Cosmological perturbations from localized sources: a gauge-invariant multipolar theory

Abstract

We develop a first-principles description of the linearized gravitational field of localized sources in spatially flat Friedmann-Lema^ıtre-Robertson-Walker universes with arbitrary smooth expansion histories. The construction determines the complete retarded metric and the induced perturbation of the surrounding cosmic fluid from prescribed source stresses and a cosmologically stationary past field. It includes all multipole orders and accounts for curvature tails, through which the field depends on the source's past history inside the light cone. Generalizing the Blanchet-Damour multipolar formalism, we identify gauge-invariant mass and current moments together with a canonical cosmic fluid moment that determines an independent scalar response. This response vanishes in Minkowski and pure de Sitter spacetime, although curvature tails persist in de Sitter. The gauge-invariant multipoles are obtained directly in position space. This gives an exterior description without the nonlocal spatial projection through the source interior required by the usual scalar-vector-tensor decomposition. We also relate the canonical multipole metric to the standard variables of cosmological perturbation theory. The retarded construction requires neither a slow-motion nor a geometric-optics approximation. We define effective multipolar sources by truncating the moments of slowly varying sub-Hubble extended sources at a prescribed Cartesian multipole order, recovering the earlier quadrupolar and octupolar models in de Sitter spacetime. We recover the known Minkowski and de Sitter multipolar fields and obtain an analytical representation for gravitational propagation in the $Λ$CDM model including radiation.