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A CMB Polarization Primer

Wayne Hu1

Institute for Advanced Study, Princeton, NJ 08540

Martin WhiteEnrico Fermi Institute, University of Chicago, Chicago, IL 60637

ABSTRACT

We present a pedagogical and phenomenological introduction to the study of cosmicmicrowave background (CMB) polarization to build intuition about the prospects andchallenges facing its detection. Thomson scattering of temperature anisotropies on thelast scattering surface generates a linear polarization pattern on the sky that can besimply read off from their quadrupole moments. These in turn correspond directlyto the fundamental scalar (compressional), vector (vortical), and tensor (gravitationalwave) modes of cosmological perturbations. We explain the origin and phenomenologyof the geometric distinction between these patterns in terms of the so-called electricand magnetic parity modes, as well as their correlation with the temperature pattern.By its isolation of the last scattering surface and the various perturbation modes, thepolarization provides unique information for the phenomenological reconstruction of thecosmological model. Finally we comment on the comparison of theory with experimentaldata and prospects for the future detection of CMB polarization.

1Alfred P. Sloan Fellow

1

1. Introduction

Why should we be concerned with the polarizationof the cosmic microwave background (CMB) anisotro-pies? That the CMB anisotropies are polarized is afundamental prediction of the gravitational instabilityparadigm. Under this paradigm, small fluctuations inthe early universe grow into the large scale structure wesee today. If the temperature anisotropies we observeare indeed the result of primordial fluctuations, theirpresence at last scattering would polarize the CMB an-isotropies themselves. The verification of the (partial)polarization of the CMB on small scales would thus rep-resent a fundamental check on our basic assumptionsabout the behavior of fluctuations in the universe, inmuch the same way that the redshift dependence of theCMB temperature is a test of our assumptions aboutthe background cosmology.

Furthermore, observations of polarization provide animportant tool for reconstructing the model of the fluc-tuations from the observed power spectrum (as distinctfrom fitting an a priori model prediction to the observa-tions). The polarization probes the epoch of last scat-tering directly as opposed to the temperature fluctua-tions which may evolve between last scattering and thepresent. This localization in time is a very powerfulconstraint for reconstructing the sources of anisotropy.Moreover, different sources of temperature anisotropies(scalar, vector and tensor) give different patterns inthe polarization: both in its intrinsic structure and inits correlation with the temperature fluctuations them-selves. Thus by including polarization information, onecan distinguish the ingredients which go to make up thetemperature power spectrum and so the cosmologicalmodel.

Finally, the polarization power spectrum providesinformation complementary to the temperature powerspectrum even for ordinary (scalar or density) pertur-bations. This can be of use in breaking parameter de-generacies and thus constraining cosmological parame-ters more accurately. The prime example of this is thedegeneracy, within the limitations of cosmic variance,between a change in the normalization and an epoch of“late” reionization.

Yet how polarized are the fluctuations? The degreeof linear polarization is directly related to the quadru-pole anisotropy in the photons when they last scatter.While the exact properties of the polarization dependon the mechanism for producing the anisotropy, severalgeneral properties arise. The polarization peaks at an-

gular scales smaller than the horizon at last scatteringdue to causality. Furthermore, the polarized fractionof the temperature anisotropy is small since only thosephotons that last scattered in an optically thin regioncould have possessed a quadrupole anisotropy. The frac-tion depends on the duration of last scattering. For thestandard thermal history, it is 10% on a characteristicscale of tens of arcminutes. Since temperature aniso-tropies are at the 10−5 level, the polarized signal is at(or below) the 10−6 level, or several µK, representing asignificant experimental challenge.

Our goal here is to provide physical intuition for theseissues. For mathematical details, we refer the readerto Kamionkowski et al. (1997), Zaldarriaga & Seljak(1997), Hu & White (1997) as well as pioneering workby Bond & Efstathiou (1984) and Polnarev (1985). Theoutline of the paper is as follows. We begin in §2 byexamining the general properties of polarization forma-tion from Thomson scattering of scalar, vector and ten-sor anisotropy sources. We discuss the properties ofthe resultant polarization patterns on the sky and theircorrelation with temperature patterns in §3. These con-siderations are applied to the reconstruction problem in§4. The current state of observations and techniques fordata analysis are reviewed in §5. We conclude in §6 withcomments on future prospects for the measurement ofCMB polarization.

2. Thomson Scattering

2.1. From Anisotropies to Polarization

The Thomson scattering cross section depends on po-larization as (see e.g. Chandrasekhar 1960)

dσT

dΩ∝ |ǫ · ǫ′|2 , (1)

where ǫ (ǫ′) are the incident (scattered) polarization di-rections. Heuristically, the incident light sets up oscilla-tions of the target electron in the direction of the elec-tric field vector E, i.e. the polarization. The scatteredradiation intensity thus peaks in the direction normalto, with polarization parallel to, the incident polariza-tion. More formally, the polarization dependence of thecross section is dictated by electromagnetic gauge in-variance and thus follows from very basic principles offundamental physics.

If the incoming radiation field were isotropic, orthog-onal polarization states from incident directions sepa-rated by 90 would balance so that the outgoing radia-tion would remain unpolarized. Conversely, if the inci-

2

Quadrupole Anisotropy

Thomson Scattering

e–

Linear Polarization

ε'

ε'

ε

Fig. 1.— Thomson scattering of radiation with a quad-rupole anisotropy generates linear polarization. Bluecolors (thick lines) represent hot and red colors (thinlines) cold radiation.

dent radiation field possesses a quadrupolar variation inintensity or temperature (which possess intensity peaksat 90 = π/2 separations), the result is a linear polar-ization of the scattered radiation (see Fig. 1). A reversalin sign of the temperature fluctuation corresponds to a90 rotation of the polarization, which reflects the spin-2 nature of polarization.

In terms of a multipole decomposition of the ra-diation field into spherical harmonics, Y m

ℓ (θ, φ), thefive quadrupole moments are represented by ℓ = 2,m = 0,±1,±2. The orthogonality of the spherical har-monics guarantees that no other moment can generatepolarization from Thomson scattering. In these spheri-cal coordinates, with the north pole at θ = 0, we call aN-S (E-W) polarization component Q > 0 (Q < 0) anda NE-SW (NW-SE) component U > 0 (U < 0). Thepolarization amplitude and angle clockwise from northare

P =√

Q2 + U2, α =1

2tan−1(U/Q) . (2)

Alternatively, the Stokes parameters Q and U repre-sent the diagonal and off diagonal components of thesymmetric, traceless, 2 × 2 intensity matrix in the po-

larization plane spanned by (eθ, eφ),

E∗

i Ej −1

2δijE

2 ∝ Qσ3 + Uσ1 , (3)

where σi are the Pauli matrices and circular polarizationis assumed absent.

If Thomson scattering is rapid, then the randomiza-tion of photon directions that results destroys any quad-rupole anisotropy and polarization. The problem of un-derstanding the polarization pattern of the CMB thusreduces to understanding the quadrupolar temperaturefluctuations at last scattering.

Temperature perturbations have 3 geometrically dis-tinct sources: the scalar (compressional), vector (vorti-cal) and tensor (gravitational wave) perturbations. For-mally, they form the irreducible basis of the symmetricmetric tensor. We shall consider each of these belowand show that the scalar, vector, and tensor quadru-pole anisotropy correspond to m = 0,±1,±2 respec-tively. This leads to different patterns of polarizationfor the three sources as we shall discuss in §3.

m=0

v

Scalars (Compression)

hot

hot

cold

Fig. 2.— The scalar quadrupole moment (ℓ = 2, m =0). Flows from hot (blue) regions into cold (red), v ‖ k,produce the azimuthally symmetric pattern Y 0

2 depictedhere.

