Cosmic Microwave Background Polarization

Home , Photon polarization

Arthur Kosowsky

Department of Physics

Enrico Fermi Institute The University of Chicago Chicago IL

and

NASAFermilab Astrophysics Center Fermi National Accelerator Laboratory Batavia IL

astro-ph/9501045 3 Feb 1995

Decemb er

Polarization of the cosmic microwave background though not yet detected provides a source

of information ab out cosmological parameters complementary to temp erature uctuations This

pap er provides a complete theoretical treatment of p olarization uctuations After a discussion of

the physics of p olarization the Boltzmann equation governing the evolution of the photon density

matrix is derived from quantum theory and applied to microwave background uctuations resulting

in a complete set of transp ort equations for the Stokes parameters from b oth scalar and tensor

metric p erturbations This approach is equivalent at lowest order in scattering kinematics to classical

radiative transfer and provides a general framework for treating the cosmological evolution of density

matrices The metric p erturbations are treated in the physically app ealing longitudina l gauge

Expressions for various temp erature and p olarizatio n correlation functions are derived Detection

prosp ects and theoretical utility of microwave background p olarization are briey discussed

I INTRODUCTION

Since the initial announcement by the COBE team of the detection of cosmic microwave background temp erature

anisotropies a great deal of exp erimental activity has resulted in nearly a dozen more anisotropy detections on a

wide range of angular scales Simultaneously detailed numerical analysis has sharp ened theoretical exp ectations

for the temp erature anisotropy and its dep endence on a variety of cosmological parameters primarily in the context of

theories with initial adiabatic p erturbations of which Cold Dark Matter is a sp ecial case but also in cosmological

defect mo dels While much work remains to b e done the fo cus of microwave background research has shifted from

simply detecting anisotropies to creating a detailed picture b oth exp erimental and theoretical of the anisotropies on

all angular scale and using this picture to constrain cosmological mo dels

The cosmological information in the microwave background is enco ded not only in temp erature uctuations but

also in its p olarization Since as discussed b elow in Sec I I the source term for generating p olarization is uctuations

in radiation intensity generally p olarization uctuations are exp ected to b e somewhat smaller than temp erature

uctuations Numerical calculations have conrmed this rough exp ectation giving p olarization uctuations no larger

than of the temp erature anisotropies Greater exp erimental sensitivity is required to measure p olarization

than temp erature uctuations and so far only upp er limits have b een established However p olarization uctuation

measurements also have certain exp erimental advantages over temp erature uctuation measurements making rst

detection of p olarization within the next few years a reasonable p ossibility

Currently the b est p olarization limit comes from the Saskato on exp eriment with a condence level upp er

limit of K b etween two orthogonal linear p olarizations at angular scales of ab out a degree corresp onding to

of the mean temp erature An earlier exp eriment mapp ed a large p ortion of the sky with a b eam to

place upp er limits of in linear p olarization and in circular p olarization for quadrup ole and o ctup ole

variations while a later measurement put limits of around on b oth linear and circular p olarization

at arcminute scales The COBE satellite made p olarized measurements and can in principle achieve a limit of

around at angular scales from to quadrup ole although doing so would require reanalysis of the entire data set

While the sensitivity of the Saskato on exp eriment is very go o d it is designed primarily to measure temp erature

uctuations A new exp eriment planned for is optimized to measure p olarization at and scales and aims

for a sensitivity at or b elow

On the theory side microwave background p olarization has b een discussed for many years Detailed

calculations for CDM mo dels have given linear p olarization as large as of the temp erature anisotropy at medium

Current Address HarvardSmithsoni an Center for Astrophysics Mail Stop Garden Street Cambridge MA

akosowskycfaharvardedu

angular scales with a strong dep endence on the ionization history of the universe Another recent calculation

has mapp ed exp ected correlation patterns b etween temp erature uctuations and p olarization

The aim of this pap er is a detailed investigation of the theory of microwave background p olarization In contrast to

previous work employing classical radiative transfer theory the evolution of p olarization is derived from a photon

description this approach has previously b een applied to temp erature uctuations in systematic investigations of

secondorder eects The usual classical Boltzmann equation adequate for describing temp erature uctuations

must b e generalized to a density matrix describing the photon p olarization state The formalism for this generalized

Boltzmann equation was recently develop ed in the context of neutrinoavor evolution Here the appropriate

equation is derived b eginning with the fundamental description of the relevant Compton scattering pro cess the

techniques easily generalize to give a Boltzmann equation for any particular density matrix The advantage of this

approach is that it treats photons in a general manner like other particle sp ecies describ ed by a Boltzmann equation

and can easily b e applied to other p olarized distributions eg electrons in a magnetic eld It also gives a systematic

p erturbative expansion in the relevant small quantities and thus provides the framework for an investigation of all

secondorder p olarization eects which may b e of particular interest in the case of early reionization for which the

p olarization contribution is the largest This pap er also serves as a review which presents many derivations of

relevant formulas which remain unpublished

Section II provides an overview of the physics of p olarization and its application to the microwave background

A simple calculation demonstrates that only quadrup olar variations in radiation intensity on a scatterer pro duce

p olarization The Stokes parameters are dened and their connection to the photon density matrix made explicit

Section I I I derives the general formula for the time evolution of a density matrix in terms of an interaction Hamiltonian

This section is rather formal the relevant result is Eq Section IV sp ecializes this result to the evolution of

the photon density matrix including Compton scattering The calculation in this section is straightforward but long

the ultimate result is Eq which is equivalent to the usual classical equations of radiative transfer

The particular application to a cosmological context b egins in Section V which derives the general relativistic

Liouville equation for a p erturb ed FriedmannRob ertsonWalker spacetime This collisionless part of the Boltzmann

equation describ es the photon geo desics in a homogeneous and isotropic universe with small scalar and tensor metric

p erturbations The scalar p erturbations are describ ed with the physically app ealing longitudinal gauge instead

of the more traditional synchronous gauge Section VI gives the nal evolution equations for the Stokes parameters

describing the microwave background The temp erature and p olarization uctuations must b e expressed in terms of

their statistical prop erties for comparison with exp erimental results Section VI I derives expressions for the p ower

sp ectra of various correlations and crosscorrelations b etween temp erature and p olarization Finally the concluding

section briey discusses what detailed p olarization measurements may eventually reveal ab out cosmology

This pap er employs natural units throughout in which h c G k Section II uses gaussian units

for electromagnetic quantities The metric signature is Spinor normalizations and other eld theory

conventions in Section IV conform to Mandl and Shaw

I I PHYSICS OF POLARIZATION

This section gives a qualitative overview of the physics of p olarization in the context of the microwave background

