Stability borders and classification of density perturbation propagations in deDonder-gauge on a cosmological background

Abstract

We investigate the propagation and the stability borders of density and metric perturbations on a cosmological background in linear perturbation theory in deDonder-gauge. We obtain the algebraic equations for the generally time-dependent stability borders by setting the typical time for perturbation contrasts infinite in the set of differential equations, while all other typical times stay finite. In dD-gauge there are in general three stability borders whereas in synchronous gauge there is only one. In the limiting cases of radiation perturbations and “dustlike” perturbations we obtain in deDonder-gauge no stability border resp. only one stability border (the ordinary Jeans limit). The first case is in contrast to the synchronous gauge and means that radiation perturbations cannot become unstable. During the recombination there could be three stability borders. We classify the propagation solutions and the systems of differential equations governing them by comparing the characteristic times in the original general system of differential equations, in deDonder-gauge and synchronous gauge. The greatest differences for the propagation of density contrasts arise from the presence of a gravitational wave time scale in deDonder-gauge. This becomes significant if the density perturbations are relativistic with respect to the velocity of sound. Gravitational retardation effects are the origin of the 6-dimensionality of the solution space for density contrasts. This reflects the necessity and physical meaning of gauge solutions.