Modelling galaxy stellar mass evolution from z~0.8 to today

Abstract

We apply the empirical method built for z=0 in the previous work of Wang et al. to a higher redshift, to link galaxy stellar mass directly with its hosting dark matter halo mass at z~0.8. The relation of the galaxy stellar mass and the host halo mass M_infall is constrained by fitting both the stellar mass function and the correlation functions at different stellar mass intervals of the VVDS observation, where M_infall is the mass of the hosting halo at the time when the galaxy was last the central galaxy. We find that for low mass haloes, their residing central galaxies are less massive at high redshift than those at low redshift. For high mass haloes, central galaxies in these haloes at high redshift are a bit more massive than the galaxies at low redshift. Satellite galaxies are less massive at earlier times, for any given mass of hosting haloes. Fitting both the SDSS and VVDS observations simultaneously, we also propose a unified model of the M_stars-M_infall relation, which describes the evolution of central galaxy mass as a function of time. The stellar mass of a satellite galaxy is determined by the same M_stars-M_infall relation of central galaxies at the time when the galaxy is accreted. With these models, we study the amount of galaxy stellar mass increased from z~0.8 to the present day through galaxy mergers and star formation. Low mass galaxies gain their stellar masses from z~0.8 to z=0 mainly through star formation. For galaxies of higher mass, the increase of stellar mass solely through mergers from z=0.8 can make the massive galaxies a factor ~2 larger than observed at z=0. We can also predict stellar mass functions of redshifts up to z~3, and the results are consistent with the latest observations. Comment: 12 pages, 10 figures, accepted for publication in MNRAS

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arXiv:0911.1864v1 [astro-ph.CO] 10 Nov 2009
Mon. Not. R. Astron. Soc. 000, 1–11 (2009) Printed 10 November 2009(MN LATEX style file v2.2)
Modelling galaxy stellar mass evolution from z ∼ 0.8 to
today
Lan Wang1,2⋆, Y.P. Jing2
1MPA/SHAO Joint Center for Astrophysical Cosmology, Shanghai Astronomical Observatory, Nandan Road 80, Shanghai 200030, China
2Key Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical Observatory,
Chinese Academy of Sciences, Nandan Road 80, Shanghai 200030, China
Accepted 2009 ???? ??. Received 2009 ???? ??; in original form 2009 ???? ??
ABSTRACT
We apply the empirical method built for redshift z = 0 in the previous work of
Wang et al. to a higher redshift, to link galaxy stellar mass directly with its hosting
dark matter halo mass at redshift of around 0.8. The Mstars-Minfallrelation of the
galaxy stellar mass Mstars and the host halo mass Minfall is constrained by fitting
both the stellar mass function and the correlation functions at different stellar mass
intervals of the VVDS observation, where Minfall is the mass of the hosting halo
at the time when the galaxy was last the central galaxy. We find that for low mass
haloes, their residing central galaxies are less massive at high redshift than those at
low redshift. For high mass haloes, central galaxies in these haloes at high redshift are
a bit more massive than the galaxies at low redshift. Satellite galaxies are less massive
at earlier times, for any given mass of hosting haloes. Fitting both the SDSS and
VVDS observations simultaneously, we also propose a unified model of the Mstars-
Minfallrelation, which describes the evolution of central galaxy mass as a function of
time. The stellar mass of a satellite galaxy is determined by the same Mstars-Minfall
relation of central galaxies at the time when the galaxy is accreted and becomes a
sub-component of a larger group. With these models, we study the amount of galaxy
stellar mass increased from z∼ 0.8 to the present day through galaxy mergers and star
formation. Low mass galaxies (< 3×1010h−1M⊙) gain their stellar masses from z∼ 0.8
to z = 0 mainly through star formation. For galaxies of higher mass, we find that the
increase of stellar mass solely through mergers from z = 0.8 can make the massive
galaxies a factor ∼ 2 larger than observed at z = 0, unless the satellite stellar mass is
scattered to intra-cluster stars by gravitational tidal stripping or to the extended halo
around the central galaxy that is not counted in the local observation. We can also
predict stellar mass functions of redshifts up to z ∼ 3, and the results are consistent
with the latest observations. Future more precise observational data will allow us to
better constrain our model.
