new constraints on the evolution of the stellar- to-dark matter … · evolution of the...

New Constraints on the evolution of the Stellar-

to-Dark Matter Connection

Alexie LeauthaudLBNL & Berkeley Center for

Cosmological Physics

J. Tinker, K.Bundy, P.Behroozi, R.Massey, J.Rhodes, M.George, J.P.Kneib, A.Benson, R.Wechsler, M.Busha,

P.Capak, O.Ilbert, A.Koekemoer, O.Le Fevre, S.Lilly, H.McCracken, M.Salvato, T.Schrabback, N.Scoville, T.Smith,

J.Taylor, and the COSMOS collaboration

OUTLINE

Motivation for combining dark matter probes

Theoretical Framework

Application to COSMOS data

Results : Form and Evolution of the Stellar-to-

Halo Mass Relation (SHMR).

Motivation for this project

How are galaxies related to the dark matter density field?

We have several tools for this purpose ...

?

Hubble deep field Millenium simulationSpringel et al. 2005

Three Dark Matter Probes

log10( stellar mass)

Num

ber p

er v

olum

e

The galaxy stellar mass function :• Number of galaxies per unit volume• “easy” to calculate• Typically modelled through “abundance matching”

1

θ [arcseconds]

w(θ

) [d

imen

sionl

ess]

Galaxy auto correlation function :• Excess probability above random of finding two

galaxies with a given separation• Typically modelled through HOD models

2

Transverse distance R [Mpc]0.1 1.0Δ

Σ : e

xces

s sur

face

m

ass d

ensit

y Galaxy-galaxy lensing :• Measures the galaxy-matter correlation function• Weak signal that is difficult to measure• Tells us directly about the galaxy-dark matter

connection3

Combining multiple Dark Matter Probes

Most studies so far have only ever used one type of probe to study the link between galaxies and dark matter.

Nonetheless, each probe contains different information. Combining probes makes sense in order to understand the beast :

1

2

3

The galaxy-dark matter connection• Building a more robust probe• Galaxy formation• Informing semi-analytic models

Cosmological parameters: Ωm, σ8

Yoo et al. 2006, Cacciato et al. 2009 the combination of lensing and clustering is a cosmological probe.

Modified gravity as an alternative to Dark EnergyΦ: dynamicsΨ+ Φ: lensing of light around galaxies Screening mechanisms on linear, quasi linear scales. Need to understand the galaxy-dark matter connection.

Motivation for combining dark matter probes

0.01 0.10 1.00Physical transverse distance, R [ Mpc h72

-1 ]

0.1

1.0

10.0

100.0

1000.0

[

h72

M

O • p

c -2 ]

Stellar mass: M*=1011 MO •

0.01 0.10 1.00Physical transverse distance, R [ Mpc h72

-1 ]

0.1

1.0

10.0

100.0

1000.0

[

h72

MO •

pc -2 ]

Stellar mass: M*=1010 MO • Point source ( stellar)One halo centralOne halo satelliteTwo halo ( 2h)Combined signal ( tot)

The Galaxy-Galaxy Lensing Signal

Central

Satellite

ΔΣ: surface mass density contrast

ΔΣ [

Msu

n pc

-2]

Theoretical Framework

Halo Occupation Distributions (HOD)

We have developed the theoretical framework necessary to combine galaxy-galaxy lensing, galaxy clustering, and the galaxy stellar mass function

Our model is an extension to the HOD framework

Our model enables us to:

- self consistently model the three observables- Fit for the stellar-to-halo mass relation (SHMR)- account for intrinsic scatter and measurement error- fit data from multiple probes while allowing independent binning schemes for each probe

The Stellar-to-Halo mass relation (SHMR)

Stellar mass

Hal

o m

ass •• • •• • •••• •

• •• • •• ••••••

••••••••••• ••••• ••••••••• •••

P(M*|Mh) is lognormal with :- a mean relation: fSHMR

- a logarithmic scatter: σlogM*

σlogM*

8 A. Leauthaud

TABLE 3Binning scheme for the g-g lensing

g-g bin1 g-g bin2 g-g bin3 g-g bin4 g-g bin5 g-g bin6 g-g bin7

z1 = [0.22, 0.48] min log10(M!) 11.12 10.89 10.64 10.3 9.82 9.2 8.7max log10(M!) 12.0 11.12 10.89 10.64 10.3 9.8 9.2


z3 = [0.74, 1.0] min log10(M!) 11.35 11.16 10.97 10.74 10.39 9.8 nonemax log10(M!) 12.0 11.35 11.16 10.97 10.74 10.39 none

In Paper I we described a HOD-based model that canbe used to analytically predict the SMF, g-g lensing,and clustering signals. A key component of this modelis the SHMR which is modelled as a log-normal prob-ability distribution function with a log-normal scatter3

noted !logM!and with a mean-log relation that is noted

as M! = fshmr(Mh).For a given parameter set and cosmology, fshmr and

!logM!can be used to determine the central and satel-

lite occupations functions, !Ncen" and !Nsat". These arethen in turn used to predict the SMF, g-g lensing, andclustering signals.

4.1. The stellar-to-halo mass relation

Following Behroozi et al. (2010) (hereafter “B10”),fshmr(Mh) is mathematically defined via its inverse func-tion:

log10(f"1shmr(M!)) = log10(Mh) =

log10(M1) + " log10

!

M!

M!,0

"

+

#

M!

M!,0

$!

1 +#

M!

M!,0

$"" #1

2

(13)where M1 is a characteristic halo mass, M!,0 is a charac-teristic stellar mass, " is the faint end slope, and # and $control the massive end slope. We refer to B10 for a moredetailed justification of this functional form. Briefly, weexpect that at least 4 parameters are required to modelthe SHMR: a normalization, break, a faint end slope anda bright end slope. In addition, B10 have suggested thatthe SHMR displays sub-exponential behaviour at largeM!. This is modelled by the # parameter which leads toa total of 5 parameters. Figure 3 illustrates the influenceof each parameter on the shape on the SHMR and fur-ther details on the role of each parameter can be foundin § 2.1 of Paper I.

