Inflation in general covariant theory of gravity
  In this paper, we study inflation in the framework of the nonrelativistic
general covariant theory of the Ho\v{r}ava-Lifshitz gravity with the
projectability condition and an arbitrary coupling constant $\lambda$. We find
that the Friedmann-Robterson-Walker (FRW) universe is necessarily flat in such
a setup. We work out explicitly the linear perturbations of the flat FRW
universe without specifying to a particular gauge, and find that the
perturbations are different from those obtained in general relativity, because
of the presence of the high-order spatial derivative terms. Applied the general
formulas to a single scalar field, we show that in the sub-horizon regions, the
metric and scalar field are tightly coupled and have the same oscillating
frequencies. In the super-horizon regions, the perturbations become adiabatic,
and the comoving curvature perturbation is constant. We also calculate the
power spectra and indices of both the scalar and tensor perturbations, and
express them explicitly in terms of the slow roll parameters and the coupling
constants of the high-order spatial derivative terms. In particular, we find
that the perturbations, of both scalar and tensor, are almost scale-invariant,
and the spectrum indices are the same as those given in GR, but the ratio of
the scalar and tensor power spectra depends on the high-order spatial
derivative terms, and can be different from that of GR significantly.
