Separating $E$ and $B$ types of polarization on an incomplete sky

Detection of magnetic-type ($B$-type) polarization in CMB radiation plays a crucial role in probing the relic gravitational wave background. In this paper, we propose a new method to deconstruct a polarization map on an incomplete sky in real space into purely electric and magnetic polarization-type maps, $\mathcal{E}(\widehat{\gamma })$ and $\mathcal{B}(\widehat{\gamma })$, respectively. The main properties of our approach are as follows: First, the fields $\mathcal{E}(\widehat{\gamma })$ and $\mathcal{B}(\widehat{\gamma })$ are constructed in real space with minimal loss of information. This loss of information arises due to the removal of a narrow edge of the constructed map in order to remove various numerical errors, including those arising from finite pixel size. Second, this method is fast and can be efficiently applied to high resolution maps due to the use of the fast spherical harmonics transformation. Third, the constructed fields, $\mathcal{E}(\widehat{\gamma })$ and $\mathcal{B}(\widehat{\gamma })$, are scalar fields. For this reason various techniques developed to deal with temperature anisotropy maps can be directly applied to analyze these fields. As a concrete example, we construct and analyze an unbiased estimator for the power spectrum of the $B$ mode of polarization $C_{\ell }^{BB}$. Basing our results on the performance of this estimator, we discuss the relic gravitational wave detection ability of two future ground-based CMB experiments, the Q/U Imaging Experiment and Polarization of Background Microwave Radiation.

Figure 1

The $\tilde{\mathcal{B}}$ map constructed from an input model with no magnetic polarization (in $\mu \mathrm{K}$).

Figure 2

The pure magnetic field ${\mathcal{B}}_{\mathrm{rec}}(\widehat{\gamma })$. The edge of the map with $W(\widehat{\gamma })<0.03$ has been removed. The left panel is scaled similarly to Fig. 1, while the right panel has the scaling magnified in order to show the residual leakage. Both panels use the units $\mu \mathrm{K}$.

Figure 3

The pseudo power spectrum $D_{\ell }^{\tilde{\mathcal{B}}\tilde{\mathcal{B}}}$ (black line, i.e. line 1) of the $\tilde{\mathcal{B}}$ field and the pseudo power spectrum $D_{\ell }$ (red line, i.e. line 2) of the ${\mathcal{B}}_{\mathrm{rec}}$ field.

Figure 4

The red line (i.e. line 2) shows the pseudo power spectrum $D_{\ell }$ calculated for the edge removal at $W=0.03$, while the green line (i.e. line 3) shows the $D_{\ell }$ for edge removal at $W=0.1$. For comparison, the black line (i.e. line 1) shows the pseudo power spectrum $D_{\ell }^{\tilde{\mathcal{B}}\tilde{\mathcal{B}}}$. Note that the black line and the red line are identical to those in Fig. 3.

Figure 5

The red line (i.e. line 2) shows the pseudo power spectrum $D_{\ell }$ calculated for $N_{\mathrm{side}}=512$, while the green line (i.e. line 3) shows $D_{\ell }$ calculated for $N_{\mathrm{side}}=1024$. For comparison, the black line (line 1) shows the pseudo power spectrum $D_{\ell }^{\tilde{\mathcal{B}}\tilde{\mathcal{B}}}$. Note that the black line and the red line are identical to those in Fig. 3.

Figure 6

The polarization power spectra generated by density perturbations (dp), gravitational waves (gw) with $r=0.1$, and cosmic lensing (lens).

Figure 7

The red line (i.e. line 2) shows the pseudo power spectrum $D_{\ell }$, calculated for a map with ${\theta }_F={30}^{\prime }$, while the green line (i.e. line 4) shows $D_{\ell }$ calculated with ${\theta }_F={10}^{\prime }$. The black line (i.e. line 1) shows $D_{\ell }^{\tilde{\mathcal{B}}\tilde{\mathcal{B}}}$ calculated for ${\theta }_F={30}^{\prime }$, and the blue line (i.e. line 3) shows $D_{\ell }^{\tilde{\mathcal{B}}\tilde{\mathcal{B}}}$ for ${\theta }_F={10}^{\prime }$. Note that the black line and the red line are identical to those in Fig. 3.

Figure 8

The averaged pseudo estimator $D_{\ell }$ from 1000 simulations. The red line (i.e. line 2) shows the result for an input model with $r=0.1$ and with a contribution from cosmic lensing. The green line (i.e. line 3) shows the result for an input model with no magnetic polarization (i.e. $C_{\ell }^{BB}=0$). For comparison, the black line (i.e. line 1) shows the averaged estimator $D_{\ell }^{\tilde{\mathcal{B}}\tilde{\mathcal{B}}}$ for an input model with no magnetic polarization.