In contrast to B10, however, we do not model the red-shift evolution of this functional form. Instead, we binthe data into three redshift bins and check for redshiftevolution in the parameters a posteriori. We also assumethat Equation 13 is only relevant for central galaxies.

4.2. Scatter between stellar and halo mass

The measured scatter in stellar mass at fixed halo masshas an intrinsic component (noted !i

logM!

), but also in-cludes a stellar mass measurement error due to redshift,photometry, and modeling uncertainties (noted !m

logM!

).Ideally, we would measure both components but unfortu-nately we can only constrain the quadratic sum of these

3 Scatter is quoted as the standard deviation of the logarithmbase 10 of the stellar mass at fixed halo mass.

two sources of scatter. Nonetheless, given a model for!m

logM!

, we could in principle extract !ilogM!

from !logM!.

Previous work suggests that !logM!is independent of

halo mass. For example, Yang et al. (2009) find that!logM!

= 0.17 dex and More et al. 2009 find a scat-ter in luminosity at fixed halo mass of 0.16 ± 0.04 dex.Both Moster et al. (2010) and B10 are able to fit theSDSS galaxy SMF assuming !logM!

= 0.15 dex and!logM!

= 0.175 dex respectively. However, these resultsare derived with spectroscopic samples of galaxies. Incontrast to these surveys, we expect a larger measure-ment error for the COSMOS stellar masses due to theuse of photometric redshifts. In addition, since photozerrors increase for fainter galaxies, we might also expectthat !m

logM!

(and thus !logM!) will depend on M!.

To test if the assumption that !logM!is constant has

any impact on our results, we implement two models for!logM!

. In the first case (called “sig mod1”), !logM!is

assumed to be constant (this is our base-line model). Inthe second case (called “sig mod2”), we explicitly model!m

logM!

to reflect stellar mass measurement errors. Notethat the goal of this exercise is not to perform a carefuland thorough error analysis, but simply to build a realis-tic enough model to asses whether or not a M! dependanterror has any strong impact on our conclusions.

For the sig mod2 model, we consider three contribu-tions to the stellar mass error budget. The first is called“model error”: this is measured by the 68% confidenceinterval of the mass probability distribution determinedfor each galaxy by the mass estimator. It represents therange of model templates (each with its own M/L ratio)that provide reasonable fits to the observed SED. Thisrange is determined both by degeneracies in the grid ofmodels used to fit the data, as well as by the photometricuncertainty in the observed SED. The second term is thephoto-z error, which derives from the uncertainty in theluminosity distance owing to the error on a given photo-metric redshift. The final component is the photometricuncertainty from the observed K-band magnitude, whichtranslates into an uncertainty in luminosity and thereforestellar mass. The total measurement error, !m

logM!

is thesum in quadrature of these three sources of error. Theresults are shown in Figure 4 for the three redshift bins.

As detailed in § 5, however, we find that our re-sults are largely unchanged, regardless of which formwe adopt for !logM!

. This can be explained as fol-lows. Since the data are binned by M!, the observ-ables are in fact sensitive to the scatter in halo massat fixed stellar mass, noted !logMh

. Given a model forthe SHMR, !logMh

can be mathematically derived from!logM!

. Further details on the mathematical connec-tion between between !logM!

and !logMhcan be found

Adopted from Behroozi et al. 2010

For central galaxies :

109 1010 1011

1011

1012

1013

1014

1015

Hal

o M

ass

M20

0b

[ M

O • ] M1=12.19M1=12.51M1=12.83

109 1010 1011

1011

1012

1013

1014

1015 M*0=10.63M*0=10.91M*0=11.19

109 1010 1011

Stellar Mass M* [ MO • ]

1011

1012

1013

1014

1015 =0.31=0.46=0.61

109 1010 1011

1011

1012

1013

1014

1015

Hal

o M

ass

M20

0b

[ M

O • ]

=0.24=0.48=0.72

109 1010 1011

1011

1012

1013

1014

1015 =-0.96=0.34=1.64

109 1010 1011

10

100

1000

10000

M20

0b/M

*

109 1010 1011

10

100

1000

10000

109 1010 1011


10

100

1000

10000

109 1010 1011

10

100

1000

10000

M20

0b/M

*

109 1010 1011

10

100

1000

10000

M1 : y-axis (halo mass) normalizationM*0 : x-axis (stellar mass) normalizationβ : low mass slopeδ : controls high mass sub-exponential behaviourγ : controls transition regime

σlogM*

The Stellar-to-Halo mass relation (SHMR)

M1 M*0 β γδ

Ten Parameter Model

+ Tinker et al. 2008 halo mass function+ Tinker et al. 2010 bias function+ Halo exclusion

Parametric form for the stellar-to-halo mass relation (M1, M*0, β, δ, γ)

8 A. Leauthaud

TABLE 3Binning scheme for the g-g lensing

g-g bin1 g-g bin2 g-g bin3 g-g bin4 g-g bin5 g-g bin6 g-g bin7



z3 = [0.74, 1.0] min log10(M!) 11.35 11.16 10.97 10.74 10.39 9.8 nonemax log10(M!) 12.0 11.35 11.16 10.97 10.74 10.39 none

In Paper I we described a HOD-based model that canbe used to analytically predict the SMF, g-g lensing,and clustering signals. A key component of this modelis the SHMR which is modelled as a log-normal prob-ability distribution function with a log-normal scatter3

noted !logM!and with a mean-log relation that is noted

as M! = fshmr(Mh).For a given parameter set and cosmology, fshmr and

!logM!can be used to determine the central and satel-

lite occupations functions, !Ncen" and !Nsat". These arethen in turn used to predict the SMF, g-g lensing, andclustering signals.

