UCSC Cosmology & Galaxy Research Group 2017 – Prof. …physics.ucsc.edu/~joel/DEEP-Theory 23Jan2017.pdf · · 2017-04-15UCSC Cosmology & Galaxy Research Group 2017 – Prof. Joel

UCSC Cosmology & Galaxy Research Group 2017 – Prof. Joel Primack UCSC Grad Students: Christoph Lee [email protected] 707-338-9543: Galaxy-Halo Connections, LSS,

Halo Properties vs. Density, Stripping, SAM, Deep Learning & Galaxies Viraj Pandya [email protected] 831-459-5722 (ISB 255): Galaxies and Cosmology

SIP students supervised by Christoph & Joel: Austin Tuan (Phillips Academy, Andover, MA) [email protected] 408-831-8787: How

Does Halo Radial Profile depend on stripping, environment, and other halo properties? Max Untrecht (Woodside HS) [email protected]: Halo Properties vs. Mass & Web

Undergraduate Astrophysics Students Radu Dragomir (UCSC) [email protected]: Galaxy Properties from SDSS, Abundance

Matching Dependence on Environment Elliot Eckholm (UCSC) [email protected] 619-993-2120: 3D Viz of Cosmic Web & Halos Sean Larkin (UCSC) [email protected] 949-439-7775: Deep Learning & Galaxy Simulations Yifei Luo (Nanjing U, China) [email protected]: Identifying SDSS PDGs (Disk

Galaxies Without Classical Bulges) and Determining Their Mass, Environment, etc. Tze Goh (Columbia) [email protected]: Halo & Galaxy Properties in Small Cosmic Walls

(Like Milky Way and Andromeda) and Other Cosmic Web Environments Graham Vanbenthuysen (UCSC) [email protected]: Galaxy Size vs. Local Density

Other Research Group Members Miguel Aragon-Calvo (UNAM-Ensenada) [email protected]: Spine of the

Cosmic Web of Bolshoi-Planck simulation and SDSS Peter Behroozi (UC Berkeley) [email protected]: Galaxies, Galaxy-Halo Connection Doug Hellinger [email protected]: DM Density & V per Voxel, Voids, Protoclusters Nina McCurdy (Scientific Computing and Imaging Institute, U Utah) [email protected]:

Visualizing Forming Galaxies & Dark Matter Aldo Rodriguez-Puebla (UNAM) [email protected]: LSS, Galaxies Paul Sutter (Ohio State) [email protected]: Voids in SDSS and in Simulations Vivian Tang (UCSC) [email protected]: Shapes of Simulated Galaxies

Dekel Research Group, Hebrew University, Jerusalem

Avishai Dekel (HU) [email protected]: Galaxy Formation Santi Roca-Fabrega (HU postdoc) [email protected]: galaxy simulations, and streams Jonathan Freundlich (starting HU postdoc) observations of gas in high-z galaxies, theory Hangzhou Jiang (starting HU postdoc) working on halos and subhalos Nir Mandelker (Yale postdoc) [email protected]: galaxyVDI, stream instability

UCSC Observational Galaxy Research Group

Guillermo Barro (Berkeley) [email protected]: Galaxy Formation & Evolution, Compaction Zhu Chen [email protected]: Galaxy Formation & Evolution Sandra Faber [email protected]: Galaxy Formation & Evolution Yicheng Guo [email protected]: Galaxy Formation & Evolution, Clumps Marc Huertas-Company (Observatoire de Paris) [email protected]: Galaxy Image Analysis

with Deep Learning David Koo [email protected]: Galaxy Formation & Evolution Hassen Yesuf [email protected]: Galaxy Formation & Evolution, Winds

DEEP-Theory Meeting 16 & 23 Jan 2017TOPICS TODAY

DM halo properties vs. density paper in press; halo stripping and halo radial profile papers being drafted (with Christoph Lee, Doug Hellinger)

Galaxy Reff = Const (spin parameter)(halo radius) predicts smaller Reff in regions of low environmental density.How to measure Reff?

Constraining the Galaxy Halo Connection: Star Formation Histories, Galaxy Mergers, and Structural Properties, by Aldo Rodriguez-Puebla, Joel, and others (in nearly final form)

Abundance Matching is Independent of Cosmic Environment Density, based on Radu Dragomir’s UCSC senior thesis, advised by Aldo and Joel (we’re drafting this now)

Analyzing VELA mock images for clumps (Yicheng Guo); measuring GALFIT statistics a, b, axis ratio b/a, Sersic index of CANDELized images (Yicheng and Vivian Tang) compared with high resolution images (Liz McGrath). Reff for SDSS galaxies as a function of density (Christoph, Graham Vanbenthuysen).

Preparing information for deep learning (DL) about the simulations using yt analysis of the saved timesteps (Sean Larkin, Fernando Caro, Christoph Lee) and using other methods (Nir Mandelker, Santi Roca-Fabrega) to see whether giving the deep learning code this information in addition to mock images will allow the code to determine some of these phenomena from the images at least in the best cases of inclination, resolution, and signal/noise (Marc Huertas-Company and team). What data about the simulations should we give DL? Can we make sufficient progress by HST Cycle 24 deadline April 8?

Galaxy Reff = Const (spin parameter)(halo radius) predicts smaller Reff in regions of low environmental density.How to measure Reff?

The galaxy data used in the new Somerville+2017 paper to measure r_*,3D came from GAMA Data Release 2, which gave 13,771 galaxies after cuts eliminating Sersic indexes n < 0.3 and n > 10 and eliminating galaxies with sizes r_e < 0.7 arc seconds, according to Section 3.1. Section 3.3 says that the conversion to r_*,3D from r_e,obs = the observed projected effective radius of the light in the same rest-frame waveband is given by

r_e,obs = f_p f_k r_*,3D

where f_p corrects for projection and and f_k is the structural k-correction. The paper quotes f_p = 1 for an edge-on disk, f_p = 0.68 for n = 4, and f_p = 0.61 for n = 1. It summarizes the literature as saying f_k ~ 1.12 to 1.5. The paper says it adopts (f_pf_k)_disk = (1x1.2) = 1.2 and (f_pf_k)_spheroid = (0.68 x 1.15) = 0.78 for spheroids.

Viraj, could you please clarify what rest-frame waveband was used in the Somerville+2017 paper? Presumably the reason you say you need the half-light radius in all 5 SDSS bands u, g, r, i, z is to convert to a fixed rest-frame band, right? Since we are only going out to z ~ 0.15, we may not need to use more than two wavebands. Viraj, what source do you suggest we use for the SDSS data we need? (We can also use the GAMA data, but there may not be enough galaxies once we separate into mass and density bins. Still, it would be good to check that we recover the same results in the Somerville+2017 paper using the GAMA data, and bin at least the lower-mass galaxies into a few density bins to see if there is an offset to smaller sizes at lower densities.)

Aldo has suggested using the GIM2D catalog based on SDSS DR7 by Luc Simard+2011 (ApJS 196, 11 with machine-readable tables online at http://vizier.cfa.harvard.edu/viz-bin/VizieR?-source=J/ApJS/196/11 and in http://adsabs.harvard.edu/abs/2011yCat..21960011S). The Simard tables include three different determinations of half-light radii (both a = semimajor axis half-light radius and circularized half-light radius = \sqrt(ab), where b is the minor axis half-light radius), one each for g and r band. Simard also gives the ellipticity e = 1 - b/a for each band. Simard’s Table 1 uses n_b = 4 and n_disk free, Table 2 uses both n_bulge and n_disk free, and Table 3 is a single-Sersic fit. Aldo, were you suggesting using Table 3, or what? Viraj, where should we get the spheroid vs. disk vs. edge-on disk data from? To get the R_halo for each galaxy, the Somerville+2017 paper uses the Behroozi-Wechsler-Conroy 2013 stellar halo mass relation (SHMR) to assign a stellar mass to every halo in the z = 0.1 halo catalog from the Bolshoi-Planck simulation. (As I emphasized to Rachel, this is inconsistent since BWC13 was based on the Bolshoi simulation with WMAP5/7 cosmological parameters, while Bolshoi-Planck used the rather different Planck parameters which lead to 20-40% more halos at the same Vmax. I thought we had agreed to wait for the updated Planck SHMR which Peter Behroozi promised to send soon, but Rachel instead submitted the paper with this inconsistent use of cosmological parameters; perhaps we can fix this when we respond to the referee. But in our new paper on halo radius vs. environmental density we can consistently use the Planck parameters, using either Peter’s new SHMR or the one from the new paper that Aldo is leading.)

Let’s discuss this by email and also at the DEEP-Theory meeting 3-5 pm today (Monday 1/16) in the CfAO Conference Room.

Joel

On Jan 13, 2017, at 6:42 PM, Viraj Pandya <[email protected]> wrote:Hi Joel,Here is a list of things I’d need from the SDSS database to estimate the 3D half-mass radii of galaxies. The first two are crucial, and the last two might come in handy but aren’t 100% necessary right now:

1 half-light radius in u, g, r, i and z bands [necessary]2 Sersic index in all 5 bands if available, else only in r-band (which SDSS galaxies are selected in) [necessary]3 axis ratio q:=b/a where b=semi-minor axis and a=semi-major axis [only if easily available]4 absolute magnitudes in u, g, r, i and z bands [only if easily available]

Viraj

http://vizier.cfa.harvard.edu/viz-bin/VizieR?-source=J/ApJS/196/11

http://vizier.cfa.harvard.edu/viz-bin/VizieR?-source=J/ApJS/196/11

http://adsabs.harvard.edu/abs/2011yCat..21960011S

mailto:[email protected]

Mon. Not. R. Astron. Soc. 000, 1–?? (20??) Printed 15 January 2017 (MN LATEX style file v2.2)

Constraining the Galaxy-Halo Connection Over The Last13.3 Gyrs: Star Formation Histories, Galaxy Mergers andStructural Properties

Aldo Rodrıguez-Puebla1,2⋆, Joel Primack3 et al.1 Department of Astronomy & Astrophysics, University of California at Santa Cruz, Santa Cruz, CA 95064, USA2 Instituto de Astronomıa, Universidad Nacional Autonoma de Mexico, A. P. 70-264, 04510, Mexico, D.F., Mexico3Physics Department, University of California, Santa Cruz, CA 95064, USA

