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 mode-mode 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 nonlinear interaction and allows it to be quantified statistically. The phase mapping technique circumvents the difficulty of the circular characteristic of φk and illustrates the statistical significance of phase difference at the same time. This generalized method on phases allows us to detect weak coupling of phases on any ∆k scales.