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 squared 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 squared 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/H 2-5 times10-10) joined to an inverse-linear inflationary plateau. The last N* 50 -- 60 e-folds are scheme independent, with scalar tilt n s 1-4/(3N*), 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 times10 6 conformally coupled scalars or 3 times10 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 0.008 with 0.973 n s 0.978, within reach of upcoming CMB polarization surveys.