THE ASTROPHYSICAL JOURNAL, 482 :6È16, 1997 June 101997. The American Astronomical Society. All rights reserved. Printed in U.S.A.(

MEASURING POLARIZATION IN THE COSMIC MICROWAVE BACKGROUND

UROS— SELJAK

Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138Received 1996 August 21 ; accepted 1997 January 8

ABSTRACTPolarization induced by cosmological scalar perturbations leads to a typical anisotropy pattern, which

can best be analyzed in a Fourier domain. This allows one to distinguish a cosmological signal of polar-ization unambiguously from other foregrounds and systematics, as well as from polarization induced bynonscalar perturbations. The precision with which polarization and cross-correlation power spectra canbe determined is limited by cosmic variance, noise, and foreground residuals. The choice of estimator cansigniÐcantly improve our capability of extracting a cosmological signal, and in the noise-dominated limitthe optimal power spectrum estimator reduces the variance by a factor of 2 compared to the simplestestimator. If foreground residuals are important, then a di†erent estimator can be used, which eliminatessystematic e†ects from foregrounds so that no further foreground subtraction is needed. A particularcombination of Stokes Q and U parameters vanishes for scalar-induced polarization, thereby allowing adirect determination of tensor modes. Theoretical predictions of polarization in standard models showthat one typically expects a signal at the level of 5È10 kK on small angular scales and around 1 kK onlarge scales (l\ 200). Satellite missions should be able to reach sensitivities needed for an unambiguousdetection of polarization, which would help to break the degeneracies in the determination of some ofthe cosmological parameters.Subject headings : cosmic microwave background È cosmology : theory È methods : data analysis È

polarization

1. INTRODUCTION

Anisotropies in the cosmic microwave background (CMB) are now widely accepted as the best probe of the early universe,which can potentially provide information over a whole range of cosmological parameters et al.(Jungman 1996 ; Zaldarriaga,Spergel, & Seljak The main advantage of CMB anisotropies as opposed to other, more local probes is that they are1997).sensitive to the universe in the linear regime, where statistical properties can easily be calculated, starting from ab initiotheoretical models, and compared to the observations. A dozen or so cosmological parameters could be extracted from theobservations produced by the future satellite and ground-based experiments. There are two potential problems in thisprogram. The Ðrst is the somewhat uncertain amount of Galactic and extragalactic foregrounds, which could severely limitour ability to extract a cosmological signal from the data. The second consists of the degeneracies among some of thecosmological parameters, which allow only certain combinations to be determined accurately but do not allow breaking thedegeneracies between them et al. et al. et al.(Bond 1995 ; Jungman 1996 ; Zaldarriaga 1997).

It is therefore important to investigate other independent conÐrmations of results produced from CMB anisotropies, and ithas long been recognized that polarization in the microwave sky might provide such an independent test (Rees 1968 ;

& Efstathiou Davis, & Steinhardt Polnarev, & ColesPolnarev 1985 ; Bond 1987 ; Crittenden, 1993 ; Frewin, 1994 ; Coulson,Crittenden, & Turok Coulson, & Turok Like temperature anisotropy, polarization probes the1994 ; Crittenden, 1995).universe in the linear regime and so can provide information useful to determine cosmological parameters. It is speciallyimportant for determining parameters that are only weakly constrained by the CMB anisotropies alone, such as the epoch ofreionization or the presence of tensor perturbations. In both of these cases, cosmic variance is the limiting factor in ourcapability of extracting the parameters, so measuring polarization would increase the amount of information and allow for amore accurate determination of the parameters. Both polarization-polarization and polarization-temperature correlation givean independent set of power spectra ; these have di†erent sensitivity to di†erent parameters, so combining them results in amuch larger amount of information about the underlying cosmological model and can signiÐcantly increase our capability ofextracting useful information from the CMB measurements.

The main disadvantage of polarization is that it is predicted to be of rather low amplitude, of the order of 10% oftemperature anisotropy, so measuring it represents an experimental challenge that has yet to be met. Currently there are nopositive detections of polarization, with the best upper limits of the order of 25 kK et al. This situation need(Wollack 1993).not continue in the future, however, as the experimental sensitivity increases and new larger and better experiments are beingplanned. Moreover, as shown in this paper, cosmologically induced polarization has a unique signature in the data thatcannot be mimicked by other foregrounds and provides a clear test of the searched signal. An experiment at Brown (Timbie

plans to reach sensitivities of a few microkelvins, which should be sufficient to detect the polarization signal if our1996)current expectations are correct. Interferometer observations could extend this to much larger areas of the sky and producemaps of polarization both in real space and in frequency space. Unfortunately, at present none of the planned interferometerexperiments is including polarization, although, as argued in this paper, frequency space has many advantages in the searchfor the unique signature of cosmological polarization. Finally, all-sky satellite maps of polarization are also planned by theMAP and Planck satellites and may eventually provide us with very high accuracy maps of polarization pattern in the sky.

6

MEASURING POLARIZATION IN CMB 7

The outline of this paper is as follows. In statistical properties of polarization parameters are derived in the small-scale° 2limit. The special nature of polarization induced by scalar perturbations allows one to use statistical methods developed in thecontext of weak lensing In various two-point estimators are presented in both Fourier and real space,(Kaiser 1992). ° 3together with their variances and covariances. is devoted to the foregrounds and possible methods of theirSection 4elimination. Polarization induced by tensor modes is discussed in and the di†erence between the two types of pertur-° 5,bations is highlighted. In theoretical predictions for polarization are computed for a variety of cosmological models, and° 6the ability to extract them with the satellite missions is discussed. This is followed by conclusions in ° 7.

2. POLARIZATION IN THE SMALL-SCALE LIMIT

In this section we derive the small-scale limit of temperature and polarization anisotropies. This limit is of special interest,because one can replace the general spherical expansion with the Fourier expansion, and the expressions simplify consider-ably. The analysis in this section will be restricted to polarization generated by scalar perturbations. Tensor perturbations arediscussed in ° 5.

In general one can describe CMB polarization as a 2] 2 temperature perturbation tensor Stokes parameters Q and UTij.

