Narrow-band parametrization for the stochastic gravitational wave background

In light of the nonperturbative resonance effects that may occur during inflation, we introduce a parametrization for the power spectrum of the stochastic gravitational wave background characterized by narrow-band amplification. We utilize the universal ${\mathrm{\Omega }}_{\mathrm{GW}}\propto k^3$ infrared limit, applicable to a wide array of gravitational wave sources, to devise a robust yet straightforward parametrization optimized for Markov chain Monte Carlo analyses. This parametrization is demonstrated through select examples where its application is pertinent, and we discuss the advantages of this approach over traditional parametrizations for narrow-band scenarios. To evaluate the sensitivity of our proposed model parameters, we apply a mock likelihood based on the CMB-Stage4 data. Furthermore, we explicate the computational process for the mapping relationship between the foundational model parameters and our parametrized framework, using a two-field inflation model that resonantly amplifies gravitational waves as an example.

Figure 1

The upper panel illustrates the comparison between the numerical results obtained from [79] and the three distinct parametrizations: narrow band (5), broken power law (13), and log-normal cutoff (14). Assuming negligible curvature perturbation amplification along with the resonance, we assert that our narrow-band parametrization outperforms the others in this case. The broken power law introduces an extra spike when the peak is narrow and generates a non-negligible error in the BB-mode CMB anisotropy, as demonstrated in the lower panel. The log-normal cutoff regime encounters challenges in performing effectively in the MCMC process. We omitted the CMB BB-mode power spectrum corresponding to the log-normal cutoff and the numerical results in the lower panel because their curves show little distinguishable variation, as seen in the upper panel. Blue dashed lines approximately mark the characteristic zero points of $f(k)=1$, which we will utilize in Sec. 4 for the prior setting of MCMC procedure. The parameter settings are as follows. The slow-roll part (11), $A_{s}r=2.5\times {10}^{-16}$, $k_{\text{pivot}}={10}^{-6}$, $n_t=-0.05$; narrow-band parametrization, $h_R={10}^{4.8}$, $g=2$, $k_{*}={10}^{-3.35}$ for red dots and $h_R={10}^{4.3}$, $g=1.9$, $k_{*}={10}^{-2.8}$ for black dots; broken power law, $A={10}^6$, $\alpha =3$, $\beta =20$, $k_{*}={10}^{-3.15}$ and $A={10}^{5.5}$, $\alpha =3$, $\beta =20$, $k_{*}={10}^{-2.5}$; log-normal cutoff, $h_R={10}^{4.8}$, $g=1.35$, $k_{*}={10}^{-3.35}$, $\mathrm{\Delta }=1.06$ and $h_R={10}^{4.4}$, $g=1.2$, $k_{*}={10}^{-2.7}$, $\mathrm{\Delta }=1.04$.

Figure 2

The parametrization of the SIGW spectrum from [67]. The dotted lines represent the original SIGW spectra, while the solid lines represent our parametrizations. From a practical perspective, we leave the parameter $k_{*}$ free from the value fixed by original spectra when carrying out parametrization fitting.

Figure 3

MCMC results using CMB-S4 mock data. In the figure, lg represents ${\log }_{10}$. Orange lines represent the fiducial values of parameters. Dark and light red denote the $1\sigma$ and $2\sigma$ confidence intervals. The adopted prior setting and assumptions are detailed in the main text. We fix the standard cosmology parameters ${\omega }_b$, ${\omega }_{\mathrm{cdm}}$, $n_s$, $A_s$, $h$, ${\tau }_{\mathrm{reio}}$ according to Planck results, given our assumption of an unchanged prediction on the primordial scalar power spectrum, as is the case for our primary example.