Key words: galaxies: masses – galaxies: high-redshift – galaxies: haloes – cosmology:
dark matter – cosmology: large-scale structure
1 INTRODUCTION
To study how galaxies form and evolve in their hosting
dark matter haloes, a lot of efforts have been made to
link galaxy properties with the properties of dark matter
haloes which they reside in. The usual methods used in-
clude galaxy kinematics(Erickson et al. 1987) and galaxy
lensing(Mandelbaum et al. 2005, 2006), which measure the
mass of hosting dark matter haloes directly. Semi-analytic
models trace the gas cooling, star formation, and feed-
⋆Email: wanglan@mpa-garching.mpg.de
back processes ‘ab initio’ to get the properties of galaxies
of present day(de Lucia & Blaizot 2007; Bower et al. 2006).
Halo occupation distribution models study the galaxy-
halo connection empirically, to model galaxy properties
using certain assumed formula to describe the galaxy-
halo relation(Jing et al. 1998; Berlind & Weinberg 2002;
Yang et al. 2003; Vale & Ostriker 2004; Conroy et al. 2006;
Wang et al. 2006).
For the models that describe the formation and
evolution history of galaxies,
commonlyused toconstrain
number statistics such as number density, luminosity
the statistics that are
thesemodels include:
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L.Wang, Y.P. Jing
function and stellar mass function(Bullock et al. 2002;
Zehavi et al. 2005; Moster et al. 2009), spatial clustering
properties described by correlation functions(Jing et al.
1998; Yang et al. 2003; Zehavi et al. 2005), void prob-
ability distribution(Vale & Ostriker
velocity dispersion(Jing et al.
relation(Yang et al.2003),
galaxies andthe clustering
colour(van den Bosch et al. 2003;
Zheng 2004; Wang et al. 2007).
The current models of galaxy-halo connection mainly
focus on low redshift study, particularly of the present day,
simply because we can handle well the observational statis-
tics only at low redshifts. With the development of large
scale galaxy redshift surveys, observations are obtained not
only for the local Universe, but also toward higher red-
shifts. Surveys aiming at studying the properties of high
redshift galaxies include DEEP2 survey(Davis et al. 2003),
the COMBO-17 survey(Wolf et al. 2004), the VIMOS-VLT
Deep Survey(VVDS)(Le F` evre et al. 2005), the Cosmic Evo-
lution Survey (COSMOS)(Scoville et al. 2007) and zCOS-
MOS survey(Lilly et al. 2009). With the help of these
surveys, luminosity functions of different types of galax-
ies are obtained up to redshift z > 7(Reddy et al. 2008;
Bouwens et al. 2008). Correlation functions in luminosity
bins reaches redshift of z ∼ 1(Coil et al. 2006; Pollo et al.
2006). Stellar mass functions have been detected for galax-
ies up to redshift of z ∼ 5(Drory et al. 2005; Fontana et al.
2006; Elsner et al. 2008). Correlation functions in bins of
stellar mass have been studied for galaxies of redshift z ∼
1(Meneux et al. 2008, 2009) for VVDS and zCOSMOS ob-
servations.
Based on the observational data obtained at high red-
shifts, several works have been done to model the proper-
ties of galaxies at early epoch. Some works use the HOD
models to study high z galaxy properties(Cooray 2005),
as well as galaxy clustering properties(Bullock et al. 2002;
Yan et al. 2003; Cooray & Ouchi 2006; Conroy et al. 2006;
White et al. 2007), which focus mainly on the clustering de-
pendence of galaxy luminosity. Most recently, Zheng et al.
(2007) uses the HOD method to model the clustering of
DEEP2 galaxies as a function of luminosity, which reaches
redshift of z ∼ 1. While luminosity is the most studied
and easily got property of a galaxy, stellar mass is never-
theless a more fundamental property, since luminosity may
be affected a lot by dust attenuation. Moster et al. (2009)
uses a statistical approach to determine the relation between
galaxy stellar mass and its hosting dark matter halo, and
constrains the evolution of galaxy stellar mass by fitting
the stellar mass functions at different redshifts taken from
Drory et al. (2004).