Released 20?? Xxxxx XX

ABSTRACT

We present new determinations of the stellar-to-halo mass relation at z = 0 − 10that match the evolution of the galaxy stellar mass function, the SFR − M∗ rela-tion, and the cosmic star formation rate. We utilize a large compilation of more than40 observational studies from the literature and correct them for potential biasesthat could affect our determinations. Using our robust determinations of the SHMRand the halo mass assembly, we study the star formation histories, merger rates andthe structural properties of galaxies. Our findings are: (1) The transition halo massabove/below which galaxies are observed to be statistically star-forming/quenched iswhen sSFR/sMAR∼ 1, where sMAR is the specific halo mass accretion rate. (2) Thistransition halo mass depends on redshift, at z ∼ 0 it is Mvir ∼ 1012M⊙ while at z ∼ 3it is Mvir ∼ 1013M⊙, presumably due to cold streams being more efficient at highredshift while virial shocks became more relevant at lower redshifts, as theoreticallyexpected. (3) Unexpectedly, the ratio sSFR/sMAR has a peak value, which occursaround Mvir ∼ 2×1011M⊙. (4) The mass density within 1 kpc, Σ1, is a good indicatorof the global sSFR. (5) galaxies are statistically quenched once they reached a max-imum in Σ1, consistent with theoretical expectations of the gas compaction model.(6) This maximum of Σ1 depends on redshift. (7) Galaxies grow primarily due toin-situ star formation but massive galaxies could have assembled ∼ 20% of their massthrough galaxy mergers. (8) While minor mergers are more frequent, major mergershave contributed ∼ 80% of the accreted stellar mass in massive galaxies but only∼ 40% in MW sized galaxies. (9) The marked change in the slope of the size–massrelation when galaxies became quenched, from d log Reff/d log M∗ ∼ 0.35 to ∼ 2.5,could be the result of dry minor mergers.

Key words: galaxies: evolution - galaxies: haloes - galaxies: luminosity function -galaxies: mass function - galaxies: star formation - cosmology: theory

1 INTRODUCTION

There is no doubt about the remarkable recent progress inassembling large galaxy samples from multiwavelength skysurveys. Moreover, these advances are not just for obser-vations of local galaxies but also for very distant galaxies,resulting in reliable samples which contain hundreds of thou-sands of galaxies at z ∼ 0.1, tens of thousands betweenz ∼ 0.2 − 4, hundreds of galaxies between z ∼ 6 − 8 and

⋆ [email protected]

few tens of galaxies confirmed as distant as z ∼ 9 − 10.1

Thus, statistical analyses of the properties of the galaxiesare now possible with unprecedented detail over a wide red-shift range. Naturally, we have benefited from these advancesby having robust determinations of the luminosity functions(LF) for a very wide redshifts range.

In parallel, substantial progress has been made on stel-lar population synthesis (SPS) modelling (for a recent re-view, see Conroy 2013), allowing the determination of phys-

1 These achievements are all the more impressive when one real-izes that the Universe was only ∼ 500 Myrs old at z = 10.

c⃝ 20?? RAS

PREVIEW





ABSTRACT

We present new determinations of the stellar-to-halo mass relation at z = 0 − 10that match the evolution of the galaxy stellar mass function, the SFR − M∗ rela-tion, and the cosmic star formation rate. We utilize a large compilation of more than40 observational studies from the literature and correct them for potential biasesthat could affect our determinations. Using our robust determinations of the SHMRand the halo mass assembly, we study the star formation histories, merger rates andthe structural properties of galaxies. Our findings are: (1) The transition halo massabove/below which galaxies are observed to be statistically star-forming/quenched iswhen sSFR/sMAR∼ 1, where sMAR is the specific halo mass accretion rate. (2) Thistransition halo mass depends on redshift, at z ∼ 0 it is Mvir ∼ 1012M⊙ while at z ∼ 3it is Mvir ∼ 1013M⊙, presumably due to cold streams being more efficient at highredshift while virial shocks became more relevant at lower redshifts, as theoreticallyexpected. (3) Unexpectedly, the ratio sSFR/sMAR has a peak value, which occursaround Mvir ∼ 2×1011M⊙. (4) The mass density within 1 kpc, Σ1, is a good indicatorof the global sSFR. (5) galaxies are statistically quenched once they reached a max-imum in Σ1, consistent with theoretical expectations of the gas compaction model.(6) This maximum of Σ1 depends on redshift. (7) Galaxies grow primarily due toin-situ star formation but massive galaxies could have assembled ∼ 20% of their massthrough galaxy mergers. (8) While minor mergers are more frequent, major mergershave contributed ∼ 80% of the accreted stellar mass in massive galaxies but only∼ 40% in MW sized galaxies. (9) The marked change in the slope of the size–massrelation when galaxies became quenched, from d log Reff/d log M∗ ∼ 0.35 to ∼ 2.5,could be the result of dry minor mergers.


1 INTRODUCTION







c⃝ 20?? RAS





ABSTRACT

We present new determinations of the stellar-to-halo mass relation at z = 0 − 10that match the evolution of the galaxy stellar mass function, the SFR − M∗ rela-tion, and the cosmic star formation rate. We utilize a large compilation of more than40 observational studies from the literature and correct them for potential biasesthat could affect our determinations. Using our robust determinations of the SHMRand the halo mass assembly, we study the star formation histories, merger rates andthe structural properties of galaxies. Our findings are: (1) The transition halo massabove/below which galaxies are observed to be statistically star-forming/quenched iswhen sSFR/sMAR∼ 1, where sMAR is the specific halo mass accretion rate. (2) Thistransition halo mass depends on redshift, at z ∼ 0 it is Mvir ∼ 1012M⊙ while at z ∼ 3it is Mvir ∼ 1013M⊙, presumably due to cold streams being more efficient at highredshift while virial shocks became more relevant at lower redshifts, as theoreticallyexpected. (3) Unexpectedly, the ratio sSFR/sMAR has a peak value, which occursaround Mvir ∼ 2×1011M⊙. (4) The mass density within 1 kpc, Σ1, is a good indicatorof the global sSFR. (5) galaxies are statistically quenched once they reached a max-imum in Σ1, consistent with theoretical expectations of the gas compaction model.(6) This maximum of Σ1 depends on redshift. (7) Galaxies grow primarily due toin-situ star formation but massive galaxies could have assembled ∼ 20% of their massthrough galaxy mergers. (8) While minor mergers are more frequent, major mergershave contributed ∼ 80% of the accreted stellar mass in massive galaxies but only∼ 40% in MW sized galaxies. (9) The marked change in the slope of the size–massrelation when galaxies became quenched, from d logReff/d logM∗ ∼ 0.35 to ∼ 2.5,could be the result of dry minor mergers.


1 INTRODUCTION







c⃝ 20?? RAS

Constraining the Galaxy Halo Connection: Star Formation Histories, Galaxy Mergers, and Structural Properties, by Aldo Rodriguez-Puebla, Joel, and others (in nearly final form) PREVIEW8

Table 1. Observational data on the galaxy stellar mass function

Author Redshifta Ω [deg2] Corrections

Bell et al. (2003) z ∼ 0.1 462 I+SP+CYang, Mo & van den Bosch (2009a) z ∼ 0.1 4681 I+SP+CLi & White (2009) z ∼ 0.1 6437 I+P+CBernardi et al. (2010) z ∼ 0.1 4681 I+SP+CBernardi et al. (2013) z ∼ 0.1 7748 I+SP+CRodriguez-Puebla et al. in prep z ∼ 0.1 7748 SDrory et al. (2009) 0 < z < 1 1.73 SP+CMoustakas et al. (2013) 0 < z < 1 9 SP+D+CPerez-Gonzalez et al. (2008) 0.2 < z < 2.5 0.184 I+SP+D+CTomczak et al. (2014) 0.2 < z < 3 0.0878 CIlbert et al. (2013) 0.2 < z < 4 2 CMuzzin et al. (2013) 0.2 < z < 4 1.62 I+CSantini et al. (2012) 0.6 < z < 4.5 0.0319 I+CMortlock et al. (2011) 1 < z < 3.5 0.0125 I+CMarchesini et al. (2009) 1.3 < z < 4 0.142 I+CStark et al. (2009) z ∼ 6 0.089 ILee et al. (2012) 3 < z < 7 0.089 I+SP+CGonzalez et al. (2011) 4 < z < 7 0.0778 I+CDuncan et al. (2014) 4 < z < 7 0.0778 CSong et al. (2015) 4 < z < 8 0.0778 IThis paper, Appendix D 4 < z < 10 0.0778 –

Notes: aIndicates the redshift used in this paper. I=IMF; P= photometry corrections; S=Surface Brightness correction; D=Dustmodel; NE= Nebular Emissions: SP = SPS Model: C = Cosmology.

ies (Bernardi et al. 2010, 2013, 2016) have found that themeasurements of the light profiles based on the standardSDSS pipeline photometry could be underestimated due tosky subtraction issues. This could result in a underestima-tion of the abundance of massive galaxies up to a factor offive. While new algorithms have been developed for obtain-ing more precise measurements of the sky subtraction andthus to improve the photometry (Blanton et al. 2011; Simardet al. 2011; Meert, Vikram & Bernardi 2015) there is not yeta consensus. For this paper, we decided to ignore this correc-tion that we may study in more detail in future works. Nev-ertheless, we apply photometric corrections to the GSMFreported in Li & White (2009). These authors used stellarmasses estimations based on the SDSS r−band Petrosianmagnitudes. It is well known that using Petrosian magni-tudes could result in a underestimation of the total light byan amount that could depend on the surface brightness pro-file of the galaxy and thus results in the underestimation ofthe total stellar mass. This will result in an artificial shiftof the GSMF towards lower masses. In order to account forthis shift for the Li & White (2009) GSMF, we apply a con-stant correction of 0.04 dex to all masses. As reported byGuo et al. (2010), this correction gives an accurate repre-sentation of the GSMF when the total light is considered,instead.