2.2. Scalar Perturbations

The most commonly considered and familiar types ofperturbations are scalar modes. These modes represent

3

êφ

êθ

n

θ=π/2θ=π/4θ=0

(a) (b)

θ

Fig. 3.— The transformation of quadrupole anisotropies into linear polarization. (a) The orientation of the quadru-pole moment with respect to the scattering direction n determines the sense and magnitude of the polarization. Itis aligned with the cold (red, long) lobe in the eθ⊗eφ tangent plane. (b) In spherical coordinates where n·k = cos θ,the polarization points north-south (Q) with magnitude varying as sin2 θ for scalar fluctuations.

perturbations in the (energy) density of the cosmolog-ical fluid(s) at last scattering and are the only fluctu-ations which can form structure though gravitationalinstability.

Consider a single large-scale Fourier component ofthe fluctuation, i.e. for the photons, a single plane wavein the temperature perturbation. Over time, the tem-perature and gravitational potential gradients cause abulk flow, or dipole anisotropy, of the photons. Botheffects can be described by introducing an “effective”temperature

(∆T/T )eff = ∆T/T + Ψ , (4)

where Ψ is the gravitational potential. Gradients inthe effective temperature always create flows from hotto cold effective temperature. Formally, both pressureand gravity act as sources of the momentum density ofthe fluid in a combination that is exactly the effectivetemperature for a relativistic fluid.

To avoid confusion, let us explicitly consider the caseof adiabatic fluctuations, where initial perturbations tothe density imply potential fluctuations that dominateat large scales. Here gravity overwhelms pressure inoverdense regions causing matter to flow towards den-sity peaks initially. Nonetheless, overdense regions areeffectively cold initially because photons must climb outof the potential wells they create and hence lose energyin the process. Though flows are established from coldto hot temperature regions on large scales, they still

go from hot to cold effective temperature regions. Thisproperty is true more generally of our adiabatic assump-tion: we hereafter refer only to effective temperaturesto keep the argument general.

Let us consider the quadrupole component of thetemperature pattern seen by an observer located in atrough of a plane wave. The azimuthal symmetry inthe problem requires that v ‖ k and hence the flow isirrotational ∇ × v = 0. Because hotter photons fromthe crests flow into the trough from the ±k directionswhile cold photons surround the observer in the plane,the quadrupole pattern seen in a trough has an m = 0,

Y 02 ∝ 3 cos2 θ − 1 , (5)

structure with angle n·k = cos θ (see Fig. 2). The oppo-site effect occurs at the crests, reversing the sign of thequadrupole but preserving the m = 0 nature in its localangular dependence. The full effect is thus describedby a local quadrupole modulated by a plane wave inspace, −Y 0

2 (n) exp(ik · x), where the sign denotes thefact that photons flowing into cold regions are hot. Thisinfall picture must be modified slightly on scales smallerthan the sound horizon where pressure plays a role (see§3.3.2), however the essential property that the flows are

parallel to k and thus generate an m = 0 quadrupoleremains true.

The sense of the quadrupole moment determines thepolarization pattern through Thomson scattering. Re-call that polarized scattering peaks when the tempera-

4

π/20

0

π/2

ππ 3π/2 2πφ

θl=2, m=0

E, B

Fig. 4.— Polarization pattern for ℓ = 2, m = 0, note the azimuthal symmetry. The scattering of a scalarm = 0 quadrupole perturbation generates the electric E (yellow, thick lines) pattern on the sphere. Its ro-tation by 45 represents the orthogonal magnetic B (purple, thin lines) pattern. Animation (available athttp://www.sns.ias.edu/∼whu/polar/scalaran.html): as the line of sight n changes, the lobes of the quadru-pole rotate in and out of the tangent plane. The polarization follows the orientation of the colder (red) lobe in thetangent plane.

ture varies in the direction orthogonal to n. Considerthen the tangent plane eθ ⊗ eφ with normal n. Thismay be visualized in an angular “lobe” diagram suchas Fig. 2 as a plane which passes through the “origin”of the quadrupole pattern perpendicular to the line ofsight. The polarization is maximal when the hot andcold lobes of the quadrupole are in this tangent plane,and is aligned with the component of the colder lobewhich lies in the plane. As θ varies from 0 to π/2 (poleto equator) the temperature differences in this plane in-crease from zero (see Fig. 3a). The local polarization atthe crest of the temperature perturbation is thus purelyin the N-S direction tapering off in amplitude towardthe poles (see Fig. 3b). This pattern represents a pureQ-field on the sky whose amplitude varies in angle asan ℓ = 2, m = 0 tensor or spin-2 spherical harmonic

Q = sin2 θ, U = 0. (6)

In different regions of space, the plane wave modulationof the quadrupole can change the sign of the polariza-tion, but not its sense.

This pattern (Fig. 4, yellow lines) is of course not theonly logical possibility for an ℓ = 2, m = 0 polarizationpattern. Its rotation by 45 is also a valid configuration(purple lines). This represents a pure NW-SE (and bysign reversal NE-SW), or U -polarization pattern. We

return in §3.3 to consider the geometrical distinctionbetween the two patterns, the electric and magnetic

modes.

trough

crest

m=1

v

Vectors (Vorticity)

Fig. 5.— The vector quadrupole moment (ℓ = 2,m = 1). Since v ⊥ k, the Doppler effect generates aquadrupole pattern with lobes 45 from v and k that isspatially out of phase (interplane peaks) with v.

5

φl=2, m=1 π/20

0

π/2

ππ 3π/2 2π

θ

E, B

Fig. 6.— Polarization pattern for ℓ = 2, m = 1. The scattering of a vector (m = 1) quadrupole perturbationgenerates the E pattern (yellow, thick lines) as opposed to the B, (purple, thin lines) pattern. Animation

(available at http://www.sns.ias.edu/∼whu/polar/vectoran.html): same as for scalars.

2.3. Vector Perturbations

Vector perturbations represent vortical motions ofthe matter, where the velocity field v obeys ∇ · v = 0and ∇× v 6= 0, similar to “eddies” in water. There isno associated density perturbation, and the vorticity isdamped by the expansion of the universe as are all mo-tions that are not enhanced by gravity. However, theassociated temperature fluctuations, once generated, donot decay as both ∆T and T scale similarly with theexpansion. For a plane wave perturbation, the veloc-ity field v ⊥ k with direction reversing in crests andtroughs (see Fig. 5). The radiation field at these ex-trema possesses a dipole pattern due to the Dopplershift from the bulk motion. Quadrupole variations van-ish here but peak between velocity extrema. To see this,imagine sitting between crests and troughs. Looking uptoward the trough, one sees the dipole pattern projectedas a hot and cold spot across the zenith; looking downtoward the crest, one sees the projected dipole reversed.The net effect is a quadrupole pattern in temperaturewith m = ±1

Y ±12 ∝ sin θ cos θe±iφ. (7)

The lobes are oriented at 45 from k and v since the lineof sight velocity vanishes along k and at 90 degrees to k

here. The latter follows since midway between the crestsand troughs v itself is zero. The full quadrupole distri-bution is therefore described by −iY ±1

2 (n) exp(ik · x),where i represents the spatial phase shift of the quad-

rupole with respect to the velocity.

Thomson scattering transforms the quadrupole tem-perature anisotropy into a local polarization field as be-fore. Again, the pattern may be visualized from theintersection of the tangent plane to n with the lobe pat-tern of Fig. 5. At the equator (θ = π/2), the lobes areoriented 45 from the line of sight n and rotate into andout of the tangent plane with φ. The polarization pat-tern here is a pure U -field which varies in magnitude assin φ. At the pole θ = 0, there are no temperature vari-ations in the tangent plane so the polarization vanishes.Other angles can equally well be visualized by viewingthe quadrupole pattern at different orientations givenby n.

