Primordial gravitational waves assisted by cosmological scalar perturbations

Abstract

Primordial gravitational waves are a crucial prediction of inflation theory, and their detection through their imprints on the cosmic microwave background is actively being pursued. However, these attempts have not yet been successful. In this paper, we propose a novel approach to detect primordial gravitational waves by searching for a signal of second-order tensor perturbations. These perturbations were produced due to nonlinear couplings between the linear tensor and scalar perturbations in the early universe. We anticipate a blue-tilted tensor spectral index, and suggest that the tensor-to-scalar ratio can potentially be measured with high precision using a detector network composed of the ground-based Einstein Telescope and the space-borne LISA project on a decade timescale.

1 Introduction

Primordial gravitational waves, originated from quantized tensor modes of perturbed metric in the very early universe, are one of the most important predictions of cosmic inflation theory [1,2,3,4,5,6]. On large scales comparable to the whole scale of observable universe, imprints of primordial tensor perturbations on the cosmic microwave background (CMB) have been proposed before two decades [7,8,9,10], but have not been observed yet. Recent studies have established upper limits on the spectral amplitude of primordial tensor perturbations [11,12,13,14,15]. The tensor-to-scalar ratio has been shown to be less than 0.032 at the 95% confidence level, based on precise measurements of anisotropies and polarization in the CMB by the Planck satellite and BICEP/Keck Array [15].

Efforts have been made to detect primordial tensor perturbations on small scales, which are detectable by space-borne and ground-based gravitational-wave interferometers [16,17,18,19,20,21,22]. However, models of canonical single-field slow-roll inflation predict a red-tilted tensor spectrum, with the spectral index exhibiting a consistency relation of $n_t=-r/8$ [23]. This makes it particularly challenging for these detectors to measure such a spectrum. Further, a blue-tilted tensor spectrum would imply a violation of the null-energy condition in the effective field theory of single-field inflation models [24,25,26]. To generate a blue-tilted tensor spectrum, additional assumptions, such as higher-derivative operators [27] and strong deviations from single-field slow-roll [28, 29], are necessarily involved.

Considering the absence of measurements of primordial tensor perturbations on large scales and the difficulties in generating a blue-tilted tensor spectrum on small scales, it is important to give serious consideration to any new mechanisms that can enhance the tensor spectral amplitude without requiring extraordinary assumptions.

Our proposal suggests that during the early universe, the linear scalar perturbations could have modulated the primordial tensor perturbations, resulting in the production of second-order tensor perturbations with a significantly blue-tilted power spectrum. This anticipated signal can potentially be detected by ongoing and planned ground-based detectors such as the Advanced LIGO, Virgo and KAGRA (LVK) [30,31,32], ET [33] and Cosmic Explorer (CE) [34]. Furthermore, scalar perturbations are believed to contribute to the formation of primordial black holes (PBHs), which are considered as a viable candidate for dark matter [35, 36]. Additionally, they are expected to produce scalar-induced gravitational waves, which can be detected by planned space-borne detectors such as the Laser Interferometer Space Antenna (LISA) [37, 38], big bang observer (BBO) [39, 40], or Deci-hertz Interferometer Gravitational wave Observatory (DECIGO) [41, 42]. If both the modulated primordial and scalar-induced gravitational waves are detected simultaneously, it would provide valuable insights into the mechanism of cosmic inflation and the nature of dark matter.

This paper investigates the theory of second-order tensor perturbations and the possible multi-band measurements of modulated primordial and scalar-induced gravitational waves using a future detector network consisting of the ground-based ET and the space-borne LISA. The main objective of this study is to achieve a high-precision measurement of the tensor-to-scalar ratio r with an accuracy of $\Delta {r}\sim \mathcal {O}(10^{-4})$, based on a fiducial model with $r=10^{-2}$ and a bumpy scalar power spectrum with amplitude $\mathcal {A}_{\zeta }=10^{-3}$.

2 Primordial tensor perturbations modulated by cosmological scalar perturbations

The perturbed Friedman–Robertson–Walker metric in the Newtonian gauge is $\textrm{d}s^{2} = a^{2}(\eta ) \{ -( 1+2\phi )\textrm{d}\eta ^{2} + [ (1-2\phi )\delta _{ij} + h_{ij} +\tilde{h}_{ij}/2 ] \textrm{d}x^{i}\textrm{d}x^{j} \}$, where $a(\eta )$ is the scale factor at the conformal time $\eta$, $\tilde{h}_{ij}$ denotes the second-order tensor perturbation sourced by the linear scalar perturbation $\phi$, and the linear tensor perturbation $h_{ij}$. The scalar perturbation in Fourier space is given by $\phi _{\textbf{k}}(\eta )=(2/3)\zeta _\textbf{k}T_{s}(k\eta )$, where $\zeta _{\textbf{k}}$ is the initial comoving curvature perturbation with power spectrum $\langle \zeta _{\textbf{k}}\zeta _{\textbf{k}^\prime }\rangle =(2\pi ^{2}/k^{3})\mathcal {P}_s(k)\delta (\textbf{k}+\textbf{k}^\prime )$, and the scalar transfer function during the radiation-dominated era is $T_{s}(k\eta )=3(\sin {\frac{k\eta }{\sqrt{3}}}/\frac{k\eta }{\sqrt{3}}-\cos {\frac{k\eta }{\sqrt{3}}})/(\frac{k\eta }{\sqrt{3}})^{2}$ [43]. The tensor perturbation in Fourier space is decomposed into two components, i.e., $h_{\textbf{k},ij}=h^{+}_{\textbf{k}}\epsilon ^+_{\textbf{k},ij}+h^{\times }_{\textbf{k}}\epsilon ^{\times }_{\textbf{k},ij}$, where the polarization tensors are defined as $\epsilon ^+_{\textbf{k},ij} = ( e_i e_j - \overline{e}_i \overline{e}_j ) /\sqrt{2}$ and $\epsilon ^\times _{\textbf{k},ij} = ( e_i \overline{e}_j + \overline{e}_i e_j ) /\sqrt{2}$ with $e_{i}$ and $\overline{e}_{i}$ being orthonormal vectors that are transverse to $\textbf{k}$. It is given by $h^{\lambda }_{\textbf{k}}(\eta )=H^{\lambda }_{\textbf{k}}T_{t}(k\eta )\ (\lambda =+,\times )$, where $H^{\lambda }_{\textbf{k}}$ is the initial tensor perturbation with the power spectrum $\langle H^{\lambda }_{\textbf{k}}H^{\lambda ^\prime }_{\textbf{k}^\prime } \rangle =(2\pi ^{2}/k^{3})\mathcal {P}_{t}(k)\delta ^{\lambda \lambda ^\prime }\delta (\textbf{k}+\textbf{k}^\prime )$ and the tensor transfer function is $T_{t}(k\eta )=\sin (k\eta )/(k\eta )$ [43]. Similarly, we decompose the second-order tensor perturbation in Fourier space into two polarization components, and further decompose each component into three terms, i.e., $\tilde{h}_{\textbf{k}}^{\lambda }=\tilde{h}_{\textbf{k}}^{\lambda }{}^{ss}+\tilde{h}_{\textbf{k}}^{\lambda }{}^{st}+\tilde{h}_{\textbf{k}}^{\lambda }{}^{tt}$, where the superscripts ${}^{s}$ and ${}^{t}$ stand for contributions from the linear scalar and tensor perturbations, respectively.

