  We consider gauge invariant cosmological perturbations in UV-modified, z=3
Horava gravity with one scalar matter field, which has been proposed as a
renormalizable gravity theory without the ghost problem in four dimensions. In
order to exhibit its dynamical degrees of freedom, we consider the Hamiltonian
reduction method and find that, by solving "all" the constraint equations, the
degrees of freedom are the same as those of Einstein gravity: One scalar and
two tensor (graviton) modes when a scalar matter field presents. However, we
confirm that there is no extra graviton modes and general relativity is
recovered in IR, which achieves the consistency of the model. From the
UV-modification terms which break the detailed balance condition in UV, we
obtain scale-invariant power spectrums for "non"-inflationary backgrounds, like
the power-law expansions, without knowing the details of early expansion
history of Universe. This could provide a new framework for the Big Bang
cosmology. Moreover, we find that "tensor and scalar fluctuations travel
differently in UV, generally". We present also some clarifying remarks about
confusing points in the literatures.