4.1. The stellar-to-halo mass relation

Following Behroozi et al. (2010) (hereafter “B10”),fshmr(Mh) is mathematically defined via its inverse func-tion:

log10(f"1shmr(M!)) = log10(Mh) =

log10(M1) + " log10

!

M!

M!,0

"

+

#

M!

M!,0

$!

1 +#

M!

M!,0

$"" #1

2

(13)where M1 is a characteristic halo mass, M!,0 is a charac-teristic stellar mass, " is the faint end slope, and # and $control the massive end slope. We refer to B10 for a moredetailed justification of this functional form. Briefly, weexpect that at least 4 parameters are required to modelthe SHMR: a normalization, break, a faint end slope anda bright end slope. In addition, B10 have suggested thatthe SHMR displays sub-exponential behaviour at largeM!. This is modelled by the # parameter which leads toa total of 5 parameters. Figure 3 illustrates the influenceof each parameter on the shape on the SHMR and fur-ther details on the role of each parameter can be foundin § 2.1 of Paper I.

In contrast to B10, however, we do not model the red-shift evolution of this functional form. Instead, we binthe data into three redshift bins and check for redshiftevolution in the parameters a posteriori. We also assumethat Equation 13 is only relevant for central galaxies.

4.2. Scatter between stellar and halo mass

The measured scatter in stellar mass at fixed halo masshas an intrinsic component (noted !i

logM!

), but also in-cludes a stellar mass measurement error due to redshift,photometry, and modeling uncertainties (noted !m

logM!

).Ideally, we would measure both components but unfortu-nately we can only constrain the quadratic sum of these

3 Scatter is quoted as the standard deviation of the logarithmbase 10 of the stellar mass at fixed halo mass.

two sources of scatter. Nonetheless, given a model for!m

logM!

, we could in principle extract !ilogM!

from !logM!.

Previous work suggests that !logM!is independent of

halo mass. For example, Yang et al. (2009) find that!logM!

= 0.17 dex and More et al. 2009 find a scat-ter in luminosity at fixed halo mass of 0.16 ± 0.04 dex.Both Moster et al. (2010) and B10 are able to fit theSDSS galaxy SMF assuming !logM!

= 0.15 dex and!logM!

= 0.175 dex respectively. However, these resultsare derived with spectroscopic samples of galaxies. Incontrast to these surveys, we expect a larger measure-ment error for the COSMOS stellar masses due to theuse of photometric redshifts. In addition, since photozerrors increase for fainter galaxies, we might also expectthat !m

logM!

(and thus !logM!) will depend on M!.

To test if the assumption that !logM!is constant has

any impact on our results, we implement two models for!logM!

. In the first case (called “sig mod1”), !logM!is

assumed to be constant (this is our base-line model). Inthe second case (called “sig mod2”), we explicitly model!m

logM!

to reflect stellar mass measurement errors. Notethat the goal of this exercise is not to perform a carefuland thorough error analysis, but simply to build a realis-tic enough model to asses whether or not a M! dependanterror has any strong impact on our conclusions.

For the sig mod2 model, we consider three contribu-tions to the stellar mass error budget. The first is called“model error”: this is measured by the 68% confidenceinterval of the mass probability distribution determinedfor each galaxy by the mass estimator. It represents therange of model templates (each with its own M/L ratio)that provide reasonable fits to the observed SED. Thisrange is determined both by degeneracies in the grid ofmodels used to fit the data, as well as by the photometricuncertainty in the observed SED. The second term is thephoto-z error, which derives from the uncertainty in theluminosity distance owing to the error on a given photo-metric redshift. The final component is the photometricuncertainty from the observed K-band magnitude, whichtranslates into an uncertainty in luminosity and thereforestellar mass. The total measurement error, !m

logM!

is thesum in quadrature of these three sources of error. Theresults are shown in Figure 4 for the three redshift bins.

As detailed in § 5, however, we find that our re-sults are largely unchanged, regardless of which formwe adopt for !logM!

. This can be explained as fol-lows. Since the data are binned by M!, the observ-ables are in fact sensitive to the scatter in halo massat fixed stellar mass, noted !logMh

. Given a model forthe SHMR, !logMh

can be mathematically derived from!logM!

. Further details on the mathematical connec-tion between between !logM!

and !logMhcan be found

Central occupation function (σlog(M*))

Satellite occupation function (Bcut, Bsat, βcut, βsat)

Dark matter probes 5

occupation functions for binned samples are trivially de-rived from the occupation function for threshold samplesvia:

!Ncen(Mh|M t1! , M t2

! )" = !Ncen(Mh|M t1! )"#!Ncen(Mh|M t2

! )"(5)

and

!Nsat(Mh|M t1! , M t2

! )" = !Nsat(Mh|M t1! )"#!Nsat(Mh|M t2

! )".(6)

3.2. Functional form for !Ncen"For a threshold sample of galaxies, !Ncen(Mh|M t1

! )" isfully specified given !c(M!|Mh) according to:

!Ncen(Mh|M t1! )" =

! "

Mt1!

!c(M!|Mh)dM!. (7)

To begin with, let us consider the most simple modelin which !logM!

is constant. Because !c is parameterizedas a log-normal, the central occupation function can beanalytically and conveniently derived from Equation 7by considering the cumulative distribution function ofthe Gaussian:

!Ncen(Mh|Mt1! )" =

1

2

"

1# erf

#log10(Mt1

! ) # log10(fshmr(Mh))$2!logM!

$%

(8)

where erf is the error function defined as:

erf(x) =2$"

! x

0e#t2dt (9)

It is important to note that Equation 8 is only validwhen !logM!

is constant. In the more general case where!logM!

varies with Mh, !Ncen" can nonetheless be calcu-lated by numerically integrating Equation 7. In PaperII, we will consider cases in which !logM!

varies due tothe e"ect of stellar mass dependant measurement errors.