Expanding the Einstein field equations up to second order using the xPand [44] package, we derive the equation of motion for the second-order tensor perturbation. The evolution of $\tilde{h}^{\lambda \alpha \beta }_{\textbf{k}}$ with $\alpha \beta =ss,st,tt$ is governed by

$$\begin{aligned} \ddot{\tilde{h}}_{\textbf{k}}^{\lambda }{}^{\alpha \beta } + 2 \mathcal {H}\dot{\tilde{h}}_{\textbf{k}}^{\lambda }{}^{\alpha \beta } + k^2 \tilde{h}_{\textbf{k}}^{\lambda }{}^{\alpha \beta } = 4 \mathcal {S}_{\textbf{k}}^{\lambda }{}^{\alpha \beta }, \end{aligned}$$

(1)

where an overdot denotes a derivative with respect to $\eta$, $\mathcal {H}=\dot{a}/a$ is the comoving Hubble parameter, and $\mathcal {S}_{\textbf{k}}^{\lambda }{}^{\alpha \beta }$, as formulated in Eqs. (A.1a–A.1c), is the source term for $\tilde{h}^{\lambda \alpha \beta }_{\textbf{k}}$.

We solve Eq. (1) with the Green’s function method and obtain $\tilde{h}_{\textbf{k}}\propto \int ^{\eta }d\tilde{\eta }\sin (k\eta -k\tilde{\eta })[a(\tilde{\eta })/a(\eta )]\mathcal {S}_{\textbf{k}}(\tilde{\eta })$ [45, 46], where $a(\eta )\propto \eta$ in the radiation-dominated universe. The power spectrum of gravitational waves is defined as the two-point correlation function, i.e.,

$$\begin{aligned} \langle \tilde{h}_\textbf{k}^{\lambda \alpha \beta } \tilde{h}_{\textbf{k}'}^{\lambda '\alpha \beta } \rangle = \frac{2\pi ^2}{k^3} \mathcal {P}^{\alpha \beta }_{\tilde{h}}(k)\ \delta ^{\lambda \lambda '} \delta (\textbf{k}+ \textbf{k}'), \end{aligned}$$

(2)

where $\langle \ldots \rangle$ denotes the ensemble average. The dimensionless energy-density spectrum of the second-order tensor perturbations, i.e., the energy density per logarithmic frequency normalized with the critical energy density of the early universe, is given by [47]

$$\begin{aligned} \Omega _\textrm{gw}^{\alpha \beta }(\eta ,k) =\frac{1}{24} \left( \frac{k}{\mathcal {H}} \right) ^2 \overline{\mathcal {P}^{\alpha \beta }_{\tilde{h}}(\eta , k)}, \end{aligned}$$

(3)

where the overbar denotes the oscillation average and the two polarization modes have been summed over. After tedious but straightforward calculations, we obtain

$$\begin{aligned} \Omega _\textrm{gw}^{\alpha \beta }(\eta ,k) = \int _0^{\infty } \textrm{d}u \int _{|1-u |} ^{|1+u |} \textrm{d}v \ \{\ldots \}_{\alpha \beta } \mathcal {P}_{\alpha }(uk) \mathcal {P}_{\beta }(vk),\nonumber \\ \end{aligned}$$

(4)

where $\{\ldots \}_{\alpha \beta }$ is a function of u and v, and the explicit expression of $\Omega _\textrm{gw}^{\alpha \beta }$ is formulated in Eqs. (A.15a–A.15c), where the limit $k\eta \rightarrow \infty$ has been used, implying that the tensor perturbations are deeply within the horizon. The total spectrum is $\Omega _{\textrm{gw}}=\Omega _{\textrm{gw}}^{ss}+\Omega _{\textrm{gw}}^{st}+\Omega _{\textrm{gw}}^{tt}$. Since the energy density of gravitational waves decays as radiation, the present-day physical energy-density spectrum for the second-order tensor perturbations is approximated by [48]

$$\begin{aligned} h^2\Omega _\textrm{gw,0}^{\alpha \beta }(k)=h^2\Omega _\textrm{r,0}\times \Omega _\textrm{gw}^{\alpha \beta }(\eta ,k), \end{aligned}$$

(5)

where the corresponding one for photons and neutrinos is $h^2\Omega _\textrm{r,0}=4.15\times 10^{-5}$, with h being the dimensionless Hubble constant [49].

Before delving into the precision of detection, we present a featured asymptotic behavior of $\Omega _{\textrm{gw}}^{st}$ in the following. In particular, we remind that the scalar power spectrum on large scales follows a power-law with amplitude $\mathcal {A}_{\zeta ,0.05}\simeq 2.1\times 10^{-9}$ and index $n_{s}\simeq 0.96$ at the pivot scale $k_{p}=0.05\ \textrm{Mpc}^{-1}$ [49]. However, the formation of primordial black holes necessitates an enhanced scalar spectral amplitude of $\sim 10^{-2}$ on small scales (see Ref. [50] for a review). We model the scalar power spectrum on small scales as a normal distribution of $\ln k$ with mean $k_\zeta$, standard deviation $\sigma _\zeta$ and spectral amplitude $A_\zeta$ at the scale $k_{\zeta }$, i.e., [51]

$$\begin{aligned} \mathcal {P}_{s}(k)=\frac{\mathcal {A}_{\zeta }}{\sqrt{2\pi }\sigma _{\zeta }} \exp \left[ {-\frac{\ln ^{2} (k/k_\zeta )}{2 \sigma ^2_{\zeta }}}\right] . \end{aligned}$$

(6)

On the other hand, we assume that the tensor power spectrum follows a sudden-broken power-law distribution of k throughout the entire scale, i.e.,

$$\begin{aligned} \mathcal {P}_{t}(k) = r \mathcal {A}_{\zeta ,0.05} \left( \frac{k}{k_{p}}\right) ^{n_{t}} \Theta \left( k_{\textrm{reh}}-k\right) , \end{aligned}$$

(7)

where r and $n_{t}$ represent the tensor-to-scalar ratio and tensor spectral index, respectively, $k_{\textrm{reh}}$ is the high-frequency end of the spectrum due to reheating at the end of inflation, and $\Theta (x)$ is the Heaviside function with variable x. In models of canonical single-field slow-roll inflation, the consistency relation $n_{t}=-r/8$ holds [23]. The current upper bound on the tensor-to-scalar ratio is $r<0.032$ at the 95% confidence level [15], indicating a slightly red-tilted tensor spectrum. The reheating frequency $f_{\textrm{reh}}=k_{\textrm{reh}}/(2\pi )$ is related to the reheating temperature $T_{\textrm{reh}}$ and the effective number of relativistic degrees of freedom $g_{*,\textrm{reh}}$ during reheating, with $f_\textrm{reh}\simeq 0.027 \ \textrm{Hz}\ (T_\textrm{reh}/10^6\textrm{GeV})\ (g_\mathrm {*,reh}/106.75)^{1/6}$ [43]. Noticing that the contribution from $g_{*,\textrm{reh}}$ may be negligible due to the small value of the power-law index, thus the reheating frequency is approximately determined by the reheating temperature.

