The statistical expectation values of the temperature fluctuations of cosmic
microwave background (CMB) are assumed to be preserved under rotations of the
sky. This assumption of {\em statistical isotropy} (SI) of the CMB anisotropy
should be observationally verified since detection of violation of SI could
have profound implications for cosmology. We propose a set of measures,
$\kappa_\ell$ ($\ell=1,2,3, ...$) for detecting violation of statistical
isotropy in an observed CMB anisotropy sky map indicated by non zero
$\kappa_\ell$. We define an estimator for the $\kappa_\ell$ spectrum and
analytically compute its cosmic bias and cosmic variance. The results match
those obtained by measuring $\kappa_\ell$ using simulated sky maps. Non-zero
(bias corrected) $\kappa_\ell$ larger than the SI cosmic variance will imply
violation of SI. The SI measure proposed in this paper is an appropriate
statistics to investigate preliminary indication of SI violation in the
recently released WMAP data.Comment: 9 pages, 1 figure, Style file changed, figure enhanced and references
  update

Amir Hajian

Lachieze-Rey M.

Tarun Souradeep

English

arXiv

Crossref

Measuring the Statistical Isotropy of the Cosmic Microwave Background Anisotropy

Hajian, Amir

Souradeep, Tarun

Publications of the IAS Fellows

L5
The Astrophysical Journal, 597:L5–L8, 2003 November 1 
 2003. The American Astronomical Society. All rights reserved. Printed in U.S.A.
MEASURING THE STATISTICAL ISOTROPY OF THE COSMIC MICROWAVE BACKGROUND ANISOTROPY
Amir Hajian and Tarun Souradeep
Inter-University Centre for Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune 411 007, India
Received 2003 August 3; accepted 2003 September 15; published 2003 October 9
ABSTRACT
The statistical expectation values of the temperature fluctuations of the cosmic microwave background (CMB)
are assumed to be preserved under the rotations of the sky. This assumption of the statistical isotropy (SI) of
the CMB anisotropy should be observationally verified since detection of a violation of SI could have profound
implications for cosmology. We propose a set of measures, ( , 2, 3, …), for detecting a violation of SIk p 1
in an observed CMB anisotropy sky map indicated by nonzero . We define an estimator for the spectrumk k 
and analytically compute its cosmic bias and cosmic variance. The results match those obtained by measuring
using simulated sky maps. Nonzero (bias-corrected) larger than the SI cosmic variance will imply a violationk k 
of SI. The SI measure proposed in this Letter is an appropriate statistic to investigate a preliminary indication
of SI violation in the recently released Wilkinson Microwave Anisotropy Probe data.
Subject headings: cosmic microwave background — cosmology: observations
On-line material: color figure
Cosmic microwave background (CMB) anisotropy is a very
powerful observational probe of cosmology. In standard cos-
mology, CMB anisotropy is expected to be statistically isotro-
pic; i.e., the statistical expectation values of the temperature
fluctuations are preserved under the rotations of the sky.ˆDT(q)
In particular, the angular correlation function ′ˆ ˆC(q, q ){
is rotationally invariant for Gaussian fields. In′ˆ ˆADT(q)DT(q )S
spherical harmonic space, where , thisˆ ˆDT(q)p  a Y (q)lm lmlm
translates to a diagonal , where , the∗Aa a Sp C d d C′ ′ ′ ′lm l m l ll mm l
widely used angular power spectrum of CMB anisotropy, is a
complete description of (Gaussian) CMB anisotropy. Hence, it
is important to be able to determine whether the observed CMB
sky is a realization of a statistically isotropic process or not.1
We propose a set of measures ( , 2, 3, …) that fork p 1
nonzero values indicate and quantify a violation in statistical
isotropy (SI) in a CMB map. A null detection of will be ak
direct confirmation of the assumed SI of the CMB sky. It will
also justify a model comparison based on the angular power
spectrum only (Bond, Pogosyan, & Souradeep 1998, 2000a,Cl
