Halo concentration and the dark matter power spectrum

Abstract

We explore the connection between halo concentration and the dark matter power spectrum using the halo model. We fit halo model parameters to non-linear power spectra over a large range of cosmological models. We find that the non-linear evolution of the power spectrum generically prefers the concentration at non-linear mass scale to decrease with the effective slope of the linear power spectrum, in agreement with the direct analysis of the halo structure in different cosmological models. Using these analyses, we compute the predictions for the non-linear power spectrum beyond the current resolution of N-body simulations. We find that the halo model predictions are generically below the analytical non-linear models, suggesting that the latter may overestimate the amount of power on small scales.

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arXiv:astro-ph/0301341v1 16 Jan 2003
Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 2 February 2008(MN LATEX style file v2.2)
Halo Concentration and the Dark Matter Power Spectrum
Kevin M. Huffenberger & Uroˇ s Seljak
Department of Physics, Jadwin Hall
Princeton University, Princeton, NJ 08544
January 2003
ABSTRACT
We explore the connection between halo concentration and the dark matter power
spectrum using the halo model. We fit halo model parameters to non-linear power
spectra over a large range of cosmological models. We find that the non-linear evolu-
tion of the power spectrum generically prefers the concentration at non-linear mass
scale to decrease with the effective slope of the linear power spectrum, in agreement
with the direct analysis of the halo structure in different cosmological models. Using
these analyses, we compute the predictions for non-linear power spectrum beyond the
current resolution of N-body simulations. We find that the halo model predictions
are generically below the analytical non-linear models, suggesting that the latter may
overestimate the amount of power on small scales.
1 INTRODUCTION
In recent years, computational N-body simulations have
provided important insights into the non-linear formation of
structure by the dark matter. Navarro et al. (1997) (NFW)
observed a ‘universal’ density profile for dark matter haloes,
suggesting that the profiles depend only on the mass of the
halo, with relatively little scatter between them. They also
noticed that the halo concentration parameter, describing
the degree of profile steepness (defined more precisely be-
low) decreases with increasing halo mass, though the exact
relation depends on the cosmology at hand. Several formulae
have been suggested for this relation (Navarro et al. 1997;
Bullock et al. 2001; Eke et al. 2001) which rely on fitting
some mechanism to haloes culled from N-body simulations.
These studies found that the concentration dependence on
the power spectrum is caused primarily by the slope of the
linear power spectrum. Both this and the mass dependence
of concentration can be explained by the epoch of the halo
formation, since less massive haloes form at higher redshift
when the universe was denser and this leads to a more con-
centrated profile.
Another relation which appears to be universal is that of
the mass function (Jenkins et al. 2001). The distribution of
halo masses in N-body simulations appears to be universal
in the sense that when mass is related to the fluctuation
amplitude of the linear power spectrum, the mass function
has a universal form. This relation has also been explored in
detail in recent work (Sheth & Tormen 1999; Jenkins et al.
2001; White 2001, 2002), defining precisely what is meant
by the mass and confirming the universality of the mass
function for a large set of cosmological models.
While the above approaches have focused on individ-
ual haloes in the simulations by counting them and explor-
ing their structure, for many cosmological applications all
that we need is the dark matter power spectrum. This can
be computed from N-body simulations as well, but com-
putational limitations prevent one from exploring a large
range of cosmological models or from achieving a high reso-
lution on small scales. For this reason many fitting formulae
have been developed with increasing accuracy over the years
(Hamilton et al. 1991; Jain et al. 1995; Peacock & Dodds
1996; Smith et al. 2002), although these are still significantly
limited on small scales by the dynamical range of simula-
tions.
The close connection between the two descriptions
has been highlighted recently with the revival of the halo
model (Seljak 2000; Peacock & Smith 2000; Ma & Fry 2000;
Scoccimarro et al. 2001). For a few cosmological models, it
was shown that with an appropriate choice of concentra-
tion mass dependence and mass function one can use the
halo model to accurately predict the non-linear dark mat-
ter power spectrum. One of the questions we explore in this
paper is whether this agreement can be extended to a wider
range of models and whether the trends seen in the analy-
sis of individual haloes are confirmed also with the power
spectrum analysis. There are several reasons why this is of
interest: since the selection of individual haloes is somewhat
subjective it is possible that those left out may be system-
atically different. For example, they could be less relaxed,
more recently formed and so less concentrated. An opposite
effect is that because the power spectrum is a pair-weighted
statistic, if there is a scatter in the mass-concentration re-
lation then pair weighting increases the mean concentration
relative to the simple particle weighting.
