arXiv subjects

Risa H. Wechsler

Publications and source records attributed to Risa H. Wechsler.

At least 91 records · Page 5Linked to original sources

arXiv2021

Searching for Dwarf Galaxies in ${\it Gaia}$ DR2 Phase-Space Data Using Wavelet Transforms

We present a wavelet-based algorithm to identify dwarf galaxies in the Milky Way in ${\it Gaia}$ DR2 data. Our algorithm detects overdensities in 4D position--proper motion space, making it the first search to explicitly use velocity information to search for dwarf galaxy candidates. We optimize our algorithm and quantify its performance by searching for mock dwarfs injected into ${\it Gaia}$ DR2 data and for known Milky Way satellite galaxies. Comparing our results with previous photometric searches, we find that our search is sensitive to undiscovered systems at Galactic latitudes~$\lvert b\rvert>20^{\circ}$ and with half-light radii larger than the 50% detection efficiency threshold for Pan-STARRS1 (PS1) at (${\it i}$) absolute magnitudes of =$-7<M_V<-3$ and distances of $32$ kpc $< D < 64$ kpc, and (${\it ii}$) $M_V< -4$ and $64$ kpc $< D < 128$ kpc. Based on these results, we predict that our search is expected to discover $5 \pm 2$ new satellite galaxies: four in the PS1 footprint and one outside the Dark Energy Survey and PS1 footprints. We apply our algorithm to the ${\it Gaia}$ DR2 dataset and recover $\sim 830$ high-significance candidates, out of which we identify a "gold standard" list of $\sim 200$ candidates based on cross-matching with potential candidates identified in a preliminary search using ${\it Gaia}$ EDR3 data. All of our candidate lists are publicly distributed for future follow-up studies. We show that improvements in astrometric measurements provided by ${\it Gaia}$ EDR3 increase the sensitivity of this technique; we plan to continue to refine our candidate list using future data releases.

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arXiv2021

Simultaneous Estimation of Large-Scale Structure and Milky Way Dust Extinction from Galaxy Surveys

The high cosmological precision offered by the next generation of galaxy surveys hinges on improved corrections for Galactic dust extinction. We explore the possibility of estimating both the dust extinction and large-scale structure from a single photometric galaxy survey, making use of the predictable manner in which Milky Way dust affects the measured brightness and colors of galaxies in a given sky location in several redshift bins. To test our method, we use a synthetic catalog from a cosmological simulation designed to model the Vera C. Rubin Observatory Legacy Survey of Space and Time. At high Galactic latitude ($|b|\gtrsim20^\circ$) and a resolution of $1^\circ$ ($7'$), we predict the uncertainty in the measurement of dust extinction, $E(B-V)$, to be $0.005\ \mathrm{mag}$ ($0.015\ \mathrm{mag}$). This is similar to the uncertainty of existing dust maps, illustrating the feasibility of our method. Simultaneous estimation of large-scale structure is predicted to recover the galaxy overdensity $δ$ with a precision of $\sim0.01$ ($\sim0.05$) at $1^\circ$ ($7'$) resolution. We also introduce a Bayesian formalism that combines prior information from existing dust maps with the likelihood of Galactic dust extinction determined from the excursion of observed galaxy properties.

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arXiv2021

UniverseMachine: Predicting Galaxy Star Formation over Seven Decades of Halo Mass with Zoom-in Simulations

We apply the empirical galaxy--halo connection model UniverseMachine to dark matter-only zoom-in simulations of isolated Milky Way (MW)--mass halos along with their parent cosmological simulations. This application extends \textsc{UniverseMachine} predictions into the ultra-faint dwarf galaxy regime ($ 10^{2}\,\mathrm{M_{\odot}} \leqslant M_{\ast} \leqslant 10^{5}\,\mathrm{M_{\odot}}$) and yields a well-resolved stellar mass--halo mass (SMHM) relation over the peak halo mass range $10^8\,\mathrm{M_{\odot}}$ to $10^{15}\,\mathrm{M_{\odot}}$. The extensive dynamic range provided by the zoom-in simulations allows us to assess specific aspects of dwarf galaxy evolution predicted by \textsc{UniverseMachine}. In particular, although UniverseMachine is not constrained for dwarf galaxies with $M_* \lesssim 10^{8}\,\mathrm{M_{\odot}}$, our predicted SMHM relation is consistent with that inferred for MW satellite galaxies at $z=0$ using abundance matching. However, UniverseMachine predicts that nearly all galaxies are actively star forming below $M_{\ast}\sim 10^{7}\,\mathrm{M_{\odot}}$ and that these systems typically form more than half of their stars at $z\lesssim 4$, which is discrepant with the star formation histories of Local Group dwarf galaxies that favor early quenching. This indicates that the current UniverseMachine model does not fully capture galaxy quenching physics at the low-mass end. We highlight specific improvements necessary to incorporate environmental and reionization-driven quenching for dwarf galaxies, and provide a new tool to connect dark matter accretion to star formation over the full dynamic range that hosts galaxies.

