Statistics of cosmic microwave background polarization
Marc Kamionkowski
*
Department of Physics, Columbia University, 538 West 120th Street, New York, New York 10027
Arthur Kosowsky
†
Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, Massachusetts 02138
and Department of Physics, Lyman Laboratory, Harvard University, Cambridge, Massachusetts 02138
Albert Stebbins
‡
NASA/Fermilab Astrophysics Center, Fermi National Accelerator Laboratory, Batavia, Illinois 60510-0500
~Received 18 November 1996!
We present a formalism for analyzing a full-sky temperature and polarization map of the cosmic microwave
background. Temperature maps are analyzed by expanding over the set of spherical harmonics to give multi-
pole moments of the two-point correlation function. Polarization, which is described by a second-rank tensor,
can be treated analogously by expanding in the appropriate tensor spherical harmonics. We provide expressions
for the complete set of temperature and polarization multipole moments for scalar and tensor metric perturba-
tions. Four sets of multipole moments completely describe isotropic temperature and polarization correlations;
for scalar metric perturbations one set is identically zero, giving the possibility of a clean determination of the
vector and tensor contributions. The variance with which the multipole moments can be measured in idealized
experiments is evaluated, including the effects of detector noise, sky coverage, and beam width. Finally, we
construct coordinate-independent polarization two-point correlation functions, express them in terms of the
multipole moments, and derive small-angle limits. @S0556-2821~97!05012-1#
PACS number~s!: 98.70.Vc, 98.80.Cq
I. INTRODUCTION
With the advent of a new generation of balloon-borne and
ground-based experiments @1# and satellite missions @2,3#,
the cosmic microwave background ~CMB! will provide an
unprecedented window to the early Universe. In addition to
determining the origin of large-scale structure, it has been
argued that CMB temperature maps may determine cosmo-
logical parameters and the ionization history of the Universe,
and perhaps probe long-wavelength gravitational waves
@4–9#.
Any mechanism which produces temperature anisotropies
will invariably lead to polarization as well @10–14#. Tem-
perature fluctuations are the result of perturbations in the
gravitational potentials, which contribute directly to the fluc-
tuations via gravitational redshifting ~the Sachs-Wolfe effect
@15#! and which drive acoustic oscillations of the primordial
plasma @5#. These processes result in temperature fluctua-
tions which are the same order of magnitude as the metric
perturbations. In contrast, polarization is not directly gener-
ated by metric perturbations: a net polarization arises from
Compton scattering only when the incident radiation field
possesses a nonzero quadrupole moment @16,13#, but only
monopole and dipole fluctuations are possible as long as the
photons in the Universe remain tightly coupled to the
charged electrons. Polarization is only generated very near
the surface of last scattering as the photons begin to decouple
from the electrons and generate a quadrupole moment
through free streaming @17#. Since by this time most of the
electrons have recombined into neutral hydrogen, the num-
ber of scatterers available to produce polarization is reduced,
so CMB polarization fluctuations are characteristically at a
part in 10
6
, an order of magnitude below the temperature
fluctuations.
A polarization map will provide information that comple-
ments that from a temperature map. For example, polariza-
tion may help distinguish the gravitational-potential and
peculiar-velocity contributions to the acoustic peaks in the
temperature-anisotropy power spectrum @11#. In models with
reionization, some of the information lost from damping of
temperature anisotropies will be regained in the polarization
spectrum @18#. Perhaps most importantly, the density-
perturbation and gravitational-wave or vorticity contributions
to the anisotropy can be geometrically decomposed with a
polarization map @16,19–21#. Furthermore, although these
nonscalar signals are expected to be small, they will not be
swamped by cosmic variance from scalar modes ~as dis-
cussed further below!. Detection of gravity waves is impor-
tant for testing inflation and for learning about the inflaton
potential which drove inflation @22#.
