Reheating constraints to inflationary models

Evidence from the BICEP2 experiment for a significant gravitational-wave background has focussed attention on inflaton potentials $V(\phi) \propto \phi^\alpha$ with $\alpha=2$ ("chaotic" or "$m^2\phi^2$" inflation) or with smaller values of $\alpha$, as may arise in axion-monodromy models. Here we show that reheating considerations may provide additional constraints to these models. The reheating phase preceding the radiation era is modeled by an effective equation-of-state parameter $w_{\rm re}$. The canonical reheating scenario is then described by $w_{\rm re}=0$. The simplest $\alpha=2$ models are consistent with $w_{\rm re} = 0$ for values of $n_s$ well within the current $1\sigma$ range. Models with $\alpha=1$ or $\alpha=2/3$ require a more exotic reheating phase, with $-1/3<w_{\rm re}<0$, unless $n_s$ falls above the current $1\sigma$ range. Likewise, models with $\alpha=4$ require a physically implausible $w_{\rm re}>1/3$, unless $n_s$ is close to the lower limit of the $2\sigma$ range. For $m^2\phi^2$ inflation and canonical reheating as a benchmark, we derive a relation $\log_{10}\left(T_{\rm re}/10^6\,{\rm GeV} \right) \simeq 2000\,(n_s-0.96)$ between the reheat temperature $T_{\rm re}$ and the scalar spectral index $n_s$. Thus, if $n_s$ is close to its central value, then $T_{\rm re}\lesssim 10^6$~GeV, just above the electroweak scale. If the reheat temperature is higher, as many theorists may prefer, then the scalar spectral index should be closer to $n_s\simeq0.965$ (at the pivot scale $k=0.05\,{\rm Mpc}^{-1}$), near the upper limit of the $1\sigma$ error range. Improved precision in the measurement of $n_s$ should allow $m^2\phi^2$, axion-monodromy, and $\phi^4$ models to be distinguished, even without precise measurement of $r$, and to test the $m^2\phi^2$ expectation of $n_s\simeq0.965$.