The spatial and velocity bias of linear density peaks and proto-haloes in the Lambda cold dark matter cosmology

Abstract

We use high resolution N-body simulations to investigate the Lagrangian bias
of cold dark matter haloes within the LCDM cosmology. Our analysis focuses on
"proto-haloes", which we identify in the simulation initial conditions with the
subsets of particles belonging to individual redshift-zero haloes. We then
calculate the number-density and velocity-divergence fields of proto-haloes and
estimate their auto spectral densities. We also measure the corresponding cross
spectral densities with the linear matter distribution. We use our results to
test a Lagrangian-bias model presented by Desjacques and Sheth which is based
on the assumption that haloes form out of local density maxima of a specific
height. Our comparison validates the predicted functional form for the
scale-dependence of the bias for both the density and velocity fields. We also
show that the bias coefficients are accurately predicted for the velocity
divergence. On the contrary, the theoretical values for the density bias
parameters do not accurately match the numerical results as a function of halo
mass. This is likely due to the simplistic assumptions that relate virialized
haloes to density peaks of a given height in the model. We also detect
appreciable stochasticity for the Lagrangian density bias, even on very large
scales. These are not included in the model at leading order but correspond to
higher order corrections.

Full text

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arXiv:1111.4211v1 [astro-ph.CO] 17 Nov 2011
Mon. Not. R. Astron. Soc. 000, 1–10 (2011) Printed 21 November 2011 (MN LATEX style file v2.2)
The spatial and velocity bias of linear density peaks and
proto-haloes in the LCDM cosmology
Anna Elia⋆, Aaron D. Ludlow and Cristiano Porciani
Argelander Institut f¨ ur Astronomie der Universit¨ at Bonn, Auf dem H¨ ugel 71, D-53121 Bonn, Germany
21 November 2011
ABSTRACT
We use high resolution N-body simulations to investigate the Lagrangian bias of cold
dark matter haloes within the LCDM cosmology. Our analysis focuses on “proto-
haloes”, which we identify in the simulation initial conditions with the subsets of
particles belonging to individual redshift-zero haloes. We then calculate the number-
density and velocity-divergence fields of proto-haloes and estimate their auto spectral
densities. We also measure the corresponding cross spectral densities with the linear
matter distribution. We use our results to test a Lagrangian-bias model presented
by Desjacques and Sheth which is based on the assumption that haloes form out of
local density maxima of a specific height. Our comparison validates the predicted
functional form for the scale-dependence of the bias for both the density and velocity
fields. We also show that the bias coefficients are accurately predicted for the velocity
divergence. On the contrary, the theoretical values for the density bias parameters do
not accurately match the numerical results as a function of halo mass. This is likely
due to the simplistic assumptions that relate virialized haloes to density peaks of a
given height in the model. We also detect appreciable stochasticity for the Lagrangian
density bias, even on very large scales. These are not included in the model at leading
order but correspond to higher order corrections.
Key words: cosmology: theory, large-scale structure – galaxies: haloes – methods:
analytical, N-body simulation.
1 INTRODUCTION
Galaxy redshift surveys are powerful probes of cosmology.
The main observables able to constrain cosmological param-
eters are the overall shape of the galaxy power spectrum
at wavenumbers k < 0.1Mpc−1and the baryonic acoustic
oscillations within it. These are treated as proxies for the
matter power spectrum for which we can make robust the-
oretical predictions. Galaxies, however, are biased tracers
of the cosmic mass distribution and many features appear-
ing in their power spectrum depend on how a specific ob-
servational sample was selected. To reconstruct the matter
power spectrum we thus need an accurate bias model whose
free coefficients should be used as nuisance parameters and
marginalized over. In the era of precision cosmology, where
measurements of the matter power spectrum with per cent
accuracy are required, this task is particularly demanding.