At z ∼ 0.1 we use the GSMF derived in Rodriguez-Puebla et al. (in prep.) that has been corrected for the frac-tion of missing galaxies due to surface brightness limits bycombining the SDSS NYU-VAGC low-redshift sample andthe SDSS DR7 based on the methodology described in Blan-ton et al. (2005b). Following Baldry et al. (2012), we correctthe GSMF for the distances based on Tonry et al. (2000). Wefound that including missing galaxies due to surface bright-ness incompleteness could increase the number of galaxies

up to a factor of ∼ 2 − 3 at the lowest masses, see FigureC1, and therefore have a direct impact on the SHMR.

4.1.3 The Evolution of the GSMF

Appendix D describes our inference of the GSMF from z ∼ 4to z ∼ 10. In short, we use several UV LFs reported in theliterature together with stellar mass-UV luminosity relationsfrom Duncan et al. (2014); Song et al. (2015); Dayal et al.(2014) to derive the evolution of the GSMF from z ∼ 4 toz ∼ 10. We assume a survey area of 0.0778 deg2s as in theCANDELS survey (e.g., Song et al. 2015).

Figure 2 shows the evolution of the GSMF from z ∼ 0.1to z ∼ 10. The filled circles show the mean of the ob-served GSMFs that we use through this paper in variousredshift bins, while the errors bars represent the propaga-tion of the individual errors from the GSMF. Alternatively,we also compute standard deviations from the set of GSMF.We calculated the mean, and the standard deviation of theobserved GSMFs by using the bootstrapping approach byresampling with replacement. We use the bootstrapping ap-proach since it will allow us to empirically derive the dis-tribution of current observations on the GSMFs and thusrobustly infer the mean evolution of the GSMFs. Method-ologically, we start by choosing various intervals in redshiftas indicated in the labels in Figure 2. For each redshiftbin, we create 30, 000 bootstrap samples based on the ob-served distribution of all the GSMFs for that redshift bin,φgobs

(M∗, z), and then compute the median and its cor-responding standard deviation from the distribution for agiven stellar mass interval.

A few features of the mean evolution of the observedGSMF are worth mentioning at this point. At high redshiftsthe GSMF is described by a Schechter function, as has been

c⃝ 20?? RAS, MNRAS 000, 1–??

8

Table 1. Observational data on the galaxy stellar mass function

Author Redshifta Ω [deg2] Corrections

Bell et al. (2003) z ∼ 0.1 462 I+SP+CYang, Mo & van den Bosch (2009a) z ∼ 0.1 4681 I+SP+CLi & White (2009) z ∼ 0.1 6437 I+P+CBernardi et al. (2010) z ∼ 0.1 4681 I+SP+CBernardi et al. (2013) z ∼ 0.1 7748 I+SP+CRodriguez-Puebla et al. in prep z ∼ 0.1 7748 SDrory et al. (2009) 0 < z < 1 1.73 SP+CMoustakas et al. (2013) 0 < z < 1 9 SP+D+CPerez-Gonzalez et al. (2008) 0.2 < z < 2.5 0.184 I+SP+D+CTomczak et al. (2014) 0.2 < z < 3 0.0878 CIlbert et al. (2013) 0.2 < z < 4 2 CMuzzin et al. (2013) 0.2 < z < 4 1.62 I+CSantini et al. (2012) 0.6 < z < 4.5 0.0319 I+CMortlock et al. (2011) 1 < z < 3.5 0.0125 I+CMarchesini et al. (2009) 1.3 < z < 4 0.142 I+CStark et al. (2009) z ∼ 6 0.089 ILee et al. (2012) 3 < z < 7 0.089 I+SP+CGonzalez et al. (2011) 4 < z < 7 0.0778 I+CDuncan et al. (2014) 4 < z < 7 0.0778 CSong et al. (2015) 4 < z < 8 0.0778 IThis paper, Appendix D 4 < z < 10 0.0778 –

Notes: aIndicates the redshift used in this paper. I=IMF; P= photometry corrections; S=Surface Brightness correction; D=Dustmodel; NE= Nebular Emissions: SP = SPS Model: C = Cosmology.

ies (Bernardi et al. 2010, 2013, 2016) have found that themeasurements of the light profiles based on the standardSDSS pipeline photometry could be underestimated due tosky subtraction issues. This could result in a underestima-tion of the abundance of massive galaxies up to a factor offive. While new algorithms have been developed for obtain-ing more precise measurements of the sky subtraction andthus to improve the photometry (Blanton et al. 2011; Simardet al. 2011; Meert, Vikram & Bernardi 2015) there is not yeta consensus. For this paper, we decided to ignore this correc-tion that we may study in more detail in future works. Nev-ertheless, we apply photometric corrections to the GSMFreported in Li & White (2009). These authors used stellarmasses estimations based on the SDSS r−band Petrosianmagnitudes. It is well known that using Petrosian magni-tudes could result in a underestimation of the total light byan amount that could depend on the surface brightness pro-file of the galaxy and thus results in the underestimation ofthe total stellar mass. This will result in an artificial shiftof the GSMF towards lower masses. In order to account forthis shift for the Li & White (2009) GSMF, we apply a con-stant correction of 0.04 dex to all masses. As reported byGuo et al. (2010), this correction gives an accurate repre-sentation of the GSMF when the total light is considered,instead.

At z ∼ 0.1 we use the GSMF derived in Rodriguez-Puebla et al. (in prep.) that has been corrected for the frac-tion of missing galaxies due to surface brightness limits bycombining the SDSS NYU-VAGC low-redshift sample andthe SDSS DR7 based on the methodology described in Blan-ton et al. (2005b). Following Baldry et al. (2012), we correctthe GSMF for the distances based on Tonry et al. (2000). Wefound that including missing galaxies due to surface bright-ness incompleteness could increase the number of galaxies

up to a factor of ∼ 2 − 3 at the lowest masses, see FigureC1, and therefore have a direct impact on the SHMR.

4.1.3 The Evolution of the GSMF

Appendix D describes our inference of the GSMF from z ∼ 4to z ∼ 10. In short, we use several UV LFs reported in theliterature together with stellar mass-UV luminosity relationsfrom Duncan et al. (2014); Song et al. (2015); Dayal et al.(2014) to derive the evolution of the GSMF from z ∼ 4 toz ∼ 10. We assume a survey area of 0.0778 deg2s as in theCANDELS survey (e.g., Song et al. 2015).

Figure 2 shows the evolution of the GSMF from z ∼ 0.1to z ∼ 10. The filled circles show the mean of the ob-served GSMFs that we use through this paper in variousredshift bins, while the errors bars represent the propaga-tion of the individual errors from the GSMF. Alternatively,we also compute standard deviations from the set of GSMF.We calculated the mean, and the standard deviation of theobserved GSMFs by using the bootstrapping approach byresampling with replacement. We use the bootstrapping ap-proach since it will allow us to empirically derive the dis-tribution of current observations on the GSMFs and thusrobustly infer the mean evolution of the GSMFs. Method-ologically, we start by choosing various intervals in redshiftas indicated in the labels in Figure 2. For each redshiftbin, we create 30, 000 bootstrap samples based on the ob-served distribution of all the GSMFs for that redshift bin,φgobs

(M∗, z), and then compute the median and its cor-responding standard deviation from the distribution for agiven stellar mass interval.

A few features of the mean evolution of the observedGSMF are worth mentioning at this point. At high redshiftsthe GSMF is described by a Schechter function, as has been

c⃝ 20?? RAS, MNRAS 000, 1–??

10

Table 2. Observational data on the star formation rates

Author Redshifta SFR Estimator Corrections Type

Chen et al. (2009) z ∼ 0.1 Hα/Hβ S AllSalim et al. (2007) z ∼ 0.1 UV SED S AllNoeske et al. (2007) 0.2 < z < 1.1 UV+IR S AllKarim et al. (2011) 0.2 < z < 3 1.4 GHz I+S+E AllDunne et al. (2009) 0.45 < z < 2 1.4 GHz I+S+E AllKajisawa et al. (2010) 0.5 < z < 3.5 UV+IR I AllWhitaker et al. (2014) 0.5 < z < 3 UV+IR I+S AllSobral et al. (2014) z ∼ 2.23 Hα I+S+SP SFReddy et al. (2012) 2.3 < z < 3.7 UV+IR I+S+SP SFMagdis et al. (2010) z ∼ 3 FUV I+S+SP SFLee et al. (2011) 3.3 < z < 4.3 FUV I+SP SFLee et al. (2012) 3.9 < z < 5 FUV I+SP SFGonzalez et al. (2012) 4 < z < 6 UV+IR I+NE SFSalmon et al. (2015) 4 < z < 6 UV SED I+NE+E SFBouwens et al. (2011) 4 < z < 7.2 FUV I+S SFDuncan et al. (2014) 4 < z < 7 UV SED I+NE SFShim et al. (2011) z ∼ 4.4 Hα I+S+SP SFSteinhardt et al. (2014) z ∼ 5 UV SED I+S SFGonzalez et al. (2010) z = 7.2 UV+IR I+NE SFThis paper, Appendix D 4 < z < 8 FUV I+E+NE SF

Notes aIndicates the redshift used in this paper. I=IMF; S=Star formation calibration; E=Extinction; NE= Nebular Emissions;SP=SPS Model

galaxies as a reference to compare with our model and thusgain more insights on how galaxies evolve from active topassive as well as on their structural evolution (discussed inSection 7). For the fraction of quiescent galaxies fQ we usethe following relation:

fQ(M∗, z) =1

1 + (M∗/Mchar(z))α, (44)

where Mchar is the transition stellar mass at which the frac-tions of blue star forming and red quenched galaxies are both50%. Figure 1 shows Mchar as a function of redshift fromobservations and previous constraints. The solid black lineshows the relation log(Mchar(z)/M⊙) = 10.2 + 0.6z that wewill employ in this paper, and the gray solid lines show theresults when shifting (Mchar(z)/M⊙) by 0.1 dex above andbelow. We will use this shift as our uncertainty in the def-inition for log(Mchar(z)/M⊙). The red (blue) curves in thefigure show the stellar mass vs. redshift where 75% (25%) ofthe galaxies are quenched.

Finally, we will assume that α = −1.3. The transitionstellar mass is such that at z = 0 log(Mchar(z)/M⊙) = 10.2and at z = 2 log(Mchar(z)/M⊙) = 11.4.

