Cosmic microwave background anisotropies in the cold dark matter model: A covariant and gauge-invariant approach

We present a fully covariant and gauge-invariant calculation of the evolution of anisotropies in the cosmic microwave background radiation. We use the physically appealing covariant approach to cosmological perturbations, which ensures that all variables are gauge-invariant and have a clear physical interpretation. We derive the complete set of frame-independent linearized equations describing the (Boltzmann) evolution of anisotropy and inhomogeneity in an almost Friedmann-Robertson-Walker cold dark matter (CDM) universe. These equations include the contributions of scalar, vector, and tensor modes in a unified manner. Frame-independent equations for scalar and tensor perturbations, which are valid for any value of the background curvature, are obtained straightforwardly from the complete set of equations. We discuss the scalar equations in detail, including the integral solution and relation with the line-of-sight approach, analytic solutions in the early radiation-dominated era, and the numerical solution in the standard CDM model. Our results confirm those obtained by other groups, who have worked carefully with noncovariant methods in specific gauges, but ours are derived here in a completely transparent fashion.