On the propagation of gravitational waves in a $\Lambda$CDM universe

We study here how the presence of non-zero matter density and a cosmological constant could affect the observation of gravitational waves in Pulsar Timing Arrays. Conventionally, the effect of matter and cosmological constant is included by considering the redshift in frequency due to the expansion. However, there is an additional effect due to the change of coordinate systems from the natural ones in the region where waves are produced to the ones used to measure the pulsar timing residuals. This change is unavoidable as the strong gravitational field in a black hole merger distorts clocks and rules. Harmonic waves produced in such a merger become anharmonic when detected by a cosmological observer. The effect is small but appears to be observable for the type of gravitational waves to which PTA are sensitive and for the favoured values of the cosmological parameters.