Fig. 1

Present-day physical energy-density spectra $h^2 \Omega _\textrm{gw,0}^{ss}$ (dashed lines) and $h^2 \Omega _\textrm{gw,0}^{st}$ (solid lines) for $\sigma _\zeta \rightarrow 0$ (blue), $\sigma _\zeta =0.5$ (red) and $\sigma _\zeta =1$ (green). The vertical lines from left to right denote $f_\textrm{reh}=27/270/2700$ Hz. Other parameters are given as $\mathcal {A}_{\zeta }=10^{-3}$, $f_\zeta =2.7$ mHz, $r=0.01$ and $n_t=-r/8$. The shaded regions show the sensitivities of LISA (orange), LIGO (purple) and ET (blue). The horizontal short line (black) denotes the upper limit of $h^2 \Omega _\textrm{gw,0}(25 \textrm{Hz})$ for LIGO O3, using the power-law model marginalizing over the spectral index with a log-uniform prior [52]

Figure 1 demonstrates that $\Omega _{\textrm{gw},0}^{st}(k)\propto k^{2+n_{t}}$ as $k_{\zeta }\ll k < k_{\textrm{reh}}$. The enhancement results from the leading term $q^2\phi _{\textbf{k}-\textbf{q}}h_\textbf{q}^{\lambda _1}$ of the source $\mathcal {S}_\textbf{k}^{\lambda st}$ (see Eq. (A.1b)) in the limit $|\textbf{k}-\textbf{q}|\ll q\approx k$. On the one hand, for larger momentum q of linear tensor perturbations, the source term $q^2\phi _{\textbf{k}-\textbf{q}}h_\textbf{q}^{\lambda _1}$ can be significantly enhanced by the factor $q^2$. On the other hand, for smaller momentum $|\textbf{k}-\textbf{q}|$ of linear scalar perturbation, considering $T_s(|\textbf{k}-\textbf{q}| \eta )\sim 1/(|\textbf{k}-\textbf{q}| \eta )^2$ within the horizon in the radiation-dominated era, the scalar perturbation decays slower and thus keeps the source term $q^2\phi _{\textbf{k}-\textbf{q}}h_\textbf{q}^{\lambda _1}$ important for a longer time to induce $\tilde{h}^{\lambda st}_\textbf{k}$. To make some rough estimates, we have the leading term $\{\ldots \}_{st}\propto 1/u^{4}$ approximately in the limit $u=|\textbf{k}-\textbf{q}|/k \rightarrow 0$ and $v=|\textbf{q}|/k\rightarrow 1$. For simplicity, we take the limit $\sigma _{\zeta }\rightarrow 0$ and get the scalar spectrum $\mathcal {P}_{s}(k)=\mathcal {A}_{\zeta } \delta (\ln (k/k_{\zeta }))$, therefore, the energy-density spectrum can be approximated as $\Omega _{\textrm{gw}}^{st}(k)\propto \int du \int dv \ u^{-4}\ \delta [\ln (uk/k_{\zeta })]\ k^{n_t}\propto k^{n_t} u^{-2}|_{u=k_\zeta /k} \propto k^{2+n_t}$, where $\int dv$ has been replaced with the integral width 2u. The spectral index $(2+n_t)$ remains unchanged for different values of $\sigma _{\zeta }$, while the spectral amplitude varies. Further, we can simply use $\Omega _\textrm{gw}^{st}(k) \simeq \textrm{few}\times r \mathcal {A}_{\zeta ,0.05} \mathcal {A}_{\zeta } (k/k_{\zeta })^2 (k/k_p)^{n_t} \Theta (k_\textrm{reh}-k)$ in $k\gg k_\zeta$ region for a good order estimate. The null-energy condition is not violated by this blue-tilted spectrum since second-order gravitational waves were produced during the radiation-dominated era, not the inflationary stage.

We compare physical energy-density spectra of second-order tensor perturbations (as functions of frequency) with sensitivity curves of LISA, LIGO, and ET in Fig. 1. The scalar-induced tensor perturbations with $\Omega _{\textrm{gw},0}^{ss}(k)$ have been semi-analytically studied in the literature [45, 46, 53, 54]. Due to $r<0.032$, the amplitude of $\Omega _{\textrm{gw},0}^{tt}(k)$ is too small to fit the scope of Fig. 1. However, the blue-tilted $\Omega _{\textrm{gw},0}^{st}(k)$ makes it promising to measure primordial tensor perturbations (r and $n_{t}$) and reheating physics ($T_{\textrm{reh}}$) with high-frequency gravitational-wave detectors. Therefore, we expect that multi-band measurements of second-order tensor perturbations may lead to a better understanding of the late-time stage of inflation.

3 Expected sensitivity of gravitational-wave detectors to measure the anticipated signal

We perform Fisher-matrix forecasts by considering instrumental uncertainties for detector networks composed of space-borne LISA and ground-based LIGO or ET. The Fisher matrix for second-order tensor perturbations is given by

$$\begin{aligned} F_{ab} = \sum _{i=1}^{N} T_{i} \epsilon _{i} \int df \frac{\partial _{\theta _{a}}\Omega _{\textrm{gw,0}}(k) \ \partial _{\theta _{b}}\Omega _{\textrm{gw,0}}(k)}{\Omega ^{2}_{n,i}(f)}, \end{aligned}$$

(8)

where $f=k/(2\pi )$ is the frequency of gravitational waves, $\theta =\{\ln \mathcal {A}_{\zeta },\sigma _{\zeta },\ln f_{\zeta },r,n_{t},\ln f_{\textrm{reh}}\}$ is the parameter space being determined, $\Omega _{n}(f)$ denotes the effective detector noise as a function of f, as summarized in Ref. [55], N is the number of independent detectors, T is the observing time, and $\epsilon$ is the duty circle. For LISA, we consider a single detector with 75% duty circle during a four-year observation. For LIGO (ET), we consider two (three) independent detectors with 100% duty circle during a four-year (one-year) observation. The fiducial parameters are $\mathcal {A}_{\zeta }=10^{-3}$, $\sigma _{\zeta }=0.5$, $f_{\zeta }=2.7$ mHz, $r=0.01$, $n_{t}=-r/8$, and $f_{\textrm{reh}}=27/270/2700$ Hz. The corresponding spectra have been shown in Fig. 1.

Table 1 The $1\sigma$ confident uncertainties of r, $n_t$ and $\ln {f_\textrm{reh}}$ measured by LIGO and ET for $f_\textrm{reh}=27/ 270/2700$ Hz

Fig. 2

Cross-correlations between r and $n_t$ measured by ET for $f_\textrm{reh}=27(\textrm{blue})/270(\textrm{red})/2700(\textrm{green})$ Hz. Dark and light shaded contours stand for the $1\sigma$ and $2\sigma$ confident regions, respectively. The fiducial model with $r=0.01$ and $n_t=-r/8$ (other parameters are marginalized) is marked as a star

Though multi-band measurements are performed with detector networks, the parameters of the scalar spectrum in Eq. (6) are completely determined by LISA. The results are given as $\Delta \ln \mathcal {A}_{\zeta }=7.5\times 10^{-3}$, $\Delta \sigma _{\zeta }=6.0\times 10^{-3}$, and $\Delta \ln f_{\zeta }=3.9\times 10^{-3}$, indicating (sub)percent-level measurements. On the other hand, the parameters of the tensor spectrum in Eq. (7) are completely determined by LIGO and ET. For our fiducial model, LIGO could achieve $\Delta r/r \sim \mathcal {O}(1)$ and $\Delta n_{t}\sim \mathcal {O}(10^{-2})$, while ET, with better sensitivity than LIGO, could achieve $\Delta r/r \sim \mathcal {O}(10^{-2})$ and $\Delta n_{t}\sim \mathcal {O}(10^{-4})$, allowing for more-than-$10\sigma$ confident measurements of the tensor-to-scalar ratio and a possibility to test the consistency relation $n_{t}=-r/8$ at the $2\sigma$ confidence level. The precision for measuring r and $n_{t}$ depends on the fiducial value of $f_{\textrm{reh}}$, as shown in Table 1. For higher reheating frequency, which implies wider frequency band being captured by LIGO and ET, we expect better precision for measurements of r and $n_{t}$. Figure 2 shows the marginalized $1\sigma$ and $2\sigma$ cross-correlations between r and $n_{t}$, as well as their dependence on $f_{\textrm{reh}}$. In addition, the best measurement of $f_{\textrm{reh}}$ can be performed when $f_{\textrm{reh}}$ coincides with the most sensitive frequency band of detectors, which is given as $\sim \mathcal {O}(10^{2})$ Hz for LIGO and ET. Therefore, we expect the best precision to be $\Delta \ln f_{\textrm{reh}}\sim \mathcal {O}(10^{-4})$ for LIGO and $\Delta \ln f_{\textrm{reh}}\sim \mathcal {O}(10^{-6})$ for ET. If such a measurement works in the best case, our results may provide meaningful insights for particle physics, as the reheating temperature is $\sim \mathcal {O}(10^{10})$ GeV.

