Minimal production of prompt gravitational waves during reheating

The inflationary reheating phase begins when accelerated expansion ends. As all Standard Model particles are coupled to gravity, gravitational interactions will lead to particle production. This includes the thermal bath, dark matter, and gravitational radiation. Here, we compute the spectrum of gravitational waves from the inflaton condensate during the initial phase of reheating. As particular examples of inflation, we consider the Starobinsky model and T models, all of which are in good phenomenological agreement with cosmic microwave background anisotropy measurements. The T models are distinguished by the shape of the potential about its minimum and can be approximated by $V\sim {\varphi }^k$, where $\varphi$ is the inflaton. Interestingly, the shape of the gravitational wave spectrum (when observed) can be used to distinguish among the models considered. As we show, the Starobinsky model and T models with $k=2$, provide very different spectra when compared to models with $k=4$ or $k>4$. Observation of multiple harmonics in the spectrum can be interpreted as a direct measurement of the inflaton mass. Furthermore, the cutoff in frequency can be used to determine the reheating temperature.

Figure 1

(a),(b) Feynman diagrams for graviton pair production from inflation annihilation mediated by the scalar. (c) Feynman diagrams for graviton pair production from inflation annihilation mediated by the graviton. (d) Feynman diagrams for graviton pair production from inflation annihilation by the contact interaction. The momenta $p_1$, $p_2$ are incoming and $k_1$, $k_2$ are outgoing.

Figure 2

Gravitational wave spectra in Eq. (25) resulting from the inflaton condensate pair annihilation in the Starobinksy model with $m_{\varphi }=3\times {10}^{13}\mathrm{GeV}$ and ${\rho }_{\mathrm{end}}=(5.5\times {10}^{15}\mathrm{GeV}{)}^4$. Each solid line of different colors is differentiated by the specified distinct reheating temperature. The gray dotted line shows the sensitivity of the resonant cavities proposal [90, 91].

Figure 3

Gravitational wave spectra in Eqs. (45), (46), and (51) resulting from the inflaton condensate pair annihilation in the T model with $\lambda =3.4\times {10}^{-12}(5.7\times {10}^{-13}$) for $k=4(6)$ and ${\rho }_{\mathrm{end}}=(4.8\times {10}^{15}\mathrm{GeV}{)}^4$. Each solid line of different colors is differentiated by the specified distinct reheating temperature and shows the GW spectrum for $k=6$. The $k=4$ case with the narrow frequency range in Eq. (50) is shown as the vertical black line. Solid lines show the contribution from $n=2$, and the dashed lines show the contributions for the higher modes $n=4$, 6, 8, 10, 12. The gray dotted line shows the sensitivity of the resonant cavities proposal [90, 91].

Figure 4

Gravitational wave spectra from the dominant $n=2$ mode resulting from the inflaton condensate pair annihilation in various $k$s in the T model [see Eq. (3)] for a fixed $T_{\mathrm{RH}}={10}^{10}\mathrm{GeV}$. The gray dotted line shows the sensitivity of the resonant cavities proposal [90, 91].