Spatial covariant gravity with two degrees of freedom in the presence of an auxiliary scalar field: Perturbation analysis* * Supported by the Natural Science Foundation of China (11975020)

We investigate a class of gravity theories respecting only spatial covariance, termed spatially covariant gravity, in the presence of an auxiliary scalar field. We examine the conditions on the Lagrangian required to eliminate scalar degrees of freedom, allowing only two tensorial degrees of freedom to propagate. Instead of strict constraint analysis, in this paper, we employ the perturbation method and focus on the necessary conditions to evade the scalar mode at the linear order in perturbations around a cosmological background. Beginning with a general action and solving the auxiliary perturbation variables in terms of a would-be dynamical scalar mode, we derive the condition to remove its kinetic term, thus ensuring that no scalar mode propagates. As an application of the general condition, we study a polynomial-type Lagrangian as a concrete example, in which all monomials are spatially covariant scalars containing two derivatives. We find that the auxiliary scalar field is essential, and new terms in the Lagrangian are allowed. Our analysis provides insights into constructing gravity theories with two degrees of freedom in the extended framework of spatially covariant gravity.