A RELATIVISTIC APPROACH TO GRAVITATIONAL-INSTABILITY IN THE EXPANDINGUNIVERSE - 2ND-ORDER LAGRANGIAN SOLUTIONS

A Lagrangian relativistic approach to the non-linear dynamics of cosmo logical perturbations of an irrotational collisionless fluid is consid ered. Solutions are given at second order in perturbation theory for t he relevant fluid and geometric quantities and compared with the corre sponding ones in the Newtonian approximation. Specifically, we compute the density, the volume expansion scalar, the shear, the 'electric' p art, or tide, and the 'magnetic' part of the Weyl tenser. The evolutio n of the shear and the tide beyond the linear regime strongly depends on the ratio of the characteristic size of the perturbation to the cos mological horizon distance. For perturbations on sub-horizon scales th e usual Newtonian approximation applies, at least at the considered pe rturbative order; on super-horizon scales, instead, a new picture emer ges, which we call the 'silent universe', as each fluid element evolve s independently of the environment, being unable to exchange signals w ith the surrounding matter through either sound waves or gravitational radiation. For perturbations inside the Hubble radius, particular att ention is paid to singling out non-local effects during the non-linear evolution of fluid elements. These non-local effects are shown to be carried by a traceless and divergence-free tenser, contained in the ma gnetic part of the Weyl tenser, which is dynamically generated as soon as the system evolves away from the linear regime.