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

Abstract

ABSTRACT In the standard paradigm for cosmological structure formation, clustering developsfrom initially random-phase (Gaussian) density fluctuations in the early Universe bya process of gravitational instability. The later, non-linear stages of this process in-volve Fourier mode-mode interactions that result in a complex pattern of non-randomphases. We present a novel mapping technique that reveals mode coupling inducedby this form of nonlinear interaction and allows it to be quantified statistically. Thephase mapping technique circumvents the difficulty of the circular characteristic of φkand illustrates the statistical significance of phase difference at the same time. Thisgeneralized method on phases allows us to detect weak coupling of phases on any ∆kscales.Key words: methods: data analysis – techniques: image processing – large-scalestructure of the Universe 1 INTRODUCTIONThe morphology of the large-scale structure in the Universeis that of a complex hierarchy of nodes, filaments and sheetsinterlocking large voids. The Fourier-space description ofsuch a pattern is dominated by the properties of the phasesrather than the amplitudes of the Fourier modes (Chiang2001). According to the prevailing theoretical ideas this pat-tern developed by a process of gravitational instability froman amorphous pattern of density fluctuations characterizedby a Gaussian field with random phases. Since the non-random phases of the present structure have grown fromrandom-phase initial perturbations then there is strong mo-tivation for understanding how phase information developswithin this paradigm and to construct a statistical descrip-tion of galaxy clustering that could be used as a test of thebasic idea.Unfortunately, quantifying the properties of Fourierphases is difficult for a number of technical reasons, sotheir use in statistical studies has so far been limited. Ry-den & Gramann (1991), Soda & Suto (1992) and Jain B Chiang & Coles 2000) to transferthe phases of different Fourier modes on to a bounded squareupon which simple statistical tests can be applied. In thisway, we build upon the earlier studies (Chiang & Coles 2000;Coles & Chiang 2000) to construct a method that allows usto transform the phase information in a clustering patterninto a more useful form.2 PHASE COUPLING IN THE NONLINEARREGIMEThe mathematical description of an inhomogeneous Uni-verse revolves around the dimensionless density contrast,δ(x), which is obtained from the spatially-varying matterdensity ρ(x) via