Cosmic microwave background temperature and polarization pseudo-C_ℓestimators and covariances

Abstract

We develop the pseudo-C method for reconstructing the cosmic microwave background (CMB) temperature and polarization auto-and cross-power spectra, and estimate the pseudo-C covariance matrix for a realistic experiment on the cut sky. We calculate the full coupling equations for all six possible CMB power spectra, relating the observed pseudo-C values to the underlying all-sky C values, and test the reconstruction on both full-sky and cut-sky simulated CMB data sets. In particular we consider the reconstruction of the C from upcoming ground-based polarization experiments covering areas of a few hundred deg 2 and find that the method is fast, unbiased and performs well over a wide range of multipoles from = 2 to = 2500. We then calculate the full covariance matrix between the modes of the pseudo-temperature and polarization power spectra, assuming that the underlying CMB fields are Gaussian randomly distributed. The complexity of the covariance matrix prohibits its rapid calculation, required for parameter estimation. Hence we present an approximation for the covariance matrix in terms of convolutions of the underlying power spectra. The coupling matrices in these expressions can be estimated by fitting to numerical simulations, circumventing direct and slow calculation, and further, inaccurate analytic approximations. We show that these coupling matrices are mostly independent of cosmology, and that the full covariance matrix for all six pseudo-C power spectra can be quickly and accurately calculated for any given cosmological model using this method. We compare these semi-analytic covariance matrices against simulations and find good agreement, the accuracy of which depends mainly on survey area and the range of cosmological parameters considered.

Statistical isotropy of CMB polarization maps

“…For cut-sky, E(n) and B(n) mode decomposition is not unique [6,7]. But since mixing is linear there always exist two linearly independent modes.…”

“…In the latter case, the incomplete sky coverage immediately induces a contamination of E-mode of polarization by its B-mode and vice-versa. Then the modified temperature and polarization fields is related to their actual values of full sky coverage by a window matrix [6,7] whose elements are basically window functions for temperature and polarization in harmonic space. It can be shown that the estimated BipoSH coefficients are in fact linear combinations of that for fullsky CMB maps.…”

Separating weak lensing and intrinsic alignments using radio observations

“…We estimate the recovered power spectra from the reconstructed shear and IA maps using the standard pseudo-C l approach (Brown & Battye 2011b;Hivon et al 2002;Brown et al 2005).…”

Bayesian inference on the sphere beyond statistical isotropy

“…Last maximum likelihood methods on CMB dataset were used in some parts of the Boomerang and Maxima datasets. Since N pix increased in the modern experiments, maximum likelihood methods are not a viable option and pseudo-C l methods [11][12][13] are used since then. We can explore the full posterior distribution with Monte Carlo sampling.…”