We b egin with a review of Stokes parameters the conventional metho d for describing p olarized light Then we

show how p olarization can b e generated by scattering application to pro cesses on the last scattering surface predict

distinctive correlations b etween hot and cold sp ots and p olarization direction The equivalent p olarization description

in terms of the photon density matrix is then presented with the connection to the conventional Stokes parameters

made explicit

A Review of Stokes Parameters

Polarized light is conventionally describ ed in terms of the Stokes parameters which are presented in any op

tics text Consider a nearly mono chromatic plane electromagnetic wave prop ogating in the z direction nearly

mono chromatic here means that its frequency comp onents are closely distributed around its mean frequency The

comp onents of the waves electric eld vector at a given p oint in space can b e written as

E a t cos t t E a t cos t t

x x x y y y

The requirement that the wave is nearly mono chromatic guarantees that the amplitudes a and a and the phase

x y

angles and will b e slowly varying functions of time relative to the inverse frequency of the wave If some

x y

correlation exists b etween the two comp onents in Eq then the wave is p olarized

The Stokes parameters are dened as the following time averages

I ha i ha i a

x y

Q ha i ha i b

x y

U ha a cos i c

x y x y

V ha a sin i d

x y x y

The parameter I gives the intensity of the radiation which is always p ositive The other three parameters dene

the p olarization state of the wave and can have either sign Unp olarized radiation or natural light is describ ed

by Q U V One imp ortant prop erty of the Stokes parameters is that they are additive for incoherent

sup erp ositions of waves The four parameters can b e measured with a linear p olarizer and a quarterwave plate the

rst three can b e measured with only a linear p olarizer The V parameter can also b e measured as the intensity

dierence b etween left and right circular p olarizations

The parameters I and V are physical observables indep endent of the co ordinate system but Q and U dep end on

the orientation of the x and y axes If a given wave is describ ed by the parameters Q and U for a certain orientation

of the co ordinate system then after a rotation of the x y plane through an angle the same wave is now describ ed

by the parameters

Q Q cos U sin

U Q sin U cos

From this transformation it is easy to see that the quantity Q U is invariant under rotation of the axes and the

angle

tan

transforms to under a rotation by and thus denes a constant direction The physically observable p olarization

vector P is here dened as orthogonal to the direction of wave prop ogation having magnitude Q U and p olar

angle For a wave with linear p olarization the vector P lies along the constant orientation of the electric eld

Note that since the denition is degenerate for and only the orientation of P is dened and not the

direction We take the range of to b e with the sign of the same as the sign of U While the

radiation transp ort equations b elow are most conveniently formulated in terms of the Stokes parameters the physical

interpretation of a p olarization pattern is clearest in terms of the observables I V and P

B Polarization and the Last Scattering Surface

In the early universe at redshifts greater than ab out z the baryons electrons and photons comprise a

tightly coupled uid Small metric p erturbations induce bulk velo cities of the uid and the resulting anisotropies in

the photon distribution will induce p olarization when the photons scatter o charged particles After recombination

the photons freely propagate along geo desics and any p olarization pro duced b efore recombination will remain xed

A suciently early reionization can of course generate further p olarization

A simple idealization of the last scattering surface elucidates the pro cess of p olarization generation Consider

initially unp olarized light which undergo es Thomson scattering at a given p oint and is then viewed by an observer If

the intensity of the light incident on the scattering p oint is uniform in every direction then obviously no p olarization

can result however if the incident intensity varies with direction then p olarization can b e generated Cho ose the

z axis to lie in the direction of the outgoing light which is describ ed by the Stokes parameters I Q U and V

represent the light incident on the scattering p oint by the intensity I Dene the p olarization vectors for the

outgoing b eam of light so that is p erp endicular to the scattering plane and is in the scattering plane and likewise

x y

with the incoming p olarization vectors and see Fig Also instead of dealing with I and Q it is convenient to

x y

describ e the scattering pro cess in terms of I I Q and I I Q The Thomson scattering crosssection

x y

for an incident wave with linear p olarization into a scattered wave with linear p olarization is given by

j j

where is the total Thomson cross section The incoming wave is unp olarized by assumption and thus satises

I I I The scattered intensities are

x y x∋

Θ = π − θ

x∋ I θ

y∋

FIG Denition of vectors and angles for Thomson scattering of a light b eam or photon

T T

I I I a I

x x x

x x y y

T T

I I I I cos b

y y y

x x y y

Thus the scattered wave has the Stokes parameters

I I I I cos a

x y

Q I I I sin b

x y

This calculation gives no information ab out the U or V parameters As will b e shown later the V parameter remains

zero after scattering and will not b e considered further The U parameter can b e determined by using Eq

Simply rotate the outgoing basis vectors in the ab ove calculation by and recalculate Q which will b e equal to U

in the original co ordinate system The result is U These results can alternatively b e obtained from the physical

description of the p olarization state in Rayleigh scattering Note that Eq b gives the wellknown result that

sunlight from the horizon at midday is linearly p olarized parallel to the horizon

The total scattered intensities are determined by integrating over all incoming intensities Note that the outgoing U

and Q ux from a given incoming direction must always b e rotated into a common co ordinate system using Eq

The result is

I d cos I a

Q d sin cos I b

d sin sinI c U

The outgoing p olarization state dep ends only on the intensity distribution of the unp olarized incident radiation

Expanding the incident intensity in spherical harmonics

I a Y

lm lm

leads to the following expressions for the outgoing Stokes parameters

a a a I

Q Re a b

Angular Dependence Coefficient

I cos 2ϕ Re a Q 22

sin2ϕ -Im a U

I 22

FIG The quadrup olar comp onents of the incident intensity distribution Any orientation of a quadrup olar distribution

can b e written as the sum of these two distribution s The small arrows indicate the corresp onding uid velo city in a tightly

coupled uid

U Im a c

Thus scattering generates p olarization from initially unp olarized radiation if the radiation intensity at a given p oint

as a function of direction has a nonzero comp onent of Y

This particular form for the source of p olarization leads to a correlation of the direction of the p olarization vector P

with hot and cold sp ots on the cosmic microwave sky Consider a given region on the last scattering surface with

a spherical mass overdensity the electronphoton uid will have a bulk velo city towards the center of the overdense

region with a velo city gradient away from the center material further from the center will b e falling inwards more

quickly In the frame of some particular scattering p oint away from the center the uid velo city in towards the p oint

is greater along the radial direction than p erp endicular to it resulting in a quadrup olar radiation intensity variation

with the largest intensity along the radial direction see Figure Cho ose an observation direction at a right angle to

the radial direction and take this direction to b e the p olar axis Then the radiation intensity at the scattering p oint

will have a comp onent prop ortional to cos with a p ositive co ecient where is the radial direction The

scattered Q intensity is prop ortional to the cos dep endence of the incident intensity and the scattered U intensity

is prop ortional to the sin piece by Eqs and Thus the direction of P in Eq lies along

the radial direction For the opp osite situation that of a mass underdensity all the velo cities change sign so b oth

