Hořava–Lifshitz gravity inspired Bianchi-II cosmology and the mixmaster universe

Abstract

Abstract. We study different aspects of the Hořava-Lifshitz inspired Bianchi-II cosmology and its relations with the mixmaster universe model. First, we present exact solutions for a toy model, where only the cubic in spatial curvature terms are present in the action; then we briefly discuss some exotic singularities, which can appear in this toy model. We study also the toy model where only the quadratic in spatial curvature terms are present in the action. We establish relations between our results and those obtained by using the Hamiltonian formalism. Finally, we apply the results obtained by studying Bianchi-II cosmology to describe the evolution of the mixmaster universe in terms of the Belinsky-Khalatnikov-Lifshitz formalism. Generally, our analysis gives some arguments in favour of the existence of the oscillatory approach to the singularity in a universe governed by the Hořava-Lifshitz type gravity.

Spatially homogeneous black hole solutions in $$z=4$$ Hořava–Lifshitz gravity in $$(4+1)$$ dimensions with Nil geometry and $$H^2\times R$$ horizons

“…Hořava-Lifshitz gravity has received growing interest and a large number of studies have explored the implications of this proposal in detail. For instance, the cosmological solutions of Hořava-Lifshitz gravity have been studied in [6][7][8][9][10], cosmological perturbation and the related properties have been discussed in [11][12][13][14][15][16][17][18], and some other properties of Hořava-Lifshitz gravity have been investigated in [19][20][21][22]. Particularly, much attention has been paid to black hole solutions and their thermodynamics behavior in the framework of Hořava-Lifshitz gravity [23][24][25][26][27][28][29][30][31][32].…”

“…All of these model geometries admit homogeneous metrics and show a close correspondence with the Bianchi types and Kantowski-Sachs homogeneous models [48,49]. The homogeneous spacetimes, possessing a symmetry called the spatial homogeneity [50], have been widely used in finding cosmological solution in the context of Einstein gravity [51], string theory [52][53][54][55][56], and Hořava-Lifshitz gravity theory [9,10].…”

“…9 (ii) If a set of parameters is chosen that holds 12ΛMk 2 w + 9M 2 > 0, giving r 2 < r 1 < r 4 < r 3 , the solutions are unstable at r 1 < r + < r 4 region, and then, after the divergent point r 4 , showing stable behavior at r 4 < r + < r 3 range, becomes unstable at r + > r 3 region. 10 (iii) If a set of parameters is chosen that yields 12ΛMk 2 w +9M 2 > 0, giving the real radii in the order of r 2 < r 1 < r 3 < r 4 , the solutions with T > 0 are unstable when r 1 < r + < r 3 , stable when r 3 < r + < r 4 and after divergent point r 4 become unstable. 11…”

“…First, we consider the case that n and λ are essentially arbitrary and independent, where concerning the asymptotic behavior for C 1 and C 2 dependent terms in f (r ) function (49), similar to the solutions of [29], we keep only the C 1dependent terms which has negative power of r for all values of n and λ. Using the Euclidean action (117), written in terms of the F(r ) function defined by (116), where the variation of boundary term is given by (118), the solution for the equation 9 For example with {Λ = 1 40 , M = 1 3 , k w = 4, α = 1} 10 For example with {Λ = −0.05, M = 0.5, k w = 0.6, α = 1}. 11 For example with…”

Spatially homogeneous black hole solutions in $z=4$ Hořava-Lifshitz gravity in $(4+1)$ dimensions with Nil geometry and $H^2\times R$ horizons

“…Hořava-Lifshitz gravity has received growing interest and a large number of studies have explored the implications of this proposal in detail. For instance, the cosmological solutions of Hořava-Lifshitz gravity have been studied in [6,7,8,9,10], cosmological perturbation and the related properties have been discussed in [11,12,13,14,15,16,17], and some other properties of Hořava-Lifshitz gravity have been investigated in [18,19,20,21]. Particularly, much attention has been paid to black hole solutions and their thermodynamics behavior in the framework of Hořava-Lifshitz gravity [22,23,24,25,26,27,28,29,30,31].…”