Separating E and B types of CMB polarization on an incomplete sky Wen Zhao Based on: WZ and D.Baskaran, Phys.Rev.D 82 023001 (2010) 2019/9/3

OUTLINE Polarization of the CMB fields Separating E and B types of polarization Unbiased estimators of B-mode polarization Summary 2019/9/3

E and B types of polarization of the CMB fields 2019/9/3

E-Polarization & B-Polarization Thomson Scatter and Generation of Polarization Two factors: ** Temperature Anisotropies ** Free electrons Two stages: @ recombination stage @ reionization stage E-Polarization & B-Polarization 2019/9/3

CMB power spectra 2019/9/3

Various contaminations in observations instrumental noises foreground noises including synchrotron, dust, free-free emissions cosmic weak lensing various systematics including differential gain, differential beam width, first order pointing error, differential ellipticity of the beam, differential rotation, et al. E-B mixture for the incomplete sky survey 2019/9/3

Separating E and B-modes on an incomplete sky ------- The problem: E-B mixture 2019/9/3

Separating E and B-modes on an incomplete sky -------- Difficulty of the general method 2019/9/3

Separating E and B-modes on an incomplete sky -------- A new method We adopt a different but related definition for electric and magnetic polarization maps: The relation between this definition and the general definition: 2019/9/3

Separating E and B-modes on an incomplete sky -------- Applying to the incomplete sky So, in principle, we can construct the pure E and B types of polarization in the area where the window function 2019/9/3

Separating E and B-modes on an incomplete sky -------- Separation in the pixel space We assume a toy model: (1) observe the northern hemisphere; (2) no B-mode in the input model. We adopt a typical window function: 2019/9/3

Separating E and B-modes on an incomplete sky -------- Numerical errors 2019/9/3

Separating E and B-modes on an incomplete sky -------- Numerical errors 2019/9/3

Separating E and B-modes on an incomplete sky -------- Applying small-sky surveys From now, we shall discuss the application of our method to the small partial sky surveys, which could be the case of various ground-based (i.e. QUIET, PolarBear) or balloon-borne (i.e. EBEX, PAPPER) experiments. We shall work with a small fraction of the sky characterized by the window function with (corresponding to a 3% sky survey) The input model, we assume the B-mode is generated by the cosmic weak lensing and gravitational waves (tensor perturbations) with r=0.1. 2019/9/3

Separating E and B-modes on an incomplete sky -------- Applying to small-sky surveys We simulate 1000 random samples of (Q, U), and construct the pure B-type polarization. We study the average of these samples. This figure shows the averaged power spectrum of the B-mode polarization. N.B. - adopt N_{side}=512 - edge removal at W=0.1 2019/9/3

Separating E and B-modes on an incomplete sky -------- Comparing with the analytical results 2019/9/3

Separating E and B-modes on an incomplete sky -------- Unbiased estimators for C_l^{BB} There are mainly three ways to build the unbiased estimators of the power spectra: * Estimation using the pseudo-Cl (PCL) Fast, but having larger error bars! * Maximum likelihood estimation (ML) Smaller error bars, but slow! * Unbiased hybrid estimator Combining a ML at low multipoles and PCL at high multipoles! 2019/9/3

Separating E and B-modes on an incomplete sky -------- Estimation using the pseudo-Cl 2019/9/3

Separating E and B-modes on an incomplete sky -------- Ideal case without instrumental noise Best choice of the weight function (future work)? 2019/9/3

Separating E and B-modes on an incomplete sky -------- loss of information The information loss is only caused by the edge removal (in order to decrease the numerical errors). In above case, we have removed 0.6% of the full sky, i.e. W=0.1 (conserved consideration). This information loss can be reflected by increasing the error bars of the data (blue: ideal lossless case; red: realistic case). 2019/9/3

Separating E and B-modes on an incomplete sky -------- Applying to QUIET experiment 2019/9/3

Summary We propose a new method to separate the E and B types of polarization on an incomplete sky. The characters of this method are: The E and B fields are constructed in real space with minimal loss of information. The method is fast and can be applied to high resolution maps. The constructed maps are scalar fields. So various techniques (i.e. unbiased estimator, non-Gaussian analysis and so on) developed to deal with T maps can be directly applied to analyze these fields. The method can be applied to the future observations, including Planck mission, ground-based and balloon-borne experiments. 2019/9/3

Thanks! 2019/9/3