Effect of Foregrounds on the CMBR Multipole Alignment

Abstract

We analyze the effect of foregrounds on the observed alignment of CMBR
quadrupole and octopole. The alignment between these multipoles is studied by
using a symmetry based approach which assigns a principal eigenvector (PEV) or
an axis with each multipole. We determine the significance of alignment between
these multipoles by using the Internal Linear Combination (ILC) 5 and 7 year
map s and also the maps obtained by using the Internal Power Spectrum
Estimation (IPSE) procedure. The effect of foreground cleaning is studied in
detail within the framework of the IPSE method both analytically and
numerically. By using simulated CMBR data, we study how the PEVs of the pure
simulated CMB map differ from those of the final cleaned map. We find that, in
general, the shift in the PEVs is relatively small and in random directions.
Due to the random nature of the shift we conclude that it can only lead to
misalignment rather than alignment of multipoles. We also directly estimate the
significance of alignment by using simulated cleaned maps. We find that the
results in this case are identical to those obtained by simple analytic
estimate or by using simulated pure CMB maps.

Full text

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arXiv:1007.1827v1 [astro-ph.CO] 12 Jul 2010
Effect of Foregrounds on the CMBR Multipole
Alignment
Pavan K. Aluri1, Pramoda K. Samal2,
Pankaj Jain1and John. P. Ralston3
1Department of Physics, Indian Institute of Technology, Kanpur 208016,
India
2Department of Physics, Utkal University, Bhubaneswar, 751004, India
3Department of Physics & Astronomy, University of Kansas,
Lawrence, KS - 66045, USA
Abstract
We analyze the effect of foregrounds on the observed alignment of
CMBR quadrupole and octopole. The alignment between these multi-
poles is studied by using a symmetry based approach which assigns a
principal eigenvector (PEV) or an axis with each multipole. We deter-
mine the significance of alignment between these multipoles by using
the Internal Linear Combination (ILC) 5 and 7 year maps and also
the maps obtained by using the Internal Power Spectrum Estimation
(IPSE) procedure. The effect of foreground cleaning is studied in de-
tail within the framework of the IPSE method both analytically and
numerically. By using simulated CMBR data, we study how the PEVs
of the pure simulated CMB map differ from those of the final cleaned
map. We find that, in general, the shift in the PEVs is relatively small
and in random directions. Due to the random nature of the shift we
conclude that it can only lead to misalignment rather than alignment
of multipoles. We also directly estimate the significance of alignment
by using simulated cleaned maps. We find that the results in this case
are identical to those obtained by simple analytic estimate or by using
simulated pure CMB maps.
Keywords: cosmic microwave background, methods: data analysis, meth-
ods: statistical
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1 Introduction
The standard cosmological model rests on the Cosmological Principle which
states that the universe is homogeneous and isotropic. However, there are
many indications from diverse data sets that this principle may not be appli-
cable. In particular, polarizations of radio waves coming from distant radio
galaxies (Birch 1982; Jain and Ralston 1999), the optical polarizations from
quasars (Hutsem´ ekers 1998; Hutsem´ ekers and Lamy 2001, Jain et al. 2004)
and the multipoles l = 1,2,3 of Cosmic Microwave Background Radiation
(CMBR) all indicate a universal axis pointing in the direction of the Virgo
cluster (de Oliveira-Costa et al. 2004; Ralston and Jain 2004; Schwarz et al
2004). This phenomenon has been called the Virgo Alignment (Ralston and
Jain 2004). There have also been other claims of large scale anisotropy in
the CMBR data. These include the dipole power modulation (Eriksen et al.
