Statistical Cosmology with Quadratic Density Fields

Primordial fluctuations in the cosmic density are usually assumed to take the form of a Gaussian random field that evolves under the action of gravitational instability. In the early stages, while they have low amplitude, the fluctuations grow linearly. During this phase the Gaussian character of the fluctuations is preserved. Later on, when the fluctuations have amplitude of order the mean density or larger, non-linear effects cause departures from Gaussianity. In Fourier space, non-linearity is responsible for coupling Fourier modes and altering the initially random distribution of phases that characterizes Gaussian initial conditions. In this paper we investigate some of the effects of quadratic non-linearity on basic statistical properties of cosmological fluctuations. We demonstrate how this form of non-linearity can affect asymptotic properties of density fields such as homogeneity, ergodicity, and behaviour under smoothing. We also show how quadratic density fluctuations give rise to a particular relationship between the phases of different Fourier modes which, in turn, leads to the generation of a non-vanishing bispectrum. We thus elucidate the relationship between higher-order power spectra and phase distributions.