Effect of the cosmological parameters on gravitational waves: general analysis

Some time ago it was pointed out that the presence of cosmological components could affect the propagation of gravitational waves (GW) beyond the usual cosmological redshift and that such effects might be observable in pulsar timing arrays. These analyses were done at leading order in the Hubble constant $H_0$, which is proportional to $\Lambda^{1/2}$ and $\rho_i^{1/2}$ ($\rho_i$ being the various cosmological fluid densities). In this work, we study in detail the propagation of metric perturbations on a Schwarzschild-de Sitter (SdS) background, close to the place where GW are produced, and obtain solutions that incorporate corrections linear in $\rho_i$ and $\Lambda$. At the next-to-leading order the corrections do not appear in the form of $H_0$ thus lifting the degeneracy among the various cosmological components. We also determine the leading corrections proportional to the mass of the final object; they are very small for the distances considered in pulsar timing arrays but may be of relevance in other cases. When transformed into comoving coordinates, the ones used in cosmological measurements, this SdS solution does satisfy the perturbation equations in a Friedmann-Lema\^itre-Robertson-Walker metric up to and including $\Lambda^{3/2}$ terms. This analysis is then extended to the other cosmological fluids, allowing us to consider GW sources in the Gpc range. Finally, we investigate the influence of these corrections in pulsar timing arrays observations.