CMB Spectral Distortions from the Scattering of Temperature Anisotropies

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.