Realistically, detection will present a significant experi-
mental challenge. Current results limit the magnitude of lin-
ear polarization to roughly a part in 10
5
@23#. Experiments
being planned or built will improve sensitivities by at least
an order of magnitude @24#. The MAP satellite will make
polarized measurements of the entire microwave sky in
around a million pixels with a precision of around one part in
10
5
per pixel @2#. If CMB polarization is not discovered by a
ground or balloon experiment in the next four years, this
*
Electronic address: kamion@phys.columbia.edu
†
Electronic address: akosowsky@cfa.harvard.edu
‡
Electronic address: stebbins@fnal.gov
PHYSICAL REVIEW D 15 JUNE 1997VOLUME 55, NUMBER 12
55
0556-2821/97/55~12!/7368~21!/$10.00 7368 © 1997 The American Physical Society
satellite will almost certainly make the first detection. The
Planck Surveyor ~formerly COBRAS/SAMBA! may also
make polarized measurements @3#. These experimental pros-
pects, as well as the theoretical considerations above, moti-
vate the analysis presented in this paper.
Previous theoretical treatments of CMB polarization have
relied on a small-angle approximation, which is valid when
considering patches of the sky small enough to be approxi-
mated as flat. Upcoming polarization maps will require a
more sophisticated formalism. In this paper, we develop in
detail a description of polarization on the full sky. The
Stokes parameters conventionally used to describe polariza-
tion are not invariant under rotations of the coordinate sys-
tem used to describe them, unlike temperature fluctuations,
but rather transform as a second-rank tensor @13#.Byex-
pressing the polarization in terms of a complete, orthonormal
set of tensor basis functions on the celestial sphere, power
spectra and correlation functions which are independent of
the coordinate system can be constructed. Earlier work on
small patches of the sky chose a particular reference coordi-
nate system which completely defines the polarization but
obscures the physical interpretation of the polarization pat-
tern. Also, the signal from vector and tensor perturbations is
expected to contribute to CMB polarization primarily at
large angles on the sky through gravitational effects, so the
correct full-sky analysis is essential.
Our formalism is stated in terms of differential geometry
on the sphere, using a notation widely used in general rela-
tivity. Similar calculations have recently been performed by
Seljak and Zaldarriaga @20,21#, using spin-weighted spheri-
cal harmonics @25#. Although the formalisms employed dif-
fer substantially and the calculations are quite lengthy, we
have verified that the end results are equivalent where they
overlap, giving us confidence both are correct.
After a brief review of Stokes parameters, the next section
defines the tensor spherical harmonic basis functions and
gives useful explicit expressions and formulas for decompos-
ing a polarization map into its harmonic components. Section
III covers the statistics of the expansion coefficients of the
temperature and polarization harmonics, derivations of vari-
ance estimates for the various multipole moments in ideal-
ized experiments, and a recipe for simulating a combined
polarization and temperature map given theoretical angular
power spectra. Section IV derives exact expressions for all of
the multipole moments from scalar and tensor metric fluc-
tuations, expressed in terms of the conventional Fourier com-
ponents of radiation brightnesses. Section V then treats two-
point correlation functions of the Stokes parameters in a
coordinate-independent manner and expresses the multipole
moments and correlation functions in terms of each other.
We also reproduce flat-sky results by taking small-angle lim-
its and make an explicit connection with earlier work in par-
ticular fixed coordinate systems. Finally, a summary and dis-
cussion section briefly considers detection prospects for
various polarization signals. A pair of mathematical appen-
dixes collect results from differential geometry on the sphere
and useful identities of Legendre polynomials and spherical
harmonics.
II. DESCRIPTION OF POLARIZATION
A. Review of Stokes parameters
The cosmic microwave background is characterized com-
pletely by its temperature and polarization in each direction
on the sky ~assuming its frequency spectrum is a perfect
blackbody!. Polarized radiation is described in terms of the
Stokes parameters Q, U, and V @26#. For a monochromatic
electromagnetic wave of frequency
v
0
propagating in the z
direction, the components of the wave’s electric field vector
at a given point in space can be written as
E
x
5 a
x
~
t
!
cos
@
v
0
t2
u
x
~
t
!
#
, E
y
5 a
y
~
t
!
cos
@
v
0
t2
u
y
~
t
!
#
.
~2.1!
If these two components are correlated, then the wave is said
to be polarized. The Stokes parameters are defined as the
time averages
I[
^
a
x
2
&
1
^
a
y
2
&
, ~2.2!
Q[
^
a
x
2
&
2
^
a
y
2
&
, ~2.3!
