Reheating Constraints to Inflationary Models

Evidence from the BICEP2 experiment for a significant gravitational-wave background has focused attention on inflaton potentials $V(\varphi )\propto {\varphi }^{\alpha }$ with $\alpha =2$ (“chaotic” or “$m^2{\varphi }^2$” 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 $w_{\text{re}}$. The canonical reheating scenario is then described by $w_{\text{re}}=0$. The simplest $\alpha =2$ models are consistent with $w_{\text{re}}=0$ 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 $-1/3<w_{\text{re}}<0$, unless $n_s$ falls above the current $1\sigma$ range. Likewise, models with $\alpha =4$ require a physically implausible $w_{\text{re}}>1/3$, unless $n_s$ is close to the lower limit of the $2\sigma$ range. For $m^2{\varphi }^2$ inflation and canonical reheating as a benchmark, we derive a relation ${\log }_{10}(T_{\text{re}}/{10}^6\mathrm{GeV})\simeq 2000(n_s-0.96)$ 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 $T_{\text{re}}\lesssim {10}^6\mathrm{GeV}$, 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 $n_s\simeq 0.965$ (at the pivot scale $k=0.05{\text{Mpc}}^{-1}$), near the upper limit of the $1\sigma$ error range. Improved precision in the measurement of $n_s$ should allow $m^2{\varphi }^2$, axion monodromy, and ${\varphi }^4$ models to be distinguished, even without precise measurement of $r$, and to test the $m^2{\varphi }^2$ expectation of $n_s\simeq 0.965$.

Figure 1

The evolution of the comoving Hubble scale $1/aH$. The reheating phase connects the inflationary phase and the radiation era. Compared to instantaneous reheating (thick dotted curve), a reheating equation-of-state parameter $w_{\text{re}}<1/3$ implies more postinflationary $e$-folds of expansion. Fewer postinflationary $e$-folds requires $w_{\text{re}}>1/3$ (thin dotted curve).

Figure 2

We plot $N_{\text{re}}$ (upper panels) and $T_{\text{re}}$ (lower panels) as determined from Eq. (11) and Eq. (7), respectively. Results for power-law indexes $\alpha =2/3,1,2,4$ are each shown separately. Different effective equation-of-state parameters for reheating are considered in each case: $w_{\text{re}}=-1/3$ (red dashed curve), $w_{\text{re}}=0$ (blue solid curve), $w_{\text{re}}=1/6$ (orange dash-dotted curve), and $w_{\text{re}}=2/3$ (green long-dashed curve). All curves intersect at the point where reheating occurs instantaneously. The width of each curve corresponds to a variation of the termination condition $0.1\lesssim {\epsilon }_0\lesssim 1$ and also roughly the uncertainty in $r$. The light purple regions are below the electroweak scale $T_{\text{EW}}\sim 100\mathrm{GeV}$. The dark purple regions, below 10 MeV, would ruin the predictions of big bang nucleosynthesis. Temperatures above the intersection point are unphysical as they correspond to $N_{\text{re}}<0$. The light yellow band indicates the $1\sigma$ range $n_s-1=-0.0397\pm 0.0073$ from Planck [9], and the dark yellow band assumes a projected uncertainty of ${10}^{-3}$ [8] for $n_s-1$, as expected from future experiments (assuming the central value remains unchanged).