5 CONSTRAINING THE MODEL

The galaxy population in our model is described by fourproperties: halo mass Mvir, halo mass accretion rates, stel-lar mass M∗, and star formation rate SFR. In order to con-strain the model we combine several observational data sets,including the GSMFs, the SFRs and the CSFR for all galax-ies. In this Section we describe our adopted methodology aswell as the best resulting fit parameters in our model.

In order to sample the best-fit parameters that maxi-

mize the likelihood function L ∝ e−χ2/2 we use the MCMC

Figure 2. Redshift evolution from z ∼ 0.1 to z ∼ 10 of thegalaxy stellar mass function (GSMF) derived by using 20 ob-servational samples from the literature and represented with thefilled circles with error bars. The various GSMFs have been cor-rected for potential systematics that could affect our results, seethe text for details. Solid lines are the best fit model from a set of3×105 MCMC models. These fits take into account uncertaintiesaffecting the GSMF as discussed in the text. Note that at lowerredshifts (z <

∼ 3) galaxies tend to pile up at M∗ ∼ 3 × 1010M⊙

due to the increase in the number of massive quenched galaxiesat lower redshifts.

approach, described in detail in Rodrıguez-Puebla, Avila-Reese & Drory (2013).

We compute the total χ2 as,

χ2 = χ2GSMF + χ2

SFR + χ2CSFR (45)

where for the GSMFs we define

χ2GSMF =

X

j,i

χ2φj,i

, (46)

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10



Chen et al. (2009) z ∼ 0.1 Hα/Hβ S AllSalim et al. (2007) z ∼ 0.1 UV SED S AllNoeske et al. (2007) 0.2 < z < 1.1 UV+IR S AllKarim et al. (2011) 0.2 < z < 3 1.4 GHz I+S+E AllDunne et al. (2009) 0.45 < z < 2 1.4 GHz I+S+E AllKajisawa et al. (2010) 0.5 < z < 3.5 UV+IR I AllWhitaker et al. (2014) 0.5 < z < 3 UV+IR I+S AllSobral et al. (2014) z ∼ 2.23 Hα I+S+SP SFReddy et al. (2012) 2.3 < z < 3.7 UV+IR I+S+SP SFMagdis et al. (2010) z ∼ 3 FUV I+S+SP SFLee et al. (2011) 3.3 < z < 4.3 FUV I+SP SFLee et al. (2012) 3.9 < z < 5 FUV I+SP SFGonzalez et al. (2012) 4 < z < 6 UV+IR I+NE SFSalmon et al. (2015) 4 < z < 6 UV SED I+NE+E SFBouwens et al. (2011) 4 < z < 7.2 FUV I+S SFDuncan et al. (2014) 4 < z < 7 UV SED I+NE SFShim et al. (2011) z ∼ 4.4 Hα I+S+SP SFSteinhardt et al. (2014) z ∼ 5 UV SED I+S SFGonzalez et al. (2010) z = 7.2 UV+IR I+NE SFThis paper, Appendix D 4 < z < 8 FUV I+E+NE SF


galaxies as a reference to compare with our model and thusgain more insights on how galaxies evolve from active topassive as well as on their structural evolution (discussed inSection 7). For the fraction of quiescent galaxies fQ we usethe following relation:

fQ(M∗, z) =1

1 + (M∗/Mchar(z))α, (44)

where Mchar is the transition stellar mass at which the frac-tions of blue star forming and red quenched galaxies are both50%. Figure 1 shows Mchar as a function of redshift fromobservations and previous constraints. The solid black lineshows the relation log(Mchar(z)/M⊙) = 10.2 + 0.6z that wewill employ in this paper, and the gray solid lines show theresults when shifting (Mchar(z)/M⊙) by 0.1 dex above andbelow. We will use this shift as our uncertainty in the def-inition for log(Mchar(z)/M⊙). The red (blue) curves in thefigure show the stellar mass vs. redshift where 75% (25%) ofthe galaxies are quenched.

Finally, we will assume that α = −1.3. The transitionstellar mass is such that at z = 0 log(Mchar(z)/M⊙) = 10.2and at z = 2 log(Mchar(z)/M⊙) = 11.4.


The galaxy population in our model is described by fourproperties: halo mass Mvir, halo mass accretion rates, stel-lar mass M∗, and star formation rate SFR. In order to con-strain the model we combine several observational data sets,including the GSMFs, the SFRs and the CSFR for all galax-ies. In this Section we describe our adopted methodology aswell as the best resulting fit parameters in our model.

In order to sample the best-fit parameters that maxi-

mize the likelihood function L ∝ e−χ2/2 we use the MCMC



due to the increase in the number of massive quenched galaxiesat lower redshifts.

approach, described in detail in Rodrıguez-Puebla, Avila-Reese & Drory (2013).


χ2 = χ2GSMF + χ2

SFR + χ2CSFR (45)

where for the GSMFs we define

χ2GSMF =

X

j,i

χ2φj,i

, (46)

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10



Chen et al. (2009) z ∼ 0.1 Hα/Hβ S AllSalim et al. (2007) z ∼ 0.1 UV SED S AllNoeske et al. (2007) 0.2 < z < 1.1 UV+IR S AllKarim et al. (2011) 0.2 < z < 3 1.4 GHz I+S+E AllDunne et al. (2009) 0.45 < z < 2 1.4 GHz I+S+E AllKajisawa et al. (2010) 0.5 < z < 3.5 UV+IR I AllWhitaker et al. (2014) 0.5 < z < 3 UV+IR I+S AllSobral et al. (2014) z ∼ 2.23 Hα I+S+SP SFReddy et al. (2012) 2.3 < z < 3.7 UV+IR I+S+SP SFMagdis et al. (2010) z ∼ 3 FUV I+S+SP SFLee et al. (2011) 3.3 < z < 4.3 FUV I+SP SFLee et al. (2012) 3.9 < z < 5 FUV I+SP SFGonzalez et al. (2012) 4 < z < 6 UV+IR I+NE SFSalmon et al. (2015) 4 < z < 6 UV SED I+NE+E SFBouwens et al. (2011) 4 < z < 7.2 FUV I+S SFDuncan et al. (2014) 4 < z < 7 UV SED I+NE SFShim et al. (2011) z ∼ 4.4 Hα I+S+SP SFSteinhardt et al. (2014) z ∼ 5 UV SED I+S SFGonzalez et al. (2010) z = 7.2 UV+IR I+NE SFThis paper 4 < z < 8 FUV I+E+NE SF


where Mchar is the transition stellar mass at which the frac-tion of blue star forming and red quenched galaxies is 50%.Figure 1 shows Mchar as a function of redshift from observa-tions and previous constraints. The open square with errorbars show when the observed fraction of star forming galax-ies is 50% for local galaxies as derived in Bell et al. (2003)based on the SDSS DR2 while the filled triangles showsthe same but derived in Bundy et al. (2006) based on theDEEP2 survey. Additionally, we include data from Drory &Alvarez (2008) (long dashed line, based on the FORS DeepField survey) and Pozzetti et al. (2010) (skeletal symbols,based on the COSMOS), Baldry et al. (2012) (filled square,from the GAMA survey) and Muzzin et al. (2013) (filled cir-cles, based on the COSMOS survey). The empirical resultsbased on abundance matching by Firmani & Avila-Reese(2010) are shown with the short dashed lines. The solidblack line shows the relation log(Mchar(z)/M⊙) = 10.2+0.6zthat we will employ in this paper and that is consistentwith most of the above studies. The gray solid lines showthe results when shifting log(Mchar(z)/M⊙) by 1 dex aboveand below. We will use this shift as our uncertainty in thedefinition for log(Mchar(z)/M⊙). Finally, we will assumethat α = −1.3. The transition stellar mass is such thatat z = 0 is log(Mchar(z)/M⊙) = 10.2 and at z = 2 islog(Mchar(z)/M⊙) = 11.4.


The galaxy population in our model is described by fourproperties: halo mass Mvir, halo mass accretion rates, stel-lar mass M∗ and star formation rate SFR. In order to con-strain the model we combine several observational data sets,including the GSMFs, the SFRs and the CSFR for all galax-ies. In this Section we describe our adopted methodology aswell as the best resulting fit parameters in our model.



due to the increase of massive quench galaxies at lower redshifts.

In order to sample the best-fit parameters that max-

imize the likelihood function L ∝ e−χ2/2 we use theMCMC approach and described in detail in Rodrıguez-Puebla, Avila-Reese & Drory (2013).


χ2 = χ2GSMF + χ2

SFR + χ2CSFR (46)

where for the GSMFs we define,

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The Galaxy-Halo Connection Over The Last 13.3 Gyrs 11

Figure 3. Star formation rates as a function of redshift z in five stellar mass bins. Black solid lines shows the resulting best fit modelto the SFRs implied by our model. The filled circles with error bars show the observed data as described in the text, see Section 2.

for the SFRs

χ2SFR =

X

j,i

χ2SFRj,i

, (47)

and for the CSFRs

χ2CSFR =

X

i

χ2ρi

. (48)

In all the equations the sum over j refers to different stellarmass bins while i refers to summation over different red-shifts. The fittings are made to the data points with theirerror bars of each GSMF, SFR and CSFR.

In total our galaxy model consists of eighteen pa-rameters. Thirteen are to model the redshift evolu-tion of the SHMR, Equations (27)–(31): pSHMR =ϵ0, ϵ1, ϵ2, ϵ3, MC0, MC1, MC2, α, α1, α2, δ0, δ1, δ2, γ0, γ1; andthree more to model the fraction of stellar mass growth dueto in-situ star formation: pin situ = Min situ,0, Min situ,1, β.To sample the best fit parameters in our model we run a setof 3 × 105 MCMC models.

Figure 2 shows the best-fit model GSMFs from z ∼ 0.1to z ∼ 10 with the solid lines as indicated by the labels. Thisfigure shows the evolution of the observed GSMF based inour compiled data described in Section 4.1.

Figure 3 shows the star formation rates as a function ofredshift z in five stellar mass bins. The observed SFRs fromthe literature are plotted with filled circles with error barswhile the best fit model is plotted with the solid black lines.In general, our model fits describe well the observations atall mass bins and all redshifts.

We present the best-fit model to the CSFR in the Up-per Panel of Figure 4. The observed CSFRs employed forconstraining the model are shown with the solid circles anderror bars. The Lower Panel of Figure 4 compares the cos-mic stellar mass density predicted by our model fit with thedata compiled in the review by Madau & Dickinson (2014);the agreement is impressive.

