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

Abstract

The global dynamics of a cosmological model based on Hořava-Lifshitz gravity in the presence of curvature is described by using the qualitative theory of differential equations.

Global Dynamics of the Hořava–Lifshitz Cosmological Model in a Non-Flat Universe with Non-Zero Cosmological Constant

“…The origin e 4 = (0, 0) of system ( 15) is a equilibrium point with eigenvalues 2 and 0, but it is not semi-hyperbolic because it is not isolated in the set of all equilibrium points. It is noted that the axis V = 0 is full of equilibrium points of system (15). For the positive semi-axis of V near e 4 , dV/dt > 0 means that V increases monotonically, and on the negative semi-axis of V, dV/dt < 0 indicates that V decreases monotonically.…”

“…Moreover, dU/dt = √ 6sV(V 2 + 1) around the straight line U = 0, thus U increases monotonically when sV > 0, and U decreases monotonically when sV < 0. Therefore, the local phase portrait of the semi-hyperbolic equilibrium point e 4 in system (15) is illustrated in Figure 3a when s > 0. Similarly, the local phase portrait of e 4 is shown in Figure 3b for when s < 0.…”

“…This system is exactly the same as system (9) in [15], so the global phase portrait of system (38) is shown in Figure 28. The finite equilibrium points e 13 = (1, 0) and e 14 = (−1, 0) are hyperbolic unstable nodes.…”

“…The finite equilibrium points e 13 = (1, 0) and e 14 = (−1, 0) are hyperbolic unstable nodes. Besides the line x = 0 and the infinity of the local chart U 1 are filled with equilibrium points (see [15] for more details). 5.1.3.…”

“…For the case Λ = 0, the global dynamics of the Hořava-Lifshitz scalar field cosmological model under the background of FLRW were described in [14,15], and the case of Λ = 0 with zero curvature has also been addressed in [16]. In the present paper we discuss the global dynamics of a non-flat universe with Λ = 0.…”