Relativistic conservation laws and integral constraints for large cosmological perturbations

For every mapping of a perturbed spacetime onto a background and with any vector field we construct a strongly, identically conserved covariant vector density I (), which is the divergence of a covariant antisymmetric tensor density, a ``superpotential. '' I () is linear in the energy-momentum tensor perturbations of matter, which may be large; I () does not contain the second order derivatives of the perturbed metric. The superpotential is identically zero when perturbations are absent. By integrating strongly conserved vectors over a part of a hypersurface S of the background, which spans a two-surface, we obtain integral relations between, on the one hand, initial data of the perturbed metric components and the energy-momentum perturbations on and, on the other, the boundary values on. We show that there are as many such integral relations as there are different mappings, 's, 's, and 's. For given boundary values on, the integral relations may be interpreted as integral constraints on local initial data including the energy-momentum perturbations. Strong conservation laws expressed in terms of Killing fields of the background become ``physical'' conservation laws. In cosmology, to each mapping of the time axis of a Robertson-Walker space on a de Sitter space with the same spatial topology there correspond ten conservation laws. The conformal mapping leads to a straightforward generalization of conservation laws in flat spacetimes. Other mappings are also considered. Traschen's ``integral constraints'' for linearized spatially localized perturbations of the energy-momentum tensor are examples of conservation laws with peculiar vectors whose equations are rederived here. In Robertson-Walker spacetimes, the ``integral constraint vectors'' are the Killing vectors of a de Sitter background for a special mapping.