  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.
