A Consistent Comparison of Bias Models using Observational Data

Abstract

We investigate five different models for the dark matter halo bias, ie., the
ratio of the fluctuations of mass tracers to those of the underlying mass, by
comparing their cosmological evolution using optical QSO and galaxy bias data
at different redshifts, consistently scaled to the WMAP7 cosmology. Under the
assumption that each halo hosts one extragalactic mass tracer, we use a
$\chi^2$ minimization procedure to determine the free parameters of the bias
models as well as to statistically quantify their ability to represent the
observational data. Using the Akaike information criterion we find that the
model that represents best the observational data is the Basilakos & Plionis
(2001; 2003) model with the tracer merger extension of Basilakos, Plionis &
Ragone-Figueroa (2008) model. The only other statistically equivalent model, as
indicated by the same criterion, is the Tinker et al. (2010) model. Finally, we
find an average, over the different models, dark matter halo mass that hosts
optical QSOs of: $M_h\simeq 2.7 (\pm 0.6) \times 10^{12} h^{-1} M_{\odot}$,
while the corresponding value for optical galaxies is: $M_h\simeq 6.3 (\pm 2.1)
\times 10^{11} h^{-1} M_{\odot}$.

Full text

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arXiv:1201.4878v1 [astro-ph.CO] 23 Jan 2012
Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 25 January 2012(MN LATEX style file v2.2)
A Consistent Comparison of Bias Models using
Observational Data
A. Papageorgiou1,2, M. Plionis1,3, S. Basilakos4,5, C. Ragone-Figueroa6
1Institute of Astronomy & Astrophysics, National Observatory of Athens, Palaia Penteli 152 36, Athens, Greece.
2Faculty of Physics, Department of Astrophysics, Astronomy & Mechanics University of Athens, Panepistemiopo-
lis, Athens 157 83, Greece
3Instituto Nacional de Astrof´ ısica Optica y Electr´ onica, AP 51 y 216, 72000, Puebla, M´ exico.
4Academy of Athens, Research Center for Astronomy & Applied Mathematics, Soranou Efesiou 4, 11527, Athens,
Greece
5High Energy Physics Group, Dept. ECM, Universitat de Barcelona, Av. Diagonal 647, E-08028 Barcelona, Spain
6Instituto de Astronom´ ıa Te´ orica y Experimental, IATE, CONICET-Observatorio Astron´ omico, Universidad Na-
cional de C´ ordoba, Laprida 854, X5000BGR, C´ ordoba, Argentina
25 January 2012
ABSTRACT
We investigate five different models for the dark matter halo bias, ie., the ratio of
the fluctuations of mass tracers to those of the underlying mass, by comparing their
cosmological evolution using optical QSO and galaxy bias data at different redshifts,
consistently scaled to the WMAP7 cosmology. Under the assumption that each halo
hosts one extragalactic mass tracer, we use a χ2minimization procedure to determine
the free parameters of the bias models as well as to statistically quantify their ability
to represent the observational data. Using the Akaike information criterion we find
that the model that represents best the observational data is the Basilakos & Plionis
(2001; 2003) model with the tracer merger extension of Basilakos, Plionis & Ragone-
Figueroa (2008) model. The only other statistically equivalent model, as indicated
by the same criterion, is the Tinker et al. (2010) model. Finally, we find an average,
over the different models, dark matter halo mass that hosts optical QSOs of: Mh≃
2.7(±0.6) × 1012h−1M⊙, while the corresponding value for optical galaxies is: Mh≃
6.3(±2.1) × 1011h−1M⊙.
1 INTRODUCTION
It is of paramount importance for cosmological and galaxy
formation studies the understanding of how galaxies and
other extragalactic mass-tracers relate to the underlying dis-
tribution of matter. The current galaxy formation paradigm
assumes that galaxies form within dark matter haloes, iden-
tified as high-peaks of an underlying initially Gaussian den-
sity fluctuation field, and that they trace in a biased man-
ner such a field (eg., Kaiser 1984; Bardeen et al. 1986). A
formation process of this sort can explain the difference of
the clustering amplitude between the different extragalactic
mass tracers (galaxies, groups and clusters of galaxies, AGN,
etc) as being due to the different bias among the underlying
density field and that of the dark matter (DM) halos that
host the mass tracers.
In order to quantify such a difference, one can use the so-
called linear bias parameter b, which for continuous density
fields is defined as the ratio of the fluctuations of the mass
tracer (δtr) to those of the underlying mass (δm):
b =δtr
δm
, (1)
Based on this definition one can write the bias parameter in
a number of equivalent ways: (a) as the square root of the
ratio of the two-point correlation function of the tracers to
the underlying mass:
b =
?
ξtr
ξm
?1/2
, (2)
since ξ(r) = ?δ(x)δ(x + r)?, in which case one considers
the large-scale correlation function (ie., scales∼
corresponding roughly to the so-called halo-halo term of the
DM halo correlation function (eg., Hamana, Yoshida, Suto
2002), and (b) as the ratio of the variances of the tracer
and underlying mass density fields, smoothed at some linear
scale traditionally taken to be 8 h−1Mpc (at which scale
the variance is of order unity):
b =σ8,tr
σ8,m
>1h−1Mpc),
, (3)
since σ2
8= ξ(0) = ?δ2(x)?.
A further important ingredient in theories of structure
formation, is the cosmological evolution of the DM halo bias
parameter (eg., Mo & White 1996; Tegmark & Peebles 1998,
etc). A large number of such bias evolution models have
been presented in the literature and the aim of this work is
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2 A. Papageorgiou et al.
to compare them using as a criterion how well do they fit
the observed bias, at different redshifts, of optical QSOs and
galaxies. In such a comparison we will make the simplified
assumption that each DM halo hosts one mass tracer. This
is consistent with the definition of the linear bias, where one
uses either the large-scale correlation function (which cor-
responds to the halo-halo term) or the smoothed to linear
scales variance of the fluctuation field, while any residual
non-linearities will probably be suppressed in the ratio of
the tracer to underlying mass correlation functions or vari-
ances. Further suppression of non-linearities, introduced for
example by redshift-space distortions, can be achived using
the integrated correlation function within some spatial scale;
see discussion in section 2 below.
There are two basic families of analytic bias evo-
lution models. The first, called the galaxy merging bias
model, utilizes the halo mass function and is based on
the Press-Schechter (1974) formalism, the peak-background
split (Bardeen et al. 1986) and the spherical collapse model
(Cole & Kaiser 1989; Mo & White 1996, Matarrese et
al. 1997; Moscardini et al. 1998; Sheth & Tormen 1999;
Valageas 2009; 2011). Cole & Kaiser (1989) found for the
bias evolution that
b(M,z) = 1/(1 +z)−1/[1.68(1 + z)]+1.68(1 +z)/σ2(M) ,
where σ2(M) is the variance of the mass fluctuation field,
while Mo & White (1996) derived for an Einstein-de Sitter
universe that
b(z) = 0.41 + [b(0) − 0.41](1 + z)2.
Mo, Jing & White (1997) extended the previous study in
the quasi-linear regime by taking into account high order
correlations of peaks and halos. Similarly, Matarrese et. al.
(1997) estimated the bias in a merging model where the
halo mass exceeds a certain threshold. They found that for
an Einstein-de Sitter universe:
b(z) = 0.41 + [b(0) − 0.41](1 + z)β,
while Moscardini et. al. (1998) generalized the above bias
evolution model for a variety cosmological models.
