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Polarization in the Microwave
Background
UroŸs Seljak
Harvard­Smithsonian Center For Astrophysics
I. Statistics
ffl Expansion on a sphere
ffl Power spectra
ffl Scalar, vector and tensor modes and their signatures
II. Physical Processes
ffl Physics of polarization in standard recombination
ffl Reionized models
III. Model Predictions and observational prospects
ffl Inflationary models
ffl Topological defect models
ffl Reconstruction of cosmological parameters with polarization
ffl Foregrounds, systematics and their removal

Stokes parameters
The CMB radiation field is characterized by a 2 \Theta 2 symmetric intensity
tensor I ij . Stokes parameters:
Q = (I 11 \Gamma I 22 )=4
U = I 12 =2
T = (I 11 + I 22 )=4
T is invariant under rotation, while Q and U depend on the choice of
coordinate system and transform as:
Q 0 = Q cos 2/ + U sin 2/
U 0 = \GammaQ sin 2/ + U cos 2/
or
(Q \Sigma iU) 0 ( “
n) = e \Upsilon2i/ (Q \Sigma iU)(“ n)
Choice: spherical coordinate system ( ~ `; ~ OE)

Expansion on a sphere
T ( “
n) can be expanded in spherical harmonics:
T ( “
n) = X
lm
a lm Y lm ( “
n)
Q and U are not rotationally invariant, should be expanded in spin­
weight 2 harmonics (Newman & Penrose 1966)
(Q \Sigma iU)(“ n) = X
lm
a 2;lm \Sigma2 Y lm ( “
n)
\Sigma2Y lm ( “
n) can be expressed in terms of ordinary spherical harmonics.
To obtain a rotationally invariant quantity we act twice with spin raising
(lowering operator) 0
@ ( 0
@ ) with a property
( 0
@ s f ) 0 = e \Gammai(s+1)/ 0
@ s f; ( 0
@ s f ) 0 = e \Gammai(s\Gamma1)/ 0
@ s f
0
@ 2 (Q + iU)(“ n) =
X
lm
2
6 4
(l + 2)!
(l \Gamma 2)!
3
7 5
1=2
a 2;lm Y lm ( “
n)
0
@ 2 (Q \Gamma iU)(“ n) = X
lm
2
6 4
(l + 2)!
(l \Gamma 2)!
3
7 5
1=2
a \Gamma2;lm Y lm ( “
n)
Two degrees of freedom ! two independent expansion coefficients

Linear combinations
a E;lm = \Gamma(a 2;lm + a \Gamma2;lm )=2
aB;lm = i(a 2;lm \Gamma a \Gamma2;lm )=2
have opposite signs under parity transformation: E is a scalar and B a
pseudoscalar.
There are 4 power spectra
C T l =
1
2l + 1
X
m
ha \Lambda
T;lm a T;lm i
C El =
1
2l + 1
X
m
ha \Lambda
E;lm a E;lm i
CBl =
1
2l + 1
X
m
ha \Lambda
B;lm aB;lm i
C Cl =
1
2l + 1
X
m
ha \Lambda
T;lm a E;lm i
The cross correlation between B and E or B and T vanishes because B
has the opposite parity of T and E.
Small scale limit:
T ( “
n) = (2ú) \Gamma2
Z
d 2 l T (l)e il\Delta` : (1)
Q 0 (`) = (2ú) \Gamma2
Z
d 2
l [E(l) cos(2OE l ) \Gamma B(l) sin(2OE l )]e il\Delta`
U 0 (`) = (2ú) \Gamma2
Z
d 2 l [E(l) sin(2OE l ) +B(l) cos(2OE l )]e il\Delta`

Mode decomposition
Line element: ds 2 = a 2 (Ü )[\GammadÜ 2 + (ffi ij + h ij )dx i dx j ]
h ij can be decomposed into
scalar: r i r j OE or ffi ij OE
ffl Q and U can both be expressed in terms of a single scalar field ! only
one degree of freedom ! B = 0
ffl If only scalars contribute (e.g. on small scales) ! B can be used as a
background monitor.
vector: r i (v j \Gamma r j OE) +r j (v i \Gamma r i OE)
ffl Both E and B contribute
ffl Vectors are pure vorticity ! E Ü B
ffl If we observe B ? ¸ E on intermediate scales ! vectors dominate (e.g.
topological defect models).
tensor: h ij ­ scalar ­ vector part
ffl Both E and B contribute and are of similar amplitude
ffl If only scalars and tensors contribute (e.g. in inflationary models on
large scales) ! B can be used as a pure tensor detector.

Estimators
Fisher information matrix
ff ij =
l max X
l=2
X
X;Y
@CXl
@s i
Cov \Gamma1 (CXl C Y l ) @C Y l
@s j
useful to estimate parameter sensitivity
The diagonal terms of covariance matrix
Cov( “
C 2
T l ) =
2
2l + 1
( “
C T l + w \Gamma1
T e l 2 oe 2
b ) 2
Cov( “
C 2
El ) =
2
2l + 1
( “
C El + w \Gamma1
P e l 2 oe 2
b ) 2
Cov( “
C 2
Bl ) = 2
2l + 1
( “
CBl + w \Gamma1
P e l 2 oe 2
b ) 2
Cov( “
C 2
Cl ) =
1
2l + 1
''
“
C 2
Cl + ( “
C T l + w \Gamma1
T e l 2 oe 2
b )( “
CEl + w \Gamma1
P e l 2 oe 2
b )
#
The non­zero off diagonal terms are
Cov( “
C T l
“
C El ) =
2
2l + 1
“
C 2
Cl
Cov( “
C T l
“
C Cl ) =
2
2l + 1
“
C Cl (C T l + w \Gamma1
T e l 2 oe 2
b )
Cov( “
C El
“
C Cl ) =
2
2l + 1
“
C Cl (C El + w \Gamma1
P e l 2 oe 2
b )

