Birefringence, CMB polarization, and magnetized B-mode

Massimo Giovannini1,2 and Kerstin E. Kunze1,3

1Department of Physics, Theory Division, CERN, 1211 Geneva 23, Switzerland2INFN, Section of Milan-Bicocca, 20126 Milan, Italy

3Departamento de Fısica Fundamental, Universidad de Salamanca, Plaza de la Merced s/n, E-37008 Salamanca, Spain(Received 15 December 2008; published 1 April 2009)

Even in the absence of a sizable tensor contribution, a B-mode polarization can be generated because of

the competition between a pseudoscalar background and predecoupling magnetic fields. By investigating

the dispersion relations of a magnetoactive plasma supplemented by a pseudoscalar interaction, the total

B-mode polarization is shown to depend not only upon the plasma and Larmor frequencies but also on the

pseudoscalar rotation rate. If the (angular) frequency channels of a given experiment are larger than the

pseudoscalar rotation rate, the only possible source of (frequency-dependent) B-mode autocorrelations

must be attributed to Faraday rotation. In the opposite case the pseudoscalar contribution dominates and

the total rate becomes, in practice, frequency independent. The B-mode cross correlations can be used,

under certain conditions, to break the degeneracy by disentangling the two birefringent contributions.

DOI: 10.1103/PhysRevD.79.087301 PACS numbers: 98.70.Vc, 78.20.Ek, 95.75.Hi, 98.62.En

In the�CDM paradigm1 a potential candidate for the B-mode polarization are the tensor modes of the geometryinducing a frequency-independent polarization of the cos-mic microwave background (CMB in what follows). Byfrequency-independent signal we mean that different ob-servational channels measure angular power spectra withthe same amplitude. In the opposite case the angular powerspectra will effectively depend upon the (angular) fre-quency of observation. The WMAP 5-yr data [1] constrainthe presence of a B-mode and, indirectly, rT, i.e. the ratio ofthe tensor power spectrum over the scalar power spectrum[1]. A further (frequency-independent) source of B-modepolarization is cosmic shear (see e.g. [2]). Diverse data sets(such as the ones of Quad and Capmap [3]) impose con-current limits on the B-mode polarization. Forthcomingexperiments are expected to improve the present status ofthe observations by reaching into the region rT < 0:2.

The only frequency-dependent signal investigated so faris provided by the Faraday effect which is a distinctivefeature of magnetized plasmas in different contexts [4].Large-scale magnetic fields present prior to the equalitytime are known to impact both on the temperature auto-correlations as well as on the polarization observables [5].It has been recently shown, within a dedicated numericalapproach [6], that the Faraday rotation signal induced by apredecoupling magnetic field can overwhelm the B-mode

polarization induced by the standard tensor contribution[7].The B-mode autocorrelations might not be always suffi-

cient to infer the presence of a preequality magnetic field.2

In short, the idea is the following. Consider a setup wherethe preequality plasma is birefringent because of the con-current presence of a pseudoscalar background field, be it�, and of a large-scale magnetic field. The possibility of aglobal breaking of parity because of pseudoscalar interac-tions has been experimentally investigated, for instance, bythe WMAP team (see, e.g. the third reference of [1]) aswell as by the Quad collaboration [3]. The coupling ofultralight pseudo-Nambu-Goldstone bosons to electromag-netism may also lead to a rotation of the plane of polarizedemission. Radio-astronomical implications of this type ofcosmic birefringence were investigated through varioussteps [8]. Absent any pseudoscalar background, the rota-tion rate would scale with the square of the wavelength [4]of the observational channel. In the presence of a pseudo-scalar field, the dispersion relations can be generalized andthe total rotation rate will have, both, a magnetic and apseudoscalar contribution. This is the idea already pursuedin [9]. The purpose of this paper is to develop this idea andto compute the B-mode polarization generated by thecompetition of the two aforementioned effects. We wantto scrutinize if (and when) the two effects can be, at leastpartially, disentangled. The essentials of the problem athand are usefully introduced in terms of the electromag-netic part of the action

