Understanding and analysing time-correlated stochastic signals in pulsar timing

Although it is widely understood that pulsar timing observations generally contain time-correlated stochastic signals (TCSSs; red timing noise is of this type), most data analysis techniques that have been developed make an assumption that the stochastic uncertainties in the data are uncorrelated, i.e. `white'. Recent work has pointed out that this can introduce severe bias in the determination of timing-model parameters and that better analysis methods should be used. This paper presents a detailed investigation of timing-model fitting in the presence of TCSSs and gives closed expressions for the post-fit signals in the data. This results in a Bayesian technique to obtain timing-model parameter estimates in the presence of TCSSs, as well as computationally more efficient expressions of their marginalized posterior distribution. A new method to analyse hundreds of mock data set realizations simultaneously without significant computational overhead is presented, as well as a statistically rigorous method to check the internal consistency of the results. As a by-product of the analysis, closed expressions of the rms introduced by a stochastic background of gravitational waves in timing residuals are obtained, valid for regularly sampled data. Using T as the length of the data set and h_c(1 yr^{- 1}) as the characteristic strain, this is σ _GWB^2 = h_c(1 yr^{-1})^2 (9 root 3 of {2π ^4}Γ (-10/3) / 8008) yr^{-4/3} T^{10/3}.