To enhance the detectability of primordial tensor perturbations, our results can be further improved if using fiducial models that anticipate larger amplitudes for $\Omega _{\textrm{gw},0}^{st}(k)$. This could be achieved, for example, by enhancing the amplitude of the scalar or tensor spectrum, or both, as $\Omega _{\textrm{gw},0}^{st}\propto r \mathcal {A}_{\zeta }$. In particular, LIGO could potentially measure primordial tensor perturbations by setting the fiducial value to be $\mathcal {A}_{\zeta }\sim 10^{-2}$, which is related to an interesting topic of the formation of PBHs [50]. Other alternatives include increasing the bump width of the scalar spectrum, indicating a larger value for $\sigma _{\zeta }$, or decreasing the peak frequency of the scalar spectrum, indicating a smaller value for $f_{\zeta }$, etc.

4 Conclusion

In the early universe, the linear tensor perturbations were modulated with bump-spectral scalar perturbations to produce second-order tensor perturbations. The resulting tensor spectral index was found to be $(2+n_{t})$, which may have a significant blue tilt. Currently, plans are underway to develop next-generation ground-based gravitational-wave detectors that could provide accurate measurements of the tensor-to-scalar ratio within the next decade. However, such measurements require the existence of both inflationary tensor perturbations and linear scalar perturbations with a bumpy power spectrum, making it difficult to discuss their specifics until the measurements are completed. If future multi-band measurements are able to detect the anticipated signal of second-order tensor perturbations, it could provide valuable insights into the physics of cosmic inflation and help constrain inflation models. While scientists are actively pursuing measurements of CMB B-mode polarization (see review in Ref. [56]), our proposal offers an alternative approach to accurately measure primordial tensor perturbations.

Appendix A: The Calculations of \(\Omega _\textrm{gw}^{\alpha \beta }\)

The key result in the main text is based on the calculations of $\Omega _\textrm{gw}^{\alpha \beta }$ ($\alpha \beta =ss,st,tt$), with the detailed calculations presented in this Appendix. The calculation of $\Omega _\textrm{gw}^{ss}$ has been extensively addressed in prior studies, e.g., Refs. [45, 46, 53, 54]. Previous works related to $\Omega _\textrm{gw}^{st}$ and $\Omega _\textrm{gw}^{tt}$ can be found in Refs. [57, 58]. Ref. [57] provided the expression of $\tilde{h}_\textbf{k}^{\lambda \alpha \beta }$ in the comoving and zero-shear gauge. Ref. [58] studied $\Omega _\textrm{gw}^{\alpha \beta }$ in Newtonian gauge with the monochromatic primordial power spectra $\mathcal {P}_s$ and $\mathcal {P}_t$ at the same pivot scale $k_*$, and the additional peaked structure of the energy density spectrum in $k>k_*$ region were noticed. In addition to the analogous calculations on $\Omega _\textrm{gw}^{\alpha \beta }$, our work, for the first time, provides the semi-analytical expressions for $\Omega _\textrm{gw}^{st}$ and $\Omega _\textrm{gw}^{tt}$ in Newtonian gauge, as given in Eqs. (A.15b–A.15c). We propose a novel approach to detect primordial tensor perturbations based on the non-trivial blue spectrum $\Omega _\textrm{gw}^{st}(k)\propto k^{2+n_t}$ in high frequencies. Furthermore, we clarify how this enhancement arises from the leading source term $\sim q^2\phi _{\textbf{k}-\textbf{q}}h_\textbf{q}^{\lambda _1}$.

After our paper was posed on arXiv in March 2023, a subsequent study [59] (appeared in July 2023) calculated $\Omega _\textrm{gw}^{st}$. Since the formulation of the perturbed metric in Ref. [59] differs slightly from the commonly used Newtonian gauge, they gave different the source terms of $\tilde{h}_{\textbf{k}}^{\lambda }{}^{st}$ compared to ours. However, the leading source term $\sim q^2\phi _{\textbf{k}-\textbf{q}}h_\textbf{q}^{\lambda _1}$ remains consistent.

Below, we present a detailed calculation of $\Omega _\textrm{gw}^{\alpha \beta }$ in the radiation-dominated era, and discuss the different behaviors between $\Omega _\textrm{gw}^{ss}$ and $\Omega _\textrm{gw}^{st}$ in the large- and small-momentum coupling, which may be helpful to understanding the enhancement of $\Omega _\textrm{gw}^{st}$ in high frequencies.

Given the settings in the main text, the explicit expression of the source term $\mathcal {S}_{\textbf{k}}^{\lambda }{}^{\alpha \beta }$ in Eq. (1) is obtained to be

$$\begin{aligned} \mathcal {S}_{\textbf{k}}^\lambda {}^{ss}&=\epsilon ^{\lambda ,lm}_{\textbf{k}}\int \frac{\textrm{d}^3 \textbf{q}}{(2\pi )^{3/2}}\ q_l q_m \Big [ 2\phi _{\textbf{k}-\textbf{q}}\phi _{\textbf{q}} \nonumber \\&\quad + \left( \eta \dot{\phi }_{\textbf{k}-\textbf{q}}+\phi _{\textbf{k}-\textbf{q}}\right) \left( \eta \dot{\phi }_{\textbf{q}}+\phi _{\textbf{q}}\right) \Big ] ,\end{aligned}$$

(A.1a)

$$\begin{aligned} \mathcal {S}_\textbf{k}^{\lambda }{}^{st}&= \epsilon ^{\lambda ,lm}_{\textbf{k}} \int \frac{\textrm{d}^3 \textbf{q}}{(2\pi )^{3/2}}\ \epsilon _{\textbf{q},lm}^{\lambda _1} \Big [ -3\ddot{\phi }_{\textbf{k}-\textbf{q}} h_{\textbf{q}}^{\lambda _1} -\frac{10}{\eta } \dot{\phi }_{\textbf{k}-\textbf{q}} h_{\textbf{q}}^{\lambda _1} \nonumber \\&\quad -\frac{1}{3}(5k^{2}+5q^{2}-4k^{c}q_{c}) \phi _{\textbf{k}-\textbf{q}}h_{\textbf{q}}^{\lambda _1} \Big ] ,\end{aligned}$$

(A.1b)

$$\begin{aligned} \mathcal {S}_{\textbf{k}}^{\lambda }{}^{tt}&= \frac{\epsilon ^{\lambda ,lm}_{\textbf{k}}}{2} \int \frac{\textrm{d}^3 \textbf{q}}{(2\pi )^{3/2}}\ \bigg \{\epsilon ^{\lambda _1,b}_{\textbf{k}-\textbf{q},l} \epsilon ^{\lambda _2}_{\textbf{q},bm} {k^2} \dot{h}_{\textbf{k}-\textbf{q}}^{\lambda _1} \dot{h}_{\textbf{q}}^{\lambda _2} \nonumber \\&\quad + \Big [ \epsilon ^{\lambda _1,b}_{\textbf{k}-\textbf{q},l} \epsilon ^{\lambda _2}_{\textbf{q},bm} {(k^{c} q_{c}-q^{2})} - 2 \epsilon ^{\lambda _1,bc}_{\textbf{k}-\textbf{q}} \epsilon ^{\lambda _2}_{\textbf{q},bm} \ q_c q_l \nonumber \\&\quad - \epsilon ^{\lambda _1}_{\textbf{k}-\textbf{q},mc} \epsilon ^{\lambda _2}_{\textbf{q},bl} \ (k^b-q^b) q^c - \epsilon ^{\lambda _1}_{\textbf{k}-\textbf{q},bc} \epsilon ^{\lambda _2}_{\textbf{q},lm} \ q^b q^c \nonumber \\&\quad - \frac{1}{2} \epsilon ^{\lambda _1,bc}_{\textbf{k}-\textbf{q}} \epsilon ^{\lambda _2}_{\textbf{q},bc} \ q^l q^m \Big ] h_{\textbf{k}-\textbf{q}}^{\lambda _1} h_{\textbf{q}}^{\lambda _2}\bigg \} . \end{aligned}$$

