Objects Appear Smaller as They Recede: How Proper Motions Can Directly
  Reveal the Cosmic Expansion, Provide Geometric Distances, and Measure the
  Hubble Constant
  Objects and structures gravitationally decoupled from the Hubble expansion
will appear to shrink in angular size as the universe expands. Observations of
extragalactic proper motions can thus directly reveal the cosmic expansion.
Relatively static structures such as galaxies or galaxy clusters can
potentially be used to measure the Hubble constant, and test masses in large
scale structures can measure the overdensity. Since recession velocities and
angular separations can be precisely measured, apparent proper motions can also
provide geometric distance measurements to static structures. The apparent
fractional angular compression of static objects is 15 microarcseconds per year
in the local universe; this motion is modulated by the overdensity in dynamic
expansion-decoupled structures.
  We use the Titov et al. quasar proper motion catalog to examine the pairwise
proper motion of a sparse network of test masses. Small-separation pairs (< 200
Mpc comoving) are too few to measure the expected effect, yielding an
inconclusive 8.3 +/- 14.9 microarcsec/yr. Large-separation pairs (200-1500 Mpc)
show no net convergence or divergence for z < 1, -2.7 +/- 3.7 microarcsec/yr,
consistent with pure Hubble expansion and significantly inconsistent with
static structures, as expected. For all pairs a "null test" gives -0.36 +/-
0.62 microarcsec/yr, consistent with Hubble expansion, and excludes a static
locus at ~5-10 sigma significance for z ~ 0.5-2.0. The observed
large-separation pairs provide a reference frame for small-separation pairs
that will significantly deviate from the Hubble flow. The current limitation is
the number of small-separation objects with precise astrometry, but Gaia will
address this and will likely detect the cosmic recession.
