Kurtosis, skewness, and non-Gaussian cosmological density perturbations

Cosmological topological defects as well as some nonstandard inflation models can give rise to non-Gaussian density perturbations. Skewness and kurtosis are the third and fourth moments that measure the deviation of a distribution from a Gaussian. Measurement of these moments for the cosmological density field and for the microwave background temperature anisotropy can provide a test of the Gaussian nature of the primordial fluctuation spectrum. In the case of the density field, the importance of measuring the kurtosis is stressed since it will be preserved through the weakly nonlinear gravitational evolution epoch. Current constraints on skewness and kurtosis of primeval perturbations are obtained from the observed density contrast on small scales and from recent COBE observations of temperature anisotropies on large scales. It is also shown how, in principle, future microwave anisotropy experiments might be able to reveal the initial skewness and kurtosis. It is shown that present data argue that if the initial spectrum is adiabatic, then it is probably Gaussian, but non-Gaussian isocurvature fluctuations are still allowed, and these are what topological defects provide.