(we will ignore V in the following, since it cannot be generated through Thomson scattering) are deÐned as andQ\Txx

[ Tyywhile temperature anisotropy is just its trace, The components are deÐned with respect to aU \ 2T

xy\ 2T

yx, T \ T

xx] T

yy.

Ðxed coordinate system (x, y) perpendicular to the photon direction n. The Stokes parameter Q is positive if temperatureperturbation is larger along the x-axis relative to the y-axis, while the U parameter is positive if perturbation is larger alongthe upper right diagonal relative to the upper left diagonal.

Equations of radiative transfer for polarization and temperature anisotropy simplify in Fourier space if we work in a framewith the axis deÐned parallel and perpendicular to L/Lb, where b is the angle between the wavevector k and the photondirection n. Azimuthal symmetry guarantees that the Thomson scattering preserves the diagonal form of just as in the caseI

ij,

of plane-parallel atmospheres In terms of the Stokes parameters, only Q is excited in this(Chandrasekhar 1960 ; Kaiser 1983).frame. This puts a restriction on the general form of polarization, and, as we show below, it can be used to separatecosmologically induced polarization from other sources and systematic e†ects. Although only Q is present in this frame, whenwe rotate polarization by an azimuthal angle to a Ðxed frame in the sky, we generate both Q and U. At the observerÏs[/

k,nposition the expressions for T , Q, and U are given by & Efstathiou(Bond 1987 ; Kosowsky 1996)

T (n) \P

d3k*T(k, n) ,

Q(n) \P

d3k*P(k, n) cos (2/

k,n) , (1)

U(n) \P

d3k*P(k, n) sin (2/

k,n) ,

where n) are the Fourier components of the temperature and polarization distribution function integrated over the*T,P(k,

momentum, and n is the direction of observation on the sky. The expression for rotation angle depends on both k and n,/k,nbut if we restrict our attention to the directions n around the pole, then it can be approximated with where is the[/

k, /

kazimuthal angle of the vector k. This approximation breaks down for wave modes k close to the pole but for(zü -direction),sufficiently small scales the contribution from these modes to the total power becomes negligible.

The solution for n) can be written as an integral over the sources along the line of sight (Seljak & Zaldarriaga*T,P(k,

1996a) :

*T,P(k, n) \

P0

q0dq eik Õ n(q~q0)S

T,P(k, q) , (2)

where q) are the source functions for temperature and polarization and can be expressed in terms of metric, baryon,ST,P(k,

and photon perturbations (see Seljak & Zaldarriaga for their explicit expressions). By combining equations and (2)1996a (1)we can write the complete solutions for T (n), Q(n) and U(n). The only term that depends on the direction n in isequation (2)the exponential. The expressions for polarization can therefore be simpliÐed by noting that for the directions n near the pole,

and can be written in terms of second derivatives with respect to h, the two-dimensional projection of ncos (2/k) sin (2/

k)

onto the Ðxed (x, y)-plane perpendicular to the pole. This leads to

Q(n) \ DQ(n)P

d3k*P(k, n) , D

Q(n) \ (Lh

xLh

x[ Lh

yLh

y)+h~2 ,

(3)

U(n) \ DU(n)P

d3k*P(k, n) , D

U(n) \ 2 Lh

xLh

y+h~2 ,

where is the inverse two-dimensional Laplacian with respect to h and are the components of h in the Ðxed basis on+h~2 hx, h

ythe sky. Because n) depend only on the angle between the two vectors, one can expand them in Legendre series :*T,Pl(k,

*T,P(k, n) \;

l(2l ] 1)([i)l*

T,Pl(k)Pl(k) , (4)

8 SELJAK Vol. 482

where k \ k Æ n. The rms values for can be obtained by solving the Boltzmann equation in di†erential form &*T,Pl(k) (Bond

Efstathiou or the integral solution itself & Zaldarriaga1987) (Seljak 1996a).Each of the observable quantities T (n), Q(n), and U(n) can be expanded on a sphere into spherical harmonics or their

derivatives :

T (n) \ ;lm

aTlm

Ylm

(n) ,

Q(n) \ ;lm

aPlm

DQ(n)Y

lm(n) ,

(5)U(n) \ ;

lmaPlm

DU(n)Y

lm(n) ,

aT,Plm\ 4n([i)l

Pd3kY

lm* (k)*

T,Pl(k) .

The statistical properties of the coefficients follow from above,aT,Plm equation (5)

SaTlm* a

Tl{m{T \ dll{ dmm{

Pd3k*

Tl2 (k) 4 d

ll{ dmm{CTl,

(6)

SaPlm* a

Pl{m{T \ dll{ dmm{

Pd3k*

Pl2 (k) 4 d

ll{ dmm{CPl.

The cross-correlation between temperature and polarization is given by

SaTlm* a

Pl{m{T \ dll{ dmm{

Pd3k*

Tl(k)*

Pl(k) 4 d

ll{ dmm{ CCl. (7)

Because we are only interested in n near the pole, one can approximate the sphere locally as a plane, in which case, insteadof spherical decomposition, we may use plane-wave expansion. In this limit we replace with / d2lP(l)eil Õ h (and;

lmaPlm

Ylm

(n)analogously for temperature Di†erential operators and acting on eil Õ h become simple again andanisotropy1). D

Q(n) D

U(n)

bring out and respectively, where is the direction angle of the two-dimensional vector l with amplitude l.cos (2/l) sin (2/

l), /

lThis leads to

T (h) \ (2n)~2P

d2leil Õ hT (l) ,

Q(h) \ (2n)~2P

d2leil Õ hP(l) cos (2/l) , (8)

U(h) \ (2n)~2P

d2leil Õ hP(l) sin (2/l) .

T (l) and P(l) are the Fourier components of temperature anisotropy and polarization in l-space and have the statisticalproperties

ST (l)T *(l@)T \ (2n)2CTl

dD(l [ l@) ,

SP(l)P*(l@)T \ (2n)2CPl

dD(l [ l@) , (9)

ST (l)P*(l@)T \ (2n)2CCl

dD(l [ l@) ,

where is the Dirac d function, as opposed to the Kronecker d in the discrete case, and is assumed to be adD(l [ l@) Clcontinuous function obtained by interpolation from the discrete spectrum deÐned in equation (7).