The full ℓ = 2, m = 1 pattern,

Q = − sin θ cos θeiφ, U = −i sin θeiφ (8)

is displayed explicitly in Fig. 6 (yellow lines, real part).Note that the pattern is dominated by U -contributionsespecially near the equator. The similarities and dif-ferences with the scalar pattern will be discussed morefully in §3.

2.4. Tensor Perturbations

Tensor fluctuations are transverse-traceless pertur-bations to the metric, which can be viewed as gravita-tional waves. A plane gravitational wave perturbationrepresents a quadrupolar “stretching” of space in theplane of the perturbation (see Fig. 7). As the wave

6

crest

trough

trough

m=2

Tensors (Gravity Waves)

Fig. 7.— The tensor quadrupole moment (m = 2).Since gravity waves distort space in the plane of theperturbation, changing a circle of test particles into anellipse, the radiation acquires an m = 2 quadrupolemoment.

passes or its amplitude changes, a circle of test par-ticles in the plane is distorted into an ellipse whosesemi-major axis → semi-minor axis as the spatial phasechanges from crest → trough (see Fig. 7, yellow ellipses).Heuristically, the accompanying stretching of the wave-length of photons produces a quadrupolar temperaturevariation with an m = ±2 pattern

Y ±22 ∝ sin2 θe±2iφ (9)

in the coordinates defined by k.

Thomson scattering again produces a polarizationpattern from the quadrupole anisotropy. At the equa-tor, the quadrupole pattern intersects the tangent (eθ⊗eφ) plane with hot and cold lobes rotating in and outof the eφ direction with the azimuthal angle φ. The po-larization pattern is therefore purely Q with a cos(2φ)dependence. At the pole, the quadrupole lobes lie com-pletely in the polarization plane and produces the maxi-mal polarization unlike the scalar and vector cases. Thefull pattern,

Q = (1 + cos2 θ)e2iφ, U = −2i cos θe2iφ, (10)

is shown in Fig. 8 (real part). Note that Q and U arepresent in nearly equal amounts for the tensors.

3. Polarization Patterns

The considerations of §2 imply that scalars, vec-tors, and tensors generate distinct patterns in the polar-ization of the CMB. However, although they separatecleanly into m = 0,±1,±2 polarization patterns for asingle plane wave perturbation in the coordinate sys-tem referenced to k, in general there will exist a spec-trum of fluctuations each with a different k. Thereforethe polarization pattern on the sky does not separateinto m = 0,±1,±2 modes. In fact, assuming statisticalisotropy, one expects the ensemble averaged power foreach multipole ℓ to be independent of m. Nonetheless,certain properties of the polarization patterns discussedin the last section do survive superposition of the per-turbations: in particular, its parity and its correlationwith the temperature fluctuations. We now discuss howone can describe polarization patterns on the sky arisingfrom a spectrum of k modes.

3.1. Electric and Magnetic Modes

Any polarization pattern on the sky can be separatedinto “electric” (E) and “magnetic” (B) components2.This decomposition is useful both observationally andtheoretically, as we will discuss below. There are twoequivalent ways of viewing the modes that reflect theirglobal and local properties respectively. The nomencla-ture reflects the global property. Like multipole radi-ation, the harmonics of an E-mode have (−1)ℓ parityon the sphere, whereas those of a B-mode have (−1)ℓ+1

parity. Under n → −n, the E-mode thus remains un-changed for even ℓ, whereas the B-mode changes sign asillustrated for the simplest case ℓ = 2, m = 0 in Fig. 9(recall that a rotation by 90 represents a change insign). Note that the E and B multipole patterns are45 rotations of each other, i.e. Q → U and U → −Q.Since this parity property is obviously rotationally in-variant, it will survive integration over k.

The local view of E and B-modes involves the sec-ond derivatives of the polarization amplitude (second

derivatives because polarization is a tensor or spin-2object). In much the same way that the distinction be-tween electric and magnetic fields in electromagnetisminvolves vanishing of gradients or curls (i.e. first deriva-tives) for the polarization there are conditions on thesecond (covariant) derivatives of Q and U . For an E-mode, the difference in second (covariant) derivatives

2These components are called the “grad” (G) and “curl” (C) com-ponents by Kamionkowski et al. (1997).

7

π/20

0

π/2

ππ 3π/2 2πφ

θl=2, m=2

E, B

Fig. 8.— Polarization pattern for ℓ = 2 m = 2. Scattering of a tensor m = 2 perturbation generates theE (yellow, thick lines) pattern as opposed to the B (purple, thin lines) pattern. Animation (available athttp://www.sns.ias.edu/∼whu/polar/tensoran.html): same as for scalars and vectors.

Global Parity

Flip

E

B

Local Axes

Principal Polarization

Fig. 9.— The electric (E) and magnetic (B) modes are distinguished by their behavior under a parity transfor-mation n → −n. E-modes have (−1)ℓ and B-modes have (−1)ℓ+1 parity; here (ℓ = 2, m = 0), even and oddrespectively. The local distinction between the two is that the polarization axis is aligned with the principle axesof the polarization amplitude for E and crossed with them for B. Dotted lines represent a sign reversal in thepolarization.

of U along eθ and eφ vanishes as does that for Q alongeθ + eφ and eθ − eφ. For a B-mode, Q and U are in-terchanged. Recalling that a Q-field points in the eθ oreφ direction and a U -field in the crossed direction, wesee that the Hessian or curvature matrix of the polar-ization amplitude has principle axes in the same sense

as the polarization for E and 45 crossed with it for B(see Fig. 9). Stated another way, near a maximum ofthe polarization (where the first derivative vanishes) thedirection of greatest change in the polarization is paral-lel/perpendicular and at 45 degrees to the polarizationin the two cases.

8

The distinction is best illustrated with examples.Take the simplest case of ℓ = 2, m = 0 where theE-mode is a Q = sin2 θ field and the B-mode is aU = sin2 θ field (see Fig. 4). In both cases, the ma-jor axis of the curvature lies in the eθ direction. Forthe E-mode, this is in the same sense; for the B-modeit is crossed with the polarization direction. The sameholds true for the m = 1, 2 modes as can be seen byinspection of Fig. 6 and 8.

3.2. Electric and Magnetic Spectra

Thomson scattering can only produce an E-mode lo-cally since the spherical harmonics that describe thetemperature anisotropy have (−1)ℓ electric parity. InFigs. 4, 6, and 8, the ℓ = 2, m = 0, 1, 2 E-mode patternsare shown in yellow. The B-mode represents these pat-terns rotated by 45 and are shown in purple and cannotbe generated by scattering. In this way, the scalars, vec-tors, and tensors are similar in that scattering producesa local ℓ = 2 E-mode only.

However, the pattern of polarization on the sky isnot simply this local signature from scattering but ismodulated over the last scattering surface by the planewave spatial dependence of the perturbation (compareFigs. 3 and 10). The modulation changes the amplitude,sign, and angular structure of the polarization but notits nature, e.g. a Q-polarization remains Q. Nonethe-less, this modulation generates a B-mode from the localE-mode pattern.

The reason why this occurs is best seen from thelocal distinction between E and B-modes. Recall thatE-modes have polarization amplitudes that change par-allel or perpendicular to, and B-modes in directions 45

away from, the polarization direction. On the otherhand, plane wave modulation always changes the po-larization amplitude in the direction k or N-S on thesphere. Whether the resultant pattern possesses E orB-contributions depends on whether the local polariza-tion has Q or U -contributions.

For scalars, the modulation is of a pure Q-field andthus its E-mode nature is preserved (Kamionkowski etal. 1997; Zaldarriaga & Seljak 1997). For the vectors,the U -mode dominates the pattern and the modulationis crossed with the polarization direction. Thus vectorsgenerate mainly B-modes for short wavelength fluctua-tions (Hu & White 1997). For the tensors, the compa-rable Q and U components of the local pattern implya more comparable distribution of E and B modes atshort wavelengths (see Fig. 11a).

