Relativistic Cosmology: from super--horizon to small scales

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The general relativistic non--linear dynamics of a self--gravitating collisionless fluid with vanishing vorticity is studied in synchronous and comoving -- i.e. Lagrangian -- coordinates. Writing the equations in terms of the metric tensor of the spatial sections orthogonal to the fluid flow allows an unambiguous expansion in inverse powers of the speed of light. The Newtonian and post--Newtonian approximations are derived in Lagrangian form; the non--linear evolution of the system on super--horizon scales, leading to the so--called ``silent universe", is also briefly discussed. A general formula for the gravitational waves generated by the non--linear evolution of cosmological perturbations is given: a stochastic gravitational--wave background is shown to be produced by non--linear cosmic structures, with present--day closure density Ωgw 10⁻⁵ -- 10⁻⁶ on the scale of 1 -- 10 Mpc.