Post-Newtonian Cosmological Dynamics in Lagrangian coordinates

Abstract

The non-linear dynamics of irrotational dust in General Relativity is studied in synchronous and comoving coordinates. All the equations are written in terms of the metric tensor of spatial sections orthogonal to the flow, which allows an unambiguous expansion in powers of $1/c^2$. To lowest order, the Newtonian approximation in Lagrangian form is derived. At this level the evolution is governed by the Raychaudhuri equation for the Lagrangian-to-Eulerian Jacobian matrix. The Lagrangian spatial metric reduces to that of Euclidean 3-space in time-dependent curvilinear coordinates. A Lagrangian version of the Bernoulli equation for the evolution of the `velocity potential' is also given. At the post-Newtonian (PN) level, an exact and general formula is derived for gravitational-wave emission from non-linear perturbations. It is shown that, in the anisotropic collapse of homogeneous ellipsoids, the ratio of the PN tensor modes to the Newtonian metric tends to diverge like the mass density. It is finally argued that a stochastic gravitational-wave background is produced, with present-day closure density $Ω_{gw} \sim 10^{-5}$ -- $10^{-6}$ on $1$ - $10$ Mpc scales.