Effect of a non null pressure on the evolution of perturbations in the matter dominated epoch

We analyze the effect of pressure on the evolution of perturbations of an Einstein-de Sitter Universe in the matter dominated epoch assuming an ideal gas equation of state. For the sake of simplicity the temperature is considered uniform. The goal of the paper is to examine the validity of the linear approximation. With this purpose the evolution equations are developed including quadratic terms in the derivatives of the metric perturbations and using coordinate conditions that, in the linear case, reduce to the longitudinal gauge. We obtain the general solution, in the coordinate space, of the evolution equation for the scalar mode, and, in the case of spherical symmetry, we express this solution in terms of unidimensional integrals of the initial conditions: the initial values of the Newtonian potential and its first time derivative. We find that the contribution of the initial first time derivative, which has been systematically forgotten, allows to form inhomogeneities similar to a cluster of galaxies starting with very small density contrast. Finally, we obtain the first non linear correction to the linearized solution due to the quadratic terms in the evolution equations. Here we find that a non null pressure plays a crucial role in constraining the non linear corrections. It is shown, by means of examples, that reasonable thermal velocities at the present epoch (not bigger than 10 −6) make the ratio between the first non linear correction and the linear solution of the order of 10 −2 for a galaxy cluster inhomogeneity.