We note that most readers may be more familiar witha simplified version of Equation 8 that assumes thatfshmr(Mh) is a power law. We will now describe the as-sumptions made in order to obtain the more commonlyemployed equation for !Ncen" from Equation 8.

If we make the assumption that fshmr(Mh) % Mph and

we define Mmin such that M t1! = fshmr(Mmin) (in other

terms, Mmin is the inverse of the SHMR relation for thestellar mass threshold M t1

! ) then using Equation 8 wecan write that:

!Ncen(Mh|M t1! )"

=1

2

"

1 # erf

#log10(M

t1! ) # log10(fshmr(Mh))$

2!logM!

$%

=1

2

"

1 # erf

#log10(M

pmin) # log10(M

ph)$

2!logM!

$%

(10)

If we now use the fact that erf(#x) = #erf(x) and ifwe define &!logM such that &!logM & !logM!

/p we find that:

!Ncen(Mh|M t1! )" =

1

2

"

1 + erf

#log10(Mh) # log10(Mmin)$

2&!logM

$%

(11)which is a commonly employed formula for !Ncen".Firstly, it is important to note that Equation 11 is only anapproximation for !Ncen" for the case when the SHMR isa power-law and is certainly not valid over a large rangeof stellar masses. Secondly, &!logM can be interpreted asthe scatter in halo mass at fixed stellar mass if and only ifthe SHMR is a power-law and if !logM!

is constant. Sincethere is accumulating evidence that the SHMR is not asingle power law (and the same is in general true for therelationship between halo mass and galaxy luminosity),we recommend using Equation 8 instead of Equation 11.

Figure 2 illustrates the di"erence in !Ncen" when Equa-tion 8 is used instead of Equation 11. At log10(M!) !10.2, the functional form for the SHMR has a sub-exponential behaviour and as a result, !Ncen" begins todeviate from a simple erf function. Assuming that Equa-tion 8 correctly represents !Ncen", the error made onMmin can be of order 10 to 20% at log10(Mmin) ! 12if Equation 11 is used to fit !Ncen" instead of Equation8.

We note that this does not invalidate equation 11 as apossible parameterization of the central occupation func-tion. The point we wish to stress is that the common in-terpretation of the scatter constrained by this parmater-ization is not proportional to a the scatter in a lognormaldistribution of stellar mass at fixed halo mass.

3.3. Functional form for !Nsat"In addition to the five parameters introduced to model

!Ncen" and !logM!, we now introduce five new parame-

ters to model !Nsat". In order to simultaneously fit g-glensing, clustering, and stellar mass function measure-ments that employ di"erent binning schemes, we requirea model for !Nsat" that is independent of the binningscheme.

Numerical simulations demonstrate that the occupa-tion of subhalos (e.g., Kravtsov et al. 2004; Conroy et al.2006) and satellite galaxies in cosmological hydrody-namic simulations (Zheng et al. 2005) follow a power lawat high host halo mass, then fall o" rapidly when themean occupation becomes significantly less than unity.Thus we parameterize the satellite occupation functiona power of host mass with an exponential cuto":

!Nsat(Mh|M t1! )" =

'Mh

Msat

(!sat

exp

'#Mcut

Mh

((12)

where #sat represents the power-law slope of the satellitemean occupation function, Msat defines the amplitude ofthe power-law, and Mcut sets the scale of the exponentialcut-o".

Observational analyses have demonstrated that thereis a self-similarity in occupation functions such thatMsat/Mmin ' constant for luminosity-defined sam-ples (Zehavi et al. 2005; Zheng et al. 2007, 2009; TheSDSS Collaboration et al. 2010), where Mmin is takenfrom equation (11) and is conceptually equivalent tof#1shmr(M!), where M! is the stellar mass threshold of the

Dark matter probes 5

occupation functions for binned samples are trivially de-rived from the occupation function for threshold samplesvia:

!Ncen(Mh|M t1! , M t2

! )" = !Ncen(Mh|M t1! )"#!Ncen(Mh|M t2

! )"(5)

and

!Nsat(Mh|M t1! , M t2

! )" = !Nsat(Mh|M t1! )"#!Nsat(Mh|M t2

! )".(6)

3.2. Functional form for !Ncen"For a threshold sample of galaxies, !Ncen(Mh|M t1

! )" isfully specified given !c(M!|Mh) according to:

!Ncen(Mh|M t1! )" =

! "

Mt1!

!c(M!|Mh)dM!. (7)

To begin with, let us consider the most simple modelin which !logM!

is constant. Because !c is parameterizedas a log-normal, the central occupation function can beanalytically and conveniently derived from Equation 7by considering the cumulative distribution function ofthe Gaussian:

!Ncen(Mh|M t1! )" =

1

2

"

1 # erf

#log10(M


2!logM!

$%

(8)

where erf is the error function defined as:

erf(x) =2$"

! x

0e#t2dt (9)

It is important to note that Equation 8 is only validwhen !logM!

is constant. In the more general case where!logM!

varies with Mh, !Ncen" can nonetheless be calcu-lated by numerically integrating Equation 7. In PaperII, we will consider cases in which !logM!

varies due tothe e"ect of stellar mass dependant measurement errors.

We note that most readers may be more familiar witha simplified version of Equation 8 that assumes thatfshmr(Mh) is a power law. We will now describe the as-sumptions made in order to obtain the more commonlyemployed equation for !Ncen" from Equation 8.

If we make the assumption that fshmr(Mh) % Mph and

we define Mmin such that M t1! = fshmr(Mmin) (in other

terms, Mmin is the inverse of the SHMR relation for thestellar mass threshold M t1

! ) then using Equation 8 wecan write that:

!Ncen(Mh|M t1! )"

=1

2

"

1 # erf

#log10(M


2!logM!

$%

=1

2

"

1 # erf

#log10(M

pmin) # log10(M

ph)$

2!logM!

