Pulsar Timing Arrays Can Measure Gravitational Wave Source Distance

Gravitation waves manifest themselves as a tidal acceleration; correspondingly, gravitational wave detectors must be analyzed as (space-time) extended systems interacting with a (space-time) extended wave. In the case of pulsar timing arrays the detector size is of order the pulsar-Earth distance, which is in excess of 1~Kpc for over half the pulsars in the International Pulsar Timing Array (IPTA). For a gravitational wave point source the radiation wavefront curvature, associated with the finite source-Earth distance, leads to timing corrections of magnitude $(h/f)\sin(\pi{}fL^2/R)$, where $h$ is the gravitational wave strain amplitude, $L$ the pulsar-Earth distance, $R$ the source-Earth distance, and $f$ the characteristic gravitational wave frequency. The timing precision of the best IPTA pulsars are better than 100~ns~rms; correspondingly pulsar timing array observations should be capable of measuring the timing parallax distance to point gravitational wave sources at cosmological distances.