  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 work, we employ the
perturbation method and focus on the necessary conditions to evade the scalar
mode at linear order in perturbations around a cosmological background.
Starting from a general action and solving the auxiliary perturbation variables
in terms of the 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 the monomials are spatially covariant scalars
containing two derivatives. We find that the auxiliary scalar field plays a
nontrivial role, and new terms in the Lagrangian are allowed. Our analysis
sheds light on constructing gravity theories with two degrees of freedom in the
extended framework of spatially covariant gravity.