To generate a map of temperature anisotropy and polarization, one proceeds in the following way. For each pair of vectorsl, [l on a discrete mesh, one diagonalizes the correlation matrix where l is theM11 \C

Tl, M22\ C

Pl, M12 \ M21\C

Cl,

amplitude of the vector l. One then generates from a normalized Gaussian distribution two pairs of random numbers andmultiplies them with the amplitudes given by the square root of the correlation matrix eigenvalues. Rotating this vector pairback to the original frame gives a realization of T (l) and P(l) (and their complex conjugates corresponding to [l), from whichfollow and Fourier transform of T , Q, and U back into the real space gives a map ofQ(l) \P(l) cos (2/

l) U(l) \ P(l) sin (2/

l).

these quantities in the sky in the small-scale limit, with the correct auto- and cross-correlations among all the quantities. Notethat this di†ers from the prescription given in & EfstathiouBond (1987).

3. TWO-POINT ESTIMATORS

The two-point correlations can be calculated either in angular or in frequency (Fourier) space. While the two are related viaa Fourier transform, there are certain advantages to the analysis performed in frequency space. In the Ðrst subsection belowwe explore this approach, while the correlation function approach and the comparison between the two are explored in thenext subsection.

1 A somewhat better correspondence between small-scale and large-scale expressions is achieved if one uses l] 1/2 as the amplitude of a wavevector thatcorresponds to C

l(Bond 1996).

No. 1, 1997 MEASURING POLARIZATION IN CMB 9

3.1. Power Spectrum AnalysisFrom a map of T (h), Q(h), and U(h) we can obtain their analogs in frequency space by Fourier transform,

X(l) \ / d2he~il Õ hX(h), where X stands for T , Q, or U. Using the expressions given in the previous section, one obtains thetwo-point functions of these quantities :

ST (l)T *(l@)T \ (2n)2CTl

dD(l [ l@) ,

ST (l)Q*(l@)T \ (2n)2CCl

cos (2/l)dD(l [ l@) ,

ST (l)U*(l@)T \ (2n)2CCl

sin (2/l)dD(l [ l@) , (10)

SQ(l)Q*(l@)T \ (2n)2CPl

cos2 (2/l)dD(l [ l@) ,

SU(l)U*(l@)T \ (2n)2CPl

sin2 (2/l)dD(l [ l@) .

We see that the correlations in polarization give rise to a very characteristic anisotropy pattern in l-space. This arises from thefact that polarization was generated not from a general mechanism but rather through a process of Thomson scattering,which cannot generate U and V components in the k-dependent frame deÐned in the previous section. This characteristicanisotropy can therefore be used to separate true signal from instrumental artifacts and foregrounds, which is discussed inmore detail in the next section. Alternatively, if foregrounds can be kept under control, one can use the characteristicanisotropy to separate scalar-induced polarization from that induced by vector or tensor modes, which do not obey the sameanisotropy pattern (° 5).

To estimate the sensitivity that it is possible to achieve in a measurement of polarization power spectra, let us assume themeasurements are given on a square grid of N pixels with a total solid angle ). The antenna beam smearing will be describedby b(l). In the case of single-dish observations with a Gaussian beam this is given by where is theb(l) \ exp (l2 p

b2/2), p

bGaussian size of the beam. In the case of interferometers b(l) is either 1 or 0, depending on whether the particular frequency isobserved or not. We will assume that di†erent measurements are uncorrelated by taking mesh spacing large enough to ignorecorrelations induced by Ðnite window & Magueijo In the case of single-dish experiments, each pixel in real(Hobson 1996).space has a noise contribution with rms noise amplitudes for temperature and both components of polarization,p

T, p

Prespectively (we assume for simplicity that Q and U are being measured equal amounts of time). We will also assume thatnoise is uncorrelated between di†erent pixels and between di†erent polarization components Q and U. This is only thesimplest possible choice, and more complicated noise correlations arise if all the components are obtained from a single set ofobservations. In the case of Brown polarization experiment the polarization measurements will be made by(Timbie 1996)rotating the antenna axis by 45¡, each time measuring directly the di†erence between the two orthogonal polarizations. Thiswould provide a direct measurement of Q and U components with no noise mixing between them. In the case of interferome-ters, each pixel in l-space is measured directly and the noise is uncorrelated between individual pixels in frequency space.Following we will introduce a pixel-independent measure of noise for single-dish experimentsKnox (1995), w

T,P~1 \ )pT,P2 /N

and for interferometers (see & Magueijo for expressions that relate to the receiver sensitivity).wT,P~1 \ p

T,P2 Hobson 1996 pT,PFor cross-correlation between temperature and polarization there are two simple cases to be considered. In one the cross-

correlation is being made with two di†erent maps, in which case noise is uncorrelated. In the second case both temperatureand polarization are obtained from the same experiment by adding and di†erentiating the two polarization states. In this casenoise in temperature and noise in polarization are related via In both cases noise in temperature is uncorrelatedp

T2 \ p

P2/2.

with noise in polarization components, so the Ðnal expressions are identical.The Ðrst step is to construct a discrete Fourier transform of the map, Using the observedXŒ (l) \ N~1 ; e~il Õ hX(h).

quantities and we can form power spectrum estimates. The simplest one is for temperature anisotropy, which forTΠ(l), QΠ(l) UΠ(l)single dish observations is given by

CŒTl

\C;l

)N

l[TΠ(l)TΠ*(l)][ w

T~1Db~2(l) , (11)

where the term b~2(l) accounts for beam smearing and is the number of independent modes around l. The expression forNlinterferometers is the same without the beam smearing term and summing only over the observed modes in frequency space.

