Bifurcations and Chaos in Hořava-Lifshitz Cosmology

Abstract

The nature of generic spacelike singularities in general relativity is connected with first principles, notably Lorentzian causal structure, scale invariance and general covariance. To bring a new perspective on how these principles affect generic spacelike singularities, we consider the initial singularity in spatially homogeneous Bianchi type VIII and IX vacuum models in Hořava-Lifshitz gravity, where relativistic first principles are replaced with anisotropic scalings of Lifshitz type. Within this class of models, General Relativity is shown to be a bifurcation where chaos becomes generic. To describe the chaotic features of generic singularities in Hořava-Lifshitz cosmology, we introduce symbolic dynamics within Cantor sets and iterated function systems.

Chaos in spatially homogeneous Hořava-Lifshitz subcritical cosmologies

“…Our goal is to prove that the Kasner map is chaotic for v ∈ (0, 1/2), which thereby completes the program in [10] and rigorously shows that GR is a critical case associated with a bifurcation where chaos becomes generic. Even though irregular and non-hyperbolic maps impose difficulties in proving that a given map is chaotic, see [17], it is the fact that the Kasner map is multivalued that complicates the analysis.…”

“…Similarly for B N 2 , B N 3 . For more details, see [10]. Thus, the type II solutions in B N 1 induce a map from the α-limit set to the ω-limit set of heteroclinics in…”

“…In this regard, vacuum spatially homogeneous cosmologies, the Bianchi models, are expected to play a key role in the dynamical asymptotic behaviour, see [3,4,9,21,22]. Similarly, the Bianchi models derived in [10] are expected to describe generic singularities in Hořava gravity. In particular, it is heuristically argued that there is an 'asymptotically dominant' curvature term in (1.2d) toward the initial singularity, yielding a respective dominant Bianchi model.…”

“…For the supercritical case, v ∈ (1/2, 1), the three arcs are pairwise disjoint and their union does not cover K ; its complement is a set consisting of stable fixed points of K and thus it is called the stable set S. The map K is well-defined and it was shown that the set of points that never reaches the set S is Cantor set of measure zero, in which the map K is chaotic. See [10]. For v ∈ (0, 1/2), the map K is multivalued for points in the overlap regions A α ∩ A β and uniquely determined for the remaining points.…”

“…Moreover, certain choices of the multivalued map is discontinuous at ∂(A α ∩ A β ), but even when lateral limits exist, the derivative is 1 which makes the map weakly hyperbolic. For this reason, little is known in this case and the map was conjectured to be chaotic in [10].…”