1.3 The Cosmological Standard Model and Extensions
2.1.1 The Vlasov Equation
The outline of this chapter is as follows: In Sect. 2.1, we derive the basic equations governing cosmological structure formation and present a general perturbative solution scheme to these nonlinear equations. We present the concept and properties of n-point correlation functions in Sect. 2.2. In particular, we show the perturbative results for the two-, three- and four-point correlation function in lowest order of the density field. As perturbation theory breaks down in the nonlinear regime, we give fitting functions for the dark matter power spectrum and the dark matter bispectrum in Sect. 2.3 . Finally, in Sect. 2.4 we show the effect of radiation on structure formation, and introduce the transfer function which describes the transition of the radiation-dominated era to the matter-dominated era.
2.1 Perturbation Theory
This section provides the basic framework of cosmological perturbation theory in the matter-dominated regime. First, we derive the fundamental equations governing the process of structure formation, namely the fluid equations, from the collisionless Boltzmann equation which describes the conservation of the number of dark matter particles in a given phase-space element. The resulting fluid equations are highly nonlinear and are in general not analytically solvable. Nevertheless, it is possible to find a closed solution in the linear regime. For the nonlinear regime we apply a perturbative ansatz around this linear solution. Most of the presented results and further issues can be found in the review paper by Bernardeau et al. (2002) which provides a thorough introduction to this field.
where ηµν is the Minkowski metric of flat space-time (see Eq. A.1) and hµν is a small perturbation to this metric. Using this perturbed Minkowski metric, we can compute the components of the Einstein equation (1.9) assuming an ideal gas for the energy-momentum tensor in Eq. (1.11). As a result, we obtain the Poisson equation from the zero-zero-component of the Einstein equation:
∇2rφ = 4πGN
ρ+3p c2
−Λ, (2.2)
where φ is the gravitational potential and ∇r denotes the nabla operator with respect to r, i.e., ∇r=∂/∂r. The first term is the result we would also get using Newtonian physics from the start. In addition, we get two terms describing the influence of the pressure p and the cosmological constant Λ. In this section, we analyze cosmological structure formation after recombination, i.e., we are considering the limit p ρc2. Then the Poisson equation simplifies to
∇2rφ = 4πGNρ−Λ. (2.3)
One can show that the weak field approximation implies the non-relativistic limit. Now we want to derive the classical potential and equation of motion noting that the result of Eq. (2.3) is the Newtonian result plus a constant.
Newtonian approach
The Universe after recombination can be described as a system ofN particles of massm interacting only gravitationally. The particles themselves can be dark matter particles like WIMPs1 or bound systems like dark matter halos. Note that it is not necessary to assume a certain particle species because the final equations are independent of the particle mass. The equation of motion of a single particle at position r is given by a superposition of the forces from all particles
dv
dt =GNm
N
X
i=1
ri −r
|ri−r|3 , (2.4)
where v denotes the proper particle velocity and ri the position of the i-th particle.
Note that we assumed all particles to have the same mass m. By introducing the Newtonian gravitational potential
φ(r) =−GN Z
d3r0 ρ(r0)
|r0 −r|, (2.5)
we can write the equation of motion in the compact form dv
dt =−∇rφ(r). (2.6)
1Weakly Interacting Massive Particles.
We already showed in Sect. 1.1.2 that the expansion of the Universe is governed by a time-dependent scale factor. It is a good practice to adopt comoving coordinates that stay constant for the normal Hubble expansion. They are defined as
r(t) = a(t)x, (2.7)
where xdenotes the comoving distance, and a(t) is the scale factor of the Universe. In addition, we introduce the conformal time
dt=a(τ)dτ , (2.8)
for notational convenience. Using conformal time coordinates the Friedmann equations (1.12) and (1.13) change to
Kc2 = [Ωm(τ) + ΩΛ(τ)−1]H2, (2.9) dH
dτ =
ΩΛ(τ)− Ωm(τ) 2
H2, (2.10)
where we have defined the conformal Hubble rateH ≡dlna/dτ =Ha. The dimension-less density parameters change accordingly to
Ωm(τ) = 8πGN
3H2 ρ¯= 8πGNa2
3H2 ρ ,¯ (2.11)
ΩΛ(τ) = Λ
3H2 =a2 Λ
3H2 . (2.12)
We define the dark matter density contrast
δ(x, τ)≡ ρ(x, τ)−ρ(τ¯ )
¯
ρ(τ) , (2.13)
whereρ¯is the mean matter density of the Universe. Applying comoving coordinates, the Poisson equation (2.3) changes to
∇2φ=a2(4πGNρ−Λ) =H2 3
2Ωm(τ)(1 +δ)−3ΩΛ(τ)
, (2.14)
where we used Eqs. (2.11), (2.12) and (2.13) in the second step. Here and in the following we will use the nabla operator in comoving coordinates, i.e., ∇x≡ ∇= ∂/∂x.
