Issue |
A&A
Volume 519, September 2010
|
|
---|---|---|
Article Number | A104 | |
Number of page(s) | 7 | |
Section | Astronomical instrumentation | |
DOI | https://doi.org/10.1051/0004-6361/201014739 | |
Published online | 20 September 2010 |
E/B decomposition of CMB polarization pattern of incomplete sky: a pixel space approach
J. Kim - P. Naselsky
Niels Bohr Institute & Discovery Center, Blegdamsvej 17, 2100 Copenhagen, Denmark
Received 6 April 2010 / Accepted 26 June 2010
Abstract
CMB polarization pattern may be decomposed into gradient-like (E)
and curl-like (B) mode, which provide invaluable information
respectively.
We have investigated E/B decomposition in pixel space. We find E/B
mixing due to incomplete sky is localized in pixel-space, and
negligible in the regions far away from the masked area. By estimating
the expected local leakage power, we have diagnosed ambiguous pixels.
Our criteria for ambiguous pixels (i.e. )
is associated with the tensor-to-scalar ratio of B mode power spectrum, which the leakage power is comparable to.
By setting
to a lower value, we may reduce leakage level, but reduce sky fraction at the same time.
Therefore, we have solved
,
and obtained the optimal
,
which minimizes the estimation uncertainty, given a foreground mask and
noise level. We have applied our method to a simulated map blocked by a
foreground (diffuse + point source) mask. Our simulation shows leakage
power is smaller than primordial (i.e. unlensed) B mode power spectrum
of tensor-to-scalar ratio
at wide range of multipoles (
), while allowing us to retain sky fraction
0.48.
Key words: methods: data analysis - cosmic background radiation
1 Introduction
Over the past years, CMB polarization has been measured by several experiments and is being measured by the Planck surveyor (Kovac et al. 2002; Leitch et al. 2002; Pryke et al. 2002a,b Halverson et al. 2002; Brown & et al. 2009; Hinderks et al. 2009; Leitch et al. 2005; Pryke & et al. 2009; The Planck Collaboration 2006; Ade & et al. 2008). CMB polarization pattern may be considered as the sum of gradient-like E mode and curl-like B mode (Zaldarriaga & Seljak 1997; Kamionkowski et al. 1997). In the standard model, B mode polarization is not produced by scalar perturbation, but solely by tensor perturbation. Therefore, measurement of B mode polarization makes it possible to probe the universe on the energy scale at inflationary period (Dodelson 2003; Zaldarriaga & Seljak 1997; Mukhanov 2005; Kamionkowski et al. 1997; Liddle & Lyth 2000). In most inflationary models, tensor-to-scalar ratio r is much smaller than one, and the WMAP 7 year data imposes an upper bound on r<0.36 at
Besides instrument noise, there are complications, which limits
detectability of tensor perturbation. Imperfection in removing
foreground and gravitational lensing imposes observational limit on
tensor-scalar-ratio:
and
respectively (Seljak & Hirata 2004; Tucci et al. 2005).
Due to the nature of the observation or heavy foreground contamination,
reliable of estimation on CMB polarization signal is not available over
a whole sky.
Incomplete sky coverage leads to E/B mixing, and very significantly
limit our capacity to measure tensor perturbation as well (Bunn et al. 2003).
Therefore, there have been various efforts to understand and reduce E/B mixing (Bunn et al. 2003; Lewis 2003; Kim 2007b; Smith 2006; Lewis et al. 2002; Kim 2007a).
It is best to implement E/B decomposition in map space, since diffuse
foregrounds and point sources are well-localized in map space, and
their spatial information are known relatively better than other
properties. In this paper, we investigate E/B decomposition in
pixel space.
Our investigation shows that E/B mixing is highly localized in
pixel space.
Therefore, we may reduce E/B mixing effectively by excluding the
ambiguous pixels. We have applied our method to simulated maps
partially blocked by a foreground (diffuse + point source) mask.
After excluding ambiguous pixels, we find that leakage power in
retained pixels (sky fraction 0.48) is smaller than primordial (i.e. unlensed) B mode power spectrum of tensor-to-scalar ratio
at wide range of multipoles (
).
The outline of this paper is as follows. In Sect. 2, we discuss all-sky analysis of CMB polarization. In Sect. 3, we derive E/B decomposition in pixel space. In Sect. 4, we discuss the application to cut sky, and the method to diagnose ambiguous pixels. In Sects. 5 and 6, we present our simulation result. In Sect. 7, we summarize our investigation. In Appendix A, we discuss error analysis of pseudo Cl estimation, and show interpixel noise correlation may be neglected.
2 Stokes parameters
The state of polarization is described by Stokes parameter (Rohlfs & Wilson 2003; Kraus 1986).
Since Thompson scattering does not generate circular polarization,
Stokes parameter Q and U are sufficient to describe CMB polarization (Dodelson 2003).
Stokes parameter Q and U transform under rotation of an angle
on the plane perpendicular to direction
(Zaldarriaga & Seljak 1997; Zaldarriaga 1998):
Therefore, all-sky Stokes parameters may be decomposed into spin

