The difficulty of finding an analytic model for the statistics of dark matter halos is that it has to deal with highly non-linear objects. The usual approach is to use numerical simulations in order to describe the formation of halos. However, there exists a special case, in which the non-linear evolution of the density contrast can be solved explicitly. If one considers the evolution of a spherical matter overdensity embedded in a homogeneous Universe, it is possible to calculate the density contrast, at which an object can be considered as virialized. This description is referred to as spherical collapse model and was studied first by Gunn and Gott [28] in 1972, who considered the collapse from an initially top-hat density perturbation. In addition to estimating the value for the virial density contrast, the spherical collapse model in the linear regime is used to determine a characteristic time-scale, at which bound objects form. In the following, we will derive the two important quantities of the spherical collapse model, namely the virial density contrast ∆vir and the linear density contrast of a collapsed object δsc.
We consider the evolution of an overdense spherical perturbation of comoving scale R in a matter-dominated Universe. The evolution of the overdense sphere will oc-cur independently of the rest of the Universe. At some initial time ti the spherical
4.2 The spherical collapse model 65
perturbation of radius Ri will have the average density ρ(Ri, ti) = ¯ρ
1 + ¯δ(Ri, ti)
, (4.33)
where ¯δ(Ri, ti) = R33 i
R dR R2δ(R, ti) is the average density contrast within the volume Vi and ¯ρ is the comoving background density of the Universe. The dynamics of the collapse is governed by the mass M enclosed in the sphere, which is conserved as long as different spheres do not cross each other1, i.e.
M(R, t) = M(Ri, ti) = 4
3πR3iρ[1 + ¯¯ δ(Ri, ti)]. (4.34) If one additionally assumes ¯δ(Ri)1, one finds a simple relation between the initial Lagrangian radius of the sphere Ri and the evolved Eulerian radius R:
Ri R
3
= (1 + ¯δ(R)). (4.35)
Eq. (4.35) will be used, when we want to relate the linear density contrast to the non-linear one and whenever a transformation from Langrangian to Eulerian coordinates is performed. Note that in the following we will denote the density contrast ¯δ(R) simply byδ for notational convenience.
Changing to physical coordinates, the dynamics of a gravitating sphere with radiusR and mass M can be described with Newtonian physics and thus follows
d2R
dt2 =−GM
R2 , (4.36)
where we restrict ourselves to one dimension due to the spherical symmetry of the problem. The total energy of the sphere determines its actual fate and is found by integrating the above equation:
1 2
dR dt
2
−GM
R =E (4.37)
For E ≥ 0 the shell will expand forever, whereas for E <0 it will first expand until it reaches a maximal radius Rmax and then collapse2. Since we are interested in the formation of objects, we restrict our calculation toE <0. In this case, Eq. (4.36) has the following parametric solution:
1Within the spherical collapse model one assumes that shell crossing happens only after the actual collapse
2This consideration is also valid for the whole Universe and provides a way to derive the Friedmann equations in the Newtonian limit.
R =A(1−cosθ) , (4.38)
t =B(θ−sinθ) , (4.39)
where A3 = GM B2 and θ parametrizes the evolution in time. With the parametric solution, it is possible to relate θ to the corresponding radius R and time t for each stage of the evolution. At the beginning of the process, one has θi = 0 andRi = 0. At turnaround, where the sphere reaches its maximal radius Rmax, one has θta =π, and the sphere is collapsed completely, when θcoll= 2π. An overview can be found in tab.
4.1.
4.2.1 Linear regime
In the initial stage, where θ 1, we can expand the parametric solution, i.e. Eqs.
(4.38) and (4.39), into a power series of θ:
R ≈ A 2 θ2
1− θ2
12
, (4.40)
t ≈ B 6 θ3
1− θ2
20
. (4.41)
Combining Eqs. (4.40) and (4.41) and neglecting higher-order terms, the radius be-comes
R(t)≈ A 2
6t B
2/3"
1− 1 20
6t B
2/3#
, (4.42)
whereas the mean density of the perturbation is
ρ(t) = 3M
4πR3 ≈ 1 6πGt2
"
1 + 3 20
6t B
2/3#
. (4.43)
For an EdS-Universe, the background density of the Universe is
¯
ρ= 1
6πGt2 , (4.44)
thus, from ρ= ¯ρ(1 +δ), we can identify
δlin = 3 20
6t B
2/3
(4.45)
4.2 The spherical collapse model 67
Table 4.1: Three stages of the spherical collapse of a perturbation with their characteristic values for the linear and non-linear density contrast.