Many studies have compared the prediction of the merg-
ing bias model with numerical simulations and beyond an
overall good agreement, differences have been found in the
details of the halo bias. For example, the spherical collapse
model under-predicts the halo bias for low mass halos and
fails to reproduce the dark matter halo mass function found
in simulations. To solve this problem, Sheth, Mo & Tor-
men (2001) extended their original model to include the
effects of ellipsoidal collapse. However according to Tinker
et. al. (2010), this model under-predicts the clustering of
high-peaks halos while over-predicts the bias of low mass
objects. Furthermore, Manera et al. (2009) and Manera &
Gaztanaga (2011) find that the clustering of massive halos
cannot be reproduced from their bias calculated using the
peak-background split.
Such and other differences have lead to other modi-
fications of the models, either suggesting new fitting bias
model parameters (eg., Jing 1998; Tinker et al. 2005), or
new forms of the bias model fitting function (eg. Seljak &
Warren 2004; Pillepich et al. 2010; Tinker et al. 2010) or
even a non-Markovian extension of the excursion set theory
(Ma et al. 2011). A further step was provided by de Simone,
Maggiore & Riotto (2011), who incorporated the effects of
ellipsoidal collapse to the original Ma et al. model, which is
based on spherical collapse.
The second family of bias evolution models assumes
a continuous mass-tracer fluctuation field, proportional to
that of the underlying mass, and the tracers act as “test
particles”. In this context, the hydrodynamic equations of
motion and linear perturbation theory are applied. This fam-
ily of models can be divided into two sub-families:
(a) The so-called galaxy conserving bias model uses the con-
tinuity equation and the assumption that tracers and un-
derlying mass share the same velocity field (Nusser & Davis
1994; Fry 1996; Tegmark & Peebles 1998; Hui & Parfey 2007;
Schaefer, Douspis & Aghanim 2009). Then the bias evolution
is given as the solution of a 1st order differential equation,
and Tegmark & Peebles (1998) derived:
b(z) = 1 + [b(0) − 1]/D(z) ,
where b(0) is the bias factor at the present time and D(z) the
growing mode of density perturbations. However, this bias
model suffers from two fundamental problems: the unbiased
problem ie., the fact that an unbiased set of tracers at the
current epoch remains always unbiased in the past, and the
low redshift problem ie., the fact that this model represents
correctly the bias evolution only at relatively low redshifts
z∼
this model to also include an evolving mass tracer population
in a ΛCDM cosmology.
(b) An extension of the previous model, based on the ba-
sic differential equation for the evolution of linear density
perturbations, which implicitly uses that mass tracers and
underlying mass share the same gravity field, and on the
assumptions of linear and scale-independent bias, provides
a second order differential equation for the bias. Its approx-
imate solution provides the functional form for the cosmo-
logical evolution of bias (Basilakos & Plionis 2001; 2003 and
Basilakos, Plionis & Ragone-Figueroa 2008; hereafter BPR
model). The provided solution applies to cosmological mod-
els, within the framework of general relativity, with a dark
energy equation of state parameter being independent of cos-
mic time (ie., quintessence or phantom). An extension of this
model to engulf also time-dependent dark energy equation of
state models, including modified gravity models (geometric
dark energy), was recently presented in Basilakos, Plionis &
Pouri (2011).
The outline of this paper is as follows. In section 2 we
present the data that we will use, we review the basic tech-
niques used in measuring the bias from samples of extra-
galactic objects and we will present the rescaling method
used in order to transform different bias data to the same
(WMAP7) cosmology (ie., flat ΛCDM with Ωm = 0.273 and
σ8 = 0.81). In section 3 we introduce the different bias evo-
lution theoretical models that we will investigate, while in
section 4 we present our results and discussion. The main
conclusions are presented in section 5. In the Appendix we
discuss the simulations used to fit the free parameters of the
BPR model, as well as the cosmological dependence of these
parameters.
<0.5 (Bagla 1998). Note that Simon (2005) has extended
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A Consistent Comparison of Bias Models using Observational Data3
2 MASS TRACER BIAS DATA
The mass tracers that we will use in this work are op-
tical QSOs and galaxies, for which their linear bias with
respect to the underlying mass is available as a function of
redshift. In particular, we will use:
• The 2dF-based QSO results of Croom et al. (2005),
which are based on spectroscopic data of over 20000 QSOs
covering the redshift range 0.3 ? z ? 2.2 and on a ΛCDM
cosmology with Ωm = 0.27 and σ8 = 0.84.
• The SDSS (DR5) QSO (z∼
al.(2009) based on spectroscopic data of ∼30000 QSOs and
and on a ΛCDM cosmology with Ωm = 0.237 and σ8 =
0.756.
• the SDSS (DR5) QSO results of Shen et al. (2009), who
used a homogeneous sample of ∼38000 QSOs within 0.1 ?
z ? 5 and on a ΛCDM cosmology with Ωm = 0.26 and
σ8 = 0.78. In this case we will use only their z∼
to avoid including in our analysis correlated measurements
of the bias, for the redshift range covered also by the Ross
et al. analysis.
<2.2) results of Ross et
>2.2 results
Although there are other QSO bias data available, like the
Myers et. al. (2006) analysis of 300000 photometrically clas-
sified SDSS DR4 QSOs, within 0.75 ? z ? 2.8, we do not
include them in our analysis in order to avoid, in the redshift
range studied, as much as possible correlated bias measure-
ments.
As far as galaxy data are concerned, we will use the
bias results of Marinoni et. al. (2005), which are based on
3448 galaxies from the VIMOS-VLT Deep Survey (VVDS),
cover the redshift range: 0.4 ? z ? 1.5 and use a ΛCDM
cosmology with Ωm = 0.3 and σ8 = 0.9.
Although in the next section we sketch the usual pro-
cedures used to estimate the linear bias of a sample of ex-
tragalactic mass tracers, we would like to stress that for the
QSO data used in this work, the corresponding authors, in
order to minimize non-linear effects, have estimated the in-
tegrated correlation function for scales > 1h−1Mpc, which
in the usual jargon corresponds roughly to the halo-halo
term of the DM halo correlation function. As for the VVTS
galaxy bias data, Marinoni et al., devised a procedure to
estimate the bias of a smooth galaxy density field in pencil
beam surveys, disentangling the non-linear effects, and thus
the bias values used in this work correspond to the linear
bias.
2.1 Estimating the tracer bias at different
redshifts
Although we will use the bias data provided by the previous
references, for completion we briefly present here the basic
methodology used to estimate the bias of some extragalactic
mass tracer at a redshift interval z ± δz, using any of the
basic definitions of eq.(1)-(3).
The first issue that one has to keep in mind is that
what we measure from redshift catalogues is the redshift-
space distorted value of either the tracer correlation func-
tion, ξtr(s), or the variance of the tracer density field σ2
(the index s indicates redshift-space distorted spatial sepa-
rations, while the index r indicates true spatial separations).
