Renormalization Group Approach to Generalized Cosmological models

It is widely accepted that Einstein’s general theory of relativity is an effective four-dimensional theory at lower energies and that it needs modifications at high energies. In the cosmological context, this break down of the theory results in a modification of the Friedmann equation with important consequences in the dynamical evolution at early times (see for example, [1, 2]). There has been several proposals for cosmological scenarios mostly inspired by brane-world models or by the presence of dark energy. All of them lead to modifications to the total energy density dependence of the Friedmann equation. Also, there have been studies using dynamical system approach, bringing together several of such corrections to the Friedmann equation, in order to study the conditions which lead to attractor scaling solutions [3]. In the dark energy scenarios, the energy density of the scalar field, responsible for the dark energy, is important only at late times and can account for the acceleration of the universe [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. Therefore, scaling solutions (i.e. solutions for which the energy density of the scalar field and the perfect fluid scale in the same way), which appear as late time attractors of the system of the evolution equations can play an important role in modeling dark energy scenarios [24]. The existence and behavior of these models in the scaling regime depends on the proposed modifications to the Friedmann equation, as well as on the form of the scalar field potential. In this paper we analyze asymptotic evolution of modified cosmologies with a scalar field and a barotropic perfect fluid. The modification of the standard cosmology is parameterized by a function of the total energy density, in the line of Copeland et al. [3]. The analysis of the asymptotic behavior of the equations is performed using the techniques of the renormalization group (RG), which has emerged as a powerful method for doing global and asymptotic analysis of ordinary and partial differential equations (Illinois group [25], Bricmont et al [26], Caginalp [27], Moise and Ziane [28], for a reformulation of the RG approach in terms of the classical theory of envelops see Kunihiro, [29], the theory of perturbations of an isotropic universe with dynamically evolving Newton constant and Cosmologial constant [30, 31], and [32, 33] for application of RG techniques to General Relativity). We shall show in this paper that the RG technique provides a very effective method for obtaining and analyzing such solutions. The plan of the paper is the following: In section II we describe the basic equations governing generalized cosmology. This is followed by a section where we give a very brief introduction to the RG method and apply this technique to the generalized cosmological models. Next section deals with a study of the exact scaling solutions, which are the fixed-points of the RG transformations. The stability of these exact solutions under linearized perturbations is analyzed in section V. We conclude with a brief summary.