Fast Scalar Quadratic Maximum Likelihood Estimators for the CMB B-mode Power Spectrum

The linear polarization of the CMB field can be completely described by the Stokes parameters Q and U. Along the line of sight, $\hat{n}$, the polarization field can be written as:

$$\begin{eqnarray}&&{P}_{\pm }(\hat{n})=Q(\hat{n})\pm {iU}(\hat{n}),\end{eqnarray} \tag{ 1 }$$

which behaves as a spin-(2) and a spin-(−2) field. In the simplest full sky case, these fields are expanded over spin-weighted spherical harmonics, _±s Y_ℓm , as (Seljak & Zaldarriaga 1996):

$$\begin{eqnarray}&&{P}_{\pm }(\hat{n})=\sum _{{\ell }m}{a}_{\pm 2,{\ell }m}\,{}_{\pm 2}{Y}_{{\ell }m}(\hat{n}).\end{eqnarray} \tag{ 2 }$$

Detailed expressions are given in Appendix A.

However, describing the polarization field by means of the the Stokes parameters is frame dependent. Therefore, for convenience, we write the polarization field in terms of the rotationally invariant E and B components. These components are defined in the harmonic space in terms of spin-harmonic coefficients a_{±2,ℓ m} .

$$\begin{eqnarray}\begin{array}{rcl}{a}_{E,{\ell }m} & \equiv & -\displaystyle \frac{1}{2}[{a}_{2,{\ell }m}+{a}_{-2,{\ell }m}]\\ & = & -\displaystyle \frac{1}{2}\left[\displaystyle \int {P}_{+}\left(\hat{n}\right){}_{2}{Y}_{{\ell }m}^{* }\left(\hat{n}\right)d\hat{n}+\displaystyle \int {P}_{-}\left(\hat{n}\right){}_{-2}{Y}_{{\ell }m}^{* }\left(\hat{n}\right)d\hat{n}\right],\end{array}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}\begin{array}{rcl}{a}_{B,{\ell }m} & \equiv & -\displaystyle \frac{1}{2i}[{a}_{2,{\ell }m}-{a}_{-2,{\ell }m}]\\ & = & \displaystyle \frac{i}{2}\left[\displaystyle \int {P}_{+}(\hat{n}){}_{2}{Y}_{{\ell }m}^{* }(\hat{n})d\hat{n}-\displaystyle \int {P}_{-}(\hat{n}){}_{-2}{Y}_{{\ell }m}^{* }(\hat{n})d\hat{n}\right].\end{array}\end{eqnarray} \tag{ 4 }$$

We can then define the $E(\hat{n})$ and $B(\hat{n})$ sky maps as:

$$\begin{eqnarray}&&E(\hat{n})\equiv \sum _{{\ell }m}{a}_{E,{\ell }m}{Y}_{{\ell }m}(\hat{n}),\quad B(\hat{n})\equiv \sum _{{\ell }m}{a}_{B,{\ell }m}{Y}_{{\ell }m}(\hat{n}).\end{eqnarray} \tag{ 5 }$$

Finally, the power spectra can be obtained as follows

$$\begin{eqnarray}&&{C}_{{\ell }}^{{EE}}\equiv \langle {E}_{{\ell }m}{E}_{{\ell }m}^{* }\rangle ,\quad {C}_{{\ell }}^{{BB}}\equiv \langle {B}_{{\ell }m}{B}_{{\ell }m}^{* }\rangle ,\end{eqnarray} \tag{ 6 }$$

where the angular brackets denote the average over realizations.

For an incomplete sky observation, defined by the window function $W(\hat{n})$, in principle, we can also define the partial sky E- and B-mode spherical harmonic coefficients (indicated by the overhead tilde) as:

$$\begin{eqnarray}&&{\tilde{a}}_{E,{\ell }m}=-\displaystyle \frac{1}{2}\left[\int {P}_{+}\,W\,{}_{2}{Y}_{{\ell }m}^{* }d\hat{n}+\int {P}_{-}\,W\,{}_{-2}{Y}_{{\ell }m}^{* }d\hat{n}\right],\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{\tilde{a}}_{B,{\ell }m}=\displaystyle \frac{i}{2}\left[\int {P}_{+}\,W\,{}_{2}{Y}_{{\ell }m}^{* }d\hat{n}-\int {P}_{-}\,W\,{}_{-2}{Y}_{{\ell }m}^{* }d\hat{n}\right].\end{eqnarray} \tag{ 8 }$$

These coefficients ${\tilde{a}}_{E,{\ell }m}$ and ${\tilde{a}}_{B,{\ell }m}$ relate to the pure E and B coefficients a_{E,ℓ m} and a_{B,ℓ m} as follows:

$$\begin{eqnarray}\begin{array}{rcl}{\tilde{a}}_{E,{\ell }m} & = & \sum _{{\ell }^{\prime} m^{\prime} }\left[{K}_{{\ell }m{\ell }^{\prime} m^{\prime} }^{{EE}}{a}_{E,{\ell }^{\prime} m^{\prime} }+{{iK}}_{{\ell }m{\ell }^{\prime} m^{\prime} }^{{EB}}{a}_{B,{\ell }^{\prime} m^{\prime} }\right],\\ {\tilde{a}}_{B,{\ell }m} & = & \sum _{{\ell }^{\prime} m^{\prime} }\left[-{{iK}}_{{\ell }m{\ell }^{\prime} m^{\prime} }^{{BE}}{a}_{E,{\ell }^{\prime} m^{\prime} }+{K}_{{\ell }m{\ell }^{\prime} m^{\prime} }^{{BB}}{a}_{B,{\ell }^{\prime} m^{\prime} }\right].\end{array}\end{eqnarray} \tag{ 9 }$$

The coupling matrices ${K}_{{\ell }m{\ell }^{\prime} m^{\prime} }^{{XY}}$ are the mixing kernels for the partial sky observation, and the full form of these matrices can be found in Grain et al. (2009).Various methods have already been proposed to avoid the mixing by constructing the pure E-type and pure B-type fields, such as Bunn et al. (2003), Bunn (2011), Lewis (2003), Cao & Fang (2009), Louis et al. (2013), Grain et al. (2009), Smith (2006), Smith & Zaldarriaga (2007), Zhao & Baskaran (2010), Kim & Naselsky (2010), Santos et al. (2016), and Santos et al. (2017).

In this article, we mainly focus on how to construct the fast estimators of the CMB B-mode power spectrum with small errors. For CMB polarization maps with any sky coverage, Tegmark (1997) defines a QML estimator for the CMB temperature power spectrum, which is generalized for CMB polarization power spectra by Tegmark & de Oliveira-Costa (2001). In this section, we briefly review the QML estimator for polarization. In the pixel domain, we can define an input data vector, x , which consists of the temperature fluctuation field and the Stokes Q and U (with respect to a fixed coordinate system) specified at the ith pixel as

$$\begin{eqnarray}&&{{\boldsymbol{x}}}_{i}=\left(\begin{array}{c}{\rm{\Delta }}{T}_{i}\\ {Q}_{i}\\ {U}_{i}\end{array}\right)+\left(\begin{array}{c}{n}_{i}^{T}\\ {n}_{i}^{Q}\\ {n}_{i}^{U}\end{array}\right).\end{eqnarray} \tag{ 10 }$$