One needs to correct for such distortions, resulting from the
8,tr,s
peculiar velocities of the mass tracers, in order to recover the
true spatial value of either measures. Kaiser (1987) provides
such a correction procedure which entails in dividing the di-
rectly measured from the data tracer correlation function or
variance with a function F(Ωm,ΩΛ,b,z), given by (see also
Hamilton 1998 and Marinoni et al. 2005):
F(Ωm,ΩΛ,b,z) = 1 +2
3β(z) +1
5β2(z) (4)
with β(z) = Ωγ
Wang & Steinhardt 1998; Linder 2005), which implies that
β(z) = Ω6/11
m
E(z)−12/11(1 + z)18/11/b(z). Therefore the re-
lation between the redshift-space and real-space measures
used to estimate the bias parameter is:
m(z)/b(z), and γ = 6/11 for the ΛCDM (eg.,
ξtr(s,z)
ξtr(r,z)=σ2
8,tr,s(z)
σ2
8,tr,r(z)= F(Ωm,ΩΛ,b,z) (5)
Then combining equations (2) or (3) with (4) and (5)
provides the real-space bias factor according to:
b(z) =
?ξtr(s,z)
?σ2
ξm(r,z)−4Ω12/11
m
(z)
45
?1/2
?1/2
−Ω6/11
m
(z)
3
=
8,tr,s(z)
σ2
8,m(z)
−4Ω12/11
m
(z)
45
−Ω6/11
m
(z)
3
(6)
where ξm(r) and σ2
function and variance of the underlying dark matter distri-
bution, given by the Fourier transform of the spatial power
spectrum P(k) of the matter fluctuations, linearly extrapo-
lated to the present epoch:
8,m are the corresponding correlation
ξm(r,z) =D2(z)
2π2
?∞
0
k2P(k)sin(kr)
kr
dk ,(7)
and
σ2
8,m(z) =D2(z)
2π2
?∞
0
k2P(k)W2(kR8)dk ,(8)
with D(z) the normalized perturbation’s growing mode (ie.,
such that D(0) = 1), P(k) the CDM power spectrum given
by:
P(k) = P0knT2(Ωm,k) , (9)
with T(Ωm,k) being the CDM transfer function (Bardeen
et al. 1986; Sugiyama 1995; Eisentein & Hu 1998), n the
slope of the primordial power-spectrum (which according to
WMAP7 is = 0.967) and W(kR8) the Fourier transform of
the top-hat smoothing kernel of radius R = R8 = 8h−1Mpc,
given by W(kR8) = 3(sinkR8− kR8coskR8)/(kR8)3.
Now, although in the case of using eq.(3), the σ8 vari-
ance is free of non-linear effects by definition, this is not so
when using the correlation function approach (eq. 2). There-
fore, in order to minimize nonlinear effects at small separa-
tions one can replace ξtr(s) in eq.(6) with the integrated
correlation function,¯ξtr(s).
An alternative approach in order to avoid redshift-space
distortions is to resolve the redshift-space separation, s, into
two components, one perpendicular (rp) and one parallel
(π) to the line-of-sight (see Davis & Peebles 1983) and then
estimating the 2-point projected correlation function wp(rp)
along the perpendicular dimension (within some range of the
parallel dimension, say πmin < π < πmax), which is related
to the spatial correlation function, ξ(r), according to:
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4 A. Papageorgiou et al.
wp(rp) =
?πmax
πmin
ξ(rp,π)dπ =
?πmax
πmin
rξ(r)
?
r2− r2
p
dr (10)
where π = |∆d| and rp = ∆dtanθ/2, with ∆d the radial
comoving distance separation of any pair of mass tracers
and θ is angular separation on the sky of the pair members.
As before, one can use the integrated correlation function,
¯ξ, in order to minimize nonlinear effects.
Additionally, one can also use the angular two-point
correlation function, w(θ), instead of ξ(s) or wp(rp), in order
to obtain ξ(r) via Limber’s inversion, a procedure which
also avoids the peculiar velocity distortions, but is hampered
by the necessity of a priori knowing the redshift selection
function of the mass tracers.
2.2 Scaling the bias data to the same Cosmology
Since different authors have estimated the optical QSO
and galaxy bias using different cosmologies, we need to con-
vert them to the same cosmological background in order to
be able to use them consistently. As such we choose the re-
cent WMAP7 cosmology (Komatsu et al. 2011).
The procedure that we will follow uses the different σ8
power-spectrum normalizations (eq. 3). We wish to translate
the value of bias from one cosmological model, say B, to
another, say A. The definition of bias at a redshift z for
these two different cosmologies are given by:
bA(z) =σ8,tr,r,A(z)
σ8,m,A(z)
(11)
and
bB(z) =σ8,tr,r,B(z)
σ8,m,B(z)
(12)
where the numerator is the real space value of σ8(z) esti-
mated directly from the data, using also eq.(5) to correct
for redshift space distortions. Dividing now equation (11)
by (12), taking into account eq.(5), and making the fair as-
sumption that:
σ8,tr,s,A(z) ≃ σ8,tr,s,B(z) ,
since the different cosmologies enter only weakly in the ob-
servational determination of σ8,tr, through the definition of
distances, we then have:
(13)
bA(z) ≃ bB(z)σ8,m,B(z)
σ8,m,A(z)
?F(Ωm,B,ΩΛ,B,bB,z)
F(Ωm,A,ΩΛ,A,bA,z)
?1/2
. (14)
As it can be realized the required rescaled real-space bias,
bA, enters also in the right hand side of the above equa-
tion, making it rather complicated to analytically derive the
full expression (using for example eq.6). However, noting
that the redshift-space distortion correction enters in the
scaling of the bias, from one cosmology to another, as the
square-root of the ratio of the F functions, the expected de-
viation by using in the right-hand side of eq.(14) the crude
approximation bA ≃ bB, does not affect significantly this
correction. In any case, the magnitude of the relevant cor-
rection, (FB/FA)1/2, is extremely small, typically: ∼ 0.8%
at z = 0.24 dropping to ∼ 0.1% at z = 2.1, and the overall
scaling of the bias to different cosmologies is dominated by
the ratio of the corresponding σ8(z) variances.
We can facilitate our scaling procedure by using the
σ8(z = 0) power-spectrum normalizations of the different
models, a value always provided by the different authors.
We therefore translate the values of σ8(z) to that at z = 0
by using the linear growing mode of perturbations according
to: σ8(z) = σ8(0)D(z). The final scaling relation from the B
cosmology to that of A, therefore becomes:
bA(z) ≃ bB(z)σ8,m,B(0)
σ8,m,A(0)
DB(z)
DA(z)
?F(Ωm,B,ΩΛ,B,bB,z)
F(Ωm,A,ΩΛ,A,bB,z)
?1/2
.(15)
3THEORETICAL BIAS MODELS
Here we briefly describe the bias evolution models that we
are going to compare. As discussed in the introduction, we
separate the models in two families. The galaxy merging
model family, based on the Press-Schether formalism and
the peak-background split. The models that we will inves-
tigate, representing this family, are the Sheth, Mo & Tor-
men (2001) extension of the original Sheth & Tormen (1998)
model (hereafter SMT), the Jing (1998) model, the Tinker
et. al. (2010) (hereafter TRK) and the Ma et. al. (2011)
model (hereafter MMRZ). All these models provide the bias
of halos as a function of the peak-height parameter, ν, where
ν ≡ δc(z)/σ(Mh,z) ,
with Mh the halo mass, σ2(Mh,z) the variance of the mass
fluctuation field at redshift z, and δc(z) the critical linear
overdensity for spherical collapse, which has a weak redshift
dependence (see eq.18 of Weinberg & Kamionkowski 2003).
The basic free parameter of these bias models, to be
fitted by the data (although depending on the model one
more parameter may be allowed to vary - see below), is ν
and through the evolution of σ(Mh,z) we will be able to
derive the predicted bias redshift evolution, as well as the
value of Mh. The latter value will be estimated by using the
definition of σ, eqs.(8) and (9), from which we have that:
(16)
σ2(Mh) = σ2
8
?∞
0
0
dkkn+2T2(k)W2(kR)
dkkn+2T2(k)W2(kR8),
?∞
(17)
with R = (3Mh/4π¯ ρ)1/3, R8 = 8 h−1Mpc and ¯ ρ = 2.78 ×
1011Ωmh2M⊙/Mpc3.
