When Can Non-Gaussian Density Fields Produce a Gaussian Sachs-Wolfe Effect?

Abstract

The Sachs-Wolfe temperature fluctuations produced by primordial density perturbations are proportional to the potential field ϕ, which is a weighted integral over the density field δ. Because of the central limit theorem, ϕcan be approximately Gaussian even when δis non-Gaussian. Using the Wold representation for non-Gaussian density fields, δ(\rvec) = \int f(|\rvec - \rvec^\prime|) Δ(\rvec^\prime) d^3 \rvec^\prime, we find conditions on Δand f for which ϕmust have a Gaussian one-point distribution, while δcan be non-Gaussian. Sufficient (but not necessary) conditions are that the density field have a power spectrum (which determines f) of P(k) \propto k^n, with -2 < n \le +1, and that Δ(\rvec) be non-Gaussian with no long-range correlations. Thus, there is an infinite set of non-Gaussian density fields which produce a nearly Gaussian one-point distribution for the Sachs-Wolfe effect.