Maximally Symmetric Spin-Two Bitensors on $S^3$ and $H^3$

TL;DR

This paper computes maximally symmetric spin-two bitensors on $S^3$ and $H^3$, providing tools for analyzing gravitational wave correlations in cosmological models with spatial curvature.

Contribution

It derives explicit expressions for spin-two tensor bitensors on curved spaces, facilitating the study of gravitational wave correlations in open and closed cosmologies.

Findings

Explicit form of the bitensor sum over eigenmodes.

Application to gravitational wave correlation functions.

Framework for stochastic gravitational wave background analysis.

Abstract

The transverse traceless spin-two tensor harmonics on $S^3$ and $H^3$ may be denoted by $T^{(kl)}{}_{ab}$. The index $k$ labels the (degenerate) eigenvalues of the Laplacian $\square$ and $l$ the other indices. We compute the bitensor $\sum_l T^{(kl)}{}_{ab}(x) T^{(kl)}{}_{a'b'}(x')^*$ where $x,x'$ are distinct points on a sphere or hyperboloid of unit radius. These quantities may be used to find the correlation function of a stochastic background of gravitational waves in spatially open or closed Friedman-Robertson-Walker cosmologies.