  We study the spin transport of light in weakly inhomogeneous axion field in a
flat Robertson-Walker universe and derive the spin Hall effect for circularly
polarized rays. Regarding primordial quantum fluctuations of the axion field in
the de Sitter phase as the origin of the inhomogeneity, we show that the
conformal invariance of the correlator determines the root-mean-square (r.m.s)
fluctuations of the path of circularly polarized cosmic rays. We explain how
the r.m.s fluctuations can be experimentally determined.
