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We consider an extended theory of Horava-Lifshitz gravity with the detailedbalance condition softly breaking, but without the projectability condition.With the former, the number of independent coupling constants is significantlyreduced. With the latter and by extending the original foliation-preservingdiffeomorphism symmetry $ {{Diff}}(M, {\cal{F}})$ to include a local U(1)symmetry, the spin-0 gravitons are eliminated. Thus, all the problems relatedto them disappear, including the instability, strong coupling, and differentspeeds in the gravitational sector. When the theory couples to a scalar field,we find that the scalar field is not only stable in both the ultraviolet (UV)and infrared (IR), but also free of the strong coupling problem, because of thepresence of high-order spatial derivative terms of the scalar field.Furthermore, applying the theory to cosmology, we find that due to theadditional U(1) symmetry, the Friedmann-Robertson-Walker (FRW) universe isnecessarily flat. We also investigate the scalar, vector, and tensorperturbations of the flat FRW universe, and derive the general linearized fieldequations for each kind of the perturbations.

General covariant Horava-Lifshitz gravity without projectability condition and its applications to cosmology