where the decomposition coefficients

Though the quantity shown in Eq. (2) has direct association with physical observables (i.e. Stokes parameters), rotational variance leads to computational complication. Therefore, two real scalar quantities, termed ``E'' and ``B'' mode, are often built out of

where




![\begin{eqnarray*}\;\raise1.0pt\hbox{$'$ }\hskip-6pt\partial\;{}_s f(\theta,\phi)...
...tial}{\partial \phi}\right] \sin^{s}\theta\;{}_s f(\theta,\phi),
\end{eqnarray*}](/articles/aa/full_html/2010/11/aa14739-10/img39.png)
where


For a Gaussian seed fluctuation model, decomposition coefficients of E and B mode satisfy the following statistical properties:
![]() |
(8) | |
![]() |
(9) |
where




![]() |
Figure 1: The power spectrum of E ( top) and B ( bottom): no lensing, B mode power spectrum is plotted for various tensor-to-scalar ratio r. |
Open with DEXTER |
3 E/B decomposition in pixel space
In this section, we are going to derive a pixel-space analogue of E/B decomposition.
Using Eqs. (3), (6) and (7), we may easily show Eqs. (4) and (5) are equivalently given by:
where
Therefore, we may identify


![]() |
Figure 2:
Filter function:
|
Open with DEXTER |
![]() |
Figure 3:
Filter function: modulus
|
Open with DEXTER |









4 Incomplete sky coverage
Due to heavy foreground contamination, CMB polarization signal is not
estimated reliably over a whole sky.
For instance, the WMAP team have subtracted diffuse foregrounds by
template-fitting, and masked the regions that cannot be cleaned
reliably.
In Fig. 7, we show a
foreground mask, which combines the WMAP team's polarization mask with
the point source mask and shall be used for our simulation.
The E/B decomposition coefficients from a masked sky are given by:
where
![]() |
(16) |
and

Since filter functions


Using Eq. (18), we may show the expected power of
is given by:
where

and











where
Therefore, we may diagnose ambiguous pixels (i.e. heavy E/B mixing) by comparing






![]() |
(25) |
where r is the assumed tensor-to-scalar ratio of Monte-Carlo simulation, from which





![]() |
Figure 4:
Effective sky fraction
|
Open with DEXTER |

![]() |
= | ![]() |
|
![]() |
![]() |
(26) |
where Nl is noise power spectrum. Note that the leakage does not bias the B mode power spectrum estimation, but increases the variance, when the power spectrum estimation is made by a pseudo-Cl method and leakage is taken care of (Grain et al. 2009; Hivon et al. 2002a). By requiring

In Fig. 5, we plot the left and right hand side of Eq. (27) for the noise level of Planck HFI instrument, and the multipole l=86, which is the peak multipole of primordial B mode power spectrum. From Fig. 5, we find curves intersect at



