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 $\alpha (1/k)$ 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 $\chi_{\mu \nu }$ at every order of the expansion in powers of $\alpha \sim 1/115$. The dependence of the scalar spectrum on the $\chi_{\mu \nu }$ mass $m_{\chi }$, on the other hand, starts from the $\alpha ^{2}$ corrections, which are handled perturbatively in the ratio $m_{\phi }/m_{\chi }$, where $m_{\phi }$ is the inflaton mass. The predictions have theoretical errors ranging from $\alpha ^{4}\sim 10^{-8}$ to $\alpha ^{3}\sim 10^{-6}$. Nontrivial issues concerning the fakeon projection at higher orders are addressed.