Hořava-Lifshitz theory of gravity with detailed balance is plagued by the presence of a negative bare (or geometrical) cosmological constant which makes its cosmology clash with observations. We argue that adding the effects of the large vacuum energy of quantum matter fields, this bare cosmological constant can be approximately compensated to account for the small observed (total) cosmological constant ΛOBS, thus resulting in a self-contained model of gravity and particle physics. Even though we cannot address the fine-tuning problem in this way, we are able to establish a relation between the smallness of ΛOBS and the scale ℓUV at which dimension 4 corrections to the Einstein gravity become significant for cosmology. This scale turns out to be ℓUV ≃ 5 ℓP for ΛOBS ≃ 0 and we therefore argue that the smallness of ΛOBS guarantees that Lorentz invariance is broken only at very small scales. We are also able to provide a first rough estimation for the values of the parameters of the theory μ and ΛW.

We employ the semiclassical approximation to the Wheeler-DeWitt equation in the spatially flat de Sitter Universe to investigate the dynamics of a minimally coupled scalar field near the Planck scale. We find that, contrary to naïve intuition, the effects of quantum gravitational fluctuations become negligible and the scalar field states asymptotically approach plane-waves at very early times. These states can then be used as initial conditions for the quantum states of matter to show that each mode essentially originated in the minimum energy vacuum. Although the full quantum dynamics cannot be solved exactly for the case at hand, our results can be considered as supporting the general idea of asymptotic safety in quantum gravity.

We study second-order perturbations for a general non-canonical scalar field, minimally coupled to gravity, on the unperturbed FRW background, where metric fluctuations are neglected a priori . By employing different approaches to cosmological perturbation theory, we show that, even in this simplified set-up, the second-order perturbations to the the stress tensor, the energy density and the pressure display potential instabilities, which are not present at linear order. The conditions on the Lagrangian under which these instabilities take place are provided. We also discuss briefly the significance of our analysis in light of the possible linearization instability of these fields about the FRW background.

We show that the semiclassical approximation to the Wheeler–DeWitt equation for the minisuperspace of a minimally coupled scalar field in the spatially flat de Sitter universe prompts the existence of an initial power-law evolution driven by non-adiabatic terms from the gravitational wavefunction which act like radiation. This simple model hence describes the onset of inflation from a previous radiation-like expansion during which the cosmological constant is already present but subleading.