θ

Last Scat. Surface

observer

Plane Wave Modulation

hot

cold

Fig. 10.— Modulation of the local pattern Fig. 3b byplane wave fluctuations on the last scattering surface.Yellow points represent polarization out of the planewith magnitude proportional to sign. The plane wavemodulation changes the amplitude and sign of the po-larization but does not mix Q and U . Modulation canmix E and B however if U is also present.

These qualitative considerations can be quantified bynoting that plane wave modulation simply represent theaddition of angular momentum from the plane wave(Y 0

ℓ ) with the local spin angular dependence. The re-sult is that plane wave modulation takes the ℓ = 2 lo-cal angular dependence to higher ℓ (smaller angles) andsplits the signal into E and B components with ratioswhich are related to Clebsch-Gordan coefficients. Atshort wavelengths, these ratios are B/E = 0, 6, 8/13 inpower for scalars, vectors, and tensors (see Fig. 11b andHu & White 1997).

The distribution of power in multipole ℓ-space is alsoimportant. Due to projection, a single plane wave con-tributes to a range of angular scales ℓ ∼< kr where ris the comoving distance to the last scattering surface.From Fig. 10, we see that the smallest angular, largestℓ ≈ kr variations occur on lines of sight n · k = 0 orθ = π/2 though a small amount of power projects toℓ ≪ kr as θ → 0. The distribution of power in multipolespace of Fig. 11b can be read directly off the local polar-ization pattern. In particular, the region near θ = π/2shown in Fig. 11a determines the behavior of the maincontribution to the polarization power spectrum.

The full power spectrum is of course obtained bysumming these plane wave contributions with weightsdependent on the source of the perturbations and the

9

Sca

lars

Vec

tors

Ten

sors

π/2

0 π/4 π/2

π/2

φ

θ

10 100

l

0.5

1.0

0.5

1.0

0.5

1.0E

B

(a) Polarization Pattern (b) Multipole Power

Fig. 11.— The E and B components of a plane wave perturbation. (a) Modulation of the local E-quadrupolepattern (yellow) from scattering by a plane wave. Modulation in the direction of (or orthogonal to) the polarizationgenerates an E-mode with higher ℓ; modulation in the crossed (45) direction generates a B-mode with higher ℓ.Scalars generate only E-modes, vectors mainly B-modes, and tensors comparable amounts of both. (b) Distributionof power in a single plane wave with kr = 100 in multipole ℓ from the addition of spin and orbital angular momentum.Features in the power spectrum can be read directly off the pattern in (a).

dynamics of their evolution up to last scattering. Sharpfeatures in the k-power spectrum will be preserved inthe multipole power spectrum to the extent that theprojectors in Fig. 11b approximate delta functions. Forscalar E-modes, the sharpness of the projection is en-hanced due to strong Q-contributions near θ = π/2(ℓ ∼ kr) that then diminish as θ → 0 (ℓ ≪ kr). Thesame enhancement occurs to a lesser extent for vectorB-modes due to U near π/2 and tensor E-modes dueto Q there. On the other hand, a supression occurs forvector E and tensor B-modes due to the absence of Qand U at π/2 respectively. These considerations haveconsequences for the sharpness of features in the polar-ization power spectrum, and the generation of asymp-totic “tails” to the polarization spectrum at low-ℓ (see§4.4 and Hu & White 1997) .

3.3. Temperature-Polarization Correlation

As we have seen in §2, the polarization pattern re-flects the local quadrupole anisotropy at last scattering.

Hence the temperature and polarization anisotropy pat-terns are correlated in a way that can distinguish be-tween the scalar, vector and tensor sources.

There are two subtleties involved in establishing thecorrelation. First, the quadrupole moment of the tem-perature anisotropy at last scattering is not generallythe dominant source of anisotropies on the sky, so thecorrelation is neither 100% nor necessarily directly vis-ible as patterns in the map.

The second subtlety is that the correlation occursthrough the E-mode unless the polarization has beenFaraday or otherwise rotated between the last scatter-ing surface and the present. As we have seen an E-mode is modulated in the direction of, or perpendicularto, its polarization axis. To be correlated with the tem-perature, this modulation must also correspond to themodulation of the temperature perturbation. The twooptions are that E is parallel or perpendicular to crestsin the temperature perturbation. As modes of differentdirection k are superimposed, this translates into a ra-

10

dial or tangential polarization pattern around hot spots(see Fig. 12a).

On the other hand B-modes do not correlate with thetemperature. In other words, the rotation of the pat-tern in Fig. 12a by 45 into those of Fig. 12b (solid anddashed lines) cannot be generated by Thomson scatter-ing. The temperature field that generates the polariza-tion has no way to distinguish between points reflectedacross the symmetric hot spot and so has no way tochoose between the ±45 rotations. This does not how-ever imply that B vanishes. For example, for a singleplane wave fluctuation B can change signs across a hotspot and hence preserve reflection symmetry (e.g. Fig. 6around the hot spot θ = π/4, φ = 0). However superpo-sition of oppositely directed waves as in Fig. 12b woulddestroy the correlation with the hot spot.

The problem of understanding the correlation thusbreaks down into two steps: (1) determine how thequadrupole moment of the temperature at last scatter-ing correlates with the dominant source of anisotropies;(2) isolate the E-component (Q-component in k coordi-nates) and determine whether it represents polarizationparallel or perpendicular to crests and so radial or tan-gential to hot spots.

E (anti) correlation B no correlation

Fig. 12.— Temperature-polarization cross correlation.E-parity polarization perpendicular (parallel) to crestsgenerates a tangential (radial) polarization field aroundhot spots. B-parity polarization does not correlate withtemperature since the ±45 rotated contributions fromoppositely directed modes cancel.

3.3.1. Large Angle Correlation Pattern

Consider first the large-angle scalar perturbations.Here the dominant source of correlated anisotropies is

the temperature perturbation on the last scattering sur-face itself. The Doppler contributions can be up to halfof the total contribution but as we have seen in §2.3do not correlate with the quadrupole moment. Contri-butions after last scattering, while potentially strong inisocurvature models for example, also rapidly lose theircorrelation with the quadrupole at last scattering.

As we have seen, the temperature gradient associ-ated with the scalar fluctuation makes the photon fluidflow from hot regions to cold initially. Around a pointon a crest therefore the intensity peaks in the directionsalong the crest and falls off to the neighboring troughs.This corresponds to a polarization perpendicular to thecrest (see Fig. 12). Around a point on a trough thepolarization is parallel to the trough. As we superposewaves with different k we find the pattern is tangentialaround hot spots and radial around cold spots (Crit-tenden et al. 1995). It is important to stress that thehot and cold spots refer only to the temperature com-ponent which is correlated with the polarization. Thecorrelation increases at scales approaching the horizonat last scattering since the quadrupole anisotropy thatgenerates polarization is caused by flows.

For the vectors, no temperature perturbations existon the last scattering surface and again Doppler contri-butions do not correlate with the quadrupole. Thus themain correlations with the temperature will come fromthe quadrupole moment itself. The correlated signalis reduced since the strong B-contributions of vectorsplay no role. Hot spots occur in the direction θ = π/4,φ = 0 where the hot lobe of the quadrupole is pointedat the observer (see Fig. 5). Here the Q (E) componentlies in the N-S direction perpendicular to the crest (seeFig. 6). Thus the pattern is tangential to hot spot, likescalars (Hu & White 1997). The signature peaks nearthe horizon at last scattering for reasons similar to thescalars.