1The � refers to the dark energy component (assumed to be inthe form of a putative cosmological constant). The CDM refersto the (cold) dark matter contribution. In what follows thecosmological parameters will be fixed to the best fit of theWMAP 5-yr data alone, i.e. ð�b0;�c0;��; h0; ns; �Þ ¼ð0:0441; 0:214; 0:742; 0:719; 0:963; 0:087Þ. In the latter string ofparameters�X denotes the critical fraction of a given species, h0fixes the present value of the Hubble rate; ns is the spectral indexof curvature perturbations and � is the reionization optical depth.

2Following the established terminology the B-mode autocor-relations are denoted by BB. With similar notation we will talkabout the TT, TE, and EE angular power spectra meaning,respectively, the autocorrelations of the temperature, the auto-correlations of the E-mode, and their mutual cross correlations.

PHYSICAL REVIEW D 79, 087301 (2009)

1550-7998=2009=79(8)=087301(4) 087301-1 � 2009 The American Physical Society

Sem ¼ � 1

16�

Zd4x

ffiffiffiffiffiffiffi�gp

��F��F

�� � ��

MF��

~F�� þ 16�j�A�

�; (1)

where g ¼ detg�� and g�� ¼ a2ð�Þ���; F�� is the elec-

tromagnetic field strength; ~F�� ¼ ����F�=ð2 ffiffiffiffiffiffiffi�gp Þ is

the dual field strength in curved space-times. Equa-tion (1) is essentially the same action of Refs. [8,9]. InEq. (1) � is a coupling constant andM a typical mass scalewhich may take specific values, for instance, in a givenscenario [9]. In Eq. (1) j� denotes the electromagneticcurrent which can be specified in terms of the chargecarriers (i.e. electrons and ions) as j� ¼ j�i þ j�e ¼eð~niu�i � ~neu

�e Þ (recall that, for both species, g��u

�e;iu

�e;i ¼

1). The relevant set of equations can then be written, forbrevity, in their covariant form and they are

r�F�� ¼ 4�j� þ �

Mr�� ~F��; r�

~F�� ¼ 0; (2)

r�ðT��e þ T��

i þ T��EMÞ ¼ 0; T��

e;i ¼ e;iu�e;iu

�e;i; (3)

where T��EM is the energy-momentum tensor of the electro-

magnetic field and e;i denote the energy density of elec-

trons and ions. Note that r� (i.e. the covariant derivative

associated with the space-time geometry) does not onlydepend upon the scale factor but also upon the inhomoge-neities. Equation (3) summarizes schematically the evolu-tion equations of charged species whose governingequations can be more appropriately derived from theVlasov-Landau equations in curved space (or from theirlowest moments). While electrons and ions are coupledthrough Coulomb scattering, the electron-ion fluid iscoupled to the photon background. The plasma containsa large-scale magnetic field whose Fourier amplitudessatisfy

hBið ~k; �ÞBjð ~p; �Þi ¼ 2�2

k3PBðkÞPijðkÞð3Þð ~kþ ~pÞ;

PB ¼ AB

�k

kL

�nB�1

;(4)

where k2PijðkÞ ¼ ðk2ij � kikjÞ; AB is the amplitude of

the magnetic power spectrum at the (comoving) magneticpivot scale kL (equal to 1 Mpc�1 in the forthcoming nu-merical examples). The magnetic field is inhomogeneousover typical length scales which are of the order of theHubble radius rH. The Larmor radius of the electrons, onthe contrary, is rL ’ Oðvth= �!BeÞ � rH where �!Be is the

