Cataclysmic Variables and AM CVn Binaries in SRG/eROSITA + Gaia: Volume Limited Samples, X-ray Luminosity Functions, and Space Densities

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We present volume-limited samples of cataclysmic variables (CVs) and AM CVn binaries jointly selected from SRG/eROSITA eRASS1 and Gaia DR3 using an X-ray + optical color-color diagram (the ``X-ray Main Sequence"). This tool identifies all CV subtypes, including magnetic and low-accretion rate systems, in contrast to most previous surveys. We find 23 CVs, 3 of which are AM CVns, out to 150 pc in the Western Galactic Hemisphere. Our 150 pc sample is spectroscopically verified and complete down to LX = 1.3× 10²⁹ \;erg s⁻¹ in the 0.2--2.3 keV band, and we also present CV candidates out to 300 pc and 1000 pc. We discovered two previously unknown systems in our 150 pc sample: the third nearest AM CVn and a magnetic period bouncer. We find the mean LX of CVs to be LX ≈ 4.6× 10³⁰ \;erg s⁻¹, in contrast to previous surveys which yielded LX ~ 10³¹-10³² \;erg s⁻¹. We construct X-ray luminosity functions that, for the first time, flatten out at LX~ 10³⁰ \; erg s⁻¹. We find average number, mass, and luminosity densities of ρN, CV = (3.7 ± 0.7) × 10⁻⁶ pc⁻³, ρM = (5.0 ± 1.0) × 10⁻⁵ M_⁻¹, and ρLX = (2.3 ± 0.4) × 10²⁶ erg s⁻¹M_⁻¹, respectively, in the solar neighborhood. Our uniform selection method also allows us to place meaningful estimates on the space density of AM CVns, ρN, AM CVn = (5.5 ± 3.7) × 10⁻⁷ pc⁻³. Magnetic CVs and period bouncers make up $35%$ and $25%$ of our sample, respectively. This work, through a novel discovery technique, shows that the observed number densities of CVs and AM CVns, as well as the fraction of period bouncers, are still in tension with population synthesis estimates.