Implications of the $B$-mode polarization measurement for direct detection of inflationary gravitational waves

The prospects for direct measurements of inflationary gravitational waves by next generation interferometric detectors inferred from the possible detection of $B$-mode polarization of the cosmic microwave background are studied. We compute the spectra of the gravitational wave background and the signal-to-noise ratios by two interferometric detectors (DECIGO and BBO) for large-field inflationary models in which the tensor-to-scalar ratio is greater than the order of 0.01. If the reheating temperature $T_{\mathrm{RH}}$ of chaotic inflation with the quadratic potential is high ($T_{\mathrm{RH}}>7.9\times 10^6\mathrm{GeV}$ for upgraded DECIGO and $T_{\mathrm{RH}}>1.8\times 10^6\mathrm{GeV}$ for BBO), it will be possible to reach the sensitivity of the gravitational background in future experiments at $3\sigma$ confidence level. The direct detection is also possible for natural inflation with the potential $V(\varphi )={\mathrm{\Lambda }}^4[1-\cos (\varphi /f)]$, provided that $f>4.2M_{\text{pl}}$ (upgraded DECIGO) and $f>3.6M_{\text{pl}}$ (BBO) with $T_{\mathrm{RH}}$ higher than $10^8\mathrm{GeV}$. The quartic potential $V(\varphi )=\lambda {\varphi }^4/4$ with a nonminimal coupling $\xi$ between the inflaton field $\varphi$ and the Ricci scalar $R$ gives rise to a detectable level of gravitational waves for $|\xi |$ smaller than the order of 0.01, irrespective of the reheating temperature.

Figure 1

The number of $e$-foldings $N(k)$ versus the reheating temperature $T_{\mathrm{RH}}$ for the model (i) (solid line) and the model (ii) with $f=7M_{\text{pl}}$ (dashed line). We choose four different values of $k$ which are inside the observed CMB range. For increasing $T_{\mathrm{RH}}$ and decreasing $k$, $N(k)$ gets larger.

Figure 2

Two-dimensional observational constraints on the models (i) and (ii) (left), and the model (iii) (right) in the $(n_s,r)$ plane with the pivot wave number $k_{*}=0.05{\mathrm{Mpc}}^{-1}$. The gray shaded regions represent the 68% C.L. (inside) and 95% C.L. (outside) parameter spaces constrained by the joint data analysis of $\text{Planck}+\mathrm{WP}+\mathrm{BAO}+\text{high}\text{−}\ell$. We also show the 68% C.L. (inside) and 95% C.L. (outside) boundaries constrained by $\text{Planck}+\mathrm{WP}+\mathrm{BAO}+\text{high}\text{−}\ell +\mathrm{BICEP}2$ with dotted ellipses. In the left panel, the three large black points correspond to the theoretical prediction of the model (i) for $N=50,55,60$, whereas the small black points represent the prediction of the model (ii) for $f/M_{\text{pl}}=5,6,7,8,9,10$ with $N=50,55,60$. In the right panel, the theoretical curves are plotted for $N=55,58,63$ with several different values of $\xi$ between 0 and $-10^{-2}$.

Figure 3

Today’s intensity ${\mathrm{\Omega }}_{\text{GW}}$ of the gravitational wave background versus the frequency $f=k/(2\pi )$ for the four models: (i) $V(\varphi )=m^2{\varphi }^2/2$ with $\xi =0$, (ii) $V(\varphi )={\mathrm{\Lambda }}^4[1+\cos (\varphi /f)]$ with $\xi =0$ and $f=7M_{\text{pl}}$, the potential $V(\varphi )=\lambda {\varphi }^4/4$ with $\xi =0$, and (iii) $V(\varphi )=\lambda {\varphi }^4/4$ and $\xi =-5.0\times 10^{-3}$. We also show the sensitivity curves for DECIGO, upgraded DECIGO, and BBO. Each panel corresponds to the spectra around the detection sensitivity with $T_{\mathrm{RH}}=10^7\mathrm{GeV}$ (left) and the spectra in the wide range of frequencies with $T_{\mathrm{RH}}=10^9\mathrm{GeV}$ (right).

Figure 4

The SNR versus the reheating temperature computed for the model (i), the model (ii) with $f=7M_{\text{pl}}$, the potential $V(\varphi )=\lambda {\varphi }^4/4$ with $\xi =0$, and the model (iii) with $\xi =-5.0\times 10^{-3}$. Each panel corresponds to the SNR for DECIGO (top), upgraded DECIGO (middle), and BBO (bottom), respectively. The gray region shows the $1\sigma$ uncertainty in the amplitude of primordial curvature perturbations ${\mathcal{P}}_{\mathcal{R}}^{\text{prim}}=(2.200_{-0.054}^{+0.056})\times 10^{-9}$ constrained from Planck [6]. For observation time of the gravitational wave, we assume $T_{\text{obs}}=10\text{year}$. The SNR scales as $\propto \sqrt{T_{\text{obs}}/10\text{year}}$.

Figure 5

(Left) Dependence of the SNR on the parameter $f/M_{\text{pl}}$ for the model (ii). (Right) Dependence of the SNR on the nonminimal coupling $\xi$ for the model (iii). Each line is the SNR calculated with noise curves of DECIGO, upgraded DECIGO, and BBO, respectively, with $T_{\text{obs}}=10$ year. In both cases, we assume $T_{\mathrm{RH}}=10^9\mathrm{GeV}$.