(comoving) Larmor frequency and vth ’ffiffiffiffiffiffiffiffiffiffiffiffiffiffiTe=me

pis the

thermal velocity of the electrons. For a comoving fieldstrength OðnGÞ the Larmor radius is roughly 8 orders ofmagnitude smaller than the Hubble radius. The guidingcenter approximation (originally due to Alfven [10]) canbe then applied. The charged particles orbiting around themagnetic field lines will see, in practice, a constant field up

to drift corrections (going as ½Teð ~B� ~rÞB�=ðeB3Þ whereB ¼ j ~Bj) and curvature corrections (going as ½Te

~B� ð ~B �~rÞ ~B�=ðeB4Þ) which are, however, negligible when the scaleof (spatial) variation of the magnetic background is muchlarger than the gyration radius of the charge carriers. Asdiscussed in [9]. the dispersion relations can be derived bystudying the propagation of the electromagnetic waves in amagnetoactive plasma at finite density. By writing Eqs. (2)and (3) in their explicit form, the compatibility of thesystem can be ensured if detAij ¼ 0, where Aij is given

by

A ij ¼ k2ðij � kikjÞ � �!2 ��ijð �!;�Þ þ i�

M�0�mijk

m:

(5)

In Eq. (5) ��ijð �!;�Þ is the dielectric tensor and �0 denotes aderivation with respect to the conformal time coordinate �.By requiring that detAij ¼ 0 and by setting the comoving

wave numbers in such a way that kx ¼ 0 and ky ¼ k sin#

and kz ¼ k cos#, the standard form of the Appleton-Hartree equation can be easily recovered and it is [9]

sin2#

��1

�k� 1

n2

��1

n2� 1

2

�1

��þ 1

�þ

��

þ �!2�

2n2 �!2

�1

��þ 1

�þ

�� cos2#

��1

n2� 1

��

��1

n2� 1

�þ

�

� �!2�

n2!2�þ��

�þ �!�

n3 �!

�1

��� 1

�þ

�cos# ¼ 0; (6)

where � ¼ iH = �!, H ¼ a0=a, and n ¼ k= �! is the re-fractive index. Denoting with �!pe;i the comoving plasma

frequencies for electrons and ions and with �!Be;i the cor-

responding gyration frequencies, ��ð �!;�Þ and �kð �!;�Þare

��ð �!;�Þ ¼ 1� �!2pi

�!½ �!ð�þ 1Þ � �!Bi�

� �!2pe

�!½ �!ð�þ 1Þ � �!Be� ; (7)

�kð �!;�Þ ¼ 1� �!2pi

�!2ð1þ �Þ ��!2pe

�!2ð1þ �Þ : (8)

The frequency �!� ¼ ��0=M measures the rate of varia-tion of the polarization because of the presence of thepseudoscalar background. If � is not homogeneous thedispersion relations will have a different form. Accordingto Eq. (6) the refractive indices for electromagnetic propa-gation along the magnetic field (i.e. # ¼ 0) can be deducedfrom �

n2 � �!�

�!n� �þ

��n2 þ �!�

�!n� ��

�¼ 0: (9)

Since �!max > �!pe � �!Be (where �!max ¼ 2� ��max and

��max ¼ 222:617 GHz corresponds to the maximum of the

BRIEF REPORTS PHYSICAL REVIEW D 79, 087301 (2009)

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CMB spectral energy density), the rotation rate experi-enced by the linearly polarized CMB traveling parallel tothe magnetic field direction is

F ðnÞ ¼ d�

d�

¼ �!

2c

��!�

�!þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

4

��!�

�!

�2 þ �þð �!;�Þ

s

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

4

��!�

�!

