Inverting the Sachs-Wolfe Formula: an Inverse Problem Arising in Early-Universe Cosmology

The (ordinary) Sachs-Wolfe effect relates primordial matter perturbations to the temperature variations $\delta T/T$ in the cosmic microwave background radiation; $\delta T/T$ can be observed in all directions around us. A standard but idealised model of this effect leads to an infinite set of moment-like equations: the integral of $P(k) j_\ell^2(ky)$ with respect to k ($0<k<\infty$) is equal to a given constant, $C_\ell$, for $\ell=0,1,2,...$. Here, P is the power spectrum of the primordial density variations, $j_\ell$ is a spherical Bessel function and y is a positive constant. It is shown how to solve these equations exactly for ~$P(k)$. The same solution can be recovered, in principle, if the first ~m equations are discarded. Comparisons with classical moment problems (where $j_\ell^2(ky)$ is replaced by $k^\ell$) are made.