Following Tegmark & de Oliveira-Costa (2001), the optimal quadratic estimate of the power spectrum, ${y}_{{\ell }}^{r}$, is defined as :

$$\begin{eqnarray}&&{y}_{{\ell }}^{r}={{\boldsymbol{x}}}_{i}^{t}{{\boldsymbol{E}}}_{{ij}}^{r{\ell }}{{\boldsymbol{x}}}_{j}-{b}_{{\ell }}^{r},\qquad r\in \left[T,{TE},E,B\right],\end{eqnarray} \tag{ 11 }$$

where t indicates a matrix transpose operation. Here, r is the index denoting the power spectrum and i, j are indices over pixels. The data x _i at a particular pixel are a TQU component vector and ${{\boldsymbol{E}}}_{{ij}}^{r{\ell }}$ is a 3 × 3 matrix. The bias vector, ${b}_{{\ell }}^{r}$, corrects for the noise bias and is given by $\mathrm{Tr}[{{\boldsymbol{E}}}_{{\ell }}^{r}{\boldsymbol{N}}]$, when the noise, n, is uncorrelated between pixels. Note that we have assumed the summation convention. The E ^{r ℓ} matrices are computed as:

$$\begin{eqnarray}&&{{\boldsymbol{E}}}^{r{\ell }}=\displaystyle \frac{1}{2}{{\boldsymbol{C}}}^{-1}\displaystyle \frac{\partial {\boldsymbol{C}}}{\partial {{\boldsymbol{C}}}_{{\ell }}^{r}}{{\boldsymbol{C}}}^{-1}\end{eqnarray} \tag{ 12 }$$

and the covariance matrix of the data x , denoted by C , is:

$$\begin{eqnarray}\begin{array}{rcl}{{\boldsymbol{C}}}_{{ij}}=\langle {{\boldsymbol{x}}}_{i}{{\boldsymbol{x}}}_{j}^{t}\rangle & = & {\boldsymbol{R}}({\alpha }_{{ij}}){\boldsymbol{M}}({\hat{r}}_{i}\cdot {\hat{r}}_{j}){\boldsymbol{R}}{\left({\alpha }_{{ji}}\right)}^{t}+{{\boldsymbol{N}}}_{{ij}}\\ & = & \left(\begin{array}{ccc}{C}_{{ij}}^{{TT}} & \quad {C}_{{ij}}^{{TQ}} & \quad {C}_{{ij}}^{{TU}}\\ {C}_{{ij}}^{{QT}} & \quad {C}_{{ij}}^{{QQ}} & \quad {C}_{{ij}}^{{QU}}\\ {C}_{{ij}}^{{UT}} & \quad {C}_{{ij}}^{{UQ}} & \quad {C}_{{ij}}^{{UU}}\end{array}\right),\end{array}\end{eqnarray} \tag{ 13 }$$

where N is the noise covariance matrix and the rotation matrix R is given by:

$$\begin{eqnarray}{\boldsymbol{R}}({\alpha }_{{ij}})=\left(\begin{array}{ccc}1 & \quad 0 & \quad 0\\ 0 & \quad \cos (2{\alpha }_{{ij}}) & \quad \sin (2{\alpha }_{{ij}})\\ 0 & \quad -\sin (2{\alpha }_{{ij}}) & \quad \cos (2{\alpha }_{{ij}})\end{array}\right),\end{eqnarray} \tag{ 14 }$$

which performs a rotation to a global frame where the reference directions are given by the meridians. M is the covariance matrix when Q and U are defined with the reference direction being the great circle connecting the two points. So, it depends only on the angular separation between the two pixels. The explicit expression for this matrix is given by

$$\begin{eqnarray}{\boldsymbol{M}}({\hat{r}}_{i}\cdot {\hat{r}}_{j})=\left(\begin{array}{ccc}\langle {T}_{i}{T}_{j}\rangle & \quad \langle {T}_{i}{Q}_{j}\rangle \ & \quad \langle {T}_{i}{U}_{j}\rangle \\ \langle {Q}_{i}{T}_{j}\rangle & \quad \langle {Q}_{i}{Q}_{j}\rangle & \quad \langle {Q}_{i}{U}_{j}\rangle \\ \langle {U}_{i}{T}_{j}\rangle & \quad \langle {U}_{i}{Q}_{j}\rangle & \quad \langle {U}_{i}{U}_{j}\rangle \end{array}\right),\end{eqnarray} \tag{ 15 }$$

where

$$\begin{eqnarray*}\begin{array}{rcl}\langle {T}_{i}{T}_{j}\rangle & = & \sum _{{\ell }}\displaystyle \frac{2{\ell }+1}{4\pi }{C}_{{\ell }}^{{TT}}{P}_{{\ell }}(z),\\ \langle {T}_{i}{Q}_{j}\rangle & = & -\sum _{{\ell }}\displaystyle \frac{2{\ell }+1}{4\pi }{C}_{{\ell }}^{{TE}}{F}_{{\ell }}^{10}(z),\\ \langle {T}_{i}{U}_{j}\rangle & = & -\sum _{{\ell }}\displaystyle \frac{2{\ell }+1}{4\pi }{C}_{{\ell }}^{{TB}}{F}_{{\ell }}^{10}(z),\\ \langle {Q}_{i}{Q}_{j}\rangle & = & \sum _{{\ell }}\displaystyle \frac{2{\ell }+1}{4\pi }[{C}_{{\ell }}^{{EE}}{F}_{{\ell }}^{12}(z)-{C}_{{\ell }}^{{BB}}{F}_{{\ell }}^{22}(z)],\\ \langle {U}_{i}{U}_{j}\rangle & = & \sum _{{\ell }}\displaystyle \frac{2{\ell }+1}{4\pi }[{C}_{{\ell }}^{{BB}}{F}_{{\ell }}^{12}(z)-{C}_{{\ell }}^{{EE}}{F}_{{\ell }}^{22}(z)],\\ \langle {Q}_{i}{U}_{j}\rangle & = & \sum _{{\ell }}\displaystyle \frac{2{\ell }+1}{4\pi }[{F}_{{\ell }}^{12}(z)+{F}_{{\ell }}^{22}(z)]{C}_{{\ell }}^{{EB}}.\end{array}\end{eqnarray*}$$

We have used $z={\hat{r}}_{i}\cdot {\hat{r}}_{j}$ as the cosine of the angle between the ith and j^th pixels. Note that P_ℓ denotes a Legendre polynomial, and the form of functions ${F}_{{\ell }}^{10}$, ${F}_{{\ell }}^{12}$, and ${F}_{{\ell }}^{22}$ are given in Appendix A.

Considering the matrix definitions given above, one can get minimum variance estimates of the CMB power spectra using Equation (11). From Equation (13), we find that the temperature fluctuation fields are coupled with the polarization fields. It has been pointed out by Tegmark & de Oliveira-Costa (2001) that this may be problematic for realistic noisy data, where systematic errors in the ΔT measurements could contaminate or bias the estimates of E- and B-mode power spectra, which have much lower magnitudes. To avoid this potential issue, we rewrite the covariance matrix in Equation (13) as:

$$\begin{eqnarray}{\check{C}}_{{ij}}=\langle {{\boldsymbol{x}}}_{i}{{\boldsymbol{x}}}_{j}^{t}\rangle =\left(\begin{array}{ccc}{C}^{{TT}} & 0 & \quad 0\\ 0 & \quad {C}^{{QQ}} & \quad {C}^{{QU}}\\ 0 & \quad {C}^{{UQ}} & \quad {C}^{{UU}}\end{array}\right).\end{eqnarray} \tag{ 16 }$$