U[
^
2a
x
a
y
cos
~
u
x
2
u
y
!
&
, ~2.4!
V[
^
2a
x
a
y
sin
~
u
x
2
u
y
!
&
. ~2.5!
The parameter I gives the radiation intensity which is posi-
tive definite. The other three parameters can take either sign
and describe the polarization state. For unpolarized radiation,
Q5 U5 V5 0. The Stokes parameters are additive for inco-
herent superpositions of waves, which makes them natural
variables for describing polarized radiative transport.
In most applications polarization is measured in units of
intensity; however it is conventional and convenient when
studying the CMB to express polarization in terms of the
difference in brightness temperature of a particular polariza-
tion state from that of the mean brightness temperature of the
CMB. The rationale for this convention comes from the
well-known result that the spectrum of polarization induced
in the CMB is exactly the same as a temperature anisotropy,
so in brightness temperature units the polarization should be
independent of frequency.
The Stokes parameters I and V describe physical observ-
ables and are independent of the choice of coordinate sys-
tem. However, Q and U describe orthogonal modes of linear
polarization and depend on the axes in relation to which the
linear polarization is defined. From Eqs. ~2.5!, it is easy to
show that when the coordinate system is rotated by an angle
a
, the same radiation field is now described by the param-
eters
Q
8
5 Qcos
~
2
a
!
1 Usin
~
2
a
!
,
U
8
52Qsin
~
2
a
!
1 Ucos
~
2
a
!
. ~2.6!
Stated another way, under rotations of the coordinate system
around the direction of propagation, the Q and U Stokes
parameters transform like the independent components of a
two-dimensional, second rank symmetric trace-free ~STF!
tensor. Thus we can equally well describe the linear polar-
ization state by a polarization tensor P
ab
, which coincides
with the photon density matrix @13#.
B. Scalar and tensor harmonic expansions
Suppose we have an all-sky map of the CMB temperature
T(n
ˆ
) and polarization tensor P
ab
(n
ˆ
). The polarization tensor
55 7369STATISTICS OF COSMIC MICROWAVE BACKGROUND . . .
isa232 symmetric (P
ab
5P
ba
) and trace-free
(g
ab
P
ab
50) tensor, so it is specified by two real quantities.
Given the Stokes parameters Q and U measured in any co-
ordinate system, we can construct P
ab
. For example, in
spherical polar coordinates (
u
,
f
), the metric is
g
ab
5diag(1,sin
2
u
) and
P
ab
~
n
ˆ
!
5
1
2
S
Q
~
n
ˆ
!
2U
~
n
ˆ
!
sin
u
2 U
~
n
ˆ
!
sin
u
2 Q
~
n
ˆ
!
sin
2
u
D
. ~2.7!
The factors of sin
u
must be included since the coordinate
basis for (
u
,
f
) is an orthogonal, but not an orthonormal
basis. ~For more details of differential geometry on the two-
sphere, see Appendix A.! The Compton scattering process
which thermalizes the CMB and generates polarization can-
not produce any net circular polarization @27#; thus we ex-
pect V5 0 for the microwave background and do not con-
sider the V Stokes parameter further. Note the spherical polar
coordinate system adopted in this paper gives an outward
direction for the z axis, which is opposite the radiation
propagation direction. The convention with the z axis in the
direction of propagation is sometimes used, particularly in
Ref. @13#; this leads to the opposite sign for the U Stokes
parameter, but all results are unchanged.
In the usual way, we can expand the temperature pattern
T(n
ˆ
) in a set of complete orthonormal basis functions, the
spherical harmonics
T
~
n
ˆ
!
T
0
5 11
(
l51
`
(
m52l
l
a
~
lm
!
T
Y
~
lm
!
~
n
ˆ
!
, ~2.8!
where
a
~
lm
!
T
5
1
T
0
E
dn
ˆ
T
~
n
ˆ
!
Y
~
lm
!
*
~
n
ˆ
!
~2.9!
are the temperature multipole coefficients and T
0
is the mean
CMB temperature. The l5 1 term in Eq. ~2.8! is indistin-
guishable from the kinematic dipole and is normally ignored.
