Cosmological perturbation theory of primordial compact sources

TL;DR

The paper develops a position-space cosmological perturbation theory around spatially flat FRLW spacetimes to model localized primordial gravitational-wave sources without relying on the SVT decomposition. By employing a generalized harmonic gauge and STF multipole decomposition, it derives fully decoupled linearized equations, constructs a Hadamard-based retarded Green's function for power-law cosmologies (matching Chu) and provides a complete quadrupolar reconstruction of the metric perturbations, including light-cone and tail contributions. A key result is the identification of a non-conservation term in the stress-energy perturbations due to evolving background fluid, which precludes strictly compact sources and motivates a nearly localized source framework. These elements yield closed-form quadrupolar metric perturbations valid beyond the geometric optics approximation, enabling robust modeling of primordial GW perturbations across different cosmological epochs (e.g., matter- and dark-energy-dominated eras). The framework connects tensor, vector, and scalar sectors through explicit Green's-function expressions and consistency checks, offering a versatile tool for early-universe GW phenomenology and related CMB signatures.

Abstract

We construct a position-space cosmological perturbation theory around spatially flat Friedmann-Lemaître-Robertson-Walker geometries that allows to model localized primordial sources of gravitational waves. The equations of motion are decoupled using a generalized harmonic gauge, which avoids the use of a scalar-vector-tensor decomposition. We point out that sources cannot generically be defined in a compact domain due to fluctuations of the cosmic perfect fluid. For power law cosmologies, we obtain the exact Green's function necessary to solve for all metric perturbations in terms of a hypergeometric function, which matches with a Green's function derived earlier by Chu. This allows us to derive the closed form expression of the linearized metric perturbation generated by sources up to quadrupolar order in the multipolar expansion.