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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.
We relate the observable number of sources per solid angle and redshift to the underlying proper source density and velocity, background evolution and line-of-sight potentials. We give an exact result in the case of linearized perturbations assuming general relativity. This consistently includes contributions of the source density perturbations and redshift distortions, magnification, radial displacement, and various additional linear terms that are small on sub-horizon scales. In addition we calculate the effect on observed luminosities, and hence the result for sources observed as a function of flux, including magnification bias and radial-displacement effects. We give the corresponding linear result for a magnitude-limited survey at low redshift, and discuss the angular power spectrum of the total count distribution. We also calculate the cross-correlation with the CMB polarization and temperature including Doppler source terms, magnification, redshift distortions and other velocity effects for the sources, and discuss why the contribution of redshift distortions is generally small. Finally we relate the result for source number counts to that for the brightness of line radiation, for example 21-cm radiation, from the sources.
PolSpice (aka Spice) is a tool to statistically analyze Cosmic Microwave Background (CMB) data, as well as any other diffuse data pixelized on the sphere.
This Fortran90 program measures the 2 point auto (or cross-) correlation functions w(θ) and the angular auto- (or cross-) power spectra C(l) from one or (two) sky map(s) of Stokes parameters (intensity I and linear polarisation Q and U). It is based on the fast Spherical Harmonic Transforms allowed by isolatitude pixelisations such as Healpix [for Npix pixels over the whole sky, and a C(l) computed up to l=lmax, PolSpice complexity scales like Npix1/2 lmax2 instead of Npix lmax2]. It corrects for the effects of the masks and can deal with inhomogeneous weights given to the pixels of the map. In the case of polarised data, the mixing of the E and B modes due to the cut sky and pixel weights can be corrected for to provide an unbiased estimate of the "magnetic" (B) component of the polarisation power spectrum. Most of the code is parallelized for shared memory (SMP) architecture using OpenMP.