Electromagnetic Timing Priors in Gravitational Wave Forecasts for Ultralight Dark Matter

Ultralight dark matter (ULDM) can induce a slow frequency modulation of nearly monochromatic millihertz gravitational waves (GWs). Existing forecasts infer the carrier phase evolution from the GW data, leaving the modulation partially degenerate with the binary frequency derivatives. We incorporate the electromagnetic (EM) timing ephemeris of J0806/HM Cnc into a Fisher forecast. The X-ray and optical pulsations of this accreting ultracompact binary track its orbital phase, allowing the measured ephemeris to constrain the carrier phase evolution without directly constraining ULDM. At low ULDM masses, the modulation varies slowly over the observation and is partly absorbed by the timing polynomial in a GW-only analysis. The EM ephemeris reduces this degeneracy after marginalization. The projected inverse coupling reach improves by up to about two orders of magnitude for a direct linear coupling and by about one order of magnitude for a direct quadratic coupling. These are relative improvements for the same source and dark matter density. EM timing can therefore improve the sensitivity of future space-based GW searches to ULDM-induced modulation in verification binaries.