  During and after recombination, in addition to Thomson scattering with free
electrons, photons also coupled to neutral hydrogen and helium atoms through
Rayleigh scattering. This coupling influences both CMB anisotropies and the
distribution of matter in the Universe. The frequency-dependence of the
Rayleigh cross section breaks the thermal nature of CMB temperature and
polarization anisotropies and effectively doubles the number of variables
needed to describe CMB intensity and polarization statistics, while the
additional atomic coupling changes the matter distribution and the lensing of
the CMB. We introduce a new method to capture the effects of Rayleigh
scattering on cosmological power spectra. Rayleigh scattering modifies CMB
temperature and polarization anisotropies at the $\sim\!1 \%$ level at $353
{\rm GHz}$ (scaling $\propto \nu^4$), and modifies matter correlations by as
much as $\sim\!0.3\%$. We show the Rayleigh signal, especially the
cross-spectra between the thermal (Rayleigh) E-polarization and Rayleigh
(thermal) intensity signal, may be detectable with future CMB missions even in
the presence of foregrounds, and how this new information might help to better
constrain the cosmological parameters.
