General relativity limit of Horava-Lifshitz gravity with a scalar field in gradient expansion

We present a fully nonlinear study of long wavelength cosmological perturbations within the framework of the projectable Horava-Lifshitz gravity, coupled to a single scalar field. Adopting the gradient expansion technique, we explicitly integrate the dynamical equations up to any order of the expansion, then restrict the integration constants by imposing the momentum constraint. While the gradient expansion relies on the long wavelength approximation, amplitudes of perturbations do not have to be small. When the $\lambda\to 1$ limit is taken, the obtained nonlinear solutions exhibit a continuous behavior at any order of the gradient expansion, recovering general relativity in the presence of a scalar field and the "dark matter as an integration constant". This is in sharp contrast to the results in the literature based on the "standard" (and naive) perturbative approach where in the same limit, the perturbative expansion of the action breaks down and the scalar graviton mode appears to be strongly coupled. We carry out a detailed analysis on the source of these apparent pathologies and determine that they originate from an improper application of the perturbative approximation in the momentum constraint. We also show that there is a new branch of solutions, valid in the regime where $|\lambda-1|$ is smaller than the order of perturbations. In the limit $\lambda\to1$, this new branch allows the theory to be continuously connected to general relativity (plus "dark matter").