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
$n_s\simeq0.965$.
