Constraints on the reheating temperature from sizable tensor modes

Despite its importance for modeling the homogeneous hot early Universe very little is experimentally known about the magnitude of the reheating temperature, leaving an uncertainty of remarkable 18 orders of magnitude. In this paper we consider a general class of polynomial inflaton potentials up to fourth order. Employing a Monte Carlo scan and imposing theoretical and experimental constraints we derive a robust lower limit on the energy scale at the end of inflation, $V_{\text{end}}^{1/4}>3\times {10}^{15}\mathrm{GeV}$ for sizable tensor modes, $r\ge 10^{-3}$. If the reheating phase is perturbative and matter dominated, this translates into a lower bound on the reheating temperature, yielding $T_{\mathrm{rh}}>3\times {10}^8(7\times {10}^2)\mathrm{GeV}$ for gravitational inflaton decay through a generic dimension five (six) operator.

Figure 1

Scan points in terms of CMB observables. Left panel: Running (${\alpha }_s$) and running of the running (${\kappa }_s$) of the spectral index. Right panel: Spectral index $n_s$ and its running ${\alpha }_s$. Color code: blue: inflationary points; yellow: inflationary points within 2-sigma-contours regarding Planck data at $\mathrm{\Delta }\varphi =0$; green: points additionally fulfilling the constraint on $N_{*}$.

Figure 2

Scalar potentials which maximize the energy drop during the final $N_{*}$ $e$-folds of inflation for different values of the tensor-to-scalar ratio $r$ (color coded on a logarithmic scale). The orange line marks the end of inflation at $\epsilon =1$.

Figure 3

Scan points in the $N_{*}\text{−}(V_{*}/V_{\text{end}})$-plane and $r\text{−}(V_{*}/V_{\text{end}})$-plane. Color code: blue: inflationary points; yellow: inflationary points within 2-sigma-contours regarding Planck data at $\mathrm{\Delta }\varphi =0$; green: points additionally fulfilling the constraint on $N_{*}$. Dashed black curves: Prediction for monomial potentials ${\varphi }^p$ with $p=\{1/2,1,2,3,4\}$ with the constraint on $N_{*}$ fulfilled within the region marked dashed red (for $p\le 4$). In the left panel, we additionally show the most conservative analytical bound, the exponential potential.

Figure 4

Fine-tuning $\xi$ among the locally Planck allowed points in the $r\text{−}(V_{*}/V_{\text{end}})$-plane (left) and lower bound on the energy density at the end of inflation in the $r\text{−}{V}_{\text{end}}$-plane (right). The color code in the right panel is as in Fig. 3.

Figure 5

Scan points in the $r\text{−}{T}_{\mathrm{rh}}$-plane showing the lower bound on the reheating temperature for a decay via dimension five operators (left) and dimension six operator (right) with $\lambda =1$. The color code is as in Fig. 3.