Phase information and the evolution of cosmological density perturbations

The Fourier transform of cosmological density perturbations can be represen ted in terms of amplitudes and phases for each Fourier mode. We investigate the phase evolution of these modes using a mixture of analytical and numer ical techniques. Using a toy model of one-dimensional perturbations evolvin g under the Zel'dovich approximation as an initial motivation, we develop a statistic that quantifies the information content of the distribution of p hases. Using numerical simulations beginning with more realistic Gaussian r andom-phase initial conditions, we show that the information content of the phases grows from zero in the initial conditions, first slowly and then ra pidly when structures become non-linear. This growth of phase information c an be expressed in terms of an effective entropy. Gaussian initial conditio ns are a maximum entropy realization of the initial power spectrum; gravita tional evolution decreases the phase entropy. We show that our definition o f phase entropy results in a statistic that explicitly quantifies the infor mation stored in the phases of density perturbations (rather than their amp litudes), and that this statistic displays interesting scaling behaviour fo r self-similar initial conditions.