Note that this simplification is useful only when we are interested in the polarization autospectra and do not wish to compute the polarization temperature cross-spectra. This is done by simply dropping the temperature and polarization cross covariance terms in the full covariance matrix. Therefore, we can rewrite matrices ${{\boldsymbol{E}}}_{l}^{r}$ as:

$$\begin{eqnarray}&&{\check{E}}^{r{\ell }}=\displaystyle \frac{1}{2}{\check{{\boldsymbol{C}}}}^{-1}\displaystyle \frac{\partial {\boldsymbol{C}}}{\partial {C}_{{\ell }}^{r}}{\check{C}}^{-1},\end{eqnarray} \tag{ 17 }$$

following the definition in Equation (12). The expectation values of y^r _l in Equation (11) are

$$\begin{eqnarray}&&\langle {y}_{{\ell }}^{r}\rangle ={\check{F}}_{{\ell }{\ell }^{\prime} }^{{ur}}{C}_{{\ell }^{\prime} }^{u},\end{eqnarray} \tag{ 18 }$$

where we have used the Fisher matrix defined as:

$$\begin{eqnarray}&&{\check{F}}_{{\ell }{\ell }^{\prime} }^{{ur}}=\displaystyle \frac{1}{2}\mathrm{Tr}\left[\displaystyle \frac{\partial {\boldsymbol{C}}}{\partial {C}_{{\ell }^{\prime} }^{u}}{\check{{\boldsymbol{C}}}}^{-1}\displaystyle \frac{\partial {\boldsymbol{C}}}{\partial {C}_{{\ell }}^{r}}{\check{{\boldsymbol{C}}}}^{-1}\right].\end{eqnarray} \tag{ 19 }$$

Therefore, ${y}_{{\ell }}^{r}$ can give the unbiased estimates of the actual power spectra C^u _l with u ∈ [T, E, B].

When we use the redefined covariance matrix of Equation (16), the power spectra estimates from relation 11 become suboptimal. However, the penalty we pay is an increase in the error bars of the power spectra estimates. It can be seen that, by redefining the covariance matrix as in Equation (16), we can separate the scalar T field from the polarization field. This allows us to only work on the QU map for the E- and B-mode power spectra. Throughout this work, we will only work with the polarization part of the QML method. If the matrix, F , is invertible, one can define unbiased estimates of the true power spectra via

$$\begin{eqnarray}&&{\hat{C}}_{{\ell }}^{r}={\check{{\boldsymbol{F}}}}^{-1}{{\boldsymbol{y}}}^{r}.\end{eqnarray} \tag{ 20 }$$

The covariance matrix of the estimates ${y}_{{\ell }}^{r}$ is then given by:

$$\begin{eqnarray}&&\langle {y}_{{\ell }}^{r}{y}_{{\ell }^{\prime} }^{u}\rangle -\langle {y}_{{\ell }}^{r}\rangle \langle {y}_{{\ell }^{\prime} }^{u}\rangle \equiv {F}_{{\ell }{\ell }^{\prime} }^{{ru}}=2\mathrm{Tr}[{\boldsymbol{C}}{\check{{\boldsymbol{E}}}}^{r{\ell }}{\boldsymbol{C}}{\check{{\boldsymbol{E}}}}^{u{\ell }^{\prime} }],\end{eqnarray} \tag{ 21 }$$

where F ^ru is the Fisher matrix. Thus, the covariance matrix of the true power spectra estimates of Equation (20) is given by

$$\begin{eqnarray}&&\langle {\rm{\Delta }}{\hat{C}}_{{\ell }}{\rm{\Delta }}{\hat{C}}_{{\ell }^{\prime} }\rangle ={\check{{\boldsymbol{F}}}}^{-1}{\boldsymbol{F}}{\check{{\boldsymbol{F}}}}^{-1}.\end{eqnarray} \tag{ 22 }$$

In this work, the QML estimators have been implemented with the xQML ¹⁰ Python package (Vanneste et al. 2018).

From these equations, we observe that the computational requirements for this standard QML estimator for polarization, in its minimal form, scales as ${ \mathcal O }$($8{N}_{\mathrm{pix},\ \mathrm{obs}}^{3}$), where N_{pix, obs} is the number of observed pixels. This makes the QML method for polarization computationally prohibitive beyond the lowest resolution CMB maps.

In order to accelerate the computation speed of the QML estimators for CMB polarization power spectra, we introduce two new estimators. In these approaches, by applying the E–B separation methods proposed in the literature, we first construct the pure E-type or B-type polarization maps from the observed Q and U maps, which are scalar (or pseudoscalar) fields in a two-dimensional sphere. Then, we can apply the QML estimator for scalar fields (Tegmark 1997). In this article, we consider two different methods for E–B separation, which are sufficiently fast and have little information loss.

In the first approach, we consider the E–B separation method proposed in Smith (2006) and Smith & Zaldarriaga (2007). To deal with the mixture in a partial sky polarization analysis, one can construct two scalar (pseudoscalar) field quantities by using the spin-raising and spin-lowering operators (Newman & Penrose 1966), ð and $\bar{\unicode{x000F0}}$, on P_±. Specifically, we define a new set of fields ${ \mathcal E }$ and ${ \mathcal B }$ as in Smith & Zaldarriaga (2007) and Zhao & Baskaran (2010):

$$\begin{eqnarray}&&{ \mathcal E }(\hat{n})=-\displaystyle \frac{1}{2}[\bar{\unicode{x000F0}}\bar{\unicode{x000F0}}{P}_{+}(\hat{n})+\unicode{x000F0}\unicode{x000F0}{P}_{-}(\hat{n})],\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&{ \mathcal B }(\hat{n})=-\displaystyle \frac{1}{2i}[\bar{\unicode{x000F0}}\bar{\unicode{x000F0}}{P}_{+}(\hat{n})-\unicode{x000F0}\unicode{x000F0}{P}_{-}(\hat{n})].\end{eqnarray} \tag{ 24 }$$

These define two mutually orthogonal scalar and pseudoscalar fields, which are usually called the pure E and pure B fields in the literature. In this paper, we focus only on the B modes since the E-mode power spectrum can be recovered satisfactorily with existing estimators. Expanding the ${ \mathcal B }(\hat{n})$ component in spherical harmonics, we obtain:

$$\begin{eqnarray}&&{ \mathcal B }(\hat{n})\equiv \sum _{{\ell }m}{{ \mathcal B }}_{{\ell }m}{Y}_{{\ell }m}(\hat{n}).\end{eqnarray} \tag{ 25 }$$

The ${{ \mathcal B }}_{{\ell }m}$ coefficient can then be computed from the pure B-field as:

$$\begin{eqnarray}&&{{ \mathcal B }}_{{\ell }m}=\int { \mathcal B }(\hat{n}){Y}_{{\ell }m}^{* }(\hat{n})d\hat{n}.\end{eqnarray} \tag{ 26 }$$

This new pure B-mode spherical harmonic coefficient is related to the coefficient B_{ℓ m} as (Seljak & Zaldarriaga 1997):

$$\begin{eqnarray}&&{{ \mathcal B }}_{{\ell }m}={N}_{{\ell },2}{B}_{{\ell }m},\end{eqnarray} \tag{ 27 }$$

and the power spectrum becomes

$$\begin{eqnarray}&&{C}_{{\ell }}^{{ \mathcal B }{ \mathcal B }}\equiv \langle {{ \mathcal B }}_{{\ell }m}{{ \mathcal B }}_{{\ell }m}^{* }\rangle ={N}_{{\ell },2}^{2}{C}_{{\ell }}^{{BB}}\end{eqnarray} \tag{ 28 }$$

with ${N}_{{\ell },s}=\sqrt{({\ell }+s)!/({\ell }-s)!}$.

