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

We propose the most general modified first-order Hořava-Lifshitz 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 Friedmann-Robertson-Walker (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 Hořava-Lifshitz $F(R)$ 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 Hořava-Lifshitz proposal. The Hamiltonian analysis of the modified Hořava-Lifshitz $F(R)$ 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 Hořava-Lifshitz $F(R)$ gravity turns out to be much richer than for its traditional counterpart. The emergence of multiple de Sitter solutions indicates the possibility of unification of early-time inflation with late-time acceleration within the same model. Power-law $F(R)$ theories are also 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 Hořava-Lifshitz $F(R)$ gravity. Finally, a covariant proposal for (renormalizable) $F(R)$ gravity within the Hořava-Lifshitz spirit is presented.

Figure 1

Plot of the curves representing the values of the exponents of the reconstructed $F(R)$ theory corresponding to a background $H=\gamma /t$, for $w=0$ and some specific values of the parameters $\lambda$ and $\mu$. From these plots one can infer, for example, that some backgrounds of this type can only be realized in $F(R)$ theories with poles. (a) The exponents of Eq. (158) for $w=0$, $\mu =-3$, and $\lambda =-1/2$. (b) The exponents of Eq. (158) for $w=0$, $\mu =-3$, and $\lambda =1$. (c) The exponents of Eq. (158) for $w=0$, $\mu =-2/5$ and $\lambda =1/2$.