Cosmological dynamics based on Lovelock's gravity. Qualitative analysis

Abstract

We perform a complete qualitative analysis of the cosmological dynamics of Lovelock gravity in a spatially flat Friedmann-Robertson-Walker (FRW) universe filled with a perfect fluid obeying a barotropic equation of state $p=ωρ$. Starting from the generalized $N$-dimensional Friedmann equations written in terms of the independent components of the Riemann tensor, we reduce the dynamics to a single autonomous first-order equation for the Hubble parameter $H$, whose right-hand side is a ratio of polynomials in $H^2$ fixed by the order $n$ of the Lovelock polynomial, the number of dimensions $N$, and the coupling constants $α_i$. The real roots of these polynomials determine the fixed points and the singular points of the system, which govern its asymptotic behaviour. We give a complete classification of the possible evolution scenarios, identify the attractive and repulsive fixed points, locate the "phantom intervals" in which the effective matter density becomes negative, and show that fixed points are reached in infinite time whereas singular points are reached in finite time. For $N>4$ and suitable negative couplings a qualitatively new scenario arises, which we call the "Big Shock": the universe starts from a state with finite density and finite scale factor but with an infinite rate of change of the Hubble parameter, replacing the standard Big Bang. The general analysis is illustrated for $n=1$, $N=4$ (general relativity), $n=2$, $N=5$ (Einstein-Gauss-Bonnet gravity), and $n=3$, $N=7$ (cubic Lovelock gravity).

40 pages, 7 figures; revised version submitted to the European Physical Journal C