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arXiv2021

ADDGALS: Simulated Sky Catalogs for Wide Field Galaxy Surveys

We present a method for creating simulated galaxy catalogs with realistic galaxy luminosities, broad-band colors, and projected clustering over large cosmic volumes. The technique, denoted ADDGALS (Adding Density Dependent GAlaxies to Lightcone Simulations), uses an empirical approach to place galaxies within lightcone outputs of cosmological simulations. It can be applied to significantly lower-resolution simulations than those required for commonly used methods such as halo occupation distributions, subhalo abundance matching, and semi-analytic models, while still accurately reproducing projected galaxy clustering statistics down to scales of r ~ 100 kpc/h. We show that \addgals\ catalogs reproduce several statistical properties of the galaxy distribution as measured by the Sloan Digital Sky Survey (SDSS) main galaxy sample, including galaxy number densities, observed magnitude and color distributions, as well as luminosity- and color-dependent clustering. We also compare to cluster-galaxy cross correlations, where we find significant discrepancies with measurements from SDSS that are likely linked to artificial subhalo disruption in the simulations. Applications of this model to simulations of deep wide-area photometric surveys, including modeling weak-lensing statistics, photometric redshifts, and galaxy cluster finding are presented in DeRose et al (2019), and an application to a full cosmology analysis of Dark Energy Survey (DES) Year 3 like data is presented in DeRose etl al (2021). We plan to publicly release a 10,313 square degree catalog constructed using ADDGALS with magnitudes appropriate for several existing and planned surveys, including SDSS, DES, VISTA, WISE, and LSST.

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arXiv2021

Modeling Redshift-Space Clustering with Abundance Matching

We explore the degrees of freedom required to jointly fit projected and redshift-space clustering of galaxies selected in three bins of stellar mass from the Sloan Digital Sky Survey Main Galaxy Sample (SDSS MGS) using a subhalo abundance matching (SHAM) model. We employ emulators for relevant clustering statistics in order to facilitate our analysis, leading to large speed gains with minimal loss of accuracy. We are able to simultaneously fit the projected and redshift-space clustering of the two most massive galaxy samples that we consider with just two free parameters: scatter in stellar mass at fixed SHAM proxy and the dependence of the SHAM proxy on dark matter halo concentration. We find some evidence for models that include velocity bias, but including orphan galaxies improves our fits to the lower mass samples significantly. We also model the clustering signals of specific star formation rate (SSFR) selected samples using conditional abundance matching (CAM). We obtain acceptable fits to projected and redshift-space clustering as a function of SSFR and stellar mass using two CAM variants, although the fits are worse than for stellar mass-selected samples alone. By incorporating non-unity correlations between the CAM proxy and SSFR we are able to resolve previously identified discrepancies between CAM predictions and SDSS observations of the environmental dependence of quenching for isolated central galaxies.

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arXiv2021

Hierarchical Inference With Bayesian Neural Networks: An Application to Strong Gravitational Lensing