$%

(10)

If we now use the fact that erf(#x) = #erf(x) and ifwe define &!logM such that &!logM & !logM!

/p we find that:

!Ncen(Mh|M t1! )" =

1

2

"

1 + erf

#log10(Mh) # log10(Mmin)$

2&!logM

$%

(11)which is a commonly employed formula for !Ncen".Firstly, it is important to note that Equation 11 is only anapproximation for !Ncen" for the case when the SHMR isa power-law and is certainly not valid over a large rangeof stellar masses. Secondly, &!logM can be interpreted asthe scatter in halo mass at fixed stellar mass if and only ifthe SHMR is a power-law and if !logM!

is constant. Sincethere is accumulating evidence that the SHMR is not asingle power law (and the same is in general true for therelationship between halo mass and galaxy luminosity),we recommend using Equation 8 instead of Equation 11.

Figure 2 illustrates the di"erence in !Ncen" when Equa-tion 8 is used instead of Equation 11. At log10(M!) !10.2, the functional form for the SHMR has a sub-exponential behaviour and as a result, !Ncen" begins todeviate from a simple erf function. Assuming that Equa-tion 8 correctly represents !Ncen", the error made onMmin can be of order 10 to 20% at log10(Mmin) ! 12if Equation 11 is used to fit !Ncen" instead of Equation8.

We note that this does not invalidate equation 11 as apossible parameterization of the central occupation func-tion. The point we wish to stress is that the common in-terpretation of the scatter constrained by this parmater-ization is not proportional to a the scatter in a lognormaldistribution of stellar mass at fixed halo mass.

3.3. Functional form for !Nsat"In addition to the five parameters introduced to model

!Ncen" and !logM!, we now introduce five new parame-

ters to model !Nsat". In order to simultaneously fit g-glensing, clustering, and stellar mass function measure-ments that employ di"erent binning schemes, we requirea model for !Nsat" that is independent of the binningscheme.

Numerical simulations demonstrate that the occupa-tion of subhalos (e.g., Kravtsov et al. 2004; Conroy et al.2006) and satellite galaxies in cosmological hydrody-namic simulations (Zheng et al. 2005) follow a power lawat high host halo mass, then fall o" rapidly when themean occupation becomes significantly less than unity.Thus we parameterize the satellite occupation functiona power of host mass with an exponential cuto":

!Nsat(Mh|Mt1! )" = !Ncen(Mh|Mt1

! )"'

Mh

Msat

(!sat

exp

'#Mcut

Mh

(

(12)where #sat represents the power-law slope of the satellitemean occupation function, Msat defines the amplitude ofthe power-law, and Mcut sets the scale of the exponentialcut-o".

Observational analyses have demonstrated that thereis a self-similarity in occupation functions such thatMsat/Mmin ' constant for luminosity-defined sam-ples (Zehavi et al. 2005; Zheng et al. 2007, 2009; TheSDSS Collaboration et al. 2010), where Mmin is takenfrom equation (11) and is conceptually equivalent tof#1shmr(M!), where M! is the stellar mass threshold of the

Applying this model to Cosmos data

The COSMOS HST survey

HST ACSCycles 12 & 13590 orbits

✔ Weak lensing

Scoville et al. 2007

0.0 0.2 0.4 0.6 0.8 1.0Photometric Redshift

6

7

8

9

10

11

12lo

g 10(M

stella

r)Cross-overmass

Active galaxies completness limitPassive galaxies completness limit

Stellar mass distribution in COSMOS

Photoz

log 1

0(M

*)

Mock catalogs: sample variance and co-variance

9.0 9.5 10.0 10.5 11.0 11.5 12.0Log10( M* [ MO • ] )

10-5

10-4

10-3

10-2

Log 1

0( dN

/dlo

gM* [

Mpc

-3 d

ex-1

])

9.0 9.5 10.0 10.5 11.0 11.5 12.0Log10( M* [ MO • ] )

10-5

10-4

10-3

10-2

Log 1

0( dN

/dlo

gM* [

Mpc

-3 d

ex-1

])

9.0 9.5 10.0 10.5 11.0 11.5 12.0Log10( M* [ MO • ] )

10-5

10-4

10-3

10-2

Log 1

0( dN

/dlo

gM* [

Mpc

-3 d

ex-1

])

9.0 9.5 10.0 10.5 11.0 11.5 12.0Log10( M* [ MO • ] )

10-5

10-4

10-3

10-2

Log 1

0( dN

/dlo

gM* [

Mpc

-3 d

ex-1

])

9.0 9.5 10.0 10.5 11.0 11.5 12.0Log10( M* [ MO • ] )

10-5

10-4

10-3

10-2

Log 1

0( dN

/dlo

gM* [

Mpc

-3 d

ex-1

])

9.0 9.5 10.0 10.5 11.0 11.5 12.0Log10( M* [ MO • ] )

10-5

10-4

10-3

10-2

Log 1

0( dN

/dlo

gM* [

Mpc

-3 d

ex-1

])

9.0 9.5 10.0 10.5 11.0 11.5 12.0Log10( M* [ MO • ] )

10-5

10-4

10-3

10-2

Log 1

0( dN

/dlo

gM* [

Mpc

-3 d

ex-1

])

First order effect of sample variance is a global up/down shift in the stellar

mass function

This will affect the halo mass

normalization of the SHMR

Las Damas “consuelo” box 420 h-1, 14003 particles, particle mass=0.1x1010 Msun

Mock catalogs: sample variance and co-variance

0.1 1.0

1

10

100

[

MO •

pc -2

]

11.12<log10(M*)<12.0

0.1 1.0

1

10

100

10.89<log10(M*)<11.12

0.1 1.0

1

10

100

10.64<log10(M*)<10.89

0.1 1.0

1

10

100

10.3<log10(M*)<10.64

0.1 1.01

10

100

[

MO •

pc -2

]

