Cosmic inflation as a renormalization-group flow: the running of power spectra in quantum gravity

We study the running of power spectra in inflationary cosmology by reformulating the slow roll expansion as a re normalization-group flow from the de Sitter fixed point. The beta function is provided by the equations of the background metric. The spectra of the scalar and tensor fluctuations obey RC evolution equations with vanishing anomalous dimensions in the superhorizon limit. By organizing the perturbative expansion in terms of leading and subleading logs, we calculate the spectral indices, their runnings, the runnings of the runnings, etc., to the next-to-leading log order in quantum gravity with fakeons (i.e., the theory R + R-2 + C-2 with the fakeon prescription/projection for C-2). We show that these quantities are related to the spectra in a universal way. We also compute the first correction to the relation r = - 8n(T) and provide a number of quantum gravity predictions that can be hopefully tested in the forthcoming future.