with a de Sitter solution. This kind of solutions consists on an exponential expansion of the scale factor, which yields a constant Hubble parameter. In the case of Bianchi type-I and Kantowski-Sachs metrics $(\\alpha=0)$ in (\\ref{metrictype3}), we may assume an exponential expansion for each spatial direction,\n\\begin{equation}\nA=A_0 {\\rm e}^{a t}\\ \\ B=B_0 {\\rm e}^{b t}\\ \\ C=C_0 {\\rm e}^{c t}\\ ,\n\\label{D2}  \n\\end{equation}\nand the rates of the expansion for each direction can be defined as,\n\\begin{equation}\nH_x=\\frac{\\dot{A}}{A}=H_{x0}\\ \\ H_{y}=\\frac{\\dot{B}}{B}=H_{y0}\\ \\ H_c=\\frac{\\dot{C}}{C}=H_{z0}\\ ,\n\\label{D3}\n\\end{equation}\nwhere $\\{H_{x0}=a,H_{y0}=b,H_{z0}=c\\}$ are constants. The torsion scalar defined in (\\ref{torsionScalar1}) is given by,\n\\begin{equation}\nT_0=-2\\left(H_{x0}H_{y0}+H_{x0}H_{z0}+H_{y0}H_{z0}\\right)\\ .\n\\label{D4}\n\\end{equation}\nThen, by assuming $p_x=p_y=p_z=p$ and an equation of state $p=w\\rho$, the conservation equation (\\ref{D1}) can be easily solved for the ansatz  (\\ref{D2}),\n\\begin{equation}\n\\rho=\\rho_0{\\rm e}^{-(H_{x0}+H_{y0}+H_{z0})(1+w)t}\\ .\n\\label{D5}\n\\end{equation}\nHence, the field equations (\\ref{densitytype1})-(\\ref{tangentialpressure2type1}) become,\n\\begin{eqnarray}\n16\\pi \\rho_0{\\rm e}^{-(H_{x0}+H_{y0}+H_{z0})(1+w)t}=f(T_0)+4\\left[H_{x0}H_{y0}+H_{z0}(H_{x0}+H_{y0})\\right]f_T(T_0)\\ , \\label{D6} \\\\\n-16\\pi w\\rho_0{\\rm e}^{-(H_{x0}+H_{y0}+H_{z0})(1+w)t}=f(T_0)+2(H_{y0}+H_{z0})(H_{x0}+H_{y0}+H_{z0})f_T(T_0)\\ , \\label{D7}\\\\\n-16\\pi w\\rho_0{\\rm e}^{-(H_{x0}+H_{y0}+H_{z0})(1+w)t}=f(T_0)+2(H_{x0}+H_{z0})(H_{x0}+H_{y0}+H_{z0})f_T(T_0)\\ , \\label{D8}\\\\\n-16\\pi w\\rho_0{\\rm e}^{-(H_{x0}+H_{y0}+H_{z0})(1+w)t}=f(T_0)+2(H_{x0}+H_{y0})(H_{x0}+H_{y0}+H_{z0})f_T(T_0)\\ . \\label{D9}\n\\end{eqnarray}\nNote that the only possible solution in the presence of a perfect fluid is one with $w=-1$ as the r.h.s. of equations (\\ref{D6})-(\\ref{D9}) is independent of time, according to the expression of the scalar torsion for a pure de Sitter solution (\\ref{D4}), unless $H_{x0}+H_{y0}+H_{z0}=0$, which would imply a decelerating expansion in a particular direction, being $H_{i0}<0$. Moreover, for a particular $f(T)$ action, the system of equations (\\ref{D4})-(\\ref{D9}) reduces to an algebraic system of equations for the variables $\\{H_{x0},H_{y0},H_{z0}\\}$. Since  the system of equations (\\ref{D4})-(\\ref{D9}) are composed by four equations, while there are only three variables,\nthe above 4-equations system has to be reduced. However, even in the case of  Kantowski-Sachs metric, where $B(t)=C(t)\\rightarrow H_{y0}=H_{z0}$, the system (\\ref{D4})-(\\ref{D9}) still posses three independent equations with two variables. Hence, the only possible solution imposes,\n\\begin{equation}\n A(t)=B(t)=C(t)\\rightarrow H_{x0}=H_{y0}=H_{z0}=H_0\\ ,\n \\label{D10}\n\\end{equation}\nAnd the metric (\\ref{metrictype3}) reduces to the well known Friedmann-Lema\\^itre-Robertson-Walker metric with an exponential expansion, $A(t)=A_0{\\rm e}^{H_0\\ t}$. Hence, the only solution for a pure de Sitter expansion in  Bianchi type-I and Kantowski-Sachs metrics gives a FLRW universe\\footnote{Recall that we have assumed here that the pressures are equal, $p_x=p_y=p_z$.}, and the system of equations (\\ref{D4})-(\\ref{D9}) reduces now to a unique independent equation,\n \\begin{equation}\n 16 \\pi \\rho_0=f(T_0)+12H_{0}^2f_T(T_0)\\ .\n \\label{D11}\n \\end{equation}\n Then, the roots of the algebraic equation (\\ref{D11}) give the de Sitter points of a particular $f(T)$ action. In order to illustrate such possibility, let us consider the action,\n \\begin{equation}\n f(T)=\\left(-T\\right)^n\\ ,\n \\label{D12}\n \\end{equation}\n where $n$ is a real constant. Then, the equation (\\ref{D11}) is rewritten as,\n \\begin{equation}\n 16\\pi \\rho_0=(1-2n)(6H_{0}^2)^n\\ ,\n \\label{D13}\n \\end{equation}\n whose solution is given by,\n \\begin{equation}\n H_{0}^2=\\frac{1}{6}\\left(\\frac{16\\pi\\rho_0}{1-2n}\\right)^{1\/n}\\ .\n \\label{D14}\n\\end{equation}\n Hence, the only physical solution ($\\rho_0,H_0^2\\geq0$) imposes $n\\leq1\/2$. Then, the de Sitter solution is a direct consequence of the energy density $\\rho_0$, which can be interpreted as a cosmological constant according to the condition imposed above for its equation of state, $w=-1$. Nevertheless, in vacuum the equation (\\ref{D13}) reduces to $0=(1-2n)(6H_{x0}^2)^n$, whose only solution is given by $n=1\/2$, rising to $f(T)=\\sqrt{-T}$ that posses an infinite number of de Sitter points. Moreover, we may consider in vacuum the action,\n \\begin{equation}\n f(T)=C_1 T+C_2 \\left(-T\\right)^n\\ ,\n \\label{D15}\n \\end{equation}\nwhere $\\{C_1,C_2\\}$ are the coupling constants and $n$ is a real constant. The field equation (\\ref{D11}) in vacuum yields,\n \\begin{equation}\n0= C_1 6H_{0}^2+C_2(1-2n)(6H_{0}^2)^n\\ .\n\\label{D16}\n\\end{equation}\nSo the roots of this equation give the dS points allowed by the class of theories expressed in (\\ref{D15}). Note that now, the exponential expansion is a direct consequence of the action instead of the contribution of a kind of cosmological constant as in the case shown above. For instance, $n=2$, it yields the solution,\n\\begin{equation}\nH_{0}=\\sqrt{\\frac{C_1}{18C_2}}\\ . \n\\label{D17}\n\\end{equation}\nWhile for higher powers of $n$, more de Sitter points can be obtained for the action (\\ref{D15}). Note that in $f(R)$ theories, dS points constitutes the critical points of the dynamical system, which may be (un)stable, and could explain both the inflationary and dark energy epochs (see \\cite{Cognola:2008zp}), which may be the case also in $f(T)$ gravity.\n\n\\subsection{Power law solutions}\n\nLet us now explore a cosmological evolution described by a power law in each direction of the space expansion. In such case, the scale parameters for the Bianchi type-I and Kantowski-Sachs  metric (\\ref{metrictype3}), where we set $(\\alpha=0)$, can be expressed as,\n\\begin{equation}\nA(t)=A_0t^a\\ , \\ B(t)=B_0t^b\\ , \\ C(t)=C_0t^c\\ ,\n\\label{D18}\n\\end{equation}\nwhere $\\{a,b,c\\}$ and $\\{A_0,B_0,C_0\\}$ are constants to be determined by the field equations, and initial conditions respectively. The expansion rates are given by,\n\\begin{equation}\nH_x=\\frac{a}{t}\\ , \\ H_y=\\frac{b}{t}\\ , \\ H_z=\\frac{c}{t}\\ .\n\\label{D19}\n\\end{equation}\nWhile the expression for the torsion scalar (\\ref{torsionScalar1}) yields,\n\\begin{equation}\nT=-2\\left(\\frac{ab}{t^2}+\\frac{ac}{t^2}+\\frac{bc}{t^2}\\right)\\ .\n\\label{D20}\n\\end{equation}\nThen, introducing the above quantities in the field equations (\\ref{densitytype1})-(\\ref{tangentialpressure2type1}), we get the following system of differential equations in $f(T)$,\n\\begin{eqnarray}\n16\\pi \\rho(T)=f(T)-2Tf_T(T)\\ , \\label{D21} \\\\\n-16\\pi w\\rho(T)=f(T)+\\frac{(b+c)(1-a-b-c)}{bc+a(b+c)}Tf_T(T)+2\\frac{(b+c)}{bc+a(b+c)}T^2f_{TT}(T)\\ , \\label{D22}\\\\\n-16\\pi w\\rho(T)=f(T)+\\frac{(a+c)(1-a-b-c)}{bc+a(b+c)}Tf_T(T)+2\\frac{(a+c)}{bc+a(b+c)}T^2f_{TT}(T)\\ ,\\label{D23} \\\\\n-16\\pi w\\rho(T)=f(T)+\\frac{(a+b)(1-a-b-c)}{bc+a(b+c)}Tf_T(T)+2\\frac{(a+b)}{bc+a(b+c)}T^2f_{TT}(T)\\ , \\label{D24}\n\\end{eqnarray}\nwhere we have assumed for simplicity that $p_x=p_y=p_z=p$ and an EoS $p=w\\rho$. Hence, the system (\\ref{D21})-(\\ref{D24}) is a set of differential equations in f(T) with the torsion scalar $T$ as the independent variable. \\\\ \n\nFirstly let us consider the vacuum case, or in other words, the homogeneous part of the first equation (\\ref{D21}), which becomes $f(T)-2Tf_T(T)=0$ and whose solution is given by,\n\\begin{equation}\nf(T)=C_1 \\sqrt{-T}\\ ,\n\\label{D26}\n\\end{equation}\nwhere $C_1$ is an integration constant. In order to satisfy the rest of the equations (\\ref{D22})-(\\ref{D24}), the condition $a=b=c$ must be imposed, so that the Hubble parameters (\\ref{D19}) reduce to the usual FLRW cosmology reproducing  power law solution. \\\\\n\nIn the presence of a kind of isotropic perfect fluid $p=w\\rho$, we can first solve the continuity equation (\\ref{D1}) in order to obtain $\\rho=\\rho(T)$, which yields,\n\\begin{equation}\n\\rho=\\rho_0 t^{-(a+b+c)(1+w)}=\\rho_0 \\left(-\\frac{T}{2(ab+ac+bc)}\\right)^{\\frac{(a+b+c)(1+w)}{2}}\\ .\n\\label{D25}\n\\end{equation}\nHence, the solution for the set of equations (\\ref{D21})-(\\ref{D24}) is given by $f(T)=f_h(T)+f_p(T)$, where $f_h(T)$ is the solution of the homogeneous equation, which coincides with the vacuum solution (\\ref{D26}), and $f_p(T)$ is the particular solution. Then, by using (\\ref{D25}) in the equation (\\ref{D21}), the particular solution can be easily found,\n\\begin{equation}\nf_{p}(T)=\\chi\\ T^{\\frac{(1+w)(a+b+c)}{2}}\\ , \n\\label{D27} \n\\end{equation}\n where $\\chi$ is a constant given by\n\\begin{equation}\n \\chi=\\frac{2^{4-(1+w)(a+b+c)\/2}\\pi \\rho_0}{\\left[-1+(1+w)(a+b+c)\\right] \\left[-bc-a(b+c)\\right]^{\\frac{(1+w)(a+b+c)}{2}}}\\ .\n \\label{D28}\n\\end{equation}\nNote that the condition $(1+w)(a+b+c)=2\\ n$ with $n$ being any natural number,  has to be imposed in order to avoid a complex gravitational action that would lacks any physical sense, recall  that $T<0$ for an expanding universe according to (\\ref{D20}).   In order to satisfy the complete set of equations (\\ref{D21})-(\\ref{D24}), we introduce the solution (\\ref{D27}) into the field equations (\\ref{D22})-(\\ref{D24}), and the following solutions for the parameters  $\\{a,b,c\\}$ are found,\n\\begin{enumerate}[ i.]\n\\item $\\ c=\\frac{1-w(a+b)}{w}\\ $, where $w\\neq 0$. This provides an anisotropic solution in $f(T)$ gravity with $A(t)$, $B(t)$ and $C(t)$ being different functions in (\\ref{D18}), and recalling that the perfect fluid assumed is an isotropic fluid. This does not hold in GR or Teleparallel Theory (TT) but is possible here due to the presence of second derivatives of the function $f(T)$ with respect to the torsion scalar $T$ in (\\ref{D21})-(\\ref{D24}). Note that field equations may be rewritten as the usual equations in TT, \n\\[\nH_xH_y+H_xH_z+H_yH_z=16\\pi(\\rho +\\rho_{f(T)})\\ ,\\; \\;\n-\\dot{H}_y-H_{y}^2-\\dot{H}_z-H_{z}^2-H_y\\ H_z=8\\pi (w\\rho +p_{f(T)}^x)\\ ,\n\\]\n\\begin{equation}\n-\\dot{H}_x-H_{x}^2-\\dot{H}_z-H_{z}^2-H_x\\ H_z=8\\pi (w\\rho +p_{f(T)}^y)\\ ,\\; \\;\n-\\dot{H}_x-H_{x}^2-\\dot{H}_y-H_{y}^2-H_x\\ H_y=8\\pi (w\\rho +p_{f(T)}^z)\\ .\n\\end{equation}\nHere, the extra terms coming from $f(T)$ are defined as an energy density $\\rho_{f(T)}$ and pressures $\\{p_{f(T)}^x, \\ p_{f(T)}^y, \\ p_{f(T)}^z\\}$, which are the origin of the anisotropic evolution. In this case, we have to fix $C_1=0$  in (\\ref{D26}) in order to satisfy the whole system.\n\\item $a=b=c$. The cosmological evolution expressed by \\ref{D18} reduces to a FLRW metric as in the homogeneous part of the equations, so that $C_1\\neq0$. \n\\item $c=-ab\/(a+b)$. In spite of this satisfies equations (\\ref{D21})-(\\ref{D24}) once (\\ref{D27}) is substituted in the equations, it gives $T=0$, and the r.h.s of equations (\\ref{D21})-(\\ref{D24}) become null while the l.h.s. are not, since $\\rho=\\rho(t)$ as given in (\\ref{D25}), so this is not a real solution.\n\n\\end{enumerate}\ufffd\n\\par\nTherefore,  we have obtained a complete set of power law solutions for Bianchi type-I  universe and Kantowski-Sachs metrics in the context of $f(T)$ gravity. Nevertheless, the action is clearly dependent on the EoS parameter $w$. Note also that in vacuum, the only possible solution reduces to a FLRW metric.\n\n\\subsection{General exponential solutions}\n\nIn this subsection we consider a more general exponential expansion for each spatial direction by \n\\begin{eqnarray}\nA=A_0e^{g_x(t)}\\,\\,\\,, \\quad B=B_0e^{g_y(t)}\\,\\,\\,,\\quad C=C_0e^{g_z(t)}\\,\\,\\,,\n\\end{eqnarray}\nwhere the function $g_i(t)$ is assumed as \n\\begin{eqnarray}\ng_i(t)=h_i(t)\\ln{\\left(t\\right)}\\,\\,,\\quad i=x,y,z\\;\\;,\n\\end{eqnarray}\n and $A_0$, $B_0$ and $C_0$ are positive constants. Note that the previous cases, the de Sitter solutions and power law solutions can be recovered from this one by setting $h_i(t)=a_it\/(\\ln{(t)})$ and $h_i(t)=a_i$, respectively, where $\\{a_i\\}=\\{a,b,c\\}$. In what follows, we will use an adiabatic approximation for the expansion in each spatial direction and neglect the derivatives of $h_i(t)$, i.e., setting $(\\dot{h}_i\\sim \\ddot{h}_i\\sim 0)$. The expansion rates in this case are given by\n\\begin{eqnarray}\nH_x=\\frac{h_x(t)}{t}\\,\\,,\\quad H_y=\\frac{h_y(t)}{t}\\,\\,,\\quad H_z=\\frac{h_z(t)}{t}\\,\\,.\n\\label{steph62}\n\\end{eqnarray}\nThus, the torsion scalar (\\ref{torsionScalar1}) becomes\n\\begin{eqnarray}\nT=-2\\left[\\frac{h_x(t)h_y(t)}{t^2}+\\frac{h_x(t)h_z(t)}{t^2}+\\frac{h_y(t)h_z(t)}{t^2}\\right]\\,\\,.\\label{steph63}\n\\end{eqnarray} \nThe acceleration in each direction is given by\n\\begin{eqnarray}\n\\ddot{A}=\\frac{h_x(h_x-1)}{t^2}A\\,\\,,\\quad \\ddot{B}=\\frac{h_y(h_y-1)}{t^2}B\\,\\,,\\quad \\ddot{C}=\\frac{h_z(h_z-1)}{t^2}C\\,\\,.\\label{steph64}\n\\end{eqnarray}\nSince $A$, $B$ and $C$ are positives, the acceleration is guaranteed in each direction when $h_i>1$, while for $0<h_i<1$, the universe  is in deceleration.\\par\nThe simplest example of $h_i(t)$ is\n\\begin{eqnarray}\nh_i(t)=\\frac{h_{i\\,in}+h_{i\\,out}\\,qt^2}{1+qt^2}\\,\\,\\,,\\label{steph65}\n\\end{eqnarray}\nwhere $h_{i\\,in}$, $h_{i,out}$ and $q$ are positive constants, and the $q$ is assumed to be enough small in order to make $h_i(t)$ varying slowly. Thus, the torsion scalar is always negative. From (\\ref{steph65}), we see that at early time $t=0$, $h_i\\rightarrow h_{i\\,in}$ and for late universe, $h_i\\rightarrow h_{i\\,out}$. By using \n(\\ref{steph65}) and (\\ref{steph63}), one gets the following equation\n\\begin{eqnarray}\nq^2Tt^6+\\left(2qT+2q^2X\\right)t^4+\\left(T+2qY\\right)t^2+2Z=0\\,\\,\\,,\\\\\nX=h_{x\\,out}h_{y\\,out}+h_{x\\,out}h_{z\\,out}+h_{y\\,out}h_{z\\,out}\\,\\,,\\quad\nY=h_{x\\,in}h_{y\\,out}+h_{x\\,out}h_{y\\,in}+h_{x\\,in}h_{z\\,out}+\\nonumber\\\\\nh_{x\\,out}h_{z\\,in}+h_{y\\,in}h_{z\\,out}+h_{y\\,out}h_{z\\,in}\\,\\,,\\quad \nZ=h_{x\\,in}h_{y\\,in}+h_{x\\,in}h_{z\\,in}+h_{y\\,in}h_{z\\,in}\\,\\,,\\nonumber\n\\end{eqnarray}\nwhose solutions read\n\\begin{eqnarray}\nt^2&=&\\left\\{\\Psi_0(T),\\;\\;\\Psi_{\\pm}(T)\\right\\}\\,,\\label{steph67}\\\\\n\\Psi_0(T)&\\equiv& \\alpha^{1\/3}+\\beta_0\\alpha^{-1\/3}+\\beta_1\\,\\;\\;\\Psi_{\\pm}(T)\\equiv e^{\\mp 2i\\pi\/3}\\alpha^{1\/3}+e^{\\pm 2i\\pi\/3}\\alpha^{-1\/3}+\\beta_1\\,\\,,\\;\\;\\,\\nonumber\\\\ \\alpha&=&\\frac{\\sqrt{\\alpha_1}}{q^2T^2}-\\alpha_2\\,\\,,\\;\n\\beta_0=\\frac{qT(8X-6Y)+4q^2X^2+T^2}{9q^2T^2}\\,\\,,\\quad \\beta_1=-\\frac{2qX+2T}{3qT}\\,\\,,\\nonumber\\\\\n\\alpha_1&=&\\frac{1}{27qT}\\Big[27qT^2Z^2+[(-36q^2TX-36qT^2)Y+16q^3X^3+48q^2TX^2+30qT^2X-2T^3]Z\\nonumber\n\\\\&+&8q^2TY^3+(-4q^3X^2-8q^2TX+8qT^2)Y^2+(-4q^2TX^2-8qT^2X+2T^3)Y-qT^2X^2-2T^3X\\Big]\\,\\,,\\nonumber\\\\\n\\alpha_2&=&\\frac{qT^2(27Z-18Y+15X)+q^2T(24X^2-18XY)+8q^3X^3-T^3}{27q^3T^3}\\,\\,.