9.82<log10(M*)<10.3

0.1 1.0Physical transverse distance R [ Mpc ]

1

10

100

[ M

O • pc -2 ]

9.2<log10(M*)<9.82

0.1 1.0

1

10

100

[ M

O • pc -2 ]

8.7<log10(M*)<9.2

0.10.20.30.40.50.60.70.80.91.011.12<log10(M*)<12.0

1.5 1.0 0.5 0.0

1.5

1.0

0.5

0.0

0.10.20.30.40.50.60.70.80.91.010.3<log10(M*)<10.64

1.5 1.0 0.5 0.0log10( R [ Mpc ])

1.5

1.0

0.5

0.0

0.10.20.30.40.50.60.70.80.91.08.7<log10(M*)<9.2

1.5 1.0 0.5 0.0

1.5

1.0

0.5

0.0

Effects of sample variance on galaxy-galaxy lensing

Correlation coefficient matrix:

Physical transverse distance R [Mpc ]

ΔΣ

[ M

sun

pc-2]

Centrals Satellites

Results : Form and Evolution of the

Stellar-To-Halo Mass Relation

10 100 1000

0.01

0.10

1.00

10.00 a)w bin1

High M*

10 100 1000

0.01

0.10

1.00

10.00 b)w bin2

10 100 1000 [arcsec]

0.01

0.10

1.00

10.00 c)w bin3

10 100 1000 [arcsec]

0.01

0.10

1.00

10.00

w

()

d)w bin4

10 100 1000

0.01

0.10

1.00

10.00

w

()

e)w bin5

10 100 1000

0.01

0.10

1.00

10.00 f)w bin6

Low M*

10-2 0.1 1

0.1

1

10

102High M* h)

g-g bin1

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

9 10 11 12Log10 (M* [MO • ])

10-6

10-5

10-4

10-3

10-2

Log 1

0 dN

/dlo

gM* [

Mpc

-3 d

ex-1

]

g)

SDSS, Panter et al. 2004.SDSS, Baldry et al. 2008SDSS, Li et al. 2009 COSMOS, this work

10-2 0.1 1

0.1

1

10

102

[

M

O • p

c -2 ] i)

g-g bin2

10-2 0.1 1

0.1

1

10

102

j)g-g bin3

10-2 0.1 1

0.1

1

10

102

k)g-g bin4

10-2 0.1 10.1

1

10

102l)

g-g bin5

10-2 0.1 1Physical transverse distance R [ Mpc ]

0.1

1

10

102

m)g-g bin6

10-2 0.1 1

0.1

1

10

102

n)g-g bin7

Low M*

Gal

axy-

gala

xy le

nsin

gC

lust

erin

g

W(θ

)

Stellar mass function

ΔΣ

[h 7

2 M

sun

pc-2]

Physical transverse distance R [Mpc h72-1]z=[0.22,0.48]

10 100 1000

0.01

0.10

1.00

10.00 a)w bin1

High M*

10 100 1000

0.01

0.10

1.00

10.00 b)w bin2

10 100 1000 [arcsec]

0.01

0.10

1.00

10.00 c)w bin3

10 100 1000 [arcsec]

0.01

0.10

1.00

10.00

w

()

d)w bin4

10 100 1000

0.01

0.10

1.00

10.00 e)w bin5

Low M*

0 2 4 6 8 100

2

4

6

8

10

z = [0.48,0.74]

10-2 0.1 1

0.1

1

10

102g)

g-g bin1High M*

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

9 10 11 12Log10 (M* [MO • ])

10-6

10-5

10-4

10-3

10-2

Log 1

0 dN

/dlo

gM* [

Mpc

-3 d

ex-1

]

f)

10-2 0.1 1

0.1

1

10

102

[

M

O • p

c -2 ] h)

g-g bin2

10-2 0.1 1

0.1

1

10

102

i)g-g bin3

10-2 0.1 1

0.1

1

10

102

j)g-g bin4

10-2 0.1 10.1

1

10

102k)

g-g bin5


0.1

1

10

102

l)g-g bin6

10-2 0.1 1

0.1

1

10

102

m)g-g bin7

Low M*

Gal

axy-

gala

xy le

nsin

gC

lust

erin

g

W(θ

)


ΔΣ

[h 7

2 M

sun

pc-2]


10 100 1000

0.01

0.10

1.00

10.00 a)w bin1

High M*

10 100 1000 [arcsec]

0.01

0.10

1.00

10.00 b)w bin2

10 100 1000 [arcsec]

0.01

0.10

1.00

10.00 c)w bin3

10 100 1000 [arcsec]

0.01

0.10

1.00

10.00

w

()

d)w bin4

Low M*

0 2 4 6 8 100

2

4

6

8

10

0 2 4 6 8 100

2

4

6

8

10

z = [0.74,1.0]

10-2 0.1 1

0.1

1

10

102f)

g-g bin1High M*

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

9 10 11 12Log10 (M* [MO • ])

10-6

10-5

10-4

10-3

10-2

Log 1

0 dN

/dlo

gM* [

Mpc

-3 d

ex-1

]

e)

10-2 0.1 1

0.1

1

10

102

[

M

O • p

c -2 ] g)

g-g bin2

10-2 0.1 1

0.1

1

10

102

h)g-g bin3

10-2 0.1 1

0.1

1

10

102

i)g-g bin4

10-2 0.1 10.1

1

10

102j)

g-g bin5


0.1

1

10

102

k)g-g bin6

Low M*

Gal

axy-

gala

xy le

nsin

gC

lust

erin

g

W(θ

)


ΔΣ

[h 7

2 M

sun

pc-2]


