The Lagrangian Approach to the Gravitational Instability in an Expanding Universe

Abstract

In order to remedy a defect found by Field and Shepley of Lifshitz's relativistic theory for the gravitational instability in an expanding universe, a new approach to the density perturbation mode is proposed. The defect lies in the fact that one solution of the third order differential equation derived by us for the density contrast K = δε/ε0 has no physical meaning, where ε0 is the unperturbed density of matter and radiation. The essence of our approach is to adopt the Lagrangian coordinate condition which assures the comoving nature of coordinates even in the perturbed state, in place of Lifshitz's condition which becomes the former condition only when p0 (the unperturbed pressure of matter and radiation) vanishes. It is shown that the differential equation for K is of the second order even when p0 = ε0/3 and is reduced to Bonnor's Newtonian equation when p0 = 0. It is shown further that, under some favorable condition, the growth rate of K is the highest at the stage p0 = ε0/3, i.e. K ∝t which does not necessarily hold in Lifshitz's original theory, where t is the cosmic time. We have also examined how the density perturbation depends sensitively on our selection of coordinate conditions by picking up another coordinate condition.