\\nonumber\n\\end{eqnarray}\nWe see from (\\ref{steph67}) that there are one real positive solution, $\\Psi_0(T)$, and two complex solutions $\\Psi_{\\pm}(T)$. By using (\\ref{steph62})-(\\ref{steph64}) the system of equations of motion (\\ref{densitytype1})-(\\ref{tangentialpressure2type1}) becomes\n\\begin{eqnarray}\n16\\pi\\rho&=&f-2Tf_T\\,\\,\\,,\\label{steph68}\\\\\n-16\\pi p_x&=&f+2f_T\\left[\\frac{(h_y+h_z)^2-(h_y+h_z)-h_yh_z}{\\Psi_0(T)}-\\frac{T}{2}\\right]+4\\left(\\frac{h_y+h_z}{\\Psi_0(T)}\\right){T}f_{TT}\\,\\,,\\label{steph69}\\\\\n-16\\pi p_y&=&f+2f_T\\left[\\frac{(h_x+h_z)^2-(h_x+h_z)-h_xh_z}{\\Psi_0(T)}-\\frac{T}{2}\\right]+4\\left(\\frac{h_x+h_z}{\\Psi_0(T)}\\right){T}f_{TT}\\,\\,,\\label{steph70}\\\\\n-16\\pi p_z&=&f+2f_T\\left[\\frac{(h_x+h_y)^2-(h_x+h_y)-h_xh_y}{\\Psi_0(T)}-\\frac{T}{2}\\right]+4\\left(\\frac{h_x+h_y}{\\Psi_0(T)}\\right){T}f_{TT}\\,\\,,\\label{steph71}\\\\\nh_i&=&\\frac{h_{i in}+h_{i out}q\\Psi_0(T)}{1+q\\Psi_0(T)}\\,\\,\\,.\\label{steph72}\n\\end{eqnarray} \n\\subsubsection{A special case}\\label{numero1}\n\nHere, we assume that the expansion rates are equal in the three spatial directions ($h_x=h_y=h_z$), and the system (\\ref{steph68})-(\\ref{steph71}) reduces to\n\\begin{eqnarray}\n16\\pi\\rho&=&f-2Tf_{T}\\,\\,,\\label{steph73}\\\\\n-16\\pi p_x&=&f-2f_T\\left(T+\\sqrt{\\frac{-2T}{3\\Psi_0(T)}}\\right)+4T\\sqrt{\\frac{-2T}{3\\Psi_0(T)}}f_{TT}\\,\\,,\\label{steph74}\\\\\np_x&=&p_y=p_z\\,\\,\\,,\\nonumber\n\\end{eqnarray}\nwhich means that the assumption of having the same rate in the three spatial direction leads to an isotropic matter content. By combining (\\ref{steph73}) and (\\ref{steph74}), one gets \n\\begin{eqnarray}\n4T\\sqrt{\\frac{-2T}{3\\Psi_0(T)}}f_{TT}-2f_T\\left[(1+\\omega)T+\\sqrt{\\frac{-2T}{3\\Psi_0(T)}}\\,\\right]+(1+\\omega)f=0\\,\\,\\,,\\label{steph75}\n\\end{eqnarray}\nwhere we used the barotropic equation $p_x=\\omega\\rho$. Let us consider an asymptotic analysis, looking for the early universe (small time) and the late time one (large time), for which the function $h_i(t)$ yields $h_{i\\,in}$ and $h_{i\\,out}$, respectively. Thus, for the early universe Eq. (\\ref{steph75}) reduces to \n\\begin{eqnarray}\n4T^2f_{TT}+6Tf_{T}\\left[3h_{i\\,in}(\\omega+1)-1\\right]-3h_{i\\,in}(\\omega+1)f=0\\,\\,\\,,\\\\\nh_{i\\,in}=h_{x\\,in}=h_{y\\,in}=h_{z\\,in}\\,\\,\\,,\\nonumber\n\\label{steph76}\n\\end{eqnarray}\nwhose general solution reads \n\\begin{eqnarray}\nf(T)=C_3T^{\\gamma_+}+C_4T^{\\gamma_-}\\,\\,,\\quad \\gamma_{\\pm}=\\frac{5-9h_{x\\,in}(1+\\omega)\\pm\\sqrt{25-78h_{x\\,in}(1+\\omega)+81h^2_{x\\,in}(1+\\omega)^2}}{4}\\,\\,,\\label{steph77}\n\\end{eqnarray}\nwhere $C_3$ and $C_4$ are integration constants. From (\\ref{steph77}), by writing the radicand as $\\left[5-9h_{x\\,in}(1+\\omega)\\right]^2+12h_{x\\,in}(1+\\omega)$, one observes that, for any $\\omega>-1$, $\\gamma_{+}>0$ and $\\gamma_{-}<0$. Moreover, in this context of asymptotic analysis, we observe from (\\ref{steph63}) that for small $t$, the torsion scalar $T$ is large, while for large $t$, the torsion is small. Thus, for small $t$ with $h_{x\\,in}>1$, corresponding to the inflation, the algebraic expression of $f(T)$ is given by\n\\begin{eqnarray}\nf(T)=C_3T^{\\gamma_{+}}\\,\\,\\,.\n\\end{eqnarray}\nSince $h_x(t)$ reduces to $h_{x\\,out}$ in the late universe, the model corresponding to the late accelerated universe can be obtained by replacing $h_{x\\,in}$ by $h_{x\\,out}$. Precisely, for large $t$, the torsion scalar is small, and for $h_{x\\,out}>1$, the dominate term in (\\ref{steph77}), corresponding to the model of late time universe, is\n\\begin{eqnarray}\nf(T)=C_4T^{\\gamma_{-}'}\\,\\,\\,,\\;\\;\\gamma_{\\pm}'=\\frac{5-9h_{x\\,out}(1+\\omega)\\pm\\sqrt{25-78h_{x\\,out}(1+\\omega)+81h^2_{x\\,out}(1+\\omega)^2}}{4}\\,\\,.\n\\end{eqnarray}\nThis model is equivalent to the teleparallel gravity for $C_4=1$ and $h_{x\\,out}=2\/(5+5\\omega)$. It is easy to see from this that, for any ordinary matter, i.e., $\\omega>0$, one gets $h_{x\\,out}<1$, meaning that the teleparallel gravity without cosmological constant cannot provide the late acceleration of the universe (remembering that the acceleration is guaranteed for $h_{x\\,out}>1$, and $0<h_{x\\,out}<1$ characterising a decelerated expanding universe). Thus, the contribution of the $f(T)$ terms plays the role of the dark energy.\\par\nLooking for the expression of $f(T)$ for large cosmic time $t$, i.e., the expression (\\ref{steph77}) (replacing $h_{x\\,in}$ by $h_{x\\,out}$), a similarity can be observed with a result of Bamba et al in \\cite{bambamyr}. In this work, they undertook $f(T)$ theory in the FLRW metric, first assuming a power-law expression for $f(T)$ in the Eq.~(4.6) for investigating what type of finite future time singularities may appear. Later, they introduced a correction term, still in the form of power-law, their Eq.~(4.22), in order to analyse the possible avoidance of the singularities. Then, they obtained the global expression (4.23) of \\cite{bambamyr}. Note that this expression is equivalent to our Eq.~(\\ref{steph77}). Moreover, they shown in their ``TABLE II\" that the Big Rip and the Big Freeze can be removed if the product of the exponents is negative. This is exactly our result, since $\\gamma_{+}'$ and $\\gamma_{-}'$ are always positive and negative, respectively ($\\gamma_{\\pm}'$ are obtained from $\\gamma_{\\pm}$ by replacing $h_{x\\,in}$ by $h_{x\\,out}$). This means that for both Bianchi type-I and KS, if the expansion of the universe occurs with the same rate in all direction, models that can realize the late time accelerated expansion of the universe, and able to prevent the Big Rip and the Big Freeze can be reconstructed.\n\n\\subsubsection{Using an auxiliary scalar field}\\label{numero2}\nIn this subsection, we would like to use the reconstruction scheme, for which an auxiliary scalar field is introduced in the action of the theory. By this way, the functional form of $f(T)$ can be found through two other scalar functions $P$ and $Q$\n(for more clarification on the method, see the case of $f(R)$ gravity \\cite{odintsov-asf} and other of $f(T)$ gravity \\cite{daouda3}).\nHere, we return to the general system (\\ref{steph68})-(\\ref{steph72}), where we introduce the auxiliary scalar field $\\phi$, by defining the algebraic function $f(T)$ by\n\\begin{eqnarray}\nf(T)=P(\\phi)T+Q(\\phi)\\,\\,\\,. \\label{steph78}\n\\end{eqnarray}\nBy using (\\ref{steph78}) and varying the action (\\ref{action}) with respect to the scalar field $\\phi$, one gets\n\\begin{eqnarray}\nP'(\\phi)T+Q'(\\phi)=0\\,\\,\\,,\n\\end{eqnarray}\nwhich may be solved with respect to $\\phi$, yielding  $\\phi=\\phi(T)$. Here, the prime (') denotes the derivative with respect to $\\phi$. By using (\\ref{steph78}), one obtains\n\\begin{eqnarray}\nf_T(T)=P(\\phi(T)) \\,\\,\\,,\\quad f_{TT}(T)=P_T(\\phi(T))\\,\\,\\,. \\label{steph80}\n\\end{eqnarray}\nMaking use of (\\ref{steph80}), one can rewrite the system (\\ref{steph68})-(\\ref{steph71}) as \n\\begin{eqnarray}\n16\\pi\\rho&=&P(\\phi)T+Q(\\phi)-2TP(\\phi)\\,\\,\\,,\\label{steph81}\\\\\n-16\\pi p_x&=&P(\\phi)T+Q(\\phi)+2P(\\phi)\\left[\\frac{(h_y+h_z)^2-(h_y+h_z)-h_yh_z}{\\Psi_0(T)}-\\frac{T}{2}\\right]\\nonumber\\\\&+&4\\left(\\frac{h_y+h_z}{\\Psi_0(T)}\\right){T}P_T(\\phi)\\,\\,,\\label{steph82}\n\\end{eqnarray}\n\\begin{eqnarray}\n-16\\pi p_y&=&P(\\phi)T+Q(\\phi)+2P(\\phi)\\left[\\frac{(h_x+h_z)^2-(h_x+h_z)-h_xh_z}{\\Psi_0(T)}-\\frac{T}{2}\\right]\\nonumber\\\\\n&+&4\\left(\\frac{h_x+h_z}{\\Psi_0(T)}\\right){T}P_T(\\phi)\\,\\,,\\label{steph83}\\\\\n-16\\pi p_z&=&P(\\phi)T+Q(\\phi)+2P(\\phi)\\left[\\frac{(h_x+h_y)^2-(h_x+h_y)-h_xh_y}{\\Psi_0(T)}-\\frac{T}{2}\\right]\\nonumber\\\\\n&+&4\\left(\\frac{h_x+h_y}{\\Psi_0(T)}\\right){T}P_T(\\phi)\\,\\,\\label{steph84} \\,\\,.\n\\end{eqnarray}\nLet us consider that $p_y=p_z$. Then, by equating (\\ref{steph83}) with (\\ref{steph84}), one gets\n\\begin{eqnarray}\nTP_T(\\phi)=-\\frac{1}{2}P(\\phi)\\left(h_x+h_y+h_z-1\\right)\\,\\,\\,.\\label{steph85}\n\\end{eqnarray}\nUsing (\\ref{steph85}), the system (\\ref{steph81})-(\\ref{steph84}) reduces to\n\\begin{eqnarray}\n16\\pi\\rho&=&Q(\\phi)-TP(\\phi)\\,\\,\\,,\\label{steph86}\\\\\n-16\\pi p_x&=&Q(\\phi)+TP(\\phi)\\,\\,\\,,\\label{steph87}\\\\\np_x&=& p_y=p_z \\,,\\,\\omega_x=\\omega_y=\\omega_z\\label{steph88}\\,.\n\\end{eqnarray} \nBy using (\\ref{steph86}), one can determine $Q(\\phi)$ as\n\\begin{eqnarray}\nQ(\\phi)=16\\pi~\\rho+TP(\\phi)\\,\\,\\,,\\label{steph90}\n\\end{eqnarray}\nwhich, substituted in (\\ref{steph87}) and (\\ref{steph88}), yields\n\\begin{eqnarray}\n-16\\pi(\\omega_x+1)\\rho=2TP(\\phi)\\label{steph91}\\,.\n\\end{eqnarray}\nAs one can redefine the scalar field properly, we may choose $\\phi=t$, then $P(\\phi)=P(t)=\\bar{P}(T)$. Note that in the case where $\\omega_x=-1$, one gets $\\bar{P}(T)=0$, and the algebraic function $f(T)=Q(t)=16\\pi\\rho=\\bar{Q}(T)$. But this requires to have the complete expression of $\\rho$ depending on $T$, i.e., solving the equation of continuity. Let us consider in general the case where $\\omega_x\\neq-1$ and try to solve the equation of continuity, which will help us to determine  $\\bar{P}(T)$ and $\\bar{Q}(T)$, leading to the reconstruction of the algebraic function $f(T)$. We choose the case of KS geometry ($h_y=h_z$), where the equation of continuity is written as\n\\begin{eqnarray}\n\\dot{\\rho}+\\rho(1+\\omega_x)(H_x+2H_y)=0\\,\\,\\,,\n\\end{eqnarray}\nwhich can be solved, giving\n\\begin{eqnarray}\n\\rho(t)&=&C_5\\exp{\\left[G_1(t)+G_2(t)\\right]}\\label{steph93}\\,\\,\\,,\\\\\nG_1(t)&=&\\frac{(1+\\omega_x)(h_{x\\,in}+2h_{y\\,in})}{t}\\,\\,\\,,\\\\\nG_2(t)&=&\\sqrt{q}\\left[(1+\\omega_x)(h_{x\\,in}-h_{x\\,out}+2h_{y\\,in}-2h_{y\\,out})\\right]\n\\arctan{\\left(\\sqrt{q}t\\right)}\\,\\,\\,,\n\\end{eqnarray}\nwhere $C_5$ is an integration constant.\nMaking use of (\\ref{steph67}), one can cast $\\rho(t)$ in terms of $T$. Also, from Eq.~(\\ref{steph91}), one obtains $P(t)$ in functions of $T$, and from (\\ref{steph90}), one gets $Q(t)$ in terms of $T$. Therefore, we can reconstruct the algebraic function  $f(T)$, given in  (\\ref{steph78}), as\n\\begin{eqnarray}\nf(T)&=&-16\\pi C_3 \\omega_x \\exp\\left[\\frac{g_1}{\\sqrt{\\Psi_0 (T)}}+g_2 \\arctan\\left(\\sqrt{q\\Psi_0 (T)}\\right)\\right]\\label{steph103}\\,,\n\\end{eqnarray}\nwhere $g_1=(1+\\omega_x)(h_{x\\,in}+2h_{y\\,in})$ and $g_2=\\sqrt{q}(1+\\omega_x)(h_{x\\,in}-h_{x\\,out}+2h_{y\\,in}-2h_{y\\,out})$.\n\\par\nIn principle, with (\\ref{steph103}), some cosmological models can now be reproduced. Let us focus our attention to the  early universe (which may be characterized by the inflation), and the late time universe (characterized by its accelerated expansion) in the three spatial direction. \\par\nAt early time, i.e., for small $t$  (or  large $T$), $\\Psi_0(T)$ is very small, and the corresponding algebraic function is\n\\begin{eqnarray}\nf(T)=-16\\pi C_3\\omega_x\\exp{\\left[\\frac{g_1}{\\sqrt{\\Psi_0 (T)}}\\right]}\\,\\,\\,.\n\\end{eqnarray}\nAt late time, the time $t$ is large (corresponding to small torsion scalar and large $\\Psi_0(T)$), and the algebraic function reads\n\\begin{eqnarray}\nf(T)=-16\\pi C_3\\omega_x\\exp{\\left[g_2 \\arctan\\left(\\sqrt{q\\Psi_0 (T)}\\right)\\right]}\\,\\,\\,.\n\\end{eqnarray}\nThus, we see that models corresponding to the inflation and the late time accelerated universe can be reconstructed within KS metrics where the matter content is partially isotropic ($p_y=p_z$). Since in this work we fall in the situation where $\\omega_x=\\omega_y=\\omega_y$ one could use the WMAP data and try to check if they fit with this anisotropic model. Because of having ultimately $p_x=p_y=p_z$ and $h_y=h_z$, one can just use the first two equations of motion of KS case, i.e., (\\ref{densityks})-(\\ref{radialpressureks}). In order to put out the contribution of the modified part of the algebraic function $f(T)$, we cast it into the form $f(T)=T+j(T)$, as the teleraparallel term plus the modified additive algebraic function $j(T)$. Thus, the equations (\\ref{densityks}) and (\\ref{tangentialpressureks}) become\n\\begin{eqnarray}\n8\\pi \\rho_{eff}&=& \\left(\\frac{\\dot{B}}{B}\\right)^2+2\\frac{\\dot{A}\\dot{B}}{AB}\\,\\,,\\label{effectivedensity}\\\\\n-8\\pi p_{x\\,eff}&=& 2 \\frac{\\ddot{B}}{B}+\\left(\\frac{\\dot{B}}{B}\\right)^2\\,\\,,\\label{effectivepressure}\n\\end{eqnarray}\nwhere $\\rho_{eff}$ and $p_{x\\,eff}$ are the effective energy density and effective pressure, respectively, and defined by\n\\begin{eqnarray}\n\\rho_{eff}&=&\\rho-\\frac{1}{16\\pi}\\left\\{j+4j_T\\left[ \\left(\\frac{\\dot{B}}{B}\\right)^2+2\\frac{\\dot{A}\\dot{B}}{AB}\\right]\\right\\}\\,\\,\\,,\\\\\np_{x\\,eff}&=&p_{x}+\\frac{1}{16\\pi}\\left\\{j+4j_T\\left[\\frac{\\ddot{B}}{B}+\\left(\\frac{\\dot{B}}{B}\\right)^2+\\frac{\\dot{A}\\dot{B}}{AB}\\right]+4\\frac{\\dot{B}}{B}\\dot{T}j_{TT}\\right\\}\\,\\,\\,.\n\\end{eqnarray}\nBy dividing (\\ref{effectivepressure}) by (\\ref{effectivedensity}), using $\\omega_{eff}=p_{x\\,eff}\/\\rho_{eff}$, one gets\n\\begin{eqnarray}\n\\omega_{eff}=\\frac{2-3h_y}{h_y+2h_x}\\label{steph104}\\,.\n\\end{eqnarray}\nSince the observational data from the WMAP project do not depend on the spatial direction, i.e., are not based on an anisotropic geometry, we have to impose $h_x=h_y$ in (\\ref{steph104}), for getting a suitable cosmological feature. Thus, assuming that the accelerated expansion of the universe is realized with the same rate in all directions, $h_x=h_y=h_z>1$, and that the universe is essentially filled  by the dark energy, where we can neglect the usual matter content such that $\\omega_{eff}\\sim \\omega_{DE}$. Therefore, using (\\ref{steph104}), we have the possibility to regain the well known range of values  allowed by the current 7-year WMAP data for the parameter of the equation of state of the  dark energy, $\\omega_{DE}= -1.1 \\pm 0.14$  WMAP \\cite{wmap}.