Minimum Requirements for Detecting a Stochastic Gravitational Wave Background Using Pulsars

We assess the detectability of a nanohertz gravitational wave (GW) background in a pulsar timing array (PTA) program by considering the shape and amplitude of the cross-correlation function summed over pulsar pairs. The distribution of correlation amplitudes is found to be non-Gaussian and highly skewed, which significantly influences detection and false-alarm probabilities. When only white noise combines with GWs in timing data, our detection results are consistent with those found by others. Contamination by red noise from spin variations and from any uncorrected interstellar plasma effects significantly increases the false-alarm probability. The number of arrival times (and thus the observing cadence) is important only as long as the residuals are dominated by white noise. When red noise and GWs dominate, the statistical significance of the correlation estimate can be improved only by increasing the number of pulsars. We characterize plausible detection regimes by evaluating the number of millisecond pulsars (MSPs) that must be monitored in a high-cadence, five-year timing program to detect a GW background spectrum hc (f) = A(f/f 0)-2/3 with f 0 = 1yr-1 and A = 10-15. Our results indicate that a sample of 20 super-stable MSPs - those with rms timing residuals σr ≲ 20 ns(A/10 -15) from red-noise contributions over a five-year span - will allow detection of the GW background and study of its spectrum. However, a timing program on ≳ 50-100 MSPs is likely needed for a complete PTA program, particularly if red noise is generally present in MSPs.