Covariant cosmic microwave background anisotropies; 2, Nonlinear dynamics

We develop a new approach to local nonlinear effects in cosmic microwave background anisotropies, and discuss the qualitative features of these effects. We find new couplings of baryonic velocity to radiation multipoles through Thomson scattering, and a new nonlinear shear effect on small angular scales. This is the second paper of a series which develops a covariant and gauge-invariant approach to the Einstein-Boltzmann-hydrodynamic equations in an inhomogeneous universe. The first paper developed the covariant algebra of anisotropies. Here we present the covariant dynamics, i.e. the nonlinear evolution and constraint equations that govern the radiation multipoles and the local quantities describing inhomogeneity. Our nonlinear equations provide a direct basis for a covariant second-order analysis of temperature anisotropies in a universe that is close to the Friedmann models. We identify the nonlinearities that arise from inhomogeneity and relative motions, and describe their effect on radiation anisotropies. The nonlinear equations are also specialized to the case of nearly isotropic radiation in universes where the anisotropy and inhomogeneity are not limited to be small, which includes the case when the background spacetime is a Bianchi model. Finally, we covariantly and gauge-invariantly linearize the equations about a Friedmann background, which is the basis for a covariant analysis of scalar, vector and tensor perturbations. Such an analysis, and other applications of the equations given here, are taken up in further papers of the series