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 FLRW metric, representing an expanding isotropic homogeneous universe. We find the analogous solution for a slightly anisotropic background, represented by 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 a 3.2% difference in inferred luminosity distance compared to what would be inferred under the assumption of the FLRW background.