Nonlinear dynamics of reheating in the primordial universe: Insights from the quartic model

We study the dynamics of the inherently unstable inflaton field that spontaneously decays into radiation during both inflation and reheating. To model this, we formulate a set of coupled rate equations describing the depletion of inflaton energy density and the concurrent buildup of radiation, ensuring strict energy conservation. These are further coupled with the Friedmann equation with backreaction from the produced radiation. Strong nonlinearities in these coupled set of differential equations preclude any analytical solution. Starting with slow-roll initial conditions and allowing for $\sim 60$ e-folds of inflation, numerical computation reveals that radiation begins to accumulate even during inflation. Once inflation ends, the Universe transitions smoothly to a reheating phase driving an efficient growth in radiation energy density, reaching a maximum of ${\rho }_r=5.2019\times 10^{50}{\mathrm{GeV}}^4$, with a reheating temperature of $T_r=1.9618\times 10^{12}$ GeV. Importantly, the Universe achieves a graceful exit, satisfying the Kofman-Yi criterion for successful reheating, ensuring a nearly complete transfer of inflaton energy into radiation. This sets the stage for a hot, radiation-dominated epoch, aligning with the observed thermal history of the early Universe. While this framework provides a robust dynamics for the graceful exit mechanism, the predicted tensor-to-scalar ratio, $r=0.2609$, exceeds current bounds from BICEP/Keck and Planck. This value arises from the quartic shape of the inflaton potential and is independent of the quartic coupling. Reconciling the graceful exit with CMB constraints would require alternative inflaton potentials.