Supermassive black hole binaries, cosmic strings, relic gravitational waves from inflation, and first order phase transitions in the early universe are expected to contribute to a stochastic background of gravitational waves in the 10^(-9) Hz-10^(-7)
Hz frequency band. Pulsar timing arrays (PTAs) exploit the high precision timing of radio pulsars to detect signals at such frequencies. Here we present a time-domain implementation of the optimal cross-correlation statistic for stochastic background searches in PTA data. Due to the irregular sampling typical of PTA data as well as the use of a timing model to predict the times-of-arrival of radio pulses, time-domain methods are better suited for gravitational wave data analysis of such data. We present a derivation of the optimal cross-correlation statistic starting from the likelihood function, a method to produce simulated stochastic background signals, and a rigorous derivation of the scaling laws for the signal-to-noise ratio of the cross-correlation statistic in the two relevant PTA regimes: the weak signal limit where instrumental noise dominates over the gravitational wave signal at all frequencies, and a second regime where the gravitational wave signal dominates at the lowest frequencies.
Cyclic spectroscopy is a signal processing technique that was originally developed for engineering applications and has recently been introduced into the field of pulsar astronomy. It is a powerful technique with many attractive features, not least o
f which is the explicit rendering of information about the relative phases in any filtering imposed on the signal, thus making holography a more straightforward proposition. Here we present methods for determining optimum estimates of both the filter itself and the statistics of the unfiltered signal, starting from a measured cyclic spectrum. In the context of radio pulsars these quantities tell us the impulse response of the interstellar medium and the intrinsic pulse profile. We demonstrate our techniques by application to 428 MHz Arecibo data on the millisecond pulsar B1937+21, obtaining the pulse profile free from the effects of interstellar scattering. As expected, the intrinsic profile exhibits main- and inter-pulse components that are narrower than they appear in the scattered profile; it also manifests some weak, but sharp features that are revealed for the first time at low frequency. We determine the structure of the received electric-field envelope as a function of delay and Doppler-shift. Our delay-Doppler image has a high dynamic-range and displays some pronounced, low-level power concentrations at large delays. These concentrations imply strong clumpiness in the ionized interstellar medium, on AU size-scales, which must adversely affect the timing of B1937+21.
