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High-order corrections to inflationary perturbation spectra in quantum gravity

We compute the inflationary perturbation spectra and the quantity $r+8n_{T}$ to the next-to-next-to-leading log order in quantum gravity with purely virtual particles (which means the theory $R+R^{2}+C^{2}$ with the fakeon prescription/projection for $C^{2}$). The spectra are functions of the inflationary running coupling $α(1/k)$ and satisfy the cosmic renormalization-group flow equations, which determine the tilts and the running coefficients. The tensor fluctuations receive contributions from the spin-2 fakeon $χ_{μν}$ at every order of the expansion in powers of $α\sim 1/115$. The dependence of the scalar spectrum on the $χ_{μν}$ mass $m_{χ}$, on the other hand, starts from the $α^{2}$ corrections, which are handled perturbatively in the ratio $m_{ϕ}/m_{χ}$, where $m_{ϕ}$ is the inflaton mass. The predictions have theoretical errors ranging from $α^{4}\sim 10^{-8}$ to $α^{3}\sim 10^{-6}$. Nontrivial issues concerning the fakeon projection at higher orders are addressed.