\n\n\\subsection{On Bianchi type-III solutions}\n\nIn this section, we propose to present some comments on Bianchi type-III solutions. This case is quite exceptional since we do not have the freedom of making cosmological reconstruction as in the case of Bianchi type-I and KS, due to the constraints equations \n(\\ref{constraint1}) and (\\ref{constraint2}). \\par\nFrom (\\ref{constraint2}), since the parameter is different from zero and the algebraic function cannot be a constant, one gets\n\\begin{eqnarray}\n\\frac{\\dot{A}}{A}=\\frac{\\dot{B}}{B}\\,\\,\\,,\n\\end{eqnarray}\nwhich, injected in (\\ref{constraint1}) leads to\n\\begin{eqnarray}\n\\dot{T}f_{TT}=0\\,\\,\\,,\\label{constraint3}\n\\end{eqnarray}\nmeaning that one has $\\dot{T}=0$ or $f_{TT}=0$. The first case, $\\dot{T}=0$ implies that one has a constant torsion scalar, i.e.,\n\\begin{eqnarray}\n\\frac{\\dot{A}^2}{A^2}+2\\frac{\\dot{A}\\dot{C}}{AC}=K\\,\\,,\\label{steph111}\n\\end{eqnarray}\nwhere $K$ is positive constant. Let us consider $A=C^n$, with $n$ bigger than zero or less  than $-2$. Thus, Eq.~(\\ref{steph111}) can be solved yielding\n\\begin{eqnarray}\nC(t)=C_0'\\exp{\\left(\\pm\\sqrt{\\frac{K}{n(n+2)}}\\;t\\;\\right)}\\,\\,\\,,\\label{soluC}\n\\end{eqnarray}\nleading to \n\\begin{eqnarray}\nA(t)=B(t)=(C_0')^n\\exp{\\left(\\pm n\\sqrt{\\frac{K}{n(n+2)}}\\;t\\;\\right)}\\,\\,\\,,\n\\end{eqnarray}\nwhere $C_0'$ is a positive constant. It is important to note that in order to guarantee the expansion of the universe, one need to have\n\\begin{eqnarray}\\label{vincent18}\nA(t)=B(t)=\\left\\{\\begin{array}{ll}\n(C_0')^n\\exp{\\left(- n\\sqrt{\\frac{K}{n(n+2)}}\\;t\\;\\right)}\\,\\,\\,,\\quad \\mbox{for}\\,\\,\\, n<-2 \\\\\n(C_0')^n\\exp{\\left( n\\sqrt{\\frac{K}{n(n+2)}}\\;t\\;\\right)}\\,\\,\\,, \\;\\;\\;\\quad \\mbox{for}\\,\\,\\, n>0\\,\\,\\,.\\end{array}\\right.\n\\end{eqnarray}\n\nIn this case, we see that the rate of expansion is constant for the three spatial direction: this is the de Sitter universe.\\par\nNow we can perform the reconstruction of the algebraic function $f(T)$.  One can cast Eqs.~(\\ref{densitytype3})-(\\ref{tangentialpressure2type3}) in the following system \n\\begin{eqnarray}\n16\\pi\\rho&=&f+4f_T\\left(K-\\frac{\\alpha^2}{2A^2}\\right)\\,\\,\\,,\\label{jesuis}\\\\\n-16\\pi\\omega_x\\rho&=&f+2Kf_T\\left[\\frac{2n^2+3n+1}{n(n+2)}\\right]\\,\\,\\,,\\label{tues}\\\\\n-16\\pi\\omega_z\\rho&=&f+4Kf_T\\left(\\frac{2n+1}{n+2}\\right)-2\\frac{\\alpha^2}{A^2}f_T\\;\\;,\\label{ilest}\\\\\np_x&=&p_y\\,\\,\\,.\n\\end{eqnarray}\nBy combining (\\ref{jesuis}) with (\\ref{ilest}), one can eliminate the term containing $\\alpha$, obtaining\n\\begin{eqnarray}\n-16\\pi(\\omega_z+1)\\rho=\\frac{4K(n-1)}{n+2}f_T \\,\\,.\\label{noussommes}\n\\end{eqnarray}\nThe energy density $\\rho$ can be eliminated by combining (\\ref{tues}) with (\\ref{noussommes}) yielding the following differential equation\n\\begin{eqnarray}\n2K\\left[2n(n-1)\\omega_x-(\\omega_z+1)(2n^2+3n+1)\\right]f_T-n(n+2)(\\omega_z+1)f=0\\,\\,\\,,\n\\end{eqnarray}\nwhose general solution is \n\\begin{eqnarray}\nf(T)&=&C_6\\exp{\\left[R(n)T\\right]}\\,\\,\\,,\\\\\nR(n)&=&\\frac{n(n+2)(\\omega_z+1)}{2K\\left[2n(n-1)\\omega_x-(\\omega_z+1)(2n^2+3n+1)\\right]}\\,\\,,\\nonumber\n\\end{eqnarray}\nwhere $C_6$ is an integration constant. Note that for $n=1$ and $\\omega_x=\\omega_z$, Eq.~(\\ref{D10}) is recovered.\n\\par\nThe second case from (\\ref{constraint3}), $f_{TT}=0$, implies that $f_{T}$ is constant, that we choose to be minus two times the cosmological constant $\\Lambda$, then, $f(T)$ is written as\n\\begin{eqnarray}\nf(T)=T-2\\Lambda\\,\\,\\,,\n\\end{eqnarray}\nwhich is the teleparallel gravity with cosmological constant.\n\n\n\\section{Conclusion} \\label{conclusions}\n\nAlong the paper, the Bianchi type-I, Kantowski-Sachs, and Bianchi type-III metrics have been studied in the context of $f(T)$ gravities. Particularly, we have shown the reconstruction of some important cosmological solutions, obtaining the corresponding $f(T)$ action. We have assumed initially a particular choice of coordinates and tetrads, specifically cartesian coordinates and a diagonal set of tetrads have been imposed in order to avoid the well known constraint $f_{TT}=0$, which reduces trivially to the action of teleparallel gravity (see Ref.~\\cite{Tamanini:2012hg}).\\par\nThen, several important cosmological solutions have been considered. In particular, dS solutions, where the scale factor is an exponential function of the cosmic time, has been considered for Bianchi type-I and Kantowski-Sachs metrics by imposing a particular exponential expansion in each direction of the space.  We have shown that the only possible solution turns out to the FLRW metric, such that no possible dS anisotropic evolution can be found in $f(T)$, unless one considers an anisotropic fluid. Nevertheless, in the case of power law solutions, we have found that in the presence of a perfect isotropic fluid, an anisotropic cosmological evolution can be found for a particular choice of the action $f(T)$, while in vacuum the action reduces to FLRW metric.\\par\nMoreover, we have extended the cosmological reconstruction scheme to a general exponential solutions, from which the above de Sitter law and power law solutions are particular cases. We have assumed an adiabatic approximation for the expansion in each spatial direction. We undertook two cases, a special case and a second where an auxiliary field is used. In the both cases, we shown that the models can realize the early accelerated universe, characterized by the inflation, and the late time acceleration of our current universe. In the special case, the model presents an interesting aspect because it ensures the avoidance of the Big Rip and the Big Freeze. In the case where the auxiliary field is used, the model corresponding to the late time accelerated universe fits with the 7-year WMAP data, confirming the consistency of the result.\n\\par\nThe Bianchi type-III case  presents some constraints from which only two forms of the algebraic function $f(T)$ can be obtained. The first is the well known teleparallel gravity with cosmological constant, and the second is a de Sitter type solution.\n\n\\vspace{0,25cm}\n{\\bf Acknowlegments} The authors thank Prof. J. D. Barrow for useful discussions. MER thanks  UFES for the hospitality during the development of this work. MJSH thanks CNPq\/FAPES for financial support. DSG acknowledges support from a postdoctoral contract from the University of the Basque Country (UPV\/EHU) under the program ``Specialization of research staff'', and support from the research project FIS2010-15640, and also by the Basque Government through the special research action KATEA and UPV\/EHU under program UFI 11\/55.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nMultiferroics\\textemdash materials exhibiting a coexistence of both magnetic and ferroelectric orders \\cite{Kim-2003a,Che-2007}\\textemdash have attracted substantial technological and scientific interest recently. The technological interest stems from the multifunctional properties exhibited by multiferroics, which make them potentially useful in device applications such as magnetoelectric memories and switches. Multiferroics are scientifically interesting, in part, because they exhibit a variety of microscopic mechanisms that can result in an interesting interplay between ferroelectric and magnetic orders; \\cite{Che-2007} among other consequences, this interplay can spawn interesting dynamical properties in multiferroic materials, including electromagnons, i.e., hybrid excitations involving a coupling between optical phonons and spin waves via the magnetoelectric interaction, \\cite{Bar-1970, Pim-2006, Kid-2008a, Kid-2008b, Sus-2008, Val-2009, Ste-2009, Moc-2010, Tiw-2010, Har-2011, Jon-2014, Cao-2015} and magnetodielectric effects. \\cite{Kim-2003b, Law-2003, Yan-2012} \n\nMaterials in which geometric frustration leads to non-collinear spin order and strong spin-lattice coupling are particularly rich material environments to find novel magnetoelectric behavior. \\cite{Kim-2003a, Got-2004} Transition-metal-oxide spinel materials (\\textit{AB}$_2$O$_4$), for example, exhibit both non-collinear spin orders and strong spin-lattice coupling that can lead to magnetoelectric coupling, because the presence of magnetic ions on the \\textit{B}-site pyrochlore lattice of the spinel structure often leads to strong geometric frustration and consequent non-collinear orders that can generate multiferroic phenomena. \\cite{Che-2007} Magnetoelectric effects are indeed realized in some \\textit{A}Cr$_2$O$_4$\\, spinels (\\textit{e.g.}, \\textit{A}=Co$^{2+}$ and Fe$^{2+}$), in which the competition among the various exchange interactions, J$_\\text{A-A}$, J$_\\text{A-Cr}$, and J$_\\text{Cr-Cr}$, involving the \\textit{A}$^{2+}$ ions and the Cr$^{3+}$ $S=3\/2$ spins lead to complex magnetic orders. \\cite{Bor-2009, Sin-2011} \n\n$\\text{CoCr}_2\\text{O}_4$, in particular, exhibits a succession of magnetic orders, including ferrimagnetic order below $T_C \\sim 94$~K, incommensurate conical spiral order below $T_S\\sim 26$ K, commensurate order below $T_L \\sim 14$ K, \\cite{Tom-2004, Tsu-2013} as well as spin-driven multiferroic behavior and dielectric anomalies below $T_S$. \\cite{Yam-2006, Law-2006, Cho-2009} Yet, the nature and origin of magnetoelectric behavior in $\\text{CoCr}_2\\text{O}_4$\\,remains uncertain. Multiferroicity in $\\text{CoCr}_2\\text{O}_4$\\,has been associated with the spin-current mechanism \\cite{Kat-2005} involving cycloidal spin order, \\cite{Yam-2006} in which the induced electric polarization is generated by the non-collinear spins \\cite{Mos-2006} via the inverse Dzyaloshinskii-Moriya interaction, $\\boldsymbol P \\sim \\boldsymbol e_{ij}\\times(\\boldsymbol S_i\\times \\boldsymbol S_j)$. However, evidence for multiferroicity, \\cite{Sin-2011, Yan-2012} structural distortion, \\cite{Yan-2012} and magnetodielectric behavior \\cite{Yan-2012} have also been reported above $T_S$ in the ferrimagnetic state of $\\text{CoCr}_2\\text{O}_4$, raising questions about the origin of multiferroic behavior in this material. Yang \\textit{et al.}, for example, have suggested that magnetodielectric behavior in $\\text{CoCr}_2\\text{O}_4$\\,results from the presence of multiferroic domains that are reoriented in the presence of a magnetic field. \\cite{Yan-2012} But magnetodielectric behavior in magnetic materials can also arise from magnetic fluctuations that induce shifts in optical phonon frequencies via strong spin-lattice coupling. \\cite{Law-2003}\n\nUnfortunately, a lack of microscopic information regarding spin-lattice coupling has prevented a clear identification of the mechanism for magnetodielectric behavior in $\\text{CoCr}_2\\text{O}_4$. The intersublattice exchange magnon has been observed in $\\text{CoCr}_2\\text{O}_4$\\,using infrared and terahertz spectroscopies \\cite{Tor-2012, Kam-2013} and optical phonons in $\\text{CoCr}_2\\text{O}_4$\\,have been identified using Raman scattering \\cite{Kus-2009, Pta-2014, Eft-2015} and optical absorption \\cite{Tor-2012} measurements. However, to our knowledge, there have been no microscopic studies of spin-lattice coupling in $\\text{CoCr}_2\\text{O}_4$\\,that could clarify the origin of magnetodielectric behavior in this material. The application of pressure \\cite{Kan-1988, Tam-1993, Che-2013} would be a useful means of studying spin-lattice coupling and its role in magnetoelectric behavior in spinels such as $\\text{CoCr}_2\\text{O}_4$; indeed, \\textit{ab initio} calculations predict that pressure should enhance the macroscopic polarization in the multiferroic regime of $\\text{CoCr}_2\\text{O}_4$. \\cite{Eft-2015} However, the effects of pressure on the magnetoelectric behavior and spin-lattice coupling in $\\text{CoCr}_2\\text{O}_4$\\,have not yet been experimentally investigated.\n\nRaman scattering is a powerful tool for studying magnons, \\cite{Gle-2014, Gim-2016} strong spin-lattice coupling \\cite{Gle-2014, Byr-2016} and electromagnons \\cite{Caz-2008, Sin-2008, Rov-2010, Rov-2011} in complex oxide materials. When used in conjunction with pressure and magnetic-field tuning, Raman scattering can provide pressure- and magnetic-field-dependent information about the energy and lifetime of phonons, magnons, and spin-phonon coupling effects. In this paper, we report an inelastic light (Raman) scattering study of magnon and phonon excitations in $\\text{CoCr}_2\\text{O}_4$\\,as simultaneous functions of temperature, pressure, and magnetic field. Below $T_C=94$ K, we report the development in $\\text{CoCr}_2\\text{O}_4$\\,of  a\\, $\\sim$ $16 \\,\\text{cm}^{-1}$ (2 meV) $\\boldsymbol q=0$ magnon excitation with T$_{1g}$\\,symmetry. The anomalously large Raman scattering susceptibility associated with the T$_{1g}$\\,symmetry magnon in $\\text{CoCr}_2\\text{O}_4$\\,is indicative of a large magneto-optical response arising from large magnetic fluctuations that couple strongly to the dielectric response; this coupling is likely associated with the dielectric anomalies \\cite{Sin-2011} observed in the ferrimagnetic phase of $\\text{CoCr}_2\\text{O}_4$. We also show that the Raman intensity of the T$_{1g}$-symmetry magnon in $\\text{CoCr}_2\\text{O}_4$\\,exhibits a strong suppression with increasing magnetic field, suggesting that the dramatic magneto-dielectric behavior \\cite{Muf-2010, Yan-2012} observed in $\\text{CoCr}_2\\text{O}_4$\\,results from the magnetic-field-induced suppression of magnetic fluctuations that are strongly coupled to phonons. \\cite{Law-2003} Using applied pressure to increase the magnetic anisotropy in $\\text{CoCr}_2\\text{O}_4$\\,results in a decreased magnetic field-dependence of the T$_{1g}$-symmetry magnon Raman intensity in $\\text{CoCr}_2\\text{O}_4$, suggesting that pressure or epitaxial strain can be used to control magnetodielectric behavior and the magneto-optical response in $\\text{CoCr}_2\\text{O}_4$\\,by suppressing magnetic fluctuations. \n\n\\section{Experimental Methods}\n\n\\subsection{Crystal Growth and Characterization}\n$\\text{CoCr}_2\\text{O}_4$\\,crystals were grown by chemical vapor transport (CVT) following a procedure described by Ohgushi et al.