Constraining the density distribution of the cold Circumgalactic Medium of quasars at z>2 through Lya, HeII and Ha emission

Extended Lya emission is routinely detected around high-redshift quasars, revealing large reservoirs of cool circumgalactic medium (CGM). Its resonant nature complicates interpretation of gas density, ionization state, and kinematics, making non-resonant tracers such as HeII and Ha essential. Extending previous analytical models, we use CLOUDY photo-ionization simulations to predict HeII/Ha in quasar-illuminated gas over a broad parameter range, including log-normal density distributions. The ratio constrains density fluctuations, commonly quantified by the clumping factor C. We compare the models with extended Lya, HeII, and Ha observations around quasars at z 2.1-4.5. At z 2.2, we observed 13 quasars with Keck/KCWI and targeted the systems with the most prominent extended Lya and HeII emission with Keck/MOSFIRE, providing the first statistical quasar-CGM sample with combined Lya, HeII, and Ha measurements. Within matched apertures out to 35 pkpc, stacked spectra give Lya/Ha=9.1, HeII/Ha=0.54, and HeII/Lya=0.06. Single-density models require very high CGM densities (n>10 cm-3) or implausibly soft AGN spectra to reproduce the low HeII/Ha ratio. Broad log-normal density distributions instead reproduce it at lower median densities but require a substantial emissivity-weighted high-density tail, corresponding formally to C 10 3-10 4 in our parameterization. We also analyze 26 VLT/MUSE quasars at 3.1<z<4.5, where Ha is inaccessible from the ground, finding 14 systems with co-spatial extended Lya and HeII emission. The detected, emission-weighted HeII/Lya component shows no evidence for strong evolution between z 2.2 and z 3.7, though this does not imply an invariant underlying density PDF. These constraints provide a basis for comparison with theoretical models and hydrodynamical simulations of the physical processes shaping quasar CGM at high redshift.