Damping time and stability of density fermion perturbations in the expanding universe

Abstract

The classic problem of the growth of density perturbations in an expanding Newtonian universe is revisited following the work of Bisnovatyi-Kogan and Zel'dovich. We propose a more general analytical approach: a system of free particles satisfying semi-degenerate Fermi-Dirac statistics on the background of an exact expanding solution is examined in the linear approximation. This differs from the corresponding work of Bisnovatyi-Kogan and Zel'dovich where classical particles fulfilling Maxwell-Boltzmann statistics were considered. The solutions of the Boltzmann equation are obtained by the method of characteristics. An expression for the damping time of a decaying solution is discussed and a zone in which free streaming is hampered is found, corresponding to wavelengths less than the Jeans one. In the evolution of the system, due to the decrease of the Jeans length, those perturbations may lead to gravitational collapse. At variance with current opinions, we deduce that perturbations with lambda >=lambda_(J Max)/1.48 are able to generate structures and the lower limit for substructures mass is M = M_(J max)/(1.48)^3 ~ M_(J max)/3, where M_(J max) is the maximum value of the Jeans mass.