Q and U change sign and the direction of P changes by When the dominant contribution to the temp erature

uctuations is a gravitational p otential dierence SachsWolfe eect a mass overdensity corresp onds to a cold

sp ot in the microwave background in this case cold sp ots will have radially correlated p olarization and hot sp ots

tangentially correlated p olarization in agreement with the result of Ref For adiabatic acoustic oscillations the

density and velo city p erturbations are out of phase so no sp ecic correlation results

C Photon Description

The Stokes parameters can b e dened equivalently in terms of a quantummechanical description The p olarization

state space of a photon is spanned by a pair of basis vectors which we take to b e the orthogonal linear p olarizations

j i and j i For a photon prop ogating in the zdirection the basis states j i and j i are oriented like the x and y

axes resp ectively An arbitrary state is given by

i i

2 1

j i j i a e ji a e

The quantummechanical op erators in the linear basis corresp onding to each Stokes parameter are given by

I j ih j j ih j a

Q j ih j j ih j b

U j ih j j ih j c

V ij ih j ij ih j d

The singleparticle state exp ectation values of these op erators repro duce the denitions For photons in a

general mixed state dened by a density matrix the exp ectation value for the I Stokes parameter is given by

hI i tr I tr

and similarly for the other three parameters These relations thus give the density matrix in the linear p olarization

basis in terms of the Stokes parameters as

I Q U iV

U iV I Q

I Q U V

where is the identity matrix and are the Pauli spin matrices Thus the density matrix for a system of photons

contains the same information as the four Stokes parameters and the time evolution of the density matrix gives the

time evolution of the systems p olarization

I I I EVOLUTION EQUATION FOR THE NUMBER OPERATOR

This section considers the quantum numb er op erator for a system of particles and derives its evolution equation

including lo cal particle interactions Taking the exp ectation value of the op erator equation gives the Boltzmann

equation for the systems density matrix which is a generalization of the usual classical Boltzmann equation for

particle o ccupation numb ers the diagonal elements of the density matrix The derivation here applies techniques

previously develop ed in the context of neutrino mixing

We adopt secondquantized formalism with creation and annihilation op erators for photons and electrons ob eying

the canonical commutation relations

a p a p p p p

s ss

fb q b q g q q

r r r

where s lab els the photon p olarization and r lab els the electron spin b old momentum variables represent three

momenta while plain momentum variables represent fourmomenta The density op erator describing a system of

photons is given by

d p

p a p a p

ij j

where is the density matrix The particular op erator for which we want the equation of motion is the photon

numb er op erator

D k a ka k

ij j

The exp ectation value of D is prop ortional to the density matrix as seen by direct calculation

d p

hpjD kjpi k k hD ki tr D k

ij ij ij ij

The last equality results from rep eated application of the commutation relation Eq the innite delta function

results from the innite quantization volume necessary with continuous momentum variables and cancels out of all

physical results

The time evolution of the op erator D considered in the Heisenb erg picture is

D i H D

ij ij

where H is the full Hamiltonian We write the Hamiltonian as a sum of the free eld piece plus an interaction term

H H H

where the interaction piece is a functional of the full elds in the problem Our goal is to express the right side of

Eq as a p erturbation series in the interaction Hamiltonian H We make the usual assumption of scattering

theory that in a given interaction the elds b egin as free elds and end as other free elds and the interactions are

isolated from each other Consider the evolution of an op erator through a single interaction b eginning at t b efore

this time the elds can b e taken as free to a go o d approximation at t the interaction Hamiltonian b egins to turn

on and the interaction nishes at some later time after which the elds can b e taken as free once again Then the

time dep endence of an arbitrary op erator to rst order in the interaction Hamiltonian can b e expressed as

t t i dt H t t t

where t is the freeeld op erator with initial condition and H is the interaction Hamiltonian as a

functional of the free elds

Equation can b e proven as follows The time derivative of b oth sides gives

d d d

H t t t t t i H t i dt

i I

dt dt dt

The time derivatives in the rst and third term on the right side can b e replaced by commutators with H t using

the Heisenb erg equation But these two terms dep end only on free elds which are evolved with the free Hamiltonian

H t H Equation b ecomes

t i H t i H t

and so to rst order in H this just gives the Heisenb erg equation for the op erator

Now we can express the time evolution of D in terms of free eld op erators Applying Eq to the commutator

on the right side of Eq gives

D k i H t D k dt H t t H t D k

I ij I I ij

The integral on the right side can b e cast in a more practical form by making the following physical assumption the

duration of each collision the time interval over which the interaction Hamiltonian is nonnegligible on the order of

the inverse energy transfer is small compared to the time scale for variation of the density matrix on the order of

the inverse collision frequency The collision pro cess relevant to the microwave background is Compton scattering o

electrons and for the cosmological ep o ch of interest the electron density is always low enough for this condition to

b e easily satised Then the time step t in Eq can b e chosen large compared with a single collision and small

compared to the time scale for density matrix evolution After extending the time integral to innity and taking the

exp ectation value of b oth sides we nd

k k k D t D t H k i H dt H

ij ij I I I

Here the integral from zero to innity has b een replaced by an integral over all time the dierence is a principle part

integral which is secondorder in the interaction Hamiltonian and thus ignored

Equation is the Boltzmann equation for the density matrix The rst term on the right side is a forward

scattering term which is resp onsible for the MSW eect in a neutrino ensemble for photons this term is zero

as will b e shown b elow The second term on the right side is the usual collision term The time integral over the

free eld time dep endence enforces energy conservation in each collision The interaction Hamiltonian will in most

cases dep end on background elds for example in the case of the microwave background the Compton scattering

collisions are essentially fourp oint interactions quadratic in b oth the photon eld and the electron eld In principle

a second coupled equation for the electron density matrix must b e solved simultaneously However in many physical

situations the background elds may b e assumed to have a xed distribution generally thermal In the early universe

the electrons maintain a thermal distribution to a very high approximation and the evolution of their density matrix

b ecomes trivial

The derivation of Eq has b een completely general In the appropriate limit the classical equations of radiative

transfer are repro duced the advantage to the current approach is that it provides the same formal framework for

treating p olarized photons as for treating neutrinos or any other particle sp ecies governed by the Boltzmann equation

It also gives a systematic metho d for analyzing all higherorder eects This Boltzmann equation has previously b een

applied to neutrinos interacting through b oth charged and neutral current pro cesses in sup ernovae

IV APPLICATION TO COMPTON SCATTERING OF PHOTONS

In principle the complete evolution of the cosmic microwave background is determined by Eq generalized

slightly to include spatial dep endence of all quantities In this work this space dep endence will simply b e put in

by hand when taking exp ectation values and assumed implicitly more formally it can b e included through Wigner

functions describing a joint spacemomentum distribution All that remains to b e done is substitution of the

correct interaction Hamiltonian and simplication of the right side General relativistic terms emerge from the total

time derivative on the left side these will b e treated in detail in Sec V

Microwave background photons interact with all charged particles However the rate of scattering varies with

the mass of the charged particle as the inverse mass squared thus it is an excellent approximation to consider only

