Lorentzian Vacuum Transitions in Ho\v{r}ava-Lifshitz Gravity

TL;DR

This paper investigates vacuum transition probabilities in Hořava-Lifshitz gravity for a universe with positive curvature, comparing results with general relativity and exploring implications for initial singularity resolution.

Contribution

It provides analytic expressions for vacuum transition probabilities in Hořava-Lifshitz gravity considering scalar fields with different dependencies, extending previous Hamiltonian approaches.

Findings

Infrared limit matches general relativity for scalar fields depending on all spacetime variables.

Ultraviolet limit shows opposite behavior to general relativity.

Scalar field dependence on time only yields similar behavior to general relativity in both limits.

Abstract

The vacuum transition probabilities for a Friedmann-Lema\^itre-Robertson-Walker universe with positive curvature in Ho\v{r}ava-Lifshitz gravity in the presence of a scalar field potential in the Wentzel-Kramers-Brillouin approximation are studied. We use a general procedure to compute such transition probabilities using a Hamiltonian approach to the Wheeler-DeWitt equation presented in a previous work. We consider two situations of scalar fields, one in which the scalar field depends on all the spacetime variables and other in which the scalar field depends only on the time variable. In both cases analytic expressions for the vacuum transition probabilities are obtained and the infrared and ultraviolet limits are discussed for comparison with the result obtained by using general relativity. For the case in which the scalar field depends on all spacetime variables we obtain that in the infrared limit it is possible to obtain a similar behavior as in general relativity, however in the ultraviolet limit the behavior found is completely opposite. Some few comments about possible phenomenological implications of our results are given. One of them is a plausible resolution of the initial singularity. On the other hand for the case in which the scalar field depends only on the time variable, the behavior coincides with that of general relativity in both limits, although in the intermediate region the probability is slightly altered.