Bias models can be divided into two broad classes. Eu-
lerian biasing schemes relate the galaxy density contrast,
δg(x,t), to the matter density distribution, δ, evaluated
at the same time t (but not necessarily at the same spa-
⋆E-mail: elia@astro.uni-bonn.de
tial location). After smoothing the fields on large scales, so
that |δ| is typically much smaller than unity, one can write
(Fry & Gazta˜ naga 1993)
δg(x)=B0+
?
d3x1d3x2B2(x − x1,x − x2)δ(x1)δ(x2) +
d3x1B1(x − x1)δ(x1) + (1)
+
1
2
... ,
?
+
where all fields are evaluated at the same time t and the
details of the bias model are specified by the kernel func-
tions, Bi. If one further assumes that biasing is local (i.e.
that all kernels can be written as products of Dirac delta
distributions), this reduces to
δg(x) = b0+ b1δ(x) +b2
2δ2(x) + ... , (2)
where now the bias coefficients bi are real numbers.
In Lagrangian bias models, on the other hand, one con-
siders the regions in the initial conditions that will collapse
to form galaxies (or their hosting dark-matter haloes) at
time t and writes their density contrast, δL
of the linear density contrast, δ0(q). Large-scale expansions
g(q), in terms
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A. Elia, A. Ludlow and C. Porciani
analogous to eqs. (1) and (2) can also be written in this case.
As a second step, one must determine the final position,
x(q,t), of a fluid element initially located at q, and com-
pute δg(x(q,t),t) out of δL
mapping (LEM) accounts for gravitationally induced mo-
tions that determine the final position of the objects.
Local Eulerian and Lagrangian bias schemes are not
equivalent; they generate a different shape for the galaxy
bispectrum (Catelan et al. 2000) and are not compatible
within the framework of perturbation theory (Matsubara
2011). In fact, Catelan et al. (1998) have shown that a local
Lagrangian biasing scheme generates a non-linear, non-local
and stochastic bias in Eulerian space. On the other hand,
non-local Eulerian and Lagrangian schemes are equivalent
and can be seen as different mathematical representations
of the same physical process (Matsubara 2011).
Due to its simplicity, the local Eulerian model is by
far the most widely used in practical applications, such
as perturbative calculations. However, it is purely phe-
nomenological and does not have a strong theoretical mo-
tivation. Detailed comparison with numerical simulations
has also evidenced its limited validity (e.g. Roth & Porciani
2011, Pollack, Smith & Porciani 2011). Physical models of
bias are generally given in the Lagrangian framework as
conditions on galaxy (halo) formation are more easily
imposed onto the linear density field using some model
for the collapse of density perturbations. Mo & White
(1996, hereafter MW) used a Press-Schechter-like argument
(Press & Schechter 1974) to compute the bias coefficients
of a local Lagrangian scheme as a function of halo mass.
The same authors also showed how these parameters can be
combined to calculate the bias coefficients of a local Eule-
rian scheme assuming that large-scale density perturbations
follow the spherical collapse model. The effect of non-linear
shear on the LEM was discussed by Catelan et al. (1998)
using the Zel’dovich approximation (Zel’dovich 1970). For
halo masses M > M∗(z) (where M∗(z) is the character-
istic mass for collapse at redshift z) the MW formula for
b1 is in good agreement with the predictions of N-body
simulations (Mo, Jing & White 1997). At lower masses,
however, the agreement rapidly deteriorates (Jing 1998).
Porciani, Catelan & Lacey (1999) and Jing (1999) showed
that the discrepancy between the N-body simulations and
the analytical predictions is already present in Lagrangian
space and should thus be attributed to the limitations of the
Press-Schechter formalism rather than to the approximated
treatment of the LEM.
The Lagrangianbias
tended Press-Schechter model (Bond et al. 1991) was
first derived by Porciani et al. (1998), rediscussed in
Scannapieco & Barkana (2002) and tested against simu-
lations by Scannapieco & Thacker (2005). This approach
follows correlated trajectories of δ0 at different Lagrangian
locations as a function of the smoothing scale and looks
for correlations in the first-crossing scales of a density
threshold.