Compton scattering o electrons and ignore baryons This section pro ceeds with evaluation of the right side of

Eq for Compton scattering

A Interaction Hamiltonian

The interaction Hamiltonian density for the fundamental threep oint interaction of QED is given by

H x e xA x x

QED

where is the electron eld op erator A is the photon eld op erator a slash indicates contraction with and the

colons signify normal ordering of the op erator pro duct The interaction Hamiltonian is the density integrated over all

of space

H t d xH x

QED QED

The scattering matrix describing all scattering pro cesses in QED is given in terms of the interaction Hamiltonian by

X X

S S d x d x TfH x H x g

n QED QED n

n n

where T signies a timeordered pro duct The nth term in the series represents all scattering pro cesses with n

interaction vertices Compton scattering is thus contained in the n term of the scattering matrix Comparing the

n term with the n term gives the interaction Hamiltonian for secondorder scattering pro cesses

Z Z

dt S dt TfH tH t g

QED QED

i dtH t

Using Wicks theorem to simplify the timeordered pro duct and denoting the piece of H describing Compton

scattering by H yields

Z Z

h i

A xA x A x A x x H t e dt x S x x d x

I F

where S is the Feynman propagator for the electron and and A A are linear in absorption creation

op erators of electrons and photons resp ectively Fourier transforms of the elds and propagator are dened using the

following conventions

d k

ik x y ik x

A x a k k e a k k e a

s s s

m d k

y ik x

b k u k e b x

m d k

ik x

b k u k e c x

r r

d k k m

ik x

S x e d

k m i

where u k is a spinor solution to the Dirac equation with spin index r and k are photon p olarization

fourvectors chosen to b e real with index s lab eling the physical transverse p olarizations of the photon The

summation convention over rep eated spin and p olarization indices is always implied The Fourierspace interaction

Hamiltonian is obtained by substituting Eqs into Eq The distributional identity

ik x

d xe k

allows trivial integration over the fourmomentum of the electron propagator The resulting interaction Hamiltonian

h i

H t dq dq dp dp q p q p exp itq p q p

h i

y y

b q a p M M a pb q

0 0

s r

r s

M M M a

p p q m pu q q u

r r

s s

M q r p s q r ps e b

p q

q u p q p m pu q

r r

s s

c M q r p s q r ps e

p q

with the abbreviations

d q m d p

dq dp

q p

for electrons and photons resp ectively All of the op erators in Eq are freeeld op erators so this is the prop er

expression to substitute into the left side of Eq

B Forward Scattering Term

We now pro ceed to evaluate the rst term on the left side of Eq First we display op erator exp ectation values

needed here and in the following subsection using op erator denitions and the commutation relations Eq

ha a b b i ha a i hb b i a

ha p a pi p p p p b

n mn

hb q b q i q q n q c

n mn e

q q

y y

0 0

q q q q n q n q hb q b q b q b q i

0 0

e e

r r r r r r

r r

1 1 2 2 1 2

1 2

q q

0 0

q q n q n q d q q

e e

r r r r

1 2

2 1

y y

0 0

p i p p p p p p p p p a p a ha p a

0 0

s s

s s s s

2 1

1 2

s s

1 2

2 1

0 0 0

p p p p p p p p e

s s s s s s

2 1 1

1 2 2

The last relationship neglects the correlation term b etween all four op erators when p p p p The

exp ectation values for electron op erators assumes a particular form for the electron density matrix appropriate to

thermal equilibrium with equal p opulations in each spin state and no correlations b etween the states n q represents

the numb er density of electrons of momentum q p er unit volume This assumed form for the electron density matrix

will not b e valid if substantial magnetic elds are present

Using the denitions and and the commutation relations the commutator in the forward scattering

term b ecomes

H D k dq dq dp dp q p q p M M

I ij

y y

b q b q a p a k p p k

0 0

r j is

r s

y y

b p k k a pp q b q a

s j s r

r i

On using the ab ove exp ectation values it follows that

ie n q

0 0

i H D k dq k k

is is s j j s

I ij

k q

k k q m k k q k m u q k u q

r s r

s s s

where the integrals have b een p erformed with the delta functions All of the terms involvingk cancel out on using

the gammamatrix identity A B A B B A and the p olarization vector prop erties k k and

i i j ij

For the remaining terms we use the identity

u q u q u q q q m u q q m

0 0

r r r s r

s s s s s

u q q u q

r s r

q q

s s

u q q m u q

r r

s s

where the second equality follows from the Dirac equation and the third equality uses the Gordon identity Thus we

have

i H D k

I ij

and the forward scattering term do es not contribute to the photon density matrix evolution

C Scattering Term

The scattering term is considerably more cumb ersome to evaluate b eing quadratic in the interaction Hamiltonian

After substituting the expressions for H and D and taking the exp ectation value the scattering term reads

I ij

Z Z

dt H t H D k dq dq dp dp dq dq dp dp

I I ij

q p q p q p q p MM

D ED E

y y y y

p p k b q b q b q b q a p a p a p a k

0 0 0 0

is r r s

r r s s

1 2 1 2

2 1 2 1

D ED E

y y y y

p p k b q b q b q b q a p a k a p a p

0 0 0 0

is r r s

r r s s

1 1 2 2

1 2 1 2

D ED E

y y y y

p k b q b q b q b q a p a p a k a p p

0 0 0

j s r r s s

r r s

1 2 1 2 1

2 1 2

D ED E

y y y y

p k b q b q b q b q a k a p a p a p p

0 0 0

j s r r s s

r r s

1 1 2 1 2

1 2 2

The energy delta function comes from the time integral on using Eq The arguments of the matrix element

indicates the subscript to b e attached to all dep endent variables in Eq and of course summation over all spin

and p olarization indices is implied

Substitution of the exp ectation values Eq into the ab ove expression and p erforming the integrals over q

p q and p using the various delta functions yields

dt H t H D k

I I ij

dq dq dp q p q k Mq r p s q r k s Mq r k s q r p s

0 0 0 0

p n q k n q

e e

s s s j s s is j is s

2 1 1 2 1 2 1 2

dq dq dp q k q pMq r k s q r ps Mq r ps q r k s

0 0 0 0

p k n q n q

e e

s j s is s j s s s is

2 2 1 1 2 1 1 2

The subscript on all momentum variables has b een dropp ed for notational simplicity All terms quadratic in the

electron phasespace density have b een dropp ed since for all cosmological scenarios this numb er is negligible compared

to unity all terms quadratic in the photon density matrix cancel exactly By relab eling the integration variables and

spin indices implicitly summed over in the second integral Eq reduces to

k D t H dt H

ij I I

dq dq dp q p q k Mq r p s q r k s Mq r k s q r p s

0 0 0 0 0

n q k k n q p

e e

s s is s j j s is is j s s s

2 1 1 2 2 1 1 2 1 2

This equation is an essentially exact expression for the collision term in the case of the microwave background the

approximation that the duration of the Compton scattering b e small compared to the time b etween scatterings is

eminently satised for any cosmological scenario and assuming the electrons to b e unp olarized is essentially exact

unless magnetic elds b ecome imp ortant at some ep o ch

Evaluating the matrix elements and p erforming the integrals in Eq is a straightforward pro cess This pap er

is concerned with the rstorder p erturbations away from a p erfectly homogeneous and isotropic universe and the

scattering term will b e explicitly calculated to rst order Evaluating the matrix elements involves standard techniques

of quantum eld theory and yields the familiar Compton crosssection to lowest order