\\cite{Ohg-2008} Polycrystalline powder samples of $\\text{CoCr}_2\\text{O}_4$\\,were first synthesized using cobalt nitrate hexahydrate (Strem Chemicals 99\\%) and chromium nitrate nonahydrate (Acros 99\\%). The nitrates were combined in stoichiometric amounts and dissolved in water. The solution was heated to $350^\\circ$ C and stirred using a magnetic stir bar at $300$ rpm until all of the liquid evaporated. The remaining powder was heated in an alumina crucible at $900^\\circ$ C for $16$ hours and then air quenched. Crystal samples of $\\text{CoCr}_2\\text{O}_4$\\,were grown by CVT using CrCl$_3$ as a transport agent. $2.0$ g of polycrystalline samples and $0.04$ g of CrCl$_3$ were sealed in an evacuated quartz ampoule, which was placed inside a three-zone furnace having $950 ^\\circ $C at the center with a temperature gradient of $10 ^\\circ $C\/cm for one month. Crystals with typical dimensions of $2 \\times 2 \\times 2$\\,mm$^3$ were obtained.\n\n\\begin{figure}\n\\includegraphics[height=7.4cm,width=8.5cm]{xray.pdf}\n\\caption{\\label{fig1}X-ray diffraction pattern and Rietveld fit of $\\text{CoCr}_2\\text{O}_4$\\,at 298 K. The Miller indices for a cubic unit cell with cell parameter $a=8.334(1) \\text{\\AA}$ are also shown.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[scale=0.4]{MPMS.pdf}\n\\caption{\\label{fig2}Molar susceptibility of $\\text{CoCr}_2\\text{O}_4$\\,as a function of temperature measured in an applied field of 100 Oe.}\n\\end{figure}\n\nThe $\\text{CoCr}_2\\text{O}_4$\\,crystals were characterised using x-ray diffraction and magnetization measurements. Crystals of $\\text{CoCr}_2\\text{O}_4$\\,were ground to a powder to obtain the x-ray diffraction pattern using a Siemens-Bruker D5000 diffractometer using Cu-K$\\alpha$ radiation shown in Fig.~\\ref{fig1}. Rietveld refinement of the $\\text{CoCr}_2\\text{O}_4$\\,cell to the XRD data was performed using XND Rietveld, \\cite{xnd} and indicates a pure sample with $Fd \\bar{3} m$ symmetry and a lattice constant of $8.334(1)$\\AA , which agrees with the established structure. \\cite{Tor-2012} The $<110>$ reflections from a single crystal of $\\text{CoCr}_2\\text{O}_4$\\,were measured, and no evidence of twinning imperfections was found. The field-cooled dc magnetization data on the $\\text{CoCr}_2\\text{O}_4$\\,powder from which our crystal sample was obtained was collected using a Quantum Design MPMS-3 and is shown as a function of temperature in Fig.~\\ref{fig2}. Our results are similar to existing data. \\cite{Law-2006} In particular, the sudden increase in the molar susceptibility, $\\chi_{\\text m}$ at $T\\sim 94$ K marks the onset of ferrimagnetic ordering. The change in slope of the graph at $T\\sim26$ K and an additional small anomaly at $T\\sim14$ K correspond to the incommensurate and commensurate spiral ordering, respectively, in $\\text{CoCr}_2\\text{O}_4$. \n\n\\subsection{Raman Scattering Measurements}\nRaman scattering measurements were performed using the $647.1\\,\\text{nm}$ excitation line from a Kr$^+$ laser. The incident laser power was limited to $5-10$ mW, and was focused to a $\\sim \\text{\\SI{50}{\\micro\\meter}}$-diameter spot to minimize laser heating of the sample. Sample heating by the laser was estimated to be in the range $5-7$ K, and this estimated laser heating is included in the temperatures given in the results section. The scattered light from the samples was collected in a backscattering geometry, dispersed through a triple stage spectrometer, and then detected with a liquid-nitrogen-cooled CCD detector. The samples were inserted into a continuous He-flow cryostat, which was horizontally mounted in the open bore of a superconducting magnet. \\cite{Kim-2011} This experimental arrangement allows Raman scattering measurements under the simultaneous conditions of low temperature ($3-300$ K), high magnetic fields ($0-9$ T), and high pressures ($0-100$ kbar). To determine the symmetries of the measured Raman excitations in zero magnetic field, linearly polarized incident and scattered light were used for various crystallographic orientations of the sample. In the magnetic field measurements, circularly polarized light was used to avoid Faraday rotation of the light polarization.\n\n\n\\begin{figure}\n\\subfloat{\\includegraphics[height=5.2cm,width=8.5cm]{exa.pdf} \\label{exa}}\\\\\n\\subfloat{\\includegraphics[height=3.8cm,width=4.1cm]{exb.pdf} \\label{exb}} \n\\subfloat{\\includegraphics[height=3.8cm,width=4.1cm]{excc.pdf} \\label{exc}}\n\\caption{\\label{fig3}Illustrations showing the experimental arrangements used for different high-magnetic-field and high temperature Raman scattering experiments at low temperatures in this study. \\cite{Kim-2011} (a) Configuration for high-magnetic field measurements in the Faraday ($\\boldsymbol k \\parallel \\boldsymbol H$) geometry, where $\\boldsymbol k$ is the wavevector of the incident light and $\\boldsymbol H$ is the applied magnetic field direction. (b) Configuration for high-magnetic-field measurements in the Voigt ($\\boldsymbol k \\perp \\boldsymbol H$) geometry. (c) Configuration for high pressure measurements using a diamond anvil cell.}\n\\end{figure}\n\nMagnetic field measurements were performed in both Voigt ($\\boldsymbol k \\perp \\boldsymbol M \\parallel \\boldsymbol H$) and Faraday ($\\boldsymbol k \\parallel \\boldsymbol M \\parallel \\boldsymbol H$) geometries, where $\\boldsymbol k$ is the wavevector of the incident light and $\\boldsymbol M$ is the magnetization direction. \\cite{Kim-2011} Because of the very small anisotropy field in $\\text{CoCr}_2\\text{O}_4$\\,($H_A\\leq0.1$ T), \\cite{Kam-2013} the net magnetization $\\boldsymbol M$ was assumed to follow the applied field $\\boldsymbol H$ in all experiments performed. To verify this, we confirmed that the field-dependence of the Raman spectrum was independent of the crystallographic orientation of the applied field. The field measurements in the Faraday geometry were performed by mounting the sample at the end of the insert, as illustrated in Fig.~\\subref*{exa}, so that the wavevector of the incident light is parallel to the applied field. The Voigt geometry was achieved by mounting the sample on an octagon plate, which was mounted sideways on the sample rod, as illustrated in Fig.~\\subref*{exb}. The incident light was guided to the sample surface with a 45$^\\circ$ mirror mounted on the sample rod. This sample mounting arrangement allows the magnetic field to be applied perpendicular to the wavevector of the incident light,  $\\boldsymbol k \\perp \\boldsymbol M \\parallel \\boldsymbol H$. \n\n High pressure measurements were performed using a miniature cryogenic diamond anvil cell (MCDAC) to exert pressure on the sample via an argon liquid medium. The high-pressure cell was inserted into the cryostat as illustrated in Fig.~\\subref*{exc}, allowing the pressure to be changed \\textit{in situ} at low temperatures without any extra warming\/cooling procedure. This arrangement also allows simultaneous high-pressure and high-magnetic field measurements in the Faraday  ($\\boldsymbol k \\parallel \\boldsymbol M \\parallel \\boldsymbol H$) geometry, as illustrated in Fig.~\\subref*{exc}. \\cite{Kim-2011} The pressure was determined from the shift in the fluorescence line of a ruby chip loaded in the cell along with the sample piece.\n\n\\section{Temperature dependence of the magnetic excitation at \\textit{P}=0 and \\textit{B}=0}\n\\subsection{Results}\n\\begin{figure}\n\\includegraphics[scale=0.4]{T1.pdf}\n\\caption{\\label{fig4}Temperature-dependence of the Raman scattering intensity, $S(\\omega)$, for $\\text{CoCr}_2\\text{O}_4$\\,at 10 K and 130 K, showing the phonon modes above 150 cm$^{-1}$ and the T$_{1g}$\\,symmetry magnon near 16 cm$^{-1}$ that evolves for $T<90$ K. Inset shows the polarization dependence of the magnon in $\\text{CoCr}_2\\text{O}_4$; the presence of this mode only in the depolarized geometry for all crystallographic orientations is indicative of the T$_{1g}$\\,symmetry, which transforms like an axial vector.}\n\\end{figure}\n\nFig.~\\ref{fig4} shows the $T=10$ K and $T=130$ K Raman spectra of $\\text{CoCr}_2\\text{O}_4$\\,between 0-700 cm$^{-1}$ in a scattering geometry with circularly polarized incident light and unanalyzed scattered light. The $T=10$ K spectrum exhibits the five Raman-active phonon modes expected and previously observed \\cite{Kus-2009, Pta-2014, Eft-2015} for $\\text{CoCr}_2\\text{O}_4$, including phonon modes at 199 cm$^{-1}$, 454 cm$^{-1}$, 518 cm$^{-1}$, 609 cm$^{-1}$, and 692 cm$^{-1}$ (at $T=10$ K). In addition to the phonon modes, the $T=10$ K spectrum in Fig.~\\ref{fig4} has an additional mode that develops near 16 cm$^{-1}$ ($\\sim$2 meV) below $T=90$ K. The inset of Fig.~\\ref{fig4} shows that the 16 cm$^{-1}$ mode intensity is present only in the ``depolarized\" scattering geometry, \\textit{i.e.}, only when the incident and scattered light polarizations are perpendicular to one another, independent of the crystallographic orientation. This polarization dependence indicates that the 16 cm$^{-1}$ mode symmetry transforms like the fully antisymmetric representation, T$_{1g}$, which has the symmetry properties of an axial vector, characteristic of a magnetic excitation. \\cite{Cot-1986, Ram-1991} Consequently, we identify the 16 cm$^{-1}$ excitation as a $\\boldsymbol q=0$ T$_{1g}$ \\,symmetry magnon in $\\text{CoCr}_2\\text{O}_4$. This interpretation is supported by the temperature-dependence of the 16 cm$^{-1}$ T$_{1g}$ -symmetry mode Raman scattering susceptibility, $\\text{Im}\\,\\chi(\\omega)$ (see Fig.~\\subref*{T2a}), where $\\text{Im}\\,\\chi(\\omega)=S(\\boldsymbol q=0,\\omega)\/[1+n(\\omega,T)]$, $S(\\boldsymbol q=0,\\omega)$ is the measured Raman scattering response, and $[1+n(\\omega,T)]$ is the Bose thermal factor with $n(\\omega,T)=[e^{\\hbar\\omega\/k_BT})-1]^{-1}$. Fig.~\\subref*{T2b} shows that the $\\sim16\\,\\text{cm}^{-1}$ T$_{1g}$\\,symmetry mode energy (solid squares) decreases in energy (``softens\") with increasing temperature toward $T_C$\\textemdash consistent with the temperature-dependence of the Co$^{2+}$ sublattice magnetization \\cite{Kam-2013}\\textemdash indicative of a single-magnon excitation. \\cite{Cot-1986} Fig.~\\subref*{T2b} also shows that the amplitude of the Raman susceptibility, $\\text{Im}\\,\\chi(\\omega)$, associated with the 16 cm$^{-1}$ T$_{1g}$ -symmetry magnon mode (solid circles) is relatively insensitive to temperature and is  comparable to that of the 199 cm$^{-1}$ T$_{2g}$ phonon. Notably, the 16 cm$^{-1}$ T$_{1g}$\\,symmetry magnon we observe in $\\text{CoCr}_2\\text{O}_4$\\,has a similar energy and temperature dependence to that of the exchange magnon observed previously in terahertz \\cite{Tor-2012} and infrared spectroscopy \\cite{Kam-2013} measurements of $\\text{CoCr}_2\\text{O}_4$. Nevertheless, it is unlikely that the 16 cm$^{-1}$ T$_{1g}$\\, symmetry magnon we observe in $\\text{CoCr}_2\\text{O}_4$\\,is the same as the intersublattice exchange mode reported in infrared measurements, because T$_{1g}$\\,is not an infrared-active symmetry. Note in this regard that the spinel structure of $\\text{CoCr}_2\\text{O}_4$\\,is expected to exhibit six $\\boldsymbol q=0$ magnon modes with 5 closely spaced optical branches, \\cite{Kap-1953, Bri-1966, Sah-1974, Tor-2012} so we are likely observing a different optical magnon that is close in energy to that observed in infrared measurements. \\cite{Tor-2012, Kam-2013}\n\n\\subsection{Discussion and Analysis}\nThe finite $\\boldsymbol q=0$ energy of the $\\omega \\sim$ 16 \\text{cm}$^{-1}$ (2 meV) T$_{1g}$-symmetry magnon in $\\text{CoCr}_2\\text{O}_4$\\,primarily reflects the finite exchange, $H_E$, and anisotropy, $H_A$, fields in $\\text{CoCr}_2\\text{O}_4$, according to $\\omega=\\gamma(2H_AH_E + {H_A}^2)^{1\/2}$, where $\\gamma$ is the gyromagnetic ratio $g\\mu_B\/\\hbar$. \\cite{Ram-1991} Fig. ~\\ref{fig5} also shows that the 16 cm$^{-1}$ T$_{1g}$\\, symmetry magnon in $\\text{CoCr}_2\\text{O}_4$\\,is apparent to temperatures as high as $T\\sim60$ K, indicating that the T$_{1g}$\\,symmetry magnon in $\\text{CoCr}_2\\text{O}_4$\\,is dominated by the Co$^{2+}$ sublattice spins, which order at a significantly higher temperature (94 K) than the Cr$^{3+}$ sublattice (49 K). \\cite{Kam-2013}\n\n\\begin{figure}\n\\subfloat{\\includegraphics[height=3.6cm,width=3.6cm]{T2a.pdf}\n\\label{T2a}}\n\\quad\n\\subfloat{\\includegraphics[height=3.6cm,width=4cm]{T2b.jpg}\n\\label{T2b}}\n\\caption{\\label{fig5}(a) Raman scattering susceptibility, $\\text{Im}\\,\\chi(\\omega)$, of the T$_{1g}$-symmetry magnon of $\\text{CoCr}_2\\text{O}_4$\\,as a function of temperature. (b) Summary of the temperature dependence of the T$_{1g}$\\,symmetry magnon energy (filled squares). Also shown in filled circles is a summary of the temperature dependence of the T$_{1g}$\\,symmetry magnon Raman susceptibility amplitude normalized to the susceptibility amplitude of the 199 cm$^{-1}$ T$_{2g}$ optical phonon, $\\text{Im}\\,\\chi_{mag}(\\omega)\/\\text{Im}\\,\\chi_{ph}(\\omega)$.}\n\\end{figure}\n\nImportantly, the Raman susceptibility of the 16 cm$^{-1}$ T$_{1g}$\\, symmetry magnon at T=10 K (for $H=0$ T and $P=0$ kbar) (see Fig.~\\ref{fig5}) reflects the degree to which this magnon modulates the dielectric response, $\\epsilon=4\\pi\\chi_E$ (where $\\chi_E$ is the electric susceptibility). \\cite{Dem-1987, Kum-2011} Consequently, while Raman scattering from magnons is generally much weaker than Raman scattering from phonons, \\cite{Cot-1986} Figs.~\\ref{fig4} and \\ref{fig5} show that Raman intensity of the T$_{1g}$-symmetry magnon is comparable to that of the Raman-active phonons in $\\text{CoCr}_2\\text{O}_4$, indicative of a strong influence of this magnon on the dielectric response of $\\text{CoCr}_2\\text{O}_4$.