2004; Eriksen et al. 2007; Hoftuft et al. 2009), as well as, the detection of an
anomalously cold spot (Cruz et al. 2005). These anomalies have been studied
in great detail (Bielewicz et al. 2004; Hansen et al. 2004; Katz and Weeks
2004; Bielewicz et al. 2005; Prunet et al 2005; Bernui et al. 2006; Copi et
al. 2006; Copi et al. 2007; de Oliveira-Costa and Tegmark 2006; Wiaux et
al. 2006; Freeman et al. 2006; Helling et al. 2006; Bernui et al. 2007; Land
and Magueijo 2007; Magueijo and Sorkin 2007). Meanwhile some studies do
not find any violation of statistical isotropy in CMB (Hajian et al. 2005;
Hajian and Souradeep 2006). Furthermore, the cluster peculiar velocities
(Kashlinsky et al. 2008, 2009), as well as the large scale galaxy distribution
(Itoh et al. 2009), also indicate deviations from isotropy. The preferred axis
for the cluster peculiar velocities with a high redshift cut is again found to
be close to the direction of Virgo cluster. In the galaxy distribution surveys,
the preferred axis appears to depend strongly on the cut made on the data.
There have been a variety of proposals in the literature explaining the
possible origin of this alignment anomaly, such as, anisotropic space-times
(Berera et al. 2004; Kahniashvili et al. 2008), foreground contaminations
(Slosar and Seljak 2004; Abramo et al. 2006; Rakic et al. 2006), noise bias
or systematics (Naselsky et al. 2008), vector like dark energy (Armendariz-
Picon 2004), spontaneous breaking of isotropy (Gordon et al. 2005; Land
and Magueijo 2006, Erickcek et al. 2009), inhomogeneous universe (Mof-
fat 2005; Land and Magueijo 2006), inhomogeneous inflation (Carroll et al.
2010), violation of rotational invariance during inflation (Ackerman et al.
2009), anisotropic perturbations due to dark energy (Battye and Moss 2006),
anisotropic inflation (Buniy et al. 2006), local voids (Inoue and Silk 2006)
and anisotropic dark energy (Koivisto and Mota 2008; Rodrigues 2008).
Another interesting effect associated with quadrupole is its low power in
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comparison to the best fit LCDM model (Bennett et al. 2003). One may
speculate that the observed quadrupole-octopole alignment is related to the
observed low quadrupole power. It has been shown that in random realiza-
tions of pure CMB maps there is no correlation between the low quadrupole
power and quadrupole-octopole alignment (Raki´ c and Schwarz 2007; Sarkar
et al. 2010).
In the present paper, we analyze the alignment of CMBR quadrupole
(l = 2) and octopole (l = 3) moments more closely. In (Slosar and Seljak
2004), it has been suggested that this alignment may be caused by the galactic
foregrounds. These authors argue that most of the power of quadrupole and
octopole lies in the most contaminated region of the sky. Hence, it is entirely
possible that these multipoles may be significantly contaminated and the
alignment may be caused due to poor cleaning.
In the Internal Power Spectrum Estimation (IPSE) procedure (Saha et
al. 2006, 2008; Samal et al. 2010) the cleaning is performed by a totally
blind procedure without making any explicit model for foregrounds. The
foregrounds are removed by adding maps, in harmonic space, at different
frequencies with suitable weights, chosen so as to minimize the foreground
power. The Internal Linear Combination (ILC) map provided by the WMAP
team (Bennett et al. 2003) also uses a similar cleaning procedure. Here the
weighted maps are added directly in pixel space. Both of these procedures are
highly effective in removing foregrounds. However some residual foregrounds
may still contaminate the data. Besides this, (Saha et al. 2008; Samal et
al. 2010) pointed out the existence of bias in the low l multipoles extracted
from foreground cleaned maps. This bias leads to a significant reduction of
extracted power in these multipoles. In fact, the bias completely explains
the observed low power at l = 2, in comparison to the best fit theoretical
LCDM model. It is clearly important to determine what the low l bias might
imply for the alignment of these multipoles. It is possible that the power gets
eliminated in just the precise manner in order to cause alignment. Here we
study whether any of these residual effects, present in the foreground cleaned
data, may cause the observed quadrupole-octopole alignment.
It is important to note that the observed alignment is only getting better
with more data. A larger data sample implies smaller uncertainties due to
detector noise. Hence the signal becomes more prominent as the fluctuations
due to detector noise get suppressed.