Mq r ps q r k s Mq r k s q r ps

r r

q k q p

0 0 0 0 0 0

e k p p k k p p k

s s s s s s s s s s s s

1 1 2 2 1 2 1 2 1 2 1 2

q p q k

0 0 0 0 0 0

k p p k k p p k

s s s s s s s s s s s s

1 1 2 2 1 2 1 2 1 2 1 2

The following subsection then obtains the general Boltzmann equation for the photon density matrix to rst order in

terms of the photon energy and p olarization vectors

D Scattering Term to First Order

Now we pro ceed to evaluate Eq to lowest order in scattering kinematics After substituting the matrix

element Eq into Eq the Boltzmann equation now explicitly including spatial dep endence

b ecomes

d e m

x k E q k p p E q k dq dp

dt m k E q k p

0 0 0 0 0

n x q x p x k n x q x k

e e

s s j s s s is s is j j s is

2 1 1 2 1 2 1 2 1 2

q k q p

0 0 0 0 0 0

p p k k p k k p

s s s s s s s s s s s s

1 2 1 2 1 2 2 1 2 1 2 1

q k q p

0 0 0 0 0 0

p p k k p k k p

s s s s s s s s s s s s

1 2 1 2 1 2 2 1 2 1 2 1

where E q q m is the energy of an electron with momentum q The electrons are describ ed by an

unp olarized thermal MaxwellBoltzmann distribution

q mv x

n x q n x exp

e e

mT mT

e e

with T the electron temp erature and v x the electron bulk velo city Useful integrals of the electron distribution are

d q

n x q n x a

e e

d q

q n x q mv xn x b

i e i e

For relevant cosmological situations the kinetic energies of the electrons and photons are negligible compared to the

electron mass implying that the energy transfer in a Compton scattering event is small compared to the characteristic

photon energy p m q m with obvious abbreviations p p jpj and q jqj Furthermore if the electron

and photon temp eratures are comparable p q We expand the various functions in Eq in terms of pq and

q m using the following asymptotic expansions

q q Q Q

E q Q m a

m m m

Q q mv Q

n q Q n q b

e e

mT mT

e e

k p q k p

k p E q E q k p k p c

m p

where in the last expression the derivative of the delta functional is dened through integration by parts Writing

out the p olarization sums explicitly yields the equation

Z Z

k p e n x d d

k p k p v x x k dp p

dt m k p

k p

x k p k p k x k p k p k x k

ij i j i j

k p

p k

x p x p

k p

p k

k p k p k p k p x p x p

i j i j

k p

k p k p k p k p x p x p

i j i j

k p k p x p k p k p x p

i j i j

Here the photon momentum integral has b een rewritten as an energy integral and an angular integral over the

momentum direction This is the basic equation describing the evolution of the photon density matrix to rst order

in the kinematic variables By rewriting the momentum and p olarization vectors in a spherical co ordinate basis and

incorp orating the velo citydep endent term into the lefthand side the equation b ecomes equivalent to Chandrasekhars

radiative transfer formalism cf Chapter Eq of Ref Before p erforming the nal angular integrals we

must consider the left side of the equation for the particular spacetime geometry in which we are interested which

determines the azimuthal dep endence of The left side of the equation will b e analyzed in the following section and

then in Section VI we p erform the remaining momentum integrals to complete the evaluation of the right side

V THE GENERALRELATIVISTIC LIOUVILLE EQUATION

The last section has analyzed the right side of the Boltzmann equation Eq we now turn to the left side

describing the prop ogation of photons in the background spacetime The Boltzmann equation with no collision term

on the right side is the Liouville equation describing the evolution of a collisionless systems phase space distribution

Writing the equation has already assumed denition of a set of spacelike hyp ersurfaces that is the equation contains

an explicit time derivative The background spacetime here will b e the canonical FriedmannRob ertsonWalker

FRW spacetime In this pap er only the at case will b e considered techniques p ertaining to op en universes have

also b een extensively develop ed Scalar and tensor metric p erturbations are added to the at background space

time we neglect vector p erturbations which kinematically decay and are unimp ortant unless a continual source of

vector p erturbations such as top ological defects exists The metric we consider is

g x t g

g a t x t h x t

ij ij ij

In this section Greek subscripts refer to spacetime indices running from to while Roman subscripts refer to

spatial indices running from to The function at is the usual cosmological scale factor The scalar p erturbations

dened in the longitudinal gauge are given by the two scalar function and The metric p erturbations h

are dened in the transversetraceless gauge and are sub ject to the constraints

h h

ij i

With the p erturbations dened in this way no residual gauge freedom remains in contrast to the more conventional

synchronous gauge condition A second advantage of the present denition is that in the Newtonian limit the metric

p erturbation simply corresp onds to the Newtonian p otential The inverse metric to rst order in the p erturbations

g x t g

ij ij ij

x t h x t g

a t

In this section we will consider the Liouville equation to rst order in the metric p erturbations secondorder treatments

have b een undertaken in Ref

Photons are describ ed by spacetime co ordinate x and fourmomentum k Our co ordinate system has x t

the photon momentum satises

i i

k dx

k dt

The photons ob ey the geo desic equation

dx dx d x

d d d

dx dx

d d

where is an ane parameter along the photon geo desic which may b e dened so that dtd dx d k thus

i i

dx d k using Eq Therefore using the denition of the Christoel symb ol

g g k k dk

dt x x k

with the geo desic condition k k

Liouvilles equation for any phasespace distribution function f is

dx dk k dk f f f f f df

dt x dt k dt t x k k dt

We change to a convenient choice of momentum variables the photon energy in a lo cal orthonormal frame k

i i

k k and the unit vector k A lo cal observer at rest with resp ect to the cosmic co ordinate system will measure

the photon to have energy k this is the energy app earing in the collision term on the right side of the Boltzmann