For the tensors, both the temperature and polariza-tion perturbations arise from the quadrupole moment,which fixes the sense of the main correlation. Hotspots and cold spots occur when the quadrupole lobeis pointed at the observer, θ = π/2, φ = π/2 and 3π/2.The cold lobe and hence the polarization then pointsin the E-W direction. Unlike the scalars and vectors,the pattern will be mainly radial to hot spots (Critten-den et al. 1995). Again the polarization and hence thecross-correlation peaks near the horizon at last scatter-ing since gravitational waves are frozen before horizoncrossing (Polnarev 1985).

11

π 2π 3π 4πks

0

1

–1

0

1

–1

(a) adiabatic

(b) isocurvature

(∆T/T)eff v/ 3

cross

Fig. 13.— Time evolution of acoustic oscillations.The polarization is related to the flows v (red) whichform quadrupole anisotropies such that its product(green) with the effective temperature (blue) reflectsthe temperature-polarization cross correlation. As de-scribed in the text the adiabatic (a) and isocurvature(b) modes differ in the phase of the oscillation in allthree quantities. Temperature and polarization are an-ticorrelated in both cases at early times or large scalesks ≪ 1.

3.3.2. Small Angle Correlation Pattern

Until now we have implicitly assumed that the evolu-tion of the perturbations plays a small role as is gener-ally true for scales larger than the horizon at last scat-tering. Evolution plays an important role for small-scale scalar perturbations where there is enough timefor sound to cross the perturbation before last scatter-ing. The infall of the photon fluid into troughs com-presses the fluid, increasing its density and temperature.For adiabatic fluctuations, this compression reverses thesign of the effective temperature perturbation when thesound horizon s grows to be ks ≈ π/2 (see Fig. 13a).This reverses the sign of the correlation with the quad-rupole moment. Infall continues until the compressionis so great that photon pressure reverses the flow when

ks ≈ π. Again the correlation reverses sign. This pat-tern of correlations and anticorrelations continues attwice the frequency of the acoustic oscillations them-selves (see Fig. 13a). Of course the polarization is onlygenerated at last scattering so the correlations and an-ticorrelations are a function of scale with sign changesat multiples of π/2s∗, where s∗ is the sound horizon atlast scattering. As discussed in §3.2, these fluctuationsproject onto anisotropies as ℓ ∼ kr.

Any scalar fluctuation will obey a similar patternthat reflects the acoustic motions of the photon fluid.In particular, at the largest scales the ks∗ ≪ π/2, thepolarization must be anticorrelated with the tempera-ture because the fluid will always flow with the tem-perature gradient initially from hot to cold. However,where the sign reversals occur depend on the acousticdynamics and so is a useful probe of the nature of thescalar perturbations, e.g. whether they are adiabatic orisocurvature (Hu & White 1996). In typical isocurva-ture models, the lack of initial temperature perturba-tions delays the acoustic oscillation by π/2 in phase sothat correlations reverse at ks∗ = π (see Fig. 13b).

For the vector and tensor modes, strong evolutioncan introduce a small correlation with temperature fluc-tuations generated after last scattering. The effect isgenerally weak and model dependent and so we shallnot consider it further here.

4. Model Reconstruction

While it is clear how to compare theoretical predic-tions of a given model with observations, the recon-struction of a phenomenological model from the datais a more subtle issue. The basic problem is that inthe CMB, we see the whole history of the evolution inredshift projected onto the two dimensional sky. The re-construction of the evolutionary history of the universemight thus seem an ill-posed problem.

Fortunately, one needs only to assume the very basicproperties of the cosmological model and the gravita-tional instability picture before useful information maybe extracted. The simplest example is the combina-tion of the amplitude of the temperature fluctuations,which reflect the conditions at horizon crossing, andlarge scale structure today. Another example is theacoustic peaks in the temperature which form a snap-shot of conditions at last scattering on scales below thehorizon at that time. In most models, the acoustic sig-nature provides a wealth of information on cosmolog-ical parameters and structure formation (Hu & White

12

100

10-1

10-2

101

101 102 103

102

103

reion. τ=0.1

l

Po

wer

(µK

2 )

temperature

cross

E-pol.

Fig. 14.— Temperature, polarization, and temperature-polarization cross correlation predictions and sensitivityof MAP for a fiducial model Ω0 = 1, ΩB = 0.1, h = 0.5 cold dark matter. The raw MAP satellite sensitivity(1 σ errors on the recovered power spectrum binned in ℓ) is approximated by noise weights of w−1

T = (0.11µK)2

for the temperature and w−1P = (0.15µK)2 for the polarization and a FWHM beam of 0.25. Note that errors

between the spectra are correlated. While the reionized model (purple τ = 0.1) is impossible to distinguishfrom the fiducial model from temperature anisotropies alone, its effect on polarization is clearly visible at lowℓ. Dashed lines for the temperature-polarization correlation represent anticorrelation. Animation (available athttp://www.sns.ias.edu/∼whu/polar/tauan.html): Variations in the spectra as τ is stepped from 0 − 1.

13

1996). Unfortunately, it does not directly tell us thebehavior on the largest scales where important causaldistinctions between models lie. Furthermore it may beabsent in models with complex evolution on small scalessuch as cosmological defect models.

Here we shall consider how polarization informationaids the reconstruction process by isolating the last scat-tering surface on large scales, and separating scalar, vec-tor and tensor components. If and when these proper-ties are determined, it will become possible to establishobservationally the basic properties of the cosmologicalmodel such as the nature of the initial fluctuations, themechanism of their generation, and the thermal historyof the universe.

4.1. Last Scattering

The main reason why polarization is so useful to thereconstruction process is that (cosmologically) it canonly be generated by Thomson scattering. The polar-ization spectrum of the CMB is thus a direct snapshot ofconditions on the last scattering surface. Contrast thiswith temperature fluctuations which can be generatedby changes in the metric fluctuations between last scat-tering and the present, such as those created by gravi-tational potential evolution. This is the reason why themere detection of large-angle anisotropies by COBE didnot rule out wide classes of models such as cosmologicaldefects. To use the temperature anisotropies for the re-construction problem, one must isolate features in thespectrum which can be associated with last scattering,or more generally, the universe at a known redshift.

Furthermore, the polarization spectrum has poten-tial advantages even for extracting information fromfeatures that are also present in the temperature spec-trum, e.g. the acoustic peaks. The polarization spec-trum is generated by local quadrupole anisotropiesalone whereas the temperature spectrum has compa-rable contributions from the local monopole and dipoleas well as possible contributions between last scatteringand the present. This property enhances the promi-nence of features (see Fig. 14) as does the fact thatthe scalar polarization has a relatively sharp projectiondue to the geometry (see §3.2). Unfortunately, theseconsiderations are mitigated by the fact that the polar-ization amplitude is so much weaker than the tempera-ture. With present day detectors, one needs to measureit in broad ℓ-bands to increase the signal–to–noise (seeFig. 14 and Table 2).

4.2. Reionization

Since polarization directly probes the last scatter-ing epoch the first thing we learn is when that oc-curred, i.e. what fraction of photons last scattered atz ≈ 1000 when the universe recombined, and what frac-tion rescattered when the intergalactic medium reion-ized at zri ∼> 5.

Since rescattering erases fluctuations below the hori-zon scale and regenerates them only weakly (Efstathiou1988), we already know from the reported excess (overCOBE) of sub-degree scale anisotropy that the opticaldepth during the reionized epoch was τ ∼< 1 and hence

zri ∼< 100

(

Ω0h2

0.25

)1/3 (

ΩBh2

0.0125

)−2/3

. (11)

It is thus likely that our universe has the interestingproperty that both the recombination and reionizationepoch are observable in the temperature and polariza-tion spectrum.