In the past few years, approximate Bayesian Neural Networks (BNNs) have demonstrated the ability to produce statistically consistent posteriors on a wide range of inference problems at unprecedented speed and scale. However, any disconnect between training sets and the distribution of real-world objects can introduce bias when BNNs are applied to data. This is a common challenge in astrophysics and cosmology, where the unknown distribution of objects in our Universe is often the science goal. In this work, we incorporate BNNs with flexible posterior parameterizations into a hierarchical inference framework that allows for the reconstruction of population hyperparameters and removes the bias introduced by the training distribution. We focus on the challenge of producing posterior PDFs for strong gravitational lens mass model parameters given Hubble Space Telescope (HST) quality single-filter, lens-subtracted, synthetic imaging data. We show that the posterior PDFs are sufficiently accurate (i.e., statistically consistent with the truth) across a wide variety of power-law elliptical lens mass distributions. We then apply our approach to test data sets whose lens parameters are drawn from distributions that are drastically different from the training set. We show that our hierarchical inference framework mitigates the bias introduced by an unrepresentative training set's interim prior. Simultaneously, given a sufficiently broad training set, we can precisely reconstruct the population hyperparameters governing our test distributions. Our full pipeline, from training to hierarchical inference on thousands of lenses, can be run in a day. The framework presented here will allow us to efficiently exploit the full constraining power of future ground- and space-based surveys.

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arXiv2021

Mock Lightcones and Theory Friendly Catalogs for the CANDELS Survey

We present mock catalogs created to support the interpretation of the CANDELS survey. We extract halos along past lightcones from the Bolshoi Planck dissipationless N-body simulations and populate these halos with galaxies using two different independently developed semi-analytic models of galaxy formation and the empirical model UniverseMachine. Our mock catalogs have geometries that encompass the footprints of observations associated with the five CANDELS fields. In order to allow field-to-field variance to be explored, we have created eight realizations of each field. In this paper, we present comparisons with observable global galaxy properties, including counts in observed frame bands, luminosity functions, color-magnitude distributions and color-color distributions. We additionally present comparisons with physical galaxy parameters derived from SED fitting for the CANDELS observations, such as stellar masses and star formation rates. We find relatively good agreement between the model predictions and CANDELS observations for luminosity and stellar mass functions. We find poorer agreement for colors and star formation rate distributions. All of the mock lightcones as well as curated "theory friendly" versions of the observational CANDELS catalogs are made available through a web-based data hub.

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arXiv2020

The SAGA Survey. II. Building a Statistical Sample of Satellite Systems around Milky Way-like Galaxies

We present the Stage II results from the ongoing Satellites Around Galactic Analogs (SAGA) Survey. Upon completion, the SAGA Survey will spectroscopically identify satellite galaxies brighter than $ M_{r,o} = -12.3 $ around 100 Milky Way (MW) analogs at $ z \sim 0.01 $. In Stage II, we have more than quadrupled the sample size of Stage I, delivering results from 127 satellites around 36 MW analogs with an improved target selection strategy and deep photometric imaging catalogs from the Dark Energy Survey and the Legacy Surveys. We have obtained 25,372 galaxy redshifts, peaking around $ z = 0.2 $. These data significantly increase spectroscopic coverage for very low redshift objects in $ 17 < r_o < 20.75 $ around SAGA hosts, creating a unique data set that places the Local Group in a wider context. The number of confirmed satellites per system ranges from zero to nine, and correlates with host galaxy and brightest satellite luminosities. We find that the number and the luminosities of MW satellites are consistent with being drawn from the same underlying distribution as SAGA systems. The majority of confirmed SAGA satellites are star forming, and the quenched fraction increases as satellite stellar mass and projected radius from the host galaxy decrease. Overall, the satellite quenched fraction among SAGA systems is lower than that in the Local Group. We compare the luminosity functions and radial distributions of SAGA satellites with theoretical predictions based on cold dark matter simulations and an empirical galaxy-halo connection model and find that the results are broadly in agreement.

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arXiv2020

RedMaPPer: Evolution and Mass Dependence of the Conditional Luminosity Functions of Red Galaxies in Galaxy Clusters

We characterize the luminosity distribution, halo mass dependence, and redshift evolution of red galaxies in galaxy clusters using the SDSS Data Release 8 RedMaPPer cluster sample. We propose a simple prescription for the relationship between the luminosity of both central and satellite galaxies and the mass of their host halos, and show that this model is well-fit by the data. Using a larger galaxy cluster sample than previously employed in the literature, we find that the luminosities of central galaxies scale as $\langle \log L \rangle \propto A_L \log (M_{200b})$, with $A_L=0.39\pm0.04$, and that the scatter of the central--galaxy luminosity at fixed $M_{200b}$ ( $σ_{\log L|M}$) is $0.23 ^{+0.05}_{-0.04}$ dex, with the error bar including systematics due to miscentering of the cluster finder, photometry, and photometric redshift estimation. Our data prefers a positive correlation between the luminosity of central galaxies and the observed richness of clusters at a fixed halo mass, with an effective correlation coefficient $d_{\rm{eff}}=0.36^{+0.17}_{-0.16}$. The characteristic luminosity of satellites becomes dimmer from $z=0.3$ to $z=0.1$ by $\sim 20\%$ after accounting for passive evolution. We estimate the fraction of galaxy clusters where the brightest galaxy is not the central to be $P_{\rm{BNC}} \sim 20\%$. We discuss implications of these findings in the context of galaxy evolution and the galaxy--halo connection.

