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.

Figures (5)

• Figure 1: (Left) Hubble parameter as a function of proper time. The vertical dashed line represents the proper time when the first slow-roll parameter satisfies $\epsilon=1$, i.e. at the end of slow-roll inflation. (Right) Ricci scalar as a function of proper time.
• Figure 2: Evolution of the scale factor $a(t)$ as a function of cosmic time $t$ in log-log scale. (Left) Dynamics of the universe during inflation (blue region), characterized by an almost exponential expansion, followed by the reheating phase (red region). (Right) Zoom-in on the reheating phase, showing oscillations induced by the inflaton dynamics. The red dashed line corresponds to $a(t) \propto t^{2/3}$, consistent with an effective equation of state $w = 0$, typical of a matter-dominated regime.
• Figure 3: Behavior of the term responsible for particle production at the onset of reheating. The initial time corresponds to the first oscillation of the Ricci scalar.
• Figure 4: Evolution of the radiation and effective energy densities during reheating.
• Figure 5: Evolution of the source term \ref{['Ueta']} in the Einstein frame. The background equations in the Einstein frame were solved using the initial conditions of Dorsch:2024nan. The amplitude of the oscillations decreases with time, in sharp contrast to Fig. \ref{['fig:U_evolution']}.