The second family contains the so-called galaxy conserv-
ing models and their extensions. These models are based on
the hydrodynamical equations of motion and linear pertur-
bation theory while the most general such model, that we
will investigate, is that of Basilakos & Plionis (2001; 2003),
extended to included a correction for halo merging in Basi-
lakos, Plionis & Ragone-Figuera (2008).
Below we present the functional form of the bias evolu-
tion for each of the models that we will investigate:
3.1BPR
Basilakos and Plionis (2001; 2003) using linear pertur-
bation theory and the Friedmann-Lemaitre solutions derived
a second-order differential equation for the evolution of bias,
assuming that the mass-tracer population is conserved in
time and that the tracer and the underlying mass share the
same dynamics.
The solution of their differential equation, for a flat cos-
mology, was found to be (Basilakos & Plionis 2001):
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A Consistent Comparison of Bias Models using Observational Data5
b(z) = C1E(z) + C2E(z)I(z) + 1 (18)
where E(z) =?Ωm(1 + z)3+ ΩΛ?1/2and
I(z) =
z
E3(x)
?∞
(1 + x)3
dx .(19)
The constants of integration depend on the halo mass, as
shown in BPR, and they are given by:
C1(Mh) ≈ α1
?
?
Mh
1013h−1M⊙
?β1
?β2
(20)
C2(Mh) ≈ α2
Mh
1013h−1M⊙
(21)
and the values of α1,2 and β1,2 where estimated originally
from a ∼WMAP1 ΛCDM numerical simulation in BPR. We
have since run a WMAP7 ΛCDM simulation, the details
of which can be found in Appendix A1, and from which
we have determined the new values of the α1,2 and β1,2
parameters (see Table A1). The cosmological dependence of
these parameters is also discussed in Appendix A2.
In BPR it was found that the original Basilakos & Plio-
nis model could well reproduce the bias evolution for z < 3,
but not at higher redshifts, indicating the necessity to extend
the model to include the contribution of an evolving mass-
tracer population. Such an extension was presented in BPR
and it was based on a phenomenological approach, although
the functional form for the effects of merging was based on
physically motivated arguments (see Appendix A2 of BPR).
To this end they introduced to the continuity equation an
additional time-dependent term, Ψ(t), associated with the
effects of merging of the mass tracers, which depends on the
tracer number density, its logarithmic derivative and on δtr.
They parameterized this term using a standard evolutionary
form:
Ψ(z) = AH0(1 + z)µ
(22)
where µ and A are positive parameters which engulf the
(unknown) physics of galaxy merging. The bias evolution is
now given by:
b(z) = C1E(z) + C2E(z)I(z) + yp(z) + 1 (23)
where the additional halo-merging factor, yp(z), is given by:
yp(z) = E(z)
?z
0
τ(x)I(x)dx− E(z)I(z)
?z
0
τ(x)dx(24)
with τ(z) = f(z)E2(z)/(1 + z)3and f(z) = A(µ − 2)(1 +
z)µE(z)/D(z). The values of both A and µ have been fitted
using ΛCDM numerical simulations (see BPR) and it was
found that µ ≃ 2.5−2.6 independent of the halo mass, while
A increases with decreasing halo mass, with A ≃ 0.006 and 0
for intermediate (ie., 1013∼
mass halos, respectively. Evidently, the bias factor at z = 0
is provided by:
<Mh∼
<1013.8h−1M⊙) and higher
b(z) = C1+ C2I(z) + 1 .(25)
Therefore in the current analysis we will leave Mh as a
free parameter to be fitted by the data (BPR model) but we
will also allow (a) the parameter α1 to be fitted by the data,
keeping A equal to its simulation based value (A = 0.006,
BPR-I model), as well as the parameter A to be fitted by the
data keeping α1 equal to its simulation based value (α1 =
4.53, BPR-II model).
3.2SMT
In Sheth et. al. (2001) the original work of Sheth & Tor-
men (1999) was extended for the case of an ellipsoidal, rather
than a spherical collapse. This new ingredient reduces the
difference between theoretical expectations and simulation
DM halo data. Considering ellipsoidal collapse the density
threshold required for collapse, contrary to the spherical col-
lapse case, depends on the mass of the final object.
Using the ratio of the halo power spectrum to that of
the underlying mass, they derived the functional form for
the bias as:
1
√aδc(z)
b(ν) = 1 +
?√a(aν2) +√ab(aν2)1−c− f(ν)?
with f(ν) =
(aν2)c+ b(1 − c)(1 − c/2), (26)
where the free parameters where evaluated using N-body
simulations to have values: a = 0.707,b = 0.5 and c = 0.6.
In particular the value of a was found to depend mostly on
how the simulation DM halos were identified. In the case of
a Friends of Friends (FoF) algorithm the value a = 0.707
corresponds to the standard linking length of 0.2 times the
mean inter-particle separation. Decreasing the linking length
would increase the value of a and vice-versa (see discussion
in SMT). Therefore, beyond the value of the DM halo mass,
Mh (which will be estimated from the resulting value of
σ(Mh) via eq.(17), we will also allow the parameter a to be
fitted by the data.
(aν2)c
3.3JING
Jing (1998) used the clustering of simulation DM halos to
derive an expression for the bias which is independent of the
shape of the initial power-spectrum, being CDM or power-
law. His corresponding expression is:
b(ν) =
?0.5
ν4+ 1
?(0.06−0.02n)?
1 +ν2− 1
δc
?
, (27)
where n is the linear power spectrum index at the halo scale
(ie., n = dlnP(k)/dlnk ≃ −2 for Mh ≃ 1013h−1M⊙). The
only free parameter of this model, to be fitted by the data, is
the halo mass, Mh (which will be estimated from the fitted
value of σ(Mh) via eq.17).
3.4 TRK
Tinker et. al. (2010) measure the clustering of dark matter
halos based on a large series of collisionless N-body simu-
lations of the ΛCDM cosmology. DM halos were identified
using the spherical overdensity algorithm by which halos are
considered as isolated peaks in the density field such that
the mean density is ∆ times the density of the background.
Their bias fitting function reads as:
b(ν) = 1 − A
νa
νa+ δa
c
+ Bνb+ Cνc
(28)
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6 A. Papageorgiou et al.
where y = log10∆. For the WMAP7 ΛCDM model the value
which corresponds to the virialization limit is ∆ΛCDM ≃
355. The rest of the parameters of the model are: A =
1 + 0.24y exp[−(4/y)4], B = 0.183, C = 0.019 + 0.107y +
0.19exp[−(4/y)4], a = 0.44y − 0.88, b = 1.5, c = 2.4.
Therefore, we will fit the observational data using as a
single free parameter the DM halo mass (Mh, derived via
σ(Mh) in eq.17) and using with y = log10(∆ΛCDM). How-
ever, we will also allow the latter parameter to be fitted by
the data, simultaneously with Mh.
3.5 MMRZ
Ma et. al. (2011) extended the original Press-Schether ap-
proach incorporating a non-Markovian extension with a
stochastic barrier, where they assume that the critical value
for spherical collapse is itself a stochastic variable, whose
scatter reflects a number of complicated aspects of the
underlying dynamics. Their model contains two parame-
ters: κ, which parameterizes the degree of non-Markovianity
and whose exact value depends on the shape of the filter
function used to smooth the density field, and α, the so-
called diffusion coefficient, which parameterizes the degree
of stochasticity of the barrier. Taking into account the non-
Markovianity and the stochasticity of the barrier, the bias
takes the form:
?
?1 − ακ +ακ
where α = (1 + DB)−1, with DB the diffusion coefficient,
and Γ(0,x) the incomplete gamma function. Without the
stochasticity of the barrier one has DB = 0 → α = 1.
