Measuring the parameters of massive black hole binary systems with pulsar timing array observations of gravitational waves

The observation of massive black hole binaries with pulsar timing arrays (PTAs) is one of the goals of gravitational-wave astronomy in the coming years. Massive (≳10⁸M_⊙) and low-redshift (≲1.5) sources are expected to be individually resolved by upcoming PTAs, and our ability to use them as astrophysical probes will depend on the accuracy with which their parameters can be measured. In this paper we estimate the precision of such measurements using the Fisher-information-matrix formalism. For this initial study we restrict ourselves to “monochromatic” sources, i.e. binaries whose frequency evolution is negligible during the expected ≈10yr observation time, which represent the bulk of the observable population based on current astrophysical predictions. In this approximation, the system is described by seven parameters and we determine their expected statistical errors as a function of the number of pulsars in the array, the array sky coverage, and the signal-to-noise ratio (SNR) of the signal. At fixed SNR (regardless of the number of pulsars in the PTA), the gravitational-wave astronomy capability of a PTA is achieved with ≈20 pulsars; adding more pulsars (up to 1000) to the array reduces the source error box in the sky ΔΩ by a factor ≈5 and has negligible consequences on the statistical errors on the other parameters, because the correlations among parameters are already removed to a large extent. If one folds in the increase of coherent SNR proportional to the square root of the number of pulsars, ΔΩ improves as 1/SNR² and the other parameters as 1/SNR. For a fiducial PTA of 100 pulsars uniformly distributed in the sky and a coherent SNR=10, we find ΔΩ≈40deg², a fractional error on the signal amplitude of ≈30% (which constrains only very poorly the chirp mass—luminosity distance combination M^5/3/D_L), and the source inclination and polarization angles are recovered at the ≈0.3rad level. The ongoing Parkes PTA is particularly sensitive to systems located in the southern hemisphere, where at SNR=10 the source position can be determined with ΔΩ≈10deg², but has poorer (by an order of magnitude) performance for sources in the northern hemisphere.