In Appendix A we discuss the impact of the differentassumptions employed in the modelling. The best fitting pa-rameters to our model are:

log(ϵ(z)) = −1.763 ± 0.034+P(0.047 ± 0.095,−0.073 ± 0.018, z) ×Q(z)+P(−0.039 ± 0.010, 0, z),

(49)

log(M0(z)) = 11.543 ± 0.041+P(−1.615 ± 0.154,−0.134 ± 0.032, z) ×Q(z),

(50)

α(z) = 1.970 ± 0.032+P(0.505 ± 0.162, 0.014 ± 0.020, z) ×Q(z),

(51)

δ(z) = 3.411 ± 0.238+P(0.687 ± 0.510,−0.561 ± 0.101, z) ×Q(z),

(52)

γ(z) = 0.496 ± 0.039 + P(−0.198 ± 0.094, 0, z) ×Q(z), (53)

log(Min situ(z)) = 12.953 ± 0.251+P(4.050 ± 1.300, 0, z),

(54)

β(z) = 1.251 ± 0.223. (55)

For our best fitting model we find that χ2 = 520.4 froma number of Nd = 488 observational data points. Since ourmodel consist of Np = 18 free parameters the resulting re-duced χ2 is χ2/d.o.f. = 1.1.

6 THE GALAXY-HALO CONNECTION

6.1 The Stellar-to-Halo mass relation from z ∼ 0.1to z ∼ 10

The upper panel of Figure 5 shows the constrained evolu-tion of the SHMR while the lower panel shows the stellar-to-halo mass ratio from z ∼ 0.1 to z ∼ 10. Recall that inthe case of central galaxies we refer to Mhalo as the virialmass Mvir of the host halo, while for satellites Mhalo refersto the maximum mass Mpeak reached along the main pro-genitor assembly history. Consistent with previous resultsthe SHMR appears to evolve only very slowly below z ∼ 1.This situation is quite different between z ∼ 1 and z ∼ 7,where at a fixed halo mass the mean stellar mass is lower athigher redshifts. The middle panel of the same figure shows

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for the SFRs

χ2SFR =

X

j,i

χ2SFRj,i

, (47)

and for the CSFRs

χ2CSFR =

X

i

χ2ρi

. (48)

In all the equations the sum over j refers to different stellarmass bins while i refers to summation over different red-shifts. The fittings are made to the data points with theirerror bars of each GSMF, SFR and CSFR.

In total our galaxy model consists of eighteen pa-rameters. Thirteen are to model the redshift evolu-tion of the SHMR, Equations (27)–(31): pSHMR =ϵ0, ϵ1, ϵ2, ϵ3, MC0, MC1, MC2, α, α1, α2, δ0, δ1, δ2, γ0, γ1; andthree more to model the fraction of stellar mass growth dueto in-situ star formation: pin situ = Min situ,0, Min situ,1, β.To sample the best fit parameters in our model we run a setof 3 × 105 MCMC models.

Figure 2 shows the best-fit model GSMFs from z ∼ 0.1to z ∼ 10 with the solid lines as indicated by the labels. Thisfigure shows the evolution of the observed GSMF based inour compiled data described in Section 4.1.

Figure 3 shows the star formation rates as a function ofredshift z in five stellar mass bins. The observed SFRs fromthe literature are plotted with filled circles with error barswhile the best fit model is plotted with the solid black lines.In general, our model fits describe well the observations atall mass bins and all redshifts.

We present the best-fit model to the CSFR in the Up-per Panel of Figure 4. The observed CSFRs employed forconstraining the model are shown with the solid circles anderror bars. The Lower Panel of Figure 4 compares the cos-mic stellar mass density predicted by our model fit with thedata compiled in the review by Madau & Dickinson (2014);the agreement is impressive.

In Appendix A we discuss the impact of the differentassumptions employed in the modelling. The best fitting pa-rameters to our model are:

log(ϵ(z)) = −1.763 ± 0.034+P(0.047 ± 0.095,−0.073 ± 0.018, z) ×Q(z)+P(−0.039 ± 0.010, 0, z),

(49)

log(M0(z)) = 11.543 ± 0.041+P(−1.615 ± 0.154,−0.134 ± 0.032, z) ×Q(z),

(50)

α(z) = 1.970 ± 0.032+P(0.505 ± 0.162, 0.014 ± 0.020, z) ×Q(z),

(51)

δ(z) = 3.411 ± 0.238+P(0.687 ± 0.510,−0.561 ± 0.101, z) ×Q(z),

(52)

γ(z) = 0.496 ± 0.039 + P(−0.198 ± 0.094, 0, z) ×Q(z), (53)

log(Min situ(z)) = 12.953 ± 0.251+P(4.050 ± 1.300, 0, z),

(54)

β(z) = 1.251 ± 0.223. (55)

For our best fitting model we find that χ2 = 520.4 froma number of Nd = 488 observational data points. Since ourmodel consist of Np = 18 free parameters the resulting re-duced χ2 is χ2/d.o.f. = 1.1.

6 THE GALAXY-HALO CONNECTION

6.1 The Stellar-to-Halo mass relation from z ∼ 0.1to z ∼ 10

The upper panel of Figure 5 shows the constrained evolu-tion of the SHMR while the lower panel shows the stellar-to-halo mass ratio from z ∼ 0.1 to z ∼ 10. Recall that inthe case of central galaxies we refer to Mhalo as the virialmass Mvir of the host halo, while for satellites Mhalo refersto the maximum mass Mpeak reached along the main pro-genitor assembly history. Consistent with previous resultsthe SHMR appears to evolve only very slowly below z ∼ 1.This situation is quite different between z ∼ 1 and z ∼ 7,where at a fixed halo mass the mean stellar mass is lower athigher redshifts. The middle panel of the same figure shows

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pointed out in previous papers (see e.g., Grazian et al. 2015;Song et al. 2015; Duncan et al. 2014). At high redshifts, thefaint-end slope becomes steeper (see e.g., Song et al. 2015;Duncan et al. 2014). As the galaxy population evolves, mas-sive galaxies tend to pile up around M∗ ∼ 3 × 1010M⊙ dueto the increasing number of massive quenched galaxies atlower redshifts (see e.g., Bundy et al. 2006; Faber et al. 2007;Peng & et al. 2010; Pozzetti et al. 2010; Muzzin et al. 2013).These represent a second component that is well describedby a Schechter function, and thus the resulting GSMF at lowredshifts is better described by a double Schechter function.

4.2 Star Formation rates

In this paper, we use a compilation of 19 studies from theliterature for the observed SFRs as a function stellar massat different redshifts. Table 2 lists the references that weutilize.

As for the GSMFs, in order to directly compare thedifferent SFR samples we applied some calibrations. To doso, we follow Speagle et al. (2014) who used a compilationto study star formation from z ∼ 0 to z ∼ 6 by correctingfor different assumptions regarding the IMF, SFR indica-tors, SPS models, dust extinction, emission lines and cos-mology. The reader is referred to that paper for details ontheir calibrations. In Table 2 we indicate the specific calibra-tions applied to the data. Note that in order to constrain ourmodel we use observations of the SFRs for all galaxies. Com-plete samples, however, for all galaxies are only available atz < 3. Therefore, here we decided to include SFRs samplesfrom star forming galaxies, especially at high z > 3. Usingstar forming galaxies at high redshift is not a big source ofuncertainty since most of the galaxies at z > 3 are actuallystar forming, see e.g., Figure 1. Table 2 indicates the typeof the data, namely, if the sample is for all galaxies or forstar forming galaxies, and the redshift range.

In addition to the compiled sample for z > 3, here wecalculate average SFRs using again the UV LFs described inAppendix D. We begin by correcting the UV rest-frame ab-solute magnitudes for extinction using the Meurer, Heckman& Calzetti (1999) average relation

⟨AUV⟩ = 4.43 + 1.99⟨β⟩, (41)

where ⟨β⟩ is the average slope of the observed UV con-tinuum. We use the following relationship independent ofredshift: ⟨β⟩ = −0.11 × (MUV + 19.5) − 2, which is consis-tent with previous determination of the β slope (see e.g.,Bouwens et al. 2014). Then we calculate UV SFRs using theKennicutt (1998) relationship

SFRM⊙ yr−1

(LUV) =LUV/erg s−1 Hz−1

13.9 × 1027. (42)

We subtract -0.24 dex to be consistent with a Chabrier(2003) IMF. Finally, we calculate the average SFR as a func-tion of stellar mass as

⟨log SFR (M∗, z)⟩ =

Z

P (M∗|MUV, z) log SFR(MUV)×

φUV(MUV, z)dMUV. (43)

The probability distribution function P (M∗|MUV, z) is de-scribed in detail in Appendix D. We use the following inter-vals of integration: MUV ∈ [−17,−22.6] at z = 4; MUV ∈

Figure 1. Transition stellar mass Mchar(z) at which the fractionsof blue star forming and red quenched galaxies are both 50%.The open square with error bars shows the transition mass forlocal galaxies as derived in Bell et al. (2003) based on the SDSSDR2, while the filled triangles show the transition mass derived inBundy et al. (2006) based on the DEEP2 survey. Drory & Alvarez(2008) based on the FORS Deep Field survey is indicated with thelong dashed line; observations from Pozzetti et al. (2010) basedon the COSMOS survey are indicated with the skeletal symbols;observations from Baldry et al. (2012) based on the GAMA surveyare shown with filled square; and observations from Muzzin et al.(2013) based on the COSMOS survey, are shown as filled circles.The empirical results based on abundance matching by Firmani& Avila-Reese (2010) are shown with the short dashed lines. Thesolid black line shows the relation log(Mchar(z)/M⊙) = 10.2 +0.6z, employed in this paper and that is consistent with mostof the above studies. The gray solid lines show the results whenshifting (Mchar(z)/M⊙) 0.1 dex higher and lower. The red (blue)curves show the stellar mass vs. z where 75% (25%) of the galaxiesare quenched.

[−16.4,−23] at z = 5; MUV ∈ [−16.75,−22.5] at z = 6;MUV ∈ [−17,−22.75] at z = 7 and MUV ∈ [−17.25,−22] atz = 8.

