Spatially covariant gravity with two 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 two 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.
