Testing the Cosmological Principle: Astrometric Limits on Systemic
  Motion of Quasars at Different Cosmological Epochs
  A sample of $60,410$ bona fide optical quasars with astrometric proper
motions in Gaia EDR3 and spectroscopic redshifts above 0.5 in an oval 8400
square degree area of the sky is constructed. Using orthogonal Zernike
functions of polar coordinates, the proper motion fields are fitted in a
weighted least-squares adjustment of the entire sample and of six equal bins of
sorted redshifts. The overall fit with 37 Zernike functions reveals a
statistically significant pattern, which is likely to be of instrumental
origin. The main feature of this pattern is a chain of peaks and dips mostly in
the R.A. component with an amplitude of 25~$\mu$as yr$^{-1}$. This field is
subtracted from each of the six analogous fits for quasars grouped by redshifts
covering the range 0.5 through 7.03, with median values 0.72, 1.00, 1.25, 1.52,
1.83, 2.34. The resulting residual patterns are noisier, with formal
uncertainties up to 8~$\mu$as yr$^{-1}$ in the central part of the area. We
detect a single high-confidence Zernike term for R.A. proper motion components
of quasars with redshifts around 1.52 representing a general gradient of 30
$\mu$as yr$^{-1}$ over $150\degr$ on the sky. We do not find any small- or
medium-scale systemic variations of the residual proper motion field as
functions of redshift above the $2.5\,\sigma$ significance level.