4.3 Cosmic Star Formation Rate

We use the CSFR data compilation from Madau & Dickin-son (2014). This data was derived from FUV and IR restframe luminosities by deriving empirical dust corrections tothe FUV data in order to estimate robust CSFRs. We ad-justed their data to a Chabrier (2003) IMF by subtracting0.24 dex from their CSFRs. Finally, for z > 3 we calculatethe CSFR using again the UV dust-corrected LFs and SFRsdescribed above and using the same integration limit as inMadau & Dickinson (2014). We find that our CSFR is con-sistent with the compilation derived in Madau & Dickinson(2014) over the same redshift range.

4.4 The Fraction of Star-Forming and Quiescent

Galaxies

In this paper we interchangeably refer to star-forming galax-ies as blue galaxies and quiescent galaxies as red galaxies. Weutilize the fraction of blue/star-forming and red/quenched

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Constraining the Galaxy Halo Connection: Star Formation Histories, Galaxy Mergers, and Structural Properties, by Aldo Rodriguez-Puebla, Joel, and others (in nearly final form) PREVIEW


24

Figure 19. Comparison with previous works that report galaxy stellar masses as a function of halo mass. Abundance matching resultsfrom Guo et al. (2010); Behroozi, Wechsler & Conroy (2013b); and Moster, Naab & White (2013) are shown respectively with violet, blueand red solid lines. The results of Yang et al. (2012) based on the evolution of the GSMF, galaxy groups counts, and galaxy clusteringare shown with the orange and olive shaded regions for their SMF1 and SMF2 cases. Constraints from combining the GSMF, galaxyclustering, and galaxy weak lensing from Leauthaud et al. (2012) and Coupon et al. (2015) are shown with the gray and magenta lines,respectively. Note the good agreement at all redshifts between the different techniques except for SMF1.

tainties in the constrained relations our SHMR is consistentwith those derived in Behroozi, Wechsler & Conroy (2013b).

In the z > 0.1 panels of Figure 19 we plot as a dashedcurve the time-independent SHMR, i.e., the SHMR obtainedat z ∼ 0.1. Note that below z ∼ 2 most of the modelsas well as our mass relations are consistent with a time-independent SHMR. Recently, Behroozi, Wechsler & Conroy(2013a) showed that assuming a time-independent SHMRcould simply explain the cosmic star formation rate sincez = 4. In a subsequent work Rodrıguez-Puebla et al. (2016b)extended that argument for studying the galaxies in themain sequence galaxy star formation by showing that thedispersion of the halo mass accretion rates matched the ob-served dispersion of star formation rates.

8.1.2 Halo mass-to-stellar mass relationship

Figure 20 shows mean virial masses as a function of galaxystellar mass, i.e., we invert the SHMR to obtain a Mvir–M∗ relationship. Because the SHMR relationship has an in-trinsic scatter, inverting this relation is not as simple asjust inverting the axis of the relation; we also need to takeinto account the scatter around the relation. This can be

done by using Bayes’ theorem by writing P (Mvir|M∗) =P (M∗|Mvir) × φvir(Mvir)/φg(M∗). Using this equation wecan thus compute mean halo masses as a function of M∗. Ingeneral, we observe that the resulting Mvir–M∗ relationshipevolves in the direction that at a fixed stellar mass galaxiestend to have lower halo masses at higher redshifts.

We now compare with recent determinations of theMvir–M∗ relationship. We begin by describing data ob-tained from galaxy weak lensing analysis. In Figure 20 weplot the results reported in Mandelbaum et al. (2006) fromthe stacked weak-lensing analysis for the SDSS DR4 atz ∼ 0.1, black open circles with error bars (95% of confi-dence intervals). Mandelbaum et al. (2006) reported halomasses separately for late- and early-type galaxies. Herewe estimated the average mass relation as: ⟨Mvir⟩(M∗) =fl(M∗)⟨Mvir⟩l(M∗)+fe(M∗)⟨Mvir⟩e(M∗), where fe(M∗) andfl(M∗) are the fraction of late- and early-type galaxies intheir sample. The corresponding values of halo masses forlate- and early-type galaxies are ⟨Mvir⟩l and ⟨Mvir⟩e. Theempty red squares in the same figure show the analysis fromvan Uitert et al. (2011) who combined image data from theRed Sequence Cluster Survey (RCS2) and the SDSS DR7to obtain the halo masses for late- and early-type galaxies

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Figure 20. Comparison with previous works that report halo mass as a function of galaxy stellar masses. Weak lensing studies fromMandelbaum et al. (2006); van Uitert et al. (2011); Velander et al. (2014); Hudson et al. (2013) and Heymans et al. (2006) are shownrespectively with the black open circles, empty red squares, empty blue triangles, open blue/red pentagons and green filled square.Galaxy clustering from Wake et al. (2011); Skibba et al. (2015), and Harikane et al. (2016) are shown with the filled blue circles, solidblack circles and red solid triangles respectively. The dotted line shows when assuming a time-independent M∗–Mvir relation.

as a function of M∗. Similarly to Mandelbaum et al. (2006)data, we also derive their mean halo masses based on theirreported fraction of early- and late-type galaxies. We also in-clude the stacked weak-lensing analysis from Velander et al.(2014) based on the CFHTLens survey, empty blue triangles.The authors derive halo masses separately for blue and redgalaxies based on the color-magnitude diagram. We againderive their mean halo masses by using the reported frac-tion of blue and red galaxies. Using the CFHTLenS surveyHudson et al. (2013) also derived halo masses as a functionof stellar masses for blue and red galaxies separately. Weshowed their results with the open blue and red pentagons.Unfortunately, the authors do not report the fraction of blueand red galaxies so we plot their mass relations separatelyfor blue and red galaxies. The green filled square in Figure20 shows the halo mass derived from galaxy weak lensing atz ∼ 0.8 from Heymans et al. (2006) by combing the ChandraDeep Field South and the Hubble Space Telescope GEMSsurvey.

Next, we discuss halo masses obtained from galaxy clus-tering. Wake et al. (2011) used the halo occupation dis-tribution (HOD) model of galaxy clustering to derive halomasses between z = 1 − 2 from the NEWFIRM MediumBand Survey (NMBS), filled blue circles. Similarly, Skibba

et al. (2015) used the HOD model and the observed stel-lar mass dependent clustering of galaxies in the PRIMUSand DEEP2 redshift survey from z ∼ 0.2 to z ∼ 1.2 toconstrain the Mvir–M∗ relationship, indicated by the solidblack circles. Martinez-Manso et al. (2015) used the Deep-Field Survey to derive the angular clustering of galaxies andobtain halo masses by modelling galaxy clustering in thecontext of the HOD. Finally, Harikane et al. (2016) esti-mated the angular distribution of of Lyman break galaxiesbetween z ∼ 4−7 from the Hubble Legacy deep Imaging andthe Subaru/Hyper Suprime-Cam data. Halo masses were es-timated using the HOD model, filled red triangles in Figure20.

Similarly to the determinations of the SHMR comparedin Figure 19, the Mvir–M∗ relationships described aboveagree very well between each other and with our mass re-lations from abundance matching, especially at z <

∼ 1. Thedotted lines show the resulting Mvir–M∗ relationship whenassuming a time-independent SHMR. Note that the evolu-tion of this relation simply reflects the fact that the ratioφvir(Mvir)/φg(M∗) is not constant with time.

We acknowledge that are other direct techniques to de-rive halo masses from galaxy samples. One example is touse the kinematics of satellite galaxies as test particles of

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Figure 11. Analytic fits to the Star formation histories as indicated by the labels. Left Panel: Lognormal fits. Right Panel: Delayed−τfits. Note that lognormal fits describes well the average star formation histories of galaxies in halos with masses around and belowMvir = 1012M⊙. Delayed−τ fits are poor fits at all masses.

Figure 12. Left Panel: Stellar-halo accretion rate coevolution (SHARC) assumption as a function of halo mass for progenitors at z = 0.The black solid lines show the trajectories for progenitors with Mvir = 1011, 1011.5, 1012, 1013, 1014 and 1015M⊙. Right Panel: Like theleft panel, but as a function of stellar mass for their corresponding halo progenitors. These figures show that the SHARC assumption isa good approximation within a factor of ∼ 2 for star-forming galaxies, which are a majority of those below the quenching curves. Recallthat in both panels, the dashed lines denote the transition below/above which galaxies are star-forming/quenched, and the upper (lower)long-dash curves show the stellar mass vs. z where 75% (25%) of the galaxies are quenched.

mass downsizing respectively (e.g., Conroy &Wechsler 2009;Firmani & Avila-Reese 2010, and references therein). Theformer implies that low-mass galaxies for some reason de-layed the stellar mass assembly with respect to their halos,while the latter implies that the more massive the galaxies,the earlier their mass growth was quenched while their haloscontinued growing. It is interesting to note that all galaxiesthat are quenched today went through a phase in which theyco-evolved with their host halo, i.e., sSFR/sMAR ∼ 1.

We note that the halo star formation efficiency peaksaround progenitors with Mvir ∼ 2 × 1011M⊙, which corre-sponds to galaxies with masses M∗ ∼ (0.8 − 3) × 109M⊙.

Those galaxies have very high values of sSFR/sMAR ∼6−10. Moreover, these galaxies spent a considerable amountof time having large values of sSFR/sMAR – of the orderof few Gyrs. Then, the halo star formation efficiency de-creases again for progenitors at z ∼ 0 with masses belowMvir ∼ 2 × 1011M⊙, implying that, at least at z ∼ 0, thehalo mass Mvir ∼ 2 × 1011M⊙ is “special”. The fact thatin more massive halos the ratio sSFR/sMAR decreases isnot surprising, this is supported by both theoretical and ob-servational studies which show that they are more likely tohost quenched galaxies, as we discussed above. Note, how-ever, that the ratio sSFR/sMAR is not always increasing as

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Figure 4. Specific star formation rates as a function of stellarmass from z ∼ 0.1 to z ∼ 6. The solid lines show our best fittingmodel while the shaded areas show the 1σ confidence intervalsusing the our set of MCMC models. The filled circles show theobservations we utilize to constrain our model.

an order of magnitude. Nonetheless, given the uncertaintieswhen deriving the GSMFs at high redshifts z > 4, this resultshould be taken with caution. For comparison, the dashedlines in both panels show the cosmic baryon fraction im-plied by the Planck Collaboration et al. (2016) cosmology,fb = ΩB/ΩM ≈ 0.16.

