Stable singularity-free cosmological solutions in nonprojectable Hořava-Lifshitz gravity

We find stable singularity-free cosmological solutions in nonflat Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime in the context of Hořava-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 M2, the effective friction coefficients in perturbation equations H and these magnitude relations |M2|/H2. 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.