Scalar-induced gravitational waves from coherent initial states

TL;DR

This work analyzes how a coherent-state initial condition for inflationary perturbations generates a non-zero space-dependent mean in the primordial curvature perturbation, violating statistical homogeneity, isotropy, and parity. The resulting scalar perturbations induce scalar-induced gravitational waves (SIGW) at second order during radiation domination, imprinting scale- and direction-dependent features, including a non-zero one-point function for the tensor sector and off-diagonal polarization correlations. By parameterizing the primordial mean with a complex function $oldsymbol{ ablaalpha}(oldsymbol{k})$, the authors show that SIGW can exhibit enhanced spectral density near the mean's characteristic scale and nontrivial chirality when anisotropy is present. The findings provide a direct GW-based probe of primordial statistics at small scales and offer observable signatures—such as chirality and cross-polarization correlations—that could help distinguish this scenario from other parity-violating mechanisms in upcoming experiments like PTA, SKA, and LISA.

Abstract

We investigate the impact of statistical inhomogeneity and anisotropy in primordial scalar perturbations on the scalar-induced gravitational waves (SIGW). Assuming inflationary quantum fluctuations originate from a coherent state, the resulting primordial scalar perturbations acquire a non-zero space-dependent mean, violating statistical homogeneity, statistical isotropy, and parity. As a consequence of statistical inhomogeneities, SIGW acquires distinct scale-dependent features in its correlation function. Statistical anisotropies further lead to possible parity violation and correlation between different polarization modes in the tensor perturbations. Therefore, detection of these signatures in the stochastic gravitational wave background would offer probes to the statistical nature of primordial scalar perturbations beyond the scales accessible to CMB observations.

Figures (3)

• Figure 1: Enhancement in the spectral density of secondary gravitational waves induced by statistical inhomogeneity in the primordial scalar perturbations. Fig. \ref{['fig:omega_by_k_vary_As']} shows $\Omega_{_{\rm GW}}^0$ as a function of wave numbers for different $A_{\rm s}$. While $\Omega_{_{\rm GW}}^0$ increases with increasing $A_{\rm s}$, the relative enhancement due to the primordial mean decreases. Fig. \ref{['fig:omega_by_As_vary_k']} presents $\Omega_{_{\rm GW}}^0$ as a function of $A_{\rm s}$ evaluated at different wave numbers near its peak. For smaller $A_{\rm s}$ the peak of $\Omega_{_{\rm GW}}^0$ varies linearly with $A_{\rm s}$, while it varies quadratically for larger $A_{\rm s}$. We have set the model parameters to be $\mathcal{P}_{_{\mathrm{S}}}(k) = A_{\rm s}$, $\alpha_0=0.1/\sqrt{A_{\rm s}}$, $\Delta = 0.1$ and $\Delta_f = 0.2$.
• Figure 2: Enhancement in the spectral density of secondary gravitational waves induced by statistical inhomogeneity in the primordial scalar perturbations. The heatmap shows $\Omega_{_{\rm GW}}( \bm k_1, \bm k_2)/ A_{\rm s}^2$ with respect to the wavenumbers $k_1, k_2$, where the wave vectors are antiparallel. We have set $\mathcal{P}_{_{\mathrm{S}}}(k) = A_{\rm s}$ with $A_{\rm s} = 10^{-9}$, other model parameters are same as in Fig. \ref{['fig:spectral-density']}. Although the amplitude in the diagonal part may be degenerate with the homogeneous contribution, or even dominated by it for higher values of $A_{\rm s}$ as in Fig. \ref{['fig:spectral-density']}, we find the off-diagonal part directly determined by the amplitude and shape of $\alpha(k)$.
• Figure 3: Enhancement in the spectral density of secondary gravitational waves induced by a scalar primordial mean with a Dirac-$\delta$ profile (cf. Eq. \ref{['eq:Dirac_delta_profile']}). The black curve shows inhomogeneous contribution to $\Omega_{_{\rm GW}}^0(\bm{k}, -\bm{k})$ obtained from the exact numerical solution Eq. \ref{['eq:delta_Ph']}, while the green and blue curves show the analytical solutions for small and large $k$ limits (Eq. \ref{['eq:delta_Ph_small_k']} and Eq. \ref{['eq:delta_Ph_large_k']}), respectively. The enhancement in the spectral density rises $\propto (k/k_0)^6 \ln^2(k/k_0)$, followed by a more gradual fall $\propto (k/k_0)^{-1}$ after peaking near $k/k_0 = 1$. We have set the model parameters to be $\mathcal{P}_{_{\mathrm{S}}}(k) = A_{\rm s} = 10^{-9}$, $\alpha_0=0.1/\sqrt{A_{\rm s}}$, and $\Delta = 0.1$.