Dynamics of cosmological perturbations in position space

We show that the linear dynamics of cosmological perturbations can be described by coupled wave equations, allowing their efficient numerical and, in certain limits, analytical integration directly in position space. The linear evolution of any perturbation can then be analyzed with the Green’s function method. Prior to hydrogen recombination, assuming tight coupling between photons and baryons, neglecting neutrino perturbations, and taking isentropic (adiabatic) initial conditions, the obtained Green’s functions for all metric, density, and velocity perturbations vanish beyond the acoustic horizon. A localized primordial cosmological perturbation expands as an acoustic wave of photon-baryon density perturbation with narrow spikes at its acoustic wave fronts. These spikes provide one of the main contributions to the cosmic microwave background radiation anisotropy on all experimentally accessible scales. The gravitational interaction between cold dark matter and baryons causes a dip in the observed temperature of the radiation at the center of the initial perturbation. We first model the radiation by a perfect fluid and then extend our analysis to account for finite photon mean free path. The resulting diffusive corrections smear the sharp features in the photon and baryon density Green’s functions over the scale of Silk damping.