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 $\alpha (1\mathrm{/}k)$ 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 $\alpha \sim 1/115$\mathrm{.}Thedependenceofthescalarspectrumonthe$\chi_{\mu \nu }$mass$m_{\chi }$,ontheotherhand,startsfromthe$\alpha ^{2}$ corrections, which are handled perturbatively in the ratio $m_{\varphi }\mathrm{/}{m}_{\chi }$, where $m_{\varphi }$ 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.