We will only present expressions for single dish here, as the modiÐcation for interferometers is obvious. Each mode is aTŒ (l)complex random variable with 0 mean and variance The variance on the estimator for a single pair ofC

Tlb2(l) ] w

T~1. CŒ

Tlmodes is therefore If we average over modes the variance is reduced by There are l2)TΠ(l), TΠ([l) CTl

b2(l) ] wT~1. N

lN

l~1@2.

d ln l/2n modes of amplitude l in an interval d ln l, which leads to the variance

Cov (CTl2 )\ C

Tl2 4n

l2) d ln lM1 ] [w

TC

Tlb2(l)]~1N2 , (12)

where If there is more than one Ðeld, the variance decreases in inverse proportion toCov (XX@)4S(XŒ [ SXŒ T)(XŒ @[ SXŒ @T)T.the square root of the number of Ðelds & Magueijo In the limit of large sky coverage, it reduces to the(Hobson 1996).expressions given by and et al.Knox (1995) Jungman (1996).

In the case of polarization there are several estimators that one can form. The simplest one is given by

CŒPl

\G)N

l;l

[QΠ(l)QΠ*(l)] UΠ(l)UΠ*(l)][ 2wP~1Hb~2(l) . (13)

10 SELJAK Vol. 482

The single-mode pair variance for this estimator is If noise is a dominant contributor to[CPl2 b4(l) ] 2C

Plb2(l)w

P~1 ] 4w

P~2]1@2.

the variance, as is likely to be the case for polarization given the small overall amplitude of the signal, then this estimator is farfrom optimal. The optimal estimator is

CŒPl

\C)N

l;l

oQΠ(l) cos (2/l) ] UΠ(l) sin (2/

l) o2[ w

P~1Db~2(l) , (14)

which has single-mode pair variance In the limit of noise-dominated variance this is 2 times smaller than theClb2(l)] w

P~1.

variance of the estimator in Averaging over all modes in d ln l givesequation (13).

Cov (CPl2 ) \ C

Pl2 4n

l2) d ln lM1 ] [w

PC

Plb2(l)]~1N2 . (15)

Finally, for cross-correlation the optimal estimator is

CŒCl

\ )2N

l;l

M[QΠ(l)TΠ*(l) ] QΠ*(l)TΠ(l)] cos (2/l)] [UΠ(l)TΠ*(l) ] UΠ*(l)TΠ(l)] sin (2/

l)Nb~2(l) , (16)

which has a single-mode pair variance As before, averaging over thesMCCl2 b4(l) ] [C

Tlb2(l)] w

T~1][C

Plb2(l) ] w

P~1]N/2t1@2.

modes reduces this variance in inverse proportion to the square root of the number of modes, and gives

Cov (CCl2 ) \ C

Cl2 2n

l2) d ln lG1 ] [C

Tlb2(l) ] w

T~1][C

Plb2(l)] w

P~1]

CCl2 b4(l)

H. (17)

For a study of cosmological parameters one also needs to include the covariance elements between various power spectrumestimators. These are given by

Cov (CTl

CPl) \ C

Cl2 (l)

4nl2) d ln l

,

Cov (CCl

CTl

) \ CCl

CTl

4nl2) d ln l

M1 ] [wT

CTl

b2(l)]~1N , (18)

Cov (CCl

CPl) \ C

ClC

Pl4n

l2) d ln lM1 ] [w

PC

Plb2(l)]~1N .

3.2. Correlation Function AnalysisIn this subsection we explore the correlation function analysis of CMB anisotropies. Taking the Fourier transforms of the

power spectra leads to the following correlation functions :

ST (0)T (h)T \P l dl

2nC

Tlb2(l)J0(lh) ,

SQ(0)Q(h)T \P l dl

4nC

Plb2(l)[J0(lh) ] cos (4/)J4(lh)] ,

SU(0)U(h)T \P l dl

4nC

Plb2(l)[J0(lh) [ cos (4/)J4(lh)] ,

(19)

SQ(0)U(h)T \P l dl

4nC

Plb2(l) sin (4/)J4(lh) ,

ST (0)Q(h)T \P l dl

4nC

Clb2(l) cos (2/)J2(lh) ,

ST (0)U(h)T \P l dl

4nC

Clb2(l) sin (2/)J2(lh) ,

where / is the direction angle of h and are the Bessel functions of order n. Although the characteristic anisotropy isJn(x)

present also in the correlation functions, it is more complicated, because there are actually two independent correlationfunctions for polarization,

CP1(h)\

P l dl4n

CPl

b2(l)J0(lh) ,

(20)

CP2(h)\

P l dl4n

CPl

b2(l)J4(lh) ,

No. 1, 1997 MEASURING POLARIZATION IN CMB 11

FIG. 1.ÈPower spectra of polarization (a), temperature-polarization cross correlation (b), temperature (c) and correlation coefficient (d). The models arestandard CDM (solid curve), open CDM (dotted curve) and reionized standard CDM with optical depth of 0.2 (dashed curve).

both of which of course depend on the same underlying power spectrum. Moreover, the prior becomes a set ofCPl

[ 0integral constraints in real space, which is more difficult to impose on the estimators obtained from the data. All this arguesfor Fourier space analysis as the method of choice in the case of polarization.

The expressions above agree with those derived by et al. and which for QQ and UU areCoulson (1994) Kosowsky (1996),also only valid in the small-scale limit, but for which this limit has not been consistently applied to all the steps, so that theirÐnal expressions look much more complicated than they actually are. They contain a term involving double summation over

in the /-dependent term, which, as shown in the Appendix, reduces to the expressions above if one consistently*Pl

, *Pl{applies the small-scale limit to their expression. All the information about polarization is therefore contained in andC

PlC

Cl.

Note that taking the Fourier transform of the QQ or UU correlation function does not result in the power spectrum of butCPlhas an additional /-dependent term that involves a double integral of and over l@ and h. Although generallyJ0(lh) J4(l@h)

smaller than this integral does not vanish in general, hence the appearance of such terms in expressions by et al.CPl

, Coulsonand This is of course not the optimal way to obtain the power spectrum from the correlation(1994) Kosowsky (1996).

function. To obtain an estimate of the underlying power spectrum, it is better to work in the Fourier domain directly,following the methods given in the ° 3.1.