According to the peak-background-split argument
(Bardeen et al. 1986; Cole & Kaiser 1989), long-wavelength
density fluctuations modulate halo formation by modifying
the collapse time of localized short-wavelength perturba-
tions. This makes it possible to generalise the calculation
of the Lagrangian MW bias coefficients to any model for
g(q). This Lagrangian-to-Eulerian
emergingfromtheex-
the halo mass function (Sheth & Tormen 1999, Tinker et al.
2010, Giannantonio & Porciani 2010) and to improve the
agreement with N-body simulations (Seljak & Warren 2004,
Tinker et al. 2005, Gao et al. 2005, Pillepich et al. 2010).
Assuming that dark-matter haloes form out of lin-
ear density peaks is the most popular alternative to the
Press-Schechter model. Tests against N-body simulations
have shown that this is a well justified assumption, espe-
cially for massive haloes (Ludlow & Porciani 2011, see also
Frenk et al. 1988). The idea dates back to Kaiser (1984) who
explained the strong clustering of Abell clusters by assuming
that they originate from the regions above a density thresh-
old in the (suitably smoothed) linear density field. The sta-
tistical properties of the local maxima in a Gaussian random
field have been extensively studied by Bardeen et al. (1986)
(see also Peacock & Heavens 1985 and Hoffman & Shaham
1985). Mo, Jing & White (1997) introduced peaks theory in
the MW formalism, while Matsubara (1999) evaluated the
level of stochasticity in the Lagrangian clustering of density
extrema.
Recently, Desjacques (2008) and Desjacques & Sheth
(2010, hereafter DS) showed that the (smoothed) distri-
bution of density peaks can be described by a simple La-
grangian biasing scheme. Due to the peak constraint, the La-
grangian peak density at a given point not only depends on
the local value of the mass density but also on its Laplacian.
Therefore, the ratio of the power spectra of the peaks and
of the mass density is scale dependent but does not involve
any non-linear coupling of Fourier modes. The scale depen-
dence of this ratio depends on the mass and height of the
peaks, and on the matter power spectrum. For high peaks,
this reduces to the results by Matsubara (1999). Moreover,
although peaks move with the matter at their positions, DS
infer the existence of a statistical velocity bias due to the fact
that local maxima can only exist at special locations. This
also leads to a bias between the velocity spectra of peaks
and matter which is predictable in quantitative terms.
In spite of the fact that the DS model provides the ini-
tial conditions for sophisticated models of the (Eulerian)
halo distribution where the LEM is based either on re-
summed perturbation theory (Elia et al. 2011) or on the
Zel’dovich approximation (Desjacques et al. 2010), its pre-
dictions for the Lagrangian clustering and velocities of the
regions that will form collapsed structures have never been
thoroughly tested against numerical simulations. This pa-
per provides such a test, which is necessary if we are to use
advanced bias models to extract useful information on the
cosmological parameters through a comparison with obser-
vations.
The structure of the paper is as follows. In Section 2
we review the bias model first presented in DS. The details
of our N-body simulations and analysis techniques are out-
lined in Section 3, and a comparison between the model and
numerical results is presented in Section 4. Finally, we sum-
marize our main results in Section 5.
2 THE DS MODEL
In this section we summarize, for completeness, the peaks
model described in Section 2 of DS. In order to do so, we
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Spatial & velocity bias of density peaks and proto-haloes
3
first introduce and define some quantities relevant for peak
statistics.
The spectral moments of the matter power spectrum
are defined as
?∞
where P(k,z) is the linear matter power spectrum at red-
shift z, and W(k,Rs) is a smoothing kernel of characteristic
length Rs. In terms of the moments we define the spectral
parameters:
σ2
n(Rs,z) =
1
2π2
0
dkk2(n+1)P(k,z)W(k,Rs)2, (3)
γn ≡
σ2
n
σn−1σn+1. (4)
There are two common choices for W(k,Rs): the Gaussian
filter
WG(k,Rs) = e−k2R2
s
2
, (5)
and the top-hat filter
WTH(k,Rs) =3[sin(kRs) − kRscos(kRs)]
k3R3
s
. (6)
The mass contained within the comoving length Rs in these
cases is
MG(Rs) = (2π)3/2¯ ρR3
s, andMTH(Rs) =4π
3¯ ρR3
s, (7)
where ¯ ρ is the mean matter density of the Universe.
