## 1. IntroductionThe Universe at short and moderate distance scales is inhomogeneous, being filled by numerous structures. In fact the clumpy structure of matter in the Universe extends for over 15 orders of magnitude in linear size, from stars to the largest clusters of galaxies, and for about 18 orders of magnitude in mass (Ostriker 1991). At scales from galaxies to clusters of galaxies and even superclusters, the Universe exhibits self-similar, fractal behavior (Mandelbrot 1983; Peebles 1993). At the largest scales, those probed by COBE observations (Smoot et al. 1991), for example, the Friedmann-Robertson-Walker (FRW) cosmology which is based on the hypotheses of homogeneity and isotropy, provides adequate description of many uncorrelated observations. Isotropy is broken by the primordial fluctuations in the matter density which subsequently become amplified by the gravitational instabilities. In this paper we explore the observed self-affinity and scaling in the density correlation function and tackle them with the renormalization group, a tool originating in quantum field theory (Gell-Mann & Low 1954), then extended to condensed matter physics (Wilson & Kogut 1974), later on extended to problems in hydrodynamics (Forster et al. 1977) and applied to surface growth (Kardar et al. 1986) and in the last few years applied to gravitation and cosmology (Pérez-Mercader et al. 1996, 1997). Our starting point will be the non-relativistic hydrodynamic equations governing the dynamical evolution of an ideal self-gravitating Newtonian fluid in a FRW background. Through a series of steps involving physically justified approximations, these equations reduce to a single evolution equation for the cosmic fluid's velocity potential. At this point one has a single deterministic hydrodynamic equation. To this we add a noise source which represents the influence of fluctuations and dissipative processes on the evolution of the fluid, arising from, but not limited to, viscosity, turbulence, explosions, late Universe phase transitions, gravitational waves, etc. The resulting equation is recognized as a cosmological variant of the Kardar-Parisi-Zhang (KPZ) equation (Kardar et al. 1986) which is the simplest archetype of nonlinear structure evolution and has been studied extensively in recent years in the context of surface growth phenomena (Barabàsi & Stanley 1995). While the essential steps involved in arriving at this KPZ equation are reviewed in Sect. 2, we refer the reader to the concise details of its complete cosmological-hydrodynamical derivation in Buchert et al. 1997. The relevance of stochastic fluctuations in structure formation in the Universe has been addressed in Berera & Fang 1994, where the rôle of the KPZ equation was emphasized. For FRW cosmologies with curved spatial sections, this equation contains a time-dependent mass-term. In Sect. 3 we discuss briefly the adiabatic approximation which allows one to treat this mass term as a constant during most of the expansion of the background cosmology. Having thus reduced the essential hydrodynamics to a single dynamical stochastic equation, we turn to the results of a dynamical renormalization group analysis to calculate the precise form of the density-density correlation function using the KPZ equation. To do so we proceed first by extracting the renormalization group equations governing the evolution of couplings appearing in the cosmological KPZ equation with respect to changes in scale and integrating out the short distance degrees of freedom. As we are interested in the behavior of the system at ever larger scales, we will run the renormalization group (RG) equations into the large scale (infrared) limit, a process known as coarse-graining. The fixed points of these RG equations determine the long-time, long-distance behavior of the system and their subsequent analysis yields both the power law form of the correlation functions as well as their explicit exponents. In particular, the value of the galaxy-galaxy correlation function exponent is calculated, and its value understood in terms of spatio-temporal correlated noise. A summary of our results and discussion are presented in the closing section. © European Southern Observatory (ESO) 1999 Online publication: March 10, 1999 |