Green's functions for gravitational waves in FRW spacetimes.

A method for calculating the retarded Green's function for the gravitational wave equation in Friedmann-Roberson-Walker spacetimes, within the formalism of linearized Einstein gravity is developed. Hadamard's general solution to Cauchy's problem for second-order, linear partial differential equations is applied to the FRW gravitational wave equation. The retarded Green's function may be calculated for any FRW spacetime, with curved or flat spatial sections, for which the functional form of the Ricci scalar curvature R is known. The retarded Green's function for gravitational waves propagating through a cosmological fluid composed of both radiation and dust is calculated analytically for the first time. It is also shown that for all FRW spacetimes in which the Ricci scalar curvatures does not vanish, R = 0, the Green's function violates Huygens' principle; the Green's function has support inside the light-cone due to the scatter of gravitational waves off the background curvature. 0 I. Introduction The Green's function is an important tool for evaluating the solution to a partial differential equation for a given set of boundary conditions. The Green's function represents the solution to Cauchy's problem: given a set of properly formulated initial data, the Green's function may be used to uniquely determine the final, evolved data. In physics, one may determine the evolution of a field according to its equations of motion, given some initial data. The techniques used in this paper have been widely applied to the calculation of Green's functions, or propagators, for quantum fields in curved spacetime. Here, those techniques will be applied to classical fields in cosmology. In this paper I will determine the evolution of a tensor field, representing gravitational waves, given a source stress-energy tensor. The goal of this paper, then, is to calculate the Green's function for gravitational waves evolving in an arbitrary FRW spacetime. The Green's functions may be useful for the study of gravitational radiation in FRW spacetimes, especially as it relates to observational cosmology (for an example, see [1]). For situations in which the stress-energy tensor of a source of gravitational waves is known, the Green's function may be used to calculate the properties of the tensor metric perturbations, and the spectrum of gravitational radiation. Examples of such sources are cosmic strings or scalar fields present in the early universe. The tensor perturbations generated by such sources may be manifest as anisotropies in the cosmic microwave background, or as a spectrum of stochastic gravitational …