  The Sachs-Wolfe temperature fluctuations produced by primordial density
perturbations are proportional to the potential field \phi, which is a weighted
integral over the density field \delta. Because of the central limit theorem,
\phi can be approximately Gaussian even when \delta is non-Gaussian. Using the
Wold representation for non-Gaussian density fields, \delta(\rvec) = \int
f(|\rvec - \rvec^\prime|) \Delta(\rvec^\prime) d^3 \rvec^\prime, we find
conditions on \Delta and f for which \phi must have a Gaussian one-point
distribution, while \delta can 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 \Delta(\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.
