Reheating Constraints to Inflationary Models
Evidence from the BICEP2 experiment for a significant gravitational-wave background has focused attention on inflaton potentials $V(\ensuremath{\phi})\ensuremath{\propto}{\ensuremath{\phi}}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}=2$ (``chaotic'' or ``${m}^{2}{\ensuremath{\phi}}^{2}$'' inflation) or with smaller values of $\ensuremath{\alpha}$, 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 ${w}_{\text{re}}$. The canonical reheating scenario is then described by ${w}_{\text{re}}=0$. The simplest $\ensuremath{\alpha}=2$ models are consistent with ${w}_{\text{re}}=0$ for values of ${n}_{s}$ well within the current $1\ensuremath{\sigma}$ range. Models with $\ensuremath{\alpha}=1$ or $\ensuremath{\alpha}=2/3$ require a more exotic reheating phase, with $\ensuremath{-}1/3<{w}_{\text{re}}<0$, unless ${n}_{s}$ falls above the current $1\ensuremath{\sigma}$ range. Likewise, models with $\ensuremath{\alpha}=4$ require a physically implausible ${w}_{\text{re}}>1/3$, unless ${n}_{s}$ is close to the lower limit of the $2\ensuremath{\sigma}$ range. For ${m}^{2}{\ensuremath{\phi}}^{2}$ inflation and canonical reheating as a benchmark, we derive a relation ${\mathrm{log}}_{10}({T}_{\text{re}}/{10}^{6}\text{ }\text{ }\mathrm{GeV})\ensuremath{\simeq}2000({n}_{s}\ensuremath{-}0.96)$ between the reheat temperature ${T}_{\text{re}}$ and the scalar spectral index ${n}_{s}$. Thus, if ${n}_{s}$ is close to its central value, then ${T}_{\text{re}}\ensuremath{\lesssim}{10}^{6}\text{ }\text{ }\mathrm{GeV}$, 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 ${n}_{s}\ensuremath{\simeq}0.965$ (at the pivot scale $k=0.05\text{ }\text{ }{\text{Mpc}}^{\ensuremath{-}1}$), near the upper limit of the $1\ensuremath{\sigma}$ error range. Improved precision in the measurement of ${n}_{s}$ should allow ${m}^{2}{\ensuremath{\phi}}^{2}$, axion monodromy, and ${\ensuremath{\phi}}^{4}$ models to be distinguished, even without precise measurement of $r$, and to test the ${m}^{2}{\ensuremath{\phi}}^{2}$ expectation of ${n}_{s}\ensuremath{\simeq}0.965$.