The linear theory
We now proceed to explore the linear regime of the Vlasov-Poisson system. One objective is to make contact with earlier stages of the gravitational dynamics and another is to introduce the notion of Green function that we will use in what follows.
The linear modes
Linearization of the equation of motion is achieved by assuming that the terms [?(x,t))uj(x,t)] i and Uj(x,t) Uj(x,t)j in the continuity and the Euler equations respectively, vanish. This occurs when both the density contrast and the velocity gradients in units of H are negligible. The linearized system is obtained in terms of the velocity divergence
so that the system now reads
after taking the divergence of the Euler equation. We have introduced here the Hubble parameter H = a/a and used the Friedman equation H2 = 8n/3 Gpc(t) together with the definition of Hm = p(t)/pc(t).
The solution of this system is now simple. It can be obtained after eliminating the velocity divergence, and one gets a second-order dynamical equation
for the density contrast. It should be noted that the spatial coordinates are here just labels: there is no operator acting of the physical coordinates. This is a specific feature of the growth of instabilities in a pressureless fluid. It implies in particular that the linear growth rate of the fluctuations will be independent of scale. The time dependence of the linear solution is given by the two solutions of
one of which is decaying and the other is growing with time. For an Einstein-de Sitter (EdS) background (a universe with no curvature and with a critical matter density), the solutions read
that is, D+dS(t) is proportional to the expansion factor. This result gives the timescale of the growth of structure. This is what permits a direct comparison between the amplitude of the metric perturbations at recombination and the density perturbation in the local universe. Note that it implies that the potential, for the corresponding mode, is constant (see the Poisson equation).
In the following, we will later see how these results can be extended to other background evolution.