In the previous work of Wang et al. (2006), an empirical
method has been used to link galaxy stellar mass directly
with its hosting dark matter halo mass. The method falls in
between the semi-analytic approach and the halo occupation
distribution approach. Positions of galaxies are predicted by
following the merging histories of halos and the trajectories
of subhaloes in the Millennium Simulation(Springel et al.
2005). The stellar mass of galaxies at redshift 0 is related
to the quantity Minfall by a double power law function.
Minfall is defined as the mass of the halo at the time when
the galaxy was last the central dominant object. Parameters
2004),
the
distribution
on
Kravtsov et al.
pairwise
Tully-Fisher1998),
colour
dependence
theof
galaxy
2004;
describing the function are constrained by fitting both the
stellar mass function and the correlation functions at dif-
ferent stellar mass intervals from SDSS observation(Li et al.
2006). The derived Mstars-Minfall relation is in excellent
agreement with the determination from galaxy-galaxy weak
lensing measurement of Mandelbaum et al. (2006). In this
study, we will apply this method to an earlier epoch. By fit-
ting the statistic results of VVDS observation, we can study
the connection between galaxy mass and its hosting halo
mass at redshift of around 0.8. Based on the relations ob-
tained both of today and of higher redshift, we can study
the evolution of galaxy stellar mass from z ∼ 0.8 to present
day.
After a galaxy falls into a larger group and becomes a
satellite, its surrounding gas is shock-heated and star forma-
tion ceases in a short timescale. The stellar mass of satellite
galaxies should remain about the same as the time when
they are accreted. In this case, the stellar mass of a satellite
galaxy is determined by the Mstars-Minfall relation of cen-
tral galaxies at the time of infall. This inspires us to describe
the Mstars-Minfallrelation for all galaxies at any redshift in
a uniform way, by modelling the evolution of Mstars-Minfall
relation of central galaxies. Assuming that satellite stellar
mass does not change after infall, the relation for satellite
galaxies follows the relation of central galaxies at an ear-
lier epoch when it is accreted. We will explore this unified
model in §4. With this model, we can also test if a significant
amount of satellite disruption by tidal forces is required by
current observations.
This paper is organised as follows: in Sec. 2, we present
the model for fitting VVDS observations to get the rela-
tion between galaxy stellar mass and the hosting halo mass
at z ∼ 0.8. Based on the models describing galaxy stellar
masses both at low and high redshifts, we analyse in Sec. 3
how galaxies gain their masses from redshift of 0.8 to to-
day. In sec. 4 we build a unified model to fit observations of
both low and high redshifts simultaneously, assuming that
satellite stellar mass is determined by the Mstars-Minfall
relation of central galaxies at the time of its accretion, and
study the mass growth of galaxies from z = 0.8 based on
this model. In sec. 5 we predict the stellar mass functions
of higher redshifts based on our two best-fit models, and
compare our results with recent observations. This work is
based on the Millennium Simulation(Springel et al. 2005).
2 MODEL
As mentioned in Sec. 1, in the previous work of Wang et al.
(2006), the relation between the galaxy stellar mass and its
hosting halo mass at infall time have been studied by fitting
both the stellar mass function and correlation functions at
different stellar mass bins from the SDSS observation. This
relation can be described by a double power law form for-
mula. To study the Mstars-Minfall relation at higher red-
shifts, as a first test, we assume that the Mstars-Minfall
relation at higher redshifts are the same as that of present
day. We test whether the resulting stellar mass functions and
correlation functions are consistent with the observation at
higher redshifts. We simply apply the best-fit Mstars-Minfall
relation at z=0 of Wang et al. (2006) to higher redshifts,
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Modelling galaxy mass evolution from z ∼ 0.8 to today
3
Figure 1. Correlation functions derived by applying the Mstars-Minfallrelation of z=0(Wang et al. 2006) to z=0.83, compared with
the observational results from VVDS of redshift of around 0.8(Meneux et al. 2008). Symbols with error bars are from observation, while
dotted lines represent the model prediction.
and derive stellar masses of all galaxies at each time. The
Mstars-Minfall relation is described as follow:
Mstars =
2
(Minfall
M0
)−α+ (Minfall
M0
)−β×k.
