## Non-circularity of beams in the CMB polarization power spectrum estimation.

##### Abstract

Precise measurements of the Cosmic Microwave Background (CMB) anisotropies have been
one of the foremost concerns in modern cosmology as it provides valuable information on the
cosmology of the Universe. However, an accurate estimation of the CMB power spectrum faces
many challenges as the CMB experiments sensitivity increases. Furthermore, for the polarization
experiments, the precision of the measurements is complicated by the fact that the polarization
signal is very faint compared to the measured total intensity and could be impossible to detect
in the presence of high level of systematics. One of the most important source of errors in CMB
polarization experiment is the beam non-circularity (asymmetry). In addition, the non-uniform
and partial sky coverage resulting from the masking of the CMB foreground contaminants as well
as point sources bias the estimation of the power spectrum. Consequently, a reasonable estimation
of the power spectrum must account for, at least, the beam asymmetry and incomplete sky
coverage. Accurate estimation of the angular power spectrum can be done using the standard
optimal Maximum Likelihood (ML), although for high resolution CMB experiments with large
data set this method is unfeasible due to the enormous computation time involved in the process.
The focus of this research is to estimate the CMB temperature anisotropy T and E-
polarization cross-power spectrum and EE polarization power spectrum using a semi-analytical
framework, and tackle the computational challenge of the TE power spectrum estimation with
the pseudo-Cl estimator in the presence of the non-circular beam and cut-sky systematics. We
examine, in the first step, the estimation of the CMB TE power spectrum by only considering
the beam non-circularity with a complete sky, and give the error estimates of the power
spectrum. Then, we will consider the more general case that includes the effect of the beam
asymmetry and cut-sky as a result of the foreground removals across the Galactic plane. The
numerical implementation of the bias matrix presents a huge computational challenge. Our
ultimate goal is to speed-up the computation of the TE bias matrix that relates the true and
observed power spectra in the case of a full sky coverage using a non-circular beam. We adopt
a model of beams obtained from a perturbative expansion of the beam around a circular (axisymmetric)
one in harmonic space and compute the bias matrix by using an efficient algorithm
for rapid computation.
We show in this work that, in the case of non-circular beams and full sky survey, a fast
computation of the TE bias matrix can be performed in few seconds using a single CPU processor
by means of precomputations and insertion of symmetry relations in the initial analytical
expression of the TE bias matrix. We present as well in the last part of this research the first
analytical results of the EE bias matrix calculations in the case of a CMB experiment using
non-circular beams and incomplete sky coverage, and derive the corresponding results for the
non-circular beams and full sky limit.