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 $f_2=2\mathrm{/}{P}_{\mathrm{M}}=0.847245\mu \mathrm{H}\mathrm{z}$. 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 $80\mu \mathrm{m}$ absolute range uncertainty, a 5-year campaign with statistically independent AM-CW phase-normal-point rate of ${\nu }_{\mathrm{e}\mathrm{f}\mathrm{f}}=500{\mathrm{y}\mathrm{r}}^{-1}$ has response-calibrated sensitivity ${\mathrm{\Omega }}_{\mathrm{g}\mathrm{w}}^{95}=5.29\times {10}^{-9}{D}_{\mathrm{c}\mathrm{o}\mathrm{v}}$; a mature implementation with ${\sigma }_R=50\mu \mathrm{m}$ gives $2.07\times {10}^{-9}{D}_{\mathrm{c}\mathrm{o}\mathrm{v}}$, where $D_{\mathrm{c}\mathrm{o}\mathrm{v}}\ge 1$ 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 $D_{\mathrm{c}\mathrm{o}\mathrm{v}}\lesssim 3.6$ and $5.4$, respectively, in the $80\mu \mathrm{m}$ case, and for $D_{\mathrm{c}\mathrm{o}\mathrm{v}}\lesssim 9.1$ and $13.7$ in the $50\mu \mathrm{m}$ 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 $D_{\mathrm{c}\mathrm{o}\mathrm{v}}$ below the discovery margins.