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arXiv2020

Preliminary Target Selection for the DESI Milky Way Survey (MWS)

The DESI Milky Way Survey (MWS) will observe $\ge$8 million stars between $16 < r < 19$ mag, supplemented by observations of brighter targets under poor observing conditions. The survey will permit an accurate determination of stellar kinematics and population gradients; characterize diffuse substructure in the thick disk and stellar halo; enable the discovery of extremely metal-poor stars and other rare stellar types; and improve constraints on the Galaxy's 3D dark matter distribution from halo star kinematics. MWS will also enable a detailed characterization of the stellar populations within 100 pc of the Sun, including a complete census of white dwarfs. The target catalog from the preliminary selection described here is public.

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arXiv2020

Line confusion in spectroscopic surveys and its possible effects: Shifts in Baryon Acoustic Oscillations position

Roman Space Telescope will survey about 17 million emission-line galaxies over a range of redshifts. Its main targets are H$α$ emission-line galaxies at low redshifts and [O III] emission-line galaxies at high redshifts. The Roman Space Telescope will estimate the redshift these galaxies with single line identification. This suggests that other emission-line galaxies may be misidentified as the main targets. In particular, it is hard to distinguish between the H$β$ and [O III] lines as the two lines are close in wavelength and hence the photometric information may not be sufficient to separate them reliably. Misidentifying H$β$ emitter as [O III] emitter will cause a shift in the inferred radial position of the galaxy by approximately 90 Mpc/h. This length scale is similar to the Baryon Acoustic Oscillation (BAO) scale and could shift and broaden the BAO peak, possibly introduce errors in determining the BAO peak position. We qualitatively describe the effect of this new systematic and further quantify it with a lightcone simulation with emission-line galaxies.

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arXiv2020

Concentrations of Dark Haloes Emerge from Their Merger Histories

The concentration parameter is a key characteristic of a dark matter halo that conveniently connects the halo's present-day structure with its assembly history. Using 'Dark Sky', a suite of cosmological $N$-body simulations, we investigate how halo concentration evolves with time and emerges from the mass assembly history. We also explore the origin of the scatter in the relation between concentration and assembly history. We show that the evolution of halo concentration has two primary modes: (1) smooth increase due to pseudo-evolution; and (2) intense responses to physical merger events. Merger events induce lasting and substantial changes in halo structures, and we observe a universal response in the concentration parameter. We argue that merger events are a major contributor to the uncertainty in halo concentration at fixed halo mass and formation time. In fact, even haloes that are typically classified as having quiescent formation histories experience multiple minor mergers. These minor mergers drive small deviations from pseudo-evolution, which cause fluctuations in the concentration parameters and result in effectively irreducible scatter in the relation between concentration and assembly history. Hence, caution should be taken when using present-day halo concentration parameter as a proxy for the halo assembly history, especially if the recent merger history is unknown.

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arXiv2020

Probing galaxy evolution in massive clusters using ACT and DES: splashback as a cosmic clock

