Consistent Hořava gravity without extra modes and equivalent to general relativity at the linearized level

We consider a Hořava theory that has a consistent structure of constraints and propagates two physical degrees of freedom. The Lagrangian includes the terms of Blas, Pujolàs, and Sibiryakov. The theory can be obtained from the general Horava’s formulation by setting $\lambda =1/3$. This value of $\lambda$ is protected in the quantum formulation of the theory by the presence of a constraint. The theory has two second-class constraints that are absent for other values of $\lambda$. They remove the extra scalar mode. There is no strong-coupling problem in this theory since there is no extra mode. We perform explicit computations on a model that put together a $z=1$ term and the IR effective action. We also show that the lowest-order perturbative version of the IR effective theory has dynamics identical to the one of linearized general relativity. Therefore, this theory is smoothly recovered at the deepest IR without discontinuities in the physical degrees of freedom.