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 {ξ} we construct a strongly, identically conserved covariant vector density I(ξ), which is the divergence of a covariant antisymmetric tensor density, a ``superpotential.'' I(ξ) is linear in the energy-momentum tensor perturbations of matter, which may be large; I(ξ) 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 {Σ} of a hypersurface S of the background, which spans a two-surface {∂}{Σ}, we obtain integral relations between, on the one hand, initial data of the perturbed metric components and the energy-momentum perturbations on {Σ} and, on the other, the boundary values on {∂}{Σ}. We show that there are as many such integral relations as there are different mappings, {ξ}'s, {Σ}'s, and {∂}{Σ}'s. For given boundary values on {∂}{Σ}, 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 ξ 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 {ξ} 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.