Multi-resolution internal template cleaning: An application to the Wilkinson Microwave Anisotropy Probe 7-yr polarization data

Abstract

Cosmic microwave background (CMB) radiation data obtained by different
experiments contain, besides the desired signal, a superposition of microwave
sky contributions. We present a fast and robust method, using a wavelet
decomposition on the sphere, to recover the CMB signal from microwave maps. An
application to \textit{WMAP} polarization data is presented, showing its good
performance particularly in very polluted regions of the sky. The applied
wavelet has the advantages of requiring little computational time in its
calculations, being adapted to the \textit{HEALPix} pixelization scheme, and
offering the possibility of multi-resolution analysis. The decomposition is
implemented as part of a fully internal template fitting method, minimizing the
variance of the resulting map at each scale. Using a $\chi^2$ characterization
of the noise, we find that the residuals of the cleaned maps are compatible
with those expected from the instrumental noise. The maps are also comparable
to those obtained from the \textit{WMAP} team, but in our case we do not make
use of external data sets. In addition, at low resolution, our cleaned maps
present a lower level of noise. The E-mode power spectrum $C_{\ell}^{EE}$ is
computed at high and low resolution; and a cross power spectrum $C_{\ell}^{TE}$
is also calculated from the foreground reduced maps of temperature given by
\textit{WMAP} and our cleaned maps of polarization at high resolution. These
spectra are consistent with the power spectra supplied by the \textit{WMAP}
team. We detect the E-mode acoustic peak at $\ell \sim 400$, as predicted by
the standard $\Lambda CDM$ model. The B-mode power spectrum $C_{\ell}^{BB}$ is
compatible with zero.

Full text

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arXiv:1106.2016v1 [astro-ph.CO] 10 Jun 2011
Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 13 June 2011(MN LATEX style file v2.2)
Multi-resolution internal template cleaning: An application
to the Wilkinson Microwave Anisotropy Probe 7-yr
polarization data
R. Fern´ andez-Cobos1,2⋆, P. Vielva1, R.B. Barreiro1, E. Mart´ ınez-Gonz´ alez1
1Instituto de F´ ısica de Cantabria, CSIC-Universidad de Cantabria, Avda. de los Castros s/n, 39005 Santander, Spain.
2Dpto. de F´ ısica Moderna, Universidad de Cantabria, Avda. los Castros s/n, 39005 Santander, Spain.
Accepted Received ; in original form
ABSTRACT
Cosmic microwave background (CMB) radiation data obtained by different experi-
ments contain, besides the desired signal, a superposition of microwave sky contribu-
tions. We present a fast and robust method, using a wavelet decomposition on the
sphere, to recover the CMB signal from microwave maps. An application to WMAP
polarization data is presented, showing its good performance particularly in very pol-
luted regions of the sky. The applied wavelet has the advantages of requiring little
computational time in its calculations, being adapted to the HEALPix pixelization
scheme, and offering the possibility of multi-resolution analysis. The decomposition
is implemented as part of a fully internal template fitting method, minimizing the
variance of the resulting map at each scale. In terms of residual levels of foregrounds,
we get better results to those obtained by the WMAP team working in real space
and with additional external data sets. Regarding the instrumental noise level in the
cleaned maps, we have obtained less noisy polarization maps at low resolution, where
full noise characteristics are available. The E-mode power spectrum CEE
at high and low resolution; and a cross power spectrum CTE
the foreground reduced maps of temperature given by WMAP and our cleaned maps
of polarization at high resolution. These spectra are consistent with the power spec-
tra supplied by the WMAP team. We detect the E-mode acoustic peak at ℓ ∼ 400,
as predicted by the standard ΛCDM model. The B-mode power spectrum CBB
compatible with zero.
ℓ
is computed
ℓ
is also calculated from
ℓ
is
Key words: methods: data analysis - cosmic microwave background
1 INTRODUCTION
Component separation is a critical aspect in the analy-
sis of cosmic microwave background (CMB) data. A good
characterization of the data is a prerequisite to the ade-
quate estimation of cosmological parameters. This need be-
comes crucial when, as happens in B-mode detection ex-
periments, foreground amplitudes are well above the sig-
nal (e.g., Tucci M. et al. 2005). Two physical galactic pro-
cesses are the major contaminants to CMB polarized sig-
nal: synchrotron radiation and thermal dust. Both appear
at large scales, are highly anisotropic and the spatial vari-
ation of their emissivity is smooth. Besides, extragalactic
emission also contaminates this cosmological signal: point
sources and clusters are compact objects, roughly isotropi-
cally distributed in the sky and every single object has a par-
⋆e-mail: cobos@ifca.unican.es
ticular frequency dependence. Most of the component sepa-
ration methods take into account only diffuse components,
assuming that we are previously masking the brightest point
sources or subtracting them by, typically, fitting approaches
(see Herranz & Vielva 2010, for a recent review).
