High-Power AM-CW Lunar Laser Ranging as a $\mu$Hz SGWB Detector

The Earth--Moon binary is a resonant detector for stochastic gravitational-wave background (SGWB) at harmonics of the lunar orbital frequency. We quantify high-power amplitude-modulated continuous-wave lunar laser ranging (AM-CW LLR) as a $\mu$Hz SGWB probe. The dominant low-eccentricity response is at $f_2=2/P_{\rm M}=0.847245\,\mu{\rm Hz}$. AM-CW LLR measures radio-frequency phase on a GHz-modulated 1064 nm optical carrier reflected by lunar corner cubes, giving range and range rate observables. With an $80\,\mu{\rm m}$ absolute range uncertainty, a 5-year campaign with statistically independent AM-CW phase-normal-point rate of $\nu_{\rm eff}=500\,{\rm yr}^{-1}$ has response-calibrated sensitivity $\Omega_{\rm gw}^{95}=5.29\times10^{-9}D_{\rm cov}$; a mature implementation with $\sigma_R=50\,\mu{\rm m}$ gives $2.07\times10^{-9}D_{\rm cov}$, where $D_{\rm cov}\ge1$ is a covariance-degradation factor for time-correlated residuals and nuisance-parameter correlations in the global solution. Anticipated first-order phase-transition and compact-binary signals lie above the nominal 5-$\sigma$ covariance-amplitude threshold for $D_{\rm cov}\lesssim3.6$ and $5.4$, respectively, in the $80\,\mu{\rm m}$ case, and for $D_{\rm cov}\lesssim9.1$ and $13.7$ in the $50\,\mu{\rm m}$ case. Thus the experiment is a sharp covariance test: absolute range carries the SGWB signal, while range rate and multi-reflector differential data determine whether nuisance correlations keep $D_{\rm cov}$ below the discovery margins.