The scatter in log(Mstars) at a given value of Minfall was
described with a Gaussian function of a width σ. The best-
fit model to the SDSS observation had the following pa-
rameters: M0 = 4.0 × 1011h−1M⊙, α = 0.29, β = 2.42,
logk = 10.35 and σ = 0.203 for central galaxies and
M0 = 4.32×1011h−1M⊙, α = 0.232, β = 2.49, logk = 10.24
and σ = 0.291 for satellite galaxies.
Once we get stellar masses of galaxies at a certain red-
shift, we can calculate the stellar mass function and also
the clustering results at different stellar mass bins at that
redshift, combined with the dynamical information of sub-
structures in the simulation. The position of each galaxy is
derived directly from Millennium Simulation by following
the evolution of substructures. Fig. 1 shows the projected
correlation functions at three different stellar mass bins at
redshift of around 0.8. Symbols with error bars are results
from VVDS observation(Meneux et al. 2008), and dashed
lines are the derived results from our simple model. The
projected correlation functions predicted by the model are
in reasonably good agreement with the observation from
VVDS.
Fig. 2 shows the observed stellar mass functions
at different redshifts(symbols), compared with the stel-
lar mass functions derived from our model(lines), with
the Mstars-Minfall relation taken the same as that of
present day. Considering the fact that different initial mass
functions(IMF) were adopted in different observations, we
convert all stellar masses to the case of the Chabrier
IMF(Chabrier 2003), in this figure and all the figures of
stellar mass functions hereafter. The galaxy mass obtained
with the Salpeter IMF(Salpeter 1955) is divided by 1.70,
and that with the Kroupa(Kroupa 2001) IMF is divided by
1.104(Cowie & Barger 2008). It is clear from Fig. 2 that the
stellar mass functions are not reproduced under the sim-
ple assumption that Mstars-Minfallrelation does not evolve
with time. Observationally, the number of galaxies with low
Figure 2.
shifts. Symbols with error bars are observational results. Black
diomends are SDSS observation of z ∼ 0.1(Li & White 2009).
Black points are VVDS results in the redshift bin of [0.7,0.9] from
Pozzetti et al. (2007). Green, red and blue points are results from
Marchesini et al. (2008), in three redshift bins. Lines are model
prediction, with black, green, red and blue ones corresponding to
results at redshifts of 0.83, 1.5, 2.07 and 3.06, respectively. Stellar
masses of galaxies are normalized to the Chabrier IMF(Chabrier
2003).
Galaxy stellar mass functions at different red-
stellar mass decreases dramatically towards higher redshifts,
while the number of high mass galaxies stays roughly un-
changed with different redshifts. The model derived results,
however, show an opposite trend. The number of low mass
galaxies evolves quite slowly, and remains almost the same
till redshift 2, while the number of high mass galaxies evolves
a lot, with a much smaller number of galaxies existing at
higher redshifts. These results show that the Mstars-Minfall
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L.Wang, Y.P. Jing
Figure 3. The best-fit model results for fitting both stellar mass function(Pozzetti et al. 2007) and correlation functions in three different
stellar mass bins(Meneux et al. 2008) from VVDS observation. Symbols are observational results, and dashed lines are the best fit model
results.
relation must vary at different redshifts. A reasonable model
should in general give more massive galaxies and fewer low
mass galaxies at higher redshifts than at the local universe.
As shown in Fig.3 of Moster et al. (2009), the stellar mass
function is more sensitive to the change of model parameters,
while the χ2of correlation functions stays flatter around
minimum in a large range of parameter sets. This explains
why the correlation functions can be well reproduced while
the stellar mass functions show such a large discrepancy.
Therefore, correlation function alone is not enough to con-
strain the Mstars-Minfall relation at z ∼ 0.8. At the end of
this section, we will show that stellar mass function alone is
not enough either to give a good constraint on the relation,
at least in fitting the current observational data results.
From Fig. 1 and Fig. 2 we know that we need to fit
both the stellar mass function and the two point correla-
tion functions to constrain the underlying Mstars-Minfall
relation at higher redshifts. Current studies of VVDS ob-
servation give both the stellar mass function(Pozzetti et al.