Similarly, we can expand the polarization tensor in terms
of a complete set of orthonormal basis functions for symmet-
ric trace-free 23 2 tensors on the two-sphere:
P
ab
~
n
ˆ
!
T
0
5
(
l52
`
(
m52l
l
@
a
~
lm
!
G
Y
~
lm
!
ab
G
~
n
ˆ
!
1a
~
lm
!
C
Y
~
lm
!
ab
C
~
n
ˆ
!
#
,
~2.10!
where the expansion coefficients are given by
a
~
lm
!
G
5
1
T
0
E
dn
ˆ
P
ab
~
n
ˆ
!
Y
~
lm
!
G ab
*
~
n
ˆ
!
,
a
~
lm
!
C
5
1
T
0
E
dn
ˆ
P
ab
~
n
ˆ
!
Y
~
lm
!
C ab
*
~
n
ˆ
!
, ~2.11!
which follow from the orthonormality properties
E
dn
ˆ
Y
~
lm
!
ab
G
*
~
n
ˆ
!
Y
~
l
8
m
8
!
G ab
~
n
ˆ
!
5
E
dn
ˆ
Y
~
lm
!
ab
C
*
~
n
ˆ
!
Y
~
l
8
m
8
!
C ab
~
n
ˆ
!
5
d
ll
8
d
mm
8
, ~2.12!
E
dn
ˆ
Y
~
lm
!
ab
G
*
~
n
ˆ
!
Y
~
l
8
m
8
!
C ab
~
n
ˆ
!
50. ~2.13!
Note that unlike scalar harmonics, the tensor harmonics only
exist for l>2 @28#.
The basis functions Y
(lm)ab
G
(n
ˆ
) and Y
(lm)ab
C
(n
ˆ
) are given
in terms of covariant derivatives of the spherical harmonics
by @28#
Y
~
lm
!
ab
G
5N
l
S
Y
~
lm
!
:ab
2
1
2
g
ab
Y
~
lm
!
:c
c
D
~2.14!
and
Y
~
lm
!
ab
C
5
N
l
2
~
Y
~
lm
!
:ac
e
c
b
1Y
~
lm
!
:bc
e
c
a
!
, ~2.15!
where
e
ab
is the completely antisymmetric tensor, the : de-
notes covariant differentiation on the two-sphere, and
N
l
[
A
2
~
l2 2
!
!
~
l1 2
!
!
~2.16!
is a normalization factor.
The existence of two sets of basis functions, labeled here
by G and C, is due to the fact that a symmetric trace-free
~STF! 23 2 tensor is specified by two independent param-
eters. In two dimensions, any STF tensor can be uniquely
decomposed into a part of the form A
:ab
2(1/2)g
ab
A
:c
c
and
another part of the form B
:ac
e
c
b
1B
:bc
e
c
a
, where A and B
are two scalar functions. This decomposition is quite similar
to the decomposition of a vector field into a part which is the
gradient of a scalar field and a part which is the curl of a
vector field; hence we use the notation G for ‘‘gradient’’ and
C for ‘‘curl.’’ Since the Y
(lm)
’s provide a complete basis for
scalar functions on the sphere, the Y
(lm)ab
G
and Y
(lm)ab
C
ten-
sors provide a complete basis for G-type and C-type STF
tensors, respectively. This G-C decomposition is also known
as the scalar-pseudoscalar decomposition @28#.
Incidentally, these tensor spherical harmonics are identi-
cal to those which appear in the theory of gravitational ra-
diation @29,30#. The propagating degrees of freedom of
gravitational field perturbations are described by a spin-2
tensor. Computing the flux of gravitational radiation from a
source requires the components of the gravitational field tan-
gent to a sphere around the source which are induced by the
motions of that source. Our G harmonics are often @29#—but
not always @30#—referred to as having ‘‘electric-type’’ par-
ity, since an electric field can be written as the gradient of a
scalar. Likewise, our C harmonics have ‘‘magnetic-type’’
parity since they are the curl of a vector field. The two vari-
eties of harmonics also correspond to electric and magnetic
multipole radiation.
Integration by parts transforms Eqs. ~2.11! into integrals
over scalar spherical harmonics and derivatives of the polar-
ization tensor:
7370 55KAMIONKOWSKI, KOSOWSKY, AND STEBBINS
a
~
lm
!