\n\nThe large Raman susceptibility of the T$_{1g}$\\, symmetry magnon reflects a large magneto-optical response in $\\text{CoCr}_2\\text{O}_4$, and is likely associated with strong magnetic fluctuations that modulate the dielectric response via strong spin-lattice coupling. \\cite{Bar-1983, Dem-1987, Bor-1988} Such large magnetic fluctuations are attributable to the weak anisotropy field in $\\text{CoCr}_2\\text{O}_4$, $H_A\\leq$0.1 T, \\cite{Kam-2013} and can contribute in several ways to fluctuations in the dielectric response:\\cite{Bar-1983, Dem-1987, Bor-1988}\n\n\\begin{equation}\n\\delta \\epsilon(m, l) = i \\, f \\delta m + g (\\delta l )^2 + a (\\delta m)^2 \n\\label{eq1}\n\\end{equation}\nwhere $\\delta\\epsilon$ is the dielectric response fluctuation, $\\delta m$ represents longitudinal fluctuations in the magnetization, $\\delta l$ represents fluctuations of the antiferromagnetic vector, and $a$, $f$, and $g$ are constants. The first term in Eq.~\\eqref{eq1} is associated with the linear magneto-optical Faraday effect, the second term is associated with linear magnetic birefringence, and the final term is an isotropic ``exchange\" mechanism for magnon scattering that is present in non-collinear antiferromagnets. \\cite{Dem-1987, Vit-1991} In non-collinear antiferromagnetic and ferrimagnetic materials with weak anisotropy\\textemdash such as $\\text{CoCr}_2\\text{O}_4$\\textemdash strong single-magnon scattering can result from large fluctuations of both $l$ and $m$. In particular, the one-magnon Raman scattering intensity, $S$, associated with large magnetic fluctuations of the antiferromagnetic vector at $H=0$ is limited only by the anisotropy field, $H_A$ (i.e., $S \\propto 1\/H_A$), \\cite{Dem-1987} which is very small in $\\text{CoCr}_2\\text{O}_4$, $H_A\\leq$0.1 T. \\cite{Kam-2013}\n\n\\section{Magnetic-field-dependence of the T$_{1g}$-symmetry magnon in $\\text{CoCr}_2\\text{O}_4$}\n\\subsection{Results}\n\\begin{figure}\n\\subfloat{\\includegraphics[scale=0.25]{F1aa.pdf}\n\\label{F1a}}\n\\quad\n\\subfloat{\\includegraphics[scale=0.25]{F1b.jpg}\n\\label{F1b}}\n\\\\\n\\subfloat{\\includegraphics[scale=0.24]{F1c.jpg}\n\\label{F1c}}\n\\quad\n\\subfloat{\\includegraphics[scale=0.24]{F1d.jpg}\n\\label{F1d}}\n\\caption{\\label{fig6} Magnetic-field-dependence of the Raman scattering susceptibility, $\\text{Im}\\,\\chi(\\omega)$, of the T$_{1g}$-symmetry magnon in $\\text{CoCr}_2\\text{O}_4$\\,at $T=10$ K in the (a) Faraday geometry ($\\boldsymbol k\\parallel\\boldsymbol M\\parallel\\boldsymbol H$) and the (b) Voigt geometry ($\\boldsymbol k \\perp \\boldsymbol M \\parallel \\boldsymbol H$). (c) Summary of the field dependences of the T$_{1g}$-symmetry magnon energy of $\\text{CoCr}_2\\text{O}_4$\\,at (filled squares) $T=10$ K and (filled circles) $T=55$ K in the Faraday geometry and at (filled triangles) $T=10$ K in the Voigt geometry. (d) Summary of the field dependences of the amplitude of the T$_{1g}$-symmetry magnon Raman susceptibility normalized to the amplitude of the 199 cm$^{-1}$ T$_{2g}$ phonon Raman susceptibility at (filled squares) $T=10$ K and (filled circles) $T=55$ K in the Faraday geometry and at (filled triangles) $T=10$ K in the Voigt geometry.}\n\\end{figure}\n\nFig.~\\ref{fig6} shows the magnetic-field-dependence of the Raman susceptibility, $\\text{Im}\\,\\chi(\\omega)$, for the T$_{1g}$-symmetry magnon of $\\text{CoCr}_2\\text{O}_4$\\,at $P=0$ kbar and $T=10$ K with an applied magnetic field in both the (Fig.~\\subref*{F1a}) Faraday ($\\boldsymbol k\\parallel\\boldsymbol M \\parallel \\boldsymbol H$) and (Fig.~\\subref*{F1b}) Voigt ($\\boldsymbol k \\perp\\boldsymbol M \\parallel \\boldsymbol H$) geometries. Fig.~\\subref*{F1c} summarizes the field-dependences of the T$_{1g}$-symmetry magnon energy at both $T=10$ K and $T=55$ K, showing that the T$_{1g}$-symmetry magnon energy exhibits a linear increase with increasing field. The shift in the T$_{1g}$-symmetry magnon energy with field, $d\\omega \/dH \\sim$ 1.1 cm$^{-1}\/$T corresponds to a dimensionless ratio $\\hbar\\omega\/\\mu_BH=2.4$. This ratio is close to the $T=4$ K value of $\\hbar\\omega\/\\mu_BH=2.5$  measured for the exchange magnon in $\\text{CoCr}_2\\text{O}_4$\\,\\cite{Kam-2013} and is consistent with the gyromagnetic ratio of 2.2 for Co$^{2+}$. \\cite{Alt-1974, Tor-2012} Fig.~\\subref*{F1d} compares the field-dependence of the normalized T$_{1g}$-symmetry magnon intensity, $\\text{Im}\\,\\chi_{mag}(\\omega)\/\\text{Im}\\,\\chi_{ph}(\\omega)$, in both the (filled circle and square) Faraday ($\\boldsymbol k \\parallel \\boldsymbol M \\parallel \\boldsymbol H$) and (filled triangle) Voigt ($\\boldsymbol k \\perp \\boldsymbol M \\parallel \\boldsymbol H$) geometries, where $\\text{Im}\\,\\chi_{mag}(\\omega)$ and $\\text{Im}\\,\\chi_{ph}(\\omega)$ are the Raman susceptibilities of the T$_{1g}$-symmetry magnon and 199 cm$^{-1}$ T$_{2g}$ phonon, respectively. Fig.~\\subref*{F1d} shows that there is a substantial decrease in the normalized T$_{1g}$-symmetry magnon intensity of $\\text{CoCr}_2\\text{O}_4$\\,with increasing field in both the Faraday ($\\boldsymbol k \\parallel \\boldsymbol M \\parallel \\boldsymbol H$) and Voigt ($\\boldsymbol k \\perp \\boldsymbol M \\parallel \\boldsymbol H$) geometries at $T=10$ K and $T=55$ K. Note that the field-dependent decrease we observe in the T$_{1g}$-symmetry magnon intensity \\textemdash which is particularly dramatic in the Faraday geometry ($\\boldsymbol k \\parallel \\boldsymbol M \\parallel \\boldsymbol H$) \\textemdash cannot be attributed to field-dependent changes in polarization or crystallographic orientation: the use of circularly polarized incident light in these experiments precludes field-dependent rotation of the incident polarization; and T$_{1g}$-symmetry modes appear in the depolarized scattering geometry independent of the crystallographic orientation of the sample.\n\n\\subsection{Discussion and Analysis}\nThe anomalously large decrease in the 16 cm$^{-1}$ T$_{1g}$-magnon Raman intensity with increasing field in the Faraday geometry ($\\boldsymbol k \\parallel \\boldsymbol M \\parallel \\boldsymbol H$) of $\\text{CoCr}_2\\text{O}_4$\\,(see Fig.~\\ref{fig6}) is quite different than the field-independent magnon Raman intensities observed in other spinel materials, such as Mn$_3$O$_4$ and MnV$_2$O$_4$. \\cite{Gle-2014} To clarify the anomalously strong field-dependence of the T$_{1g}$-symmetry magnon Raman intensity in $\\text{CoCr}_2\\text{O}_4$, note that the magnon Raman intensity in the Faraday geometry is expected to be dominated by the linear magnetic birefringence contribution to dielectric fluctuations, $\\delta\\epsilon=g (\\delta l )^2$. \\cite{Dem-1987, Bor-1988, Vit-1991} Thus, the strong decrease in the T$_{1g}$-symmetry magnon Raman intensity in the Faraday geometry likely reflects a field-induced decrease in fluctuations of the antiferromagnetic vector, $\\delta l$. A similar field-dependent decrease in the single-magnon inelastic light scattering response associated with fluctuations of the antiferromagnetic vector was also observed in the canted antiferromagnet EuTe. \\cite{Dem-1987}\n\nFig.~\\subref*{F1b}, \\subref*{F1d} shows that there is a similar, albeit less dramatic, field-dependent decrease in the T$_{1g}$-symmetry magnon Raman intensity measured in the Voigt ($\\boldsymbol k \\perp \\boldsymbol M \\parallel \\boldsymbol H$) geometry. This geometry is primarily sensitive to the Faraday ($\\delta\\epsilon=i \\, f \\delta m$) and isotropic exchange ($\\delta\\epsilon=a (\\delta m)^2$) contributions to dielectric fluctuations, which are dominated by longitudinal fluctuations in the magnetization. \\cite{Dem-1987, Bor-1988, Vit-1991} Altogether, the suppression of the T$_{1g}$-symmetry magnon Raman scattering intensities in both Faraday and Voigt geometries is indicative of a field-induced suppression of both transverse and longitudinal magnetic fluctuations in $\\text{CoCr}_2\\text{O}_4$.\n\nThe field-dependent suppression of the T$_{1g}$-symmetry magnon Raman intensity in $\\text{CoCr}_2\\text{O}_4$\\,points to a specific microscopic mechanism for the magnetodielectric response observed in $\\text{CoCr}_2\\text{O}_4$. \\cite{Sin-2011, Muf-2010, Yan-2012} Lawes \\textit{et al.} have pointed out that the field-induced suppression of magnetic fluctuations can contribute to the magnetodielectric response of a material via the coupling of magnetic fluctuations to optical phonons. \\cite{Law-2003} This spin-phonon coupling contributes to the magnetodielectric response of a material through field-induced changes to the net magnetization. \\cite{Smo-1982, Kim-2003b, Law-2003, Yan-2012} A simple phenomenological description for how the magnetization of a magnetoelectric material influences the dielectric response of the material is obtained by considering the free energy, $F$, in a magnetoelectric material with a coupling between the magnetization $\\boldsymbol M$ and polarization $\\boldsymbol P$: \\cite{Smo-1982, Kim-2003b, Yan-2012}\n\n\\begin{eqnarray}\nF(M,P) = F_0 + aP^2 &+& bP^4 - PE + cM^2 + \\nonumber \n\\\\  &&dM^4 - MH + eM^2P^2 ,\n\\label{eq2}\n\\end{eqnarray}\nwhere $F_0$, $a$, $b$, $c$, $d$, and $e$ are temperature-dependent constants, and $M$, $P$, $E$, and $H$ are the magnitudes of the magnetization, polarization, applied electric field, and applied magnetic field, respectively. The dependence of the dielectric response on magnetization in a magnetoelectric material, $\\epsilon(M)$ , can be obtained from the second derivative of the free energy with respect to polarization $P$: \\cite{Smo-1982, Kim-2003b, Yan-2012}\n\n\\begin{equation}\n[\\epsilon(M)]^{-1} \\sim (\\partial^2F\/\\partial P^2) = 2a + 12bP^2 + 2eM^2,\n\\end{equation}\nwhich, for a negligible macroscopic polarization $P$ in the material, can be written: \\cite{Kim-2003b, Yan-2012}\n\\begin{equation}\n\\epsilon(M) = 1\/[2a + 2eM^2].\n\\end{equation}\nThus, the dielectric response, $\\epsilon = 4\\pi\\chi_E$, decreases with increasing squared magnetization, $M^2$ and decreasing magnetic fluctuations. \\cite{Law-2003}\n\nThe above results suggest that both the magnetic-field-dependent decrease in the intensity of the 16 cm$^{-1}$ T$_{1g}$-symmetry magnon (see Fig.~\\ref{fig6}) and the magnetodielectric response, $\\Delta\\epsilon(H)\/\\epsilon(0)=[\\epsilon(H)-\\epsilon(0)]\/\\epsilon(0)$, in $\\text{CoCr}_2\\text{O}_4$\\,\\cite{Muf-2010, Yan-2012} reflect magnetic-field-induced changes to magnetic fluctuations\\textemdash particularly fluctuations associated with the antiferromagnetic vector\\textemdash that are strongly coupled to phonons \\cite{Law-2003} via the biquadratic contribution to the free energy, $M^2P^2$ (see Eq.~\\eqref{eq2}).\n\n\\section{Pressure dependence of the T$_{1g}$-symmetry magnon in $\\text{CoCr}_2\\text{O}_4$}\n\n\\subsection{Results}\n\nAs discussed above, the strong T$_{1g}$-symmetry magnon Raman intensity of $\\text{CoCr}_2\\text{O}_4$\\,is believed to reflect strong magnetic fluctuations that are coupled to long-wavelength phonons, which should also be associated with significant magneto-optical responses (both linear Faraday and linear magnetic birefringence) in $\\text{CoCr}_2\\text{O}_4$. Our results show that the application of a magnetic field suppresses these fluctuations, leading to the substantial magnetodielectric response observed in $\\text{CoCr}_2\\text{O}_4$. An alternative approach to suppressing magnetic fluctuations is to use applied pressure or strain to increase the crystalline anisotropy of $\\text{CoCr}_2\\text{O}_4$. To investigate this possibility, magnetic-field-dependent measurements of the T$_{1g}$-symmetry magnon in $\\text{CoCr}_2\\text{O}_4$\\,were performed for different applied pressures.\n\n\\begin{figure}\n\\subfloat{\\includegraphics[scale=0.23]{P1aa.pdf}\n\\label{P1a}}\n\\,\\,\n\\subfloat{\\includegraphics[scale=0.23]{P1bb.pdf}\n\\label{P1b}}\n\\,\\,\n\\subfloat{\\includegraphics[scale=0.23]{P1cc.pdf}\n\\label{P1c}}\n\\caption{\\label{fig7}Field-dependence in the Faraday ($\\boldsymbol k \\parallel \\boldsymbol M \\parallel \\boldsymbol H$) geometry of the T$_{1g}$-symmetry magnon Raman susceptibility of $\\text{CoCr}_2\\text{O}_4$\\,at $T=10$ K and at various applied pressures, including (a) $P=0$ kbar, (b) $P=15$ kbar, and (c) $P=21$ kbar.}\n\\end{figure}\n\nFig.~\\ref{fig7} shows the field-dependence of the T$_{1g}$-symmetry magnon spectrum of $\\text{CoCr}_2\\text{O}_4$\\,in the Faraday ($\\boldsymbol k \\parallel \\boldsymbol M \\parallel \\boldsymbol H$) geometry for different applied pressures at $T=10$ K. Fig.~\\subref*{P2a} summarizes the field dependence of the T$_{1g}$-symmetry magnon energy at $T=10$ K for different applied pressures, and Fig.~\\subref*{P2b} shows the amplitude of the T$_{1g}$-symmetry magnon Raman susceptibility (normalized by the amplitude of the 199 cm$^{-1}$ T$_{2g}$ phonon susceptibility) at $T=10$ K for different applied pressures. The inset of Fig.~\\subref*{P2b} summarizes the pressure-dependence of the T$_{1g}$-symmetry magnon energy of $\\text{CoCr}_2\\text{O}_4$\\,for $H=0$ T and $T=10$ K.\n\n\\subsection{Discussion and Analysis}\nThe inset of Fig.~\\subref*{P2b} shows that the T$_{1g}$-symmetry magnon energy increases linearly with applied pressure at a rate of $d\\omega\/dP=$0.46 cm$^{-1}$\/kbar. This increase likely reflects a systematic increase in the anisotropy field, $H_A$, with increasing pressure, according to the relationship $\\omega\\sim(2H_AH_E)^{1\/2}$. Additionally, the magnetic field dependence of the Raman spectrum of $\\text{CoCr}_2\\text{O}_4$\\,at different fixed pressures summarized in Fig.~\\subref*{P2a} shows that the field-dependent slope associated with the T$_{1g}$-symmetry magnon frequency, $d\\omega\/dH$, is insensitive to applied pressure up to roughly 21 kbar, indicating that the gyromagnetic ratio associated with Co$^{2+}$ is not strongly affected by these pressures in $\\text{CoCr}_2\\text{O}_4$.