2007) and correlation functions in different stellar mass
bins(Meneux et al. 2008) at redshift of ∼ 0.8. We there-
fore focus on constraining Mstars-Minfall relation by these
VVDS results, building models based on the simulation out-
put at redshift of 0.83. Following the method of Wang et al.
(2006), we assume that the Mstars-Minfall relation at red-
shift of ∼ 0.8 can be described by a double power law form.
The relation is determined by five parameters for central
and satellite galaxies separately, which includes in total 10
parameters in the modelling. When applying the modelling
to VVDS results, we notice that compared with the obser-
vations of the local Universe, the observational results from
higher redshift survey like VVDS provide much fewer data
points. Besides, the error bars of these data points are still
large, which yields weak constraint on the model. Therefore,
on the basis of our previous model of Wang et al. (2006), we
alter only the critical mass(and the k parameter simultane-
ously) when constructing the new model, and keep the rest
parameters describing the power law slopes and the relation
scatter the same as those for the local Universe of SDSS. In
this case, we now have four free parameters that are needed
to be constrained.
By fitting both the stellar mass function and the cor-
relation functions of VVDS observation, we get our best-fit
model. Stellar mass function is provided by Pozzetti et al.
(2007), in the redshift bin of [0.7,0.9], with a mean redshift
of 0.81. The correlation functions are from Meneux et al.
(2008), based on the galaxy sample in the redshift range
z = [0.5,1.2], with mean redshift of 0.85. The best fit is
defined as the one that makes the Ξ minimum.
Ξ =χ2(Φ)
NΦ
Ncorr
+χ2
corr
with
χ2(Φ) =
?
NΦ
[log Φ − logΦV V DS
σ(logΦV V DS)
]2
and
χ2
corr=
?
Ncorr
[log w(rp) − logw(rp)V V DS
σ(logw(rp)V V DS)
]2
NΦ = 7, is the number of points over which the stellar mass
function is measured, ranging from 109.5M⊙ to 1011.6M⊙.
Ncorr = 18, is the number of points over which the correla-
tion function is measured, ranging from 0.2 to 10.0h−1Mpc,
in three different stellar mass bins.
Our best-fit model has the parameters: M0 = 6.31 ×
1011h−1M⊙, logk = 10.48 for central galaxies and M0 =
5.03×1011h−1M⊙, logk = 9.99 for satellite galaxies. The re-
sulting Ξ = 3.07591, with χ2(Φ)/NΦ = 1.944. These param-
eter values are listed in Tab. 1 to be compared with the best-
fit parameters of modelling SDSS observation(Wang et al.
2006). Fig. 3 shows the best-fit model results. Symbols with
error bars are the VVDS observation, and dashed lines are
the model results. The stellar mass function is well fitted.
The clustering is also reasonably reproduced. In Fig. 4,
we plot the derived best-fit Mstars-Minfall relation at red-
shift of 0.83, for central(black solid line) and satellite(black
dashed line) galaxies respectively. In comparison, we over-
plotted the relations at redshift 0 in red lines(solid line for
central galaxies and dashed line for satellite galaxies). The
result shows that for a given infall mass of hosting halo, the
galaxy mass changes with redshift, and the dependence on
redshift depends on mass. For massive haloes, the central
galaxies inside these haloes are a bit more massive than the
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Modelling galaxy mass evolution from z ∼ 0.8 to today
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Figure 4.
of central galaxies(black solid line) and satellite galaxies(black
dashed line), by fitting both stellar mass function and correla-
tion functions of VVDS observation. For comparison, relations of
central(red solid line) and satellite galaxies(red dashed line) from
Wang et al. (2006) at z = 0 are also plotted.
The best-fit Mstars-Minfall relation at z = 0.83
galaxies within the same mass of haloes at z = 0. For less
massive haloes, the mass of central galaxies is smaller at
higher redshift. For satellite galaxies, however, the mass of
galaxies is much smaller toward higher redshift at all mass
scales.
In a recent paper, Moster et al. (2009) claim that stel-
lar mass function alone is enough to constrain the relation
between galaxy stellar mass and its hosting halo mass, since
the fit to correlation functions has a much wider range at
its χ2minimum than the fit to stellar mass functions. How-
ever, this may not be true for higher redshift case. In Fig. 5,
we show the derived stellar mass function and correlation
functions for the best-fit model when fitting only the stel-
lar mass function of VVDS observation. The results show
that the clustering of galaxies is generally over-predicted
in this case. Therefore, we believe that fitting both stellar
mass function and the correlation functions simultaneously
is required to get reasonable fit, at least for the current ob-
servational data we can get.