We will often characterize peaks in terms of their di-
mensionless peak height, ν, defined
ν(Rs,zc) =
δp
σ0(Rs,zc). (8)
Here δp is the smoothed overdensity at the peak location lin-
early extrapolated to z = 0, and σ0(Rs,zc) is the linear rms
mass fluctuation in spheres of radius Rs. DS linked density
maxima of height ν(Rs,z) to dark matter haloes of mass Ms
collapsing at redshift zc, assuming that δpcoincides with the
threshold for collapse, δc.
Bardeen et al. (1986) and DS computed the cross-
correlation between peaks and the underlying density field
which also corresponds to the average density profile around
density maxima. Similarly, Desjacques (2008) evaluated the
leading order expressions (on large spatial separations) for
the peak auto-correlation function and the line-of-sight
mean streaming for pairs of discrete local maxima of height
ν. Desjacques (2008) and DS showed that the full expression
for the cross-correlation and the large-scale asymptotic of
the auto-correlation function are consistent with – and thus
can be thought of as arising from – a non-linear bias relation.
The number density and the velocity of peaks, which are dis-
crete quantities, are assumed to be drawn from the continu-
ous fields δnpk and vpk, respectively. In the DS model, they
are related to the dark matter density contrast and velocity
fields, linearly extrapolated to z = 0, via:
δnpk(x|zc) = bνδS(x) − bζ∇2δS(x), (9)
and
vpk(x|zc) = vS(x) −σ2
0
σ2
1
∇δS(x). (10)
The subscript “S” indicates that the fields are smoothed on
the scale Rs, and the bias parameters, bν and bζ, are given
by
bν =
1
σ0
?ν − γ1¯ u
1 − γ2
1
?
, (11)
and
bζ =
1
σ2
?
¯ u − γ1ν
1 − γ2
1
?
. (12)
Here ¯ u is the mean curvature of the peaks, which can be
approximated by (Bardeen et al. 1986)
¯ u = γ1ν+
3(1 − γ2
1) + (1.216 − 0.9γ4
1)exp
?
2+?γ1ν
−γ1
2
?γ1ν
2
2
?2?
?
3(1 − γ2
1) + 0.45 +?γ1ν
2
?2? 1
?
.(13)
Note that bν coincides with the peak bias factor found by
Bardeen et al. (1986) after neglecting the derivatives of the
density correlation function.
Since, by definition, the gradient of the density field van-
ishes at peak locations, eq. (10) suggests that the peak and
dark matter velocity fields must be coincident there. This
implies that peaks move with the dark matter flow, yet the
spatial bias induces a statistical velocity bias. We will con-
sider the scaled velocity divergence, θ(x) = ∇×v(x)/(aHf),
rather than the velocity field. Here a is the scale factor; H
the Hubble parameter; f = dlnD/dlna, with D the linear
growth factor. In these units, both θ(x) and δ(x) are dimen-
sionless quantities. With these changes, eqs. (9) and (10)
can be rewritten in Fourier space as
δnpk(k) = (bν+ bζk2)δ(k)W(k,Rs)(14)
and
θpk(k) =?1 − bσk2?θ(k)W(k,Rs) = bvel(k)θ(k),
where we have defined
bσ =σ2
σ2
1
(15)
0
.(16)
In the limit of high peaks (ν ≫ 1) it can be shown that
the bias parameters obey the following asymptotic relations:
bν → ν/σ0 and bζ → 0. This implies that the highest
peaks are linearly biased tracers of the underlying matter
field. This is consistent with the predictions of the peak-
background split (Mo, Jing & White 1997) and DS showed
that, indeed, bν is the appropriate generalization of the con-
stant, large-scale bias for low ν. Unlike the density bias fac-
tors, bσ does not depend on ν.
