Quantum cosmology of a Ho\v{r}ava-Lifshitz model coupled to radiation

Abstract

In the present paper, we canonically quantize an homogeneous and isotropic Ho\v{r}ava-Lifshitz cosmological model, with constant positive spatial sections and coupled to radiation. We consider the projectable version of that gravitational theory without the detailed balance condition. We use the ADM formalism to write the gravitational Hamiltonian of the model and the Schutz variational formalism to write the perfect fluid Hamiltonian. We find the Wheeler-DeWitt equation for the model, which depends on several parameters. We study the case in which parameter values are chosen so that the solutions to the Wheeler-DeWitt equation are bounded. Initially, we solve it using the {\it Many Worlds} interpretation. Using wavepackets computed with the solutions to the Wheeler-DeWitt equation, we obtain the scalar factor expected value $\left<a\right>$. We show that this quantity oscillates between finite maximum and minimum values and never vanishes. Such result indicates that the model is free from singularities, at the quantum level. We reinforce this indication by showing that by subtracting one standard deviation unit from the expected value $\left<a\right>$, the latter remains positive. Then, we use the {\it DeBroglie-Bohm} interpretation. Initially, we compute the Bohm's trajectories for the scale factor and show that they never vanish. Then, we show that each trajectory agrees with the corresponding $\left<a\right>$. Finally, we compute the quantum potential, which helps understanding why the scale factor never vanishes.