G
5
N
l
T
0
E
dn
ˆ
Y
~
lm
!
*
~
n
ˆ
!
P
ab
:ab
~
n
ˆ
!
, ~2.17!
a
~
lm
!
C
5
N
l
T
0
E
dn
ˆ
Y
~
lm
!
*
~
n
ˆ
!
P
ab
:ac
~
n
ˆ
!
e
c
b
, ~2.18!
where the second equation uses the fact that
e
ab
:c
50. These
forms are useful for theoretical calculations of the multipole
moments. We don’t recommend taking second derivatives of
real data. Since T and P
ab
are real, all of the multipole must
obey the reality condition
a
~
lm
!
X
*
5
~
21
!
m
a
~
l,2m
!
X
, ~2.19!
where X5
$
T,G,C
%
.
C. Explicit form of the harmonics
In spherical polar coordinates (
u
,
f
) the tensor spherical
harmonics are given explicitly by @28,30#
Y
~
lm
!
ab
G
~
n
ˆ
!
5
N
l
2
S
W
~
lm
!
~
n
ˆ
!
X
~
lm
!
~
n
ˆ
!
sin
u
X
~
lm
!
~
n
ˆ
!
sin
u
2 W
~
lm
!
~
n
ˆ
!
sin
2
u
D
~2.20!
and
Y
~
lm
!
ab
C
~
n
ˆ
!
5
N
l
2
S
2X
~
lm
!
~
n
ˆ
!
W
~
lm
!
~
n
ˆ
!
sin
u
W
~
lm
!
~
n
ˆ
!
sin
u
X
~
lm
!
~
n
ˆ
!
sin
2
u
D
,
~2.21!
where
W
~
lm
!
~
n
ˆ
!
5
S
]
2
]
u
2
2cot
u
]
]
u
1
m
2
sin
2
u
D
Y
~
lm
!
~
n
ˆ
!
5
S
2
]
2
]
u
2
2l
~
l11
!
D
Y
~
lm
!
~
n
ˆ
!
~2.22!
and
X
~
lm
!
~
n
ˆ
!
5
2im
sin
u
S
]
]
u
2 cot
u
D
Y
~
lm
!
~
n
ˆ
!
. ~2.23!
„Note that this definition of X
(lm)
(n
ˆ
) differs from that in Ref.
@30# by a factor of sin
u
.… The exchange symmetry
$
Q,U
%
↔
$
U,2 Q
%
as G↔C indicates that Y
(lm)ab
G
and
Y
(lm)ab
C
represent polarizations rotated by 45°. By evaluating
the derivatives, these functions can be written
W
~
lm
!
~
n
ˆ
!
52
A
2l11
4
p
~
l2m
!
!
~
l1m
!
!
G
~
lm
!
1
~
cos
u
!
e
im
f
,
~2.24!
iX
~
lm
!
~
n
ˆ
!
522
A
2l11
4
p
~
l2m
!
!
~
l1m
!
!
G
~
lm
!
2
~
cos
u
!
e
im
f
,
~2.25!
where the real functions G
(lm)
6
are defined by @28#
G
~
lm
!
1
~
cos
u
!
[2
S
l2 m
2
sin
2
u
1
1
2
l
~
l2 1
!
D
P
l
m
~
cos
u
!
1
~
l1 m
!
cos
u
sin
2
u
P
l2 1
m
~
cos
u
!
, ~2.26!
G
~
lm
!
2
~
cos
u
!
[
m
sin
2
u
„
~
l2 1
!
cos
u
P
l
m
~
cos
u
!
2
~
l1 m
!
P
l2 1
m
~
x
!
…. ~2.27!
These expressions will be useful for the correlation functions
in Sec. V, and for simulating maps and data analysis.
In linear theory, scalar perturbations can produce only
G-type polarization and not C-type polarization. On the other
hand, tensor or vector metric perturbations will produce a
mixture of both types @19,20,31#. Heuristically, this is be-
cause scalar perturbations have no handedness so they cannot
produce any ‘‘curl,’’ whereas vector and tensor perturbations
do have a handedness and therefore can. Observation of a
nonzero primordial component of C-type polarization ~a non-
zero a
(lm)
C
) in the CMB would provide compelling evidence
for significant contribution of either vector or tensor pertur-
bations at the time of last scattering.