Unfortunately for the temperature spectrum, at theselow optical depths the main effect of reionization is anerasure of the primary anisotropies (from recombina-tion) as e−τ . This occurs below the horizon at lastscattering, since only on these scales has there beensufficient time to convert the originally isotropic tem-perature fluctuations into anisotropies. The uniformreduction of power at small scales has the same effectas a change in the overall normalization. For zri ∼ 5–20 the difference in the power spectrum is confined tolarge angles (ℓ < 30). Here the observations are lim-ited by “cosmic variance”: the fact that we only haveone sample of the sky and hence only 2ℓ + 1 samplesof any given multipole. Cosmic variance is the dom-inant source of uncertainty on the low-ℓ temperaturespectrum in Fig. 14.

The same is not true for the polarization. As we haveseen the polarization spectrum is very sensitive to theepoch of last scattering. More specifically, the locationof its peak depends on the horizon size at last scatteringand its height depends on the duration of last scattering(Efstathiou 1988). This signature is not cosmic variancelimited until quite late reionization, though the combi-nation of low optical depth and partial polarization willmake it difficult to measure in practice (see Fig. 14 andTable 2). Optical depths of a few percent are poten-tially observable from the MAP satellite (Zaldarriagaet al. 1997) and of order unity from the POLAR exper-iment (Keating et al. 1997).

On the other hand, these considerations imply that

14

the interesting polarization signatures from recombina-tion will not be completely obscured by reionization.We turn to these now.

4.3. Scalars, Vectors, & Tensors

There are three types of fluctuations: scalars, vec-tors and tensors, and four observables: the temperature,E-mode, B-mode, and temperature cross polarizationpower spectra. The CMB thus provides sufficient infor-mation to separate these contributions, which in turncan tell us about the generation mechanism for fluctu-ations in the early universe (see §4.5).

Ignoring for the moment the question of foregrounds,to which we turn in §5.2, if the E-mode polarizationgreatly exceeds the B-mode then scalar fluctuationsdominate the anisotropy. Conversely if the B-mode isgreater than the E-mode, then vectors dominate. Iftensors dominate, then the E and B are comparable(see Fig. 15). These statements are independent of thedynamics and underlying spectrum of the perturbationsthemselves.

The causal constraint on the generation of a quad-rupole moment (and hence the polarization) introducesfurther distinctions. It tells us that the polarizationpeaks around the scale the horizon subtends at lastscattering. This is about a degree in a flat universeand scales with the angular diameter distance to lastscattering. Geometric projection tells us that the low-ℓtails of the polarization can fall no faster than ℓ6, ℓ4

and ℓ2 for scalars, vectors and tensors (see §3.2). Thecross spectrum falls no more rapidly than ℓ4 for each.We shall see below that these are the predicted slopesof an isocurvature model.

Furthermore, causality sets the scale that separatesthe large and small angle temperature-polarization cor-relation pattern. Well above this scale, scalar and vec-tor fluctuations should show anticorrelation (tangentialaround hot spots) whereas tensor perturbations shouldshow correlations (radial around hot spots).

Of course one must use measurements at small enoughangular scales that reionization is not a source of confu-sion and one must understand the contamination fromforegrounds extremely well. The latter is especially trueat large angles where the polarization amplitude de-creases rapidly.

4.4. Adiabatic vs. Isocurvature Perturbations

The scalar component is interesting to isolate sinceit alone is responsible for large scale structure forma-

tion. There remain however two possibilities. Densityfluctuations could be present initially. This representsthe adiabatic mode. Alternately, they can be generatedfrom stresses in the matter which causally push matteraround. This represents the isocurvature mode.

The presence or absence of density perturbationsabove the horizon at last scattering is crucial for the fea-tures in both the temperature and polarization powerspectrum. As we have seen in §3.3.1, it has as strong ef-fect on the phase of the acoustic oscillation. In a typicalisocurvature model, the phase is delayed by π/2 mov-ing structure in the temperature spectrum to smallerangles (see Fig. 13). Consistency checks exist in theE-polarization and cross spectrum which should be outof phase with the temperature spectrum and oscillat-ing at twice the frequency of the temperature respec-tively. However, if the stresses are set up sufficientlycarefully, this acoustic phase test can be evaded by anisocurvature model (Turok 1996). Perhaps more impor-tantly, acoustic features can be washed out in isocurva-ture models with complicated small scale dynamics toforce the acoustic oscillation, as in many defect models(Albrecht et al. 1996).

Even in these cases, the polarization carries a robustsignature of isocurvature fluctuations. The polarizationisolates the last scattering surface and eliminates anysource of confusion from the epoch between last scat-tering and the present. In particular, the delayed gen-eration of density perturbations in these models impliesa steep decline in the polarization above the angle sub-tended by the horizon at last scattering. The polar-ization power thus hits the asymptotic limits given inthe previous section in contrast to the adiabatic powerspectra shown in Fig. 14 which have more power at largeangles. To be more specific, if the E-power spectrumfalls off as ℓ6 and or the cross spectrum as ℓ4, then theinitial fluctuations are isocurvature in nature (or an adi-abatic model with a large spectral index, which is highlyconstrained by COBE).

Of course measuring a steeply falling spectrum is dif-ficult in practice. Perhaps more easily measured is thefirst feature in the adiabatic E-polarization spectrum attwice the angular scale of the first temperature peak andthe first sign crossing of the correlation at even largerscales. Since isocurvature models must be pushed tothe causal limit to generate these features, their obser-vation would provide good evidence for the adiabaticnature of the initial fluctuations (Hu, Spergel & White1997).

15

101 102 103

100

10-1

10-2

101

102

Po

wer

(µK

2 )

l

temperature

cross

E-pol.

B-pol.

1σ MAP

Fig. 15.— Tensor power spectra. The tensor temperature, temperature-cross-polarization, E-mode, and B-modepolarization for a scale invariant intial spectrum of tensors with Ω0 = 1.0, ΩB = 0.1 and h = 0.5. The normalizationhas been artificially set (high) to have the same quadrupole as the scalar spectrum of Fig. 14. Also shown is theMAP 1σ upper limits on the B-mode due to noise, i.e. assuming no signal, showing that higher sensitivity willlikely be necessary to obtain a significant detection of B-polarization from tensors.

16

4.5. Inflation vs. Defects

One would like to know not only the nature of thefluctuations, but also the means by which they are gen-erated. We assume of course that they are not merelyplaced by fiat in the initial conditions. Let us first dividethe possibilities into broad classes. In fact, the distinc-tion between isocurvature and adiabatic fluctuations isoperationally the same as the distinction between con-ventional causal sources (e.g. defects) and those gener-ated by a period of superluminal expansion in the earlyuniverse (i.e. inflation). It can be shown that inflationis the only causal mechanism for generating superhori-zon size density (curvature) fluctuations (Liddle 1995).Since the slope of the power spectrum in E can be traceddirectly to the presence of “superhorizon size” temper-ature, and hence curvature, fluctuations at last scat-tering, it represents a “test” of inflation (Hu & White1997, Spergel & Zaldarriaga 1997). The acoustic phasetest, either in the temperature or polarization, repre-sents a marginally less robust test that should be easilyobservable if the former fails to be.

These tests, while interesting, do not tell us anythingabout the detailed physics that generates the fluctu-ations. Once a distinction is made between the twopossibilities one would like to learn about the mecha-nism for generating the fluctuations in more detail. Forexample in the inflationary case there is a well knowntest of single-field slow-roll inflation which can be im-proved by using polarization information. In principle,inflation generates both scalar and tensor anisotropies.If we assume that the two spectra come from a singleunderlying inflationary potential their amplitudes andslopes are not independent. This leads to an algebraicconsistency relation between the ratio of the tensor andscalar perturbation spectra and the tensor spectral in-dex. However information on the tensor contribution tothe spectrum is limited by cosmic variance and is easilyconfused with other effects such as those of a cosmolog-ical constant or tilt of the initial spectrum. By usingpolarization information much smaller ratios of tensorto scalar perturbations may be probed, with more ac-curacy, and the test refined (see Zaldarriaga et al. 1997for details). In principle, this extra information mayalso allow one to reconstruct the low order derivativesof the inflaton potential.