![]() |
Figure 5: Numerical solution of Eq. (27) for various r and the noise level of Planck HFI instrument: two plots represent the left hand side (LHS) and right hand side (RHS) of Eq. (27). |
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5 Application to simulated data
Using the WMAP concordance CDM model, we have simulated Stokes parameter Q and U over a whole-sky with a HEALPix pixel resolution (
)
and 10' FWHM
beam. We have made the inputmap to contain no B mode polarization.
Therefore, any non-zero values in output B map are attributed to
leakage.
We show our simulated polarization map in Fig. 6, where the orientation and length of headless arrows indicates polarization angle and amplitude respectively.
Note that the polarization map shows only gradient-like patterns, because they contain only E mode polarization.
It is well-known that E/B mixing increases with the length of cut sky boundary (Bunn et al. 2003).
We have combined the WMAP team's polarization mask with the point source mask, and prograded it to
.
In order to reduce sharp boundaries, we have smoothed the mask with
FWHM Gaussian kernel.
We have referred to the WMAP team's boundary smoothing process of Internal Linear Combination map (Hinshaw et al. 2007).
Nevertheless, it should be noted that smoothed boundary is not
essential to our method, and further improvement may be possible by
using more sophisticated smoothing kernel (Das et al. 2009). In Fig. 7, we show our smoothed mask, whose sky fraction amounts to 0.71.
![]() |
Figure 6: Input polarization map: E mode polarization only. |
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![]() |
Figure 7:
Foreground mask: smoothed with |
Open with DEXTER |
![]() |
Figure 8:
Output |
Open with DEXTER |



![]() |
Figure 9:
Filerted |
Open with DEXTER |
![]() |
Figure 10: Leakage power spectrum and primordial B mode power of various tensor-to-scalar ratio r: a blue curve correspond to the leakage power estimated without ambiguous pixel filtering (Fig. 8), a green curve to the leakage power estimated with ambiguous pixel filtering (Fig. 9). |
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6 Scale-dependence of leakage
In order to reduce leakage, we have removed ambiguous pixels in a scale-independent way. On the other hand, it is known that leakage has some dependence on scales as well as real-space. Specifically, leakage of low l extends over large area, while leakage of high l is often confined to the small area nearby the boundary. Therefore, one may argue that our method does not reduce the leakage at low l as effectively as that of high l or leads to unnecessary loss of information. However, as shown in Fig. 8, leakage is relatively localized in pixel space, and we were able to reduced leakage very significantly at wide range of multipoles, while retaining sky fraction 0.48. Besides that, our simulation shows the leakage at lowest l is reduced significantly as well. It is also possible to implement a further leakage reduction in a scale-dependent way, after ambiguous pixels are removed. Nevertheless, a hybrid method, which exploits scale and position dependence simultaneously, may be most optimal. A wavelet approach may be promising for such implementation, since wavelet functions are, in general, well-localized in harmonic space as well as pixel space. We defer a rigorous investigation to a separate publication.7 Discussion
We have investigated E/B decomposition in pixel space, and shown
that we may produce E/B decomposed maps by convolving polarization maps
with certain filter functions of a sharp peak. We find that E/B mixing
due to incomplete sky is localized in pixel-space, and negligible in
the regions far away from masked area. By estimating the expected local
leakage power and comparing it with the expected pure mode power, we
have diagnosed ambiguous pixels and excluded them.
Our criteria for ambiguous pixels (i.e. )
is associated with the tensor-to-scalar ratio of B mode power spectrum, which the leakage power is comparable to.
The estimation error
may increases with lower
,
because sky fraction decreases.
Therefore, we have solved
and obtained the optimal
,
which minimizes the estimation error, given a foreground mask and noise level.
We have applied our method to simulated maps blocked by a foreground mask.
Simulation shows that leakage power is subdominant in comparison with unlensed B mode power spectrum of
at wide range of multipoles (
), while pixels of sky fraction 0.48 are retained.
We may apply our method equally to small sky patch observation, by treating unobserved sky as masked region.
From simulation with sky patch of simple symmetric shape, we have confirmed our method reduce E/B mixing very effectively.
A rigorous investigation is deferred to a separate publication.
Noise is slightly correlated from pixel to pixel in E and B maps, even when interpixel correlation is absent in Q and U maps. However, this interpixel noise correlation induced by E/B decomposition is not confined to our method, but E/B decomposition in general. Besides that, we find interpixel noise correlation may be neglected without sacrificing the accuracy of error analysis (refer to Appendix A for details). Therefore, it does not limit the applicability of our method.
Current observations such as WMAP were unable to detect B mode polarization. Therefore, we did not attempt to apply our method to observation data. When Planck polarization data of high Signal-to-Noise-Ratio (SNR) are available in near future, we may apply our method to the data, and be able to detect B mode polarization.
AcknowledgementsWe are grateful to the anonymous referee for thorough reading and helpful comments, which leads to significant improvement of this work. We acknowledge the use of the Legacy Archive for Microwave Background Data Analysis (LAMBDA) and the HEALPix package (Gorski et al. 1999,2005). This work is supported by FNU grant 272-06-0417, 272-07-0528 and 21-04-0355. This work is supported in part by Danmarks Grundforskningsfond, which allowed the establishment of the Danish Discovery Center.
Appendix A: Error analysis
Power spectrum is usually estimated by pseudo-Cl method at high multipoles (Hivon et al. 2002b; Efstathiou 2006; Nolta et al. 2009; Larson et al. 2010; Wandelt et al. 2001).
According to pseudo-Cl method, we may estimate power spectrum as follows (Hivon et al. 2002b; Wandelt et al. 2001):
![]() |
(1) |
where
![]() |
(2) |
and
![]() |
(3) |
with



