Post-Newtonian cosmological dynamics in Lagrangian coordinates

The non-linear dynamics of irrotational dust in General Relativity is studiedin synchronous and comoving coordinates. All the equations are written in termsof the metric tensor of spatial sections orthogonal to the flow, which allowsan unambiguous expansion in powers of $1/c^2$. To lowest order, the Newtonianapproximation in Lagrangian form is derived. At this level the evolution isgoverned by the Raychaudhuri equation for the Lagrangian-to-Eulerian Jacobianmatrix. The Lagrangian spatial metric reduces to that of Euclidean 3-space intime-dependent curvilinear coordinates. A Lagrangian version of the Bernoulliequation for the evolution of the `velocity potential' is also given. At thepost-Newtonian (PN) level, an exact and general formula is derived forgravitational-wave emission from non-linear perturbations. It is shown that, inthe anisotropic collapse of homogeneous ellipsoids, the ratio of the PN tensormodes to the Newtonian metric tends to diverge like the mass density. It isfinally argued that a stochastic gravitational-wave background is produced,with present-day closure density $\Omega_{gw} \sim 10^{-5}$ -- $10^{-6}$ on $1$- $10$ Mpc scales.Comment: Revised version accepted for publication in MNRAS. 1 figure added. Latex file, 38 page