In order to test this model against N-body simulations,
we will make use of the cross-spectra between the peak
and dark matter densities and velocities (denoted as Pmp
and Pmp, respectively) and of the corresponding peak auto-
spectra (Pp and Pp). From eqs. (14) and (15) we obtain
Pmp(k)=(bν+ bζk2)P(k)W(k,Rs),
(bν+ bζk2)2P(k)W2(k,Rs),Pp(k)≃(17)
Pmp(k)≃ (1 − bσk2)P(k)W(k,Rs),
(1 − bσk2)2P(k)W2(k,Rs),Pp(k)≃(18)
where P(k) and P(k) are the matter density and veloc-
ity divergence auto-spectra, respectively. We remind the
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A. Elia, A. Ludlow and C. Porciani
reader that the expression for Pp(k) is only valid to first
order in P(k) as k → 0, and that higher order correc-
tions should be included to improve its accuracy (see e.g.
Desjacques et al. 2010). On the contrary, the expression for
the cross-spectrum Pmp is exact, as shown in Bardeen et al.
(1986) and in the Appendix A of DS. Eq. (17) has the same
functional form as eq. (57) in Matsubara (1999) who stud-
ied the clustering of density extrema. The DS coefficients bν
and bζ match those in Matsubara (1999) only in the limit
ν ≫ 1, for which nearly all extrema are density maxima.
3 NUMERICAL ISSUES
In this section we provide a brief description of the main
numerical issues relevant for this work. This includes a brief
description of our numerical simulations in Section 3.1, our
main analysis techniques in Section 3.2, and a characteriza-
tion of halo collapse barriers in Section 3.3.
3.1 N-body simulations
Our analysis focuses on two high-resolution N-body simu-
lations of structure formation in the standard LCDM cos-
mology. The cosmological parameters for our runs were cho-
sen to be consistent with the fifth-year WMAP data release
(Komatsu et al. 2009). These are h = 0.701, σ8 = 0.817,
ns = 0.96, Ωm = 0.279, Ωb = 0.0462 and ΩΛ = 1 − Ωm =
0.721. Each simulation was run with a lean version of the
Tree-PM code Gadget-2 (Springel 2005), and followed the
dark matter using 10243collisionless particles. One simula-
tion had a box side-length of Lbox = 1200h−1Mpc and a
particle mass of mpart = 1.246 · 1011h−1M⊙; the other used
Lbox = 150h−1Mpc and had mpart = 2.433 · 108h−1M⊙.
The initial redshifts of the simulations were zin = 50 and
zin = 70 for the larger and smaller box, respectively. Using
these simulations we are able to probe a wide range of halo
masses, spanning 8·1010h−1M⊙ < Mh< 1014h−1M⊙. These
simulations were first studied in Pillepich et al. (2010), and
later by Ludlow & Porciani (2011), and we refer the reader
to those papers for further details.
Haloes were identified at z = 0 using a friends-of-friends
(FOF) algorithm (Davis et al. 1986) with a linking length of
0.2 times the mean interparticle distance. Proto-haloes were
identified by tracing backward to the initial redshift all sub-
sets of the particles belonging to z = 0 FOF haloes. We use
the centre of mass of each proto-halo as a proxy for its spatial
location; the mass-weighted linear velocity provides an esti-
mate of the proto-halo’s motion. As a test of the sensitivity
of our results to the adopted halo finder, we also generated
a spherical overdensity (SO) halo catalogue with an over-
density threshold of 200 times the critical density, ρc. For a
fixed halo mass, the two halo-finders produce results consis-
tent within 10 per cent (in terms of all the bias coefficients)
and so, in what follows, we will focus on results obtained for
the FOF haloes, and only consider those containing at least
100 particles.
