Gravitational Waves from Matter Perturbations of Spectator Scalar Fields

We compute the stochastic gravitational wave background sourced at second order by a spectator scalar field $\chi$ coupled to the inflaton $\phi$ through a portal interaction $\sigma\phi^2\chi^2$ and with quartic self-interaction $\lambda_\chi\chi^4/4!$. In the large portal coupling regime ($\sigma/\lambda \gg 1$, with $\lambda$ the inflaton normalization), parametric resonance during reheating amplifies the spectator power spectrum by many orders of magnitude near the resonance band until Hartree backreaction from the quartic coupling detunes the instability, while the large inflationary effective mass suppresses superhorizon power and ensures compatibility with CMB isocurvature bounds. We focus on the direct field-gradient source $\partial_a\chi\,\partial_b\chi$ in the second-order Einstein equations and derive a master formula that factorizes into a spectral integral over the frozen, vacuum-subtracted spectator spectrum and a time integral encoding the post-inflationary expansion history. For our benchmark reheating history we obtain analytic scaling relations, including a peak amplitude $\Omega_{\rm GW}\propto T_{\rm reh}^{8/3}$, strong dependence on the portal strength, and weak sensitivity to $m_\chi$. We validate the framework against nonlinear lattice simulations, demonstrating complementarity: the Hartree treatment captures superhorizon evolution inaccessible to the lattice, while the lattice resolves rescattering and fragmentation near the spectral peak. For $\sigma/\lambda \simeq 10^4$ and $T_{\rm reh}=2 \times 10^{14}\,\mathrm{GeV}$, the signal reaches $\Omega_{\rm GW}h^2\sim 10^{-11}$ at $f\sim10^{7}$-$10^{8}\,\mathrm{Hz}$. Increasing $\lambda_\chi$ at fixed $\sigma$ has a non-monotonic effect: small values enhance the signal via rescattering, whereas larger values suppress it by detuning the resonance.