From the Old to the New Starobinsky Inflation

Abstract

Starobinsky's original model of inflation, driven by the trace anomaly of conformal fields, and its $R+R^2$ (scalaron) formulation are often treated separately. The original anomaly-driven evolution connects an early de Sitter phase supported by the Euler anomaly to a scalaron regime, where the observable modes leave the horizon. Building on this known background connection, we compute the scalar and tensor spectra on the exact background and find that the anomaly corrections are controlled by $aN_*/(18b)$, where $a$ and $b$ are the Euler-anomaly and $R^2$ coefficients and $N_*$ the number of e-folds after the pivot exits the horizon; scalaron inflation is recovered as this parameter vanishes. The observed tilt at $N_*=60$ then requires only $b/a\simeq35$, rather than the $b/a\gtrsim10^9$ of early realizations. At this benchmark the scalar spectrum is redder than in scalaron inflation at fixed $N_*$, and the tensor-to-scalar ratio is about 17% lower at fixed tilt, a quantitative signature of the anomaly. For minimal gravitational reheating, $N_*\simeq53$, the tilt remains within $1σ$ of the Planck value for $b/a\gtrsim35$. In the de Sitter analysis, a finite range of the Weyl coupling excludes an extra real spin-two pole and keeps the causal response decaying. A companion paper gives the full analysis.