We measure the projected number density profiles of galaxies and the splashback feature in clusters selected by the Sunyaev--Zeldovich (SZ) effect from the Advanced Atacama Cosmology Telescope (AdvACT) survey using galaxies observed by the Dark Energy Survey (DES). The splashback radius for the complete galaxy sample is consistent with theoretical measurements from CDM-only simulations, and is located at $2.4^{+0.3}_{-0.4}$ Mpc $h^{-1}$. We split the sample based on galaxy color and find significant differences in the profile shapes. Red galaxies and those in the green valley show a splashback-like minimum in their slope profile consistent with theoretical predictions, while the bluest galaxies show a weak feature that appears at a smaller radius. We develop a mapping of galaxies to subhalos in $N$-body simulations by splitting subhalos based on infall time onto the cluster halos. We find that the location of the steepest slope and differences in the shapes of the profiles can be mapped to differences in the average time of infall of galaxies of different colors. The minima of the slope in the galaxy profiles trace a discontinuity in the phase space of dark matter halos. By relating spatial profiles to infall time for galaxies of different colours, we can use splashback as a clock to understand galaxy quenching. We find that red galaxies have on average been in their clusters for over $3.2 ~\rm Gyrs$, green galaxies about $2.2 ~\rm Gyrs$, while blue galaxies have been accreted most recently and have not reached apocenter. Using the information from the complete radial profiles, we fit a simple quenching model and find that the onset of galaxy quenching in clusters occurs after a delay of about a gigayear, and that galaxies quench rapidly thereafter with an exponential timescale of $0.6$ Gyr.

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arXiv2020

The Universe at z>10: Predictions for JWST from the UniverseMachine DR1

The James Webb Space Telescope (JWST) is expected to observe galaxies at $z>10$ that are presently inaccessible. Here, we use a self-consistent empirical model, the UniverseMachine, to generate mock galaxy catalogues and lightcones over the redshift range $z=0-15$. These data include realistic galaxy properties (stellar masses, star formation rates, and UV luminosities), galaxy-halo relationships, and galaxy-galaxy clustering. Mock observables are also provided for different model parameters spanning observational uncertainties at $z<10$. We predict that Cycle 1 JWST surveys will very likely detect galaxies with $M_*>10^7 M_\odot$ and/or $M_{1500}<-17$ out to at least $z\sim 13.5$. Number density uncertainties at $z>12$ expand dramatically, so efforts to detect $z>12$ galaxies will provide the most valuable constraints on galaxy formation models. The faint-end slopes of the stellar mass/luminosity functions at a given mass/luminosity threshold steepen as redshift increases. This is because observable galaxies are hosted by haloes in the exponentially falling regime of the halo mass function at high redshifts. Hence, these faint-end slopes are robustly predicted to become shallower below current observable limits ($M_\ast < 10^7M_\odot$ or $M_\mathrm{1500}>-17$). For reionization models, extrapolating luminosity functions with a constant faint-end slope from $M_{1500}=-17$ down to $M_{1500}=-12$ gives the most reasonable upper limit for the total UV luminosity and cosmic star formation rate up to $z\sim 12$. We compare to three other empirical models and one semi-analytic model, showing that the range of predicted observables from our approach encompasses predictions from other techniques. Public catalogues and lightcones for common fields are available online.

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arXiv2020

Signatures of Velocity-Dependent Dark Matter Self-Interactions in Milky Way-mass Halos

We explore the impact of elastic, anisotropic, velocity-dependent dark matter (DM) self-interactions on the host halo and subhalos of Milky Way (MW)--mass systems. We consider a generic self-interacting dark matter (SIDM) model parameterized by the masses of a light mediator and the DM particle. The ratio of these masses, $w$, sets the velocity scale above which momentum transfer due to DM self-interactions becomes inefficient. We perform high-resolution zoom-in simulations of an MW-mass halo for values of $w$ that span scenarios in which self-interactions either between the host and its subhalos or only within subhalos efficiently transfer momentum, and we study the effects of self-interactions on the host halo and on the abundance, radial distribution, orbital dynamics, and density profiles of subhalos in each case. The abundance and properties of surviving subhalos are consistent with being determined primarily by subhalo--host halo interactions. In particular, subhalos on radial orbits in models with larger values of the cross section at the host halo velocity scale are more susceptible to tidal disruption owing to mass loss from ram pressure stripping caused by self-interactions with the host. This mechanism suppresses the abundance of surviving subhalos relative to collisionless DM simulations, with stronger suppression for larger values of $w$. Thus, probes of subhalo abundance around MW-mass hosts can be used to place upper limits on the self-interaction cross section at velocity scales of $\sim 200\ \rm{km\ s}^{-1}$, and combining these measurements with the orbital properties and internal dynamics of subhalos may break degeneracies among velocity-dependent SIDM models.

