Maximally symmetric spin-two bitensors on ${\mathrm{S}}^3$ and ${\mathrm{H}}^3$

The transverse traceless spin-two tensor harmonics on ${\mathrm{S}}^3$ and ${\mathrm{H}}^3$ may be denoted by ${\mathrm{T}}_{\mathrm{ab}}^{\mathrm{}\left(\mathrm{kl}\right)\mathrm{}}$. The index k labels the (degenerate) eigenvalues of the Laplacian √ and l the other indices. We compute the bitensor ${\sum }_{\mathrm{t}}$${\mathrm{T}}_{\mathrm{ab}}^{\mathrm{}\left(\mathrm{kl}\right)\mathrm{}}$(x)${\mathrm{T}}_{\mathrm{a}\prime \mathrm{b}\prime }^{\mathrm{}\left(\mathrm{kl}\right)\mathrm{}}$(x′${)}^{*}$ where x,x′ are distinct points on a three-dimensional 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 close Friedmann-Robertson-Walker cosmologies.