Entanglement, Discord, and Residual Coherence in Scalar-Induced Gravitational Waves

Abstract

Scalar-induced gravitational waves (SIGWs) are usually characterized by their power spectrum, although their quadratic scalar source also carries higher-order correlation information. We investigate whether phase-sensitive correlations of a primordial Gaussian scalar state can survive decoherence and be transferred to the induced tensor sector. The scalar state is described in a fixed oscillator basis by the occupation $N_k^ζ$ and anomalous moment $M_k^ζ$, while the transfer is formulated through the unequal-time scalar Wightman function. The late-time tensor moments $d_k$ and $γ_k$ define an effective Gaussian reference state and its discord, whereas the exact SIGW state is generally non-Gaussian and contains an additional connected fourth-order cumulant. We therefore introduce the baseline-subtracted observable $Δκ_{\rm coh}=κ_N[G^>_{\rm coh}]-κ_N[G^>_{\rm ref}]$. For phase damping with $M_k^ζ\propto e^{-D_k}$, the Gaussian-reference contribution conditionally scales as $|γ_k|\propto e^{-2D_k}$ and $κ_N^{\rm G}\propto e^{-4D_k}$. This provides a framework in which the SIGW spectrum fixes the signal band, while higher-order statistics probe residual coherence beyond the power spectrum.