This problem came up while stuyding stochastic backgrounds of gravitational waves in the early universe.
I would like to invert
$$ \Omega_{\mathrm{GW}}(k) = \int_0^{1/\sqrt{3}} \int_{1/\sqrt{3}}^\infty \mathrm{d}d \, \mathrm{d}s \; T(d,s) P_\zeta\left(k\frac{\sqrt{3}}{2}(s+d)\right) P_\zeta\left(k\frac{\sqrt{3}}{2}(s-d)\right) $$
In other words if I know $T(d,s)$ and $\Omega_\mathrm{GW}(k)$, is there a way to recover $P_\zeta(k)$? I've been reading up on deconvolution techniques and playing with changes of variables for $s$ and $d$, but the presence of the $T(d,s)$ invalidates most of the Fourier deconvolution methods I've found.
$T(d,s)$ is unfortunately not a function of the difference $s-d$ :