�2 þ ��ð �!;�Þ

s �: (10)

Since �!� / �0 the contribution to the rate can be, inprinciple, either positive or negative.3 This will affect thesign of the cross correlations (e.g. the TB and EB angularpower spectra). If �!< �!�, the shift in the polarizationplane of the CMB will essentially be independent upon thechannel of observation.4 In the opposite case (i.e. �!> �!�)the magnetized and the pseudoscalar contribution concurin determining the total amount of rotation and, ultimately,the various polarization observables, i.e., according toEq. (10) c ðn; �Þ ¼ ��ð�Þ þ�Faradayðn; �Þ. The B-mode

polarization induced by the magnetoactive plasma in thepresence of a pseudoscalar background can be computedby means of an iterative approach which generalizes thecalculation of [7]. Since ��ðn; �Þ ¼ �Qðn; �Þ � i�Uðn�Þ,it transforms as a spin �2 for rotations around a planeorthogonal to the direction of propagation of the radiation.The three-dimensional rotations and the rotations on thetangent plane of the sphere at a given point combine to givea Oð4Þ symmetry group [11]. Generalized ladder operatorsraising (or lowering) the spin weight of a given functioncan then be defined as [11,12]

Ks�ðnÞ ¼ �ðsin#Þ�s

�@# � i

sin#@’

�ðsin#Þ�s;

n ¼ ð#;’Þ: (11)

In real space the E-mode and the B-mode polarization willhave spin weight s ¼ 0:

�Eðn; �Þ ¼ �12fKð1Þ� ðnÞ½Kð2Þ� ðnÞ�þðn; �Þ�

þ Kð�1Þþ ðnÞ½Kð�2Þ

þ ðnÞ��ðn; �Þ�g; (12)

�Bðn; �Þ ¼ i

2fKð1Þ� ðnÞ½Kð2Þ� ðnÞ�þðn; �Þ�

� Kð�1Þþ ðnÞ½Kð�2Þ

þ ðnÞ��ðn; �Þ�g: (13)

The heat transfer equation will then contain, in Fourierspace, a convolution. Since the polarization is generatedrather close to last scattering, the iterative procedure of [7]implies that, to zeroth order in the rotation rate, the polar-ization is given, in real space, as

�Pðn; �Þ ¼ 1

ð2�Þ3=2Z

d3k�Pðk;�; �0Þ;

SP ¼ �P0 þ �P2 þ�I2;

�Pðk;�; �0Þ ¼ 3

4ð1��2Þ

Z �0

0Kð�ÞSPðk; �Þe�ik�ð���0Þd�;

(14)

whereKð�Þ is the visibility function. In terms of �Pðn; �Þ,Eqs. (12) and (13) imply

�Eðn; �Þ ¼ �@2�fcos½2c ðn; �Þ�ð1��2Þ�Pðn; �Þg;�Bðn; �Þ ¼ @2�fsin½2c ðn; �Þ�ð1��2Þ�Pðn; �Þg;

(15)

where, as usual, � ¼ cos# and @� denotes a derivation

with respect to cos#. In the absence of the (inhomogene-ous) magnetized contribution, Eq. (15) leads to the expres-sions customarily used in standard analyses, i.e. forinstance,

CðBBÞ‘ ¼ sin22c �CðEEÞ

‘ ; CðEBÞ‘ ¼ 1

2 sin4c�CðEEÞ‘ ;

CðTBÞ‘ ¼ sin2c �CðEEÞ

‘ ;(16)

where c ðn; �Þ ¼ ��ð�Þ ’ ��ð�0Þ is fully homogeneous

and where the �CðEEÞ‘ and �CðTEÞ

‘ are computed from

�Pðn; �Þ. If also the magnetized contribution is taken intoaccount, the situation changes both qualitatively and quan-titatively. This aspect can be understood from Eq. (15). Inthe limit of small rotation rate

�Eðn; �Þ ¼ �@2�½ð1��2Þ�Pðn; �Þ�;�Bðn; �Þ ¼ 2@2�½c ðn; �Þð1��2Þ�Pðn; �Þ�;