Haloes in each simulation are split into four separate
mass bins in order to preserve their peculiar clustering prop-
erties. These bins are referred to as 1S to 4S for the small
box, and 1L to 4L for the large one. To asses the impact
of shot noise in the analysis of our small-box simulation we
consider an additional mass bin, labeled bin0S, which in-
cludes all haloes with N ? 100. The mass ranges and total
number of haloes in each bin are given in Table 1.
3.2 Analysis
We construct proto-halo density and momentum fields us-
ing cloud-in-cell grid assignment on a 5123mesh. Velocity
fields are obtained by taking the ratio of the momentum
and density fields, as described in Scoccimarro (2004). In
the case of haloes, these distributions are smoothed to pre-
clude the existence of empty cells; the smoothing scales used
are Rf = 7h−1Mpc for the large box and Rf = 1.8h−1Mpc
for the small one. These values are chosen in order to mask
the effects of the grid, but we have explicitly verified that
our results are not significantly affected by them. All power
spectra have been computed using a fast Fourier transform
technique.
The discreteness of dark-matter particles and haloes
gives rise to a shot-noise component in the spectra. For the
density fields, the estimated power spectrum,ˆP, includes a
shot-noise term which is inversely proportional to the num-
ber density of objects ¯ n (assuming Poisson sampling):
ˆP = Ptrue+1
¯ n.
(19)
Shot noise is therefore negligible for the matter spectra but
may be significant for that of the haloes. The issue is more
severe in Lagrangian space, because fluctuations in the ini-
tial conditions are small. We will consider two alternative
estimates of the protohalo bias; one is determined from the
shot-noise corrected auto-spectrum,
b(k) ≡
?
Ph(k)
P(k), (20)
and the other from the cross-spectrum,
beff(k) ≡Pmh(k)
P(k)
. (21)
Here the subscript “h” indicates the halo fields, and “m”
the matter field (the analogous fields for peaks are indicated
with the subscript “p”.) The relation between the two is
beff(k) = b(k) · r(k), (22)
where r is the linear correlation coefficient, defined as
r(k) =
Pmh(k)
?
P(k)Ph(k)
. (23)
These relations tell us that the two definitions of the bias
are equivalent only if the bias is purely deterministic, i.e.
r = 1. At leading order, the model presented by DS assumes
a perfect correlation between the fields, so that beff(k) = bν+
bζk2and b(k) = |beff(k)| (neglecting the filter function). Any
stochasticity (represented by the higher order corrections in
Pp) will degrade the correlation, yielding different estimates
for the bias. Because of this, we will consider both the cross-
and auto-spectra to check for a potential stochastic element
of the bias.
As for the density, we can define two estimates for the
velocity bias, that we denote bθ and bθeff, with a correlation
coefficient rθ. There is no conclusive way to subtract the
shot noise for the velocity divergence spectrum. Hence
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Spatial & velocity bias of density peaks and proto-haloes
5
we used our simulations to gain some insight into this
issue. We introduced an artificial shot noise in the matter
spectrum P by randomly drawing a fraction of the particles
and found that Pshot(k) ≃ A · k2asymptotically for large
k. Therefore we fitted the amplitude factor A to obtain
shot-noise corrected power spectra. Note however that
other terms could be important at smaller wavenumbers; in
this work we only consider data for which Pshot(k) < 0.1ˆP.
The DS model describes the biasing of linear density
peaks that are expected to form haloes of a given mass
at a specified redshift according to some collapse model
(which determines the value for δc). However, we analyse
the density and velocity fields for the actual FOF and SO
proto-haloes. These are the quantities of physical interest for
studying galaxy clustering in terms of halo occupation mod-
els (e.g. Cooray & Sheth 2002). Ludlow & Porciani (2011)
showed that the vast majority of haloes in our N-body sim-
ulations can be unambiguously associated with linear peaks
in the initial conditions when smoothed on the mass scale
of the halo. For example,
∼70 per cent of all haloes can
be matched with similar-mass peaks in δS(x), and
per cent for haloes with M
∼1014h−1M⊙. Note, however,
that the correspondence between the DS peaks and proto-
haloes will not be perfect. On a given mass scale, some
peaks with overdensities at the collapse threshold will evolve
into substructures contained within larger virialized haloes
(the so-called cloud-in-cloud problem), or to haloes of signif-
icantly different mass. On top of this, numerical simulations
have shown that the collapse threshold δc for haloes of a
given mass and redshift has a broad probabilistic distribu-
tion rather than a fixed value (Porciani, Dekel & Hoffman
2002; Robertson et al. 2009) and possibly also depends on
the form of the adopted smoothing kernel. We consider some
of these issues in the following section.
