Inferred Hubble parameter from gravitational waves in a perturbative Bianchi I background

It is straightforward to take the gravitational wave solution to first order in $v/c$ far from a binary source in a Minkowski background and adapt it to the Friedmann-Lemaitre-Robertson-Walker (FLRW) background, representing an expanding isotropic and homogeneous universe. We find the analogous solution for a slightly anisotropic background, which may be a more accurate description of our late universe through which gravitational waves propagate, and implications from tight CMB anisotropy constraints may not necessarily determine the level of anisotropy in the late universe in light of modified gravity models as well as the Hubble tension. We use a perturbative form of the Bianchi I metric and demonstrate how the waveform differs. Using supernova anisotropy data as a reference, we show that the assumption of a Bianchi I background could imply on average a 2.1% difference in inferred luminosity distance compared to what would be inferred under the assumption of the FLRW background. This difference can be as high as 5.9% depending on the observation direction. Therefore, the background spacetime used for the inference of the Hubble parameter from gravitational wave data should be considered carefully.