Equation The wave vector is thus given by

i n m i

k k h k k k

and the Liouville equation in the new variables is

i i

k dk f f f f dk df

dt t x k k dt dt

Now it is a straightforward matter to calculate the derivatives dk dt and dk dt Substituting k k and

Eq into the zero comp onent of Eq gives to rst order

h k a dk

j i

k k k

dt a t a x t

The derivative of k is most easily computed by dierentiating Eq and equating this result with Eq it is

straightforward to show that dk dt has no lowestorder terms Physically this is b ecause geo desics are straight lines

in the unp erturb ed metric Since f k is itself linear in the metric p erturbations the nal term of the Liouville

equation drops out to rst order Expanding the distribution function as

f x k k t f k t f x k k t

leads to the zerothorder equation

a f f

t a k

whose solution is simply f k t f k a which is just the uniform redshift of the sp ectrum with cosmic expansion

The rstorder Liouville equation is

i i

h k k f a f f f

j i

k k k k

i i

t x a a k k t a x t

Note that k in this equation is the physical not the comoving photon wave numb er

The terms in the Liouville equation dep ending directly on the metric p erturbations determine the form of the

directional dep endence of the distribution function AFourier transform over the spatial dep endence of the equation

gives for the Boltzmann equation

i a

f K k k K kf K k k k f K k k

t a a k

f k i

j i

k k K K K kK k h K C K k k

k t t a t

where C represents the collision term on the right side For scalar p erturbations the previous section shows the

right side contains source terms prop ortional to k v where v is the lo cal velo city of the electrons But for scalar

p erturbations v K so if we cho ose spherical co ordinates for k with axis in the K direction then f is

manifestly indep endent of the azimuthal angle in other words

f K k k f K k

for scalar p erturbations

Tensor p erturbations do dep end on but in a simple manner We neglect any electron velo city v arising from

tensor p erturbations as corrections to the scalarinduced velo city The dep endence of the distribution function is

determined by the p erturbation term which can b e written as

j i j i

k k h K te K h K te K k h K t k

ij ij

are p olarization tensors for the plus and cross p olarizations of the gravity wave Again cho ose and e where e

ij ij

spherical co ordinates with the z axis p ointing in the direction of K In this co ordinate system the p olarization tensors

are given by e e and e e with the other comp onents zero Contraction of the unit vectors

xx y y xy y x

with the p olarization tensors gives

j i

k e sin cos sin sin sin cos k

j i

e sin cos sin sin sin k k

Therefore for a given plane wave comp onent of a metric tensor p erturbation

f K k k f K k cos a

for the plus p olarization of the gravity wave and

f K k k f K k sin b

for the cross p olarization

VI COMPLETE POLARIZATION EQUATIONS

Now we have assembled all the ingredients for deriving the nal p olarization evolution equations Eq with

the p erturbation expansion for given by Eq and the angular dep endence of given by Eqs and

ij ij

Note the unp erturb ed photon density matrix satises and since it represents a

uniform unp olarized blackb o dy sp ectrum The lowestorder term on the right side of Eq is zero the resulting

equation for gives the uniform shift of the photon sp ectrum with scale factor in an expanding universe Eq

The rst order term gives

Z Z

p e n k d d

k p K k K k dp p

ij ij

dt m k k p

p k p k K k p k p k K k

i i

j j

p k

k p k p k p k p K p K p

i j i j

k p

k p k p k p k p K p K p

i j i j

k p k p K p k p k p K p

i j i j

p p k k k p

p p k k p v K

p k p k p

p k p k k p k p k k

i i

j j

k p k p p k p k p p

i j i j

As in the previous section K is the Fourier conjugate of x andn is the mean electron density which is constant to

lowest order The remainder of this section evaluates the remaining angular integrals in this expression and converts

the equations for the density matrix elements to equations for the brightness of each Stokes parameter

To evaluate the angular integrals most conveniently cho ose the z axis of the spherical co ordinate system to coincide

with K indep endently for each Fourier mo de Note that on transforming back to realspace co ordinates care must

b e taken b ecause the density matrix is not invariant under a change of basis see Sec I I The basis for the photon

direction and p olarization vectors is taken to b e

k sin cos k cos cos k sin

x x x

k sin sin k cos sin k cos

y y y

k cos k sin k

z z z

The same denition is used for p and its asso ciated p olarization vectors with and The angular

integral in Eq is over and and the various dot pro ducts are given by

p k sin cos cos cos sin

p k sin sin

k p cos cos cos sin sin

k p cos sin

k p cos

Two additional convenient abbreviations are v p cos and v k cos Now the angular integrals are

straightforward resulting in expressions like

Z Z

d e n d d

P k k k k

dt m

k k v k v

k k

for scalar p erturbations where dep endence on the Fourier mo de K is implicit In solving the evolution equations it

is convenient to split the densitymatrix p erturbation into two parts one due to the scalar metric p erturbations and

one due to the tensor metric p erturbation In making this split the bulk velo city v is entirely attributed to the scalar

p erturbations

For the nal set of evolution equations we change variables to comoving wave numb er q k a and convert the

density matrix elements to Stokes parameter brightness p erturbations

q q

K q K q a K q

q q

K q K q b K q

q q

K q K q K q c

q q

K q K q d K q i

where the sup erscript i stands for s or representing the three typ es of p ossible metric p erturbations scalar and

two p olarizations of tensor For linear p erturbations considered here the distribution function undergo es no sp ectral

distortions and the p erturbations are blackb o dy in this case is just the temp erature uctuation T T with

T the mean temp erature We also dene moments of these variables

i i

q P q

I l I

where P is the Legendre p olynomial of order l Note these moments are sometimes dened dierently as

i P

l l

For scalar p erturbations the brightness is governed by the set of equations

s s s s s s

v iK iK n P a

T e

I I I I Q Q

t a t a

s s s s s

iK n P b

T e

Q I Q Q Q

t a

s s

iK n c

T e

U U

t a

V s s s

iK n d

T e

V V V

t a

The evolution of the brightness thus do es not dep end on the direction of K only on its magnitude the brightness

dep ends on the direction of K only through initial conditions which factor out of the linear evolution equations The

equations for U and V have no source terms so for each K the evolution leaves U and V The co ordinate

dep endence of Eq gives a nonzero U on transforming back to x space but V remains zero

For tensor p erturbations the evolution equations take their simplest form after the co ordinate transformation

cos sin a

I I I I

cos sin b

Q Q Q Q

sin cos c

U U U U

The dep endence is determined by Eqs a and b and the dep endence is chosen to simplify the nal

equations After this change of variables the brightness equations b ecome

a iK n

T e

I I

t a t

b iK n

T e

Q Q

t a

U Q

d iK n

T e

V V

t a

I I I Q Q Q

The denition of h is given in Eq The tensor p erturbation gives same equations Again V since it

has no source term

For a given cosmological scenario which determines the metric p erturbations Eqs and must b e evolved

numerically This can b e done eciently by expanding the dep endence of the brightness functions in terms of