Physics of Polarization in Tight Coupling
Approximation
To create polarization need
a) scattering with polarization dependent cross­section ! Thomson
scattering:
doe
d\Omega
=
3oe T
8ú
jffl \Delta ffl 0 j
b) anisotropic intensity distribution \Delta T2 6= 0
Tight coupling approximation: Before recombination photons scatter
off e \Gamma ! photons and baryons coupled into a single fluid:
ffi b = 3
4 ffi fl ~v b = ~v fl \Delta T2 = 0
Solution for temperature:
\Delta T / e ik¯(Ü rec \GammaÜ 0 ) (ffi fl + \Psi + ik¯v fl )
After recombination photons start to free stream and quadrupole mo­
ment is created, but if photons do not scatter no polarization is gener­
ated: polarization amplitude is proportional to the width of recombination
epoch. Integral solution for polarization
\Delta P / e ik¯(Ü rec \GammaÜ 0 ) k\DeltaÜ rec v fl

Euler+continuity equation: ¨ ffi fl + ff —
ffi fl + (kc s ) 2 ffi fl = F (OE)
Damped forced acoustic oscillator
Pressure restoring force determined by the sound speed of photon­
baryon plasma c \Gamma2
s = 3(1 + 3
4
y b ), where y b = ¯
ae b
¯
ae fl
/\Omega b h 2 .
Force F (OE) driving the oscillations is determined by gravitational po­
tential. Two possibilities:
OE 6= 0 kÜ Ü 1: curvature (adiabatic) fluctuations ! ffi fl / cos(kÜ )
OE = 0 kÜ Ü 1: isocurvature fluctuations ! ffi fl / sin(kÜ )
velocity v fl / —
ffi fl
ffl Polarization will be small (¸ 10%)
ffl Acoustic oscillations for \Delta T and \Delta P are out of phase
ffl Structure of acoustic oscillations only depends
on\Omega b h 2
and\Omega matter h 2 .
ffl Coupling between photons and baryons not completely tight ! sup­
pression below damping (Silk) scale
ffl Curvature shifts the spectrum to small scales

Polarization in Reionized Universe
Visibility function —
Ÿe \GammaŸ has two parts: during recombination and after
reionization
First contribution to polarization are the photons coming from recom­
bination: \Delta reion
P = \Delta P e \GammaŸ
Second contribution to polarization is from photons scattered during
reionization: the source is \Delta T2 , which has been created by free­streaming
of photons
\Delta T2 = (ffi fl + \Psi)(Ü rec )j 2 [k(Ü reion \Gamma Ü rec )]
This peaks where j 2 peaks: k(Ü reion \Gamma Ü rec ) ¸ 2
Integral solution \Delta l / \Delta T2 (Ü reion )j l [k(Ü 0 \Gamma Ü reion )]
peaks at k(Ü 0 \Gamma Ü reion ) = l
New peak at horizon scale of reionization:
l ¸ 2(Ü 0 \Gamma Ü reion )=(Ü reion \Gamma Ü rec ) ¸ 2
p
z reion
The amplitude of the peak / (1 \Gamma e \GammaŸ ) ¸ Ÿ
This differs from temperature where reionization produces only e \GammaŸ sup­
pression of small versus large scales
Reionization leaves a distinct signature in polarization and can be used
to determine the epoch of reionization and its optical depth

Parameters sensitive to polarization
ffl amplitude of power spectrum
ffl slope n: C l / l n
ffl curvature: position of acoustic peaks
ffl
\Omega – : position of acoustic peaks
ffl
\Omega b h 2 : positions and amplitudes of acoustic peaks
ffl
\Omega mh 2 : amplitudes of acoustic peaks
ffl reionization: suppression by exp(\Gamma2Ÿ)+additional peak(s)
ffl tensor parameters T/S, n T (B combination)
Parameter MAP no pol. MAP w. pol. Planck no pol. Planck w pol.
\Omega 1% 1% 0:1% 0:1%
amplitude 25% 10% 10% 1%
H 0 10% 5% 0:5% 0:5%
\Omega \Lambda 50% 30% 4% 3%
\Omega b h 2 2% 1% 0:2% 0:15%
Ü ri 10% 2% 10% 0:5%
n s 6% 5% 3% 3%
T=S 50% 25% 30% 5%
n T \Gamma \Gamma \Gamma 20%
Because reionizationand tensor parameters are partially degenerate with
other cosmological parameters improves limits on all parameters !
Important to include polarization in future satellite missions

Foregrounds and systematics
ffl free­free emission: not polarized (except through subsequent Thomson
scattering)
ffl atmosphere: not polarized
ffl dust emission: up to 10% polarized, frequency dependent
ffl IR point sources: weakly polarized?
ffl synchrotron emission: up to 70% polarized, frequency dependent
ffl radio point sources: synchrotron emission; up to 15% polarized
ffl other effects: sidelobe pickup ...?
Removal
­ Fractional polarization in foregrounds is somewhat higher than the ex­
pected fraction in cosmological models
+ Fewer foregrounds are polarized
+ Most (but not all) foregrounds and systematics contribute equally to
both E and B

Conclusions
ffl Polarization of CMB traces the universe in the linear regime just like
temperature anisotropies
ffl With polarization 4 independent power spectra can be extracted !
more information than with temperature only
ffl Geometric signature of polarization allows one to separate scalar, vec­
tor and tensor modes and/or foregrounds and systematics
ffl Important for high precision determination of cosmological parame­
ters with MAP and Planck, in particular on reionization and tensor
parameters