(A.1c)

To establish a contact with primordial perturbations during the inflationary stage, we rewrite the source term $\mathcal {S}_{\textbf{k}}^{\lambda }{}^{\alpha \beta }$ above in the form of

$$\begin{aligned} \mathcal {S}_{\textbf{k}}^{\lambda ss}&= \int \frac{\textrm{d}^3 \textbf{q}}{(2\pi )^{3/2}}\ Q_{ss}^{\lambda }(\textbf{k},\textbf{q}) \ k^2 f_{ss}(|\textbf{k}-\textbf{q}|,q,\eta )\ \zeta _{\textbf{k}-\textbf{q}}\zeta _{\textbf{q}}, \end{aligned}$$

(A.2a)

$$\begin{aligned} \mathcal {S}_{\textbf{k}}^{\lambda st}&=\int \frac{\textrm{d}^3 \textbf{q}}{(2\pi )^{3/2}}\ Q_{st}^{\lambda \lambda _1}(\textbf{k},\textbf{q}) \ k^2 f_{st}(|\textbf{k}-\textbf{q}|,q,\eta )\ \zeta _{\textbf{k}-\textbf{q}} H_\textbf{q}^{\lambda _1}, \end{aligned}$$

(A.2b)

$$\begin{aligned} \mathcal {S}_{\textbf{k}}^{\lambda tt}&= \int \frac{\textrm{d}^3 \textbf{q}}{(2\pi )^{3/2}} \left( \sum _{i=1}^{5} Q_{tt,i}^{\lambda \lambda _1 \lambda _2}(\textbf{k},\textbf{q}) \ k^2 f_{tt,i}(|\textbf{k}-\textbf{q}|,q,\eta ) \right) \nonumber \\&\quad \times H_{\textbf{k}-\textbf{q}}^{\lambda _1}H_\textbf{q}^{\lambda _2}. \end{aligned}$$

(A.2c)

The projection factor $Q_{\alpha \beta }(\textbf{k},\textbf{q})$ in Eq. (A.2) describes the geometric relations between the momenta and polarization tensors of the linear perturbations, i.e.,

$$\begin{aligned} Q_{ss}^{\lambda }(\textbf{k},\textbf{q})&= \epsilon ^{\lambda ,lm}_\textbf{k}\ q_l q_m / k^2, \end{aligned}$$

(A.3a)

$$\begin{aligned} Q_{st}^{\lambda \lambda _1}(\textbf{k},\textbf{q})&= \epsilon ^{\lambda ,lm}_\textbf{k}\ \epsilon ^{\lambda _1}_{\textbf{q},lm}, \end{aligned}$$

(A.3b)

$$\begin{aligned} Q_{tt,1}^{\lambda \lambda _1 \lambda _2}(\textbf{k},\textbf{q})&=\ \epsilon ^{\lambda ,lm}_\textbf{k}\ \epsilon ^{\lambda _1,b}_{\textbf{k}-\textbf{q},l}\ \epsilon ^{\lambda _2}_{\textbf{q},bm}, \nonumber \\ Q_{tt,2}^{\lambda \lambda _1 \lambda _2}(\textbf{k},\textbf{q})&=- \epsilon ^{\lambda ,lm}_\textbf{k}\epsilon ^{\lambda _1,bc}_{\textbf{k}-\textbf{q}}\ \epsilon _{\textbf{q},bm}^{\lambda _2}\ (k-q)_l q_c/k^2\nonumber \\&\quad +\Big ( (\textbf{k}-\textbf{q}) \leftrightarrow \textbf{q}\ \textrm{term} \Big ),\nonumber \\ Q_{tt,3}^{\lambda \lambda _1 \lambda _2}(\textbf{k},\textbf{q})&=\ \epsilon ^{\lambda ,lm}_\textbf{k}\ \epsilon ^{\lambda _1}_{\textbf{k}-\textbf{q},mc}\ \epsilon _{\textbf{q},lb}^{\lambda _2}\ k^b k^c/k^2, \nonumber \\ Q_{tt,4}^{\lambda \lambda _1 \lambda _2}(\textbf{k},\textbf{q})&=\ \frac{1}{2}\ \epsilon ^{\lambda ,lm}_\textbf{k}\ \epsilon ^{\lambda _1}_{\textbf{k}-\textbf{q},bc}\ \epsilon ^{\lambda _2}_{\textbf{q},lm}\ k^b k^c/k^2\nonumber \\&\quad +\Big ( (\textbf{k}-\textbf{q}) \leftrightarrow \textbf{q}\ \textrm{term} \Big ), \nonumber \\ Q_{tt,5}^{\lambda \lambda _1 \lambda _2}(\textbf{k},\textbf{q})&=- \epsilon ^{\lambda ,lm}_\textbf{k}\ \epsilon ^{\lambda _1,bc}_{\textbf{k}-\textbf{q}}\ \epsilon _{\textbf{q},bc}^{\lambda _2}\ (k-q)_l q_m/k^2,\nonumber \\ \end{aligned}$$

(A.3c)

where $Q_{tt,i}^{\lambda \lambda _1 \lambda _2}$ is defined in a symmetric form with respect to $\textbf{k}-\textbf{q}$ and $\textbf{q}$, which facilitates the subsequent manipulation in Eq. (A.11) while keeping the integral in Eq. (A.12) unchanged. The source function $f_{\alpha \beta }(|\textbf{k}-\textbf{q}|,q,\eta )$ in Eq. (A.2) describes the time evolution of the linear perturbations, being given by

$$\begin{aligned} f_{ss}&=\frac{12}{u^2 v^2 x^6}\ \bigg \{ 18 uvx^2 \cos {\frac{ux}{\sqrt{3}}} \cos {\frac{vx}{\sqrt{3}}} \nonumber \\&\quad +[\ 54-6(u^2+v^2)x^2+u^2 v^2 x^4\ ] \sin {\frac{ux}{\sqrt{3}}} \sin {\frac{vx}{\sqrt{3}}} \nonumber \\&\quad +2\sqrt{3}\ ux(v^2 x^2-9) \cos {\frac{ux}{\sqrt{3}}} \sin {\frac{vx}{\sqrt{3}}} \nonumber \\&\quad +2\sqrt{3}\ vx(u^2 x^2-9) \sin {\frac{ux}{\sqrt{3}}} \cos {\frac{vx}{\sqrt{3}}} \bigg \}, \end{aligned}$$

(A.4a)

$$\begin{aligned} f_{st}&= -\frac{1}{3 u^3 v x^6}\ \bigg \{ \left[ ux^3 \left( u^2{-}3 v^2{-}3\right) {-}18\ u x \right] \cos {\frac{u x}{\sqrt{3}}}\ \sin {v x} \nonumber \\&\quad + \left[ \ 3 \sqrt{3}x^2 \left( -u^2+v^2+1\right) +18 \sqrt{3}\ \right] \sin {\frac{u x}{\sqrt{3}}}\ \sin {v x} \bigg \}, \end{aligned}$$

(A.4b)

$$\begin{aligned} f_{tt,1}&= -\frac{\left( u^2+v^2-3\right) \ \sin {u x} \ \sin {v x}}{4 u v x^2}, \nonumber \\ f_{tt,2}&=f_{tt,3}=f_{tt,4}=2f_{tt,5} =-\frac{\sin {ux}\ \sin {vx}}{2uvx^2}. \end{aligned}$$

(A.4c)

Note that we have substituted the variables $|\textbf{k}-\textbf{q}|$, q, and $\eta$ with corresponding dimensionless variables rescaled by k (i.e., $u= |\textbf{k}-\textbf{q}|/k$, $v= q/k$, and $x= k\eta$) in Eq. (A.4) and subsequent calculations. However, for simplicity, we still use the original names of the functions (i.e., $f_{\alpha \beta }(|\textbf{k}-\textbf{q}|,q,\eta )$ is substituted by $f_{\alpha \beta }(u,v,x)$).