2000b; Souradeep 2000). The detection of an SI violation can
have exciting and far-reaching implications for cosmology. In
particular, an SI violation in CMB anisotropy is the most ge-
neric signature of nontrivial geometrical and topological struc-
ture of space on ultralarge scales. Nontrivial cosmic topology
is a theoretically well-motivated possibility that is only recently
being observationally probed on the largest scales (Ellis 1971;
Lachieze-Rey & Luminet 1995; Starkman 1998; Levin 2002).
For a statistically isotropic CMB sky, the correlation function
1
ˆ ˆ ˆ ˆ ˆ ˆC(n , n ){ C(n · n )p dR C(Rn , Rn ), (1)1 2 1 2  1 228p
where denotes the direction obtained under the action of aˆRn
rotation on , and is a volume element of the three-ˆR n dR
dimensional rotation group. The invariance of the underlying
statistics under rotation allows the estimation of usingˆ ˆC(n · n )1 2
the average of the temperature product between all′˜ ˜ˆ ˆDT(n)DT(n )
1 Statistical isotropy of CMB anisotropy and its measurement have been
discussed in the literature (Ferreira & Magueijo 1997; Bunn & Scott 2000).
pairs of pixels with the angular separation v. In particular, for a
CMB temperature map defined on a discrete set of points˜ ˆDT(q )i
on a celestial sphere (pixels) ( ),qˆ ip 1, … , Ni p
Np
˜ ˜ ˜ˆ ˆ ˆ ˆC(v)p DT(q )DT(q )d(cos v q · q ) (2) i j i j
i, jp1
is an estimator of the correlation function of an underlyingC(v)
SI statistic.2
In the absence of SI, is estimated by a single product′ˆ ˆC(q, q )
and hence is poorly determined from a single′˜ ˜ˆ ˆDT(q)DT(q )
realization. Although it is not possible to estimate each element
of the full correlation function , some measures of the′ˆ ˆC(q, q )
statistical anisotropy of the CMB map can be estimated through
suitably weighted angular averages of . The an-′˜ ˜ˆ ˆDT(q)DT(q )
gular averaging procedure should be such that the measure
involves averaging over a sufficient number of independent
“measurements,” but it should ensure that the averaging does
not erase all of the signature of statistical anisotropy (as would
happen in eq. [1] or eq. [2]). Another important desirable prop-
erty is that the measures be independent of the overall orien-
tation of the sky. Based on these considerations, we propose
a set of measures of SI violation given byk
2(2 1)′ ′
ˆ ˆk p dQ dQ dR x (R )C(Rq, Rq ) , (3)[ ]    28p
where is the two-point correlation between and′ˆ ˆ ˆC(Rq, Rq ) Rq
obtained by rotating and by an element of the rotation′ ′ˆ ˆ ˆRq q q R
group. The measures involve an angular average of the cor-k
relation weighed by the characteristic function of the rotation
group , where are the Wigner D- x (R )p  D (R ) D ′ MM MMM
functions (Varshalovich, Moskalev, & Khersonskii 1988). When
is expressed as rotation by an angle q (where )R 0 ≤ q ≤ p
about an axis , the characteristic functionrˆ(V, F) x (R ){
2 This simplified description does not include optimal weights to account
for observational issues, such as instrument noise and nonuniform coverage.
However, this is well studied in the literature, and therefore we avoid discussing
them here in order to keep our presentation clear.
L6 HAJIAN & SOURADEEP Vol. 597
is completely determinedx (q)p sin [(2 1)q/2]/ sin (q/2)
by q, and the volume element of the three-dimensional rotation
group is given by . Using the2dRp 4 sin q/2 dq sin V dV dF
identity , equation (3) can be sim-′ ′ ′dR x (R )x (RR )p x (R )∫   
plified to
(2 1) ′ ′ ′
ˆ ˆ ˆ ˆk p dQ dQ C(q, q ) dR x (R )C(Rq, Rq ),    28p
(4)
containing only one integral over the rotation group. For a
statistically isotropic model, is in-ˆ ˆ ˆ ˆC(Rq , Rq ){ C(q , q )1 2 1 2
variant under rotation, and equation (4) gives be-0k p k d 0
cause of the orthonormality of . Hence, defined inx (q) k 
equation (3) is a measure of SI.
The measure has a clear interpretation in harmonic space.k
The two-point correlation can be expanded in terms′ˆ ˆC(q, q )
of the orthonormal set of bipolar spherical harmonics as
′ M ′
