Hence, in the spherical collapse model, virialized objects are characterized by a density that is roughly 180 times larger than the mean background density. The linear density contrast at collapse is easily calculated because Eq. (3.8) is still applicable. Inserting the time of collapse tcoll = 2πB yields
δlin(coll)= 3
20(12π)2/3 '1.686. (3.17)
Thus, a linear density contrast of 1.69 marks roughly the point where the virialization of a spherical halo occurs. We will label the corresponding linear density contrast at collapse with δc in the following.
In summary, the spherical collapse model provides two important numbers: the linear density contrast at the time of collapse δc and the density ratio of a virialized halo
∆vir=ρvir/ρ. If we consider virialized objects today we take¯ ρ¯=ρcritΩm which is the mean matter density today. For an EdS Universe we obtained for both parameters
δc(EdS)'1.686, (3.18)
∆(EdS)vir = 18π2 '178. (3.19)
Note that for a general cosmological model both parameters are redshift- and cosmology-dependent. Calculations for a flat cosmological ΛCDM model in Henry (2000), where Ωm+ ΩΛ = 1, give the following results
δc(z) =δ(EdS)c [1−0.0123 ln(1 +x3)], (3.20)
∆vir(z) = ∆(EdS)vir (1 + 0.4093x2.71572), (3.21) where the redshift and cosmology dependence is encoded in the parameter x with
x≡ (Ω−1m −1)1/3
1 +z . (3.22)
Note that the value of the thresholdδcis only weakly dependent on cosmology. Therefore we use the EdS value in Eq. (3.18)for the subsequent calculations. However, for the density ratio we use the redshift and cosmology dependence as given in Eq. (3.21) in the following which is valid for a flat ΛCDM model.
First we present the pioneering work by Press-Schechter (see Press & Schechter 1974) on the mass function in Sect. 3.2.1 which showed good agreement with observations at that time. However, high-resolution N-body simulations showed that the Press-Schechter mass function while capturing the rough features of the simulation fails in detail. In the advent of precision cosmology we need more reliable models for the mass function. We present such improved models in Sect. 3.2.2 including the most used parametrization, namely the Sheth-Tormen mass function (Sheth & Tormen 1999).
3.2.1 Press-Schechter Model
The Press-Schechter approach to calculate the mass function of virialized objects is based on the assumption that objects form in the so-called hierarchical clustering or bottom-up picture. In this case small structures form first and then subsequently merge to form larger structures.
We can assign a characteristic length scale R to a halo of mass m defined as the comoving radius of a homogeneous sphere with mean matter density ¯ρ
4π
3 R3ρ¯=m , R(m) = 3m
4πρ¯ 1/3
. (3.23)
The mean density contrast within this volume needs to be larger than δc to form a virialized structure. Therefore, we will smooth the density contrast δ over the characteristic scale R with a filter function. We are free to choose the form of this filter function. Here we will adopt a top-hat function defined in Eq. (B.22). Appendix B.5 provides a complete description of the smoothing of Gaussian density fields. For an initially Gaussian random field, as assumed for simple inflationary models, the probability of finding the smoothed density contrast δ is given by
pR(δ) = exp[−δ2/2σR2(m)]
p2πσR2(m) , (3.24)
where the variance of scaleR is denoted byσR2(m) and is given by σ2R(m) =
Z dk k
k3Ppt(k, z = 0)
2π2 |WR(k)|2, (3.25)
whereWR(k) is the filter function in Fourier space andPpt(k, z= 0) is the linear power spectrum at redshift z = 0. Note that σ2R(m) is a monotonically decreasing function of the halo mass m.
The fraction of the Universe contained in virialized objects with mass larger thanm (δ > δc) is estimated as
F>(m) = Z ∞
δc
pR(δ)dδ = 1 2erfc
δc
√2σR(m)
, (3.26)
where erfc(x) is the complementary error function. However, there is a serious problem with this approach as the considered mass of bound objects goes to zero. In this case the total fraction should be 1, whereas the Press-Schechter approach gives a fraction of 1/2, i.e.,
F>(0) = 1 2erfc
δc
√2σR=0(m= 0)
= 1
2, (3.27)
where we used the fact that the top-hat function is WR(0) = 1 for R = 0 and thus σR2(m)→ ∞. To solve this problem, Press & Schechter simply multiplied the final result in Eq. (3.29) below with the missing factor of 2. We convert the fraction (3.26) to the fraction of the cosmic volume filled with halos with masses between m and m+ dm using
f(m) = −∂F>
∂mdm , (3.28)
where we introduced a minus sign since F>(m) is a decreasing function of m. The number density n(m) per comoving volume is obtained by dividing Eq. (3.28) with the mean occupied volume V of a halo with mass m
n(m)dm = f(m)
V =−ρ¯∂F>
∂m dm
m , (3.29)
where ρ¯is the mean comoving density. Finally, after performing the partial derivative in Eq. (3.29) we find the halo mass function
dn
dm ≡nPS(m) =−2ρ¯ m
δc σ2
e−δc2/2σ2
√2π dσ dm = ρ¯
m
e−δ2c/2σ2
√2π δc σ3
dσ2 dm
, (3.30)
where we emphasized the equality of dn/dmandn(m) in the first step as both notations are used in the literature. In addition, we use from now on the compact notationσR≡σ and only write the mass dependence when necessary. The ad-hoc factor of 2 can be derived from extended Press-Schechter theory (e.g., Zentner 2007). The dependence of the mass function on cosmological parameters enters through the variance σ2 which depends on the linear power spectrum (see Eq. 3.25) and through the mean density of the Universe which depends on Ωm.
