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

Research output: Contribution to journalArticle

Original languageEnglish
Pages (from-to)1393-1404
JournalInverse problems
Issue number6
Publication statusPublished - 1 Dec 1999


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.

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