Cosmological Perturbations in Ho\v{r}ava-Lifshitz Gravity
  We study cosmological perturbations in Ho\v{r}ava-Lifshitz Gravity. We
consider scalar metric fluctuations about a homogeneous and isotropic
space-time. Starting from the most general metric, we work out the complete
second order action for the perturbations. We then make use of the residual
gauge invariance and of the constraint equations to reduce the number of
dynamical degrees of freedom. After introducing the Sasaki-Mukhanov variable,
the combination of spatial metric fluctuation and matter inhomogeneity for
which the action in General Relativity has canonical form, we find that this
variable has the standard time derivative term in the second order action, and
that the extra degree of freedom is non-dynamical. The limit $\lambda \to 1$ is
well-behaved, unlike what is obtained when performing incomplete analyses of
cosmological fluctuations. Thus, there is no strong coupling problem for
Ho\v{r}ava-Lifshitz gravity when considering cosmological solutions. We also
compute the spectrum of cosmological perturbations. If the potential in the
action is taken to be of "detailed balance" form, we find a cancelation of the
highest derivative terms in the action for the curvature fluctuations. As a
consequence, the initial spectrum of perturbations will not be scale-invariant
in a general spacetime background. As an application, we consider fluctuations
in an inflationary background and draw connections with the "trans-Planckian
problem" for cosmological perturbations. In the special case in which the
potential term in the action is of detailed balance form and in which $\lambda
= 1$, the equation of motion for cosmological perturbations in the far UV takes
the same form as in GR. However, in general the equation of motion is
characterized by a modified dispersion relation.
