Spatially covariant gravity with 2 degrees of freedom: Perturbative analysis

We revisit the problem of building the Lagrangian of a large class of metric theories that respect spatial covariance, which propagate at most 2 degrees of freedom and in particular no scalar mode. The Lagrangians are polynomials built of the spatially covariant geometric quantities. By expanding the Lagrangian around a cosmological background and focusing on the scalar modes only, we find the conditions for the coefficients of the monomials in order to eliminate the scalar mode at the linear order in perturbations. We find the conditions up to d = 4, with d the total number of derivatives in the monomials, and determine the explicit Lagrangians for the cases of d = 2, d = 3 as well as the combination of d = 2 and d = 3. We also expand the Lagrangian of d = 2 to the cubic order in perturbations, and find additional conditions for the coefficients such that the scalar mode is eliminated up to the cubic order. This perturbative analysis can be performed order by order, and one expects to determine the final Lagrangian at some finite order such that the scalar mode is fully eliminated. Our analysis provides an alternative and complimentary approach to building spatially covariant gravity with only tensorial degrees of freedom. The resulting theories can be used as alternatives to the general relativity to describe the tensorial gravitational waves in a cosmological setting.