ˆ ˆ ˆ ˆC(q, q )p A {Y (q) Y (q )} , (5)′ ′ ll l l M′ll M
where are the coefficients of the expansion. These coef-MA ′ll
ficients are related to an “angular momentum” sum over the
covariances as∗Aa a S′ ′lm l m
′M ∗ m MA p Aa a S(1)  , (6)′ ′ ′ ′ ′ll lm l m lml m′mm
where are Clebsch-Gordan coefficients. The bipolar func-M ′ ′lml m
tions transform just like an ordinary spherical harmonic func-
tion under rotation (Varshalovich et al. 1988). SubstitutingYM
the expansion equation (5) into equation (3), we can show that
M 2k p FA F ≥ 0 (7)′ ll′ll M
is positive semidefinite and can be expressed in the form
2 1 ∗ ∗ Rk p dR x (R ) Aa a SAa a S , (8)′ ′ ′ ′   lm l m lm l m2 ′ ′8p lml m
where is computed in a frame rotated by . When SIRA…S R
holds, , implying∗ M lAa a Sp C d d A p (1) C (2l′ ′ ′ ′ ′lm l m l ll mm ll l
. The coefficients represent the statistically iso-1/2 001) d d d A′ll 0 M0 ll
tropic part of a general correlation function. The coefficients
are the inverse transform of the two-point correlationMAl l1 2
M ′ ′ ′ ∗
ˆ ˆ ˆ ˆA p dQ dQ C(n, n ){Y (n) Y (n )} . (9)l l   l l M1 2 1 2
The symmetry implies′ ′ˆ ˆ ˆ ˆC(n, n )p C(n , n)
M (l l ) M M M1 2A p (1) A , A p A d , kp 0, 1, 2, … .l l l l ll ll , 2k2 1 1 2
(10)
Recently, the Wilkinson Microwave Anisotropy Probe
(WMAP) has provided high-resolution (almost) full sky maps of
CMB anisotropy (Bennet et al. 2003) from which can bek
measured. Given a single independent CMB map, , we˜ ˆDT(q)
need to look for a violation of SI. Formally, the estimation pro-
cedure involves averaging the product of temperature at pairs of
pixels obtained by rotating a given pair of pixels by an angle q
around a sufficiently large sample of rotation axes. The integral
in the brackets in equation (3) is estimated by summing up the
terms for different values of q weighed by the characteristic
function. We can define an estimator for ask
B
˜ ˜k p k  ,  
N Np w(2 1)B ˜ ˜
˜ ˆ ˆk p DT(q )DT(q ) x (w )  i j  m28p i, jp1 mp1
Nr
˜ ˜ˆ ˆ# DT(R q )DT(R q ), mn i mn j
np1
(11)
where, as described below, accounts for the “cosmicB˜ { Ak S 
bias” for the biased estimator . As with the sky, the rotationBk˜
group is also discretized as , where is anR mp 1, … , Nmn w
index of equally spaced intervals in rotation angle w and
indexes a set of equally spaced directions in thenp 1, … , Nr
sky. While we have also implemented this real-space compu-
tation, practically, we find it faster to estimate in the harmonick
space by taking advantage of fast methods of the spherical har-
monic transform of the map. In harmonic space, we first define
an unbiased estimator for the bipolar harmonic coefficients based
on equation (6) and then estimate using equation (7),k
2
M M M˜ ˜F F˜A p a a  , k p A  . (12)′ ′ ′ ′ ′ ′ ll lm l m lml m  ll ′ ′mm ll M
Assuming Gaussian statistics of the temperature fluctuations,
the cosmic bias is given by (A. Hajian & T. Souradeep 2003,
in preparation)
B ∗ ∗
˜Ak S k p [Aa a SAa a S′ ′    l m l m l m l m1 1 1 1 2 2 2 2′ ′l , l m , m m , m1 2 1 1 2 2
∗ ∗ Aa a SAa a S]′ ′l m l m l m l m1 1 2 2 2 2 1 1
M M#   . (13)′ ′ l m l m l m l m1 1 2 2 1 1 2 2
M
Given a single CMB sky map, the individual elements of
the covariance are poorly determined. So we can cor-∗Aa a S′ ′lm l m
rect for the bias that arises from the SI part of correlation
function where
l1
B 
˜ { Ak S p (2 1) C C [1 (1) d ].   SI l l l l1 2 1 2
l l pFl F1 2 1
(14)
Hence, for an SI correlation, the estimator is unbiased, i.e.,k˜
.˜Ak Sp 0
Assuming Gaussian CMB anisotropy, the cosmic variance
of the estimators and can be obtained analytically forM˜ ˜A k′ll 
full sky maps. The cosmic variance of the bipolar coefficients
2 M˜j (A )p [Aa a SAa a S′ ′ l l l m l m l m l m1 2 1 1 1 1 2 2 2 2′ ′m , m m , m1 1 2 2
 Aa a SAa a S]′ ′l m l m l m l m1 1 2 2 2 2 1 1
M M#   , (15)′ ′l m l m l m l m1 1 2 2 1 1 2 2
No. 1, 2003 STATISTICAL ISOTROPY OF CMB L7
Fig. 1.—Bias-corrected “measurement” of of an SI CMB sky with ak
flat-band power spectrum smoothed by a Gaussian beam [l(l 1)C pl
