Searches for continuous gravitational waves from rapidly spinning neutron stars normally assume that the star rotates about one of its principal axes of moment of inertia, and hence the gravitational radiation emits only at twice the spin frequency of the star, 2f⋆. The superfluid interior of a star pinned to the crust along an axis nonaligned with any of its principal axes allows the star to emit gravitational waves at both f⋆ and 2f⋆, even without free precession, a phenomenon not clearly observed in known pulsars. The dual-harmonic emission mechanism motivates searches combining the two frequency components of a signal to improve signal-to-noise ratio. We describe an economical, semicoherent, dual-harmonic search method, combined with a maximum likelihood coherent matched filter, F-statistic, and improved from an existing hidden Markov model (HMM) tracking scheme to track two frequency components simultaneously. We validate the method and demonstrate its performance through Monte Carlo simulations. We find that for sources emitting gravitational waves at both f⋆ and 2f⋆, the rate of correctly recovering synthetic signals (i.e., detection efficiency), at a given false alarm probability, can be improved by ∼10%–70% by tracking two frequencies simultaneously compared to tracking a single component only. For sources emitting at 2f⋆ only, dual-harmonic tracking only leads to minor sensitivity loss, producing ≲10% lower detection efficiency than tracking a single component. In directed continuous-wave searches where f⋆ is unknown and hence the full frequency band is searched, the computationally efficient HMM tracking algorithm provides an option of conducting both the dual-harmonic search and the conventional single frequency tracking to obtain optimal sensitivity, with a typical run time of ∼10^3 core-hr for one year’s observation
