Observational signatures of the parametric amplification of gravitational waves during reheating after inflation

We study the evolution of gravitational waves (GWs) during and after inflation as well as the resulting observational consequences in a Lorentz-violating massive gravity theory with one scalar (inflaton) and two tensor degrees of freedom. We consider two explicit examples of the tensor mass $m_g$ that depends either on the inflaton field $\varphi$ or on its time derivative $\dot{\varphi }$, both of which lead to parametric excitations of GWs during reheating after inflation. The first example is Starobinsky’s $R^2$ inflation model with a $\varphi$-dependent $m_g$, and the second is a low energy–scale inflation model with a $\dot{\varphi }$-dependent $m_g$. We compute the energy density spectrum ${\mathrm{\Omega }}_{\mathrm{GW}}(k)$ today of the GW background. In the Starobinsky’s model, we show that the GWs can be amplified up to the detectable ranges of both cosmic microwave background and DECi-hertz Interferometer Gravitational wave Observatory, but the bound from the big bang nucleosynthesis is quite tight to limit the growth. In low-scale inflation with a fast transition to the reheating stage driven by the potential $V(\varphi )=M^2{\varphi }^2/2$ around $\varphi \approx M_{\mathrm{pl}}$ (where $M_{\mathrm{pl}}$ is the reduced Planck mass), we find that the peak position of ${\mathrm{\Omega }}_{\mathrm{GW}}(k)$ induced by the parametric resonance can reach the sensitivity region of advanced LIGO for the Hubble parameter of order 1 GeV at the end of inflation. Thus, our massive gravity scenario offers exciting possibilities for probing the physics of primordial GWs at various different frequencies.

Figure 1

The physical scales of our interest at which the tensor power spectrum changes its shape are depicted in the space-time diagram. At $t>t_f$, the amplitude of the tensor modes $v(k,t)$ outside the Hubble horizon is nearly constant in time, while the modes inside the Hubble horizon behave as $\propto a^{-1}$. At $t_f<t<t_i$, the modes at $a/k>m_g^{-1}$ evolve as $a^{-3/2}$, while those at $a/k<m_g^{-1}$ behave as $a^{-1}$.

Figure 2

The primordial tensor power spectra at the onset of reheating ($t=t_i$) for the potential (5.1) with $M=1.3\times {10}^{-5}{M}_{\mathrm{pl}}$. Each curve corresponds to (i) the massless tensor ($m_g^2=0$), (ii) the tensor mass squared (2.12) with $n=2$, $b=4$, $\lambda =5\times 10^{-7}$, and (iii) the tensor mass squared (2.12) with $n=1$, $b=4$, $\lambda =5\times 10^{-7}$. The suppression of GWs induced by the heavy tensor mass during inflation tends to be less significant for larger values of $n$ and $b$.

Figure 3

Evolution of the tensor power spectrum in Starobinsky inflation with $M=1.3\times {10}^{-5}{M}_{\mathrm{pl}}$ for the tensor mass squared (2.12) with $\lambda =4.8\times {10}^{-7}$, $n=2$, and $b=2$. The three curves correspond to the modes which crossed the Hubble radius at $N=55$, 16, 3 $e$-folding before the end of inflation, respectively.

Figure 4

The tensor power spectra at $t={10}^2{t}_i$ and $t={10}^4{t}_i$ for the potential (5.1) with $M=1.3\times {10}^{-5}{M}_{\mathrm{pl}}$ in the presence of the tensor mass squared (2.12) with $\lambda =4.8\times {10}^{-7}$, $n=2$, and $b=2$. We also show the spectrum at the onset of reheating ($t=t_i$). After the amplitude of perturbations reaches the maximum around the time $t\simeq 80t_i$, it decreases until the moment $t_{\mathrm{\Gamma }}$ at which the inflaton decays to the radiation.

Figure 5

Evolution of the tensor power spectrum during reheating in low-scale inflation with $M=4.1\times {10}^{-11}{M}_{\mathrm{pl}}={10}^8\mathrm{GeV}$ for the tensor mass squared (2.13) with $\mu =1.523\times {10}^4$. We integrate Eq. (3.13) from the onset of reheating ($a_i=1$) with the initial conditions ${\varphi }_i=\sqrt{2}{M}_{\mathrm{pl}}$ and ${\dot{\varphi }}_i=-\sqrt{2/3}M{M}_{\mathrm{pl}}$. Each curve corresponds to the evolution of ${\mathcal{P}}_T$ for $k=0.067M$, $3.341M$, and $16.75M$, respectively.

Figure 6

The tensor power spectrum at $t={10}^3/M$ in low-scale inflation with the tensor mass squared (2.13) for the same model parameters used in Fig. 5.

Figure 7

Today’s spectral energy density ${\mathrm{\Omega }}_{\mathrm{GW}}$ of the GW background versus the frequency $f=k/(2\pi )$ in the Starobinsky model with the $\varphi$-dependent tensor mass squared (2.12) for $M=1.3\times {10}^{-5}{M}_{\mathrm{pl}}$, $\lambda =4.785\times {10}^{-7}$, $n=2$, and $b=2$. We also show the sensitivity curves for DECIGO [27], upgraded DECIGO [43], A-LIGO [26], LISA [44], SKA [45], and SRI [46], together with the upper bound from the BBN [47]. For a recent BBN status report, see, e.g., Ref. [48]. Each line corresponds to the case in which the inflaton decay to the radiation occurs at $t={10}^2/M$ (red) and $t={10}^4/M$ (blue).

Figure 8

${\mathrm{\Omega }}_{\mathrm{GW}}$ versus $f$ in low-scale inflation with the $\dot{\varphi }$-dependent tensor mass squared (2.13). The sensitivity curves of GW experiments are the same as shown in Fig. 7. Each line corresponds to the models with (i) $M={10}^8\mathrm{GeV}$, $\mu =1.523\times {10}^4$, $k_1=0.067M$, and $t_{\mathrm{\Gamma }}={10}^3/M$ (red) and (ii) $M=1\mathrm{GeV}$, $\mu =4.78\times {10}^4$, $k_1=0.067M$, and $t_{\mathrm{\Gamma }}={10}^3/M$ (blue).