Impact of the observation frequency coverage on the significance of a gravitational wave background detection in pulsar timing array data

Pulsar timing srray (PTA) collaborations gather high-precision timing measurements of pulsars, with the aim of detecting gravitational wave (GW) signals. A major challenge lies in the identification and characterisation of the different sources of noise that may hamper their sensitivity to GWs. The presence of time-correlated noise that resembles the target signal might give rise to degeneracies that can directly impact the detection statistics. In this work, we focus on the covariance that exists between a 'chromatic' dispersion measure (DM) noise and an 'achromatic' stochastic gravitational wave background (GWB). The term 'chromatic' associated with the DM noise means that its amplitude depends on the frequency of the incoming pulsar photons measured by the radio telescopes. Multi-frequency coverage is then required to accurately characterise its chromatic features and when the coverage of incoming frequency is poor, it becomes impossible to disentangle chromatic and achromatic noise contributions. In this paper, we explore this situation by injecting realistic GWB into 100 realisations of two mock versions of the second data release (DR2) of the European PTA (EPTA), characterised by different types of frequency coverage. The first dataset is a faithful copy of DR2, in which the first half of the data is dominated by only one frequency channel for the observations; the second one is identical, except for a more homogeneous frequency coverage across the full dataset. We show that for 91% of the injections, a better frequency coverage leads to an improved statistical significance (approximate to 1.3 dex higher log Bayes factor on average) of the GWB and a better characterisation of its properties. We propose a metric to quantify the degeneracy between DM and GWB parameters. We show that it is correlated with a loss of significance for the recovered GWB, as well as with an increase in the GWB bias towards a higher and flatter spectral shape. In the second part of the paper, this correlation between the loss of GWB significance, the degeneracy between the DM and GWB parameters, and the frequency coverage is further investigated using an analytical toy model.