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Supermassive black hole binaries, cosmic strings, relic gravitational wavesfrom inflation, and first order phase transitions in the early universe areexpected to contribute to a stochastic background of gravitational waves in the10^(-9) Hz-10^(-7) Hz frequency band. Pulsar timing arrays (PTAs) exploit thehigh precision timing of radio pulsars to detect signals at such frequencies.Here we present a time-domain implementation of the optimal cross-correlationstatistic for stochastic background searches in PTA data. Due to the irregularsampling typical of PTA data as well as the use of a timing model to predictthe times-of-arrival of radio pulses, time-domain methods are better suited forgravitational wave data analysis of such data. We present a derivation of theoptimal cross-correlation statistic starting from the likelihood function, amethod to produce simulated stochastic background signals, and a rigorousderivation of the scaling laws for the signal-to-noise ratio of thecross-correlation statistic in the two relevant PTA regimes: the weak signallimit where instrumental noise dominates over the gravitational wave signal atall frequencies, and a second regime where the gravitational wave signaldominates at the lowest frequencies.

Time-domain Implementation of the Optimal Cross-Correlation Statistic for Stochastic Gravitational-Wave Background Searches in Pulsar Timing Data