]. The cosmic error , obtained using 50 independent re-2 2exp (l /18 ) j(k )
alizations of the CMB (full) sky map, match the analytic results shown
by the lower dotted curve with stars. The upper dotted curves separately
outline the cosmic error envelope for odd multipoles (filled triangles) and
for even multipoles (open triangles). Violation of SI will be indicated by
nonzero measured in an observed CMB map in excess of givenk j(k ) 
by the of the map. The lower dashed curve (filled squares) shows theCl
cosmic error for an ideal unit flat-band power spectrum [ ]l(l 1)C p 1l
with no beam smoothing. The curve falls off roughly at 1/ at large .
[See the electronic edition of the Journal for a color version of this figure.]
which, for an SI correlation, further simplifies to
2 M ˜j (A )p C C [1 (1) d ]SI l l l l l l1 2 1 2 1 2
m m M M1 2# (1)   .′ ′ l m l m l m l m1 1 2 2 1 1 2 2
m , m1 2
(16)
Note that for , the cosmic variance is zero for oddl p l 1 2
as a result of equation (10) arising from the symmetry of
.
′
ˆ ˆC(q, q )
A similar but more tedious computation of 105 terms of the
eight-point correlation function yields an analytic expression
for the cosmic variance of (A. Hajian & T. Souradeep 2003,k˜
in preparation). For the SI correlation, the cosmic variance for
is given by 1 0
2(2 1)2 4 
˜j (k )p 4C {2  (1) (2 1)SI  l 2l 1l : 2l≥
  [1 2(1) ]F }ll
l1
2 2  4C C [(2 1) F ]  l l l l1 3 1 3
l l pFl F1 3 1
l 212(2 1) 2 8 C C [ ]l l1 22l  1l l pFl F1 2 11
l12(2 1)
 3 16(1) C C , (17)  l l1 22l  1l : 2l ≥ l pFl F1 1 2 11
where
l l 1 3
 M MF p C C  l l l m l m l m l m1 3 1 1 3 3 1 2 3 4′m m pl m m pl M, Mp1 2 1 3 4 3
′ ′M M# C C . (18)l m l m l m l m3 4 1 1 3 3 1 2
Numerically, it is advantageous to rewrite in a series involvingF ′ll
9-j symbols. The expressions for variance and bias are valid for
full sky CMB maps. For observed maps, one has to contend
with incomplete or nonuniform sky coverage. In such cases, one
could estimate the cosmic bias and variance by averaging over
many independent realizations of simulated CMB sky from the
same underlying correlation function. Figure 1 shows the mea-
surement of in an SI model with a flat-band power spectrum.k
The bias and variance are estimated by taking measurements of
50 independent random full sky maps using the HEALPix soft-
ware package.3 The cosmic bias and variance obtained from these
realizations match the analytical results. Just as in the case of
cosmic bias, the cosmic variance of at odd multipoles isk
smaller. The figure clearly shows that the envelope of cosmic
variance for odd and even multipoles converge at large . For
a constant angular power spectrum, the falls˜l(l 1)C j (k )l SI 
off with . (The absence of the dipole and monopole in the maps
affects for , leading to the apparent rise in cosmic variancek  ! 4
at seen in Fig. 1.) ! 4
The bias and cosmic variance depend on the total SI angular
power spectrum of the signal and noise, . Hence,S NC p C  Cl l l
where possible, prior knowledge of the expected signal shouldk
be used to construct multipole space windows to weigh down
3 Publicly available at http://www.eso.org/science/healpix (Go´rski, Hivon,
& Wandelt 1999).
the contribution from the region of multipole space where the
SI violation is not expected, e.g., the generic breakdown of SI
due to cosmic topology. The underlying correlation patterns in
the CMB anisotropy in a multiply connected universe is related
to the symmetry of the Dirichlet domain (Wolf 1984; Vinberg
1993). In a companion paper, we study the signal expectedk
in flat, toroidal models of the universe and connect the spectrum
to the principle directions in the Dirichlet domain (Hajian &
Souradeep 2003). The SI violation arising from cosmic topology
is usually limited to low multipoles. A wise detection strategy
would be to smooth CMB maps to a low angular resolution.
When searching for a specific form of SI violation, linear com-
binations of can be used to optimize the signal-to-noise ratio.k