Given a polarization map of even a small part of the sky,
one could in principle test for vector or tensor contribution
by computing the combination of derivatives of the polariza-
tion field given by P
ab
:bc
e
c
a
which will be nonzero only for
C-type polarization. Of course, taking derivatives of noisy
data is problematic. We discuss more robust probes of this
signal below.
III. STATISTICS OF THE MULTIPOLE COEFFICIENTS
A. Statistical independence of the coefficients
We now have three sets of multipole moments, a
(lm)
T
,
a
(lm)
G
, and a
(lm)
C
, which fully describe the temperature or
polarization map of the sky. Statistical isotropy implies that
^
a
~
lm
!
T
*
a
~
l
8
m
8
!
T
&
5C
l
T
d
ll
8
d
mm
8
,
^
a
~
lm
!
G
*
a
~
l
8
m
8
!
G
&
5C
l
G
d
ll
8
d
mm
8
,
^
a
~
lm
!
C
*
a
~
l
8
m
8
!
C
&
5C
l
C
d
ll
8
d
mm
8
,
^
a
~
lm
!
T
*
a
~
l
8
m
8
!
G
&
5C
l
TG
d
ll
8
d
mm
8
,
^
a
~
lm
!
T
*
a
~
l
8
m
8
!
C
&
5C
l
TC
d
ll
8
d
mm
8
,
^
a
~
lm
!
G
*
a
~
l
8
m
8
!
C
&
5C
l
GC
d
ll
8
d
mm
8
, ~3.1!
where the angle brackets are an average over all realizations.
For Gaussian theories, the statistical properties of a tempera-
ture or polarization map are specified fully by these six sets
of multipole moments. In fact, the scalar spherical harmonics
Y
(lm)
and the G tensor harmonics Y
(lm)ab
G
have parity
(2 1)
l
, but the C harmonics Y
(lm)ab
C
have parity (21)
l11
.
Therefore, symmetry under parity transformations requires
that C
l
TC
5 C
l
GC
5 0, which will also be demonstrated explic-
itly in the following section. Measurement of nonzero cos-
mological values for these moments would be quite extraor-
dinary, demonstrating a handedness to primordial
perturbations. In practice, these two sets of moments can be
used to monitor foreground emission. Furthermore, as men-
55 7371STATISTICS OF COSMIC MICROWAVE BACKGROUND . . .
tioned above and demonstrated explicitly in Sec. IV,
C
l
C
5 0 for scalar metric perturbations @19,20#. At small an-
gular scales where the contribution from tensor and vector
perturbations is expected to be negligible, C
l
C
can also be
pressed into duty as a foreground monitor. Exact expressions
for these multipole moments in terms of the photon bright-
nesses usually calculated by early-Universe Boltzmann
codes are derived below.
B. Map simulation
For the case of Gaussian statistics, realizations of tem-
perature or polarizations maps are easy to generate using
standard techniques. Since the only cross correlation be-
tween mode coefficients, given by C
l
TG
, correlates only
a
(lm)
T
and a
(lm)
G
with the same l and m, the total correlation
matrix is block diagonal with the largest blocks being only
23 2 matrices. In particular, set
a
~
lm
!
T
5
z
1
~
C
l
T
!
1/2
,
a
~
lm
!
G
5
z
1
C
l
TG
~
C
l
T
!
1/2
1
z
2
S
C
l
G
2
~
C
l
TG
!
2
C
l
T
D
1/2
,
a
~
lm
!
C
5
z
3
~
C
l
C
!
1/2
, ~3.2!
where for each value of l and m. 0 choose three complex
numbers (
z
1
,
z
2
,
z
3
) drawn from a Gaussian distribution
with unit variance, i.e., both
A
2 Re(
z
i
) and
A
2 Im(
z
i
) are
drawn from a normal distribution. For m5 0 the same equa-
tions hold but the
z
i
should be real and normally distributed;
for m, 0 the coefficients are given by Eq. ~2.19!. Note that
in all cases C
l
G
C
l
T
>(C
l
TG
)
2
. This set of coefficients can be
combined with Eqs. ~2.7!, ~2.10!, ~2.23!, and ~2.22! to obtain
the explicit expressions
Q
~
n
ˆ
!