Similar considerations apply to causal generation offluctuations without inflation. Two possibilities are amodel with “passive evolution” where initial stress fluc-tuations move matter around and “active evolution”

where exotic but causal physics continually generatesstress-energy perturbations inside the horizon. All cos-mological defect models bear aspects of the latter class.The hallmark of an active generation mechanism is thepresence of vector modes. Vector modes decay with theexpansion so in a passive model they would no longerbe present by recombination. Detection of (cosmologi-cal) vector modes would be strong evidence for defectmodels. Polarization is useful since it provides a uniquesignature of vector modes in the dominance of the B-mode polarization (Hu & White 1997). Its level forseveral specific defect models is given in Seljak, Pen,Turok (1997).

5. Phenomenology

5.1. Observations

While the theoretical case for observing polariza-tion is strong, it is a difficult experimental task to ob-serve signals of the low level of several µK and be-low. Nonetheless, polarization experiments have onepotential advantage over temperature anisotropy exper-iments. They can reduce atmospheric emission effectsby differencing the polarization states on the same patchof sky instead of physically chopping between differentangles on the sky since atmospheric emission is thoughtto be nearly unpolarized (see §5.2). However, to be suc-cessful an experiment must overcome a number of sys-tematic effects, many of which are discussed in Keatinget al. (1997). It must at least balance the sensitivity ofthe instrument to the orthogonal polarization channels(including the far side lobes) to nearly 8 orders of mag-nitude. Multiple levels of switching and a very carefuldesign are minimum requirements.

To date the experimental upper limits on polariza-tion of the CMB have been at least an order of mag-nitude larger than the theoretical expectations. Theoriginal polarization limits go back to Penzias & Wil-son (1965) who set a limit of 10% on the polarizationof the CMB. There have been several subsequent up-per limits which have now reached the level of ∼ 20µK(see Tab. 1), about a factor of 5–10 above the predictedlevels for popular models (see Tab. 2).

5.2. Foregrounds

Given that the amplitude of the polarization is sosmall the question of foregrounds is even more impor-tant than for the temperature anisotropy. Unfortu-nately, the level and structure of the various foregroundpolarization in the CMB frequency bands is currently

17

Reference Frequency (GHz) Ang. Scale ∆T (µK) (CL)Penzias & Wilson (1965) 4.0 – 105 95%Nanos (1979) 9.3 15 beam 2 × 103 90%Caderni et al. (1978) 100-600 0.5 ∼< θ ∼< 40 2 × 103(40/θ)1/2 65%Lubin & Smoot (1981) 33 ℓ ∼< 3 200 95%Partridge et al. (1988) 5 1′ ∼< θ ∼< 3′ 100 95%Wollack et al. (1993) 26-36 50 ∼< ℓ ∼< 100 25 95%Netterfield et al. (1995) 26-46 50 ∼< ℓ ∼< 100 16 95%

Table 1: Experimental upper limits on the polarization of the CMB. The values are converted from the quoted∆T/T by multiplying by T = 2.728K, thus these numbers only approximate the bandpowers used in Tab. 2.

Cosmological Parameters ℓ: 2–20 (µK) ℓ: 100-200 (µK) ℓ: 700–1000 (µK)Ω0 ΩB zri T/S E TE B E TE B E TE B

1.0 0.05 0 0 0.02 (−)0.42 0 0.68 (+)3.6 0 3.7 (+)4.8 01.0 0.10 0 0 0.017 (−)0.40 0 0.71 (+)4.2 0 3.4 (+)3.3 01.0 0.05 5 0 0.03 (−)0.59 0 0.68 (+)3.6 0 3.7 (+)4.8 01.0 0.05 10 0 0.06 (−)0.78 0 0.68 (+)3.6 0 3.6 (+)4.8 01.0 0.05 20 0 0.14 (−)1.1 0 0.66 (+)3.5 0 3.5 (+)4.7 01.0 0.05 0 0.1 0.02 (−)0.40 0.01 0.65 (+)3.4 0.05 3.5 (+)4.6 0.010.4K 0.05 0.1 0 0.01 (−)0.18 0 0.44 (+)2.3 0 2.8 (+)5.5 00.4Λ 0.05 0 0 0.02 (−)0.37 0 0.65 (+)4.0 0 4.1 (+)6.4 0

Table 2: Bandpowers for a selection of cosmological models all with h = 0.5 and scale invariant spectra save forthe T/S = 0.1 model where T/S = 7(1 − ns). The low-Ω0 models are indicated as 0.4K for the open model and0.4Λ for the model with ΩΛ = 1 − Ω0 = 0.6. The bandpower is defined in units of the COBE quadrupole over therange of ℓ indicated. The sign of the cross correlation is indicated in parentheses and note that the band averagedsignal may be suppressed from cancellation of correlated and anticorrelated angular regimes.

not well known. We review some of the observationsin the adjacent radio and IR bands (a more completediscussion can be found in Keating et al. 1997). At-mospheric emission is believed to be negligibly polar-ized (Keating et al. 1997), leaving the main astrophysi-cal foregrounds: free-free, synchrotron, dust, and pointsource emissions. Of these the most important fore-ground is synchrotron emission.

Free-free emission (bremsstrahlung) is intrinsicallyunpolarized (Rybicki & Lightman 1979) but can be par-tially polarized by Thomson scattering within the HIIregion. This small effect is not expected to polarize theemission by more than 10% (Keating et al. 1997). Theemission is larger at low frequencies but is not expectedto dominate the polarization at any frequency.

The polarization of dust is not well known. In princi-ple, emission from dust particles could be highly polar-

ized, however Hildebrand & Dragovan (1995) find thatin their observations the majority of dust is polarizedat the ≈ 2% level at 100µm with a small fraction ofregions approaching 10% polarization. Moreover Keat-ing et al. (1997) show that even at 100% polarization,extrapolation of the IRAS 100µm map with the COBE

FIRAS index shows that dust emission is negligible be-low 80GHz. At higher frequencies it will become thedominant foreground.

Radio point sources are polarized due to synchrotronemission at < 20% level. For large angle experiments,the random contribution from point sources will con-tribute negligibly, but may be of more concern for theupcoming satellite missions.

Galactic synchrotron emission is the major concern.It is potentially highly polarized with the fraction de-pendent on the spectral index and depolarization from

18

Faraday rotation and non-uniform magnetic fields. Thelevel of polarization is expected to lie between 10%-75%of a total intensity which itself is approximately 50µKat 30GHz. This estimate follows from extrapolating theBrouw & Spoelstra (1976) measurements at 1411 MHzwith an index of T ∝ ν−3.

Due to their different spectral indices, the minimumin the foreground polarization, like the temperature, liesnear 100GHz. For full sky measurements, since syn-chrotron emission is more highly polarized than dust,the optimum frequency at which to measure intrin-sic (CMB) polarization is slightly higher than for theanisotropy. Over small regions of the sky where one orthe other of the foregrounds is known a priori to beabsent the optimum frequency would clearly be differ-ent. However as with anisotropy measurements, withmultifrequency coverage, polarized foregrounds can beremoved.

It is also interesting to consider whether the spatialas well as frequency signature of the polarization canbe used to separate foregrounds. Using angular powerspectra for the spatial properties of the foregroundsis a simple generalization of methods already used inanisotropy work. For instance, in the case of synchotronemission, if the spatial correlation in the polarizationfollows that of the temperature itself, the relative con-tamination will decrease on smaller angular scales dueto its diffuse nature. Furthermore the peak of the cos-mic signal in polarization occurs at even smaller angularscales than for the anisotropy.