\n\n\\begin{figure}\n\\subfloat{\\includegraphics[scale=0.25]{P2aa.pdf}\n\\label{P2a}}\n\\quad\n\\subfloat{\\includegraphics[scale=0.25]{P2b.pdf}\n\\label{P2b}}\n\\caption{\\label{fig8}(a) Summary of the field dependences in the Faraday ($\\boldsymbol k \\parallel \\boldsymbol M \\parallel \\boldsymbol H$) geometry of the T$_{1g}$-symmetry magnon energy of $\\text{CoCr}_2\\text{O}_4$\\,at (filled squares) $T=10$ K and at various pressures, including (filled squares) $P=0$ kbar, (filled circles) $P=4.5$ kbar, (filled triangles) $P=15$ kbar, and (filled diamonds) $P=21$ kbar. (b) Summary of the field dependences of the amplitude of the T$_{1g}$-symmetry magnon Raman susceptibility normalized to the amplitude of the 199 cm$^{-1}$ T$_{2g}$ phonon Raman susceptibility at $T=10$ K and various pressures, including (filled squares) $P=0$ kbar, (filled circles) $P=4.5$ kbar, (filled triangles) $P=15$ kbar, and (filled diamonds) $P=21$ kbar. ((b) inset) Summary of the pressure-dependence of the T$_{1g}$-symmetry magnon energy in $\\text{CoCr}_2\\text{O}_4$\\,at $T=10$ K and $H=0$ T.}\n\\end{figure}\n\nOn the other hand, Figs.~\\ref{fig7} and \\ref{fig8} also show that $H=0$ T$_{1g}$-symmetry magnon Raman intensity, $\\text{Im}\\,\\chi(\\omega)$, systematically decreases relative to the T$_{2g}$ phonon intensity, illustrating that increasing pressure suppresses the magnetic fluctuations and the magneto-optical response in $\\text{CoCr}_2\\text{O}_4$\\,by increasing the anisotropy field. Additionally, Fig.~\\subref*{P2b} shows that increasing pressure reduces the strong suppression of the T$_{1g}$-symmetry magnon intensity with increasing magnetic field in the Faraday geometry ($\\boldsymbol k \\parallel \\boldsymbol M \\parallel \\boldsymbol H$), providing evidence that the magnetodielectric response of $\\text{CoCr}_2\\text{O}_4$\\,decreases with increasing pressure. Altogether, these results show that, by tuning magnetic anisotropy and reducing magnetic fluctuations of the Co$^{2+}$ spins, pressure and epitaxial strain can be used as effective tuning parameters for controlling the magnetodielectric response of $\\text{CoCr}_2\\text{O}_4$.\n\n\\section{Summary and Conclusions}\n\nIn this paper, we showed that the $\\boldsymbol q=0$ T$_{1g}$-symmetry magnon in $\\text{CoCr}_2\\text{O}_4$\\,exhibits an anomalously large Raman scattering intensity, which reflects a large magneto-optical response that likely results from large magnetic fluctuations that couple strongly to the dielectric response. The strong suppression of the T$_{1g}$-symmetry magnon Raman intensity in an applied field is consistent with the magnetodielectric response observed previously in this material, \\cite{Muf-2010, Yan-2012} and suggests that the strong magnetodielectric response in $\\text{CoCr}_2\\text{O}_4$\\,results from the magnetic-field-induced suppression of magnetic fluctuations that are strongly coupled to phonons. \\cite{Law-2003} Using pressure to increase the magnetic anisotropy in $\\text{CoCr}_2\\text{O}_4$, we found that we can suppress the magnetic field-dependence of the T$_{1g}$-symmetry magnon Raman intensity by suppressing magnetic fluctuations, demonstrating that pressure or epitaxial strain should be an effective means of controlling magnetodielectric behavior and the magneto-optical response in $\\text{CoCr}_2\\text{O}_4$. This Raman study also reveals conditions that are conducive for the substantial magneto-optical responses and magneto-dielectric behaviors in materials, including the presence of strong spin-orbit coupling and weak magnetic anisotropy, both of which create favorable conditions for large magnetic fluctuations that strongly modulate the dielectric response.\n\n\n\\acknowledgements{\nResearch was supported by the National Science Foundation under Grant NSF DMR 1464090. RDM and DPS thank the Illinois Department of Materials Science and Engineering for support. X-ray diffraction and magnetic susceptibility measurements were performed in the Frederick Seitz Materials Research Laboratory.}\n\n\\bibliographystyle{apsrev}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAdequate semigroups form a class of semigroups in which the cancellation properties\nof elements are reflected in the cancellation properties of idempotents.\nThey form a natural common generalisation of inverse semigroups and cancellative\nmonoids. Their importance was first recognised by Fountain \nin the 1970's \\cite{Fountain79}, but for many years their study was restricted by\na lack of applicable methods. In the last few years, interest has been\nreawakened by the development of several new techniques and results \n(see for example \\cite{Araujo11,Branco09a,Gomes09,\nK_freeadequate,K_onesidedadequate}).\n\n\\textit{Free algebras} form a natural focus of attention when studying any class of algebras\nin which they exist; indeed, an understanding of the free objects in a class of algebras usually\nyields considerable information about the class as a whole. In the case of adequate semigroups,\nthe fact that free adequate semigroups of every rank exist follows from\nelementary principles of universal algebra (see for example \\cite[Proposition~VI.4.5]{Cohn81}),\nbut an explicit description proved elusive until recently. In \\cite{K_freeadequate}, the first\nauthor gave a concrete geometric realisation of the free adequate semigroups (and monoids),\ninspired by Munn's celebrated representation of the free inverse semigroups, in terms of directed,\nlabelled, birooted trees under a natural combinatorial multiplication operation. In \\cite{K_onesidedadequate} he showed further\nthat the certain natural subsemigroups are the free objects in the\nrelated categories of \\textit{left adequate} and \\textit{right adequate}\nsemigroups (and monoids). The free left, right and two-sided adequate semigroups also\nturn out to be free objects in the larger classes of left, right and two-sided\n\\textit{Ehresmann semigroups} \\cite{Branco09a,Gomes09,Lawson91}.\n\nThis representation immediately gave rise to a non-deterministic\npolynomial-time\nalgorithm for the\nword problem in finite rank free adequate semigroups and\nmonoids (and hence also in finite rank free left adequate and right adequate\nsemigroups and monoids). Since a relation holds in a free algebra in a category\nexactly if the corresponding identity holds in \\textit{all} algebras in the\ncategory, this also yields an algorithm to check whether a given identity\nholds in all adequate (or left adequate or right adequate) semigroups or\nmonoids.\nThis algorithm has proved surprisingly practical for human application to \nshort words, with the intuitive geometric nature of the representation \noften allowing an effective use of guesswork to circumvent the issue\nof non-determinism. However, a non-deterministic algorithm is clearly\nnot well-suited to computer implementation for larger words, and it would\nalso be more satisfactory for theoretical reasons to know the precise\nasymptotic complexity of the problem.\n\nIn this paper, we apply some ideas from constraint satisfaction theory\nto refine the algorithm into a deterministic form, thus showing that the\nword problems for free adequate, free left adequate and free adequate\nsemigroups and monoids, and hence also the problem of checking whether identities hold\nfor all adequate semigroups or monoids, are decidable in quadratic (in the\nRAM model of computation) time. Moreover, we show how to efficiently (again, in quadratic\ntime in the RAM model) compute normal forms (either trees\nor words) for elements of the free adequate semigroup or monoid.\n\n\\section{Preliminaries}\\label{sec_prelim}\n\nIn this section we very briefly recall the definitions of (left, right and two-sided) adequate\nsemigroups, and the first author's characterisation of the free (left, right\nand two-sided) adequate semigroups monoids. The reader seeking a more complete\nintroduction with examples is referred to \\cite{Fountain79} for adequate semigroups in general and\n\\cite{K_freeadequate} for free adequate semigroups and monoids.\n\nLet $S$ be a semigroup whose idempotent elements commute. Denote by $S^1$ the\nmonoid consisting of $S$ with a new identity element $1$ adjoined. Then $S$ is\ncalled \\textit{left adequate} if for every element $x \\in S$ there\nis an idempotent element $x^+ \\in S$ such that\n$a x = bx \\iff a x^+ = b x^+$ for all $a, b \\in S^1$. If $S$ is left adequate then\nthe choice of $x^+$ is uniquely determined by $x$, and it is usual to\nconsider $S$ as a $(2,1)$-algebra with the binary operation of multiplication\nand the unary operation $x \\mapsto x^+$. In particular, we restrict attention\nto morphisms respecting both operations.\nDually, $S$ is \\textit{right adequate} if for every $x \\in S$ there is an\nidempotent $x^*$ with $xa = xb \\iff x^* a = x^* b$ for all $a, b \\in S^1$;\nright adequate semigroups are also $(2,1)$-algebras. The semigroup $S$ is\ncalled \\textit{(two-sided) adequate} if it is both left adequate and right\nadequate; the two maps $x \\to x^+$ and $x \\to x^*$, which in general will\nbe different, make $S$ into a $(2,1,1)$-algebra.\n\nNow let $\\Sigma$ be an alphabet. A \\textit{$\\Sigma$-tree} (or just\na \\textit{tree} if the alphabet $\\Sigma$ is clear) is a finite\ndirected graph with edges labelled by letters from $\\Sigma$, whose\nunderlying undirected graph is a tree, together\nwith two distinguished\nvertices (the \\textit{start} vertex and the \\textit{end} vertex) such that\nthere is a (possibly empty) directed path\nfrom the start vertex to the end\nvertex. The (unique) simple path from the start vertex to the end vertex\nis termed the \\textit{trunk} of the tree; vertices and edges lying on it are\ncalled \\textit{trunk vertices} and \\textit{trunk edges} respectively. If\n$e$ is an edge in such a tree, we denote by $\\alpha(e)$,\n$\\omega(e)$ and $\\lambda(e)$ respectively the source vertex, target vertex\nand label of $e$. We say that a vertex $v$ is a \\textit{descendant} of a vertex\n$u$ if the unique simple undirected path between $v$ and the start vertex passes\nthrough $u$.\n\nAs a notational convenience, we let $\\Sigma' = \\lbrace j' \\mid j \\in \\Sigma \\rbrace$\nbe an alphabet disjoint from and in bijective correspondence with $\\Sigma$, and say\nthat a $\\Sigma$-tree $X$ has \\textit{an edge from $u$ to $v$ labelled $j'$} to mean that\nit has an edge from $v$ to $u$ labelled $j$. (Intuitively, the elements of $\\Sigma'$ can\nbe thought of as labelling directed edges when read ``in the wrong direction''. This\nnotation will allow a unified consideration of labels and directions of edges;\nsince label and direction play similar roles as obstructions to a morphism mapping one\nedge to another, considering them together simplifies our arguments in several places.)\n\nA \\textit{morphism} $\\rho : X \\to Y$ of $\\Sigma$-trees $X$ and $Y$ is a\nmap taking edges to edges and vertices to vertices which commutes with\n$\\alpha$, $\\lambda$ and $\\omega$ and maps the start and end vertex of $X$\nto the start and end vertex of $Y$ respectively. An \\textit{isomorphism} is a\nmorphism which is bijective on both edges and vertices. A \\textit{retraction} is\nan idempotent morphism from a $\\Sigma$-tree to itself; its image is called a \\textit{retract}.\nA tree is called \\textit{pruned} if it does not admit a\nnon-identity retraction. (Structures without retractions are often called\n\\textit{cores} in graph theory.)\n\nThe $\\Sigma$-tree with a single vertex and no edges is called \\textit{trivial}.\nThe set of all isomorphism types of $\\Sigma$-trees (including the trivial\n$\\Sigma$-tree) is denoted $UT^1(\\Sigma)$\nwhile the set of isomorphism types of non-trivial $\\Sigma$-trees is\ndenoted $UT(\\Sigma)$. The set of all isomorphism types of pruned trees\n[respectively, non-trivial pruned trees] is denoted $T^1(\\Sigma)$\n[respectively, $T(\\Sigma)$]. For any $X \\in UT^1(\\Sigma)$ there is a unique\n$Y \\in T^1(\\Sigma)$ which is isomorphic to a retract of $X$ \\cite[Proposition~3.5]{K_freeadequate};\nwe denote this pruned tree $\\ol{X}$ and call it the \\textit{pruning} of $X$.\n\nIf $X, Y \\in UT^1(\\Sigma)$ then the \\textit{unpruned product} $X \\times Y$ is (the isomorphism type of)\nthe tree obtained by glueing together $X$ and $Y$, identifying the end vertex\nof $X$ with the start vertex of $Y$ and keeping all other vertices and\nall edges distinct; this is a well-defined, associative binary operation\n\\cite[Proposition~4.2]{K_freeadequate}. If\n$X \\in UT^1(\\Sigma)$ then $X^{(+)}$ is (the isomorphism type of) the tree with\nthe same labelled graph and start vertex of $X$, but with end vertex of $X^{(+)}$ the\nstart vertex of $X$.\nDually, $X^{(*)}$ is the isomorphism type of the idempotent tree with the\nsame underlying graph and end vertex as $X$, but with start vertex the end vertex of $X$.\nWe define corresponding \\textit{pruned operations} on $T^1(\\Sigma)$ by\n$XY = \\ol{X \\times Y}$, $X^* = \\ol{X^{(*)}}$ and $X^+ = \\ol{X^{(+)}}$.\n\nA tree with a single edge and distinct start and end vertices is called a\n\\textit{base tree}; we identify each base tree with the label of its\nedge, thus viewing $\\Sigma$ itself as a set of $\\Sigma$-trees.\nThe main\nresult of \\cite{K_freeadequate} is that $T^1(\\Sigma)$ is the free adequate monoid\non $\\Sigma$, being freely generated under pruned multiplication, $*$ and $+$ by the base $\\Sigma$-trees\n\\cite[Theorem~5.16]{K_freeadequate}. The map\n$X \\to \\ol{X}$ is a $(2,1,1,0)$-morphism from $UT^1(\\Sigma)$ onto $T^1(\\Sigma)$\n\\cite[Theorem~4.5]{K_freeadequate}.\nMoreover, the submonoid of $T^1(\\Sigma)$ generated by the base trees under pruned\nmultiplication and $*$ [respectively, $+$] is the free left adequate [respectively,\nright adequate] monoid on $\\Sigma$ \\cite[Theorem~3.18]{K_onesidedadequate}. Free\nadequate, left adequate or right adequate semigroups can all be obtained by discarding\nthe trivial tree (which is the identity element) in the corresponding monoids\n(see \\cite[Proposition~2.2]{K_freeadequate} and \\cite[Proposition~2.6]{K_onesidedadequate}).\n\n\\section{Computing with Formulas and Trees}\\label{sec_exptree}\n\nIn this section we study the computational complexity of converting between\nwell-formed formulas, over a generating set $\\Sigma$ and the binary and unary\noperations in an adequate semigroup, and $\\Sigma$-trees. This will allow\nus, in later sections, to use algorithms operating on $\\Sigma$-trees to\nsolve computational problems involving formulas.\n\nFor our complexity\nanalysis throughout this paper, we shall work in the RAM\nmodel of computation, in which integer operations and indirection\n(finding a value stored at a known position in an array) take unit time.