We solve Eq. (1) following the Green’s function method. The Green’s function $G_\textbf{k}(\eta ,\overline{\eta })$ is defined as the solution of the equation

$$\begin{aligned} \frac{\partial ^2}{\partial \eta ^2}G_\textbf{k}(\eta ,\overline{\eta }) +\left( k^2-\frac{1}{a(\eta )}\frac{\partial ^2 a(\eta )}{\partial \eta ^2}\right) G_\textbf{k}(\eta ,\overline{\eta }) =\delta (\eta ,\overline{\eta }).\nonumber \\ \end{aligned}$$

(A.5)

In the radiation-dominated era with $a(\eta )\propto \eta$, the solution of Eq. (A.5) is

$$\begin{aligned} G_\textbf{k}(x,\overline{x}) =\sin {(x-\overline{x})}/k. \end{aligned}$$

(A.6)

Therefore, $\tilde{h}_{\textbf{k}}^{\lambda }{}^{\alpha \beta }$ in Eq. (1) can be written as

$$\begin{aligned} \tilde{h}_{\textbf{k}}^{\lambda }{}^{\alpha \beta }(\eta ) =4 \int ^{\eta }d\overline{\eta } \ \frac{a(\overline{\eta })}{a(\eta )} k G_\textbf{k}(\eta ,\overline{\eta }) \mathcal {S}_{\textbf{k}}^{\lambda }{}^{\alpha \beta }(\overline{\eta }). \end{aligned}$$

(A.7)

Substituting Eq. (A.2) into Eq. (A.7), we can recast $\tilde{h}_{\textbf{k}}^{\lambda }{}^{\alpha \beta }$ as

$$\begin{aligned} \tilde{h}_\textbf{k}^{\lambda ss}&=4 \int \frac{\textrm{d}^3 \textbf{q}}{(2\pi )^{3/2}}\ Q_{ss}^{\lambda }(\textbf{k},\textbf{q}) I_{ss}(|\textbf{k}-\textbf{q}|,q,\eta )\ \zeta _{\textbf{k}-\textbf{q}}\zeta _{\textbf{q}}, \end{aligned}$$

(A.8a)

$$\begin{aligned} \tilde{h}_\textbf{k}^{\lambda st}&=4 \int \frac{\textrm{d}^3 \textbf{q}}{(2\pi )^{3/2}}\ Q_{st}^{\lambda \lambda _1}(\textbf{k},\textbf{q}) I_{st}(|\textbf{k}-\textbf{q}|,q,\eta )\ \zeta _{\textbf{k}-\textbf{q}} H_\textbf{q}^{\lambda _1}, \end{aligned}$$

(A.8b)

$$\begin{aligned} \tilde{h}_\textbf{k}^{\lambda tt}&=4 \int \frac{\textrm{d}^3 \textbf{q}}{(2\pi )^{3/2}} \left( \sum _{i=1}^{5} Q_{tt,i}^{\lambda \lambda _1 \lambda _2}(\textbf{k},\textbf{q}) I_{tt,i}(|\textbf{k}-\textbf{q}|,q,\eta ) \right) \nonumber \\&\quad \times H_{\textbf{k}-\textbf{q}}^{\lambda _1}H_\textbf{q}^{\lambda _2}, \end{aligned}$$

(A.8c)

where the kernel function $I_{\alpha \beta }(|\textbf{k}-\textbf{q}|,q,\eta )$ is defined as

$$\begin{aligned} I_{\alpha \beta }(|\textbf{k}-\textbf{q}|,q,\eta )= & {} \int _0^{\eta }\textrm{d}\overline{\eta }\ \frac{a(\overline{\eta })}{a(\eta )} \ k G_\textbf{k}(\eta ,\overline{\eta })\nonumber \\{} & {} \times f_{\alpha \beta }(|\textbf{k}-\textbf{q}|,q,\eta ). \end{aligned}$$

(A.9)

The kernel function $I_{\alpha \beta }(|\textbf{k}-\textbf{q}|,q,\eta )$ encodes the time evolution of the second-order tensor perturbation $\tilde{h}_\textbf{k}^{\lambda \alpha \beta }$, where $a(\overline{\eta })/a(\eta )$ describes the red-shift effect due to the expansion of the Universe, $G_\textbf{k}(\eta ,\overline{\eta })$ describes the propagation of second-order tensor perturbation, and $f_{\alpha \beta }(|\textbf{k}-\textbf{q}|,q,\eta )$ describes the evolution of the source terms. Substituting Eqs. (A.4) and (A.6) into Eq. (A.9), we obtain the explicit expression of the kernel function $I_{\alpha \beta }$, namely,

$$\begin{aligned}&I_{ss}(u,v,x) \nonumber \\&\quad = \frac{3\cos {x}}{4u^3v^3 x} \ \bigg \{ (u^2+v^2-3)^2 \bigg [ -\textrm{Si} \left( \left( 1-\frac{v-u}{\sqrt{3}} \right) x \right) \nonumber \\&\qquad +\textrm{Si} \left( \left( 1-\frac{v+u}{\sqrt{3}} \right) x \right) -\textrm{Si} \left( \left( 1+\frac{v-u}{\sqrt{3}} \right) x \right) \nonumber \\&\qquad +\textrm{Si} \left( \left( 1+\frac{v+u}{\sqrt{3}} \right) x \right) \bigg ] \bigg \} \nonumber \\&\qquad -\frac{3\sin {x}}{4u^3v^3 x} \bigg \{ 4u v(u^2+v^2-3) -(u^2+v^2-3)^2 \nonumber \\&\qquad \times \bigg [ \textrm{Ci} \left( \left( 1-\frac{v-u}{\sqrt{3}} \right) x \right) -\textrm{Ci} \left( \left| 1- \frac{v+u}{\sqrt{3}} \right| x \right) \nonumber \\&\qquad +\textrm{Ci} \left( \left( 1+\frac{v-u}{\sqrt{3}} \right) x \right) -\textrm{Ci} \left( \left( 1+\frac{v+u}{\sqrt{3}} \right) x \right) \nonumber \\&\qquad +\ln \left( \left| \frac{3-(u+v)^2}{3-(u-v)^2}\right| \right) \bigg ] \bigg \} +\mathcal {O}\left( \frac{1}{x^2} \right) , \end{aligned}$$

(A.10a)

$$\begin{aligned}&I_{st}(u,v,x)\nonumber \\&\quad =\frac{\cos {x}}{24u^3vx} \bigg \{ \sqrt{3} \left[ u^2-3 \left( v-1\right) ^2\right] \nonumber \\&\qquad \times \left[ u^2-3 \left( v+1\right) ^2\right] \bigg [\ \textrm{Si} \left( \left( 1-\frac{u}{\sqrt{3}}+v \right) x \right) \nonumber \\&\qquad -\textrm{Si} \left( \left( 1- \frac{u}{\sqrt{3}}-v \right) x \right) -\textrm{Si} \left( \left( 1+ \frac{u}{\sqrt{3}}+v \right) x \right) \nonumber \\&\qquad +\textrm{Si} \left( \left( 1+ \frac{u}{\sqrt{3}}-v \right) x \right) \bigg ] \bigg \} +\frac{\sin {x}}{24u^3vx} \bigg \{ 4 u v \left[ u^2-9\right. \nonumber \\&\qquad \times \left. \left( v^2-1\right) \right] {+}\sqrt{3} \left[ u^2{-}3 \left( v{-}1\right) ^2\right] \left[ u^2-3 \left( v{+}1\right) ^2\right] \nonumber \\&\qquad \times \bigg [ \textrm{Ci} \left( \left( 1- \frac{u}{\sqrt{3}}+v \right) x \right) -\textrm{Ci} \left( \bigg |1- \frac{u}{\sqrt{3}}-v \bigg |x \right) \nonumber \\&\qquad -\textrm{Ci} \left( \left( 1+ \frac{u}{\sqrt{3}}+v \right) x \right) +\textrm{Ci} \left( \bigg |1+ \frac{u}{\sqrt{3}}-v \bigg |x \right) \nonumber \\&\qquad +\ln \left| \frac{3-(u+\sqrt{3}\ v)^2}{3-(u-\sqrt{3}\ v)^2}\right| \bigg ] \bigg \} +\mathcal {O}\left( \frac{1}{x^2} \right) , \end{aligned}$$