The problem of testing violation of isotropy in the CMBR signal has been
addressed by many authors. The CMBR anisotropy spectrum is generally
assumed to be isotropic in a statistical sense. The assumption of statistical
isotropy of the CMBR signal means that the spectral coefficients, alm, are
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uncorrelated for different m and l, i.e. the ensemble average,
< a∗
lmal′m′ >= δll′δmm′Cl,(1)
where Clis the standard power. A multitude of statistics have been proposed
in order to test for deviation from statistical isotropy in the CMBR data
(Hajian et al. 2005; Hajian and Souradeep 2006; Copi et al. 2006, 2007).
Here we shall use the method introduced in (Ralston and Jain 2004) and
further developed in (Samal et al. 2008, 2009). In this procedure, one assigns
three orthogonal unit vectors for each multipole. The orientation of these
vectors as well as the power associated with each vector contains information
about possible violation of statistical isotropy. This information is encoded in
two entropy measures, the power entropy and alignment entropy, defined in
(Samal et al. 2008). Using this method one can test for violation of isotropy
for each individual multipoles or collectively over a range of multipoles. We
present an outline of our analysis procedure in the next section followed by
results from cosmological data and simulations and conclusions in the end.
2Analysis Methodology
The primary objective of the present paper is to determine whether the align-
ment of quadrupole and octopole can be caused by foreground contamination.
The alignment of different multipoles can be quantified by associating a frame
with each multipole (Ralston and Jain 2004; Samal et al. 2008). In order to
analyse the effect of foreground contamination we first perform an analytic
calculation of the power tensor, introduced in (Ralston and Jain 2004; Samal
et al. 2008), within the framework of IPSE technique (Saha et al. 2006,
2008; Samal et al. 2010). Here we confine ourselves to the simple case of one
foreground component following rigid frequency scaling and two frequency
bands. This computation is useful to determine if the low l bias also leads
to violation of statistical isotropy. We then numerically determine the effect
of this bias and residual foregrounds on the alignment of quadrupole and oc-
topole. We generate many random realizations of the CMBR maps including
foregrounds and detector noise. We use the Planck Sky Model (PSM)3for
foregrounds and use all the five WMAP frequency bands. These simulated
raw maps are cleaned using the IPSE technique. We study the difference
in the extracted principal eigenvectors of cleaned maps and simulated pure
CMB maps. This allows us to determine if the residual foregrounds lead to
any preferential alignment of these extracted vectors.
3http://www.planck.fr/heading79.html
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In the next subsections, we briefly review the extraction of frames and
the IPSE technique.
2.1 Covariant frames
The present study utilizes a symmetry based statistic (Samal et al. 2008)
to test for statistical isotropy in the CMBR data. The method associates a
wave function ψk
m, defined as,
ψk
m(l) =
1
?l(l + 1)?l,m|Jk|a(l)? (2)
with every multipole l. Here |a(l)? contains information about the spectral
moments, such that,
alm= ?l,m|a(l)? (3)
where |lm? are the eigenstates of the angular momentum operators?J2, Jzin
the spin-l representation. The wave function ψk
frame, i.e. three orthonormal vectors, at each l. The frame may be extracted
by making a singular value decomposition of the wave function.
It is also convenient to define the power tensor,
mis useful since it assigns a
Aij(l) =
?
m
ψi
mψj
m
∗
=
1
l(l + 1)
?
m,m′
a∗
lm(JiJj)mm′alm′ . (4)
The three eigenvectors, eα
ated with each multipole. The index, i = 1,2,3, labels the three components
of each eigenvector. The corresponding eigenvalues, (Λα)2, contain informa-
tion about the power associated with each eigenvector. The sum of the three
eigenvalues equals the total power, Cl. A large dispersion in the eigenvalues
indicates violation of statistical isotropy for a particular mode.
We define the principal pigenvector (PEV), ˆ nl, as the eigenvector cor-
responding to the maximum eigenvalue. Here we note that the extracted
PEVs are headless, meaning, these vectors specify only an axis and not a di-
rection. The P-value or the significance of alignment of the quadrupole and
octopole PEV may be estimated analytically by the formula P = 1−ˆ n2·ˆ n3=
1 − cos(δΘ). Here δΘ is the angle between the PEVs ˆ n2and ˆ n3, which cor-
respond to l = 2 and l = 3 respectively. Alternatively, one can estimate the
significance directly by numerical simulations, as described below.
i, α = 1,2,3, of this matrix define the frame associ-
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