For an incomplete sky observation defined by the window function $W(\hat{n})$, the partial sky harmonic coefficients of the pure E and B fields are defined as (Efstathiou 2004),

$$\begin{eqnarray}&&{\tilde{{ \mathcal E }}}_{{\ell }m}=\int W(\hat{n}){ \mathcal E }(\hat{n}){Y}_{{\ell }m}^{* }(\hat{n})d\hat{n},\end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray}&&{\tilde{{ \mathcal B }}}_{{\ell }m}=\int W(\hat{n}){ \mathcal B }(\hat{n}){Y}_{{\ell }m}^{* }(\hat{n})d\hat{n}.\end{eqnarray} \tag{ 30 }$$

Following the SZ method detailed in Smith (2006), we write the partial sky pure-field harmonic coefficients as:

$$\begin{eqnarray}\begin{array}{rcl}{\tilde{{ \mathcal E }}}_{{\ell }m} & = & -\displaystyle \frac{1}{2}\displaystyle \int d\hat{n}\left\{{P}_{+}(\hat{n}){\left[\bar{\unicode{x000F0}}\bar{\unicode{x000F0}}\left(W(\hat{n}){Y}_{{\ell }m}(\hat{n})\right)\right]}^{* }\right.\\ & & +\left.{P}_{-}(\hat{n}){\left[\unicode{x000F0}\unicode{x000F0}\left(W(\hat{n}){Y}_{{\ell }m}(\hat{n})\right)\right]}^{* }\right\},\end{array}\end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray}\begin{array}{rcl}{\tilde{{ \mathcal B }}}_{{\ell }m} & = & -\displaystyle \frac{1}{2i}\displaystyle \int d\hat{n}\left\{{P}_{+}(\hat{n}){\left[\bar{\unicode{x000F0}}\bar{\unicode{x000F0}}\left(W(\hat{n}){Y}_{{\ell }m}(\hat{n})\right)\right]}^{* }\right.\\ & & -\left.{P}_{-}(\hat{n}){\left[\unicode{x000F0}\unicode{x000F0}\left(W(\hat{n}){Y}_{{\ell }m}(\hat{n})\right)\right]}^{* }\right\}.\end{array}\end{eqnarray} \tag{ 32 }$$

These expressions can be expanded and simplified further for implementation. Full expressions can be found in Appendix A. Once the coefficients ${\tilde{{ \mathcal E }}}_{{\ell }m}$ and ${\tilde{{ \mathcal B }}}_{{\ell }m}$ are derived, the scalar fields in our observation window $W(\hat{n}){ \mathcal E }(\hat{n})$ and $W(\hat{n}){ \mathcal B }(\hat{n})$ can be directly obtained by inverting the relations in Equations (29) and (30).

Similar to Equation (9), the partial sky pure harmonic coefficients ${\tilde{{ \mathcal E }}}_{{\ell }m}$ and ${\tilde{{ \mathcal B }}}_{{\ell }m}$ are related to the full sky harmonic coefficients E_{ℓ m} and B_{ℓ m} as follows,

$$\begin{eqnarray}&&{\tilde{{ \mathcal E }}}_{{\ell }m}=\sum _{{\ell }^{\prime} m^{\prime} }[{{ \mathcal K }}_{{\ell }m,{\ell }^{\prime} m^{\prime} }^{{EE}}{a}_{{\ell }^{\prime} m^{\prime} }^{E}+i{{ \mathcal K }}_{{\ell }m,{\ell }^{\prime} m^{\prime} }^{{EB}}{a}_{{\ell }^{\prime} m^{\prime} }^{B}],\end{eqnarray} \tag{ 33 }$$

$$\begin{eqnarray}&&{\tilde{{ \mathcal B }}}_{{\ell }m}=\sum _{{\ell }^{\prime} m^{\prime} }[-i{{ \mathcal K }}_{{\ell }m,{\ell }^{\prime} m^{\prime} }^{{BE}}{a}_{l^{\prime} m^{\prime} }^{E}+{{ \mathcal K }}_{{\ell }m,{\ell }^{\prime} m^{\prime} }^{{BB}}{a}_{{\ell }^{\prime} m^{\prime} }^{B}],\end{eqnarray} \tag{ 34 }$$

where ${{ \mathcal K }}_{{\ell }^{\prime} m^{\prime} {\ell }m}^{{ru}}$ are the pure-field mixing kernels, which in general can be all different, nonvanishing, and nondiagonal in both ℓand m. Just as in the case of Equation (9), ${{ \mathcal K }}_{{\ell }^{\prime} m^{\prime} {\ell }m}^{{EB}}$ and ${{ \mathcal K }}_{{\ell }^{\prime} m^{\prime} {\ell }m}^{{BE}}$ are the mixing terms between the two polarization modes. However, it can be shown that for pure E and pure B construction, the mixing terms are orders of magnitude smaller than for the standard case of Equation (9). This indicates that the pure fields are nearly orthogonal with very small mixing between the two polarization modes.

In the previous subsection, the redefined covariance matrix in Equation (16) indicates that the QML estimator for the scalar temperature field can be separated from the polarization part. By using the pure E/B fields, we can construct the two scalar (pseudoscalar) polarization modes. Thus, it is conceivable to use the scalar QML method to estimate the power spectrum of the decoupled scalar (pseudoscalar) polarization fields. Let us briefly outline the scalar QML implementation, which we use to estimate the power spectrum of the pure B-field.

Let ${x}_{i}^{{ \mathcal S }}$ denote the ith pixel value in the scalar map ${x}_{i}^{{ \mathcal S }}={{ \mathcal S }}_{i}+{n}_{i}^{{ \mathcal S }}$, where ${n}_{i}^{{ \mathcal S }}$ is the noise in the individual pixel. The covariance matrix ${{\boldsymbol{C}}}_{{ \mathcal S }}$ of input data ${x}_{i}^{{ \mathcal S }}$ can be written as:

$$\begin{eqnarray}&&{C}_{s,{ij}}=\langle {x}_{i}^{{ \mathcal S }}{\left({x}_{j}^{{ \mathcal S }}\right)}^{t}\rangle =\sum _{{\ell }}\displaystyle \frac{2{\ell }+1}{4\pi }{C}_{{\ell }}^{{ \mathcal S }{ \mathcal S }}{P}_{{\ell }}(z)+{N}_{{ \mathcal S },{ij}},\end{eqnarray} \tag{ 35 }$$

where ${C}_{{\ell }}^{{ \mathcal S }{ \mathcal S }}$ means the power spectrum corresponding to the signal of scalar field map ${ \mathcal S }(\hat{n})$ and ${{\boldsymbol{N}}}_{{ \mathcal S }}$ is the noise variance matrix.

