Extended phase-space analysis of the Hořava–Lifshitz cosmology

Abstract

We examine the phase space of Hořava-Lifshitz cosmology for a wide range of self-interacting potentials for the scalar field under the detailed-balance condition and without imposing it, by means of the powerful method of fdevisers. A compactification approach is performed for the exponential potential and for potentials beyond the exponential one, extending the previous findings in the literature. By using this approach it is possible to describe the finite region of the phase space and the region where the phase-space variables becomes infinity. Furthermore, we present several results concerning the stability of the de Sitter solution in Hořava-Lifshitz cosmology using Center Manifold theory. The advantages of this procedure are unveiled immediately when it is compared with the Normal Forms Calculations presented before in the literature.

Global dynamics of Hořava–Lifshitz cosmology with non-zero curvature and a wide range of potentials

“…Since the usual scalar field potential V (φ) admits multiple representations, the potential function f (s) also has various mathematical expressions (see the papers [4,17,22,23] and the references therein). In this paper we will investigate the monomial potential V (φ) = (μφ) 2n /2n with a positive constant μ and a natural number n. Thus the power-law potential is f (s) = −s 2 /(2n), and we obtain ds/dt = √ 6xs 2 /n.…”

“…In order to study the local phase portraits of the finite and infinite equilibrium points, and the global phase portraits of system (7) in the region G, which is the meaningful region for cosmology, see again [4] or [14]. We start discussing the phase portraits on its invariant planes and surface…”

“…In recent years Hořava [1] proposed a spacetime asymmetric gravitational theory similar to Lifshitz's scalar field theory, also known as Hořava-Lifshitz gravity. This theory has inspired a great deal of research for its applications in cosmology and black hole physics (see [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] or the review articles [19,20] and the references therein).…”

“…With or without detailed-balance condition, few researchers have published several papers on Hořava-Lifshitz gravity through phase space analysis, see for instance [3][4][5][6][7]17]. Among these papers there are also some of them about the case of non-zero space curvature k. When the terms related to k are dominant, Friedmann solution will be generated.…”

“…On the contrary, when the effective Hořava radiation dominates, the corresponding cosmology gives a solution of "radiation-dominated like" [6]. According to whether the cosmological constant is zero in flat (k = 0) and non-flat (k = 0) universe, Leon et al [3][4][5]14] divided the Hořava-Lifshitz gravity into four cases: (1) k = 0, = 0; (2) k = 0, = 0; (3) k = 0, = 0; (4) k = 0, = 0, and analyzed these phase spaces. More precisely, they either studied the global dynamics of the planar case of Hořava-Lifshitz gravity with exponential potential by using the twodimensional Poincaré compactification, or discussed only local dynamics of this cosmological model with powerlaw potential but without investigate the dynamics close to infinity.…”