Current and future experiments (Rubi˜ no-Mart´ ın et al.
2008; Brown et al. 2009; Sievers et al. 2009; Arnold et al.
2010; Kogut et al.2006;
Charlassier et al. 2008) are able to measure CMB po-
larization anisotropies with such precision that foreground
contamination have become the major limitation when we
try to analyze the data. This is the principal reason to invest
effort and time in developing new techniques for separating
components. The goal of all the proposed methods is to
separate or, at least, to identify CMB anisotropies from the
other components. The range of proposals includes internal
linear combinations (ILC), Bayesian methods and indepen-
Grainger et al.2008;
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2 R. Fern´ andez-Cobos et al.
dent component analysis (see Delabrouille & Cardoso 2007,
for a recent review).
There is abundantliterature
plicationsof thevarious
polarizationexperiments in
PLANCK
(Leach et al.2008;
Betoule et al. 2009; Baccigalupi et al. 2004) and WMAP
(Gold et al. 2011; Delabrouille J. et al. 2009; Kim et al.
2009; Bonaldi et al. 2007; Maino et al. 2007).
The method that we present in this paper is situated
in the context of the internal linear combinations and it is
a bet for a template cleaning in which coefficients are fitted
in the space of a particular wavelet that enables a multi-
resolution analysis. A fitting by scales allows, in practice,
some effective variation of the coefficients in the sky, which
is an advantage over the template cleaning in real space. This
is a fast procedure that especially shows its effectiveness in
polluted regions, such as those that appear in polarization
experiments.
This paper is structured as follows. The methodology
is described in detail in Section 2. We set out an analysis of
the low-resolution polarization WMAP data in Section 3. In
section 4, we show the treatment for high-resolution WMAP
data in order to obtain the CEE
ℓ
we present the conclusions and discussion in section 5.
that
related
as,
Efstathiou et al.
includesap-
methods
vogue
to
instance,
2009;
some
for
and CTE
ℓ
spectra. Finally,
2 METHODOLOGY
In this work, we present a multi-resolution internal template
cleaning (MITC) method for foreground removal. This is
the initial step of the map cleaning process in the SEVEM
method (Mart´ ınez-Gonz´ alez et al. 2003; Leach et al. 2008)
to the case of polarization.
For many purposes, it is a key point to have CMB maps
at several frequencies instead of a single map. For instance,
it would serve as a consistency check to verify whether any
detected feature of the data is actually monochromatic or
not (as, for instance, the case for non-gaussianity analysis).
Another advantage of the method is that we do not
need a thorough knowledge of foregrounds, because we ob-
tain all the information to construct different templates
from the data. Furthermore, this procedure preserves the
original resolution of the CMB component. But the down-
side is that the internal templates are noisy, so we in-
crease the total noise level when we remove them from
the data. This circumstance results, for instance, in an in-
crease in the error bars of the power spectrum at high mul-
tipoles. An alternative would be to incorporate external
templates, created from data from other independent ob-
servations or based on theoretical arguments. However, the
current knowledge of foreground emissions, in polarization,
is not substantiated with suitable ancillary data set, and
for that reason, this option is not considered in this case.
This situation may change in the future with the informa-
tion expected to be provided by experiments like PLANCK
(Tauber et al. 2010), C-BASS (King et al. 2011) or QUI-
JOTE (Rubi˜ no-Mart´ ın et al. 2008).
Figure 1. Outline of construction of the detail coefficients at
resolution j (di) as the substraction of the approximation coeffi-
cients at resolution j-1 (yi), from the approximation coefficients
at resolution j (xi).
2.1 The HEALPix wavelet
Wavelets are a powerful tool in signal analysis and
are extensively used in many astrophysics applications.
Several examples of implementation of component sep-
aration methods which employ very diverse wavelets
can be found in the literature (e.g., Ghosh et al. 2011;
Delabrouille J. et al.2009;
Vielva et al. 2003; Hansen et al. 2006). They are localized
wave functions, that allow for a multi-resolution treatment
of the data. This fact represents an advantage over other
component separation methods because it allows us to vary
the effective emissivity of foregrounds.
We usethe so-called
(Casaponsa B. et al. 2011), a discrete and orthogonal
wavelet that provides a multi-scale decomposition on the
sphere adapted to the HEALPix pixelization (G´ orski et al.