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arXiv2020

Constraints on Dark Matter Microphysics from the Milky Way Satellite Population

Alternatives to the cold, collisionless dark matter (DM) paradigm in which DM behaves as a collisional fluid generically suppress small-scale structure. Herein we use the observed population of Milky Way (MW) satellite galaxies to constrain the collisional nature of DM, focusing on DM-baryon scattering. We first derive analytic upper limits on the velocity-independent DM-baryon scattering cross section by translating the upper bound on the lowest mass of halos inferred to host satellites into a characteristic cutoff scale in the linear matter power spectrum. We then confirm and improve these results through a detailed probabilistic inference of the MW satellite population that marginalizes over relevant astrophysical uncertainties. This yields $95\%$ confidence upper limits on the DM-baryon scattering cross section of $2\times10^{-29}\ \rm{cm}^2$ ($6\times 10^{-27}\ \rm{cm}^2$) for DM particle masses $m_χ$ of~$10\ \rm{keV}$ ($10\ \rm{GeV}$); these limits scale as $m_χ^{1/4}$ for $m_χ\ll 1\ \rm{GeV}$ and $m_χ$ for~$m_χ\gg 1\ \rm{GeV}$. This analysis improves upon cosmological bounds derived from cosmic-microwave-background anisotropy measurements by multiple orders of magnitude over a wide range of DM masses, excluding regions of parameter space previously unexplored by other methods, including direct-detection experiments. Our work reveals a mapping between DM-baryon scattering and other alternative DM models, and we discuss the implications of our results for warm and fuzzy DM scenarios.

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arXiv2020

The binary-host connection: astrophysics of gravitational wave binaries from their host galaxy properties

Gravitational waves produced from the merger of binary neutron stars (BNSs) are accompanied by electromagnetic counterparts, making it possible to identify the associated host galaxy. We explore how properties of the host galaxies relate to the astrophysical processes leading to the mergers. It is thought that the BNS merger rate within a galaxy at a given epoch depends primarily on the galaxy's star-formation history as well as the underlying merger time-delay distribution of the binary systems. The stellar history of a galaxy, meanwhile, depends on the cosmological evolution of the galaxy through time, and is tied to the growth of structure in the Universe. We study the hosts of BNS mergers in the context of structure formation by populating the Universe Machine simulations with gravitational-wave events~ according to a simple time-delay model. We find that different time-delay distributions predict different properties of the associated host galaxies, including the distributions of stellar mass, star-formation rate, halo mass, and local and large-scale clustering of hosts. BNSs that merge today with short delay times prefer to be in hosts that have high star-formation rates, while those with long delay times live in dense regions within massive halos that have low star formation. We show that with ${\mathcal O}(10)$ events from current gravitational-wave detector networks, it is possible to make preliminary distinctions between formation channels which trace stellar mass, halo mass, or star-formation rate. We also find that strategies to follow up gravitational-wave events with electromagnetic telescopes can be significantly optimized using the clustering properties of their hosts.

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arXiv2019

Probing the galaxy-halo connection with total satellite luminosity

We demonstrate how the total luminosity in satellite galaxies is a powerful probe of dark matter halos around central galaxies. The method cross-correlates central galaxies in spectroscopic galaxy samples with fainter galaxies detected in photometric surveys. After background subtraction, the excess galaxies around the central galaxies represent faint satellite galaxies within the dark matter halo. Using abundance matching models, we show that the the total galaxy luminosity, L_sat, scales linearly with host halo mass, making L_sat an excellent proxy for M_h. L_sat is also sensitive to the formation time of the halo, as younger halos have more substructure at fixed M_h. We demonstrate that probes of galaxy large-scale environment can break this degeneracy. Although this is an indirect probe of the halo, it can yield a high-S/N measurement for galaxies expected to occupy halos at $<10^{12}$ M_sol, where other methods suffer from larger errors. In this paper we focus on observational and theoretical systematics in the L_sat method. We test the robustness of our method of finding central galaxies and our methods of estimating the number of background galaxies. We implement this method on central galaxies in SDSS data, with satellites identified in faint imaging from the DESI Legacy Imaging Surveys. We find excellent agreement between our theoretical predictions and the observational measurements. Finally, we compare our L_sat measurements to weak lensing estimates of M_h for red and blue subsamples. In the stellar mass range where the measurements overlap, we find consistent results, where red galaxies live in larger halos. However, the L_sat approach allows us to probe significantly lower mass galaxies. At these masses, the L_sat values are equivalent. This example shows the potential of L_sat as a probe of dark halos. (Abridged)

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