Next we study the integral stellar conversion efficiency,defined as η = f∗/fb. This is shown in the left panel of Figure7 for progenitors at z = 0 of dark matter halos with massesbetween Mvir = 1011M⊙ and Mvir = 1015M⊙. Dark mat-ter halos are most efficient when their progenitors reachedmasses between Mvir ∼ 5× 1011M⊙ − 2× 1012M⊙ at z < 1,and the stellar conversion efficiency is never larger than

η ∼ 0.2. Theoretically, the characteristic mass of 1012h−1M⊙

is expected to mark a transition above which the stellarconversion efficiency becomes increasingly inefficient. Thereason is that at halo masses above 1012h−1M⊙ the effi-ciency at which the virial shocks can heat the gas increases(e.g., Dekel & Birnboim 2006). Additionally, in such massivegalaxies the gas can be kept from cooling by the feedbackfrom active galactic nuclei (Croton et al. 2006; Cattaneoet al. 2008; Henriques et al. 2015; Somerville & Dave 2015,and references therein). Central galaxies in massive halos areexpected to become passive systems roughly at the epochwhen the halo reached the mass of 1012h−1M⊙, thus theterm halo mass quenching.

The right panel of Figure 7 shows the stellar conversionefficiency for the corresponding stellar mass growth histo-ries of the halo progenitors discussed above. The range ofthe transition stellar mass Mtrans(z), defined as the stellarmass at which the fraction of star forming is equal to thefraction of quenched galaxies (see Figure 1 and Section 7),is shown by the dashed lines. Below these lines galaxies aremore likely to be star forming. Note that the right panel ofFigure 7 shows that Mtrans(z) roughly coincides with whereη is maximum, especially at low z. This reflects the fact thathalo mass quenching is part of the physical mechanisms thatquench galaxies in massive halos. We will come back to thispoint in Section 8.2.

Finally, Figure 8 shows the trajectories for the M∗/Mvir

ratios of progenitors of dark matter halos with masses be-tween Mvir = 1011M⊙ and Mvir = 1015M⊙ at z = 0. Notethat all galaxies in halos above Mvir = 1012M⊙ had a max-imum followed by a decline of their M∗/Mvir ratio.

6.2 Galaxy Growth and Star-Formation Histories

Figure 9 shows the predicted star formation histories forprogenitors of average dark matter halos at z = 0 withmasses between Mvir = 1011M⊙ and Mvir = 1015M⊙. Panela) shows the resulting 3D surface for the redshift evolutionof the stellar-to-halo mass relation for progenitors of dark

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Figure 4. Specific star formation rates as a function of stellarmass from z ∼ 0.1 to z ∼ 6. The solid lines show our best fittingmodel while the shaded areas show the 1σ confidence intervalsusing the our set of MCMC models. The filled circles show theobservations we utilize to constrain our model.

an order of magnitude. Nonetheless, given the uncertaintieswhen deriving the GSMFs at high redshifts z > 4, this resultshould be taken with caution. For comparison, the dashedlines in both panels show the cosmic baryon fraction im-plied by the Planck Collaboration et al. (2016) cosmology,fb = ΩB/ΩM ≈ 0.16.

Next we study the integral stellar conversion efficiency,defined as η = f∗/fb. This is shown in the left panel of Figure7 for progenitors at z = 0 of dark matter halos with massesbetween Mvir = 1011M⊙ and Mvir = 1015M⊙. Dark mat-ter halos are most efficient when their progenitors reachedmasses between Mvir ∼ 5× 1011M⊙ − 2× 1012M⊙ at z < 1,and the stellar conversion efficiency is never larger than

η ∼ 0.2. Theoretically, the characteristic mass of 1012h−1M⊙

is expected to mark a transition above which the stellarconversion efficiency becomes increasingly inefficient. Thereason is that at halo masses above 1012h−1M⊙ the effi-ciency at which the virial shocks can heat the gas increases(e.g., Dekel & Birnboim 2006). Additionally, in such massivegalaxies the gas can be kept from cooling by the feedbackfrom active galactic nuclei (Croton et al. 2006; Cattaneoet al. 2008; Henriques et al. 2015; Somerville & Dave 2015,and references therein). Central galaxies in massive halos areexpected to become passive systems roughly at the epochwhen the halo reached the mass of 1012h−1M⊙, thus theterm halo mass quenching.

The right panel of Figure 7 shows the stellar conversionefficiency for the corresponding stellar mass growth histo-ries of the halo progenitors discussed above. The range ofthe transition stellar mass Mtrans(z), defined as the stellarmass at which the fraction of star forming is equal to thefraction of quenched galaxies (see Figure 1 and Section 7),is shown by the dashed lines. Below these lines galaxies aremore likely to be star forming. Note that the right panel ofFigure 7 shows that Mtrans(z) roughly coincides with whereη is maximum, especially at low z. This reflects the fact thathalo mass quenching is part of the physical mechanisms thatquench galaxies in massive halos. We will come back to thispoint in Section 8.2.

Finally, Figure 8 shows the trajectories for the M∗/Mvir

ratios of progenitors of dark matter halos with masses be-tween Mvir = 1011M⊙ and Mvir = 1015M⊙ at z = 0. Notethat all galaxies in halos above Mvir = 1012M⊙ had a max-imum followed by a decline of their M∗/Mvir ratio.

6.2 Galaxy Growth and Star-Formation Histories

Figure 9 shows the predicted star formation histories forprogenitors of average dark matter halos at z = 0 withmasses between Mvir = 1011M⊙ and Mvir = 1015M⊙. Panela) shows the resulting 3D surface for the redshift evolutionof the stellar-to-halo mass relation for progenitors of dark

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SFR DATA & FITS

)

The integral stellar conversion η = f/fb, where f = M/Mvir and fb = ΩB/ΩM

The star formation rate (SFR) as a function of redshift and (left panel) Mvir and (right panel) M

\

This figure shows that quenching is correlated with sSFR/sSMR = thalo/t, since sSFR/sSMR and quenching curves are nearly parallel. sSFR/sSMR - first rises, reaching a peak ~2 at z ~ 3 for 1013 halos, a peak ~7 for 1012 halos at z~1.5, and 1011 halos are still at peak sSFR/sSMR ~ 10 - then declines along all Mvir and M* progenitor tracks toward z=0.

This figure shows that the SHARC approximation is rather well satisfied until quenching, the SHARC ratio RSHARC = (SFR / MAR) / (dMvir/dlog M*) having a value of about 1 to 2 along the progenitor trajectories, and then dropping after quenching. This shows quenching is correlated with RSHARC :

- the fraction of quenched galaxies is ~ 50% when RSHARC ~ 1 to 1.5, and the quenched fraction is > 75% when RSHARC drops to ~1 - like sSFR/sSMR, RSHARC first rises along all progenitor curves, reaches a peak at higher z for higher mass (Mvir or M*), and then declines - unlike sSFR/sSMR, the peak SHARC ratio is nearly constant between 1.5 and 2 (the SHARC ratio peaks at about 2 for both 1011.5 halos at z ~ 0.5 and

1015 halos at z ~ 3, and at about 1.5 for intermediate mass halos). Note: the SHARC formula is SFR = (dM/dMvir) MAR where MAR = dMvir/dt. Define RSHARC = (SFR / MAR) / (dM/dMvir), so SHARC ==> RSHARC = 1.

RSH

AR

C =

(SFR

/MA

R) /

(dM

/d

Mvi

r)

RSH

AR

C =

(SFR

/MA

R) /

(dM

/d

Mvi

r)


This figure (and the left panel below) shows that ∑1 reaching a maximum correlates with quenching:- ∑1 rises steadily toward z = 0 along all progenitor tracks- ∑1 at the quenching transition rises steadily with Mvir and reaches its maximum at lower redshifts for lower Mvir — “quenching downsizing” - The fact that the progenitor tracks are parallel to the trajectory curves shows that ∑1 remains constant after it reaches its maximum

The right panel shows that Reff steadily rises along halo trajectories, and quenching occurs when Reff ≈ 3 kpc. Although ∑1 is flat afterquenching, the middle panel shows that ∑eff declines after quenching as Reff increases.

20

Figure 16. Circularized effective radius for blue star-forming galaxies, left panel, and red quiescent galaxies, right panel. The filledcircles show the circularized effective radius as function of redshift from van der Wel et al. (2014) based on multiwavelength photometryfrom the 3D-HST survey and HST/WFC3 imaging from CANDELS. Solid lines show the redshift dependence for blue and red galaxiesof the local relation by Mosleh, Williams & Franx (2013) based on the MPA-JHU SDSS DR7. We utilize the above redshift dependencesas an input to derive average galaxy’s radial mass distribution as a function of stellar mass by assuming that blue/star-forming galaxieshave a Sersic index n = 1 while red/quenched galaxies have a Sersic index n = 4 (see text for details).

Figure 17. Average, evolution of the radial distribution of stellar mass for galaxies in halo progenitors at z = 0 with Mvir =1011, 1011.5, 1012, 1013, 1014 and 1015M⊙. These radial distributions can be imagined as stacking all the density profiles of galaxiesat a given z, no matter whether galaxies are spheroids or disks or a combination of both.

the effective radius Reff ,7 both at z = 0 and at higher red-

shifts (see e.g., Bell et al. 2012; Patel et al. 2013; van derWel et al. 2012). For a Sersic law r1/n, the Sersic index n is aparameter that controls the slope of the curvature for the ra-dial distribution of light/mass. Observational results basedon the SDSS have shown that when fitting the global lightdistribution to a Sersic law r1/n most of the galaxies have in-dices n between 0.5 and 8 (see e.g., Simard et al. 2011; Meert,

7 The effective radius is defined as the radius that encloses halfthe luminosity or stellar mass of the galaxy.

Vikram & Bernardi 2015). However, when dividing galaxiesinto two main clases – e.g., as early- and late-types – theradial distribution of the light/mass is fairly well describedwith n = 4 (de Vaucouleurs 1948) and n = 1 respectively.In this section, we will assume for simplicity that all late-type galaxies are blue/star-forming systems with a Sersicindex n = 1 while all early-type morphologies correspond tored/quiescent galaxies with Sersic index n = 4. Hereafter, wewill use these galaxy classifications interchangeably. Whilethis is an oversimplification of a more complex reality, forour purpose it is accurate enough since the relatively small

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Figure 17. Average evolution of the radial distribution of stellar mass for galaxies in halo progenitors at z = 0 with Mvir = 1011,1011.5,1012,1013, 1014 and 1015M⊙. These radial distributions can be imagined as stacking all the density profiles of galaxies at a given z, no matter whether galaxies are spheroids or disks or a combination of both.


