where Γ(x) is the Gamma function. Note that the amplitude only depends on the parameter p. For our fiducial choice of p= 0.3 the amplitude is A= 0.322. The general form of the Sheth-Tormen mass function contains the Press-Schechter mass function as a special case choosing the parameters p= 0 and q = 1. In this case we get A = 0.5 resembling the result in Eq. (3.35).
• Hernquist (see Hernquist 1990)
ρ(r) = ρs
(r/rs)[1 + (r/rs)]3 , (3.44)
• Einasto (see e.g., Gao et al. 2008) ρ(r) =ρsexp
−2 α
r rs
α
−1
, (3.45)
whereρsis the central density parameter andrsthe scale radius. The scale radius divides the density profile into its inner and outer part that have a different power-law behavior.
The NFW and M99 density profiles differ only on small scales (r .rs). For large radii (r rs, i.e., the outer profile) they have the same asymptotic behavior ρ(r) ∝ r−3. Most of the results from simulations agree on this scaling behavior of the outer profile.
However, the form of the inner profile is still under debate as high-resolution simulations are needed. The newest results from the Millennium Run simulation prefer the Einasto parametrization for the inner part of the profile and they constrain the parameter α in (3.45) to be approximately 0.15 with a weak mass dependence (Neto et al. 2007).
The four different profiles are plotted in Fig. 3.2 as a function of the radius. We can clearly see the different scaling of the inner and outer part of the profiles. The difference between the NFW profile and the Einasto profile is most significant for the inner part of the profiles.
The density profiles are usually expressed in terms of the halo mass, which can be derived from the density through
m= Z
d3r ρ(r) = Z rvir
0
dr4πr2ρ(r), (3.46)
where we assumed a spherical symmetric profile in the second step. In addition, we introduced a cutoff radiusrvir, the so-called virial radius which defines the virial mass of a halo, i.e., the mass of a halo in virial equilibrium. The virial radius can be easily computed from
rvir(z) = 3
4π m
∆vir(z) ¯ρ 1/3
, (3.47)
where ∆vir is the ratio of the density of a virialized halo to the mean density and is calculated using the spherical collapse model. For a ΛCDM model we found that ∆vir is redshift dependent as given in Eq.(3.21). In the literature one can find several different definitions of the cutoff radius and therefore of a virial mass of a halo. Instead of using
∆vir in Eq. (3.47) one can find the EdS inspired values (compare with Eq. 3.16)
∆200 = 200, ∆180 = 180, (3.48)
independent of cosmology and redshift. The NFW paper even used the combination
∆200ρcrit instead of the product of ∆vir with the mean background density. All these
10-8 10-6 10-4 10-2 100 102 104
10-2 10-1 100 101 102
ρ/ρs
r/rs
NFW M99 Hernquist Einasto
Figure 3.2:Different functional forms of the dark matter density profile in terms of the scaled densityρ/ρs as a function of the scaled radius r/rs. We plot the NFW- (solid line), the M99-(long-dashed line), the Hernquist- (short-dashed line) and the Einasto profile (dotted line) as given in Eqs. (3.42),(3.43),(3.44)and (3.45), respectively. Current numerical simulations favor the NFW-profile or the Einasto profile. At r/rs = 1 the transition from the power-law behavior of the inner to the outer part of the profile is clearly visible.
different combinations show that it is up to now not exactly clear what is the best way to define a halo. Hence, caution is needed before interpreting results from simulations because one needs to know their virial radius definition.
Formally one needs to integrate Eq.(3.46) up to infinity, but the mass of the M99 and the NFW profile is logarithmically divergent for large radii. Therefore one needs to introduce an arbitrary cutoff radius. Here we will adopt rvir as a cutoff radius to be consistent with the results from the spherical collapse model. Strictly speaking the results of different simulations for the density profile are only tested for radii smaller than the virial radius. The Hernquist profile is constructed in a way that it cures this problem. For large r the density is proportional to r−4 so that it is not divergent.
The integration in Eq.(3.46) can be done analytically for the NFW and M99 profiles yielding
• NFW
m= 4πρsr3s
ln(1 +c)− c 1 +c
≡ 4πρsr3s
f(c) , (3.49)
• M99
m= 4πρsr3s
2 ln(1 +c3/2) 3
≡ 4πρsr3s
g(c) , (3.50)
where we have defined the halo concentration parameter
c≡rvir/rs. (3.51)
For convenience we defined the function in square brackets as 1/f(c) and 1/g(c), respectively. Finally, we rewrite the NFW profile defined in Eq. (3.42) as
ρ(r, m) = m 4π
c3f(c) rvir3
1
x(1 +x)2 , (3.52)
where we replaced ρs using Eq.(3.49) and defined x≡c r/rvir. Hence, we parametrized the density profile with the virial mass (or equivalently the virial radius) and the concentration parameter. For the formulation of the halo model correlation functions it is convenient to define the normalized density profile
u(r, m)≡ ρ(r, m)
m , (3.53)
where R
d3r u(r, m) = 1.
