Abstract: We present a fully covariant and gauge-invariant calculation of the evolution of anisotropies in the cosmic microwave background (CMB) 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, linearised equations describing the (Boltzmann) evolution of anisotropy and inhomogeneity in an almost Friedmann-Robertson-Walker (FRW) 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 non-covariant methods in specific gauges, but are derived here in a completely transparent fashion.
Credit: Lewis, Antony; Challinor, Anthony
Reason: Updated code entry.
* Improved speed and accuracy of lensed CMB calculation at L>5000
* More accurate Limber approximation for lensing potential
* A few internal changes for greater consistency with CAMB sources
* Updated high-L template file for Planck parameters, non-linear lensing and higher L
Note that this version changes numerical results compared to the March version used for
the first Planck analysis, though only at a non-significant level (importance weights
typically 1 to with 10%).