Cosmological perturbations in Hořava-Lifshitz gravity

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 nondynamical. The limit $\lambda \to 1$ is well behaved, unlike what is obtained when expanding about Minkowski space-time. 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 cancellation 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 space-time 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 general relativity. However, in general the equation of motion is characterized by a modified dispersion relation.

Figure 1

Evolution of the mode function with the scale factor $a$. The red curve corresponds to the mode function in Hořava theory, and the black curve shows the standard mode function in GR. The dashed vertical line denotes the sound horizon. The parameters were chosen to be $c_s=1$, $k=H=10$, $M=0$, $\Xi =0.1$.

Figure 2

Power spectrum on super Hubble scales. The red line denotes the spectrum of perturbations in Hořava theory, and the black line denotes the standard GR result. The parameters were chosen to be $c_s=1$, $\Xi =0.1$, $\nu =1.52$, $H=10$. The spectra are evaluated at the conformal time $\eta =-0.001$.