3.3. Predicting Polarization from Temperature MapsOne can use the measured temperature maps to predict the polarization pattern. The estimator is

QΠ(l) \ aT (l) cos (2/l) , UΠ(l) \ aT (l) sin (2/

l) , (21)

where by minimizing the variance

S[QΠ(l) [ Q(l)]2T \ cos2 (2/l)(a2C

Tl[ 2aC

Cl] C

Pl) , (22)

and similarly for U, one Ðnds so that the fractional variance in the estimator isa \ CCl/C

Tl,

S[QΠ(l) [ Q(l)]2TQ(l)2 \ 1 [ Corr (T , P)

l2 . (23)

12 SELJAK Vol. 482

The correlation coefficient is deÐned as Corr (T , shows that the correlation coefficientP)l\ C

Cl/(C

TlC

Pl)1@2. Figure 1d

typically ranges between [ 0.5 and 0.5, and so the fractional variance will be at best around 0.8 or so in l-space and evenlarger than that in real space, where one averages over positive and negative cross-correlations in the power spectrum. This isnot very impressive in terms of predicting where to look for large polarization amplitude, although in a statistical sense one isstill much more likely to Ðnd a high signal at a high-p peak in the temperature map than at a random point et al.(Coulson

On large angular scales (l \ 10) the correlation coefficient can be much larger and approaches unity in some models, so1994).that the expressions in this limit would actually give a good correspondence between observed and predicted polarization.Unfortunately, the amplitude of polarization is extremely small on these scales, and there is little hope to measure it in thenear future even with the help of this ““ matching Ðlter ÏÏ technique.

4. REMOVAL OF FOREGROUNDS AND OTHER SYSTEMATICS

Although several Galactic and extragalactic foregrounds are signiÐcant in the case of temperature measurements, only fewof these are polarized and need to be considered for polarization measurements. Radiation from Earth atmosphere andbremsstrahlung emission are not polarized at millimeter wavelengths and need not be discussed further (although atmo-spheric emission does have an e†ect by increasing the e†ective temperature of the receiver and adding a Ñuctuating o†set).

Extragalactic radio sources have synchrotron radiation as the dominant emission mechanism and can be 20% polarized.Their contribution to the polarization signal will be similar to their contribution to the temperature anisotropy signal, and soanalysis of point-source e†ects on the CMB can be directly applied to the case of polarization as well. This is discussed inmore detail by Franceschini et al. and & Efstathiou Point-source contamination depends on the(1989, 1991) Tegmark (1996).observed frequency, angular scale, and Ñux cut above which point sources can be identiÐed and eliminated. A Poissondistribution produces a white-noise spectrum, and at large angular scales radio point sources do not pose a signiÐcantproblem. For example, on angular scales above 1¡ their contribution is less than 1 kK at 30 GHz which is(Timbie 1996),below the expected amplitude of the signal and is even less than that at higher frequencies. On smaller angular scales pointsources become more important and more ambitious Ñux cuts are needed, which limits the area of the sky that can beobserved. While in the case of temperature anisotropy this can be the main limitation of an experiment (and indeed of thewhole CMB Ðeld), in the case of polarization we do have an additional constraint that allows us to separate cosmologicallyinduced signal from the foregrounds. This is discussed in more detail below.

On large angular scales the main foreground contribution comes from our Galaxy, where both dust and synchrotronemission can be polarized. Dust emission in the far-infrared is polarized up to 10% et al. The contribution(Hildebrand 1995).of dust to the lower frequency channels (where most HEMT-based polarization measurements are being planned) is small. Atfrequencies below 100 GHz the most important source of polarization is likely to be Galactic synchrotron emission. Its linearpolarization can reach 70%, so that polarization amplitudes of 50 kK are expected around 30 GHz & Spoelstra(Cortiglioni

This drops signiÐcantly at higher frequencies, and only a few microkelvins of synchrotron contamination in polariza-1995).tion is expected around 100 GHz Nevertheless, this is of the same order as the expected signal, so that some(Timbie 1996).further foreground rejection is needed. One possibility is to use only clean parts of the sky where synchrotron emission is low,such as at high Galactic latitudes. In addition, multifrequency CMB observations can be used to remove the foregrounds,which, in the case of only one important foreground with approximately known frequency dependence, can be very e†ective

et al. Accuracy of 1 kK can be achieved with only two frequency channels if the noise level is around 1 kK(Brandt 1994).pixel~1 Another possibility is multifrequency removal using the more sensitive temperature maps. This would(Timbie 1996).be a useful strategy, for example in the case of the Planck satellite, where only lower frequency channels will have polarizationcapabilities but all frequency channels could be used to determine the local contribution of various foregrounds. This way onecould e†ectively remove multicomponent foregrounds from polarization even if only a few channels actually measured it.

While each of the foregrounds above can in principle be removed from the data, in practice this may not always be possibleat levels of 1 kK, and it would be useful to have an additional test of the presence of cosmological signal. Characteristicanisotropy of polarization in Fourier space provides such a test and gives a unique signature of polarization induced by scalarperturbations. To test whether the signal is cosmological, one needs to compare the quantities

E(l) \ Q(l) cos (2/l) ] U(l) sin (2/

l) (24)

and

B(l) \ [Q(l) sin (2/l) ] U(l) cos (2/

l) . (25)

The Ðrst quantity contains all of the polarization signal, and its estimator is given in while the second quantityequation (14),vanishes even in the presence of cosmological polarization induced by scalar perturbations. This is true not just statistically,but for each Fourier mode individually. Most of the foregrounds should contribute on average the same amount to bothvariables. This is certainly valid for uncorrelated point sources, but even in the case of synchrotron and dust emission thealignment is preferentially determined by magnetic Ðelds, which are not scalar in nature and will not exhibit the characteristicanisotropy in Fourier space. Hence the di†erence between and can be taken as a measure of the cosmological signal asEŒ BŒcompared to the foregrounds and/or instrumental o†sets. If foregrounds and not noise are expected to be the main limitation,then one may subtract the power spectrum of from the one for in and no further foreground removal isEŒ BŒ equation (14),needed. One could therefore use this technique to measure polarization even at frequencies below 50 GHz, where foregroundcontribution is large but could be averaged over if a sufficient number of channels are being measured. Note that this test doesnot depend on the temperature anisotropies at all and can be applied directly to the measurements of Q and U. To obtain a