0.6 0.8

sat

M*,0

11

11.05

11.1

0.44

0.46

0.48

0.4

0.6

0.8

1.5

2

2.5

logM

*

0.2

0.25

0.3

B cut

1234

B sat

8

10

12

cut

0.50

0.51

M1

sat

12.6512.712.7512.8

0.60.70.80.9

M*,0

11 11.05 11.10.44 0.46 0.48 0.4 0.6 0.8 1.5 2 2.5

logM*

0.2 0.25 0.3Bcut2 4

Bsat8 10 12

cut

0.5 0 0.5 1

Constraints on 10 parametersThe SHMR

Scatter

Satellites

107 108 109 1010 1011 1012

Stellar Mass M* [ -2 MO • ]

1010

1011

1012

1013

1014

1015

Hal

o M

ass

M

200b

[

-1 M

O • ]

Typical systematic error in stellar masses(between different surveys)

107 108 109 1010 1011 1012

1010

1011

1012

1013

1014

1015WL, COSMOS this paper, z=0.37

WL, Mandelbaum et al. 2006, z=0.1

WL, Leauthaud et al. 2010, z=0.3

WL, Hoekstra et al. 2007, z~0.2

AM, Moster et al. 2010, z=0.1

AM, Behroozi et al. 2010, z=0.1

SK, Conroy et al. 2007, z~0.06

SK, More et al. 2010, z~0.05

TF, Geha et al. 2006, z=0

TF, Pizagno et al. 2006, z=0

TF, Springob et al. 2005, z=0

107 108 109 1010 1011 1012

Stellar Mass M* [ -2 MO • ]

1

10

100

1000

10000

M20

0b/M

*

bin7 bin6 bin5 bin4 bin3 bin2 bin1

Stellar mass M* [Msun]

Hal

o m

ass

M20

0b [

Msu

n]M

200b

/M*

Star

form

atio

n ef

ficie

ncy

Stellar mass vs Halo mass

Pivot masses: location of the minimum of M200b/M* marks the halo mass in which accumulated stellar growth of the

central galaxy has been the mode efficient

Redshift evolution in the SHMR

109 1010 1011


1011

1012

1013

1014

1015

Hal

o M

ass

M

200b

[

MO • ]

Behroozi et al. 2010 z=0.1COSMOS z=0.37COSMOS z=0.66COSMOS z=0.88

Typical systematic error in stellar masses(when comparing COSMOS to SDSS)

109 1010 1011


1

10

100

1000

10000

M20

0b/M

*

Downsizingin Mh

piv and M*piv

Stellar mass M* [Msun] Stellar mass M* [Msun]

Hal

o m

ass

Mh

[M

sun]

Mha

lo /

M*

M*piv

Pivot ratio

Evolution of the “pivot masses”We find a downsizing evolution for the pivot stellar mass, and the pivot halo

mass. The pivot ratio however remains constant.

Pivot halo mass

0.0 0.2 0.4 0.6 0.8 1.0Redshift

6.0•1011

8.0•1011

1.0•1012

1.2•1012

1.4•1012

1.6•1012

1.8•1012

2.0•1012

Mhpi

v

[ MO • ]

Pivot stellar mass

0.0 0.2 0.4 0.6 0.8 1.0Redshift

1•1010

2•1010

3•1010

4•1010

5•1010

6•1010

M*pi

v

[ MO • ]

COSMOSBehroozi et al 2010Moster et al 2010

Pivot ratio

0.0 0.2 0.4 0.6 0.8 1.0Redshift

1020

30

40

50

60

70

(Mh/M

*)piv

Most previous results see a downsizing trend in Mhpiv

Normalization difference between Moster et al. 2010 and Behroozi et al. 2010 and our results

Pivot halo mass Pivot stellar mass Pivot ratio

First study to see a clear downsizing trend for M*piv

Systematic errors between stellar mass estimates in distinct surveys: ~ 0.25 dex

Moster et al 2010 do not account for these errors. Behroozi et al 2010 do account for these errors

The evolution of Mhpiv and M*piv leaves the pivot ratio constant

Evolution of the “pivot masses”We find a downsizing evolution for the pivot stellar mass, and the pivot halo

mass. The pivot ratio however remains constant.

Pivot halo mass

0.0 0.2 0.4 0.6 0.8 1.0Redshift

6.0•1011

8.0•1011

1.0•1012

1.2•1012

1.4•1012

1.6•1012

1.8•1012

2.0•1012

Mhpi

v

[ MO • ]

Pivot stellar mass

0.0 0.2 0.4 0.6 0.8 1.0Redshift

1•1010

2•1010

3•1010

4•1010

5•1010

6•1010

M*pi

v

[ MO • ]


Pivot ratio

0.0 0.2 0.4 0.6 0.8 1.0Redshift

1020

30

40

50

60

70

(Mh/M

*)piv


Pivot halo mass

0.0 0.2 0.4 0.6 0.8 1.0Redshift

6.0•1011

8.0•1011

1.0•1012

1.2•1012

1.4•1012

1.6•1012

1.8•1012

2.0•1012

Mhpi

v

[ MO • ]

Pivot stellar mass

0.0 0.2 0.4 0.6 0.8 1.0Redshift

1•1010

2•1010

3•1010

4•1010

5•1010

6•1010

M*pi

v

[ MO • ]


Pivot ratio

0.0 0.2 0.4 0.6 0.8 1.0Redshift

1020

30

40

50

60

70

(Mh/M

*)piv

P.Behroozi using our data

Stellar growth versus halo growth

1011 1012 1013

Halo Mass M200b [ MO • ]

0.01

0.10

M*,

cen/M

200b

Mh

0> 1

1

0Mh

0< 1

1

0

Mh < M* Mh > M*

z=0.8

8

z=0.3

7

The relative growth of the stellar mass has been stronger than the growth of the dark matter: λM* > λMh

• Halo mass has grown by a factor of λMh :

Mh(tlow) = λMh x Mh(thigh)

• Stellar mass has grown by a factor of λM*:

M*(tlow) = λM* x M*(thigh)