(A.10b)

$$\begin{aligned}&I_{tt,1}(u,v,x) = -\frac{\sin {x}}{4x}+\mathcal {O}\left( \frac{1}{x^2} \right) , \nonumber \\&I_{tt,2}(u,v,x)\nonumber \\&\quad = I_{tt,3}(u,v,x) =\ I_{tt,4}(u,v,x) =2 I_{tt,5}(u,v,x) \nonumber \\&\quad = \frac{\cos {x}}{8uvx} \bigg \{ \textrm{Si}\Big ((1-u+v)x\Big ) -\textrm{Si}\Big ((1-u-v)x\Big ) \nonumber \\&\qquad +\textrm{Si}\Big ((1+u-v)x\Big ) -\textrm{Si}\Big ((1+u+v)x\Big ) \bigg \} \nonumber \\&\qquad -\frac{\sin {x}}{8uvx}\ \bigg \{ \textrm{Ci}\Big ((1-u+v)x\Big ) +\textrm{Ci}\Big ((1+u-v)x\Big )\nonumber \\&\qquad -\textrm{Ci}\Big (\left| 1-u-v\right| x\Big ) -\textrm{Ci}\Big ((1+u+v)x\Big ) \nonumber \\&\qquad +\ln \left| {\frac{1-(u+v)^2}{1-(u-v)^2}}\right| \ \bigg \}\ +\mathcal {O}\left( \frac{1}{x^2} \right) , \end{aligned}$$

(A.10c)

where $\textrm{Si}(x)$ and $\textrm{Ci}(x)$ are defined as $\textrm{Si}(x)=\int _0^x \textrm{d}y\ (\sin {y}/y)$ and $\textrm{Ci}(x)=\ -\int _x^\infty \textrm{d}y\ (\cos {y}/y)$. In the limit of $x \rightarrow \infty$, the $\mathcal {O}(1/x^2)$ can be neglected and we can use $\lim _{x\rightarrow \infty } \textrm{Si}(Ax)=\textrm{sgn}(A)\pi /2$ and $\lim _{x\rightarrow \infty }\textrm{Ci}(Ax)=0$ to simplify the expressions in Eq. (A.10).

Then, we can calculate $\Omega _\textrm{gw}^{\alpha \beta }$. By neglecting the non-Gaussianity of the primordial linear perturbations, we have

$$\begin{aligned}&\Omega _\textrm{gw}^{ss} \propto \langle \zeta _{\textbf{k}-\textbf{q}} \zeta _{\textbf{q}} \zeta _{\textbf{k}'-\textbf{q}'} \zeta _{\textbf{q}'} \rangle \nonumber \\&\quad = \Big [ \delta (\textbf{q}+\textbf{q}') +\delta (\textbf{q}+\textbf{k}'-\textbf{q}') \Big ] \delta (\textbf{k}+\textbf{k}')\nonumber \\&\qquad \times \frac{2\pi ^2\mathcal {P}_s(|\textbf{k}-\textbf{q}|)}{|\textbf{k}-\textbf{q}|^3} \frac{2\pi ^2\mathcal {P}_s(q)}{q^3}, \end{aligned}$$

(A.11a)

$$\begin{aligned}&\Omega _\textrm{gw}^{st} \propto \langle \zeta _{\textbf{k}-\textbf{q}} H^{\lambda _1}_{\textbf{q}} \zeta _{\textbf{k}'-\textbf{q}'} H^{\lambda _1'}_{\textbf{q}'} \rangle \nonumber \\&\quad =\delta ^{\lambda _1 \lambda _1'} \delta (\textbf{q}+\textbf{q}') \delta (\textbf{k}+\textbf{k}')\ \frac{2\pi ^2\mathcal {P}_s(|\textbf{k}-\textbf{q}|)}{|\textbf{k}-\textbf{q}|^3} \frac{2\pi ^2\mathcal {P}_t(q)}{q^3}, \end{aligned}$$

(A.11b)

$$\begin{aligned}&\Omega _\textrm{gw}^{tt} \propto \langle H_{\textbf{k}-\textbf{q}}^{\lambda _1} H_{\textbf{q}}^{\lambda _2} H_{\textbf{k}'-\textbf{q}'}^{\lambda _1'} H_{\textbf{q}'}^{\lambda _2'} \rangle \nonumber \\&\quad = \left[ \delta ^{\lambda _1 \lambda _1'}\delta ^{\lambda _2 \lambda _2'} \delta (\textbf{q}+\textbf{q}') +\delta ^{\lambda _1 \lambda _2'}\delta ^{\lambda _2 \lambda _1'} \delta (\textbf{q}+\textbf{k}'-\textbf{q}') \right] \nonumber \\&\qquad \times \delta (\textbf{k}+\textbf{k}')\ \frac{2\pi ^2\mathcal {P}_t(|\textbf{k}-\textbf{q}|)}{|\textbf{k}-\textbf{q}|^3} \frac{2\pi ^2\mathcal {P}_t(q)}{q^3}. \end{aligned}$$

(A.11c)

We can finally obtain the general expression of $\Omega _\textrm{gw}^{\alpha \beta }$ after straightforward calculations, i.e.,

$$\begin{aligned} \Omega _\textrm{gw}^{\alpha \beta }(\eta ,k)= & {} \int _0^{\infty } \textrm{d}u \int _{|1-u |} ^{|1+u |} \textrm{d}v \ \frac{k^2}{6\mathcal {H}^2 u^2v^2}\nonumber \\{} & {} \times Q_{\alpha \beta }^2(u,v) \overline{I_{\alpha \beta }^{2}(u,v,x\rightarrow \infty )}\ \mathcal {P}_{\alpha }(uk) \mathcal {P}_{\beta }(vk),\nonumber \\ \end{aligned}$$

(A.12)

In Eq. (A.12), $Q_{ss}^2$ and $Q_{st}^2$ are defined as the sum of the polarizations over two projection factors $Q_{ss}^{\lambda }(\textbf{k},\textbf{q})$ and $Q_{st}^{\lambda \lambda _1}(\textbf{k},\textbf{q})$ in Eq. (A.3), i.e.,

$$\begin{aligned} Q_{ss}^2(u,v)&= \delta ^{\lambda \lambda '} Q_{ss}^{\lambda }(\textbf{k},\textbf{q}) Q_{ss}^{\lambda '}(\textbf{k},\textbf{q})\nonumber \\&= \bigg (\frac{4 v^2-\left( 1+v^2-u^2\right) ^2}{4}\bigg )^2, \end{aligned}$$

(A.13a)

$$\begin{aligned} Q_{st}^2(u,v)&= \delta ^{\lambda \lambda '} \delta ^{\lambda _1\lambda _1'} Q_{st}^{\lambda \lambda _1}(\textbf{k},\textbf{q}) Q_{st}^{\lambda '\lambda _1'}(\textbf{k},\textbf{q})\nonumber \\&= \frac{1}{4}+\frac{3 \left( 1+v^2-u^2\right) ^2}{8v^2}+\frac{\left( 1+v^2-u^2\right) ^4}{64v^4}, \end{aligned}$$