\nFor simplicity we will analyse \nthe complexity of problems for a fixed rank semigroup, say on an alphabet\n$\\Sigma$, rather than the uniform complexity as the rank grows.\nIn places where it is necessary to be formal, we shall regard formulas as\nwords over the alphabet $\\Omega$ consisting of generators from $\\Sigma$ plus\nthe symbols\n$($, $)$, $*$ and $+$ with the obvious meaning. We denote by $\\Omega^*$\nthe set of all words over the alphabet $\\Omega$, including the empty word\nwhich we denote $\\epsilon$. Our measure of the size of an expression will\nbe its length as a word over $\\Omega$.\n\nWe assume\n$\\Sigma$-trees are by default stored as a natural number representing the\nstart vertex, a natural number representing the end vertex\nand a list of edges (in no particular order), each\nbeing a triple consisting of a label from $\\Sigma$ and two natural numbers\nencoding its start vertex and its end vertex. Sometimes it\nwill be expedient to convert trees to an alternative\nrepresentation. Note that the same abstract $\\Sigma$-tree can admit\nmultiple representations, by numbering the vertices and ordering the edges\ndifferently. Our measure of the size of a $\\Sigma$-tree will be the\nnumber of edges.\n\n\\begin{proposition}\\label{prop_exptotree}\nGiven a well-formed formula $\\omega$, one can compute in quadratic time\nthe (unpruned) $\\Sigma$-tree which is its evaluation in $UT^1(\\Sigma)$.\nMoreover, this tree has size linear in the length of $\\omega$.\n\\end{proposition}\n\\begin{proof}\nA formula of length $n$ can be evaluated by a depth-first traversal\nof a parse tree; this will clearly involve performing at most $n$ unpruned\noperations with trees whose size is $O(n)$.\nClearly the unpruned $(+)$ and $(*)$ operations on trees can be performed\nin constant time. Unpruned multiplication of trees can be performed in\ntime linear in the number of edges in the trees, by first relabelling\nthe vertices in the second tree (so that all references to its start\nvertex become the end vertex of the first tree, and all its other vertices\nare distinct from those in the first tree) and then concatenating the edge\nlists and setting start and end vertices appropriately.\n\nThus, the $O(n)$ unpruned operations can each be performed in $O(n)$ time,\nand the evaluation of the expression takes time $O(n^2)$.\nMoreover, the resulting tree clearly has exactly one edge for each\noccurrence of a generator in the expression, and hence has size linear\nin the size of the expression.\n\\end{proof}\n\nFor our present purpose, the computations we wish to perform with\ntrees will all take quadratic time, so there is no particular benefit in\nbeing able to compute the trees in faster than quadratic time.\nHowever, we remark that the complexity of the algorithm given above can be\nimproved by a more sophisticated approach, using what is known in the computer science literature as a\n``Union Find'' algorithm. Under this approach, when\nperforming multiplication, instead of merging the end vertex of one\ntree with the start vertex of another, we keep them separate (allowing\nthe data structure to become a forest, rather than a tree) and\nmaintain another data structure recording which vertices are to be\nmerged at the end. An efficient implemention of this algorithm is extremely\nclose to being linear time; see \\cite[Section~21.3]{Cormen01}\nfor more details.\n\nNext, we shall show how a (not necessarily pruned) $\\Sigma$-tree can be efficiently\nconverted into an well-formed formula.\nWe will define a function $\\sigma : UT^1(\\Sigma) \\to \\Omega^*$ such\nthat for each tree $X$, $\\sigma(X)$ is a well-formed formula which\nevaluates to $X$\nin $UT^1(X)$, and then show that this function can be computed in quadratic time.\nNote that since $\\sigma$ is a function defined on abstract trees, the\nalgorithm produces a formula depending only on the abstract tree,\nand not on its representation. We shall exploit this in\nSection~\\ref{sec_normalforms}\nbelow\nto compute normal forms (as formulas) for elements of the free adequate\nsemigroup.\nTo do this, we shall need a linear order on the set of all\nformulas; for now, we will assume that we have such an order fixed. We\nwill discuss the choice and implementation of this order when we come to\nanalyse the complexity of the algorithm.\n\nLet $X$ be a tree. We begin by defining a function $\\rho$\nfrom the vertex set of $X$ to $\\Omega^*$; this is done inductively\nby downwards induction on the distance of the vertex from the trunk.\nLet $v$ be a vertex, and suppose $\\rho$ is already defined on all vertices\nstrictly further from the trunk than $v$. Let\n$v_1, \\dots, v_p$ be the vertices adjacent to $v$ and strictly further from\nthe trunk, noting that $\\rho(v_i)$ is already defined for each $i$.\nFor each $i$, let $e_i$ be the edge connecting $v$ to $v_i$,\nand let $a_i \\in \\Sigma$ be its label. Define a formula $\\tau_i \\in \\Omega^*$\nby:\n$$\\tau_i = \\begin{cases}\n (a_i \\rho(v_i))+ &\\textrm{ if $e_i$ is orientated away from $v$} \\\\\n (\\rho(v_i) a_i)* &\\textrm{ if $e_i$ is orientated towards $v$}\n\\end{cases}$$\nNow we define $\\rho(v) \\in \\Omega^*$ to be the word obtained by sorting the words\n$\\tau_i$ according to our ordering of formulas, and then concatenating.\n(If $p = 0$, that is, if $v$ is a ``leaf'', this means $\\rho(v) = \\epsilon$.)\n\nNow let $t_0, \\dots t_q$ be\nthe trunk vertices of $X$ and $b_1, \\dots, b_q$ the labels of the edges between\nthem, both in the obvious order. We define\n$$\\sigma(X) = \\rho(t_0) b_1 \\rho(t_1) b_2 \\dots \\rho(t_{q-1}) b_q \\rho(t_q).$$\n\nA simple but tedious inductive argument, akin to those in \\cite{K_freeadequate},\nshows that $\\sigma(X)$ evaluates to the tree $X$ in $UT^1(\\Sigma)$, and that\nthe number of characters in $\\sigma(X)$ is at most four times the number\nof edges in $X$.\n\nTo compute $\\sigma(X)$, we start by precomputing adjacency matrices for $X$\ncorresponding to each possible edge label and direction; it is easily seen\nthat this can be done in $O(n^2)$ time where $n$ is the number of edges in $X$.\nIt is immediate from the inductive method of definition how to compute\n$\\sigma(X)$ by a simple depth first traversal (following non-trunk edges)\nfrom each of the trunk vertices; this involves\nconsidering each of $O(n)$ vertices once.\n\nAt each vertex, the only non-trivial operation is to sort the words $\\tau_i$ into order\nand then concatenate; the complexity of this of course depends on the choice of order. The\nsum length of all the words $\\tau_i$ is clearly $O(n)$. If\nwe choose the order to be lexicographic order (with respect to some arbitrary linear\norder on $\\Omega$), then a careful implementation of radix sort gives us a lexicographically\nsorted list of formulas in $O(n)$ time, and concatenation is\nclearly also $O(n)$.\n\nThus, the total time required for the algorithm is $O(n^2)$, and we have established:\n\n\\begin{proposition}\\label{prop_treetoexp}\nGiven an unpruned $\\Sigma$-tree $X$, we can in quadratic time compute\na well-formed formula which evaluates to $X$ in $UT^1(\\Sigma)$. Moreover,\nthe formula has size linear in the size of $X$, and depends only\non the isomorphism type of $X$ and not on its representation.\n\\end{proposition}\n\n\\section{The Word Problem}\\label{sec_wp}\n\nRecall that the \\textit{word problem} for an algebra $A$ with a given generating set\nis the algorithmic problem of determining, given as input two well-formed formulas\nover the generating set and the operations of the algebra, whether the formulas\nrepresent the same element of the algebra. The word problem for free objects\nin a variety of algebras is of particular importance, since it is trivially\nequivalent\nto the problem of testing whether a given identity holds in all algebras of\nthe variety.\n\nIn this section, we shall exhibit a quadratic time algorithm to solve\nthe word problem in a free adequate monoid $T^1(\\Sigma)$. In fact in\nSection~\\ref{sec_normalforms} below, we shall see that it is also possible\nto compute normal forms of elements of $T^1(\\Sigma)$ in quadratic time;\nthis automatically yields another algorithm for the word\nproblem (by computing normal forms and comparing), of the same asymptotic\ncomplexity. However, we present an explicit word problem algorithm first\nsince this is simpler, potentially easier to implement, and illustrates in\na simple context some of the ideas we will need in Section~\\ref{sec_normalforms}.\n\nBy Proposition~\\ref{prop_exptotree} we can efficiently convert\nwell-formed formulas in the free adequate monoid into unpruned\n$\\Sigma$-trees of comparable\nsize. It follows that to test\n(efficiently) whether two given expression $x$ and $y$ represent the same\nelement of the free adequate monoid, that is, to solve the word problem,\nit suffices to compute corresponding $\\Sigma$-trees\n$X, Y \\in UT^1(\\Sigma)$,\nand then check (efficiently) if $\\ol{X} = \\ol{Y}$ in $T^1(\\Sigma)$.\n\nTo solve\nthis latter problem, we begin with an elementary proposition, which\nreduces it a constraint satisfaction problem (formulated in terms of\nmorphisms between structures, in the manner usual in the literature\nof areas such as graph theory and universal algebra --- see for example\n\\cite{Hell04}).\n\n\\begin{proposition}\\label{prop_equiv}\nLet $X$ and $Y$ be $\\Sigma$-trees. Then the following are equivalent:\n\\begin{itemize}\n\\item[(i)] $\\ol{X} = \\ol{Y}$;\n\\item[(ii)] $X$ and $Y$ admit isomorphic retracts;\n\\item[(iii)] there is a morphism from $X$ to $Y$ and a morphism from $Y$ to $X$.\n\\end{itemize}\n\\end{proposition}\n\\begin{proof}\nThe equivalence of (i) and (ii) follows from \\cite[Proposition~3.5]{K_freeadequate}, so\nit suffices to establish the equivalence of (ii) and (iii).\n\nIf (ii) holds then, in particular, some retract of $X$ is isomorphic to a\nsubstructure of $Y$; composing the retraction of $X$ with the isomorphism\nyields a morphism from $X$ to $Y$. By symmetry of assumption there is also\na morphism from $Y$ to $X$, so (iii) holds.\n\nNow suppose (iii) holds, say $\\sigma : X \\to Y$ and $\\tau : Y \\to X$ are\nmorphisms. Then the compositions $\\tau \\circ \\sigma : X \\to X$ and\n$\\sigma \\circ \\tau : Y \\to Y$ are maps on finite sets, and it follows that\nwe may choose $n$ such that both\n$(\\tau \\circ \\sigma)^n : X \\to X$ and $(\\sigma \\circ \\tau)^n : Y \\to Y$\nare idempotent, that is, are retractions of $X$ and $Y$ respectively.\nLet $X'$ and $Y'$ be the retracts which are the respective images of\nthese retractions. Now it is easily verified that \n$\\sigma$ and $\\tau \\circ (\\sigma \\circ \\tau)^{n-1}$ restrict to mutually\ninverse isomorphisms between the retracts $X'$ and $Y'$, showing that\n(ii) holds.\n\\end{proof}\n\nProposition~\\ref{prop_equiv} implies that to check if two $\\Sigma$-trees\nare equivalent, and hence by the preceding arguments to solve the word problem\nfor the free adequate semigroup on $\\Sigma$, it suffices to check whether each\n$\\Sigma$-tree admits a morphism to the other. Our main goal in the rest of\nthis section, then, is an efficient algorithm to test, given an ordered pair of\n$\\Sigma$-trees, whether there is a morphism from the first to the second.\nOur approach is essentially a constraint propagation\nalgorithm, with the correctness of the result being shown by an \narc consistency argument utilising the tree-like nature of\nour geometric representatives for elements. The ideas behind the proof\nare well known in the fields of constraint satisfaction and artificial\nintelligence (see for example\n\\cite{Dechter87}), but for the benefit of semigroup theorists who may not be\nfamiliar with these fields we present the algorithm in an elementary form:\n\n\\begin{algorithm}\\label{algH} \\ \\\\\n\n\\noindent\\textbf{Input:} Two $\\Sigma$-trees $T_1$ and $T_2$ on $n$ and $m$ vertices respectively.\n\n\\noindent\\textbf{Output:} ``Yes'' if there exists a homomorphism from $T_1$ to $T_2$. ``No'' otherwise.\n\n\\begin{enumerate}\n\\item Consider the start vertex of $T_1$, label this vertex $1$, and then use\na depth-first traversal (ignoring direction of edges) to label the remaining vertices\nfrom $2$ to $n$ in ascending order.\n\n\\item For each $i$ in $\\{1,\\dots,n\\}$, let $B_i$ be the set of vertices\nin $T_2$.\n\n\\item For the start [end] vertex $i$ set $B_i$ to be the singleton set containing the\nstart [end] vertex of $T_2$\n\\item For $i$ descending from $n$ to $1$, and each vertex $j > i$ adjacent to $i$, do the following:\n\\begin{itemize}\n\\item[(i)] Let $a \\in \\Sigma \\cup \\Sigma'$ be the label of the edge from $i$ to $j$ in $T_1$;\n\\item[(ii)] Let $B_i:= B_i \\cap B_j^\\star$ where\n\\begin{align*}\nB_j^\\star = \\lbrace x \\mid & \\textrm{ $T_2$ has an edge labelled $a$ } \\\\\n&\\textrm{ from $x$ to some $y \\in B_j$ } \\rbrace.\n\\end{align*}\n\\end{itemize}\n\\item If $B_1=\\emptyset$, output ``No''; otherwise output ``Yes.''\n\\end{enumerate}\n\\end{algorithm}\n\n\\begin{proposition}\\label{prop_wpcorrect}\nAlgorithm~\\ref{algH} is correct, that is, $B_1$ is non-empty on completion of the\nalgorithm if and only if there is a morphism from $T_1$ to $T_2$.\n\\end{proposition}\n\\begin{proof}\nFor brevity, we identify the vertices with the labels from $1$ to $n$\nassigned in the algorithm.\nSuppose first that there is a morphism $\\sigma : T_1 \\to T_2$. We claim\nthat $B_i$ contains $\\sigma(i)$ for all $i$, from which it follows in\nparticular that $B_1$ contains $\\sigma(1)$ so that $B_1$ is\nnon-empty as required. Indeed, if not, choose $i$ maximal such that\n$\\sigma(i) \\notin B_i$. Clearly $\\sigma(i)$ was in $B_i$ after Step 2 of the\nalgorithm and, because $\\sigma$ preserves start and end vertices, also after\nStep 3; therefore, it must have been removed during Step 4.\nFor this to have happened, there must have been a $j > i$ and an edge from $i$\nto $j$ (labelled $a \\in \\Sigma \\cup \\Sigma'$, say) such that\n$\\sigma(i) \\notin B_j^\\star$.\nBy the definition of $B_j^\\star$, this means there was (at the time of removal)\nno edge labelled $a$ from $\\sigma(i)$ to any $y \\in B_j$.