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 idealised model of this effect leads to an infinite set of moment-like equations: the integral of P (k) j2 ` (ky) with respect to k (0 < k < ∞) is equal to a given constant, C`, for ` = 0, 1, 2, . . .. Here, P is the power spectrum of the primordial density variations, j` 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 j2 ` (ky) is replaced by k`) are made. In Press Inverse Problems 1999 ∗E-mail: [email protected], [email protected]; address after October 1, 1999, Department of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, Scotland †E-mail:[email protected]; address after September 1, 1999, Mathematics Department, Colorado School of Mines, Golden CO 80401, USA