On the other hand, if the analysis of individual haloes
is in agreement with the power spectrum analysis over the
range where both are reliable, then the same analysis can be
extended to smaller scales which are not resolved by cosmo-
logical N-body simulations that compute the power spec-
trum. The reason for this is that the resolution scale for the
halo structure can be extended significantly by resimulating
the representative regions of the haloes with a higher resolu-
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2 K. Huffenberger & U. Seljak
tion simulation zoomed on that halo. On the other hand, a
power spectrum calculation requires a large simulation vol-
ume, so that the largest scales are in the linear regime, which
then implies that the mass and force resolution cannot be
as high as in the simulations which focus on single haloes.
This paper is structured in the following way. We sum-
marize the halo model and give relevant background infor-
mation in §2. In §3, we give our method and results. In §4, we
compare our results to other models and in §5 we summarize
our conclusions.
2 BACKGROUND
2.1 The Halo Model
We begin by reviewing the formalism of the halo model to
compute the non-linear dark matter power spectra, following
the notation in Seljak (2000). The haloes are characterized
by their mass M, density profile ρ(r,M), number density
n(M) and bias b(M). N-body simulations suggest a family
of density profiles:
ρs
(r/rs)−α(1 + r/rs)3+α,
ρ(r) =
(1)
where ρsis a characteristic density, rsis the radius where the
profile has an effective power law index of −2, and −1.5 <
α < −1 (Navarro et al. 1997; Moore et al. 1998). Here we
use α = −1, since power spectra are not sensitive to the
inner parts of the halo.
As is conventional, we re-parametrize ρsand rsin equa-
tion (1) in terms of a halo mass and concentration. We
define the virial radius rvir of a halo to be the radius of
a sphere with some characteristic mean density, discussed
below. Insisting that the mass contained inside rvir is M
fixes ρs for a given rs. We introduce the concentration pa-
rameter, the ratio c ≡ rvir/rs. There are several defini-
tions of virial radius used in the literature. In this paper,
the virial radius is defined as the radius of a halo-centreed
sphere which has a mean density 180 times the mean den-
sity of the universe. This may be denoted r180Ω. Other au-
thors set rvir to the radius inside which the mean density is
200 times the critical density (r200), a measure independent
of Ω (Navarro et al. 1997, for example). Still others use a
spherical collapse model to estimate the radius of a virial-
ized halo (r∆). These lead to different concentrations and
masses. For Ω = 0.3 and Λ = 0.7, r200 < r∆ < r180Ω, so for
the same halo with c180Ω = 10 one has c200 ∼ 6, c∆ ∼ 8,
M200/M180Ω ∼ 0.72 and M∆/M180Ω ∼ 0.87. Assuming the
NFW halo profile it is straightforward to translate values
for concentrations and masses between these conventions.
We choose c180Ω as it permits the use of the universal mass
function (White 2002). To avoid cumbersome notation, in
this paper we refer to c180Ω as c and M180Ω as M.
For the mass dependence of concentration we choose a
power law parametrization
c = c0(M/M∗)β, (2)
where c0and β are free parameters, and M∗is the non-linear
mass scale, which we will define shortly. This is of course not
the most general parametrization, but as we will show below
it suffices for the current dynamical range.
The halo number density is written in terms of the mul-
tiplicity function via
n(M)dM =
¯ ρ
Mf(ν)dν, (3)
where ¯ ρ is the mean matter density of the universe and mass
is written in terms of peak height
ν ≡
?
δc(z)
σ(M)
?2
, (4)
where δc(z) is the spherical over-density which collapses at
z (δc ≈ 1.68) and σ(M) is the rms fluctuation in the matter
density smoothed on a scale R = (3M/4π¯ ρ)1/3. ν(M∗) = 1
defines the non-linear mass scale. A cosmology with Ω0 =
0.3, Λ = 0.7, Γ = 0.2, and σ8 = 0.9 has M∗ ≈ 1013h−1M⊙.
The power spectrum has two terms. The first corre-
sponds to correlations in density between pairs of points
where each member of the pair lies in a different halo, and
so is named the “halo-halo” (hh) term. The second corre-
sponds to correlations between pairs in the same halo, and is
known as the one-halo or Poisson (P) term. For convenience
in calculating convolutions, we work in Fourier space, intro-
ducing the Fourier transform of the halo profile, normalized
by the virial mass,
y(M,k) =
1
M
?
