The Equation of State and Duration to Radiation Domination After Inflation

Abstract

We calculate the equation of state after inflation and provide an upper bound on the duration before radiation domination by taking the nonlinear dynamics of the fragmented inflaton field into account. A broad class of single-field inflationary models with observationally consistent flattening of the potential at a scale $M$ away from the origin, $V(\phi)\propto |\phi|^{2n}$ near the origin, and where the couplings to other fields are ignored are included in our analysis. We find that the equation of state parameter $w\rightarrow 0$ for $n=1$ and $w\rightarrow 1/3$ (after sufficient time) for $n\gtrsim 1$. We calculate how the number of $e$-folds to radiation domination depends on both $n$ and $M$ when $M\sim m_{\rm pl}$, whereas when $M\ll m_{\rm pl}$, we find that the duration to radiation domination is negligible. Our results are explained in terms of a linear instability analysis in an expanding universe, scaling arguments, and are supported by detailed 3+1 dimensional lattice simulations. We show how our work significantly reduces the uncertainty in inflationary observables, even after including couplings to additional light fields.