Bulk flows in the local Universe and the importance of relativistic effects

Bulk flow velocities are typically estimated in the idealised picture where observers are moving within a perfectly homogeneous and isotropic space-time. This picture is consistent within standard perturbation theory up to relativistic effects that lead to correction terms of order $v z$, where $z$ is the redshift of observation, and $v$ is the amplitude of the bulk flow. The dominant relativistic contributions at scales $z \lesssim 1$ are caused by gravitational redshift and time evolution of the velocity field. We include these effects within a broadly applicable weak-field approximation, and provide a cosmographic formula for estimating bulk flows at a high precision. Based on this formula, we judge that recent bulk flow estimates are biased toward larger values by $\sim 10\%$. This theoretical bias surpasses the measurement biases of the same estimates, and it will become still more important to account for the relativistic effects as the scales at which bulk flows are estimated to increase.