4πr2ρ(r)sin(kr)
kr
dr. (5)
The mass of the NFW profile is logarithmically diver-
gent, so in order to evaluate this integral, we must impose
a cutoff. This does not have to be at the virial radius, since
we know that haloes are not completely truncated there. In-
stead, NFW profile typically continues to 2 − 3rvir. In this
regime there is already some overlap between the haloes, so
for the purpose of correlations one can count the same mass
element in more than one halo. Thus the mass function inte-
grated over all the haloes may even exceed the mean density
of the universe (but it can also be below it, since it is not
required that all the matter should be inside a halo).
The halo-halo contribution is
Phh(k) = Plin(k)
??
dνf(ν)dν b(ν)y(M(ν),k)
?2
,(6)
where b(ν) is the (linear) bias of a halo of mass M(ν), for
which we use the N-body fit of Sheth & Tormen (1999).
Since we want this term to reproduce the linear power spec-
trum on large scales we impose this constraint onto the form
for b(ν) (Seljak 2000). The Poisson contribution is
PP(k) =
1
(2π)3
?
dνf(ν)M(ν)
¯ ρ
|y(M(ν),k)|2. (7)
If a cutoff beyond the virial radius is used, the mass weight-
ing should reflect the increase. The treatment of the Poisson
term on large scales is only approximate. Mass and momen-
tum conservation require that on very large scales the non-
linear term should scale as k4, rather than as a constant
implied by equation 7, so the contribution from this term
on large scales is overestimated for k ≪ r−1
compensates the increase in power from the matter outside
the virial radius and for this reason we chose to use rvir as
the radial cutoff for the halo. The halo-halo term is only ap-
proximate, since we do not include the exclusion of haloes,
vir. This partially
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Halo Concentration and the Dark Matter Power Spectrum3
?
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Figure 1. Using the halo model, several power spectra were gen-
erated from a cosmology with Ω0 = 0.3, Λ = 0.7, Γ = 0.2, and
σ8= 0.9 at z = 0. In the upper panel, c0varies and β = −0.2. In
the lower panel, β varies and c0= 10.
which would suppress the term. Because of these approxi-
mations one would not expect the halo model to be perfect,
especially in the transition between the linear and non-linear
regime.
The total power spectrum is the sum of the two contri-
butions, P(k) = Phh(k)+PP(k). On small scales, the Pois-
son term dominates. On larger scales, the halo-halo term
dominates and reduces to the linear power spectrum. From
here forward, we express the power spectrum in dimension-
less form, ∆2(k) = 4πk3P(k).
The effect on the power spectrum of varying c0 and β in
a typical ΛCDM cosmology is shown in Figure 1. The con-
centration controls how tightly matter is correlated within a
single halo. Therefore, higher concentration means a larger
one-halo term. Increasing c0 increases the concentration at
the non-linear mass, affecting the amplitude of the power
spectrum where it is dominated by one-halo term.
Varying β keeping c0 constant produces a tilt in the
non-linear power spectrum around the fiducial value, which
is set by the scale where the non-linear mass dominates the
power spectrum contribution (k ∼ 30–40 h Mpc−1). Haloes
less massive than the non-linear mass dominate at higher
k. Steeper (more negative) β means that haloes less mas-
sive than the non-linear mass will have their concentrations
enhanced, leading to an enhanced one-halo term, and more
power at high k. Haloes more massive than the non-linear
mass will have their concentration reduced. These dominate
at intermediate k in the non-linear regime, so power there is
reduced. The changes in β shown in the figure may not have
much effect on the power spectrum at k < 100 h Mpc−1,
which is the range where N-body simulations are reliable.
On the other hand, they have a substantial effect on the
concentration of typical-sized haloes. For the same concen-
tration at the non-linear mass (∼ 1013h−1M⊙), a halo at
1012h−1M⊙ has a concentration 50% lower with β = −0.1
than with β = −0.4. These masses do not dominate the
power spectrum until k > 100 h Mpc−1, so this change does
not make as much of a difference on the power spectrum at
scales larger than this. This discussion suggests that while
power spectrum analysis cannot provide strong constraints
on the mass dependence of concentration, one can use con-
centration mass relation to predict the power spectrum at
small scales which are not resolved by simulations.
3 METHOD AND RESULTS
We consider two sets of cases which have been extensively
simulated, self-similar initial conditions and more realistic
cold dark matter initial conditions. In both cases we use the
fitting formulae for non-linear ∆2(k) given by Smith et al.
(2002).
3.1 Self-Similar Case
The simplest case to consider is Ω = 1 Einstein-de Sitter
universe with a power law linear power spectrum. In this
case, β has an analytic form, provided we assume that once
a halo collapses, the scale radius rs is fixed in proper coordi-
nates. This is suggested by the simple model which assumes
the haloes remain unchanged once formed, which seems to
hold in numerical simulations (Bullock et al. 2001). We fol-
low the evolution of a single halo, as it traces out a portion
of the c(M) relation.
