Return mapping of phases and the analysis of the gravitational clustering hierarchy

In the standard paradigm for cosmological structure formation, clustering develops from initially random-phase (Gaussian) density fluctuations in the early Universe by a process of gravitational instability. The later, non-linear stages of this process involve Fourier modemode interactions that result in a complex pattern of non-random phases. We present a novel mapping technique that reveals mode coupling induced by this form of non-linear interaction and allows it to be quantified statistically. The phase mapping technique circumvents the difficulty of the circular characteristic of k and illustrates the statistical significance of phase difference at the same time. This generalized phase method allows us to detect weak coupling of phases on any k scales.