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

We present volume-limited samples of cataclysmic variables (CVs) and AM CVn binaries jointly selected from SRG/eROSITA eRASS1 and \textit{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 $L_X = 1.3\times 10^{29} \;\textrm{erg s}^{-1}$ 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 $L_X$ of CVs to be $\langle L_X \rangle \approx 4.6\times 10^{30} \;\textrm{erg s}^{-1}$, in contrast to previous surveys which yielded $\langle L_X \rangle \sim 10^{31}-10^{32} \;\textrm{erg s}^{-1}$. We construct X-ray luminosity functions that, for the first time, flatten out at $L_X\sim 10^{30} \; \textrm{erg s}^{-1}$. We find average number, mass, and luminosity densities of $\rho_\textrm{N, CV} = (3.7 \pm 0.7) \times 10^{-6} \textrm{pc}^{-3}$, $\rho_M = (5.0 \pm 1.0) \times 10^{-5} M_\odot^{-1}$, and $\rho_{L_X} = (2.3 \pm 0.4) \times 10^{26} \textrm{erg s}^{-1}M_\odot^{-1}$, respectively, in the solar neighborhood. Our uniform selection method also allows us to place meaningful estimates on the space density of AM CVns, $\rho_\textrm{N, AM CVn} = (5.5 \pm 3.7) \times 10^{-7} \textrm{pc}^{-3}$. 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.