  We study primordial cosmology with two scalar fields that participate in
inflation at the same time, by coupling quantum gravity (i.e., the theory
$R+R^{2}+C^{2}$ with the fakeon prescription/projection for $C^{2}$) to a
scalar field with a quadratic potential. We show that there exists a
perturbative regime that can be described by an asymptotically de Sitter,
cosmic RG flow in two couplings. Since the two scalar degrees of freedom mix in
nontrivial ways, the adiabatic and isocurvature perturbations are not RG
invariant on superhorizon scales. It is possible to identify the correct
perturbations by using RG invariance as a guiding principle. We work out the
resulting power spectra of the tensor and scalar perturbations to the NNLL and
NLL orders, respectively. An unexpected consequence of RG invariance is that
the theory remains predictive. Indeed, the scalar mixing affects only the
subleading corrections, so the predictions of quantum gravity with single-field
inflation are confirmed to the leading order.