Stage θ r t δlin ∆nl
Initial time 0 0 0 0 ∞
Turnaround π 2A πB 1.06 9π2/16
Collapse 2π 0 2πB 1.686 ∞
as the linear density contrast of a spherical object. The scaling t2/3 of the density contrast is exactly what one would expect in case of an EdS-Universe. Although the linear solution is a poor approximation for the real density contrast as soon as θ &1 (see Fig. 4.2), its value at collapse can be used to define a characteristic time-scale for virialized objects. This becomes e.g. important when deriving the halo abundance at different redshifts in section 4.3. The exact value for the linear density contrast is obtained by inserting the time of collapsetcoll = 2πB into Eq. (4.45):
δsc = 3
20(12π)2/3 ≈1.686, (4.46)
where we the use the subscript ‘sc’ in order to indicate the origin from the spherical collapse model.
The redshift dependence of the density contrast can also be incorporated into the consideration. If one assumes e.g. that an object has formed in the recent past, one can set the present time for the collapse epoch and calculate the initial linear density contrast at redshift z with the help of the growth factor according to
δsc(z) = G(z)δsc(zcoll = 0) = 1.686
1 +z , (4.47)
where the last equality is only valid for an EdS-Universe.
4.2.2 Non-linear regime
As stated in the beginning, the evolution of a gravitating sphere can also be solved analytically in the non-linear regime by Eqs. (4.38) and (4.39). In analogy to the linear case, one can use these solutions to determine the average density and density contrast:
ρnl(t) = 3M
4πR3nl(t) = 3 4πGt2
(θ−sinθ)2
(1−cosθ)3 , (4.48)
δnl(t) = ¯ρ[1 +δnl(t)] = 9 2
(θ−sinθ)2
(1−cosθ)3 −1, (4.49)
where ¯ρ denotes the background density in an EdS cosmology as given in Eq. (4.44).
We can now reconsider the most important stages of a collapsing sphere and focus on how the non-linear density contrast evolves with respect to the background density of the Universe. At first the spherical perturbation expands from a zero radius until it reaches a maximum atRta. The turnaround marks the point of time, where the actual collapse begins. At this moment, the ratio of the density of the sphere to that of the background density is
∆ta ≡ ρ(tta)
¯
ρ(tta) = 9π2
16 ≈5.552, (4.50)
which can be derived using the non-linear density as given in Eq. (4.48). For the time of the actual collapse, non-linear theory predicts the sphere to have a radiusRcoll = 0 and an infinite density contrast, whereas the linear consideration approximatesδsc'1.686.
In reality the spherical perturbation will not become singular, since this occurs only if the collapse is exactly symmetric. Instead the perturbation virializes in a process of violent relaxation to a finite radius. With the help of the virial theorem, one finds the radius of the relaxed state to beRvir=Rta/2. Thus, the sphere is in a virialized state if the radius reaches half of the maximum expansion at turnaround, which results in a density
ρ(tvir) = 3M
4πGR3vir = 8 3M
4πGR3ta = 8ρ(tta). (4.51) Numerical simulations of the process suggest instead that the relaxation occurs at about the time of collapse. Assuming this time-scale, one finds
¯
ρ(tvir) = (6πGt2coll)−1 = ¯ρ(tta)/4. (4.52) Combing these last two equations determined for the time of virialization with the result in Eq. (4.50), one finds the virial density contrast to be
∆vir ≡δvir+ 1 = 32ρ(tta)
¯
ρ(tta) = 18π2 ≈177.7 (4.53) in the non-linear regime. Note that this consideration is only valid for an EdS-Universe and the collapse of spherical objects. A more realistic treatment would assume an elliptical mass distribution of gravitating perturbation instead. Consequences of an elliptical collapse are discussed in detail e.g. in [8, 53, 78]. Fitting formulae for other cosmologies can be found in [23, 31, 55, 60].