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

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 $\mathrm{\square }$ and $l$ the other indices. We compute the bitensor ${\sum }_l{T}^{(kl)}{}_{ab}(x)T^{(kl)}{}_{a^{\mathrm{\prime }}{b}^{\mathrm{\prime }}}(x^{\mathrm{\prime }}{)}^{\ast }$ where $x,x^{\mathrm{\prime }}$ 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.