According to the QML approach, we can construct the optimal estimator by making use of C_S,ij and ${x}_{i}^{{ \mathcal S }}$,

$$\begin{eqnarray}&&{y}_{{\ell }}^{{ \mathcal S }}={x}_{i}^{{ \mathcal S }}{x}_{j}^{{ \mathcal S }}{E}_{S,{ij}}^{{\ell }}-{b}_{{\ell }}^{{ \mathcal S }}.\end{eqnarray} \tag{ 36 }$$

The matrices ${{\boldsymbol{E}}}_{{ \mathcal S }}^{{\ell }}$ have a similar form to Equation (12)

$$\begin{eqnarray}&&{{\boldsymbol{E}}}_{{ \mathcal S }}^{{\ell }}=\displaystyle \frac{1}{2}{{\boldsymbol{C}}}_{{ \mathcal S }}^{-1}\displaystyle \frac{\partial {{\boldsymbol{C}}}_{{ \mathcal S }}}{\partial {C}_{{ \mathcal S },{\ell }}}{{\boldsymbol{C}}}_{{ \mathcal S }}^{-1}.\end{eqnarray} \tag{ 37 }$$

Similarly, the Fisher matrix ${F}_{S,{\ell }{\ell }^{\prime} }$ expression becomes:

$$\begin{eqnarray}&&{F}_{S,{\ell }{\ell }^{\prime} }=\displaystyle \frac{1}{2}{Tr}\left[\displaystyle \frac{\partial {{\boldsymbol{C}}}_{{ \mathcal S }}}{\partial {C}_{{ \mathcal S },{\ell }}}{{\boldsymbol{C}}}_{{ \mathcal S }}^{-1}\displaystyle \frac{\partial {{\boldsymbol{C}}}_{{ \mathcal S }}}{\partial {C}_{{ \mathcal S },{\ell }^{\prime} }}{{\boldsymbol{C}}}_{{ \mathcal S }}^{-1}\right].\end{eqnarray} \tag{ 38 }$$

The QML estimator for a scalar power spectrum ${\hat{C}}_{{\ell }}^{{ \mathcal S }{ \mathcal S }}$ is given by

$$\begin{eqnarray}&&{\hat{C}}_{{\ell }}^{{ \mathcal S }{ \mathcal S }}={\left({\tilde{{\boldsymbol{F}}}}_{{ \mathcal S }}\right)}^{-1}{{\boldsymbol{y}}}^{{ \mathcal S }}.\end{eqnarray} \tag{ 39 }$$

We apply this scalar QML method to estimate the power spectrum of the decoupled pure B polarization field, and it is denoted as the QML-SZ estimator in this work.

In the SZ approach to separate the E and B modes, we must use a proper sky apodization, instead of the binary window function, to avoid numerical divergences in the calculation of the window function derivatives. A Gaussian smoothing kernel has been shown to induce very small leakage in the final B map (Wang et al. 2016; Kim 2011). So, this is our apodization choice for obtaining the pure B map for the QML-SZ method. For the ith pixel in the region allowed by the binary mask, the apodized window is defined as:

$$\begin{eqnarray}{W}_{i}=\left\{\begin{array}{ll}\displaystyle \frac{1}{2}+\displaystyle \frac{1}{2}\mathrm{erf}\left(\displaystyle \frac{{\delta }_{i}-\tfrac{{\delta }_{c}}{2}}{\sqrt{2}\sigma }\right), & {\delta }_{i}\lt {\delta }_{c}\\ 1, & {\delta }_{i}\gt {\delta }_{c}\end{array}\right.\end{eqnarray} \tag{ 40 }$$

where δ_i is the shortest distance between the ith observed pixel from the boundary of the allowed region, $\sigma =\mathrm{FWHM}/\sqrt{8\mathrm{ln}2}$ with FWHM denoting the full width at half maximum of the Gaussian kernel, and δ_c is the apodization length which acts as an additional adjustable parameter.

Now, we summarize the construction of the QML-SZ estimator as follows. For the given observed Q and U polarization maps, we construct a partial sky pure B-type map ${ \mathcal B }(\hat{n})$, following the SZ method with Gaussian apodization of the binary window function. Then, we use the ${ \mathcal B }(\hat{n})$ map and ${C}_{{\ell }}^{{ \mathcal B }{ \mathcal B }}$ as input parameters to replace ${x}_{i}^{{ \mathcal S }}$ and ${C}_{{\ell }}^{{ \mathcal S }{ \mathcal S }}$ in the scalar QML method to estimate the B-mode power spectrum ${C}_{{\ell }}^{{BB}}$. In comparison with traditional QML estimator, our goal with the QML-SZ method is to simplify the calculation, without compromising significantly on the accuracy or error bars. We implement the QML-SZ estimator with the modified xQML Python package.

In this subsection, we describe a similar procedure to construct a B-mode map that is leakage free, and then use the scalar QML method to estimate the power spectrum. We use the E-mode recycling method proposed in Liu et al. (2019a, 2019b) to obtain a leakage free B-mode map. This procedure essentially uses the E-mode signal in the CMB map to estimate the leakage due to the incompleteness of the sky. This template for the leakage is then used to clean the B-mode map. The mathematical principle of this idea is the following:

The true E-to-B leakage in the pixel domain is given by the following equation:

$$\begin{eqnarray}&&{\boldsymbol{L}}{\left({\boldsymbol{n}}\right)}_{\mathrm{true}}=\displaystyle \int {G}_{{EB}}({\boldsymbol{n}},{\boldsymbol{n}}^{\prime} ){\boldsymbol{P}}({\boldsymbol{n}}^{\prime} )d{\boldsymbol{n}}^{\prime} ,\end{eqnarray} \tag{ 41 }$$

where ${G}_{{EB}}({\boldsymbol{n}},{\boldsymbol{n}}^{\prime} )$ is the convolution kernel of the E-to-B leakage given in Equation (3.2) of Liu et al. (2019b), and ${\boldsymbol{P}}({\boldsymbol{n}}^{\prime} )$ includes both the Q and U Stokes parameters. Note that this convolution kernel is fixed and does not change with different realizations of the CMB sky.

The above integral should be calculated over the entire sphere. This explains why the true E-to-B leakage cannot be precisely estimated in the case of partial sky coverage. However, the above equation also tells us that if there is no additional information regarding the mission's sky region, then the estimation of the E-to-B leakage is given by

$$\begin{eqnarray}&&{\boldsymbol{L}}{\left({\boldsymbol{n}}\right)}_{\mathrm{blind}}=\int {G}_{{EB}}({\boldsymbol{n}},{\boldsymbol{n}}^{\prime} ){\boldsymbol{P}}({\boldsymbol{n}}^{\prime} ){\boldsymbol{M}}({\boldsymbol{n}}^{\prime} )d{\boldsymbol{n}}^{\prime} ,\end{eqnarray} \tag{ 42 }$$

where ${\boldsymbol{M}}({\boldsymbol{n}}^{\prime} )$ is the sky mask. The above equation can be further decomposed into the combination of two integrals:

$$\begin{eqnarray}\begin{array}{rcl}{\boldsymbol{L}}{\left({\boldsymbol{n}}\right)}_{\mathrm{blind}} & = & \int {G}_{B}({\boldsymbol{n}},{\boldsymbol{n}}^{\prime} ){\boldsymbol{M}}({\boldsymbol{n}}^{\prime} )d{\boldsymbol{n}}^{\prime} \\ & & \int {G}_{E}({\boldsymbol{n}}^{\prime} ,{\boldsymbol{n}}^{\prime\prime} ){\boldsymbol{M}}({\boldsymbol{n}}^{\prime\prime} ){\boldsymbol{P}}({\boldsymbol{n}}^{\prime\prime} )d{\boldsymbol{n}}^{\prime\prime} ,\end{array}\end{eqnarray} \tag{ 43 }$$

where ${G}_{E}({\boldsymbol{n}},{\boldsymbol{n}}^{\prime} )$ and ${G}_{B}({\boldsymbol{n}},{\boldsymbol{n}}^{\prime} )$ are the pixel domain convolution kernels of the E- and B-mode signals, respectively. Mathematically, the integrals with these two kernels are nothing but standard forward-backward spherical harmonic transforms of the E and B modes.

