Class. Quantum Grav. 14 (1997) 3391–3405. Printed in the UK PII: S0264-9381(97)82944-9 A fully covariant description of cosmic microwave background anisotropies

Abstract. Starting from the exact non-linear description of matter and radiation, a fully covariant and gauge-invariant formula for the observed temperature anisotropy of the cosmic microwave background radiation, expressed in terms of the electric (Eab) and magnetic (Hab) parts of the Weyl tensor, is obtained by integrating photon geodesics from last scattering to the point of observation today. This improves and extends earlier 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 (density) perturbations, Eab is related to the harmonic components of the gravitational potential 8k and the usual dominant Sachs–Wolfe contribution TR = NTR 8k 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 important in low density universes. We also explicitly demonstrate the validity of assuming that the perturbations are adiabatic at decoupling and show that if the surface of last scattering is correctly placed and the background universe model is taken to be a flat dust-dominated Friedmann–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 surface of last scattering on those scales. PACS numbers: 0430N, 9880 1