Phenomenologically viable gravitational theory based on a preferred foliation without extra modes

We present a gravitational field theory that implements Horava's proposal of foliation-preserving-diffeomorphisms symmetry and higher spatial curvature directly in the canonical formalism. Due to the higher spatial derivative the theory is potentially renormalizable. Since this gauge symmetry is natural in the canonical formalism, we do not require a Lagrangian of second-order in time derivatives to begin with. We define the nonzero part of the Hamiltonian and the constraints motivated by the kinetic-conformal version of the nonprojectable Horava theory. The resulting theory is an extension of the latter, in the sense that it admits more solutions. Among the additional solutions there are homogeneous and isotropic configurations governed by the Friedmann equations. The theory has the same number of propagating degrees of freedom of general relativity. At the linearized level it reproduces the tensorial gravitational waves of general relativity. We discuss how observational bounds can be satisfied.