Ma et al. (2011) have found using N-body simulations
that using α = 0.818 and κ = 0.23 they can reproduce to
a good extent both the simulation bias and the halo mass-
function as a function of ν. We will therefore use these pa-
rameter values to fit the observational bias data in order to
constrain Mh. Additionally, we will allow both α and Mhto
be fitted simultanesouly by the data, using κ = 0.44, since
this is the value for a top-hat smoothing kernel in coordinate
space. Note that the value of κ appears to be almost inde-
pendent of cosmology, as discussed in Maggiore & Riotto
(2010).
b(ν) = 1 +
αν2− 1 +ακ
√αδc
2
2 − eαν2/2Γ(0,αν2
2eαν2/2Γ(0,αν2
2)
?
2)?
(29)
4 FITTING MODELS TO THE DATA
In order quantify the free parameters of the DM halo
bias models we perform a standard χ2minimization pro-
cedure between N bias data measurements, bi(z), with the
bias values predicted by the models at the corresponding
redshifts, b(p,z). The vector p represents the free parame-
ters of the model and depending on the model their number
is one or two. This procedure makes the simplistic assump-
tion that each DM halo hosts one mass tracer, an assump-
tion which is justified from the way the QSO and galaxy
bias data have been estimated (see discussion in section 2).
The χ2function is defined as:
χ2=
N
?
i=1
?bi(z) − b(p,z)
σbi(z)
?2
, (30)
with σbi(z) is the observed bias uncertainty. We have in total
N = 22 measured bias data for the optical QSOs, spanning
from z = 0.24 to z = 4, and N = 5 for the optical galaxies,
spanning from z = 0.55 to z = 1.4.
Note that the uncertainty of the fitted parameters will
be estimated, in the case of more than one such parame-
ter, by marginalizing one with respect to the other. How-
ever, since such a procedure may hide possible degeneracies
between parameters, we will also present the 1, 2 and 3σ
likelihood contours in the parameter plane.
Furthermore, since we will attempt to compare the dif-
ferent models among them, the χ2test alone is not sufficient
for such a task, since different models may have a different
number of free parameters. Instead we will use information
criteria to compare the strengths of the different models, ac-
cording to the work of Liddle (2004), a procedure that favors
those models that give a similarly good fit to the data but
with fewer free parameters (see for example Saini et al. 2004;
Godlowski & Szydlowski 2005; Davis et al. 2007 and refer-
ences therein). To this end we will use, the relevant to our
case, corrected Akaike information criterion for small sample
size (AICc; Akaike 1974, Sugiura 1978), defined, for the case
of Gaussian errors, as:
AICc = χ2+ 2k + 2k(k − 1)/(N − k − 1)
where k is the number of free parameters, and thus when
k = 1 then AICc = χ2
cates a better model-data fit. However, small differences in
AICcare not necessarily significant and therefore, in order to
assess, the effectiveness of the different models in reproduc-
ing the data, one has to investigate the model pair difference
∆AICc = AICc,y−AICc,x. The higher the value of |∆AICc|,
the higher the evidence against the model with higher value
of AICc, with a difference |∆AICc|∼
such evidence and |∆AICc|∼
idence, while a value∼
two comparison models.
(31)
min+ 2. A smaller value of AICc indi-
>2 indicating a positive
>6 indicating a strong such ev-
<2 indicates consistency among the
4.1Optical QSO Results
Here we fit the different bias evolution models to the scaled
to the WMAP7 cosmology optical QSO data, described in
section 2. It is important to note that all the bias models
used in this work (except the BPR) have been studied as a
function of the threshold ν, eq.(16), ie., in effect as a function
of the variance of the fluctuation field and thus as a func-
tion of halo mass, while the free parameters of most models
have been fitted using z = 0 simulations. In these models
the redshift dependence of the bias comes mostly from the
redshift dependence of the peak-height, ν (see eq.16).
We will present separately the results of the models with
one free parameter, the halo mass, and the models with an
additional free parameter, as discussed in the theoretical
model presentation sections.
4.1.1One free parameter models
In Table 1 we present the best fit model parameters based
on the χ2minimization procedure, with the first and sec-
ond columns listing the fitted halo mass, Mh, derived using
eq.(17) and the value of bias at z = 0, respectively. We
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A Consistent Comparison of Bias Models using Observational Data7
Model 1012h−1M⊙
b(0)χ2
min/dfAICc
BPR
SMT
JING
TRK
MMRZ
3.0 ± 0.4
3.2 ± 0.4
2.1 ± 0.2
3.0 ± 0.4
2.2 ± 0.2
1.02
1.07
0.98
1.00
0.87
12.88/21
23.21/21
18.00/21
15.79/21
20.14/21
14.88
25.21
20.00
17.79
22.14
Table 1. Results of the χ2minimization procedure between the
optical QSO data (N = 22) and bias models with one free pa-
rameter (k = 1).
Figure 1. Main Panel: Comparison of the QSO bias data (open
circles correspond to Croom et. al. 2005; filled circles to Ross et
al. 2009 and filled triangles to the high redshift data of Shen et. al.
2010), with the one free parameter bias model fits (BPR: black
continuous line; SMT: magenta dashed line; JING: green long-
short dashed line; TRK: red short-dashed line; MMRZ: blue dot-
dashed line). Lower Panel: The relative difference between the
BPR model and all the rest, ∆bBPR(z). Inset Panel: The value
of χ2− χ2
minas a function of halo mass, Mh, for the indicated
bias models.
also present the goodness of fit statistics, as discussed pre-
viously (reduced χ2and AICc). In the main panel of Figure
1 we present the bias evolution models (different lines), us-
ing the best fit parameter values listed in Table 1 together
with the WMAP7-scaled optical QSO bias data. The inset
panel of Fig.(1) shows that the resulting Mh values cluster
around two, relatively similar, values: ∼ 3×1012h−1M⊙ and
∼ 2 × 1012h−1M⊙. In the lower panel we present the rela-
tive difference between the BPR model and all the rest, ie.,
∆bBPR(z) = [bi(z) − bBPR(z)]/bBPR(z).
Some basic conclusions that become evident, inspecting
also Table 1, are:
• Although all one free parameter bias models appear
to fit at a statistically acceptable level the optical QSO
bias data, by far the best model is the BPR, which is the
only model fitting also the highest redshifts (z∼
>3). The
MMRZ is the only model that does not fit the lowest red-
shifts (z∼
epoch, b(0) = 0.87.
• The relative bias difference of the various fitted models
with respect to that of BPR, ∆bBPR(z), indicates that the
BPR, JING and TRK models have a very similar redshift de-
pendence for z∼
models show very large such deviations for z∼
|∆bBPR|∼
models show large deviations at the lowest redshifts as well.
• Beyond the fact that the BPR model provides by far
the best fit to the QSO bias data, the second best model is
the TRK model, with ∆AICc ∼ −2.9. Furthermore, one can
distinguish that the model pairs (JING, TRK) and (JING,
MMRZ) are statistically equivalent (∆AICc∼
• The traditional SMT and the recently proposed MMRZ
models rate the worst among all the other one parameter
models, but interestingly the former provides consistent val-
ues of Mh and b(0) with those of the BPR model.
<0.3), providing an anti-biased value at the current
<2.2 (with |∆bBPR(z)|∼
>0.3 at the largest redshifts. The SMT and MMRZ
<0.05), while all the
>2.8, reaching
<2).
We attempt now to provide a robust average value of the
DM halo mass that hosts optical QSO, using an inverse-AICc
weighting of the different one parameter model results. This
procedure provides a weighted mean and combined weighted
standard deviation of the DM halo mass of:
(µMh,σMh) = (2.72,0.56) × 1012h−1M⊙
while the weighted scatter of the mean is ∼
1012h−1M⊙.