>
>
∼90
>
3.3Barrier heights for top-hat and Gaussian
filters
The peak model described in Section
defined spectral moments. However, due to its sharp bound-
ary in real space, the top-hat filter decays very slowly in
Fourier space so that the integral defining σ2
in a LCDM model. Since eqs. (11), (12) and (13) depend
on σ2, DS instead adopted a Gaussian filter and assumed
δc = δsc = 1.68. Here, δc corresponds to the critical den-
sity for the collapse of a spherical top-hat perturbation in
an otherwise unperturbed EdS universe, and this does not
necessarily apply to peaks in a smoothed Gaussian random
field. Another issue is that the validity of the simple spher-
ical collapse model is questionable, at best; the probabil-
ity of a proto-halo or a peak being spherical is null, since
it would require the three eigenvalues of the tidal tensor
being equal. Sheth, Mo & Tormen (2001) showed that the
barrier height in the more general ellipsoidal collapse model
(Bond & Myers 1996) can be approximated by
2 requires well-
2is divergent
δec(M,zc) = δ∗
?
1 + α ·
?σ2
0(Rs(M),zc)
δ2
∗
?β?
, (24)
where δ∗ = δsc is taken from the spherical collapse model,
and α = 0.47 and β = 0.615 are determined from fits to
Figure 1. Mass-dependence of the linear overdensities measured
at proto-halo centres in the initial conditions of our simulations.
The shaded regions and contours show the full halo distribution
after smoothing with a Gaussian filter; connected circles highlight
the median trend. All values of δhhave been linearly extrapolated
to z = 0. Open and filled points correspond to our 150h−1Mpc
and 1200h−1Mpc boxes, respectively. For comparison, we also
show, using boxes, the median trends obtained after smoothing
with a top-hat kernel. In both cases, the median trends are well
described by eq. (24): the dot-dashed curve shows the SMT result
(note that this is not a fit to our simulation data), and the dashed
curve shows the result for slightly different values of the free pa-
rameters: δ∗= 1.15, α = 0.37 and β = 0.515. The top panel shows
the mass dependence of the scatter about the median trend for
the Gaussian-filtered case, which is well approximated by a simple
power-law, Σ = 0.4 σ6/5
0
.
the model results. The presence of the dispersion, σ0, in eq.
(24) results in a mass-dependent barrier height: lower mass
haloes require, on average, higher overdensities for collapse
since they must hold themselves together against larger tidal
forces. It should be emphasized that eq. (24) describes the
mean barrier height; the scatter about the mean can be ap-
proximated by Σ = 0.3 σ0 (Robertson et al. 2009). These
values are valid only for the top-hat filter.
What is the appropriate barrier height corresponding to
a Gaussian filter? In Figure 1 we show the linear overdensi-
ties measured at proto-halo centres of mass after smoothing
with a Gaussian filter on the halo mass scale. The shaded
regions show the halo data, and connected circles the me-
dians of the distribution. Open points correspond to results
from our 150h−1Mpc box simulation, and solid points to
our 1200h−1Mpc box run. Squares show the median δc(M)
for the same sample of haloes, but after smoothing the lin-
ear density field with a top-hat filter instead (note the good
agreement with the result of Sheth, Mo & Tormen (2001),
shown as a dot-dashed line in Figure 1). All values have been
linearly extrapolated to z = 0.
c ? 2011 RAS, MNRAS 000, 1–10