  The Fourier transform of cosmological density perturbations can be
represented in terms of amplitudes and phases for each Fourier mode. We
investigate the phase evolution of these modes using a mixture of analytical
and numerical techniques. Using a toy model of one-dimensional perturbations
evolving under the Zel'dovich approximation as an initial motivation, we
develop a statistic that quantifies the information content of the distribution
of phases. Using numerical simulations beginning with more realistic Gaussian
random-phase initial conditions, we show that the information content of the
phases grows from zero in the initial conditions, first slowly and then rapidly
when structures become non-linear. This growth of phase information can be
expressed in terms of an effective entropy: Gaussian initial conditions are a
maximum entropy realisation of the initial power spectrum, gravitational
evolution decreases the phase entropy. We show that our definition of phase
entropy results in a statistic that explicitly quantifies the information
stored in the phases of density perturbations (rather than their amplitudes)
and that this statistic displays interesting scaling behaviour for self-similar
initial conditions.