Figure 19. Left Panel: Trajectories for progenitors of halos with Mvir = 1011, 1011.5, 1012, 1013, 1014 and 1015M⊙ at z = 0 in theΣ1−sSFR plane. Right Panel: Same progenitors but in the M∗−sSFR plane. The symbols show different redshifts as indicated by thelabels. The dashed curves show the transition below which half the galaxies are quiescent, and the upper (lower) dot-dashed curves showwhere 25% (75%) of the galaxies are quenched.

dex at the same redshift range. van Dokkum et al. (2014)reported similar trends based on the analysis of galaxy sizesfrom the SDSS, Ultra VISTA and 3D-HST survey. The au-thors considered three different scenarios: i) mass loss due tocore-core mergers, ii) mass loss due to stellar evolution, andiii) adiabatic expansion due to stellar winds (also consid-ered in Damjanov et al. 2009, and referred as the less likelyscenario for the mass-size relation). Unfortunately, our anal-ysis is not detailed enough to decide for one over the other.Nonetheless, given the subtlety of this effect we suspect thatmass loss due to stellar evolution (and the associated stellarwinds) is the most likely explanation, although more workis needed on this.

The upper left panel in Figure 20 shows the trajectoriesfor the halo progenitors discussed above in the size-mass re-lation. Note first that star-forming galaxies evolved in par-allel tracks in the size-mass relation. The second thing tonote is that the progenitors at z = 0 of quenched galaxiesevolved along two very different trajectories. When star for-mation was the dominant mode of evolution for progenitorsof quenched galaxies at z = 0, they evolved in a trajecto-ries with a slope of ∼ 0.35. This situation changed dramat-ically when star formation was suppressed: the slope of thetrajectory in the size-mass relation now became ∼ 2.5, im-plying that the most massive galaxies increased their sizeby a factor of ∼ 3 after they quenched. One of the mostpopular explanations for this upturn in the size-mass rela-tion is dry minor mergers. Indeed, Hilz, Naab & Ostriker(2013) showed that dry minor mergers of diffuse satellitesembedded in dark matter halos produced slopes ∼ 2 consis-tent with our findings. In order to investigate this further,we present the merger rate history for the progenitors dis-cussed above in the bottom panels of Figure 20. The leftpanel plots the merger rate histories from minor mergers,

µ∗ ! 1/4, while the right panel is the same but from majormergers, µ∗ " 1/4. Note finally that at all times galaxiesare, on average, growing inside-out.

Similarly to Figure 13, Figure 20 shows that the mergerrate history is dominated by minor mergers. Moreover, in thecase of the most massive halos we do observe an upturn inthe merger rate approximately at the same epoch when theupturn in the size-mass relation happened due to quenching.Therefore, we conclude that our semi-empirical results arenot in conflict with the minor merger hypothesis.

The next thing to note, and perhaps the most surpris-ing, is the fact that galaxies with sizes above Reff ∼ 3 kpcare more likely to be quenched, as can be seen by the dashedlines in the upper panel of Figure 20 (and also in Panel f) ofFigure 23). Recall that the dashed lines mark the transitionabove/below which galaxies are likely to be quench/star-forming. We believe that the above result does not indi-cate something fundamental on how galaxies quench, but isjust a coincidence. As we will discuss in Section 8.3, similartrajectories in the size-mass relation have been reported inprevious studies (e.g., Patel et al. 2013; van Dokkum et al.2013). Our findings are not in conflict with those previousempirical results.

8 DISCUSSION

The semi-empirical inferences of galaxy mass and structuralevolution presented here refer to the average behaviour ofthe whole galaxy population as a function of halo or stellarmass at z = 0. While this is clearly an oversimplification, ourresults provide relevant clues to constrain the main processesof galaxy evolution and their dependence on galaxy mass.

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Figure 20. Trajectories for progenitors of halos with Mvir = 1011, 1011.5, 1012, 1013, 1014 and 1015M⊙ at z = 0 in the size-mass relation(upper left panel) and for the minor (bottom left) and major (bottom right) merger rates. As in Figure 19, the symbols show differentredshifts as indicated by the labels. The dashed lines show the transition below which most galaxies are quiescent, and the upper (lower)dot-dashed curves show where 25% (75%) of the galaxies are quenched. Progenitors of quenched galaxies went through two phases,initially they grew in parallel trajectories as star formers in the size-mass relation, but after quenching they evolved much faster in sizethan in mass, resulting in a steeper relation at low z. Presumably the high rate of minor mergers is responsible for this rapid size growth.

We now discuss some implications and interpretations of ourresults.

8.1 Comparison with Previous Studies

In this section we compare to previous works that derivedthe SHMR at different redshifts. We divide our discussioninto two main comparisons: those studies that reported stel-lar mass as function of halo mass, SHMR, and those thathave estimated the inverse of this relation, Mvir −M∗. Theformer is typically reported in studies based on statisticalapproaches, namely indirect methods, as in our case, whilethe latter is more natural for studies based on direct deter-minations (e.g., weak gravitational lensing). All results wereadjusted to a Hubble parameter of h = 0.678 and to virialhalo masses. When required, we adjusted stellar masses toa Chabrier IMF.

8.1.1 Stellar-to-halo mass relationship

We begin by comparing mean stellar masses as a functionof virial mass, i.e., the SHMR. This is shown in Figure 21for seven different redshift bins. Our resulting SHMRs areshown with the solid black lines. The grey areas in the upperpanels show the standard deviations from the set of 3× 105

MCMC models described in Section 5 – in other words, theyrepresent the 1σ confidence level of our inferences. System-atic errors are usually of the order of ∼ 0.25 dex and mayincreases with redshift.

In Figure 21 we compare our results with those reportedin Guo et al. (2010, constrained only at z ∼ 0.1, violetsolid line), Behroozi, Wechsler & Conroy (2013b, blue solidlines) and Moster, Naab & White (2013, red solid lines).These authors used subhalo abundance matching to derivethe SHMRs. Guo et al. (2010) and Moster, Naab & White(2013) used only the GSMF as a constraint, while Behroozi,Wechsler & Conroy (2013b) included the observed specificSFRs and the CSFR for constraining the best fit parame-ters in their model. The filled circles with error bars show

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Change in slope in Reff - M diagram after quenching

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Figure 13. Left Panel: Instantaneous fraction of mass that formed ex-situ and was accreted by galaxy mergers as a function of thehalo mass at redshift z = 0. Right Panel: Cumulative fraction of mass that formed ex-situ and accreted through galaxy mergers. Notethat 20% of the final mass in host galaxies of halos with Mvir(0) = 1× 1015 was accreted by galaxy mergers.

Figure 14. Left Panel: Galaxy major merger rate for galaxies with masses above 1× 1010M∗. Solid lines show the predictions basedon our new SHMR while the different symbols show observational estimates from Conselice et al. (2003); Conselice, Rajgor & Myers(2008); Conselice, Yang & Bluck (2009); Lopez-Sanjuan et al. (2009) and Lopez-Sanjuan et al. (2010) based on galaxy asymmetries whileBundy et al. (2009) gives the merger rate fraction from galaxy pairs. Right Panel: Similarly above but for galaxies with masses above1× 1010.8M∗. Symbols are fromBluck et al. (2009) using galaxy asymmetries, Lopez-Sanjuan et al. (2012); Man et al. (2012); Man, Zirm& Toft (2016) and Williams, Quadri & Franx (2011) based on galaxy pairs.

into account the “true” time that it would take to the hostsatellite to merge with the central galaxy. Therefore, whencomparing our predicted merger rate with observations oneshould keep in mind that these results could represent anupper limit to the true merger rates of galaxies. Neverthe-less, as shown by Wetzel & White (2010) the disruption ofsubhalos occurs mainly in the inner regions of the halo wherethe tidal forces are strongest, while only a small fraction aredisrupted at larger radii.

We calculate galaxy mergers by convolving subhalo dis-ruption rates with the evolution of the SHMRs:

dRm

d log µ∗

(µ∗|Mvir, z) =

!

P (µ∗|Mpeak, z)×

dRm

d log µpeak(µpeak|Mvir, z)× d log µpeak, (56)

where dRm/d log µpeak is the disruption rate per host haloper logarithmic interval in subhalo peak mass to primaryhalo mass and per unit of redshift. We use the fitted rela-tion from Behroozi, Wechsler & Conroy (2013b). The distri-bution P (µ∗|Mpeak, z) is given by Equation 10 where µ∗ isthe observed satellite-to-central galaxy stellar mass ratio in ahalo of mass Mvir. Note that the above distribution includesuncertainties due to random errors from stellar masses inaddition to the intrinsic scatter of the SHMR.

Figure 14 shows the galaxy major merger rates calcu-lated using our SHMRs and compared to observations. Inorder to compare directly to our model predictions, we com-

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dRm

d log µ∗

(µ∗|Mvir, z) =

!


dRm




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dRm

d log µ∗

(µ∗|Mvir, z) =

!


dRm




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Figure 14. Left Panel: Galaxy major merger rate for galaxies with masses above 1 × 1010M∗. Solid lines show the predictions based on our new SHMR while the different symbols show observational estimates from Conselice et al. (2003); Conselice, Rajgor & Myers (2008); Conselice, Yang & Bluck (2009); L opez-Sanjuan et al. (2009) and L opez-Sanjuan et al. (2010) based on galaxy asymmetries while Bundy et al. (2009) gives the merger rate fraction from galaxy pairs. Right Panel: Similarly above but for galaxies with masses above 1×1010.8M∗. Symbols are fromBluck et al. (2009) using galaxy asymmetries, L opez-Sanjuan et al. (2012); Man et al. (2012); Man, Zirm & Toft (2016) and Williams, Quadri & Franx (2011) based on galaxy pairs.

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