(17)

where c ðn; �Þ and �Pðn; �Þ depend upon the same point onthe microwave sky. Equation (17) holds in real space. InFourier space the B-mode would be a convolution. Inmultipole space the angular power spectra inherit a pecu-liar form which contains a double sum involving also aknownWigner coefficient arising from the integral of threeLegendre polynomials [7].In Fig. 1 (plot at the left) the B-mode autocorrelations

are reported in the case when the magnetized backgroundcompetes with the pseudoscalar background. The B-modesignal is frequency dependent. In the plot at the right thecross correlations of the B-mode with the other CMBobservables are illustrated. The observational frequencychannel has been taken, for illustration, �� ¼ 100 GHz. InFig. 1 BL denotes the magnetic field intensity regularizedover a the pivot length scale k�1

L ’ Mpc. If the pseudosca-lar contribution is totally subleading the CBB

‘ angular

power spectrum will diminish with the frequency as ���4.Still the cross correlations of the B-mode with the tem-perature and the E-mode polarization (i.e. CTB

‘ and CEB‘ )

3The relation of �!� to an explicit model of quintessence canbe found, for instance, in [9]. Typical values of the mass of � areof the order of 10�33 eV in such a way that, between redshifts 0and 3, � can start dominating the background with energydensity m2�2

0 �4, where � ’ 10�3 eV is the energy scaleand �0 ’ MP 1018 GeV is the expectation value of �.

4Different experiments are characterized by different channelsof observations. For instance, Quad [3] employs two series ofbolometers located, respectively, at 100 and at 150 GHz.

BRIEF REPORTS PHYSICAL REVIEW D 79, 087301 (2009)

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are nonvanishing. This aspect is illustrated in Fig. 1 (plot atthe right) where CTB

‘ and CEB‘ are reported. The numerical

calculation leading to the results reported in Fig. 1 has beenperformed by including the magnetic field in the initialconditions and at every step of the Einstein-Boltzmannhierarchy and for the initial conditions corresponding tothe magnetized adiabatic mode.

In summary, if observations point towards a frequencydependence of the B-mode polarization Faraday rotation isprobably the only candidate. The cross correlations (i.e.,the EB and TB spectra) are expected to vanish in the caseof a stochastic magnetic field leaving unbroken spatialisotropy [5]. If they are observed this means that therotation rate is quasihomogeneous and a pseudoscalarbackground field may be around. In the latter case, if �!>�!� the B-mode will scale with frequency as dictated by the

Faraday effect while the EB and TB correlations will allowone to measure independently!�. In the opposite case (i.e.�!< �!�) the frequency dependence induced by theFaraday effect is overwhelmed by the (homogeneous)pseudoscalar rate. In this second case the effects of theprimordial magnetic fields will be imprinted on the EE andTT angular power spectra [5,6] but the B-mode autocorre-lations will be independent of the frequency. This demon-strates that the analysis of the B-mode autocorrelation isnecessary but might be insufficient, if taken individually, toinfer the existence of predecoupling magnetic fields.

K. E. K. is supported by the ‘‘Ramon y Cajal’’ programand by Grants No. FPA2005-04823, No. FIS2006-05319,and No. CSD2007-00042 of the Spanish Science Ministry.

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(1997).

100

101

102

103

104

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

102

103

B−mode autocorrelations (ν =100 GHz)

l

l(l+

1)C

l(BB

) /(2π

)(µK

)2

nB= 2, B

L = 5 nG, |Φσ|= 0.109

nB=1.1, B

L = 1.1 nG, |Φσ|=0.016

100

101

102

103

104

10−4

10−3

10−2

10−1

100

101

102

103

104

l

l(l+

1)C

l(XY

) /(2π

)(µK

)2

B−mode cross−correlations (ν =100 GHz)

EB correlationn

B= 1.1, B

L = 1 nG, |Φσ|= 0.016

TB correlationn

B = 2, B

L=5 nG, |Φσ|=0.109

FIG. 1 (color online). The angular power spectra of the B-mode autocorrelations (plot at the left) and the absolute values of the crosscorrelations (plot at the right) are reported in the case when the pseudoscalar background and the magnetized background aresimultaneously present. The �CDM parameters have been chosen in accordance with the best fit to the WMAP 5-yr data alone [1].

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