The Hubble Expansion is Isotropic in the Epoch of Dark Energy

The isotropy of the universal Hubble expansion is a fundamental tenet of physical cosmology, but it has not been precisely tested during the current epoch, when dark energy is dominant. Anisotropic expansion will produce a shearing velocity field, causing objects to stream toward directions of faster expansion and away from directions of slower expansion. This work tests the basic cosmological assumption of isotropic expansion and thus the isotropy of dark energy. The simplest anisotropy will manifest as a quadrupolar curl-free proper motion vector field. We derive this theoretical signature using a tri-axial expanding metric with a flat geometry (Bianchi I model), generalizing and correcting previous work. We then employ the best current data, the Titov & Lambert (2013) proper motion catalog of 429 objects, to measure the isotropy of universal expansion. We demonstrate that the Hubble expansion is isotropic to 7% (1 $\sigma$), corresponding to streaming motions of 1 microarcsecond/yr, in the best-constrained directions (-19% and +17% in the least-constrained directions) and does not significantly deviate from isotropy in any direction. The Gaia mission, which is expected to obtain proper motions for 500,000 quasars, will likely constrain the anisotropy below 1%.