Reheating after Starobinsky Inflation in the Jordan Frame

Abstract: We investigate gravitational reheating in the Starobinsky model in the Jordan frame, where inflation is driven by an $R^2$ modification of gravity with no explicit inflaton field. In this description, reheating proceeds exclusively through gravitational particle production triggered by the oscillations of the Ricci scalar after the end of inflation. We analyze the post-inflationary background evolution and show that an effective fluid emerging from the modified gravitational dynamics behaves as pressureless matter during the oscillatory phase. Including the backreaction of the produced particles, we demonstrate that the Ricci scalar oscillations acquire an exponential damping, consistently terminating particle production. Solving the coupled background and Boltzmann equations, we obtain a reheating temperature $T_{\mathrm{reh}} \sim 2 \times 10^{9}$ GeV. We finally compare the Jordan and Einstein frame descriptions and argue that, although classically equivalent, they can lead to distinct microphysical interpretations and quantitative predictions for reheating once quantum effects are taken into account.