No. 1, 1997 MEASURING POLARIZATION IN CMB 13

statistically signiÐcant measure of polarization, one only needs to show that in an rms sense is larger than If one is noiseEŒ BŒ .and not foreground limited, then it is more advantageous to use the optimal estimator in because subtractingequation (14),the power spectrum of from the optimal estimator for leads to an increase in noise by 21@2. In practice the actual analysisBŒ EŒwill depend on the level of foregrounds and other systematics (e.g., sidelobe pickup) relative to noise, and di†erent estimatorsmust be tested for consistency, but it is important to note that in the case of polarization we have the possibility of using acombination of Stokes parameters in which foregrounds can be separated from the cosmological signal, something thatcannot be achieved in the temperature measurements alone.

5. TENSOR POLARIZATION

Discussion in applies only if polarization is produced by scalar perturbations. While this is certainly a valid approx-° 4imation on small angular scales (l [ 100), on larger scales one may be able to detect polarization from nonscalar pertur-bations, induced either by vector or by tensor modes. The latter are of particular interest, because they are expected to bepresent in several inÑationary based models, although only on large angular scales and with rather small amplitudes

et al. Defect models also predict production of both tensor and vector perturbations. In the case of tensor(Crittenden 1993).perturbations, temperature anisotropy and the two Stokes parameters can be written in the small-scale limit as (Bond 1996 ;Kosowsky 1996)

T (T)(n) \P

d3k(1 [ k2)*T(T`)(k, k) ,

Q(T)(n) \P

d3k[(1] k2)*P(T`)(k, k) cos (2/

k) [ 2k*

P(TC)(k, k) sin (2/

k)] , (26)

U(T)(n) \P

d3k[(1] k2)*P(T`)(k, k) sin (2/

k) ] 2k*

P(TC)(k, k) cos (2/

k)] ,

where k) and k) are the two independent polarization states of a gravity wave with equal rms amplitudes. For*P(T`)(k, *

P(TC)(k,

convenience we deÐned the orientation of the two polarization states in the plane perpendicular to so that the localkü ,x-direction points in the direction of the pole, in which case only k) contributes to the temperature in the small-scale*

T(T`)(k,

limit. Expectation values for k) and k) can be calculated just as in the scalar case by expanding them in a*P(`)(k, *

P(TC)(k,

Legendre series and solving a system of Boltzmann equations et al. or the integral solution &(Crittenden 1993) (ZaldarriagaSeljak Because of additional k terms in a more convenient set of variables is obtained by eliminating the1996). equation (26),explicit k dependence (Kosowsky 1996) :

Bl1,v\ 2

2l ] 1[(l ] 1)*

P,l`1(Tv) ] l*P,l~1(Tv) ] ,

(27)

Bl2,v\ 1

2l ] 1C(l ] 1)(l] 2)

2l ] 3*

P,l`2(Tv) ] 26l3 ] 9l2[ l [ 2(2l [ 1)(2l ] 3)

*P,l(Tv) ] (l [ 1)l

2l [ 1*

P,l~2(Tv)D

,

where v stands for ] and ] and all the variables explicitly depend on k.One can now follow the same steps as in the case of scalar perturbations, which transform the angle into The/

k/l.

variables E(T) and B(T) deÐned in equations and (where the superscript T indicates that these are produced by tensor(24) (25)modes) are independent of the azimuthal angle, and the two tensor components decouple, so that k) contributes only*

P(T`)(k,

to E(T) and k) contributes only to B(T). Their power spectra can be expressed in terms of integrals over as*P(TC)(k, B

l1, B

l2

SB(T)(l)B(T)*(l@)T \ (2n)2d(l [ l@)P

d3k oBl1 o2(k) ,

SE(T)(l)E(T)*(l@)T \ (2n)2d(l [ l@)P

d3k oBl2 o2(k) , (28)

SE(T)(l)B(T)*(l@)T \ 0 .

The cross-correlation term vanishes because the two tensor polarization states are independent. The variable B(T) does notvanish in the case of tensor perturbations, and its power spectrum di†ers from the power spectrum of E(T). Note that di†erentcombinations of Q and U will result in di†erent power spectra, which can always be expressed in terms of the two deÐnedabove. Just as the cross term between E(T) and B(T) vanishes, so does the cross-correlation term between T (T) and B(T). There isonly one power spectrum present in the case of temperature-polarization cross-correlation :

ST (T)(l)B(T)*(l@)T \ 0 , ST (T)(l)E(T)*(l@)T \ (2n)2d(l[l@)P

d3k*Tl(T)(k)B

l2(k) . (29)

Detailed calculations of these spectra have been presented elsewhere & Zaldarriaga(Seljak 1996b).

6. MODEL PREDICTIONS

Instead of the power spectrum we will use the quantity which gives the contribution to the variance perCl

l(l ] 1)Cl/2n,

logarithmic interval of l. This is a familiar quantity in the case of temperature anisotropies, where its broadband average is

14 SELJAK Vol. 482

approximately Ñat up to the damping scale. Predictions for and are given in Figures forl(l ] 1)CPl/2n l(l ] 1)C

Cl/2n 1aand 1b

a variety of cosmological models. For comparison we also plot the usual in as well as the correlationl(l ] 1)CTl

/2n Figure 1c,coefficient in All the model predictions have been computed using the CMBFAST package by & Zaldar-Figure 1d. Seljakriaga One can see that typically the models predict very little polarization on large angular scales, below l D 200. On(1996a).smaller angular scales most of the models predict polarization at the level of 5È10 kK. There are several characteristic featuresof interest in this regime. The most important one is that the acoustic peaks are narrower than the corresponding ones intemperature anisotropy. One can understand this with the help of the tight coupling approximation & Sugiyama(Hu 1995 ;

& Harari The dominant sources of temperature anisotropy are intrinsic photon anisotropySeljak 1994 ; Zaldarriaga 1995).and velocity Both terms oscillate but are out of phase with each other. This means that they partially cancel each(*0) (*1).other, and oscillations in the temperature anisotropy are less pronounced than they would be if only one term were

contributing. The dominant source of polarization is the photon velocity so the oscillations are more pronounced than in*1,the case of temperature anisotropy. These oscillations are even more pronounced in the case of temperature-polarizationcross-correlation, which can be either positive or negative. Another characteristic of polarization is that it is not sensitive tothe integrated Sachs-Wolfe term. This term is responsible for increase in the temperature anisotropies at low l, as in modelswith cosmological constant, curvature, or if recombination occurs close to the matter-radiation equality, during whichgravitational potential is changing with time. Another class of such models are topological defect models, where a small-lspectrum is dominated by a late time integrated Sachs-Wolfe e†ect. A measurement of cross-correlation on large scales willput a signiÐcant constraint on such models.

