Directed searches for continuous gravitational waves from binary systems: Parameter-space metrics and optimal Scorpius X-1 sensitivity

We derive simple analytic expressions for the (coherent and semicoherent) phase metrics of continuous-wave sources in low-eccentricity binary systems for the two regimes of long and short segments compared to the orbital period. The resulting expressions correct and extend previous results found in the literature. We present results of extensive Monte Carlo studies comparing metric mismatch predictions against the measured loss of detection statistics for binary parameter offsets. The agreement is generally found to be within $\ensuremath{\sim}10%--30%$. For an application of the metric template expressions, we estimate the optimal achievable sensitivity of an Einstein@Home directed search for Scorpius X-1, under the assumption of sufficiently small spin wandering. We find that such a search, using data from the upcoming advanced detectors, would be able to beat the torque-balance level [R. V. Wagoner, Astrophys. J. 278, 345 (1984); L. Bildsten, Astrophys. J. 501, L89 (1998).] up to a frequency of $\ensuremath{\sim}500--600\text{ }\text{ }\mathrm{Hz}$, if orbital eccentricity is well constrained, and up to a frequency of $\ensuremath{\sim}160--200\text{ }\text{ }\mathrm{Hz}$ for more conservative assumptions about the uncertainty on orbital eccentricity.