In an Ω = 1 universe, density perturbations grow as the
scale factor a = 1/(1+z). For a power law linear spectrum,
this means Plin(k) ∝ a2kn. Top-hat smoothing at the scale
corresponding to M yields σ(M) ∝ aM−n+3
universe, δc ≈ 1.68 is constant in time, so σ(M∗) is also
n+3
6
∗
c ∝ rvir ∝ a follows from the definitions of c and rvir and
from ¯ ρ ∝ a−3. Since c ∝ M−β
∗
M is constant. In reality M also increases with a, which
decreases β somewhat, although not by more than 20%.
We calculated halo model power spectra with n = −2.0
and n = −1.5, for which β = −0.16 and −0.25 as calculated
above, with several values of c0. These spectra are shown
in Figure 2. The agreement is quite good, given all the lim-
itations of the halo model. At higher k in the n = −2.0
case, it appears that the slope of the power spectrum dis-
agrees independent of c0. For k/k∗ > 2 this model is better
fit by a power spectrum with β = −.037, c0 = 2.6. How-
ever, according to figure 12 of Smith et al. (2002), simula-
tion data do not exist for k/k∗ greater than a few tens in
the n = −2 case, so the discrepancy is not really significant.
6 . For an Ωm = 1
constant. This means M
∝ a. If rs is fixed in time,
then β ≈ −n+3
6
if we assume
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4K. Huffenberger & U. Seljak
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Figure 2. Non-linear power spectra in scale free models, divided
by the linear power. The power law linear spectra are ∆2
4πk3(k/k∗)n, where n = −2 and n = −1.5. In each case the halo
model spectra for several different c(M) are shown in solid lines,
and the fitting formula of Smith et al. (2002) is in dashed lines.
For the n = −2 case we show three models with β = . − 1667,
which is the analytically predicted value from the text. These
have c0= 3,4,5 in increasing amplitude, and fit poorly. We also
show the best-fitting β = −.037, c0= 2.6 with a thicker line. For
n = −1.5, we show the predicted β = −.25 with c0 = 6,8, and
the best-fitting β = −.245, c0= 7.1.
lin=
In the n = −1.5 case, which is tested against simulations to
k/k∗ = 100, the predicted β = −0.25 is close to the best fit-
ted value. This agreement therefore confirms the assumption
that haloes have a fixed scale radius once they formed.
3.2 Cold Dark Matter Models
Next we consider several flat CDM models (we do not con-
sider open or closed models here, since they are observation-
ally disfavored). To compare the halo model power spectrum
to the Smith et al. (2002) power spectrum, we use a simple
χ2statistic:
χ2=
N
?
i=1
?∆2
Smith(ki) − ∆2(ki)
σi
?2
. (8)
We distributed the N = 40 sample ki evenly in logk, consid-
ering 0.01 < k < 40 h Mpc−1. This range takes us from the
linear regime to some of the highest k that the fitting formula
has been tested with, according to figure 16 of Smith et al.
(2002). For the errors σi, we take 30% of the Smith et al.
(2002) power. This is somewhat arbitrary, but roughly the
size of the combined error in the mildly non-linear regime,
and is probably conservatively large in the fully non-linear
regime where the Poisson term dominates. We are ignoring
the correlations between the points, so the actual value of
χ2is just a qualitative measure of the goodness of fit.
For a variety of cosmologies, we minimized χ2over the
two-dimensional parameter space of c0 and β from equa-
tion (2). We employed a Powell minimization as described
in Press et al. (1992) to give a best-fitting c0 and β. All runs
were given the same initial point: c0 = 10 and β = −0.2, and
terminated when the the minimum changed by less than
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Figure 3. The values of c0 and β at several values of the mass
density parameter Ω0 for the fitted halo model power spectrum
and the models compared in §4. The vertical line denotes the
fiducial model with Ω0= 0.3. The other cosmological parameters
are given in §3.2.
0.1%. We place an error bar on each minimization by calcu-
lating χ2on a grid about the minimum point. We compute
an error contour where exp(−χ2/2) falls to half-maximum.
The error bars in c0 and β show the maximum extent of
this contour in each direction. With this definition, these er-
ror bars cannot properly be used to set confidence limits or
rule out models, but are included simply to gauge in which
cosmologies parameters are better or worse constrained and
to explore parameter degeneracies. Indeed, because of the
range of k we consider, c0 and β are somewhat degenerate:
making β more negative has a similar effect to lowering c0
(compare Figure 1).
