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 $\ddot{\nu }$. This $\ddot{\nu }$ 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 $\ddot{\nu }$ values, we find that we are able to recover the injected $\ddot{\nu }$, even when the noise model used to inject and recover the input parameters are different. Using simulations, we show that the measurement uncertainty on $\ddot{\nu }$ decreases with the timing baseline T as Tγ, where γ = −7/2 + α/2 for power-law noise models with shallow power-law indices α (0 < α < 4). For steep power-law indices (α > 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 $\ddot{\nu }$ probability density functions for 49 millisecond pulsars. We find a statistically significant $\ddot{\nu }$ value for PSR B1937+21 and consider possible options for its origin. Significant (95 per cent C.L.) values for $\ddot{\nu }$ 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.