In the following, we will need the expression of the density profile in Fourier space because we want to calculate the halo model power spectrum defined as the Fourier space counterpart of the two-point correlation function and also higher-order spectra.
We find for the normalized dark matter density profile
˜
u(k, m) =
R d3r ρ(r, m)eik·r
R d3r ρ(r, m) . (3.54)
For spherical symmetric profiles we can perform the angular integration and the equation simplifies to
˜
u(k, m) = Z rvir
0
dr4πr2sin(kr) kr
ρ(r, m)
m , (3.55)
where r =|r|. Note that we need to truncate the integration at the virial radius rvir to be consistent with the definition of the halo mass in Eq. (3.46). For the NFW profile in Eq. (3.52) it is then possible to find a closed solution given by
˜
u(k, m) = f(c)
sinη{Si[η(1 +c)]−Si(η)}+ cosη{Ci[η(1 +c)]−Ci(η)}
− sin(ηc) η(1 +c)
, (3.56)
where the sine- and cosine-integrals are defined as Si(x) =
Z x 0
dtsint
t , Ci(x) = − Z ∞
x
dtcost
t , (3.57)
and we introducedη =krvir/c. The profile has the asymptotic behavior u(k, m)˜ →1 for large scales which is easily verified using the limit k →0 in Eq.(3.54). Going to smaller scales the amplitude begins to decrease, i.e.,u(k, m)˜ .1. This decline begins earlier for high-mass halos compared to small-mass halos. Finally, for small scales the profile goes asymptotically as u(k, m)˜ ∝k−2. We derive this result in the following. First we take the limit of Eq. (3.56) for small halo masses, which results in large concentrations (see Eq. 3.62 below). Thus, for the limit c1 we find (Scoccimarro et al. 2001)
˜
u(k, m) = (lnc)−1[−sinηsi(η)−cosηCi(η)], (3.58) where we definedsi(x)≡Si(x)−π/2, and used the fact thatSi(∞) =π/2 andCi(∞) = 0.
Now taking the limit for small scales, i.e., η1 (note that k grows faster thanc) we find
˜
u(k, m)'(lnc)−1
sin2η
η2 + cos2η η2
= (lnc)−1η−2 ∝k−2, (3.59) where we used the asymptotic expansion of the sine- and cosine-integrals forx 1.
The expansion is given by si(x) =−cosx
x
1− 2!
x2 + 4!
x4 −. . .
− sinx x2
1− 3!
x2 + 5!
x4 −. . .
, (3.60) Ci(x) = sinx
x
1− 2!
x2 + 4!
x4 −. . .
− cosx x2
1− 3!
x2 + 5!
x4 −. . .
, (3.61)
which is found by successive partial integrations of the functions in Eq. (3.57).
We show the normalized density profile in Fourier space in Fig. 3.3 as a function of the wave-number k for six different halo masses ranging from m = 1011h−1M to m = 1016h−1M. The plot shows that massive halos contribute power only on large scales, whereas smaller halos contribute power also on small scales.
3.3.1 Halo Concentration Parameter
The halo model spectra are strongly dependent on the concentration parameter defined in Eq. (3.51). In order to get an equation for the concentration one must again resort to N-body simulations. The general result is that high-mass halos are less concentrated than low-mass halos. A good fitting formula for the mass- and redshift-dependence of the concentration of NFW halos is given by Bullock et al. (2001)
c(m, z) = c0
1 +z
m m∗(z = 0)
−α
, (3.62)
with the parameters c0 = 9 and α = 0.13. On the other hand, Takada & Jain (2003) used c0 = 10 andα = 0.2 instead which gave better results for a halo model applied to estimate higher-order weak lensing correlation functions. We already introduced the
10-3 10-2 10-1 100
10-1 100 101 102 103 104
u˜(k,m)
k [h Mpc-1]
Figure 3.3: Fourier transform of the normalized NFW dark matter density profileu(k, m) as˜ defined in Eq.(3.56)against the wave-number kfor six different halo massesm. Shown is the important halo mass range from m = 1011h−1M at the right to m = 1016h−1M at the left.
massm∗ in Eq.(3.31)which is defined byν = 1. In the next section we will demonstrate the influence ofm∗ on the halo bias.
Simulations show that different halos of the same mass have a distribution of concen-trations which is well fitted by a log-normal distribution with dispersion σlnc
p(c|m)dc= 1
p2πσ2lnc exp
−(lnc−ln ¯c)2 2σln2 c
d lnc , (3.63)
where c¯≡¯c(m, z) is the mean concentration parameter given by Eq. (3.62). The width of this distribution is obtained from simulations to be σlnc ≈ 0.2−0.4 (for example Bullock et al. 2001). Note that the width of the distribution is independent of the halo mass.