  Thomson scattering of CMBR temperature anisotropies will cause the spectrum
of the CMBR to differ from blackbody even when one resolves all anisotropies. A
formalism for computing the anisotropic and inhomogeneous spectral distortions
of intensity and polarization is derived in terms of Lorentz invariant central
moments of the temperature distribution. The formalism is non-perturbative,
requiring neither small anisotropies nor small metric or matter
inhomogeneities; but it does assume cold electrons. The low order moments are
not coupled to the higher order moments allowing one to truncate the equations
without any loss of accuracy. This formalism is applied to a standard
Lambda-CDM cosmology after reionization where the temperature anisotropies are
dominated by the Doppler effect for the bulk motion of the gas with respect to
the CMBR frame. The resultant spectral distortion is parameterized by u ~ 3e-8,
where in this case u is observationally degenerate with the Sunyaev-Zel'dovich
(SZ) y parameter. In comparison the expected thermal SZ y-distortion from the
hot IGM is expected to be >30 times larger. However at z >5 the effect
described here would have been the dominant source of spectral distortions. The
effect could be much larger in non-standard cosmologies.
