We study cosmological perturbations in Hořava-Lifshitz Gravity , a recently proposed potentially ultraviolet-complete quantum theory of 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 .
At first glance , it appears that there is an extra scalar metric degree of freedom .
However , 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 \rightarrow 1 is well-behaved , unlike what is obtained when performing incomplete analyses of cosmological fluctuations .
Thus , there is no strong coupling problem for Hořava-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 , in contrast to what happens when considering Hořava-Lifshitz matter leaving the gravitational sector unperturbed .
However , if we break the detailed balance condition , then the initial spectrum of curvature fluctuations is indeed scale-invariant on ultraviolet scales .
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 .