Inverting the Sachs-Wolfe formula: an inverse problem arising in early-universe cosmology

The (ordinary) Sachs-WoIfe 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 idealized model of this effect leads to an infinite set of moment-like equations: the integral of P(k)j(e)(2)(ky) with respect to k(0 < k < infinity) is equal to a even constant, C-l, for l = 0,1,2,.... Here, P is the power spectrum of the primordial density variations, j(l) 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 ill equations are discarded. Comparisons with classical moment problems (where j(e)(2)(ky) is replaced by k(l)) are made.