Therefore, the algorithm for the template cleaning method is:

• 1.  
  Starting with the data vector x from Equation (10) and the binary window function for the observed sky W, we obtain the spherical harmonic coefficients ${a}_{{\ell }m}^{r}$ with r ∈ [T, E, B].
• 2.  
  We reconstruct a QU map with the ${a}_{{\ell }m}^{E}{\rm{s}}$ only. In the CMB case, the power in the B mode is much smaller than that in the E-mode. So, compared to the E-mode signal, the leakage from B-to-E is negligible. Therefore, this QU map represents the actual E-mode-only CMB polarization in the observed sky patch.
• 3.  
  Next, we mask the E-mode-only QU maps again with the binary window function and obtain harmonic coefficients ${\bar{a}}_{{\ell }m}^{r}$, with r ∈ [E, B]. Since these QU maps were constructed from only E-mode information, any B modes generated by the harmonic transformation is produced by the E-to-B leakage. So we can construct a scalar B-mode leakage template by using the ${\bar{a}}_{{\ell }m}^{B}$ obtained this way.
• 4.  
  We use the original ${a}_{{\ell }m}^{B}$ to obtain a scalar B-mode map, which is contaminated by E-to-B leakage. With the contaminated B-mode map, we can obtain a linear fit of the leakage template, which is subtracted from the contaminated B-mode map in pixel space to obtain the leakage cleaned B map.

This cleaned B map is essentially the isolated B-mode pseudoscalar field, and we can apply the scalar QML estimator to reconstruct its power spectrum. However, the template cleaning method is not perfect so we will have some residual leakage that gets left in the cleaned B maps. Therefore, the cleaned B map can be treated as ${B}_{i}+{x}_{i}^{R}+{n}_{i}^{B}$, where B_i is the ith pixel value of the cleaned B modes, x^R _i is the residual leakage, and n^B _i is the noise. This residual leakage term will be treated similar to the noise contribution. In practice, we compute the residual leakage covariance matrix from simulations. Then the scalar covariance matrix shown in Equation (35) becomes:

$$\begin{eqnarray}&&{C}_{B,{ij}}=\langle {x}_{i}^{B}{\left({x}_{j}^{B}\right)}^{t}\rangle =\sum _{{\ell }}\displaystyle \frac{2{\ell }+1}{4\pi }{C}_{{\ell }}^{{BB}}{P}_{{\ell }}(z)+{N}_{{ij}}+{R}_{{ij}},\end{eqnarray} \tag{ 44 }$$

where R_ij is the covariance matrix of residual leakage.

The bias term ${b}_{{\ell }}^{B}$ is now given by ${Tr}[{{\boldsymbol{E}}}_{{\ell }}({\boldsymbol{N}}+{\boldsymbol{R}})]$. We find that, for ℓ > 5, the impact of the residual leakage can be removed by masking pixels near the mask boundary in the cleaned B map, and its impact on the final results can be ignored in most cases. In this paper, we mask all pixels 35 from the mask boundary. The results of the QML methods discussed here are also not very sensitive to the choice of the fiducial ${C}_{{\ell }}^{{BB}}$ or ${C}_{{\ell }}^{{ \mathcal B }{ \mathcal B }}$ chosen in computing the signal covariance matrix. For example, a different choice of the tensor-to-scalar ratio does not make any difference in the final results.

Using the template cleaned B-mode map as input and the covariance matrix above, we use the scalar QML (Equations (37) to (39)) to obtain the QML-TC estimator for the B-mode power spectrum. Similar to the QML-SZ estimator, the QML-TC estimator is a much simpler implementation that should have reasonably good performance at the largest scales. In our calculation, the QML-TC estimator is also implemented with the xQML Python package.

The most common method that is adopted in CMB data analysis is the so-called pseudo-C_ℓ (PCL) estimators, which are both computationally inexpensive and near optimal at high multipoles. For polarization analysis with PCL estimators, the preferred method is to work with pure E and pure B fields. In case of an incomplete sky, Equations (33) and (34) give the relation between the partial sky pure E/B-mode harmonics and the full sky E and B modes. They can be used to obtain a relation between the partial sky pure E/B power spectra and the actual E/B-mode power spectra:

$$\begin{eqnarray}&&{\tilde{C}}_{{\ell }}^{{ \mathcal E }{ \mathcal E }}=\sum _{{\ell }^{\prime} }\left[{{ \mathcal M }}_{{\ell }{\ell }^{\prime} }^{{EE}}{C}_{{\ell }^{\prime} }^{{EE}}+{{ \mathcal M }}_{{\ell }{\ell }^{\prime} }^{{EB}}{C}_{{\ell }^{\prime} }^{{BB}}\right],\end{eqnarray} \tag{ 45 }$$

$$\begin{eqnarray}&&{\tilde{C}}_{{\ell }}^{{ \mathcal B }{ \mathcal B }}=\sum _{{\ell }^{\prime} }\left[{{ \mathcal M }}_{{\ell }{\ell }^{\prime} }^{{BE}}{C}_{{\ell }^{\prime} }^{{EE}}+{{ \mathcal M }}_{{\ell }{\ell }^{\prime} }^{{BB}}{C}_{{\ell }^{\prime} }^{{BB}}\right].\end{eqnarray} \tag{ 46 }$$

In these expressions ${{ \mathcal M }}_{{\ell }{\ell }^{\prime} }^{{ru}}$ is the mixing matrix that relates the two sets of power spectra. The mixing matrices are defined as:

$$\begin{eqnarray}&&{{ \mathcal M }}_{{\ell }{\ell }^{\prime} }^{{ru}}=\displaystyle \frac{1}{2{\ell }+1}\sum _{{mm}^{\prime} }| {{ \mathcal K }}_{{\ell }m{\ell }^{\prime} m^{\prime} }^{{ru}}{| }^{2},\end{eqnarray} \tag{ 47 }$$

with r, u ∈ [E, B]. The exact expressions for the mixing matrices can be found in Grain et al. (2009). We can solve the set of Equations (45) and (46) for the E- and B-mode power spectra. These are called the pseudo-full-sky power spectra estimates. In this work, we will compare the performance of the three QML methods with the PCL method in the multipole range of interest.

When we perform a spherical harmonic transformation with binary window functions, it will lead to severe leakage and mode mixing. Therefore we have apodized our observation window with a "C2" (cosine) apodization function for the PCL estimates in this work. The weight in the ith pixel is given as (Alonso et al. 2019):

$$\begin{eqnarray}\left\{\begin{array}{ll}{W}_{i}=\displaystyle \frac{1}{2}\left[1-\cos (\pi {\delta }_{i}^{r})\right] & {\delta }_{i}^{r}\lt 1\\ 1 & \mathrm{otherwise},\end{array}\right.\end{eqnarray} \tag{ 48 }$$

where ${\delta }_{i}^{r}=\sqrt{(1-\cos {\delta }_{i})/(1-\cos {\delta }_{c})}$. We compute all PCL estimator results in this work with the C2 apodization. The PCL estimator for this work has been implemented with the Python package of NaMaster ¹¹ (Alonso et al. 2019).

In an incomplete sky, besides the mixtures between the E and B modes, we have to deal with the well-known mode mixing problem. In this subsection, we will test the QML estimator, as well as the QML-SZ and QML-TC estimators, introduced previously. As discussed, we simulate the pure B and B-mode maps directly from ${C}_{{\ell }}^{{ \mathcal B }{ \mathcal B }}$ and ${C}_{{\ell }}^{{BB}}$, so there is no E-to-B leakage residual contribution. We simulate at NSIDE of 16 and 32 for satellite and ground experiments, respectively, with the noise level set to 0.1 μK arcmin, which is close to the noise-free case. Therefore, we can only study the effect of mode mixture caused by the partial sky surveys.