Finally, we point out that since it appears that mainly
the 2 high-z bias points are the ones that give the advantage
to the BPR model with respect to the others, we perform
a more conservative comparison among the models by ex-
cluding these two high-z data points. We find that although
the resulting halo mass and b(0) are very similar to those
of Table 1, with variations of a few percent, there are now
three models that perform equivalently well, the BPR, JING
and TRK with AICc ≃ 13. The other two models perform
moderately (SMT) or significantly (MMRZ) worse, as was
the case also in the full data comparison, with ∆AICc ≃ 2
and 4, respectively.
0.44 ×
4.1.2 Two free parameter models
We now allow a second parameter to be fitted simultaneously
with the DM halo mass. Since the free parameters of the bias
models have been determined using N-body simulations, it
would be interesting to investigate if their simulation-based
value can be reproduced by real observational data. The sec-
ond free parameter that we will use is α1, A, a, y and α for
the BPR-I, BPR-II, SMT-I, TRK-I and MMRZ-I models,
respectively. Note that in the case of the BPR-II model we
will use the simulation based value of α1, with the free pa-
rameter A being the halo-merging parameter of the BPR
model (defined in section 3.5).
Table 2 presents the best fit model parameters result-
ing from the χ2minimization procedure, with the first
and second columns representing respectively the result-
ing halo mass, Mh, and the second free parameter, while
the third column the value of the bias at z = 0. In Fig-
ure 2 we compare the resulting bias evolution models with
the WMAP7 scaled QSO bias data (as in Figure 1), while
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8 A. Papageorgiou et al.
Model1012h−1M⊙
2ndparam.b(0)χ2
min/dfAICc
BPR-I
BPR-II
SMT-I
TRK-I
MMRZ-I 0.3 ± 0.1
2.2 ± 0.4
2.8 ± 0.6
27.0 ± 4.0
0.4 ± 0.1
4.64 ± 0.07
0.008 ± 0.005 1.02 12.45/20 16.66
0.40 ± 0.02 0.95 18.72/20 22.93
6.14 ± 0.43 1.11 13.37/20 17.58
1.21 ± 0.04 0.87 21.27/20 25.48
1.08 11.95/20 16.16
Table 2. Results of the χ2minimization procedure between the
optical QSO data and the bias models with 2 free parameters.
in the lower panel we present the relative difference be-
tween the BPR-II model and all the rest, ie., ∆bBPR−II(z) =
[bi(z) − bBPR−II(z)]/bBPR−II(z).
Below we list the main conclusions of the above fitting
procedure:
• A first important result is that the only model that re-
produces the simulation-based second free parameter value,
is the BPR-I model. The simulation based value is α1 = 4.53
while the fitted value, based on the QSO bias data, is
α1 = 4.64±0.07. This fact will allow us to derive the depen-
dence of the parameters of the BPR bias evolution model on
the relevant cosmological parameters (see Appendix A2).
• Fitting the BPR-II model to the QSO data provides
A = 0.008±0.005 which is almost identical to the simulation
determined value, used in the BPR case (A = 0.006). As it
is therefore expected, the fitted values of Mh and b(0) are
almost identical to those of the one parameter BPR model,
but the statistical significance of the BPR-II model is lower
than that of the BPR due to its 2 free parameters.
• Although the SMT-I and TRK-I models appear now to
fit slightly better the QSO bias data, especially the higher z
range, this happens on the expense of providing unexpected
values for the Mhand very different values of the second fit-
ted parameter with respect to their simulation based value.
For example, the SMT-I model provides a huge halo mass,
∼9 times larger than that provided by the corresponding one
parameter model. This should be attributed to the fact that
the fitted second parameter, a, is significantly smaller than
the nominal value of 0.707. Similarly, the TRK-I model pro-
vides a very small value of Mh, a factor of ∼ 9 less than of
the corresponding one parameter model, while the resulting
value y = 6.14 implies that ∆ ≃ 106, a value extremely large
and unphysical. Finally, the MMRZ-I model provides again
a very small value of Mh, while it is the only two-parameter
model that fits the data worst than the corresponding one
parameter model. This is due to the fact that we have used
κ = 0.44 and not κ = 0.23, which is used in the one free
parameter model, as suggested by MMRZ. Had we used the
latter κ value we would have found an extremely small value
of Mh≃ 1010h−1M⊙. These results probably indicate a de-
generacy between the two fitted parameters, a fact which we
indeed confirm for the SMT-I, TRK-I and MMRZ-I models,
as can be seen in Figure 3 where we plot the 1, 2 and 3σ
likelihood contours in the parameter solution plane. Con-
trary to the above models, no such degeneracy is present
for BPR-I model. Note that in Fig.3 the cross indicates the
best 2-parameter solution, while the dashed line indicates
the simulation based value of the second parameter. In the
case of the TRK-I model the latter corresponds to the virial-
Figure 2. Comparison of the QSO bias data with the two free
parameter bias models (their line types and colors are indicated in
the Figure). Lower Panel: The relative difference, ∆bBPR−II(z),
between the BPR-II and the rest of the models.
ization value (y = log10355), used in the one free parameter
fit.
• Beyond the previously mentioned fundamental flow of
the two parameter models (SMT-I, TRK-I and MMRZ-
I), they all provide relatively comparable to the BPR-II
model fits of the QSO bias data but only within the range
0.8∼
MMRZ-I model, as in the case of the one free parameter
fit, provides an anti-biased value at z∼
provides the worst overall fit to the QSO bias data.
• Finally, the BPR one parameter model scores the best
among all the one or two free parameter models and over the
whole available QSO bias redshift range, while it is statisti-
cally equivalent with the BPR-I and BPR-II models (since
|∆AICc|∼
We assess in a more quantitative manner the statistical
relevance of the different theoretical bias models in repre-
senting the observational QSO bias data by using the in-
formation theory parameter AICc and presenting in Table
3 the model pair difference ∆AICc. As previously discussed
a smaller AICc value indicates a model that better fits the
data, while a small |∆AICc| value (ie.,∼
the two comparison models represent the data at a statisti-
cally equivalent level. Due to the resulting unphysical sec-
ond free parameter, as discussed previously, we do not use
in Table 3 a comparison based on the SMT-I, TRK-I and
MMRZ-I models. It is obvious that the one free parameter
BPR model fairs the best among any model, while it is sta-
tistically equivalent, as indicated by the relevant values of
∆AICc, to the BPR-I and BPR-II models and to a slightly
lesser degree to the TRK model.
<z∼
<2.4 (see lower panel of Fig.2). Furthermore, the
<0.2, while it also
<1.8; see Table 3).
<2) indicates that
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A Consistent Comparison of Bias Models using Observational Data9
Figure 3. Contour plots of the two fitted parameter solution
space. The 1σ level is indicated by the relatively thicker red curve.
The blue dashed line indicates the simulation based value of the
y-axis parameter.
BPR-I BPR-II SMTJINGTRK MMRZ
BPR
BPR-I
BPR-II
SMT
JING
TRK
-1.3 -1.8
-0.5
-10.3
-9.0
-8.5
-5.1
-3.8
-3.3
5.2
-2.9
-1.6
-1.1
7.4
2.2
-7.3
-6.0
-5.5
3.1
-2.1
-4.3
Table 3. Results of the pair difference ∆AICc for the bias evo-
lution models fitted to the optical QSO data.
4.2 Optical VVTS galaxy Results
This model-data comparison takes place at relatively low
redshifts (z < 1.5), covering a small dynamical range in z,
and therefore we will use only the one free parameter models
to fit the galaxy bias data. An additional reason is that even
with the much larger z-dynamical range covered by the QSO
data, the second parameter could not be constrained (except
for the case of the BPR-I and BPR-II models).