\n But because $\\sigma$ is a morphism, $\\sigma(j) \\in B_j$\nis connected to $\\sigma(i)$ by such an edge, so it must be that\n$\\sigma(j)$ was not in $B_j$ at the time $\\sigma(i)$ was removed from\n$B_i$. Now since $B_j$ only gets smaller, $\\sigma(j)$ is not in $B_j$\nat the end of the algorithm. But $j > i$, so this contradicts the\nmaximality of $i$.\n\nConversely, suppose $B_1$ is non-empty at the end of the algorithm. We\ndefine a morphism $\\sigma : T_1 \\to T_2$ inductively as follows. First,\nchoose $\\sigma(1) \\in B_1$ arbitrarily. Now assume $1 < i < n$ and\nwe have defined\n$\\sigma$ on the vertices $1$, \\dots $i-1$ and all edges between them, in\nsuch a way as to preserve adjacency, labels and directions of edges, and\nthe start and end vertices if appropriate, and such that $\\sigma(p) \\in B_p$\nfor $1 \\leq p \\leq i-1$.\n\nSince $T_1$ is a tree and the edges were numbered by a depth-first \ntraversal, it follows that vertex $i$ is connected to vertex $k$ for some \nunique $k < i$; suppose $T_1$ has an edge from $k$ to $i$ labelled\n$a \\in \\Sigma \\cup \\Sigma'$. \n\nConsidering the way $B_k$ is constructed, we see that every vertex in $B_k$,\nincluding $\\sigma(k)$, is connected to some $v \\in B_i$. \nMoreover, if $i$ happens to be the start [respectively, end] vertex of $T_1$,\nthen $B_i$ was originally set to contain only the start [end] vertex of $T_2$,\nso it must be that $v \\in B_i$ is the start [end] vertex of $T_2$.\nThus, by defining $\\sigma(i) = v$ and $\\sigma(e)$ to be the \nappropriate edge, we extend $\\sigma$ to be defined on the vertices \n$1, \\dots, i$ and all edges between, with the appropriate properties.\n\\end{proof}\n\nWe now analyse the complexity of Algorithm~\\ref{algH}.\nAt the start of the algorithm, we can precompute for\neach vertex in $T_1$ a list of edges adjacent to that vertex; this can be done\nin $O(n)$ time.\n\nHaving done this, Step 1 of the algorithm (a simple depth first traversal\nof the tree $T_1$) has complexity $O(n)$. If we store the lists $B_i$ as\narrays of $m$ boolean flags then Step 2 has complexity $O(mn)$\nsince we need to initialise $mn$ values. Step 3 has complexity $O(m)$,\nsince we must reset $m-1$ values for each of the start and end vertices.\n\nThe most interesting part is the complexity of Step 4. The number of\niterations of the outer loop is clearly bounded by the number of edges\nin $T_1$, so it is $O(n)$ and the precomputed lists of edges mean there is\nno extra overhead in finding the edges in the correct order.\nIn each iteration, the fact that the corresponding edge has been found\nmeans Step 4(i) takes constant time. In Step 4(ii), computing $B_j^\\star$\ninvolves\npassing through the list of all $O(m)$ edges of $T_2$ and for each edge\nchecking (in constant time) if one of the ends lies in $B_j$ and if the\nlabel is correct; this takes\n$O(m)$ time. Computing the intersection is simply\na boolean ``and'' operation on two arrays of length $m$, and so also takes $O(m)$\ntime. Thus, Step 4 takes time $O(mn)$, and the total complexity of the\nalgorithm is $O(mn)$.\n\nCombining the above arguments with Proposition~\\ref{prop_exptotree},\nwe have established the following main result:\n\\begin{theorem}\nThe word problem for any finite rank free left adequate, free right\nadequate or free adequate semigroup is decidable in time\npolynomial (quadratic, in the RAM model of computation) in the combined\nlength of the two formulas.\n\\end{theorem}\n\n\\section{Pruned Trees and Normal Forms}\\label{sec_normalforms}\n\nIn this section, we show how to efficiently compute the minimal retract of a \n$\\Sigma$-tree. Combined with the results of Section~\\ref{sec_exptree}, this\nwill allow us to compute normal forms (as formulas) for elements of\nfree adequate monoids. Our main algorithm is the following, the first four\nsteps of which are essentially the same as in Algorithm~\\ref{algH}:\n\n\\begin{algorithm}\\label{algN} \\ \\\\\n\n\\noindent\\textbf{Input:} A $\\Sigma$-tree $T$ on $n$ vertices.\n\n\\noindent\\textbf{Output:} The vertex set of a pruned subtree of $T$,\nisomorphic to the $\\ol{T}$.\n\n\\begin{itemize}\n\\item[(1)] Consider the start vertex of $T$, label this vertex $1$, and then use\na depth-first traversal (ignoring direction of edges) to label the remaining vertices\nfrom $2$ to $n$ in ascending order.\n\n\\item[(2)] For each $i$ in $\\{1,\\dots,n\\}$, set $B_i = \\lbrace 1, \\dots, n \\rbrace$.\n\n\\item[(3)] For the start [end] vertex $i$ set $B_i = \\lbrace i \\rbrace$.\n\n\\item[(4)] For $i$ descending from $n$ to $1$ and each $j$ with $j > i$ and $i$ connected to $j$, do the following:\n\\begin{itemize}\n\\item[(i)] Let $a \\in \\Sigma \\cup \\Sigma'$ be the label of the edge in $T$\nfrom $i$ to $j$.\n\\item[(ii)] Let $B_i := B_i \\cap B_j^\\star$ where\n\\begin{align*}\nB_j^\\star = \\lbrace x \\mid & \\textrm{ $T$ has an edge labelled $a$ } \\\\\n&\\textrm{ from $x$ to some $y \\in B_j$ } \\rbrace.\n\\end{align*}\n\\end{itemize}\n\n\\item[(5)] Set $X = \\lbrace 1, \\dots, n \\rbrace$.\n\n\\item[(6)] For $w$ ascending from $1$ to $n$ and $a \\in \\Sigma \\cup \\Sigma'$,\ndo the following:\n\\begin{itemize}\n\\item[(i)] If $w \\notin X$ then go to the next $w$.\n\\item[(ii)] Otherwise, find all vertices $u$ such that $a$ labels\nan edge from $w$ to $u$ and put them in a list $K$.\n\\item[(iii)] For each $u \\in K$ such that $u > w$:\n\\begin{itemize}\n\\item[(a)] Check if $K \\cap B_u = \\lbrace u \\rbrace$.\n\\item[(b)] If \\textbf{not}, then remove $u$ from $K$, and traverse the\ntree below $u$, removing $u$ and all its descendant vertices from $X$.\n\\end{itemize}\n\\end{itemize}\n\\item[(7)] Output $X$.\n\\end{itemize}\n\\end{algorithm}\n\nOur next aim is to prove the correctness of this algorithm.\n\n\\begin{lemma}\\label{lemma_retract}\nThe subtree $X$, as computed at the end of Algorithm~\\ref{algN}, is a retract of $T$.\n\\end{lemma}\n\\begin{proof}\nWe shall show that each time a vertex and its descendants are removed from\n$X$ at Step 6(iii)(b), there is a retraction from the tree $X$ prior\nto the removal, onto the tree $X$ after the removal. Since the successive\nsubtrees $X$ form a chain under inclusion, it is clear that composing these\nretractions in the appropriate order yields a retraction from $T$ onto\nthe final tree $X$, as required.\n\nIndeed, suppose $u$ and its descendants are removed from $X$ at some point.\nLet $w$, $a$ and $K$ be as in the algorithm at that point, and let $X_1$ and $X_2$\nbe the values of $X$ immediately before and after\nthe deletion, respectively.\n\nNote that, since the identity map is a morphism, it is easily verified\nthat $i \\in B_i$ for all vertices $i$ of $T$. The\nfact that $u$ was removed means that $K \\cap B_u \\neq \\lbrace u \\rbrace$, and\nwe know $u \\in K \\cap B_u$, so we may choose some vertex $v \\in K \\cap B_u$\nwith $v \\neq u$.\n\nFirst, we follow the procedure from the proof of Proposition~\\ref{prop_wpcorrect}\nto inductively define a morphism $\\sigma : T \\to T$, but\nbeing  more careful about our choices in order to ensure that\n$\\sigma(u) = \\sigma(v) = v$.\nWe start by setting $\\sigma(i) = i$ for all $i < u$; since $i \\in B_i$ for all\n$i$ it is easily verified that this is consistent with the procedure\nin Proposition~\\ref{prop_wpcorrect}. Note in particular\nthat $w < u$, so this means $\\sigma(w) = w$. Now since $u, v \\in K$, there\nare edges from $w = \\sigma(w)$ to $u$ and $v$ both labelled $a$, so in\nfollowing the procedure of Proposition~\\ref{prop_wpcorrect}\nwe may choose to set $\\sigma(u) = v$.\nWe now continue the process from the proof of Proposition~\\ref{prop_wpcorrect}. For each vertex\n$r$ in turn, if $r$ is a descendant of $u$, then we define $\\sigma(r)$\nas in the proof of Proposition~\\ref{prop_wpcorrect}, making any choices arbitrarily. If $r$ is\nnot a descendant of $u$ then the unique vertex $k < r$ adjacent to $r$\nis also not a descendant of $u$; thus, we have already defined\n$\\sigma(k) = k$ and we may set $\\sigma(r) = r$.\n\nNow $\\sigma$ is a map on a finite set, and so has an idempotent power,\nsay $\\sigma^i$. Since $v$ is not a descendant of $u$, we have\n$\\sigma(v) = v$, and hence $\\sigma^i(u) = \\sigma^{i-1}(v) = v$, so $u$\nis not in the image of $\\sigma^i$. Since the image of $\\sigma^i$ is a\n$\\Sigma$-tree, it must contain the start\nvertex and be connected, so we deduce that no descendants of $u$ are\nin the image of $\\sigma^i$. It follows that $\\sigma^i$ maps $X_1$ to\n$X_2$. Moreover, $\\sigma$ fixes $X_2$, so restricting $\\sigma^i$ to\n$X_1$ gives the required retraction of $X_1$ onto $X_2$.\n\\end{proof}\n\n\\begin{lemma}\nThe retract $X$, as computed at the end of Algorithm~\\ref{algN}, is pruned.\n\\end{lemma}\n\\begin{proof}\nSuppose not, say $X$ admits a proper retraction $\\sigma : X \\to X$. Let\n$u$ be a vertex in $X$ but not in the image of $\\sigma$, and suppose $u$\nis minimal with respect to this condition. Then $u \\neq 1$, since $1$ labels\nthe start vertex which is fixed by every retraction. Thus, we may let $w$ be\nthe unique vertex with $w < u$ and $w$ adjacent to $u$.\n\nSince $w < u$, by the minimality\nof the choice of $u$, we have $\\sigma(w) = w$. It follows that $\\sigma(u)$\nis connected to $w$ by an edge of the same label and orientation as that\nconnecting $u$ to $w$. This means that, when considering $w$ at step\n6(iii), we would initially have had $\\sigma(u) \\in K$. Since\n$\\sigma(u)$ is in the final tree $X$, it was never removed from $K$.\nMoreover, composing the retraction of $T$ onto $X$ (given by\nLemma~\\ref{lemma_retract}) with\n$\\sigma$ gives a morphism of $T$ mapping $u$ to $\\sigma(u)$; it follows\nfrom the argument in the proof of Proposition~\\ref{prop_wpcorrect} that\n$\\sigma(u) \\in B_u$.\n\nThis means that at the time\n$u$ was considered in Step 6(iii)(a) we had\n$\\sigma(u) \\in K \\cap B_u$. But then $K \\cap B_u \\neq \\lbrace u \\rbrace$,\nso $u$ would have been removed from $X$, giving a contradiction.\n\\end{proof}\n\nTurning to the complexity of the algorithm, Steps (1)-(4) are exactly\nas in Algorithm~\\ref{algH} (except that the source and target trees for\nthe morphism are the same, so $m=n$), and by the same analysis as in\nSection~\\ref{sec_wp} take time $O(n^2)$.\n\nFor efficiency, we store the set $X$ as an array of boolean flags. The time\nrequirement for Step\n(5) is clearly $O(n)$. The loop in Step 6 is iterated at most $O(n)$ times.\nIn each such iteration, step (i) takes constant time. Step (ii) cannot\ninvolve checking more than $O(n)$ vertices, so the total contribution\nto the time required will be $O(n^2)$. In step (iii), note that each\nelement of $L$ is uniquely determined (across the entire algorithm) by\nthe ordered pair $(w,u)$ where there is always an edge between $w$ and $u$;\nthus, the number of iterations of this step across the whole algorithm is\nat most twice the number of edges in the tree, which is $O(n)$.\nWithin each iteration, each step takes $O(n)$ time, so the total contribution is\n$O(n^2)$.\n\nThus, we have established:\n\\begin{theorem}\\label{thm_pruning}\nGiven a $\\Sigma$-tree $T$, one can compute in polynomial time (quadratic\ntime in the RAM model of computation) the pruned $\\Sigma$-tree $\\ol{T}$.\n\\end{theorem}\n\nCombining with the results of Section~\\ref{sec_exptree}, Theorem~\\ref{thm_pruning}\nallows us to compute normal forms (as formulas) in the free adequate monoid.\nIndeed, given a formula $w$, by Proposition~\\ref{prop_exptotree} we may\nconvert it in quadratic time to a corresponding unpruned $\\Sigma$-tree $T$ of comparable size. By\nTheorem~\\ref{thm_pruning} we may then compute the pruned tree $\\ol{T}$ in\ntime quadratic in the size of $T$ and hence in the size of $w$. Finally, by\nProposition~\\ref{prop_treetoexp} we can convert $\\ol{T}$ into the uniquely\ndefined formula $\\sigma(\\ol{T})$ in time quadratic in the size of $\\ol{T}$; since $\\ol{T}$\nis no larger than $T$, this is also quadratic in the size of $T$, and hence\nin the size of $w$.\n\n\\begin{theorem}\nGiven a formula in the free adequate, left adequate or right adequate\nsemigroup or monoid, one can compute\na normal form in polynomial (quadratic in the RAM model of computation) time.\n\\end{theorem}\n\nWe note that the resulting language of normal forms for elements, which by\ndefinition is the set\n$$\\lbrace \\sigma(\\ol{T}) \\mid T \\textrm{ is a pruned $\\Sigma$-tree} \\rbrace,$$\ndoes not appear to have a completely elementary description without reference to trees. Of\ncourse one may check (in quadratic time) whether a given formula $w$ is a\nnormal form by following the above procedure to convert $w$ to a normal form\nand then comparing with $w$; we do not know of a fundamentally easier method.\n\nWe also note that in the case of free \\textit{inverse} monoids (and semigroups), it is known\n\\cite[Theorem 11]{Lohrey07} that the word problem is decidable in (RAM) linear \ntime. In the inverse case computations appear to be inherently simpler, as\nthe operation corresponding to computing a minimal retract (namely, computing a minimal\nmorphic image) can be performed by an iterative process of identifying vertices, where\nthe fact a pair of vertices can be identified is determined ``locally'', by looking only\nin the immediate neighbourhood of the vertices. It seems unlikely that quite such a\nfast algorithm can be obtained in the adequate case, but one might still ask whether our\nalgorithms can be significantly improved upon. Also shown in \\cite[Theorem 11]{Lohrey07}\nis that the word problem for a free inverse monoid is decidable (using a different\nalgorithm to the linear time one) in logarithmic space: the space complexity of the word\nproblem for free adequate monoids and semigroups is a natural topic for future research.\n\n\\section*{Acknowledgements}\n\nThe second author was supported by the Czech Government Grant Agency GA\\v CR\nproject 13-01832S.\nThe authors thank the organisers of the 4th Novi Sad Algebraic Conference\n(NSAC2013), which by bringing together researchers in semigroup theory\nand universal algebra catalysed this research. They also thank Victoria Gould\nfor some helpful comments on the draft, and Stuart Margolis for pointing them\nto the work of Lohrey and Ondrusch \\cite{Lohrey07}.\n\n\\bibliographystyle{plain}\n\n\\def$'$} \\def\\cprime{$'${$'$} \\def$'$} \\def\\cprime{$'${$'$}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}