Legendre p olynomials Eq giving a large set of coupled ordinary dierential equations Several detailed

co des to calculate temp erature uctuations have b een implemented using this scheme

VI I POWER SPECTRA

s s

Numerical solution of the ab ove transp ort equations gives the Fourier space brightness functions K and K

I l Ql

K for tensor p erturbations where represents the two gravity wave K and for scalar p erturbations and

U l I l

p olarization states and The temp erature uctuations in real space are then

X X

T x

iKx

l P cos e

K l

i h

K K sin sin K sin cos

I l

I l I l

where T is the mean temp erature of the microwave background and represents the same direction as

except in the co ordinate system dened by the K direction

The p olarization is more complicated b ecause for each K mo de the co ordinate system in the direction has

a dierent orientation when the Q and U brightnesses are summed up the axes must b e rotated to the orientation

in the x co ordinate system using Eq To determine this rotation angle for each K mo de refer to Fig the

needed angle is lab elled the angle b etween the vectors and Let the direction of K b e denoted by

K K

On the unit sphere the lengths of the sides of the spherical triangle AB C are just the angles they subtend The angle

is given by the law of cosines

cos cos cos sin sin cos

K K K

and the rotation angle is

sin sin sin csc

K K

Then the Q and U brightnesses are given by

X X

iKx

l P cos e

K l

K cos cos K cos sin K cos

Ql Ql

cos sin K cos cos K sin

Ql Ql

and U the same except for the replacements cos sin and sin cos From these two quantities the

p olarization vector P follows from Eq as

q q

P P Q P P Q

T j j

U Q z φ −φ k K C φ ξ K A θ^ θ^ B

θ K θ

θ y φ

φ K

FIG Angles and directions for determining the orientation of dierent spherical co ordinate bases at a given p oint

The predictions of a given cosmological scenario are only statistical The traditional statistical measure of temp er

ature uctuations is the angular p ower sp ectrum C dened by

T q T q

C q q cos

T T

where the angle brackets represent an ensemble average over initial conditions this average can b e replaced in calcu

lations with an average over space assuming ergo dicity Many analogous correlation functions for various p olarization

variables are p ossible

T q P q

C a

T T

P q P q

C b

T T

Pq Pq

c C

T T

Pq rT q

C d

T T

roughly in increasing order of diculty to measure The various combinations of Stokes parameters can also b e used

to form correlations

Qq Qq

a C

T T

T q Qq

b C

T T

and so on These correlation functions dep end on the orientation of the axes used to dene the Stokes parameters

Their advantage is that they are easier to calculate than Eqs and are easier to measure when signal to noise

is low their disadvantage is that their physical interpretation is less easily visualized Correlation functions are

commonly characterized by the co ecients C of an expansion in Legendre p olynomials

TT TT

C P cos C

and likewise for the others The l term indistinguishable from the Doppler shift from prop er motion with resp ect

to the rest frame of the microwave background is ignored Note this conventional normalization of C unfortunately

diers by a factor of from the conventional normalization of the brightness moments Eq

A Temp erature Correlation Functions

The temp erature correlation function C can b e evaluated exactly in terms of the brightness moments First

since the equations for K dep end only on jKj separate out the angular dep endence by writing the uctuations as

i i i K

K K Ke

I l I l

where i stands for s or The functions and are random variables set by the initial conditions is real and

i i i

d p ositive normalized such that K and the phase is a real numb er b etween and For gaussian

K i

initial conditions is drawn from a normal distribution and is uniformly distributed indep endently for each K

i i

mo de and each typ e of p erturbation

Substituting Eq into Eq provides the starting p oint for evaluating the temp erature correlation functions

d x where V is the sample volume and in the limit of a large volume The average value can b e replaced by V

V d K The x integral gives a delta functional the sums over K vectors can b e replaced by integrals

which eliminates one of the K integrals For scalar p erturbations the result is

T q V T q

s s

K K cos cos P l l P d K

l l

2 1

I l I l

2 1

T T

l l

1 2

Then expressing each Legendre p olynomial in terms of spherical harmonics using the spherical harmonic addition

formula and p erforming the angular piece of the K integral using the decomp osition in Eq gives the familiar

formula

TT s

C K dK j K j

l I l

For tensor p erturbations the added angular dep endence makes the calculation somewhat more involved in this

case Eq b ecomes

T q V T q

cos sin sin d K cos P l l P

l l

2 1

T T

l l

1 2

h i

cos cos K K K sin sin K K K

I l I l I l I l

1 2 1 2

The crossterms b etween the two typ es of tensor p erturbations and b etween the tensor and scalar p erturbations

cancel b ecause the integral over K contains random phase factors eg expi K i K Rearranging the terms

in brackets yields

T q T q V

d K cos sin sin cos P l l P

l l

2 1

T T

l l

1 2

cos K K K K K K

I l I l I l I l

2 1 2 1

cos K K K K K K

I l I l I l I l

2 1 2 1

Further progress can b e made with the mild and physically reasonable assumption that K K in other

I l I l

words that the p ower sp ectra for the two p olarizations of gravity waves are the same Then the two terms on the

second line of the ab ove equations give equal integrals and cancel The remaining trigonometric functions can b e

written in a form indep endent of

T q V T q

K K l l dK K

I l I l

2 1

T T

l l

1 2

q q P d P

l K l

2 1

where as in the scalar case the K integral has b een separated into its magnitude and angular dep endences and

cos etc Now the evaluation and simplication of the angular integral is a straightforward but lengthy

pro cess The various factors of may b e absorb ed into the Legendre p olynomials using the recursion relation

l xP x l P x l P x

l l l

The angular integral may b e p erformed using the same pro cedure as in the scalar case replace the Legendre p oly

nomials with spherical harmonics using the addition theorem and use the orthogonality of the spherical harmonics

to eliminate the integrals Then converting the remaining spherical harmonics back to Legendre p olynomials gives

many terms prop ortional to P q q with various indices l The original factors of q q in Eq can now b e

absorb ed into the Legendre p olynomials using the ab ove recursion relation At this p oint the terms can b e collected

together and after much algebra the formula nally simplies to

K K

V l

I l I l

I l

C dK K

l l l l l l l

The total temp erature correlation function is given by the sum of the scalar and tensor pieces as long as no correlation

exists b etween the two p erturbations as will b e the case for any inationary scenario

B Polarization Correlation Functions

Direct evaluation of correlation functions involving p olarization involves further complications For correlations

involving the p olarization vector P or its magnitude substantial simplication is not p ossible since the p olarization

vector is not linear in the Stokes parameters the x integral replacing the ensemble average cannot b e immediately

p erformed as in the temp erature correlation functions ab ove and no progress can b e made in simplifying the general

expressions for the correlation functions The evaluation of Eqs must b e p erformed numerically through for

example a Monte Carlo average over random pairs of directions separated by a xed angle at a given p oint in space

