Large-Scale Structure of the Universe and Cosmological ...

4 From Dynamics to Statistics: N-Point Results
A general approach to go from dynamics to statistics would be to solve the
Vlasov equation from initial conditions for the phase-space density function
f(x,p) given by a stochastic process such as inflation. Correlation functions
in configuration space reviewed in Chapter 3 can be trivially extended to
phase-space, and the Vlasov equation yields equations of motion for these
phase-space correlation functions. The result is a set of coupled non-linear
integro-differential equations, the so-called BBGKY hierarchy 16 , in which the
one-point density is related to the two-point phase-space correlation function,
the two-point depends on the three-point, and so forth. However, as mentioned
in Chapter 2, if we restrict ourselves to the single stream regime study of the
Vlasov equation reduces to studying the evolution of the density and velocity
fields given by the continuity, Euler and Poisson equations. Therefore, all we
have to consider in this case is the correlation functions of density and velocity
fields.
In this chapter, we review how the results discussed in Chapter 2 about the
time evolution of density and velocity fields can be used to understand the
evolution of their statistical properties, characterized by correlation functions
as summarized in the previous chapter. Most of the calculations will be done
assuming Gaussian initial conditions; in this case the main focus is in quantitative
understanding of the emergence of non-Gaussianity due to non-linear
evolution. In Sect. 4.4 we discuss results derived from non-Gaussian initial
conditions. In Chapter 5 we present, with similar structure, analogous results
for one-point statistics, with emphasis on the evolution of local moments and
PDF’s.
4.1 The Weakly Non-Linear Regime: “Tree-Level” PT
4.1.1 Emergence of Non-Gaussianity
If the cosmic fields are Gaussian, their power spectrum P(k, τ),
˜δ(k, τ) ˜ δ(k ′ , τ)
c = δD(k + k ′ )P(k, τ). (149)
(or, equivalently, their two-point correlation function) completely describes
the statistical properties. However, as we saw in Chapter 2, the dynamics
of gravitational instability is non-linear, and therefore non-linear evolution
inevitably leads to the development of non-Gaussian features.
16 after N. N. Bogoliubov, M. Born, H. S. Green, J. G. Kirkwood and J. Yvon, who
independently obtained the set of equation between 1935 and 1962. Rigorously, this
route from the Vlasov equation to the BBGKY equations is restricted to the socalled
“fluid limit” in which the number of particles is effectively infinite and there
are no relaxation effects.
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