One of the parameters of special importance for polarization is the optical depth to Thomson scattering. As photonspropagate through the universe, they scatter o† free electrons, which were ionized by UV light either from an early generationof stars or from quasars. Current limits indicate that the universe was mostly ionized up to the redshift 5, which results in anoptical depth of the order of 1% in a standard CDM universe and somewhat larger in open or high-baryon models. It is likelythat the reionization did not occur much earlier, so that the optical depth would exceed unity, because then it would suppressCMB anisotropies on small angular scales, in contrast to recent observational data et al. et al.(NetterÐeld 1995 ; Scott 1996).The exact epoch of reionization remains unknown, however, and its determination would provide an important constraint tothe models of galaxy formation. Temperature anisotropies alone will not be able to constrain this epoch signiÐcantly, becauseeven in the most optimistic scenario sensitivity to optical depth q is around 10%È20% et al. This is because(Jungman 1996).reionization is degenerate with the amplitude of Ñuctuations, which can be only broken at low l, where cosmic variance islarge. Polarization can help here both because reionization introduces new structure and also simply because the cosmicvariance is reduced as more independent realizations are observed. As shown in the e†ect of reionization is toFigure 1,increase somewhat the amplitude of polarization at low l, but not by much, and the amplitude still remains below 1È2 kK. Thee†ect is better seen in the cross-correlation spectrum and in the corresponding correlation coefficient. The latter clearlydisplays the rich structure at low l that allows one to determine the epoch of reionization and the integrated optical depth

On smaller angular scales polarization amplitude decreases with optical depth just like the temperature(Zaldarriaga 1996).anisotropy, and the ratio of the two remains constant & Efstathiou(Bond 1987).

presents a more quantitative estimate of sensitivity in the case of satellites, assuming standard CDM as theFigure 2underlying model. The middle curve shown is the underlying theoretical model, while the two curves above and below show 1standard deviation of the reconstructed spectrum from the true model. The variances were obtained using equations (15) and(17), assuming 50% sky coverage ()\ 2n) and d ln l \ 0.2. We adopted three di†erent noise characteristics. The mostoptimistic possibility is kK in beam, which could easily be achieved by the Planck satellite with theirw

P~1@2 \ 0.1 0¡.2

FIG. 2.È(a) Variance in polarization and (b) cross-correlation power spectrum for a satellite with noise characteristic We assumed 50% skywP\w

T/2.

coverage and beam in the most optimistic case). The spectra were averaged over a 20% band in l, and the bands shown are 1 standard deviation0¡.3 (0¡.2above and below the underlying model, taken to be COBE-normalized standard CDM.

No. 1, 1997 MEASURING POLARIZATION IN CMB 15

bolometric detectors. The intermediate model assumes kK and beam, which could be feasible with the MAPwP~1@2 \ 0.2 0¡.3

satellite by combining the most sensitive channels. The third model is the most conservative one and assumes kKwP~1@2 \ 0.3

and All of the sensitivities assume 1 year of observation, and longer observation periods would reduce the noise0¡.3.accordingly. From we see that only the most optimistic model is capable of constraining the polarization powerFigure 2spectrum signiÐcantly. On the other hand, for the cross-correlation spectrum the situation is much better, and all of theassumed models will give some positive detection, although of course with lower noise levels one will be able to extend this tomuch smaller angular scales. This di†erence between polarization and cross-correlation is to be expected, because noise intemperature is lower and because the cross-correlation power spectrum has more power than the polarization powerspectrum itself. Although more detailed calculations are needed to estimate the sensitivity for actual satellites et(Zaldarriagaal. it is clear that reducing the noise leads to a signiÐcant improvement in the sensitivity to polarization.1997),

7. CONCLUSIONS

Polarization in the cosmic microwave background promises to become the new testing ground for processes in the earlyuniverse, quite independent of the measurements of temperature anisotropies. The spectrum of polarization, while lower insignal than the temperature anisotropies, can be more sensitive to certain parameters such as reionization or gravity waves.Even for the determination of the standard parameters, polarization provides some advantages ; for example, acousticoscillation peaks are much more prominent and so can be more easily detected. In addition to the spectrum of polarization,one can also determine the spectrum of temperature-polarization cross-correlation. Two additional spectra can help to breaksome of the degeneracies present in the estimation of cosmological parameters. This is specially important for those param-eters whose precision is limited by cosmic variance, because polarization provides additional independent realizations ofinitial conditions in the universe.