Performing the time derivative of Eq. (2.7), yields the proper velocity in terms ofτ: v(x, τ) =Hx+u(x, τ). (2.15) The first term on the right-hand side of this equation is the Hubble flow, whereas the second term describes departures from the mean expansion of the Universe. This is
the so-called peculiar velocity defined as u ≡ x. We define the scaled cosmological˙ gravitational potential by
Φ(x, τ)≡φ(x, τ) + 1 2
dH
dτ x2. (2.16)
Using the identity
∇2Φ(x, τ) = ∇2φ(x, τ) + 3dH
dτ , (2.17)
we find that the new potential fulfills the Poisson equation
∇2Φ(x, τ) = 3
2Ωm(τ)H2(τ)δ(x, τ), (2.18) where we used the Poisson equation for φin comoving coordinates(2.14)and the second Friedmann equation (2.10). Rewriting the equation of motion (2.6) in terms of the newly introduced variables, i.e., the peculiar velocity u (see Eq. 2.15) and the potential Φ (see Eq. 2.16), we get the following result
dp
dτ =−am∇Φ(x, τ), (2.19) where we defined the peculiar momentum
p=amu. (2.20)
To study the motion of N dark matter particles analytically, one needs to solve N three-dimensional differential equations of the same type as Eq. (2.19). However, the huge number of particles makes it impossible to follow the motion of each individual particle. Therefore, we need to employ a different approach. A classical way out is to study the distribution function f(x,p, τ) of all particles rather than the individual equations. The distribution function is defined such that
dN =f(x,p, τ) d3xd3p (2.21)
is the number of particles at timeτ contained in the infinitesimal six-dimensional phase-space volume d3xd3p. The general Boltzmann equation describes the time evolution of this distribution function:
df
dt =Ccoll, (2.22)
where Ccoll contains all possible collision terms. As we are considering collisionless dark matter, we can set this term to zero. Then the Boltzmann equation simplifies to the Vlasov equation:
df dτ = ∂f
∂τ + ˙x· ∇f+ ˙p·∂f
∂p = 0. (2.23)
This equation is a special case of Liouville’s theorem which describes the conservation of the phase-space density f(x,p, τ) over time. As a consequence, particles contained
in an initial region move in phase-space such that the region will continue to occupy the same volume but with altered shape. Inserting the equations of motions (2.19) and (2.20) in the Vlasov equation yields
∂f
∂τ + p
ma· ∇f −am∇Φ· ∂f
∂p = 0. (2.24)
This equation is highly nonlinear which is induced by the potential Φ. This can be seen from the fact that the potential Φ depends through the Poisson equation (2.18) on the matter density. But the density is proportional to the integral over the distribution function f itself (see Eq. 2.25).
The common ansatz to solve the Vlasov equation(2.24) is to take velocity moments of the equation. The first three moments are defined as R
d3u, R
d3uu andR
d3u uiuj and higher-order moments can be defined accordingly. To identify the moments with physical quantities, we define the proper density as the zeroth moment:
ρ(x, τ) =ma−3 Z
d3p f(x,p, τ), (2.25) the peculiar velocity as the first moment:
hu(x, τ)i= Z
d3p p ma
f(x,p, τ) Z
d3p f(x,p, τ), (2.26) and the second velocity moment:
huiuji ≡[huiihuji+σij] = Z
d3p pi ma
pj ma
f(x,p, τ) Z
d3p f(x,p, τ), (2.27) where σij is the stress tensor which is defined as σij ≡ huiuji − huiihuji. In the single-stream approximation one sets σij and all higher-order velocity moments to zero. Multiple streams occur in cosmology at nonlinear scales, for example during the virialization process of a halo.
For notational convenience we will omit to write the average over the velocity h· · · i in the following and simply denote hui ≡u. Taking the zeroth moment of the Vlasov equation (2.24) yields
∂
∂τ Z
d3p f +∇i
Z
d3p pi
maf−ma∇Φ· Z
d3p∂f
∂p = 0, (2.28) where ∇i ≡ ∂/∂xi is the i-th component of the nabla operator and we assumed summation over multiple occurring indices. The third term vanishes because the distribution function is zero at infinity. Using the definitions (2.25) and (2.26), one obtains the continuity equation
∂δ
∂τ +∇ ·[(1 +δ)u] = 0. (2.29)
We analogously get the Euler equation by taking the first velocity moment of the Vlasov equation
∂u
∂τ +Hu+ (u· ∇)u=−∇Φ− 1
ρ∇j(ρσij), (2.30) where we used Eqs. (2.25),(2.26)and (2.27). In combination with the Poisson equation (2.18), the continuity and the Euler equation build the fundamental set of equations for our analytical study of structure formation.
A complementary approach to taking moments of the Vlasov equation is to assume an ideal fluid (see Peebles 1980), which is characterized by its density ρ and isotropic pressure p. The results of both methods are the same when using the equation of state ρσij =δijp. The Vlasov equation approach is more general in the sense that one also obtains differential equations for higher-order moments. Note that the moments of the Vlasov equation all have a similar feature: they couple the (N −1)-th moment to the N-th moment. For example, taking the second moment of the Vlasov equation one gets a differential equation for the stress tensor that is coupled to a third-order velocity tensor.
Up to now, there is no approach for solving the fluid equations for nonlinear scales, where the contribution from the stress tensor σij becomes important. Recent studies try to use renormalized perturbation theory (see Crocce & Scoccimarro 2006) and a renormalization group approach (see McDonald 2007) to solve the fluid equations in the quasilinear regime. Maybe one can extend this kind of calculations also into the nonlinear regime, where multi-streaming is important. This would be a breakthrough in the study of structure formation. But it is more likely that perturbation theory will provide results that can be used as a startup configuration for subsequent N-body simulations.