In Figures 2 and 3, we show the power spectra estimates plotted for space-based and ground-based experiments, respectively. Note that we have binned the power spectra with the following definition,

$$\begin{eqnarray}&&{{ \mathcal D }}_{{\ell }}^{{BB}}=\displaystyle \frac{1}{{\rm{\Delta }}{\ell }}\sum _{{\ell }^{\prime} ={\ell }-{\rm{\Delta }}{\ell }/2}^{{\ell }^{\prime} ={\ell }+{\rm{\Delta }}{\ell }/2}\displaystyle \frac{{\ell }^{\prime} ({\ell }^{\prime} +1)}{2\pi }{C}_{{\ell }^{\prime} }^{{BB}},\end{eqnarray} \tag{ 49 }$$

where Δℓ is the bin width. For the space-based experiment, due the large sky coverage, it is reasonable to estimate the power spectra at each multipole, hence we set Δℓ = 1. For ground-based experiments, we bin the power spectrum into bands with Δℓ = 11 to mitigate the greater mode mixing problem due to a smaller observation patch. Throughout this work, our power spectra estimates for any estimator is a mean of 1000 random simulations, and the errors are computed as the standard deviation of the samples. For comparison, we also compare our results with the simple analytical estimate of error bars, which is given by

$$\begin{eqnarray}&&{\rm{\Delta }}{{ \mathcal D }}_{{\ell },\mathrm{optimal}}^{{BB}}\cong \sqrt{\displaystyle \frac{2}{(2{\ell }+1){\rm{\Delta }}{\ell }{f}_{\mathrm{sky}}}}\left[{{ \mathcal D }}_{{\ell }}^{{BB}}+{{ \mathcal N }}_{{\ell }}^{{BB}}\right],\end{eqnarray} \tag{ 50 }$$

where ${{ \mathcal N }}_{{\ell }}^{{BB}}$ is the noise power spectrum binned using relation (49).

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As anticipated, the results in Figures 2 and 3 show that all three methods can obtain unbiased estimates for the B-mode power spectrum. Next, we will investigate the impact of f_sky on the error bars. In the case of the Planck mask, we find the rms errors of these QML methods are even smaller than for the analytical results in Equation (50), since the large-scale information in the masked region can be partly recovered by the QML analysis, which is consistent with the results in Efstathiou (2004, 2006). For smaller scales, the errors of the three methods are optimal, and all methods perform equally. Our results show that the mode mixing due to f_sky is sufficiently well corrected in all of the presented methods, and its impact on the final results is negligible for both of the two sky patches considered in this work.

Actual CMB polarization maps are usually at much higher resolution than the resolution imposed by computational limitations in calculating the QML estimator. Any analysis using QML estimators will then require a smoothing process to reduce the influence of higher multipoles, followed by a downgrading process to reduce the NSIDE of the map. The map degrading process for this work will be done with the ud_grade subroutine from HEALPix. In this subsection, we test the impact of ud_grade alone (without considering the effect of smoothing).

For this test, we simulate CMB maps at NSIDE = 512, with ℓ_max set to 32 and 96 for the space-based and ground-based cases, respectively. This is done to mimic the cases with smoothing that cuts off the modes above the set ℓ_max values. Thus, it helps to isolate the effect of the downgrading procedure only. The full sky maps at NSIDE = 512 are masked with appropriate binary masks and then downgraded to the targeted NSIDE by ud_grade. We also downgrade the masks with ud_grade and set any pixels with values <0.99 to zero. We multiply this downgraded mask by the downgraded maps, which act as the input maps for the QML estimators.

The results for the impact of ud_grade are shown in Figures 4 and 5 for the space-based and ground-based experiment cases, respectively. For both sky patches, we find that downgrading the map has a negligible impact on the mean estimates. Thus, this process does not generate any additional biases to the QML estimators. For the space-based experiment case, the error bars for ℓ ≤ 5 show a small increase for the QML estimator and a significant increase for the QML-SZ estimator. This behavior can be explained by the power spectrum of QML-SZ method, which is given by ${N}_{{\ell },2}^{2}{C}_{{\ell }}^{{BB}}$. By downgrading the map, the power from high multipoles leaks to the lower multipoles and increases the uncertainty on the large angular scales. We also find that errors for the standard QML estimator slightly increase throughout the multipole range, though the errors are still near optimal. For the ground-based experiment, we find a negligible impact of the downgrading process on the error bars.

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In every CMB experiment, we have to suitably mitigate the impact of the noise. The noise level depends on various factors, such as instrument sensitivity, survey duration, survey strategy, etc. As mentioned before, the rms noise levels can vary due to various factors, but for this work we test three cases: 10 μK arcmin, 1 μK arcmin, and 0.1 μK arcmin, the latter one acting as the noise-free limit of our estimator performance.

For this study, we simulated the different noise maps at the targeted NSIDE = 16 for the large sky fraction patch and NSIDE = 32 for the small sky patch. The rms noise levels are suitably converted to get the noise variance per pixel at one of those NSIDE values. This is used to generate Gaussian white noise. The CMB maps are simulated at the targeted NSIDE with ℓ_max of 32 and 96. The two maps are then coadded and masked to produce the input maps for the QML estimators.

In Figures 6 and 7, we show the results for the power spectra estimated with the QML methods for the large and small sky fraction patches. In these figures, we compare the errors with the cosmic variance limit that can be obtained from Equation (50) by setting ${{ \mathcal N }}_{{\ell }}^{{BB}}$ to zero. We can see that the variation in the noise level does not affect the mean of the power spectra estimates. This is expected as we have the ${b}_{{\ell }}^{r}$ term in Equation (11) to debias for the noise. We also find that the noise level will impact the error bars, but at a low noise level it almost behaves as the cosmic variance limited case. Considering the same noise level, for every tested case, all of the QML estimators behave very similar to each other.

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The space-based experiments have a major advantage of being able to observe the full sky. However, the Galactic plane must be masked out due to the strong polarized foreground contribution from our Galaxy. Even though part of the sky must be removed to avoid the astrophysical Galactic contamination, satellite experiments are our best bet for observing the largest angular modes. For the satellite case, we set our target to NSIDE = 16 and ℓ_max = 32 for all three QML estimators. The QU and scalar maps at NSIDE = 512 are downgraded to the targeted NSIDE = 16, using the ud_grade Healpix subroutine, as well as the masks, setting any pixel with values <0.99 to zero. Note that in this section, for the QML-SZ estimator, we downgrade the apodized mask instead of the binary mask, as the SZ method uses an apodized mask. We multiply this downgraded mask to the downgraded maps. These are our input maps for QML estimators. We will compare these results with the PCL estimator results.

The results for this first case are shown in Figure 8 for both r = 0.05 and r = 0. We plot the mean of the power spectra estimates from 1000 simulations with error bars given by the standard deviation of these samples. In addition, we plot the results of the pure B-mode PCL estimator for the same case. The PCL results are obtained with C2 apodization with δ_c of 6°. We find that all of the QML methods outperform the PCL results in the entire multipole range. We notice that while the standard QML method has near optimal error bars throughout the entire multipole range, the QML-SZ method has suboptimal error bars for the lowest multipoles because we downgrade the input map, as discussed in Section 3.2.

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That said, all QML estimators are tested with the same binary mask as defined in Section 3.2. However, for the QML-SZ estimator, we use the Gaussian apodized window to replace the binary mask, which causes the effective f_sky of the QML-SZ estimator to be smaller than that for the standard QML one and enlarges the uncertainties of the QML-SZ estimator for the entire multipole range. While the performance of the QML-SZ estimator is not as good as that of the standard QML method, it is still a fast and reliable solution for power spectrum estimation, except for the lowest few multipoles.