One could attempt to exploit the additional prop-erties of polarization, such as its E- and B-mode na-ture. We mentioned above that in a wide class ofmodels where scalars dominate on small angular scales,the polarization is predicted to be dominantly E-mode(Kamionkowski et al. 1997 Zaldarriaga & Seljak 1997).Seljak (1997) suggests that one could use this to helpeliminate foreground contamination by “vetoing” on ar-eas of B-mode signal. However in general one does notexpect that the foregrounds will have equal E- and B-mode contribution, so while this extra information isvaluable, its use as a foreground monitor can be com-promised in certain circumstances. Specifically, if a cor-relation exists between the direction of polarization andthe rate of change (curvature) of its amplitude, the fore-ground will populate the two modes unequally. Twosimple examples: either radial or tangential polariza-tion around a source with the amplitude of the polar-ization dropping off with radius, or polarization parallelor perpendicular to a “jet” whose amplitude drops along

the jet axis. Both examples would give predominantlyE-mode polarization.

5.3. Data Analysis

Several authors have addressed the question of theoptimal estimators of the polarization power spectrafrom high sensitivity, all-sky maps of the polarization.They suggest that one calculate the coefficients of theexpansion in spin-2 spherical harmonics and then formquadratic estimators of the power spectrum, as the av-erage of the squares of the coefficients over m, correctedfor noise bias as in the example of Fig. 14.

We shall return to consider this below, however be-fore such maps are obtained we would like to know howto analyze ground based polarization data. These dataare likely to consist of Q and U measurements fromtens or perhaps hundreds of pointings, convolved withan approximately gaussian beam on some angular scale.How can we use this data to provide constraints or mea-surements of the electric and magnetic power spectra,presumably averaged across bands in ℓ?

For a small number of points, the simplest and mostpowerful way to obtain the power spectrum is to per-form a likelihood analysis of the data. The likelihoodfunction encodes all of the information in the mea-surement and can be modified to correctly account fornon-uniform noise, sky coverage, foreground subtrac-tion and correlations between measurements. Opera-tionally, one computes the probability of obtaining themeasured points Qi ≡ Q(ni) and Ui ≡ U(ni) assum-ing a given “theory” (including a model for foregroundsand detector noise) and maximizes the likelihood overthe theories. For our purposes, the theories could begiven simply by the polarization bandpowers in E andB for example, or could be a more “realistic” modelsuch as CDM with a given reionization history. Theconfidence levels on the parameters are obtained as mo-ments of the likelihood function in the usual way. Suchan approach also allows one to generalize the analysisto include temperature information (for the cross cor-relation) if it becomes available.

Assuming that the fluctuations are gaussian, the like-lihood function is given in terms of the data and thecorrelation function of Q and U for any pair of the ndata points. The calculation of this correlation functionis straightforward, and Kamionkowski et al. (1997) dis-cuss the problem extensively. Let us assume that weare fitting only one component or have only one fre-quency channel. The generalization to multiple frequen-

19

cies with a model for the foreground is also straightfor-ward. We shall also assume for notational simplicitythat we are fitting only to polarization data, thoughagain the generalization to include temperature data isstraightforward. The construction is as follows. Wedefine a data vector which contains the Q and U infor-mation referenced to a particular coordinate system (inprinciple this coordinate system could change betweendifferent subsets of the data). Call this data vector

D = (Q1, U1, . . . , Qn, Un), (12)

which has N = 2n components. We can construct thelikelihood of obtaining the data given a theory once weknow the correlation matrix CIJ :

L(D|T ) ∝ 1√detC

exp

[

−1

2DIC

−1IJ DJ

]

(13)

All of the theory information is encoded in CIJ = CthIJ +

NIJ , where NIJ is the noise correlation matrix, to beprovided by the experiment, and Cth

IJ is a function ofthe theory parameters.

All that remains is to compute each element of CthIJ

for a given theory. Consider a pair of points i and jcorresponding to 4 entries of our data vector DI . Fol-lowing Kamionkowski et al. (1997), define Q′ and U ′

as the components of the polarization in a new coor-dinate system, where the great arc connecting n1 andn2 runs along the equator. Expressions for 〈Q′Q′〉 and〈U ′U ′〉 in terms of the E and B angular power followdirectly from the definitions of these spectra and thespin-weighted spherical harmonics. They can be foundin Kamionkowski et al. (1997) [their Eqs. (5.9,5.10)].They also give the flat sky limit of these equations.Knowing the angle φij about ni through which we mustrotate our primed coordinate system to return to thesystem in which our data is defined we can write

Qi = Q′

i cos 2φij + U ′

i sin 2φij

Ui = Q′

i sin 2φij − U ′

i cos 2φij (14)

and similarly for the jth element. Thus using theknown expressions for 〈Q′Q′〉 and 〈U ′U ′〉 we can cal-culate 〈QiQj〉, 〈UiUj〉, 〈QiUj〉 and 〈UiQj〉 for the ithand jth pixel and thus all of the elements of Cth

IJ . Sub-stitution of Cth

IJ into Eq. (13) allows one to obtain limitson any theory given the data.

Finally let us note that the method outlined here iscompletely general and thus can be applied also thethe high-sensitivity, all-sky maps which would result

from satellite experiments. For these experiments, thelarge volume of data is however an issue in the analysispipeline design, as has been addressed by several au-thors. The generalization of the above analysis proce-dure to include filtering and compression is straightfor-ward, and directly analogous to the case of anisotropy,so we will not discuss it explicitly here.

6. Future Prospects

Following a hiatus of some years, experimental ef-forts to detect CMB polarization are now underway.At the large angular scale, an experiment based in Wis-consin (Keating et al. 1997) plans to make a 30GHzmeasurement of polarization on 7 angular scales witha sensitivity of a few µK per pixel. The main goal ofthe experiment would be to look for the signature ofreionization (see Fig. 14). Several groups are currentlyconsidering measuring the signature at smaller scalesfrom recombination.

The MAP and Planck missions also plan to mea-sure polarization. Because of their all sky nature thesemissions will be the first to be able to measure the po-larization power spectrum and temperature cross cor-relation with reasonable spectral sensitivity. We showan example of the sensitivity to the which MAP shouldnominally obtain in the absence of foregrounds and sys-tematic effects in Fig. 14. As can be seen, MAP shouldcertainly detect the E-mode of polarization at mediumangular scales, and obtain a significant detection ofthe temperature polarization cross-correlation. Thesesignatures can be useful for separating adiabatic andisocurvature models of structure formation as we haveseen in §4.

However the detailed study of the polarization powerspectrum will require the improved sensitivity and ex-panded frequency coverage of Planck for detailed fea-tures and foreground removal respectively. Planck willhave the sensitivity to make a measurement of the large-angle polarization predicted in CDM models, regardlessof the epoch of reionization if foreground contaminationcan be removed. Likewise, it can potentially measuresmall levels of B-mode polarization. The separation ofthe E and B modes is of course crucial for the isolationof scalar, vector and tensor modes and so the recon-struction problem in general.

Clearly, the polarization spectrum represents anothergold mine of information in the CMB. Though signifi-cant challenges will have to be overcome, the prospectsfor its detection are bright.

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Acknowledgments: We thank D. Eisenstein, S. Staggs,M. Tegmark, P. Timbie and M. Zaldarriaga for usefuldiscussions. W.H. acknowledges support from the W.M.Keck foundation.

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whu@ias.edu

http://www.sns.ias.edu/∼whu/polar/polar.html

This 2-column preprint was prepared with the AAS LATEX macrosv4.0.

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