Cosmological matter perturbations

We investigate matter density perturbations in models of structure formation with or without causal/acausal source. Under the fluid approximation in the linear theory, we first derive full perturbation equations in flat space with a cosmological constant $\Lambda$. We then use Green-function technique to obtain analytic solutions for matter perturbations in a flat $\Lambda=0$ model. Some incorrect solutions in the literature are corrected here. A simple yet accurate extrapolation scheme is then proposed to obtain solutions in curved or $\Lambda\neq 0$ cosmologies. Some general features of these solutions are revealed. In particular, we analytically prove that the resulting matter density perturbations are independent of the way the causal source was compensated into the background contents of the universe when it was first formed. We also use our Green-function solutions to investigate the compensation mechanism for perturbations with causal seeds, and yield a mathematically and physically explicit formin interpreting it. We found that the compensation scale depends not only on the dynamics of the universe, but also on the properties of the seeds near the horizon scale. It can be accurately located by employing our Green functions.