Before ascribing the detected breakdown of statistical anisotropy
to cosmological or astrophysical effects, one must carefully ac-
count for and model into the SI simulations other mundane
sources of SI violation in real data, such as incomplete and
nonuniform sky coverage, beam anisotropy, foreground resid-
uals, and statistically anisotropic noise.
In summary, the statistics quantifies the breakdown of SIk
into a set of numbers that can be measured from the single
CMB sky available. The spectrum can be measured veryk
fast, even for high-resolution CMB maps. The statistics have
a very clear interpretation as quadratic combinations of off-
diagonal correlations between coefficients. The signal SIalm
violation is related to underlying correlation patterns. The an-
gular scale on which the off-diagonal correlations (patterns)
occur is reflected in the spectrum. As a tool for detectingk
cosmic topology (more generally, cosmic structures on ultra-
large scales), the spectrum has the advantage of being in-k
dependent of the overall orientation of the correlation pattern.
This is particularly suited for searching for cosmic topology
since the signal is independent of the orientation of the Dirichlet
domain. (However, orientation information is available in the
.) The recent all-sky CMB map from WMAP is an idealMAl l1 2
L8 HAJIAN & SOURADEEP Vol. 597
data set where one can measure the SI. Interestingly, there are
hints of SI violation in the low multipole of WMAP (Tegmark,
de Oliveira-Costa, & Hamilton 2003; de Oliveira-Costa et al.
2003; Eriksen et al. 2003). Hence, it is of great interest to make
a careful estimation of the SI violation in the WMAP data via
the spectrum. This work is in progress, and results will bek
reported elsewhere (A. Hajian et al. 2003, in preparation). This
approach complements the direct search for the signature of
cosmic topology (Cornish, Spergel, & Starkman 1998).
T. S. acknowledges enlightening discussions with Larry Wea-
ver of Kansas State University at the start of this work. T. S.
also benefited from discussions with J. R. Bond and D. Po-
gosyan on cosmic topology and related issues.
REFERENCES
Bennet, et al., C. L. 2003, ApJS, 148, 1
Bond, J. R., Pogosyan, D., & Souradeep,T. 1998, Classical Quantum Gravity,
15, 2671
———. 2000a, Phys. Rev. D, 62, 043005
———. 2000b, Phys. Rev. D, 62, 043006
Bunn, E. F., & Scott, D. 2000, MNRAS, 313, 331
Cornish, N. J., Spergel, D. N., & Starkman, G. D. 1998, Classical Quantum
Gravity, 15, 2657
de Oliveira-Costa, A., Tegmark, M., Zaldarriaga, M., & Hamilton, A. 2003,
preprint (astro-ph/0307282)
Ellis, G. F. R. 1971, Gen. Relativ. Gravitation, 2, 7
Eriksen, H. K., Hansen, F. K., Banday, A. J., Go´rski, K. M., & Lilje, P. B.
2003, preprint (astro-ph/0307507)
Ferreira, P. G., & Magueijo, J. 1997, Phys. Rev. D, 56, 4578
Go´rski, K. M., Hivon, E., & Wandelt, B. D. 1999, in Proc. MPA-ESO Cos-
mology Conf., Evolution of Large Scale Structure: From Recombination to
Garching, ed. A. J. Banday, R. K. Sheth, & L. A. N. da Costa (Enschede:
PrintPartners Ipskamp), 37
Hajian, A., & Souradeep, T. 2003, preprint (astro-ph/0301590)
Lachieze-Rey, M., & Luminet, J.-P. 1995, Phys. Rep., 25, 136
Levin, J. 2002, Phys. Rep., 365, 251
Souradeep,T. 2000, in The Universe: Visions and Perspectives, ed. N. Dadhich
& A. K. Kembhavi (Dordrecht: Kluwer), 289
Starkman, G. 1998, Classical Quantum Gravity, 15, 2529
Tegmark, M., de Oliveira-Costa, A., & Hamilton, A. 2003, preprint (astro-ph/
0302496)
Varshalovich, D. A., Moskalev, A. N., & Khersonskii, V. K. 1988, Quantum
Theory of Angular Momentum (Singapore: World Scientific)
Vinberg, E. B. 1993, Geometry II: Spaces of Constant Curvature (New York:
Springer)
Wolf, J. A. 1984, Spaces of Constant Curvature (5th ed.; Wilmington: Publish
or Perish, Inc.)


Measuring the statistical isotropy of the cosmic microwave background Anisotropy

CiteSeerX

Measuring the statistical isotropy of the cosmic microwave background anisotropy