Primordial perturbations in a rainbow universe with running Newton
  constant
  We compute the spectral index of primordial perturbations in a rainbow
universe. We allow the Newton constant $G$ to run at (super-)Planckian energies
and we consider both vacuum and thermal perturbations. If the rainbow metric is
the one associated to a generalized Horava-Lifshitz dispersion relation, we
find that only when $G$ tends asymptotically to zero can one match the observed
value of the spectral index and solve the horizon problem, both for vacuum and
thermal perturbations. For vacuum fluctuations the observational constraints
imply that the primordial universe expansion can be both accelerating or
decelerating, while in the case of thermal perturbations only decelerating
expansion is allowed.