The results of the χ2-minimization procedure are pre-
sented in Table 4, while in Figure 4 we present the model fits
to the galaxy bias data. Note that the layout of Figure 4 is
as Figure 1. It is evident that the BPR model fairs the best
providing the lowest reduced χ2and AICc parameter with
respect to the other models, while the MMRZ model fairs
the worst. However, due to the small dynamical range in
redshift, the information theory pair model characterization
parameter, ∆AICc, indicates that all the bias models are
statistically equivalent in representing the bias data, since
∆AICc∼
vide an average halo mass that hosts VVTS optical galaxies
using an AICc weighted procedure over the different one-
parameter bias models. The resulting weighted mean and
<1.8 for any model pair. As in the QSO case, we pro-
Model1011h−1M⊙
b(0)χ2
min/df AICc
BPR
SMT
JING
TRK
MMRZ
6.0 ± 2.5
6.4 ± 1.9
5.1 ± 1.3
7.8 ± 1.9
6.1 ± 1.3
0.99
0.90
0.83
0.86
0.70
0.29/4
0.45/4
0.93/4
0.74/4
2.10/4
2.29
2.45
2.93
2.74
4.10
Table 4. Results of the χ2minimization procedure between the
one free parameter models and the optical VVTS galaxy bias
data.
Figure 4. Results of the χ2minimization procedure between
the VVTS galaxy bias data and the bias models with one free
parameter.
combined weighted standard deviation are:
(µMh,σMh) = (6.3,2.1) × 1011h−1M⊙
while the weighted scatter of the mean is also ∼ 0.9 ×
1011h−1M⊙.
It is interesting to point out that the only model that
finds that at z = 0 the optical galaxies are unbiased (b0 ≃ 1),
in agreement with other studies of wide-area optical galaxy
catalogues (Verde et al. 2002; Lahav et al. 2002), is the BPR
model, while all the other models indicate that optical galax-
ies are quite anti-biased with b(0) ? 0.9.
5 CONCLUSIONS
In this work we assess the ability of five recent bias evolu-
tion models to represent a variety of observational bias data,
based either on optical QSO or optical galaxies. To this end
we applied a χ2minimization procedure between the ob-
servational bias data, after rescaling them to the WMAP7
cosmology, with the model expectations, through which we
fit the model free parameters.
In performing this comparison we assume that each halo
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10 A. Papageorgiou et al.
is populated by one extragalactic mass tracer, being a QSO
or a galaxy; an assumption which is justified since the obser-
vational data have been estimated on the basis of either the
large-scale clustering (> 1h−1Mpc), corresponding to the
halo-halo term, or the corrected for non-linear effects vari-
ance of the smoothed (on 8 h−1Mpc scales) density field.
The comparison shows that all models fit at an ac-
ceptable level the QSO data as indicated by the their re-
duced χ2values. Using the information theory characteris-
tic, AICc, which takes into account the different number of
model free parameters we find that the model that rates
the best among all the other is the Basilakos & Plionis
(2001; 2003) model with the tracer merging extension of
Basilakos, Plionis & Ragone-Figueroa (2008), which is the
only model fitting accurately the optical QSO bias data over
the whole redshift range traced (0 < z < 4). The only other
model that is statistically equivalent at an acceptable level
is that of Tinker et al. (2010). The average, over the differ-
ent bias models, DM halo mass that hosts optical QSOs is:
Mh≃ 2.7(±0.6) × 1012h−1M⊙.
Finally, all the investigated bias models fit well and
at a statistically equivalent level the VVTS galaxy bias
data, with the BPR model scoring again the best, and
the MMRZ the worst. The average, over the different bias
models, DM halo mass hosting optical galaxies is: Mh ≃
6(±2) × 1011h−1M⊙.
ACKNOWLEDGEMENTS
S.B. wishes to thank the Dept. ECM of the University of
Barcelona for hospitality, and acknowledges financial sup-
port from the Spanish Ministry of Education, within the
program of Estancias de Profesores e Investigadores Extran-
jeros en Centros Espa˜ noles (SAB2010-0118).
REFERENCES
Akaike, H., 1974, IEEE Transactions of Automatic Control, 19,
716
Bagla J. S., 1998, MNRAS, 299, 417
Bardeen, J. M., Bond, J. R., Kaiser, N., Szalay, A. S., 1986, ApJ,
304, 15
Basilakos, S., Plionis, M., 2001, ApJ, 550, 522
Basilakos, S., Plionis, M., 2003, ApJ, 593, L61
Basilakos, S., Plionis, M., Figueroa, C. R., 2008, ApJ, 678, 627
Basilakos, S., Plionis, M., Pouri, A., 2011, PhRvD, 83, 123525
Bertschinger, E., 2001, ApJS, 137, 1
Cole, S., Kaiser, N., 1989, MNRAS, 237, 1127
Cole, S., Coles, P., Matarrese, S., Moscardini, L., 1994, MNRAS,
268, 966
Croom, S. M., et. al., 2005, MNRAS, 356, 415
Davis, M., Peebles, P. J. E, 1983, ApJ, 267, 465
Davis, T.M. et al., 2007, ApJ, 666, 716
de Simon, A., Maggiore, M., Riotto, A., 2011, MNRAS, 412, 2587
Fry, J. N., 1996, ApJ, 461, L65
Godlowski, W., Szydlowski, M., 2005, Phys.Lett.B, 540, 1
Eisenstein, D. J., Hu, W., 1998, ApJ, 496, 605
Hamana, T., Yoshira, N., Suto, Y., 2002, ApJ, 568, 455
Hamilton, A.J.S., 1992, ApJ, 385, L5
Hamilton, A.J.S., 1998, ASSL, 231, 185 astro-ph/9708102
Hui L., Parfrey K. P., 2008, Phys.Rev.D, 77, 043527
Jing, Y. P., 1998, ApJ, 503, L9
Kaiser, N., 1984, ApJ, 284, L9
Kaiser, N., 1987, MNRAS, 227, 1
Komatsu, E., et al., 2011, ApJS, 192, 18
Lahav, O., et al., 2002, MNRAS, 333, 961
Liddle, A.R., 2004, MNRAS, 351, L49
Linder, E. V., 2005, Phys. Rev. D., 72, 043529
Ma, C-P., Maggiore, M., Riotto, A., Jun, Z., 2011, MNRAS, 411,
2644
Maggiore, M., Riotto, A., 2010, ApJ, 711, 907
Manera, M., Sheth, R.K., Scoccimarro, R., 2009, MNRAS, 402,
589
Manera, M., Gaztanaga, E., 2011,
Marinoni, C., et. al., 2005, A&A, 442, 801
Matarrese, S., Coles, P., Lucchin, F., Moscardini, L., 1997, MN-
RAS, 286, 115
Mo, H. J., White, S. D. M., 1996, MNRAS, 282, 347
Mo, H. J., Jing, Y. P., White, S. D. M., 1997, MNRAS, 284, 189
Moscardini, L., Coles, P., Lucchin, F., Matarrese, S., 1998, MN-
RAS, 299, 95
Myers, A. D., et. al., 2006, ApJ, 658, 85
Nusser, A., Davis, M., 1994, ApJ, 421, L1
Pillepich A., Porciani C., Han O., 2010, MNRAS, 402, 191
Press, W. H., Schechter, P., 1974, ApJ, 187, 425
Ragone-Figueroa, C., & Plionis, M. 2007, MNRAS, 377, 1785
Ross, N. P., et. al., 2009, ApJ, 697, 1634
Saini, T.D., Weller, J., Bridle, S.L., 2004, MNRAS, 348, 603
Schaefer B. M., Douspis M., Aghanim N., 2009, MNRAS, 397,
925
Seljak, U., Warren, M. S., 2004, MNRAS, 355, 129
Shen, Y.,et. al., 2009, ApJ, 697, 1656
Sheth, R. K., Tormen, G., 1999, MNRAS, 308, 119
Sheth, R. K., Mo, H. J., Tormen, G., 2001, MNRAS, 323, 1
Simon P., A&A, 2005, 430, 827
Springel, V. 2005, MNRAS, 364, 1105
Sugiura, N. 1978, Communications in Statistics A, Theory &
Methods, 7, 13
Sugiyama, N., 1995, ApJS, 100, 281
Tegmark, M., Peebles, P. J. E., 1998, ApJ, 500, L79
Tinker, J. L., Weinberg, D. H., Zheng, Z., 2005, ApJ, 631, 41
Tinker, J. L., et. al. 2010, ApJ, 724, 878
Valageas P., 2009, A&A, 508, 93
Valageas P., 2011, A&A, 525, 98
Verde L., et al., 2002, MNRAS, 335, 432
Wang, L., Steinhardt, J. P., 1998, ApJ, 508, 483
Weinberg, N.N. & Kamionkowski, M., 2003, MNRAS, 341, 251
APPENDIX A: SIMULATION BASED BPR
MODEL PARAMETER ESTIMATION
A1
ΛCDM Simulations
We have run a new WMAP7 ΛCDM N-body simulation us-
ing the GADGET-2 code (Springel 2005) with dark mat-
ter only. The size of the box simulation is 500h−1Mpc
and the number of particles is 5123. The adopted cosmo-
logical parameters are the following: Ωm = 0.273 ΩΛ =
0.727, h = 0.704, σ8 = 0.81 and the particle mass is
7.07×1010h−1M⊙, comparable to the mass of a single galaxy.