Figure 9

All of the lines are a result of averaging over 1000 simulated samples. The red line (i.e. line 2) shows the averaged pseudo estimator $D_{\ell }$ for an input model with $r=0.1$ and with a contribution from cosmic lensing. The red dashed line (i.e. line 3) shows the square root of the average of the diagonal terms in the covariance matrix $(\overline{\Delta D_{\ell }\Delta D_{\ell }}{)}^{1/2}$. The blue solid line (i.e. line 1) shows the analytical result of $⟨D_{\ell }⟩$, for a model with $r=0.1$ and with a contribution from cosmic lensing. The two contributions are shown separately with a magenta solid line (i.e. line 5) for the gravitational waves and a green solid line (i.e. line 6) for the cosmic lensing. The blue dashed line (i.e. line 4) shows the analytical approximation (41) for $⟨\Delta D_{\ell }\Delta D_{\ell }{⟩}^{1/2}$.

Figure 10

The averaged values (larger red dots) and error bars (larger red error bars) following from simulations of the unbiased estimators of the power spectra $\ell (\ell +1)C_{\ell }^{BB}(\mathrm{total})/2\pi$ (left panel) and $\ell (\ell +1)C_{\ell }^{BB}(\mathrm{gw})/2\pi$ (right panel). In both panels, the black solid line denotes the theoretical values of the underlying power spectra. For comparison, in both panels we plot the averaged values (smaller blue dots) and simulated error bars (smaller blue error bars) of the unbiased estimators for an ideal case without information loss (see text for details). In both panels, we have considered a case with no instrumental noise.

Figure 11

The ratio of practically achievable error bars (red error bars in Fig. 10) to the corresponding error bars in an information lossless case (blue error bars in Fig. 10), as a function of the multipole $\ell$.

Figure 12

The averaged values and simulated error bars of the unbiased estimators for the power spectra $\ell (\ell +1)C_{\ell }^{BB}/2\pi$ (green dots and larger green error bars) and $\ell (\ell +1)C_{\ell }^{BB}(\mathrm{gw})/2\pi$ (red dots and larger red error bars). In both panels, the solid lines denote the theoretical values for these power spectra. In this figure, we have considered the instrumental noise for the QUIET experiment. The left panel shows the results for an input cosmological model with $r=0.1$, while the right panel shows the results for an $r=0.01$ model. In both panels, the smaller error bars calculated using the analytical approximation (54) are plotted in grey.

Figure 13

The results for the POLARBEAR experiment. The averaged values and simulated error bars of the unbiased estimators for the power spectra $\ell (\ell +1)C_{\ell }^{BB}/2\pi$ (green dots and larger green error bars) and $\ell (\ell +1)C_{\ell }^{BB}(\mathrm{gw})/2\pi$ (red dots and larger red error bars) are calculated for an input model with $r=0.1$. The solid lines denote the theoretical values for these power spectra. The smaller error bars calculated using the analytical approximation (54) are plotted in grey.

Figure 14

The averaged values (smaller blue and larger red dots) and error bars (smaller blue and larger red error bars) following from simulations of the unbiased estimators of the power spectra $\ell (\ell +1)C_{\ell }^{BB}(\mathrm{total})/2\pi$ (left panel) and $\ell (\ell +1)C_{\ell }^{BB}(\mathrm{gw})/2\pi$ (right panel). In both panels, the black solid line denotes the theoretical values of the underlying power spectrum. The smaller blue dots and error bars denote the result by adopting the weight function $\mathcal{W}(\widehat{\gamma })=W(\widehat{\gamma })$, and the larger red dots and error bars denote the result by adopting a uniform weight function $\mathcal{W}(\widehat{\gamma })$. Note that, in both panels, we have considered a case with no instrumental noise. The larger red dots and error bars are identical to those in Fig. 10.

Figure 15

The red line (i.e. line 2) shows the pseudo power spectrum $D_{\ell }$ calculated for a map with ${\theta }_F={30}^{\prime }$. The green line (i.e. line 3) shows $D_{\ell }$ constructed for a map which was smoothed from ${\theta }_F={10}^{\prime }$ to ${\theta }_F={30}^{\prime }$. The blue line (i.e. line 4) shows the difference between the green and the red lines and corresponds to the leakage from $\mathcal{E}$ polarization to $\mathcal{B}$ polarization due to the map smoothing. For comparison, the black line (i.e. line 1) shows the pseudo power spectrum $D_{\ell }^{\tilde{\mathcal{B}}\tilde{\mathcal{B}}}$. Note that the black line and the red line are identical to those in Fig. 3.