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 δT/T in the cosmic microwave background radiation; δT/T can be observed in all directions around us. A standard but idealized model of this effect leads to an infinite set of moment-like equations: the integral of P(k)jl2(ky) with respect to k (0 <k < ∞) is equal to a given constant, Cl, for l = 0, 1, 2,.... Here, P is the power spectrum of the primordial density variations, jl 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 jl2(ky) is replaced by kl) are made.