Even though measurements in polarization are not likely to achieve the same level of precision as in the case of temperatureanisotropies, polarization still has some advantages that may prove crucial if the amplitude and complexity of Galacticforegrounds and extragalactic point sources have been signiÐcantly underestimated. Multifrequency subtraction is simpler inthe case of polarization, because fewer foregrounds are polarized and need to be modeled. More important, polarizationinduced by scalar perturbations has a unique signature in Fourier space, and by exploiting this, one may separate cosmo-logical signal from other sources of polarization. Although the analysis in this paper has been limited to small scales, it hasrecently been shown that the same property is also valid in the more general all-sky analysis & Seljak(Zaldarriaga 1996 ;

Kosowsky, & Stebbins This signature would be especially important if the level of foregrounds isKamionkowski, 1996).signiÐcantly larger than expected. While one would not be able to remove the foregrounds from the temperature maps, acombination of Q and U Stokes parameters would allow one to subtract statistically the e†ects of foregrounds in the case ofpolarization. This technique would be feasible both for interferometer measurements or for measurements with a largecoverage of the sky such as the forthcoming satellite missions. The sensitivity of satellites will be sufficient for an unambiguousdetermination of polarization, and indeed polarization may prove to be crucial to break some of the degeneracies present inthe parameter reconstruction from the temperature anisotropies alone et al. Finally, if foregrounds can be(Zaldarriaga 1997).controlled, then a unique signature of tensor (or vector) perturbations could be directly observed, although this would requirean exquisite understanding of noise properties, systematics, and foregrounds at the level of 0.5 kK. Given the unique nature ofinformation present in the microwave background and its simple linear dependence on the underlying cosmological param-eters, it is important to explore it at its maximum, which certainly includes polarization as one of its main components.

I would like to thank Arthur Kosowsky, Douglas Scott, David Spergel, Martin White, and especially Matias Zaldarriagafor useful discussions.

APPENDIX

In this Appendix we show how the small-scale limit of correlation functions QQ and UU follows from the(eq. [19])expression derived by et al. We start from their expression (correcting for missing factors ofCoulson (1994). 12) :

SQ(0)Q(h)T \;l

2l ] 18n

CC

PlP

l(cos h) ] cos (4/)

2C

PPlP

l4 (cos h)

D,

CPPl

\ (l [ 4) !(l ] 4) !

;l{

(2l@ ] 1)P

k2 dk*Pl{*Pl

al{l4 , (A1)

where is the Legendre polynomial and the associated Legendre function of 4th order. CoefficientsPl

Pl4 a

ll{4 \ /~11 dxPl(x)P

l{4(x)have a closed-form expression and in the large-l limit they peak at l \ l@. More important, is a rapidly(Coulson 1994), *

loscillating function of k, and for the integral over k leads to almost complete cancellation of Thus in thisl D l@ / k2 dk*Pl{*Pl

.limit one can write

CPPl

\ 2l(l [ 1)(l [ 2)(l[ 3)(l [ 4) !(l] 4) !

CPl

, (A2)

16 SELJAK

where we used the l\ l@ closed form for Furthermore, in the l] O limit Legendre functions can be writtenall{4 (Coulson 1994).

as where is the Bessel function of order m. Combining all the expressions leads toPlm (cos h)\ J

m(lh)l~m(l ] m) !/(l[ m) !, J

m(x)

the SQ(0)Q(h)T correlation function given in Other correlation functions in can be derived fromequation (19). equation (19)the expressions in et al. using similar manipulations. Numerical results presented in & SeljakCoulson (1994) Zaldarriaga

show that the small-scale approximation derived here is an excellent approximation everywhere except at very small l(1996)(l [ 20).

REFERENCESJ. R. 1996, in Cosmology and Large Scale Structure, ed. R. Schae†erBond,

et al. (Amsterdam: Elsevier Science), 469J. R., & Efstathiou, G. 1987, MNRAS, 226,Bond, 655J. R., et al. 1994, Phys. Rev. Lett., 72,Bond, 13

W. N., et al. 1994, ApJ, 424,Brandt, 1S. 1960, Radiative Transfer (New York :Chandrasekhar, Dover)

S. & Spoelstra, T. A. Th. 1995, A&A, 302,Cortiglioni, 1D. 1994, Ph.D. thesis, PrincetonCoulson, Univ.D., Crittenden, R. G., & Turok, N. 1994, Phys. Rev. Lett., 73,Coulson,

2390R. G., Coulson, D., & Turok, N. 1995, Phys. Rev. D, 52,Crittenden, 5402R. G., Davis, R., & Steinhardt, P. 1993, ApJ, 417,Crittenden, L13

A. 1989, ApJ, 344,Franceschini, 351991, A&AS, 89,ÈÈÈ. 285R. A., Polnarev, A. G., & Coles, P. 1994, MNRAS, 266,Frewin, L21

R. H., et al. 1995, in ASP Conf. Ser. 73, Airborne AstronomyHildebrand,Symposium on the Galactic Ecosystem: From Gas to Stars to Dust, ed.M. R. Haas, J. A. Davidson, & E. F. Erickson (San Francisco : ASP), 397

M. P., & Magueijo, J. 1996, preprint,Hobson, astro-ph/9603064W., & Sugiyama, N. 1995, ApJ, 436,Hu, 456

G., Kamionkowski, M., Kosowsky, A., & Spergel, D. N. 1996,Jungman,Phys. Rev. D, 54, 1332

N. 1983, MNRAS, 202,Kaiser, 11691992, ApJ, 388,ÈÈÈ. 272

M., Kosowsky, A., & Stebbins, A. 1996, preprint,Kamionkowski, astro-ph/9611125

L. 1995, Phys. Rev. D, 52,Knox, 4307A. 1996, Ann. Phys., 246,Kosowsky, 49C. B., Jarosik, N., Page, L., Wilkinson, D., & Wollack, E. 1995,NetterÐeld,

ApJ, 445, L69A. G. 1985, Soviet Astron., 29,Polnarev, 607

M. 1968, ApJ, 153,Rees, L1P. F., et al. 1996, ApJ, 461,Scott, L1U. 1994, ApJ, 435,Seljak, L87U., & Zaldarriaga, M. 1996a, ApJ, 469,Seljak, 4371996b, preprint,ÈÈÈ. astro-ph/9609169

M., & Efstathiou, G. 1996, MNRAS, 281,Tegmark, 1297P. 1996,Timbie, unpublishedE. J., et al. 1993, ApJ, 419,Wollack, L49

M. 1996, preprint,Zaldarriaga, astro-ph/9608050M., & Harari, D. 1995, Phys. Rev. D, 52,Zaldarriaga, 3276M., & Seljak, U. 1996, preprint,Zaldarriaga, astro-ph/9609170M., Spergel, D. N., & Seljak, U. 1997, preprintZaldarriaga, astro-

ph/9702157