The initial conditions were generated using the GRAFIC2
package (Bertschinger 2001). We also use a similar size sim-
ulation, generated in Ragone-Figueroa & Plionis (2007), of
a ΛCDM model with Ωm = 0.3, ΩΛ = 0.7, h = 0.72 and
σ8 = 0.9.
The dark matter haloes were defined using a FoF algo-
rithm with a linking length l = 0.17?n?−1/3, where ?n? is
the mean particle density.
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A Consistent Comparison of Bias Models using Observational Data11
Figure A1. The Parameters C1and C2derived from the WMAP7
(filled points) and WMAP1 (open points) ΛCDM simulation
(points). Continuous lines correspond to the function form of eq.
(A3), with best fit parameters shown in Table A1.
Modelα1
α2
β1
β2
WMAP1
WMAP7
3.30±0.13
4.53±0.22
-0.36±0.01
-0.41±0.02
0.34±0.04
0.37±0.04
0.32±0.04
0.36±0.04
Table A1. Results of the χ2minimization used to evaluate the
parameters that enter in the C1 and C2 constants (which depend
on DM halo mass) of the BPR bias evolution model.
We estimate the bias redshift evolution of the different
DM haloes, with respect to the underlying matter distribu-
tion, by measuring their relative fluctuations in spheres of
radius 8 h−1Mpc, according to the definition of eq.(3), ie.,
b(M,z) =σ8,h(M,z)
σ8,m(z)
, (A1)
where the subscripts h and m denote haloes and the under-
lying mass, respectively. The values of σ8,h(M,z), for haloes
of mass M, are computed at different redshifts, z, by:
σ2
8,h(M,z) =
??
N −¯ N
¯ N
?2?
−1
¯ N
, (A2)
where¯ N is the mean number of such haloes in spheres of 8
h−1Mpc radius and the factor 1/¯ N is the expected Poisso-
nian contribution to the value of σ2
at each redshift the value of the underlying mass σ8,m. In
order to numerically estimate σ2
sphere centers in the simulation volume, such that the sum
of their volumes is equal to ∼ 1/8 the simulation volume
(Nrand= 8000). This is to ensure that we are not oversam-
pling the available volume, in which case we would have
been multiply sampling the same halo or mass fluctuations.
The relevant uncertainties are estimated as the dispersion
of σ2
halo sample.Note that we do not explicitely correct for
possible non-linear effects in δ (although the density field is
indeed smoothed on linear scales - 8 h−1Mpc); we do how-
ever expect that such effects should be mostly cancelled in
the overdensity ratio definition of the bias.
We use the DM halo bias evolution, measured in the two
simulations, for different DM halo mass range subsamples in
order to constrain the constants of our bias evolution model,
ie., C1,C2. The procedure used is based on a χ2minimization
8,h. Similarly, we estimate
8,jwe randomly place Nrand
8,jover 20 bootstrap re-samplings of the corresponding
Figure A2. Correlation between (a) the best fitted α1 values
of the BPR model using the QSO bias data scaled to different
cosmologies for a grid of Ωm and σ8values and (b) the predicted
α1values, based on eq.(A4). The red points correspond to Ωm ?
0.273 while the black points to Ωm > 0.273.
of whose details are presented in BPR and thus will not be
repeated here. In Fig.A1 we present as points the simulations
based values of these parameters, for both cosmologies used,
and as continuous curves their analytic fits, which are given
in eqs.(20) and (21). The resulting values of the parameters
α1,2 and β1,2 can be found in Table A1. It is interesting to
note that the slope of the functions C1 and C2 is roughly a
constant and independent of cosmology, with a value β1 ≃
β2 ≃ 0.35(±0.06).
A2Dependence of the BPR model constants on
Cosmology
The dependence of the constants α1 and α2 of the BPR
bias model on the different cosmological parameters is an
important prerequisite for the versatile use of the model in
investigating the bias evolution of different mass tracers and
determine the mass of the dark matter halos which they in-
habit. In Basilakos & Plionis (2001) we predicted a power
law dependence of α2 on Ωm. Indeed, fitting such a depen-
dence, using the WMAP1 and WMAP7 ΛCDM simulations,
we find:
?0.273
consistent with the value n = 3/2 anticipated in Basilakos
& Plionis (2001).
Now, in order to investigate the dependence on different
cosmological parameters of the parameter a1, we have used
the optical QSO data and the procedure outlined in section
2.2 to scale the QSO bias data to different flat cosmologies,
using a grid of Ωm and σ8 values. The grid was defined as
follows: Ωm ∈ [0.18,0.5] and σ8 ∈ [0.7,0.94], both in steps
of 0.01. We then minimize the BPR bias evolution model
to the scaled bias data to different cosmologies QSO, finally
α2(Ωm) = −0.41
Ωm
?n
with n ≃ 2.8/2 (A3)
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12 A. Papageorgiou et al.
providing for each pair of (Ωm,σ8) values the best fitted α1
and Mh values.
Then using a trial and error approach to select the
best functional dependence of the derived α1(Ωm,σ8) val-
ues to the relevant cosmological parameters, we find a best
fit model of the form:
?0.81
with
?
In Fig. A2 we correlate the derived α1(Ωm,σ8) values, result-
ing from fitting the BPR bias evolution model to the scaled
QSO bias data, to those predicted by the model of eq.(A4).
It is evident that the correspondence is excellent, indicating
that indeed the above estimated cosmological dependence of
α1 is the indicated one.
α1(Ωm,σ8) ≃ 4.53
σ8
?κ1
exp[κ2(Ωm− 0.273)](A4)
(κ1,κ2) =
(12.15,0.30)
(8.70,0.37)
Ωm ? 0.273
Ωm > 0.273
(A5)
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