  We study the running of power spectra in inflationary cosmology as a
renormalization-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 RG 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.
