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 $T^\gamma$, 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 $\ddot{\nu}$ 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, like 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.