Correlated timing noise and high-precision pulsar timing: measuring frequency second derivatives as an example

We investigate the impact of noise processes on high-precision pulsar timing. Our analysis focuses on the measurability of the second spin frequency derivative <(nu)double over dot>. This <(nu)double over dot> can be induced by several factors including the radial velocity of a pulsar. We use Bayesian methods to model the pulsar times-of-arrival in the presence of red timing noise and dispersion measure variations, modelling the noise processes as power laws. Using simulated times-of-arrival that both include red noise, dispersion measure variations, and non-zero <(nu)double over dot>. values, we find that we are able to recover the injected <(nu)double over dot>, even when the noise model used to inject and recover the input parameters are different. Using simulations, we show that the measurement uncertainty on <(nu)double over dot> decreases with the timing baseline T as T-gamma, where gamma = -7/2 + alpha/2 for power-law noise models with shallow power-law indices alpha (0 < alpha < 4). For steep power-law indices (alpha > 8), the measurement uncertainty reduces with T-1/2. We applied this method to times-of-arrival from the European Pulsar Timing Array and the Parkes Pulsar Timing Array and determined <(nu)double over dot> probability density functions for 49 millisecond pulsars. We find a statistically significant <(nu)double over dot>. value for PSR B1937+21 and consider possible options for its origin. Significant (95 per cent C.L.) values for <(nu)double over dot> are also measured for PSRs J0621+1002 and J1022+1001, thus future studies should consider including it in their ephemerides. For binary pulsars with small orbital eccentricities, such as PSR J1909-3744, extended ELL1 models should be used to overcome computational issues. The impacts of our results on the detection of gravitational waves are also discussed.