3 MASS INCREASE FROM Z∼ 0.8 TO Z= 0
From high redshift to the present day, dark matter haloes
get larger through mergers, and galaxies inside them also be-
come bigger in size. The galaxies gain their masses through
either mergers with other galaxies, or by forming new stars.
Using the model we build in Sec. 2 we already know the
galaxy masses at redshift of 0.83. We also have galaxy stel-
lar masses of today according to the model built at z = 0
from Wang et al. (2006). The galaxy mass of today is a to-
tal amount of stellar component of galaxy stellar mass that
already exists at redshift of 0.83, the mass increase result-
ing from mergers with other galaxies, and in addition the
newly formed stars during the time interval. By tracing the
merger histories of haloes/subhaloes and hence the galaxies
that reside in these haloes/subhaloes, the amount of stellar
mass that was added through mergers can be calculated.
Combined with the stellar mass of galaxies at both z = 0.83
and z = 0, the stars that were newly formed during the time
interval between these two redshift epochs can be predicted.
For a galaxy that resides in a halo of given mass at
z = 0, we trace back through merger trees to its most mas-
sive progenitor at z = 0.83. We plot in Fig. 6 the median
relation between the stellar mass of the most massive pro-
genitor at z = 0.83 of a galaxy and the mass of its host
halo at present day in black solid line. Among the galaxies
that merge into this most massive progenitor, some of them
are galaxies that already exist at z = 0.83, including both
central and satellite galaxies at that time. The other galax-
ies are newly formed galaxies after z = 0.83, and merge
into the main group before the present day. From our fit-
ted model results we know that the Mstars-Minfall relation
evolves with time. Therefore, at the time of each merger,
the mass of the merged galaxy should not be the same as its
mass at the time of redshift 0.83. We get the galaxy mass
at the time of each merger by interpolating Mstars-Minfall
relation between z = 0.83 and z = 0, assuming that the
model parameters evolve linearly with redshift.
In Fig. 6, the dotted black line is the sum of the stel-
lar masses at z = 0.83 of the most massive progenitor and
of its satellites merged in. The Red solid line is the result
when the merged mass from central galaxies is added, in-
cluding both the stellar mass existing at z = 0.83 and those
newly formed since then. The contribution from the merged
central galaxies is much smaller compared with the mass
from merged satellite galaxies. Blue lines are the mass of
galaxies of present day, according to our model fit result for
SDSS observation(Wang et al. 2006). In the bottom panel
of Fig. 6, the corresponding mass ratio of each component
to the galaxy of the present day is shown.
From Fig. 6 we can tell that for galaxies that reside
in haloes of mass less than 1012h−1M⊙, the mergers since
z = 0.83 contributes to the stellar mass growth by a very
small fraction, which can even be ignored. Compared with
the galaxy mass at present day, their most massive progen-
itors contribute from about 20 percent to around 60 per-
cent of the present day mass, while the rest of the mass
of z = 0 galaxies should come from star formation of the
central galaxy itself. However, for high mass galaxies whose
hosting halo masses are more than 1012.5h−1M⊙, the story
is totally different. The mass of the most massive progen-
itor galaxy at z = 0.83 is comparable to its present day
mass. When taking into account the stellar component of
other merged galaxies, the total mass is significantly larger
than the galaxy mass of the present day. This paradoxical
result is the consequence of hierarchical merging in the cur-
rent cosmological model. Here we have adopted the merger
trees constructed by de Lucia & Blaizot (2007) for galaxies
by taking into account the dynamical time scales. We found
that the merged fraction of stellar mass does not change
when the merger time scale of Jiang et al. (2008) is adopted.
There are several possibilities to reconcile the observations
at low and high redshifts in the hierarchical model. One is
that satellite galaxies are tidally disrupted in a significant
amount of stellar mass before merged into the central galax-
ies (Yang et al. 2009; Wetzel & White 2009). However, as
we see in §4, the significant tidal disruption is not strongly
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