5 2P
uu
~
n
ˆ
!
5T
0
(
l52
`
(
m52l
l
N
l
@
a
~
lm
!
G
W
~
lm
!
~
n
ˆ
!
2a
~
lm
!
C
X
~
lm
!
~
n
ˆ
!
#
,
U
~
n
ˆ
!
522csc
u
P
uf
~
n
ˆ
!
52T
0
(
l52
`
(
m52l
l
N
l
@
a
~
lm
!
G
X
~
lm
!
~
n
ˆ
!
1a
~
lm
!
C
W
~
lm
!
~
n
ˆ
!
#
~3.3!
with W
(lm)
and X
(lm)
given by Eqs. ~2.24! and ~2.25!. Note
that polarization maps are traditionally plotted as headless
vectors with amplitude (Q
2
1 U
2
)
1/2
and orientation angle
(1/2)arctan(U/Q).
C. Estimators
One of the the main uses of a temperature/polarization
map will be to determine the multipole moments with the
best possible accuracy. From a full-sky CMB temperature
map, we can construct the following rotationally invariant
estimators ~denoted by a caret! for the multipole coefficients
C
l
T
ˆ
5
(
m52l
l
u
a
~
lm
!
T
u
2
2l11
, C
l
G
ˆ
5
(
m52l
l
u
a
~
lm
!
G
u
2
2l11
,
C
l
C
ˆ
5
(
m52l
l
u
a
~
lm
!
C
u
2
2l11
, C
l
TG
ˆ
5
(
m52l
l
a
~
lm
!
T
*
a
~
lm
!
G
2l11
. ~3.4!
Note that Eq. ~2.19! guarantees that the C
l
TG
ˆ
will be real.
When averaged over the sky ~denoted by an overbar!, the
mean square temperature anisotropy after subtracting the di-
pole is
S
DT
T
0
D
2
5
(
l5 2
`
2l1 1
4
p
C
l
T
ˆ
~3.5!
and the mean square polarization is
P
2
[Q
2
1 U
2
5 2P
ab
P
ab
5P
G
2
1P
C
2
, ~3.6!
P
G
2
[T
0
2
(
l5 2
`
2l1 1
8
p
C
l
G
ˆ
, P
C
2
[T
0
2
(
l5 2
`
2l1 1
8
p
C
l
C
ˆ
. ~3.7!
Even if no single C
l
C
ˆ
or C
l
G
ˆ
gives a significant signal, com-
bining different l’s as in
P
G
2
or P
C
2
can give a statistically
significant signal.
D. Cosmic and pixel-noise variance
The averages in Eqs. ~3.1! are over an ensemble of uni-
verses drawn from a theoretically defined statistical distribu-
tion, or assuming ergodicity, a spatial average over all ob-
server positions in the Universe. However, we can only
observe a single realization of the ensemble from a single
location. Therefore, even if we had an ideal ~full-sky cover-
age, no foreground contamination, infinite angular resolu-
tion, and no instrumental noise! experiment, the accuracy
with which the estimators in Eqs. ~3.4! could recover the
multipole moments would be limited by a sample variance
known as ‘‘cosmic variance.’’ Furthermore, a realistic ex-
periment may have limited sky coverage and angular resolu-
tion and some instrumental noise. In this section, we calcu-
late the cosmic variance with which the multipole moments
can be recovered. We also calculate the variance due to finite
sky coverage, angular resolution, and instrumental noise in
an idealized experiment. To do so, we adopt a simplified
model in which we assume a pixelized map in which the
noise in each pixel is independent and Gaussian distributed
after foregrounds have been successfully subtracted. In many
respects, our derivation follows that in Ref. @32#, and our
results agree with those in Ref. @21#.
We must first determine the contribution of pixel noise to
each multipole moment, and we begin with the temperature
moments. Consider a temperature map of the full sky
T
map
(n
ˆ
), which is pixelized with N
pix
pixels. If we assume
that each pixel subtends the same area on the sky then we
7372 55KAMIONKOWSKI, KOSOWSKY, AND STEBBINS
