Recent Developments in the Calculation of CMB Anisotropies

Abstract

We discuss three recent developments in the calculation of Cosmic Microwave Background (CMB) anisotropies. We begin with a discussion of the relativistic corrections to the Sunyaev-Zel’dovich effect. By extending the Kompaneets equation to include relativistic effects, we are able to to derive simple analytic forms for the spectral changes due to the thermal Sunyaev-Zel’dovich effect. Relativistic corrections result in a small reduction in the amplitude of the effect over the Rayleigh-Jeans region, which for a typical cluster temperature of 8 keV, amounts to a correction downwards to the value of the Hubble constant derived from combined X-ray and Sunyaev-Zel’dovich information by about 5 percent. Our second topic is a discussion of covariant kinetic theory methods for the gauge-invariant calculation of primordial CMB anisotropies. We present a covariant version of the Boltzmann equation, which we use to derive a set of covariant equations for the gauge-invariant variables in a Cold Dark Matter (CDM) model, which are independent of the background curvature and type of perturbation. Equations describing a particular type of perturbation are obtained readily from this set, as we demonstrate for scalar perturbations in a K=0 universe. By integrating the covariant Boltzmann equation along the line of sight, and using the instantaneous recombination approximation, we obtain an expression for the large scale anisotropy which agrees with an expression derived recently by Dunsby. Finally, we discuss some recent accurate calculations of the CMB anisotropy in global defect models. In such models, the effect of vorticity generation by the causal sources proves to be significant, leading to a suppression of acoustic peaks. The result is that global defect models of structure formation may already be at variance with the growing volume of CMB (and large scale structure) data.