  Supermassive black hole binaries (SMBHBs) are expected to emit continuous
gravitational waves in the pulsar timing array (PTA) frequency band
($10^{-9}$--$10^{-7}$ Hz). The development of data analysis techniques aimed at
efficient detection and characterization of these signals is critical to the
gravitational wave detection effort. In this paper we leverage methods
developed for LIGO continuous wave gravitational searches, and explore the use
of the $\mathcal{F}$-statistic for such searches in pulsar timing data. Babak &
Sesana 2012 have already used this approach in the context of PTAs to show that
one can resolve multiple SMBHB sources in the sky. Our work improves on several
aspects of prior continuous wave search methods developed for PTA data
analysis. The algorithm is implemented fully in the time domain, which
naturally deals with the irregular sampling typical of PTA data and avoids
spectral leakage problems associated with frequency domain methods. We take
into account the fitting of the timing model, and have generalized our approach
to deal with both correlated and uncorrelated colored noise sources. We also
develop an incoherent detection statistic that maximizes over all pulsar
dependent contributions to the likelihood. To test the effectiveness and
sensitivity of our detection statistics, we perform a number of monte-carlo
simulations. We produce sensitivity curves for PTAs of various configurations,
and outline an implementation of a fully functional data analysis pipeline.
Finally, we present a derivation of the likelihood maximized over the
gravitational wave phases at the pulsar locations, which results in a vast
reduction of the search parameter space.

Instrumentation and Methods for Astrophysics;General Relativity and Quantum Cosmology