where j and k refers to a DA-year combination and


The noise part is given by:
where a pixel index i runs over pixels outside a foreground mask, and Ni refers to noise at ith pixel. By central limit theorem (Riley et al. 2006; Arfken & Weber 2000),


If cross power spectra are used (i.e. ), noise does not bias estimation, and its statistical properties need to be known only for errors analysis (Hinshaw et al. 2007).
In order to understand the effect of noise on error analysis, let us consider covariance of
:
where Cl is a theoretical power spectrum and
Using Eqs. (A.4) and (A.6), we find
where A denotes terms irrelevant to noise. Therefore, we need to estimate noise covariance

The main contribution of Eq. (A.11) comes from the first term, because of cancellation through summation in the second term. Therefore, we find with good approximation:

Off-diagonal elements of noise covariance are given by:
Comparing (A.12) with Eq. (A.11), we may see the magnitude of off-diagonal elements is much smaller than that of diagonal elements, because cancellation through summation arise both in the first and the second term of Eq. (A.12). Therefore, we find noise covariance as follow:
Using Eqs. (A.8)-(A.10) and (A.13), we find covariance of

where C denotes terms irrelevant to noise. As shown in Eq. (A.14), we may neglect interpixel noise correlation in computing covariance of

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All Figures
![]() |
Figure 1: The power spectrum of E ( top) and B ( bottom): no lensing, B mode power spectrum is plotted for various tensor-to-scalar ratio r. |
Open with DEXTER | |
In the text |
![]() |
Figure 2:
Filter function:
|
Open with DEXTER | |
In the text |
![]() |
Figure 3:
Filter function: modulus
|
Open with DEXTER | |
In the text |
![]() |
Figure 4:
Effective sky fraction
|
Open with DEXTER | |
In the text |
![]() |
Figure 5: Numerical solution of Eq. (27) for various r and the noise level of Planck HFI instrument: two plots represent the left hand side (LHS) and right hand side (RHS) of Eq. (27). |
Open with DEXTER | |
In the text |
![]() |
Figure 6: Input polarization map: E mode polarization only. |
Open with DEXTER | |
In the text |
![]() |
Figure 7:
Foreground mask: smoothed with |
Open with DEXTER | |
In the text |
![]() |
Figure 8:
Output |
Open with DEXTER | |
In the text |
![]() |
Figure 9:
Filerted |
Open with DEXTER | |
In the text |
![]() |
Figure 10: Leakage power spectrum and primordial B mode power of various tensor-to-scalar ratio r: a blue curve correspond to the leakage power estimated without ambiguous pixel filtering (Fig. 8), a green curve to the leakage power estimated with ambiguous pixel filtering (Fig. 9). |
Open with DEXTER | |
In the text |
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