Precision calculations of the gravitational wave background spectrum from inflation

The spectrum of the gravitational wave background originating from quantum fluctuations during inflation is calculated numerically for various inflation models over a wide range of frequencies. We take into account four ingredients: the scalar field dynamics during inflation making no use of the slow-roll approximation, the fermionic decay of the scalar field with a small coupling constant during the reheating process, the change of the effective number of degrees of freedom $g_{*}$ in the radiation-dominated era, and the anisotropic stress of free-streaming neutrinos. By numerically solving the evolution of gravitational waves during and after inflation up to the present, all of these effects can be examined comprehensively and accurately over a broad spectrum, even at very high frequencies. We find that the spectrum shows (i) a large deviation from the spectrum less accurate obtained by Taylor expanding around the CMB scale using the slow-roll approximation, (ii) a characteristic frequency dependence due to the reheating effect, and (iii) damping due to the $g_{*}$ changes and the neutrino anisotropic stress. We suggest that future analysis of the gravitational wave background should take into consideration the fact that analytical estimates using the Taylor expansion overestimate the amplitude of the spectrum.

Figure 1

The spectrum of the gravitational wave background generated during inflation with a quadratic potential $V=m{\varphi }^2/2$, versus the gravitational frequency $f_0=k/2\pi [\mathrm{Hz}]$. The black-line spectrum includes the effects of $g_{*}$ changes and the neutrino anisotropic stress, and the light gray one does not. Two lines are shown to compare our numerical result with theoretical prediction: the dotted curve represents a scale-invariant spectrum and the dashed line is the spectrum predicted under the slow-roll approximation, which is plotted according to Eq. (22) with the spectral index $n_T(k_{\mathrm{pivot}})=-1.76\times {10}^{-2}$ and its running ${\alpha }_T(k_{\mathrm{pivot}})=-3.11\times {10}^{-4}$. The shaded area corresponds to the spectrum due to quantum zero point fluctuations (see Sec. 4a).

Figure 2

Portion of the same spectrum of Fig. 1, focusing on the features induced by the change of $g_{*}$ and the neutrino anisotropic stress. Note that ${\Omega }_{\mathrm{GW}}$ is plotted on a linear scale. The black spectrum includes both contributions, the dark gray one includes only the effect of $g_{*}$ changes, and the light gray one does not include either.

Figure 3

The spectrum for different values of the reheating temperature for chaotic inflation. The lightest gray spectrum is the same as shown in Fig. 1, i.e. the decay rate is set to be $\Gamma ={10}^{-2}m$ which corresponds to $T_{\mathrm{RH}}\simeq 2\times {10}^{14}\mathrm{GeV}$. The dark gray one shows the case of $T_{\mathrm{RH}}\simeq {10}^9\mathrm{GeV}$ and the black one shows the case of $T_{\mathrm{RH}}\simeq {10}^6\mathrm{GeV}$.

Figure 4

The spectrum for different inflation potentials.

Figure 5

The spectrum of gravitational wave background is shown for several inflation models. The solid curve shows the spectrum for the quadratic potential, the dashed curve for the quartic potential, the dotted curve for the new inflation potential with $\sigma =10m_{\mathrm{Pl}}$, and the dot-dashed curve for hybrid inflation.

Figure 6

Time evolution of ${\chi }^{\prime }(u\equiv k\tau )\equiv h_{\mathbf{k}}^{\lambda \prime }(u)/h_{\mathbf{k}}^{\lambda }(0)$ for the mode $k{\tau }_{\nu ,\mathrm{dec}}=5.0$, which entered the horizon before neutrino decoupling. The solid line is the analytical solution, ${\chi }^{\prime }=-j_1(u)$, and does not include the effect of neutrino anisotropic stress. The dotted line, which is almost indistinguishable from the analytical solution, represents our numerical result. The dashed line is an attempt to reproduce the numerical result of Ref. 22. The vertical dashed line represents the time of neutrino decoupling.

Figure 7

Evolution of the source term, i.e. the right-hand side of Eq. (c1). The upper figure shows our results and the lower shows the reproduced results of Ref. 22. The solid, dotted, and dashed lines are the same as in Fig. 6, and the short-dashed line represents the source term. The artificially introduced discontinuity in the source term in the lower panel has been chosen by trial and error to recover the evolution of Ref. 22. Note that the scale of the vertical axis for the source term is 10 times larger than that for the others.