λM* = (η0/η1) x λMh

1011 1012 1013 1014


0.01

0.10

tota

l(M*)/

M20

0b

WMAP5 b/ m

Satellites

Centrals

z = 0.37

1011 1012 1013 1014

Halo Mass M200b [ MO · ]

0.01

0.10

tota

l(M*)/

M20

0b

WMAP5 b/ m

z=1

to z

=0.2

z=1

to z

=0.2

Total stellar content as a function of halo mass

Centrals dominate the total stellar content at Mh < 2x1013 Msun

Satellites dominate the total stellar content at Mh > 2x1013 Msun

Also see Conroy and Wechsler 2009

At Mh < 1012 Msun: stellar growth outpaces halo growth

At Mh > 1012 Msun: halo growth outpaces stellar growth (smooth accretion)

Critical halo massTcool>Tdyn

• growth ~ 22 Msun yr-1

• 3.2 Gyr between bins

•λMh ~ 1.34

A rough calculation of λM* and λMh

Noeske et al. 2007

A halo of 2x1011 Msun hosts a central galaxy of M* ≈1010Msun

• SFR ~ 2-7 Msun yr-1

• 3.2 Gyr between bins

•λM* ~ 2-3

Halo growth rate :

Fakhouri et al. 2010

Star formation rate :

Using this naive calculation :

1011 1012 1013


0.01

0.10

M*,

cen/M

200b

WMAP5 b/ m

Halo mass

M*,

cen/

M20

0b

Z=0.88

λM*= 2

λM*= 3

Plausible

Does not produce downsizing

The most simple evolution scenario

log(halo mass)

log(

M*/M

h)

log(halo mass) log(halo mass)

λM* = λMh λM* > λMh λM* < λMh

log(

M*/M

h)

log(

M*/M

h)

Fixed halo mass (Mq) above which star formation is quenched

log(halo mass)

λM* > λMh

for Mh<Mq

log(

M*/M

h)

Mq

Does not produce downsizingin the pivot quantities

Evolving pivot mass (Mq) above which star formation is quenched

log(halo mass)

λM* > λMh

for Mh<Mq

log(

M*/M

h)

Mq

Can produce downsizingin the pivot quantities

But does not necessarily produce a constant pivot ratio

Fixed (M*/Mh) limit above which star formation is quenched

log(halo mass)

λM* > λMh

for Mh<Mq

log(

M*/M

h)

(M*/Mh)crit A critical pivot ratio, coupled with λM*>λMh

naturally leads to downsizing

Still need maintenance mode feedback to prevent growth in higher mass halos

Evolution of the “pivot masses”


The mechanism(s) that regulate the quenching of star formation may have a physical

dependance on M*/Mh and not simply on Mh

Pivot halo mass

0.0 0.2 0.4 0.6 0.8 1.0Redshift

6.0•1011

8.0•1011

1.0•1012

1.2•1012

1.4•1012

1.6•1012

1.8•1012

2.0•1012

Mhpi

v

[ MO • ]

Pivot stellar mass

0.0 0.2 0.4 0.6 0.8 1.0Redshift

1•1010

2•1010

3•1010

4•1010

5•1010

6•1010

M*pi

v

[ MO • ]


Pivot ratio

0.0 0.2 0.4 0.6 0.8 1.0Redshift

1020

30

40

50

60

70

(Mh/M

*)piv

P.Behroozi using our data

Disk instabilities : if the mass of the disk dominates the gravitational potential, the system is unstable to perturbations.

AGN feedback a) halo of hot gas to which AGN jets can couple b) a sufficiently large BH to offset cooling in the halo

Mechanisms that might depend on M*/Mh ?

Vmax/(GMdisk/Rdisk)0.5 < 1

From A.Benson

• M*/Mh=1/27 Lheat=37 × Lcool

• uncertainties in fgas, ε, ...others ??

Speculative !!

Lheat/Lcool ≈ 103(Δ/200)-1/3(Ωm/0.25)-1/3(fgas/0.15)-1(τcool/Gyr)(Mh/1012Msun)-2/3(ε/0.01)(M*/Mh)

Efstathiou et al. 1982

The link between halo and stellar-mass

-6 -5 -4 -3 -2log10( dN/dlogM [ Mpc-3 dex-1])

10

11

12

13

14

15

log 1

0( M

h [ M

O •])

Halo mass function

pivot halo mass

9.0 9.5 10.0 10.5 11.0 11.5 12.010

11

12

13

14

15

Stellar-to-halo mass relation (SHMR)

pivot halo mass

-6 -5 -4 -3 -2log10( dN/dlogMh [ Mpc-3 dex-1])

-5

-4

-3

-2

-1

log 1

0( dN

/dlo

gM* [

Mpc

-3 d

ex-1

])

y=x+log10(dlogMh/dlogM*)

9.0 9.5 10.0 10.5 11.0 11.5 12.0log10( M* [ MO • ])

-5

-4

-3

-2

-1

Stellar mass function(centrals only)

pivo

t ste

llar m

ass

scatter

log10(M*)log10(dN/dlog10Mh)

log 1

0(dN

/dlo

g 10M

*)lo

g 10(

Mh)

Summary

We have build a method to combine galaxy-galaxy lensing, galaxy clustering, and number densities into a more robust probe of the galaxy-dark matter connection

We have applied this technique to the COSMOS field and studied the SHMR from z=0.2 to z=1.0

We find a downsizing behaviour for the “pivot masses” but the “pivot ratio” remains remarkably constant

This coincidence raises the intriguing possibility that the quenching of star-formation may have a physical dependance on Mh/M*

Work in Progress and future plans

Calculation of the total amount of stars locked up in galaxies, implications for the baryon fraction in groups

Comparison between HOD models and abundance matching methods

Comparisons with semi-analytic models such as Galacticus with A. Benson

Application to larger data sets (e.g. BOSS, HSC)

Cosmology and GR tests

new constraints on the evolution of the stellar- to-dark matter … · evolution of the...

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