2005). The resolution of a map in the HEALPix tessellation
is given in terms of the Nside parameter, defined so that the
number of pixels needed to cover the sphere is N = 12N2
The resolution j of a map is a number such that 2j= Nside.
A CMB map is decomposed in the wavelet coefficient space
in a series of maps from the resolution of the original map
to the lowest resolution considered. All of these maps, ex-
cept the lowest resolution one, are called details. The last
one is called the approximation, and is constructed by de-
grading the original map to the appropiate resolution, i.e.,
to calculate the approximation coefficient at resolution j-1
at a given position i we take the average of the four daugh-
ter pixels at resolution j. The way that different detail maps
are built is illustrated in figure 1. At each resolution j, detail
coefficients are calculated as the substraction of the approxi-
mation coefficients at resolution j-1 from the approximation
coefficients at resolution j. Both this process and the math-
ematical formalism of this wavelet is carefully explained in
Casaponsa B. et al. (2011), where the HW is used to put
constraints on the fNL parameter from WMAP data.
In that paper, it is said that the reconstruction of a map
Gonz´ alez-Nuevo et al.2006;
HEALPix
wavelet,HW,
side.
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Multi-resolution internal template cleaning3
M(xi) can be written as
M(xi)=
nj0−1
?
4
?
p=0
λj0,pφj0,p(xi)+
+
m=1
J−1
?
j=j0
nj−1
?
p=0
γm,j,pψm,j,p(xi),
(1)
where λj0,ℓ and γm,j,ℓ are the approximation and detail co-
efficients respectively, φj,p(xi) is the scaling function and
ψm,j,p(xi) refers to the wavelet functions. The j index takes
values from the highest resolution J to the approximation
resolution j0.
The advantage of this wavelet with respect to others, in
addition to its straightforward implementation, lies in the
speed of the involved operations. The computational time
for the wavelet decomposition is of the order of the num-
ber of pixels (∼ Npix) whereas, for example, for the conti-
nous wavelet transform of the spherical Mexican hat wavelet
(Mart´ ınez-Gonz´ alez et al. 2002) or the needlets (Baldi et al.
2009) this time is of the order of ∼ N3/2
pix.
2.2Template fitting
The signal?Tj(p) at resolution j is constructed by subtract-
original signal, Tj, as follows
ing a linear combination of different templates tij from the
?Tj(p) = Tj(p) −
Nt
?
i=1
βijtij(p),
(2)
where Nt is the total number of templates and p is a pixel
index.
An internal template is formed as the difference of two
maps of the same resolution, corresponding to different fre-
quencies, in units of thermodynamic temperature.
The variance of the cleaned map is optimally minimized
at each scale to obtain the coefficients βij or, equivalently,
the quadratic quantity
?
where C−1is the inverse of the covariance matrix calculated
as the sum of contributions of the CMB and instrumental
noise (both, from the map to be cleaned and the templates).
From the previous discussion, it is obvious that the ap-
proach to produce an optimal recovery of the CMB would
require a certain knowledge of this signal, via its correla-
tions. However, a more robust estimator, without a priori
knowledge of the signal to be estimated, may be built by
considering only the instrumental noise correlations.
We have checked, however, that the gain in the CMB
recovery, by including the information related to the instru-
mental characteristics is, in practice, very little. Even more,
in some situations (as it is the case of the WMAP full resolu-
tion data, see section 4) the instrumental noise information
is limited to the autocorrelation. Therefore, in this work we
have decided to perform the internal template fitting with
uniform weights for all the pixels at each scale, which implies
to minimize the following quantity:
χ2
j=
p
??Tj(p)C−1[?Tj(p)]t?
,
(3)
Ej =
?
p
?
Tj(p) −
Nt
?
i=1
βijtij(p)
?2
.
(4)
Finally, we recover a single map performing the wavelet
synthesis. It can be written as
?T(? x) = T(? x) −
Nt
?
i=1
Nres
?
j=1
γij(? x)tij(? x),
(5)
where Nres denotes the number of involved resolutions and
γij some new coefficients given as linear combinations of βij
coefficients which are the result of the synthesis process.
3 ANALYSIS OF LOW RESOLUTION WMAP
DATA
The instrumental noise in WMAP polarization is known
to be correlated (Jarosik et al. 2011). Although the WMAP
data are typically given at a HEALPix resolution of Nside=
512, a more accurate version of the pixel-to-pixel correla-
tion is only available at low resolution, namely, Nside= 16.
Taking into account this difference, we have performed the
cleaning of the WMAP data in two cases: for low and high
resolution maps. In this section, we analyse the maps at
Nside= 16.
