The Trans-Planckian Problem in the Healthy Extension of Horava-Lifshitz
  Gravity
  Planck scale physics may influence the evolution of cosmological fluctuations
in the early stages of cosmological evolution. Because of the quasi-exponential
redshifting, which occurs during an inflationary period, the physical
wavelengths of comoving scales that correspond to the present large-scale
structure of the Universe were smaller than the Planck length in the early
stages of the inflationary period. This trans-Planckian effect was studied
before using toy models. The Horava-Lifshitz (HL) theory offers the chance to
study this problem in a candidate UV complete theory of gravity. In this paper
we study the evolution of cosmological perturbations according to HL gravity
assuming that matter gives rise to an inflationary background. As is usually
done in inflationary cosmology, we assume that the fluctuations originate in
their minimum energy state. In the trans-Planckian region the fluctuations obey
a non-linear dispersion relation of Corley-Jacobson type. In the "healthy
extension" of HL gravity there is an extra degree of freedom which plays an
important role in the UV region but decouples in the IR, and which influences
the cosmological perturbations. We find that in spite of these important
changes compared to the usual description, the overall scale-invariance of the
power spectrum of cosmological perturbations is recovered. However, we obtain
oscillations in the spectrum as a function of wavenumber with a relative
amplitude of order unity and with an effective frequency which scales
nonlinearly with wavenumber. Taking the usual inflationary parameters we find
that the frequency of the oscillations is so large as to render the effect
difficult to observe.
