Modified first-order Horava-Lifshitz gravity: Hamiltonian analysis of the general theory and accelerating FRW cosmology in power-law F(R) model

We propose the most general modified first-order Ho\v{r}ava-Lifshitz (HL) gravity, whose action does not contain time derivatives higher than the second order. The Hamiltonian structure of this theory is studied in all the details in the case of the spatially-flat FRW space-time, demonstrating many of the features of the general theory. It is shown that, with some plausible assumptions, including the projectability of the lapse function, this model is consistent. As a large class of such theories, the modified HL $F(R)$ gravity is introduced. The study of its ultraviolet properties shows that its $z=3$ version seems to be renormalizable in the same way as the original HL proposal. The Hamiltonian analysis of the modified HL $F(R)$ gravity shows that it is in general a consistent theory. The $F(R)$ gravity action is also studied in the fixed-gauge form, where the appearance of a scalar field is particularly illustrative. Then the spatially-flat FRW cosmology for this $F(R)$ gravity is investigated. It is shown that a special choice of parameters for this theory leads to the same equations of motion as in the case of traditional $F(R)$ gravity. Nevertheless, the cosmological structure of the modified HL $F(R)$ gravity turns out to be much richer than for its traditional counterpart. The emergence of multiple de Sitter solutions indicates to the possibility of unification of early-time inflation with late-time acceleration within the same model. Power-law $F(R)$ theories are investigated in detail. It is analytically shown that they have a quite rich cosmological structure: early/late-time cosmic acceleration of quintessence, as well as of phantom types. Also it is demonstrated that all the four known types of finite-time future singularities may occur in the power-law HL $F(R)$ gravity. Finally, a covariant proposal for (renormalizable) $F(R)$ gravity within the HL spirit is presented.