The WMAP data are composed by, at least, a superpo-
sition of CMB, synchrotron and thermal dust emissions. The
WMAP team proposed a template fitting in the pixel space
to clean the foreground emission in the Ka, Q, V and W
maps, using as templates the K band (for the synchrotron)
and a low resolution version of the Finkbeiner et al. (1999)
model for the thermal dust, with polarization direction de-
rived from starlight measurements (Gold et al. 2011).
In our approach, we use only a synchrotron template,
constructed as K-Ka. The reason for neglecting the thermal
dust template is because a previous analysis in real space
shows that its coefficients are much smaller than the cor-
responding ones for the synchrotron template. We clean Q
and U polarization components independently minimizing
the variance of the cleaned maps of the Q1, Q2, V1 and V2
differencing assemblies (DAs). The wavelet decomposition
is carried out down to resolution j = 3 for the data map,
thus, in addition to the approximation, we have a single de-
tail map at j = 4. Best fitting coefficients for the considered
DAs are given in table 1. We apply the WMAP polarization
analysis mask that excludes a 26% of the sky.
3.1Cleaned maps
Since the CMB polarization signal is clearly subdominant
in the WMAP low resolution data, it is hard to establish a
criterion to evaluate the goodness of the cleaning process,
and to perform comparisons with different solutions.
We have decided to evaluate this goodness by compar-
ing the cleaned map with the expected signal for a noisy sky
following the WMAP instrumental noise characteristics. In
this sense, a good compatibility with the noise properties
would indicate that foregrounds have been satisfactorily re-
duced.
We generate a set of 104simulations of the noise maps
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4 R. Fern´ andez-Cobos et al.
Frequency bandQ1
0.092
0.074
0.244
0.241
Q2
0.103
0.117
0.259
0.236
V1
0.036
0.020
0.081
0.085
V2
0.023
0.048
0.125
0.112
Detail (j = 4)
Q Stokes
U Stokes
Q Stokes
U Stokes
Approximation (j = 3)
Table 1. Template cleaning coefficients for Q and U Stokes parameters and DAs for the low resolution case.
Figure 2. Upper panels show the χ2distributions of our cleaned maps and the bottom ones present the same results for the WMAP
procedure. The solid line (red) corresponds to the theoretical curve of a χ2with as many degrees of freedom as pixels outside the mask
(i.e., the effective number of pixels in Q and U maps: 4518). The dashed line is the distribution calculated from simulations of our cleaned
maps. The vertical line shows the χ2value of the data maps in each case. The columns correspond to different frequency bands, from
left to right: Q, V and W bands.
resulting from our MITC method at Q, V and W frequency
bands, Mr(p), with r ∈ {1,...,104}, in order to construct a
χ2distribution, calculating each value as
?
where NObs is the noise correlation matrix. A number of
simulations of the order of a million is required to estimate
this matrix so that the distribution converges to the the-
orical curve of a χ2distribution with as many degrees of
freedom as pixels outside the mask in Q and U maps (in
this case, we have 4518 degrees of freedom). This distribu-
tion characterizes the expected noise level at each frequency
map. We can associate the χ2value of the data with relative
levels of signal. We can say that the cleaned maps contain
more than just noise (typically foreground residuals, since
the CMB is subdominant compared to the noise at these
scales) if the data value is much higher than typical values
of the distribution. Conversely, we can ensure that our maps
are compatible with the expected noise and that residuals
are small if the data value falls within the distribution. The
χ2
r=
p,q
Mr(p)N−1
Obs(p,q)Mt
r(q),
(6)
Frequency band
χ2
Cleaned
χ2
Forered
QVW
4762
4787
4566
4709
4453
4586
Table 2. Different values of χ2computed with our seven-
year cleaned maps per frequency band (χ2
WMAP seven-year foreground reduced maps per frequency band
(χ2
Forered).
Cleaned) and with
DA Q1
4489
Q2
4546
V1
4405
V2
4486
χ2
Cleaned
Table 3. Different values of χ2computed with our seven-year
cleaned maps per DA.
χ2values for each band are listed in table 2 and for each
DA in table 3.
Our test is based on the assumption that the CMB con-
tribution is negligible. We have tested that the CMB pro-
vides a very small contribution (a shift of ∼ 10 units of χ2)
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Multi-resolution internal template cleaning5
Figure 3. The combinations of the raw W-band maps. Theo-
retical curve of the χ2distribution is represented by a solid red
line and the χ2values are shown by successive vertical lines, from
left to right: W2-W4, W1-W2, (W1+W3)-(W2+W4), W2-W3,
W1-W4, W3-W4 and W1-W3.
by generating 104simulations with CMB and instrumen-
tal noise of the cleaned maps. These simulations have been
used to compute another χ2distribution with the matrix
that we have already calculated with only the noise compo-
nent. When distributions are compared with each other we
observe this typical deviation. Thus, the CMB contribution
to the value of the χ2of the data is negligible and, there-
fore, any significant deviation from the mean value has to
be assigned to foreground residuals.
