Hamiltonian analysis of non-projectable modified F(R) Ho\v{r}ava-Lifshitz gravity

We study a version of the recently proposed modified $F(R)$ Ho\v{r}ava-Lifshitz gravity that abandons the projectability condition of the lapse variable. We discovered that the projectable version of this theory has a consistent Hamiltonian structure, and that the theory has interesting cosmological solutions which can describe the eras of accelerated expansion of the universe in a unified manner. The usual Ho\v{r}ava-Lifshitz gravity is a special case of our theory. Hamiltonian analysis of the non-projectable theory, however, shows that this theory has serious problems. These problems are compared with those found in the original Ho\v{r}ava-Lifshitz gravity. A general observation on the structure of the Poisson bracket of Hamiltonian constraints in all theories of the Ho\v{r}ava-Lifshitz type is made: in the resulting tertiary constraint the highest order spatial derivative of the lapse $N$ is always of uneven order. Since the vanishing of the lapse (N=0) is required by the preservation of the Hamiltonian constraints under time evolution, we conclude that the non-projectable version of the theory is physically inconsistent.