# Stable Singularity-free Cosmological Solutions in non-projectable Horava-Lifshitz Gravity

## TL;DR

This paper discovers stable, singularity-free cosmological solutions within non-projectable Hořava-Lifshitz gravity, addressing scalar perturbation instabilities by relaxing projectability and analyzing background and perturbation dynamics.

## Contribution

It introduces a classification of gradient stability types and highlights the role of higher order spatial curvature terms in stabilizing solutions in HL gravity.

## Key findings

- Stable singularity-free solutions in non-projectable HL gravity.
- Classification of gradient stability into five types based on perturbation parameters.
- Higher order spatial curvature terms suppress instabilities.

## Abstract

We find stable singularity-free cosmological solutions in non-flat Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW) spacetime in the context of Ho\v{r}ava-Lifshitz (HL) theory. Although we encounter the negative squared effective masses of the scalar perturbations in the original HL theory, the behaviors can be remedied by relaxing the projectability condition. In our analysis, the effects from the background dynamics are taken into account as well as the sign of the coefficients in the quadratic action for perturbations. More specifically, we give further classification of the gradient stability/instability into five types. These types are defined in terms of the effective squared masses of perturbations $\mathcal{M}^2$, the effective friction coefficients in perturbation equations $\mathcal{H}$ and these magnitude relations $|\mathcal{M}^2|/\mathcal{H}^2$. Furthermore, we indicate that oscillating solutions possibly show a kind of resonance especially in open FLRW spacetime. We find that the higher order spatial curvature terms with Lifshitz scaling $z=3$ are significant to suppress the instabilities due to the background dynamics.