Due to the partial sky analysis, the coupling between different multipoles is inevitable. In order to quantify it, we calculate the normalized covariance matrices defined as

$$\begin{eqnarray}&&{{ \mathcal C }}_{{\ell }{\ell }^{\prime} }=\displaystyle \frac{\mathrm{cov}\left({\hat{C}}_{{\ell }}^{{BB}},{\hat{C}}_{{\ell }^{\prime} }^{{BB}}\right)}{\sqrt{\mathrm{var}({\hat{C}}_{{\ell }}^{{BB}})\mathrm{var}({\hat{C}}_{{\ell }^{\prime} }^{{BB}})}},\end{eqnarray} \tag{ 51 }$$

and present the results in Figure 9. For the space-based experiment considering homogeneous noise, the power spectra estimates only weakly couple among different multipoles for every QML method tested here, with the covariance matrices being approximately diagonal.

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In this subsection, we will study the performance of the QML methods for a satellite experiment with inhomogeneous noise. We generate inhomogeneous noise maps using the hitmap for Planck HFI 100 GHz channel (shown in Figure 10) following the prescription given in Ducout et al. (2013). We set the white noise level of these maps (σ_{isotropic noise} of Ducout et al. 2013) to 5 μK arcmin. All of the calculation steps and parameters are consistent with the last subsection. We will still compare the results for the QML methodology with the ones for the PCL estimator.

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The results for this case are shown in Figure 11, where the upper and lower panels represent r = 0.05 and r = 0, respectively. Comparing the results for the different methods with inhomogeneous noise, we can find that the different QMLs still outperform the PCL in our entire multipole range. The standard QML method still have near optimal error bars throughout the entire multipole range, while the QML-SZ method has slightly larger error bars for the lowest multipoles.

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We also show the normalized covariance matrices for the power spectra estimators with the QML methods for this case in Figure 12. Similarly, we find that the covariance matrices are approximately diagonal, which indicates that the cross-correlations between different modes are weak for the presented QML methods.

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Ground-based CMB experiments cannot account for full sky observations, since they are limited by their geographical location in terms of the total sky area available for the survey. That said, they have longer mission plans, during which they undergo instrumental upgrades that allow for higher sensitivity. For a ground-based survey, f_sky is usually quite small. For most cases, we have f_sky ≲ 10%, which allows us to choose larger NSIDE, in comparison with the ones chosen for the space-based experiment case.

For the analysis in this subsection, we downgrade the high resolution maps to NSIDE = 64 (ℓ_max = 192) for all three QML methods. As in our previous examples, we also compare these QML results with the pure B-mode PCL estimator, obtained with C2 apodization. The apodization length δ_c is set to 6° and 10° for r = 0.05 and r = 0, respectively.

The process we use for suppressing the higher multipoles in the simulated maps at high resolution is the major difference between a ground-based and space-based experiment. Instead of smoothing the map, we start by obtaining the spherical harmonic coefficients of the maps at NSIDE = 512. We set all harmonic coefficients to zero above ℓ_max values stated above. We use the a_{ℓ m} values with this cutoff to reconstruct the map at NSIDE = 512, but remove the information above ℓ_max. This methodology is applied to the QU maps at NSIDE = 512 with ℓ_max = 192, and to the scalar ${ \mathcal B }$ and B-mode maps at NSIDE = 512 with ℓ_max = 192, too. Then we downgrade the map to the targeted NSIDE of 64.

In this subsection, we consider the homogeneous noise with the 3 μK arcmin noise level to study the performance of the QML methods on a small sky patch. The results for the power spectrum estimator are shown in Figure 13, from which we find that all of the methods discussed here give unbiased estimates of the B-mode band powers for both the r = 0 and r = 0.05 cases. However, when we focus on the error bars, the r = 0 and r = 0.05 cases have obvious differences. For r = 0.05, all of the methods (QMLs and PCL) have near optimal error bars in the entire multipole range.

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However, for the r = 0 case, the CMB signal of the B-mode for ℓ < 60 is so weak that the error bars are too small, and we cannot tell which method performs better. However, for ℓ > 60, we find that all QML methods have smaller error bars than those for the PCL method. This means that, for the case with small r, QML methods perform better in reconstructing the B-mode power spectrum.In Figure 14, we have shown the normalized covariance matrices for the power spectra estimates with the three QML methods for this case. For the ground-based experiment, we also find our covariance matrices to be approximately diagonal, showing that the band power leakages have been suitably removed.

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In this subsection, we will study the performance of the QML methods with the noise profile of the AliCPT-1 experiment as a realistic example. A map of the noise standard deviation per pixel for AliCPT-1 is shown in Figure 15. The major difference between the inhomogeneous noise case and homogeneous noise case is the way we deal with the noise. QML methods require a precise knowledge of the pixel noise matrix N to compute the bias term b_l in Equation (11). For the homogeneous case, the noise covariance matrix is a diagonal matrix with all diagonal elements equal, and we can calculate N with the noise power spectrum directly. However, in the inhomogeneous case, it is difficult to characterize the noise covariance matrix by an analytical formula. Here, we estimate the noise covariance matrix from numerical simulations. We generate 1000 noise samples and use the SZ method (or the TC method) to produce scalar noise maps of the QML-SZ method (or the QML-TC method). Then, we downgrade the scalar noise maps, as well as the original noise maps, to the targeted resolution. The covariance matrices of these noise-only simulations are the N matrices that we use in this case.

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The corresponding results are shown in Figure 16. We find that all of the QML-based methods, as well as the PCL method, give unbiased estimates for the B-mode band powers for both r = 0 and r = 0.05. Let us focus on the r = 0.05 situation first. Comparing the QML results with that of the PCL method, we find that all of the QML methods have near optimal error bars in the entire multipole range, while for the PCL method, uncertainties increase rapidly with increasing angular scale. The QML methods have significant advantages on large scales for r = 0.05. For r = 0, the QML methods still keep their excellent performance on large scales, and in addition, we find that, even for the small scale, the QML methods outperforms the PCL method.

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In Figure 17, we also show the normalized covariance matrices for the power spectra estimates for the three QML methods for this case. Similarly, we find that the covariance matrices are nearly diagonal, which indicates that the correlations between different bands are negligible.

In Section 3, we discussed that the computational complexity of the QML estimator is a $O({N}_{d}^{3})$ problem, where N_d is the length of the data vector. We also discussed that N_d for the QML-SZ method or the QML-TC method is half of the N_d for the polarization part of the "reshaped" classical QML estimator for polarization. This reduces the computational requirements for the new QML methods. In Tables 1 and 2, we summarize the computational parameters for the three QML estimators for the space-based and ground-based experiment cases, respectively. We run all our computations on an Intel Xeon E2620 2.10 GHz workstation and list NSIDE, N_d , ℓ_max, RAM (in gigabytes), and computation time for a single computation.

As shown in Table 1, for the QML-SZ estimators, the computation time is only about 1/13 of that for the classic QML estimator. This happens because the data vector size, N_d , for the scalar QML method is about half of that for the classic QML method. Additionally, as both the QML-SZ and QML-TC methods are based on the scalar-mode QML method, its algorithm complexity is less than for the polarization mode, and thereby the computation is faster. In Table 2, we show the same set of parameters for the ground-based experiment case. In this example, we compute all three QML methods at NSIDE = 64. And as shown in Table 2, the two scalar QML methods save on both computation time and memory requirements. Comparing with the results listed in the Table 1, we find that as N_d is increased, the advantages of the two scalar QML methods in terms of computation time become more obvious. In the scalar QML methods there is no need to calculate the rotation angle and perform coordinate transformation of Equation (13), and the signal covariance matrix is easier to compute with less memory overhead. For the ground-based case, the two scalar QML methods are near optimal in the multipole range of interest in this case, and their computational requirements mean that they can be applied on higher resolution maps to compute the power spectrum at higher multipoles.