O5 dark-siren forecasts for modified GW propagation: background robustness of the $\Xi$ posterior

Abstract

Binary black hole mergers without electromagnetic counterparts are expected to dominate O5 gravitational-wave catalogs. Recent CHIMERA~2.0 forecasts typically fix $\Omega_m$ to a CMB-informed value and use spectroscopic hosts when available, but the corresponding sensitivity of the $\Xi$ posterior has not been assessed for pure dark sirens on the public O5 mock catalog. We analyze 300 O5-sensitivity mock events without galaxy catalogs, varying $\Omega_m$ over $[0.20,\,0.35]$ (including Planck $0.315$), and compare fixed-background inference with joint $(H_0,\,\Omega_m,\,\Xi)$ inference. The marginalized $\Xi$ posterior is $0.9783 \pm 0.3548$ and shows no change across this interval. Only GW luminosity distances enter the analysis, so the likelihood constrains $\Xi\, D_L^{\rm EM}(H_0,\,\Omega_m)$; when $\Omega_m$ is changed, $H_0$ shifts to compensate and the $\Xi$ marginal remains unchanged. Joint inference gives $\Xi^{\rm joint} = 0.9550 \pm 0.3710$, with $|\rho| \lesssim 0.05$ for the $\Omega_m$--$\Xi$ and $H_0$--$\Xi$ pairs, whereas $\rho_{H_0 \Omega_m} \simeq -0.4$ and the $H_0$ median moves by $\simeq 4.2\,{\rm km\,s^{-1}\,Mpc^{-1}}$ over the adopted $\Omega_m$ range. Galaxy-catalog analyses on the same events at fixed $\Omega_m = 0.3$ reach $\sim 7.5\%$ precision on $\Xi$, compared with $\pm 36.3\%$ here; the larger uncertainty is driven mainly by missing host redshifts. Sub-percent $\Xi$ tests will therefore still require measured redshifts even if dark sirens dominate the detection rate.