ALMA visits the QSO MUSEUM: Connecting molecular gas and the cool circumgalactic medium around 37 z~3 quasars
Extended Ly$\alpha$ emission is ubiquitous around quasars and traces the cool circumgalactic medium, providing insights into halo gas dynamics and active galactic nucleus feedback. However, its connection to the cold molecular gas of the host galaxies remains largely unexplored. We characterize the molecular gas reservoirs of quasars at cosmic noon and investigate their connection to extended Ly$\alpha$ emission using ALMA CO(4-3) observations of 37 quasars at $z\sim3$ from the QSO MUSEUM survey, previously mapped in Ly$\alpha$ with VLT/MUSE. We derive molecular gas masses and gas fractions, explore correlations with Ly$\alpha$ nebula and quasar properties, and search for CO-emitting companions. We detect 21/37 quasars in CO(4-3), with gas masses of $M_\mathrm{gas}\approx(3-40) \times10^9\,\mathrm{M_\odot}$. Quasars with the most massive molecular gas reservoirs are associated with the centrally dimmest Ly$\alpha$ nebulae, while those hosting the centrally brightest Ly$\alpha$ nebulae are generally not detected in CO. This suggests that gas and dust in the hosts regulate Ly$\alpha$ escape and consequently affect the emission from halo gas. We find evidence that lower-Eddington-ratio quasars harbor more massive gas reservoirs, while strongly accreting quasars ($\lambda_\mathrm{Edd} \gtrapprox 0.9$) likely deplete their gas; for example, through powerful quasar-driven outflows. Despite their higher molecular gas masses within the sample, CO-detected low-Eddington quasars exhibit low gas fractions with a median $M_\mathrm{gas}/M_* \sim 0.10$, below what is typically found for inactive star-forming galaxies. Six quasars are marginally resolved in CO, with effective radii up to $\sim 8\,\mathrm{kpc}$. In addition, we detect 14 high-fidelity companion galaxies, indicating overdense quasar environments with a quasar-galaxy cross-correlation length of $9.81^{+2.22}_{-2.05}\,h^{-1}\mathrm{cMpc}$.