A new model for the continuum spectra of AM CVn binaries and multi-messenger inference with normalizing flows

Future electromagnetic telescopes, such as $\textit{NewAthena}$, $\textit{CASTOR}$, and an $\textit{AXIS}$-like mission, along with milli-Hz gravitational-wave (GW) detectors such as $\textit{LISA}$, are expected to unearth the population of Galactic ultra-compact binaries (UCBs). Joint multi-messenger detections will probe the uncertain formation, evolution, and observables of mass-transferring UCBs such as AM CVns, but theoretical tools need to be advanced to anticipate future data challenges. Motivated by this, we present a new forward model for the continuum emission of AM CVn binaries that connects source binary parameters to X-ray, optical, and ultraviolet observables. The model assumes GW-driven mass transfer with physically motivated prescriptions for accretion energetics, emission geometry, absorption, and instrumental response. Combining this with $\textit{LISA}$ observations and the output of binary population synthesis enables exploration of the multi-messenger properties of AM CVns. Although uncertain, our model predicts that approximately one per $7000$ AM CVn binaries will permit a joint multi-messenger detection with $\textit{LISA}$, $\textit{CASTOR}$, and $\textit{AXIS}$. We also develop a framework for inferring binary parameters from the inverse model with a convolutional neural net and normalizing flows. Testing the trained flow with our synthetic AM CVn population, we find mean absolute fractional error on the inferred accretor mass of $0.05$ M$_{\odot}$, donor mass of $0.26$ M$_{\odot}$, orbital period of $0.1$ s, and distance of $0.2$ pc, while Spearman's rank shows strongly correlated true and predicted distributions except for the donor mass. These efforts lay a foundation for follow-up studies that will explore detailed binary astrophysics and observational requirements for effective multi-messenger scientific discovery in the coming decade.