Spatially covariant gravity with two degrees of freedom: A perturbative analysis up to cubic order

There has been considerable interest in constructing modified gravity theories that propagate only two degrees of freedom (DOFs), corresponding to the tensorial gravitational waves of general relativity. Within the framework of spatially covariant gravity (SCG), the conditions for obtaining 2-DOF theories can be derived from Hamiltonian constraint analysis, but it is generally difficult to translate those conditions into explicit SCG Lagrangians, especially when the Lagrangian depends nonlinearly on the extrinsic curvature. In this work, we adopt an alternative perturbative approach. We consider polynomial-type SCG Lagrangians up to $d=3$, where $d$ denotes the total number of derivatives in each monomial, and expand them around a cosmological background. By requiring the scalar mode to be eliminated up to cubic order in perturbations, we derive the corresponding conditions on the coefficient functions in the Lagrangian. We find five explicit Lagrangians that propagate only 2 DOFs up to cubic order in perturbations around a cosmological background. These theories therefore provide concrete candidate 2-DOF SCG models, at least at the perturbative level up to cubic order.