An indirect comparison can be made between the
WMAP procedure and our MITC method through the rela-
tive possitions of the χ2value of the data with respect to the
distribution. As seen in figure 2, we obtain that the value
of the cleaned maps is fully compatible with instrumental
noise at Q and V frequency bands. At W band the χ2value
of the cleaned data is in the tail of the distribution probably
due to the presence of foreground residuals. The deviation
is even larger when the WMAP procedure is used. A sig-
nificant improvement is also found at Q band since the χ2
value is shifted from 2σ to 0.5σ when our MITC method is
used.
In addition, although we use a template that is noisier
than the ones used by the WMAP team, the noise levels of
our cleaned maps are lower. We have measured a difference
of about a 10% in terms of the standard deviation of the
data maps (this difference is confirmed by instrumental noise
simulations).
In order to check further the apparent excess of signal
at W band obtained by the two approaches, we computed
analytically the noise covariance matrix of different com-
binations of the raw W-band DAs maps which contain, in
principle, only a combination of instrumental noise. With
these covariance matrices, based on the full-sky covariance
matrices of each DA, a χ2value of the data maps is obtained.
It is shown in figure 3 that these maps are still compatible
with the expected noise. However, it is significant that all
values are to the left of the distribution and that the most
deviated ones involve W2, followed by W4. We have also
analysed the distribution of the χ2values of the single-year
foreground reduced maps supplied by the WMAP team for
each DA at the W band, obtaining values more deviated to-
wards the tails for the W2 and W4 DAs. This may suggest a
not good enough characterization of the instrumental noise
for these DAs.
3.2 Polarization power spectra, CEE
ℓ
and CBB
ℓ
We carry out an estimation of the polarization spectrum us-
ing our cleaned maps of the Q1, Q2, V1 and V2 DAs. A
pseudo cross-power spectrumˆDAB
encing assemblies A and B can be calculated as
?
where A,B ∈ {Q1,Q2,V 1,V 2 | A ?= B}; and, in the case of
an EE power spectra,
ℓ
between any two differ-
ˆDAB
ℓ
=
ℓ′
MAB
ℓℓ′ |pℓ′|2BA
ℓ′BB
ℓ′?CAB
ℓ′ ? + ?NAB
ℓ
?,
(7)
ˆCAB
ℓ
=
1
2ℓ + 1
ℓ
?
m=−ℓ
eA
ℓmeB∗
ℓm,
(8)
where eℓm are the spherical harmonic coefficients of the E-
mode. Assuming a circular beam response, we denote the
beam of the A map as BA
HEALPix pixel by pℓ; ?NAB
ℓ
? is the noise cross-power spec-
trum. The bias introduced by this term comes from the inter-
nal template fitting procedure. It is small and controled by
simulations. Finally, the coupling kernel matrix Mℓℓ′ is de-
scribed in Hivon et al. (2002) and, for the case of the polar-
ization components, in Appendix A of Kogut et al. (2003).
This procedure is usually referred to as MASTER estima-
tion. An estimator,ˆCℓ, can be computed as a linear combi-
nation of the six different spectra weighted by the inverse of
their variances in the following way:
??
where i = AB and σ2
by the WMAP team in the LAMBDA web site1.
The resulting power spectra are shown in figure 4. From
the CEE
ℓ
spectrum we can say that most of the values are
compatible with zero, so there is almost no signal except
perhaps for low multipoles ℓ ? 6. As expected, the B-mode
spectrum CBB
ℓ
signal is compatible with zero. Both spectra
are compatible with those that the WMAP team supplies.
Our error bars are larger than those obtained by the WMAP
team, because of the use of an estimator that is not optimal,
a pseudo-spectrum, whereas the WMAP team uses a pixel-
base likelihood.
ℓ and the window function of the
ˆCℓ=
i
1
σ2
i
?−1?
i
1
σ2
i
ˆCi
ℓ,
(9)
i = σAσB. These variances are given
4 ANALYSIS OF HIGH RESOLUTION WMAP
DATA
In this section, we analyze WMAP data maps at Nside =
512. This approach allows us to study the cleaning at smaller
scales where, a priori, the correlation of the noise is less im-
portant. So then we only take into account the noise covari-
ance matrix of each pixel. In this case, the cleaning method
1http://lambda.gsfc.nasa.gov/
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