Optimal strategies for gravitational wave memory detection in pulsar timing arrays

In this work we derive two computationally efficient frequentist detection statistics that can be used in searches for gravitational-wave bursts with memory in pulsar timing data. By maximizing the likelihood ratio in two different ways we construct a coherent statistic and an incoherent statistic, which are analogs of the ${\mathcal{F}}_{e}$ and ${\mathcal{F}}_{p}$ statistics commonly used for continuous-wave searches in pulsar timing data. We show that both statistics are ${\ensuremath{\chi}}^{2}$-distributed with varying degrees of freedom and noncentrality parameters given by the signal-to-noise (SNR) ratio of the signal present in our data. The statistics can also be used to compute the maximum likelihood estimators of amplitude parameters of a possible gravitational-wave memory signal in pulsar timing data. We find that in the low-signal regime ($\mathrm{SNR}\ensuremath{\lesssim}5$), the estimators are inaccurate. However, in intermediate- to high-signal regimes, we show that these estimators can accurately determine the correct signal parameters. We also study the effects of red noise on these statistics, and find that the inclusion of red noise reduces the signal-to-noise ratio for injected memory signals.