CMB polarization TE power spectrum estimation with non-circular beam
  Precise measurements of the Cosmic Microwave Background (CMB) anisotropy 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 polarization experiments the precision
of measurement is complicated by the fact that the polarisation 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 asymmetry. For
large data set the estimation of the CMB polarization power spectrum with
standard optimal Maximum Likelihood (ML) is prohibitive for high resolution CMB
experiments due to the enormous required computation time. In this paper, we
present a semi-analytical framework using the pseudo-$C_{l}$ estimator to
compute the power spectrum TE of the temperature anisotropy and the E-component
of the polarization radiation field using non-circular beams. 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 resulting bias matrix
which relates the true power spectrum with the observed one using an efficient
algorithm for rapid computation. We show that for a multipole up to
$l_{max}=500$, the bias matrix can be computed in less than one second with a
single CPU processor at 2.53 GHz. We find that the uncertainties induced by the
beam asymmetry in the polarization power spectrum at the peak of the bias
matrix for WMAP and Planck experiments can be as large as a few 10 to 20%
(upper limit for Planck LFI 30 GHz)
