Post–Newtonian Cosmological Dynamics in Lagrangian coordinates

We study the non–linear dynamics of self–gravitating irrotational dust in a general relativistic framework, using synchronous and comoving (i.e. Lagrangian) coordinates. All the equations are written in terms of a single tensor variable, the metric tensor of the spatial sections orthogonal to the fluid flow. This treatment allows an unambiguous expansion in inverse (even) powers of the speed of light. To lowest order, the Newtonian approximation – in Lagrangian form – is derived and written in a transparent way; the corresponding Lagrangian Newtonian metric is obtained. Post–Newtonian corrections are then derived and their physical meaning clarified. A number of results are obtained: i) the master equation of Lagrangian Newtonian dynamics, the Raychaudhuri equation, can be interpreted as an equation for the evolution of the Lagrangian–to–Eulerian Jacobian matrix, complemented by the irrotationality constraint; ii) the Lagrangian spatial metric reduces, in the Newtonian limit, to that of Euclidean 3–space written in time– dependent curvilinear coordinates, with non–vanishing Christoffel symbols, but vanishing spatial curvature (a particular example of it is given within the Zel’dovich approximation); iii) a Lagrangian version of the Bernoulli equation for the evolution of the “velocity potential” is obtained. iv) The Newtonian and post–Newtonian content of the electric and magnetic parts of the Weyl tensor is clarified. v) At the Post–Newtonian level, an exact and general formula is derived for gravitational–wave emission from non–linear cosmological perturbations; vi) a straightforward application to the anisotropic collapse of homogeneous ellipsoids shows that the ratio of these post–Newtonian terms to the Newtonian ones tends to diverge at least like the mass density. vii) It is argued that a stochastic gravitational–wave background is produced by non–linear cosmic structures, with present–day closure density Ωgw ∼ 10 −5 – 10−6 on 1 – 10 Mpc scales.