However the correlation functions of the Stokes parameters themselves Eqs can b e simplied if a smallangle

approximation is invoked The additional approximation is needed b ecause in contrast with the temp erature case

the Q and U brightnesses have factors involving the rotation angle Consider the hQQi correlation function rst for

scalar p erturbations Cho ose the axis of the spherical co ordinate system to b e one of the two observation directions

As ab ove the ensemble average can b e replaced by a space integration which then eliminates one of the K integrals

resulting in the expression

Qq V Qq

s s

K cos cos K cos cos P l l P d K

l l

Q l 2 Q l 1

2 1

T T

l l

1 2

with q z which implies and Now if q and q p oint nearly in the same direction then

K K

cos cos cos using Eq Then the cosine factors in the ab ove equation can b e rewritten as

cos The rst term then lo oks just like the temp erature case considered earlier The second term can b e

and then explicitly p erforming somewhat simplied by using the spherical harmonic summation formula to rewrite P

the integral over The nal result is

C K dK K

s s

l l A cos A K K K dK

l l ll Ql Ql

1 2 2 2 1

l l

1 2

m m

x A xP dxP

l l 1 l

1 2 2

where the sum over l b egins at l Note the nal result dep ends explicitly on the azimuthal angle since the

value of Q changes if the axes are rotated

For tensor p erturbations the drill is now familiar The assumption that K K with the ab ove small

Ql Ql

angle approximation gives the expression

Qq Qq V

l l K dK K K

Q l Q l

1 2

T T

l l

1 2

d cos cos cos cos cos cos sin cos P P

K K K l l

2 1

Then the factors of cos can b e absorb ed into the Legendre p olynomials using the recursion relation The nal

expression for tensor p erturbations is

C K dK B B

l l

A cos A K K B K B K B K dK B

l l ll l l l l

1 2 2 2 1 2 1

l l

1 2

B K l K l K

l l

l l l l l l l

B K K K K

l l l

l l l l

The expressions for hQT i can b e obtained in the same way they are given in Ref For gaussian initial

p erturbations the p olarization map can b e reconstructed from the temp erature map and the correlation functions

hQQi and hQT i as shown in Ref In principle all of the correlation functions involving the p olarization vector are

obtainable from these as well although there app ear to b e no simple formulas connecting the two sets of correlation

functions

VI I I DISCUSSION

The detection of microwave background p olarization is very dicult The most optimistic scenarios predict p olar

ization no larger than of the temp erature uctuations or a few parts in a million of the temp erature Needless

to say this sensitivity is hard to attain The same backgrounds which aect the temp erature measurements will also

aect p olarization measurements One advantage of measuring p olarization which has b een realized for a long time is

that the exp eriment can chop b etween two orthogonal p olarizations on the same patch of sky which involves rotating

a p olarizer instead of mechanically rep ointing the telescop e In practice temp erature measurements can only chop at

a few Hz while p olarization measurements can chop at hundreds of Hz Any atmospheric noise is thus supressed much

more eectively A mitigating eect is that since the orientation of the horn is imp ortant side lob es from the ground

and diraction eects can add noise dierently to the two p olarization channels As with temp erature measurements

the ultimate exp erimental limit is astrophysical foreground sources particularly our own galaxy

On the theoretical side where do es p olarization t into the systematic investigation of microwave background

anisotropies In cosmological scenarios which invoke adiabatic initial p erturbations as in Cold or Mixed Dark

Matter the temp erature correlation function Eq contains features from which cosmological information may

b e extracted provided reionization did not o ccur to o early If the universe evolved according to this broad class

of mo dels measurement of the anisotropy sp ectrum will provide a detailed picture of the early linear growth of

p erturbations In this case the exp ected p olarization can b e calculated and its detection at the predicted level will

provide imp ortant conrming evidence for the theory Conversely the angular scale and amplitude for the p eak of

the p olarization correlation function likely the easiest p olarization quantity to measure can serve as an additional

piece of information when constraining theories Also precision calculations of the temp erature anisotropies in these

mo dels should incorp orate the p olarization contribution to the temp erature source term in Eqs and the

p olarization may aect the p ower sp ectrum at the few p ercent level

If the universe did not start out with adiabatic uctuations as in top ological defect mo dels or underwent early

reionization p olarization uctuations may take on added imp ortance in understanding the nature of the linear p ertur

bations b ecause less information will b e enco ded in the temp erature uctuation sp ectrum Because of computational

diculties in the cosmic string scenario currently the most viable defect mo del no detailed temp erature uctuation

sp ectra including all relevant physical pro cesses has yet b een pro duced and p olarization has not b een investigated

Owing to the more complex nature of the underlying physics it is reasonable to exp ect that cosmological information

will b e more dicult to recover from the cosmic string temp erature uctuation sp ectrum than in the adiabatic case

and then p olarization may provide useful additional information Likewise if the universe underwent early reioniza

tion prior to z much of the information in the adiabatic microwave background sp ectrum will b e washed out

In this case the level of p olarization will b e signicantly larger making it easier to detect and may partially

comp ensate for the temp erature sp ectrums loss of information Additionally in a reionization scenario substantial

secondary anisotropies in b oth temp erature and p olarization may b e pro duced via the OstrikerVishniac eect

Secondorder eects on p olarization have not yet b een systematically investigated

Polarization limits or detections may also give useful information ab out intergalactic magnetic elds at early times

Sp ecically if a net circular p olarization is ever detected and convincingly separated from foreground contamination

this would b e a strong suggestion for a magnetic eld p olarizing the free electrons Even upp er limits on p olarization

provide one of the only metho ds of constraining the primordial magnetic eld

Finally since temp erature and p olarization couple dierently to tensor and scalar metric p erturbations it may b e

p ossible to separate the two contributions with a combination of temp erature and p olarization measurements Initial

steps in this direction have b een taken in Ref which considers dierences in hQT i b etween tensor and scalar

p erturbations The overall size of the correlations are of course small but some dierences do app ear in the cross

correlation function A dierent correlation function or some other typ e of statistic optimized to lo ok for correlations

p eculiar to tensor p erturbations may give more promising results

ACKNOWLEDGMENTS

I would like to thank Scott Do delson for teaching me the Boltzmann equation approach to the microwave back

ground Gunter Sigl for patient explanations of his work and Lloyd Knox for p enetrating questions I would also like

to acknowledge helpful conversations with Stephan Meyer and Rob ert Crittenden Michael Turner has continually

provided supp ort and encouragement Thanks also to Edward Kolb Tom Witten and Rene Ong for their time and

energy This work was supp orted primarily by the NASA Graduate Student Researchers Program and in part by the

DOE at Chicago and Fermilab and by NASA through grant No NAGW at Fermilab

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