(A.13b)

while a bit differently, $Q_{tt}^2(u,v)$ are defined together with $I_{tt}^2(u,v,x)$ for convenience, given by the expression below composed of $Q_{tt,i}^{\lambda \lambda _1 \lambda _2}$ in Eq. (A.3) and $I_{tt,i}$ in Eq. (A.10), i.e.,

$$\begin{aligned} Q_{tt}^2 \times I_{tt}^2= & {} \delta ^{\lambda \lambda '} \left( \delta ^{\lambda _1 \lambda _1'} \delta ^{\lambda _2 \lambda _2'} + \delta ^{\lambda _1 \lambda _2'} \delta ^{\lambda _2 \lambda _1'} \right) \nonumber \\{} & {} \times \bigg ( \sum _{i=1}^{5}\ Q_{tt,i}^{\lambda \lambda _1 \lambda _2} I_{tt,i} \bigg ) \bigg ( \sum _{j=1}^{5}\ Q_{tt,j}^{\lambda ' \lambda _1' \lambda _2'} I_{tt,j} \bigg ),\nonumber \\ \end{aligned}$$

(A.14)

where the contractions of polarization indices in Eq. (A.14) correspond to those in Eq. (A.11). Remind that $Q_{tt,i}$ in Eq. (A.3) and $f_{tt,i}$ in Eq. (A.4) are already symmetric about $\textbf{k}-\textbf{q}$ and $\textbf{q}$, so the two types of momentum contractions in $\Omega _\textrm{gw}^{tt}$ in Eq. (A.11) yield the same result. We finally get the expressions of the fractional energy density spectrum of the second-order tensor perturbations $\Omega _\textrm{gw}^{\alpha \beta }(k)$ in the oscillation average, namely

$$\begin{aligned} \Omega _\textrm{gw}^{ss}(k)&= \int _0^{\infty } \textrm{d}u \int _{|1-u |} ^{|1+u |} \textrm{d}v \ \mathcal {P}_{s}(uk) \mathcal {P}_{s}(vk) \frac{3}{1024u^8v^8}\nonumber \\&\quad \times \Big [{4 v^2-\left( 1+v^2-u^2\right) ^2}\Big ]^{2} \left( {u^2+v^2-3}\right) ^2 \nonumber \\&\quad \times \bigg \{ \left[ -4 u v +(u^2+v^2-3)\ln \left| \frac{3-(u+v)^2}{3-(u-v)^2}\right| \ \right] ^2 \nonumber \\&\quad +\pi ^2 (u^2+v^2-3)^2\ \Theta (u+v-\sqrt{3}) \bigg \},\nonumber \\ \end{aligned}$$

(A.15a)

$$\begin{aligned} \Omega _\textrm{gw}^{st}(k)&= \int _0^{\infty } \textrm{d}u \int _{|1-u |} ^{|1+u |} \textrm{d}v \ \mathcal {P}_{s}(uk) \mathcal {P}_{t}(vk) \nonumber \\&\quad \times \frac{1}{442368u^8v^8} \left[ 16 v^4+24 v^2\left( 1+v^2-u^2\right) ^2\right. \nonumber \\&\quad \left. +\left( 1+v^2-u^2\right) ^4 \right] \bigg \{ \left[ 4 u v \left[ u^2-9\left( v^2-1\right) \right] \right. \nonumber \\&\quad +\sqrt{3} \left[ u^2-3 \left( v-1\right) ^2\right] \left[ u^2-3 \left( v+1\right) ^2\right] \nonumber \\&\quad \times \ln \left. \left| \frac{3-(u+\sqrt{3}v)^2}{3-(u-\sqrt{3}v)^2}\right| \ \right] ^2 \nonumber \\&\quad +3 \pi ^2 \left[ u^2-3 \left( v-1\right) ^2\right] ^2 \left[ u^2-3 \left( v+1\right) ^2\right] ^2\nonumber \\&\quad \times \left. \Theta \left( u^2 - 3(v - 1)^2\right) \right\} \end{aligned}$$

(A.15b)

$$\begin{aligned} \Omega _\textrm{gw}^{tt}(k)&= \int _0^{\infty } \textrm{d}u \int _{|1-u |} ^{|1+u |} \textrm{d}v \ \mathcal {P}_{t}(uk) \mathcal {P}_{t}(vk) \nonumber \\&\quad \times \frac{1}{3145728 u^8 v^8} \left[ (u-v)^2-1\right] ^2 \left[ (u+v)^2-1\right] ^2 \nonumber \\&\quad \times \bigg \{64 u^2 v^2 \left[ u^4+v^4+6 u^2 v^2+6\left( u^2+ v^2\right) +1\right] \nonumber \\&\quad -16 u v \left[ 5 \left( u^6{+}v^6\right) +11 u^2 v^2 \left( u^2{+}v^2\right) {+}11 \left( u^4{+}v^4\right) \right. \nonumber \\&\quad \left. -126 u^2 v^2+11 \left( u^2+v^2\right) +5\right] \ln \left| \frac{1-(u+v)^2}{1-(u-v)^2}\right| \nonumber \\&\quad +\left[ 25 \left( u^8+v^8\right) -4 u^2 v^2 \left( u^4+v^4\right) \right. \nonumber \\&\quad \left. +86 u^4 v^4-4 \left( u^6+v^6\right) +68 u^2 v^2 \left( u^2+v^2\right) \right. \nonumber \\&\quad \left. +86 \left( u^4+v^4\right) +68 u^2 v^2-4 \left( u^2+v^2\right) +25 \right] \nonumber \\&\quad \times \left[ \pi ^2+\ln ^2\left| \frac{1-(u+v)^2}{1-(u-v)^2}\right| \ \right] \bigg \} . \end{aligned}$$

(A.15c)

Specially, we analyse the differences between $\Omega _\textrm{gw}^{ss}$ and $\Omega _\textrm{gw}^{st}$ in the limits of large- and small-momentum couplings, i.e., $u \rightarrow 0$ and $v\rightarrow 1$. They may help to explain why $\Omega _\textrm{gw}^{st}$ is enhanced in high frequencies, while $\Omega _\textrm{gw}^{ss}$ not, as demonstrated in the main text.

(a) One of the differences is projection factors $Q_{\alpha \beta }$, which describes the geometric relations between the momenta and polarization tensors of the linear perturbations. As shown in Eq. (A.13a), we have $Q^2_{ss}(u,v) \rightarrow 0$ in $u \rightarrow 0, v \rightarrow 1$ limit, implying a suppression on the couplings between small-momentum scalar and large-momentum tensor. However, as for “scalar-tensor” mode in Eq. (A.13b), we have $Q^2_{st}(u,v) \rightarrow 2$ keeping a constant in the limit of $u \rightarrow 0, v \rightarrow 1$, which means there is always a non-vanishing “effective quadrupole moment” in “scalar-tensor” mode.

(b) Another difference is kernel function $I_{\alpha \beta }$, coming from the different transfer functions between scalar and tensor perturbations. In the limit of $u\rightarrow 0,\ v\rightarrow 1$, the “scalar-scalar” mode $I^2_{ss}\rightarrow \textrm{const}$, while the “scalar-tensor” mode $I^2_{st}\propto u^{-2}$, which provides an enhancement factor related to small-momentum scalar perturbations.

In summary, the differences between $\Omega _\textrm{gw}^{ss}$ and $\Omega _\textrm{gw}^{st}$ result from distinct behaviors of the projection factors $Q_{\alpha \beta }$ (related to geometry) and kernel function $I_{\alpha \beta }$ (related to time evolution) in the limit of $u\rightarrow 0,\ v\rightarrow 1$.