A FULLY COVARIANT DESCRIPTION OF COSMIC MICROWAVE BACKGROUND ANISOTROPIES

Starting from the exact non-linear description of matter and radiation , a fully covariant and gauge-invariant formula for the observed tempe rature anisotropy of the cosmic microwave background radiation, expres sed in terms of the electric (E-ab) and magnetic (Hob) parts of the We yl tensor, is obtained by integrating photon geodesics from last scatt ering to the point of observation today. This improves and extends ear lier work by Russ et al where a similar formula was obtained by taking first-order variations of the red-shift. In the case of scalar (densi ty) perturbations, E-ab is related to the harmonic components of the g ravitational potential Phi(k) and the usual dominant Sachs-Wolfe contr ibution delta T-R/(T) over bar(R) similar to Phi(k) to the temperature anisotropy is recovered, together with contributions due to the time variation of the potential (Rees-Sciama effect), entropy and velocity perturbations at last scattering and a pressure suppression term impor tant in low density universes. We also explicitly demonstrate the vali dity of assuming that the perturbations are adiabatic at decoupling an d show that if the surface of last scattering is correctly placed and the background universe model is taken to be a flat dust-dominated Fri edmann-Robertson-Walker model (FRW), then the large-scale temperature anisotropy can be interpreted as being due to the motion of the matter relative to the surface of constant temperature which defines the sur face of last scattering on those scales.