We varied one cosmological parameter at a time, choos-
ing the other parameters from a fiducial model with Ω0 =
0.3, Ω0 + Λ = 1, Γ = 0.2, σ8 = 0.9, and z = 0. We
examined the ranges 0.1 < Ω0 < 1.0, 0.1 < Γ < 0.4,
0.6 < σ8 < 1.5,and 0 < z < 1. We plot the variation of
c0 and β with Ω0 with solid lines in Figure 3. The dashed
lines are discussed in §4. The relation for c0 is well fit by:
c0(Ω0) = 11(Ω0/0.3)−0.35,(9)
while β is consistent with being constant around zero or
slightly negative. Note that the Ω dependence depends on
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Halo Concentration and the Dark Matter Power Spectrum5
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Figure 4. The values of c0 and β as a function of the effec-
tive spectral index at the non-linear scale neff for the fitted
halo model power spectrum and for the models compared in
§4. The line type indicates which model. We vary one param-
eter at a time. The fiducial model, given in §3.2, is marked
the vertical line at neff
≈ −1.7. Each parameter increases
as neff increases except for z, which decreases. In their re-
spective panels, the points (left to right) correspond to: Γ =
0.1,0.2,0.3,0.4, σ8 = 0.6,0.7,0.8,0.9,1.0,1.1,1.2,1.3,1.4, and
z = 1.0,0.8,0.6,0.4,0.2,0.0.
the definition of the virial radius and with a different defi-
nition there would be a different Ω dependence.
Although c0 and β vary with all of the other cosmologi-
cal parameters, it is more instructive to present its variation
with the effective power law index of the linear power spec-
trum at the non-linear scale,
neff =d(logPlin(k))
d(logk)
????
k∗
, (10)
where ∆2
is that the slope of the linear power spectrum determines the
epoch of formation of small haloes that merge into the larger
halo. For lower effective slope neff these haloes formed later,
when the density of the universe was smaller, and the final
halo concentration of the larger halo is also lower. This is
depicted with solid lines in Figure 4. Our fiducial cosmology
has neff ≈ −1.7. Varying Ω0 has no effect on neff. When
varying z, a portion of the variation of c0 and β should be
due to the change in Ω(z). The variation of c0 with neff is
consistent with
lin(k∗) = 1 defines the non-linear scale. The reason
c0 = 11(Ω(z)/0.3)−0.35?neff
while β has large errors, but is typically slightly negative.
−1.7
?−1.6
, (11)
4 COMPARISON TO HALO ANALYSIS
We now compare our results to the models of Eke et al.
(2001) and Bullock et al. (2001). These are designed to re-
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Figure 5. A comparison of the variation of the concentration pa-
rameter with mass for our fiducial ΛCDM cosmology (§3.2). The
masses and concentrations here have been translated to a common
virial radius convention, discussed in the text. The shaded region
covers power law models within our error contour, discussed in
the text, and should not be considered a limit on c(M) relations
of arbitrary shape.
produce the concentrations of haloes found in N-body sim-
ulations. Throughout we translate to our virial convention.
We begin by considering concentration as a function of
halo mass for the fiducial ΛCDM cosmology defined in §3.2.
This is plotted in Figure 5. It is clear that our results are
in agreement with those from direct halo structure analy-
sis. However, most of the information in our model comes
from the haloes with masses between 1013and 1014h−1M⊙,
because haloes smaller than this dominate at scales smaller
than the range we considered for our fit. This means we have
a relatively short lever arm with which to determine the
mass concentration relation and consequently we are unable
to strongly constrain the slope β. In direct analysis of haloes,
concentration is a decreasing function of mass, so β < 0. Our
best fitted values for β are also typically negative, although
the errors are large and positive values also cannot be ex-
cluded. The strongest confirmation of this prediction comes
from the analysis of scale-free models in previous section,
where β is negative for n = −1.5. It is clear that measuring
the mass-concentration directly is a better way to determine
the mass dependence of concentration rather than using the
power spectrum, particularly at lower masses.
Since determining mass dependence of concentration
from the power spectrum does not appear promising let
us ask the opposite question: how well can we predict the
power spectrum using the mass concentration relations mea-
sured directly from the analysis of individual haloes in
the simulations? Figure 6 shows the power spectra calcu-
lated with power law approximations to the c(M) func-
tions shown in Figure 5, as well as the Smith et al. (2002)
power spectrum. While there is good agrement between
them for k < 40hMpc−1, the halo models always predicts
less power at k > 40hMpc−1compared to the fits to the
power spectrum. However, the direct fits are not reliable
in this regime, since the N-body simulations become unre-
liable for k > 40hMpc−1, which is why the fits presented
above only use information from k < 40hMpc−1.
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