To get insight into the behavior of the mass function, it is useful to consider a scale-free power spectrum as we can then find an analytical expression for the mass function. For a scale-free power spectrum P(k)∝kn we find that the variance is given by σ2(m) = (m/m∗)−n+33 δc2. Here we write the result in terms of the nonlinear mass scale m∗ defined as
σ2(m∗) =δc2, (3.31)
which provides a natural way to divide the mass function into two regimes. We find nPS(m) = A
m2 m
m∗
(3+n)/6
exp
"
−1 2
m m∗
1+n/3#
, (3.32)
0.1 1
1
νf(ν)
ν
Press-Schechter Sheth-Tormen Jenkins
Figure 3.1: Comparison of the different halo mass function parametrizations. Shown is the multiplicity functionνf(ν) (see Eq. 3.33 for the connection to the mass function) against the dimensionless variableν. The solid line shows the Press-Schechter mass function in Eq.(3.35), the dashed line the Sheth-Tormen mass function in Eq. (3.36)and the dotted line the Jenkins mass function in Eq. (3.37).
whereA = √ρ¯
2π 3+n
3
. For mm∗ we find nPS(m)∝m−2m(3+n)/6 meaning that at low masses the mass function diverges like a power law. For large masses the exponential takes over leading to a cutoff for masses m &m∗.
Up to now the expression for the mass function is limited to the current time z = 0.
If we want to know the mass function of halos at a specific redshift z we simply need to modify the variance as σ2(m, z) =D2(z)σ2(m), where we used the redshift dependence of the linear power spectrum (see Eq. 2.48 for the expression of the growth factorD(z)).
3.2.2 General Halo Mass Function
We can write the differential number density of halos in the following compact form n(m, z) = ρ¯
m2νf(ν)d lnν
d lnm, (3.33)
where we defined the dimensionless variableν as ν = δc(z)
D(z)σ(m). (3.34)
Thus, part of the mass function can be expressed by the multiplicity functionνf(ν), which has a universal shape, i.e., independent on cosmological parameters and redshift.
The universal shape is one of the main nontrivial results from simulations. However, there is very recent indication of a redshift dependence of the multiplicity function as reported in Tinker et al. (2008).
In the literature one finds a number of different parametrizations of the multiplicity function. The most popular ones are
• Press-Schechter mass function (Press & Schechter 1974) νf(ν) =
r2
πνexp(−ν2/2). (3.35)
• Sheth-Tormen mass function (Sheth & Tormen 1999) νf(ν) = A
r2
π[1 + (qν2)−p]p
qν2exp(−qν2/2), (3.36) where the two parameters are given by q = 0.707 and p= 0.3 and A denotes the amplitude of the mass function that is fixed by mass conservation as shown in Eq. (3.38) below.
• Jenkins mass function (Jenkins et al. 2001)
νf(ν) = 0.315 exp(−|lnσ−1+ 0.61|3.8). (3.37) The Sheth-Tormen and Jenkins functions are directly fitted to numerical simulations.
For smallν we findνf(ν)∝ν0.4 for the Sheth-Tormen mass function andνf(ν)∝ν for the Press-Schechter mass function. This limit corresponds to small halo masses because the variance is largest for small masses. We present the three different multiplicity functions in Fig. 3.1. The largest difference is seen for small ν (or small m) which thus affects the small scales of the halo model spectra. This will be explained in detail in Sect. 3.3.
In the following we will focus on the Sheth-Tormen mass function as it provides the best agreement with simulations. In addition, it was shown to be connected to a physical model that describes an ellipsoidal mass collapse in contrast to a spherical mass collapse (see Sheth et al. 2001). The amplitude A in Eq. (3.36) is determined by assuming mass conservation, such that the integral over the mass function times the halo mass gives the mean density of the Universe:
1
¯ ρ
Z ∞ 0
n(m, z)mdm= Z ∞
0
f(ν) dν= 1, (3.38)
where we have used the definition of the mass function in terms of the variable ν as in Eq. (3.33) in the first step. The integral can be solved analytically forp < 1/2 and we obtain for the amplitude
A(p) =
1 + 2−pΓ 1
2 −p .√
π −1
, (3.39)
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).