Secular Extragalactic Parallax: Measurement Methods and Predictions for
  Gaia
  Secular extragalactic parallax caused by the solar system's velocity relative
to the cosmic microwave background rest frame may be observable as a dipole
proper motion field with amplitude $78~\mu$as yr$^{-1}$ Mpc. Nearby galaxies
also exhibit proper motions caused by their transverse peculiar velocities that
prevent detection of secular parallax for any single galaxy, although a
statistical detection may be made instead. Such a detection could constrain the
local Hubble parameter. We present methods to measure secular parallax using
correlated extragalactic proper motions and find a first limit on the secular
parallax amplitude using proper motions of 232 nearby galaxies from Gaia Data
Release 2. The recovered dipole has insignificant upper limit of 3500 $\mu$as
yr$^{-1}$ Mpc. This measurement will be improved by larger sample size and
reduced proper motion uncertainties in future data releases. Using the local
peculiar velocity field derived from Cosmicflows-3, we simulate galaxy proper
motions and predict that a significant detection ($5-10\sigma$) of the secular
parallax amplitude will be possible by Gaia's end of mission. The detection is
contingent on proper motions of nearby ($<5$ Mpc), bright ($G<15$ mag)
galaxies, and corresponds to an insignificant upper limit on the Hubble
parameter. We further investigate the implications of our simulations for the
study of transverse peculiar velocities, which we find to be consistent with
large scale structure theory. The peculiar velocity field additionally results
in low-multipole correlated proper motions on the order of $0.3~\mu$as
yr$^{-1}$ that may be confounded with other cosmological proper motion
measurements, such as limits on the gravitational wave background and the
anisotropy of the Hubble expansion.
