Gauss--Bonnet running and the de Sitter saddle of quadratic gravity inflation

Inflation driven purely by the quantum running of the curvature-squared couplings of quadratic gravity was shown to accommodate the high scalar tilt favored by recent cosmic microwave background observations, starting from a Euclidean four-sphere at a maximum of the running $R^2$ coupling. That maximum exists only in one renormalization scheme, and is not stationary once the Euler trace anomaly is included. We show that the standard one-loop running, with the usually neglected Gauss–Bonnet coefficient retained, recovers the missing de Sitter state: the Euler running balances the scale dependence of the $R^2$ term at a unique coupling ratio, giving a de Sitter solution that is stationary for the gravitational constraint and for the compact Euclidean action alike, and whose scalaron potential is an extremely flat hilltop ($m^2\mathrm{/}{H}^2\simeq -5\times {10}^{-10}$) joined to an inverse-linear inflationary plateau. The last $N_{\ast }\simeq 50$–$60$ $e$-folds are scheme independent, with scalar tilt $n_s\simeq 1-4\mathrm{/}(3N_{\ast })$, while the tensor-to-scalar ratio, $r$, is set by the number of matter fields that enhance the running. The current tensor bound excludes pure gravity and sets a minimum matter content, some $4\times {10}^6$ conformally coupled scalars or $3\times {10}^5$ vectors—$2.6$ times fewer than the momentum-induced scheme requires—while one-loop control to the end of inflation sets a maximum about ten times higher. Across that window the model predicts $r\gtrsim 0.008$ with $0.973\lesssim n_s\lesssim 0.978$, within reach of upcoming CMB polarization surveys.