Table Of ContentAstron.Nachr./ANxxx(xxxx)x,xxx–xxx
Effects of cosmological magnetic helicity on the CMB
TINA KAHNIASHVILI1,2
1 DepartmentofPhysics,KansasStateUniversity,Manhattan,KS,USA
2 CenterforPlasmaAstrophysics,AbastumaniAstrophysicalObservatory,Tbilisi,Georgia
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Abstract. Inthis talkI present a short review of primordial magnetic helicityeffects on Cosmic Microwave Background
3 (CMB)temperatureandpolarizationanisotropies.Theseeffectsallowustotestforcosmologicalmagnetichelicity,however,
v veryaccurateCMBfluctuationdataisrequired.Thisschemeformagnetichelicitydetectionisvalidonlyforacosmological
1 magneticfieldwithapresentamplitudelargerthan10−9−10−10Gauss.
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Keywords:cosmologicalmagneticfields,magnetichelicity,CMBtemperatureandpolarizationanisotropies
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(cid:13)c0000WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
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h 1. Introduction (CMB photons), taking into account that CMB anisotropy
p measurementshavethesameorderofaccuracy,10−6−10−5,
-
o Recentastrophysicalobservationsindicatethatthemagnetic we expect that such a cosmological magnetic field would
r fields in the Sun and some galaxies and clusters of galax- leave detectable traces in CMB temperature or polarization
t
s iesmighthaveanhelicalstructure(forreviewssee(Widrow anisotropies, see (Lewis 2004; Giovannini 2005) and refer-
a
2002; Valle´e 2004)). One promising possibility to explain encestherein.
:
v the observed magnetic helicity is assuming the presence of
i HereIfocusontheeffectsonCMBtemperatureandpo-
X seed helicalmagneticfields. Severalmechanismshavebeen
larizationanisotropiesinducedbymagnetichelicity.Thistalk
proposed for generating such a primordial helical magnetic
r isbasedonresultsobtainedincollaborationwithC.Caprini,
a field duringan early epoch of the universe (Cornwall1997;
R. Durrer, G. Gogoberidze, A. Kosowsky, G. Lavrelashvili,
Giovannini & Shaposhnikov 1998; Giovannini 2000; Field
A. Mack, and B. Ratra. (Mack, Kahniashvili & Kosowsky
& Carroll 2000; Vachaspati 2001; Sigl 2002; Semikoz &
2002;Caprini,Durrer&Kahniashvili2004;Kosowskyetal.
Sokoloff2005;Campanelli&Giannotti2005).Magnetiche-
2005;Kahniashvili&Ratra2005;Kahniashvili,Gogoberidze
licity can be also be generatedvia helicalturbulentmotions
&Ratra2005).Wefindthatmagnetichelicitygeneratesvec-
if there is an inverse cascade (Brandenburg2001;Chistens-
torandtensormetricperturbationsandasaresultaffectsall
son,Hindmarsh&Brandenburg2005;Verma&Ayyer2003;
CMBfluctuations.Inparticular:(i)magnetichelicityreduces
Boldyrev & Cattaneo 2004). On the other hand, primordial
the amplitudes of parity-evenCMB fluctuation power spec-
helicityinfluencesmagneto-hydrodynamicalprocessesinthe
traincomparisontothecaseofanon-helicalmagneticfield
early plasma as well as cosmological perturbation dynam-
(Caprini et al. 2004; Kahniashvili & Ratra 2005); (ii) the
ics(Vichniac&Cho2001;Brandenburg2001;Kleorinetal.
Faraday rotation of the CMB polarization plane is strongly
2003; Subramanian 2002; Vishniac, Lazarian & Cho 2003;
dependentontheaverageenergydensityofthemagneticfield
Subramanian & Brandenburg 2004; Banerjee & Jedamzik
and is independent of magnetic helicity, see Kosowsky et
2004;Subramanian&Shukurov2005).
al. (2005); (iii) magnetic helicity induces parity-odd cross-
To preserve large-scale isotropy, a seed magnetic field
correl ations of the CMB fluctuations, which vanish for the
(and so magnetic helicity) has to be small enough to allow
case ofa magneticfield withouthelicity(Pogosian,Vachas-
treatingthe averageenergydensityandmeanhelicityof the
pati & Winitski 2002; Caprini et al. 2004; Kahniashvili &
magneticfieldasfirstorderinperturbationtheory.
Ratra2005)1;and(iv)magnetichelicitygeneratescircularly
Eveniftheenergydensityofaprimordialmagneticfield
B2/(8π) is five or six magnitude less than that of radiation
1 This is not true for a homogeneous magnetic field, Scoccola,
Correspondenceto:[email protected] Harrari&Mollerach(2004)
3 CMBANISOTROPIES
polarized stochastic gravitational waves, Kahniashvili et al. infrareddivergenceofmagneticfieldenergydensity).Finite-
(2005). nessofthemagneticfieldaveragehelicityrequiresn >−4;
H
this is automatically satisfied as a consequence of n >
H
n >−3.
2. Magneticsource formetricperturbations B
To obtain the magnetic field source terms in the equa-
tions for vector (transverse peculiar velocity) and tensor
We assume the existence of a cosmological magnetic field
(gravitationalwaves)metricperturbationsweneedtoextract
generated during or prior to the radiation-dominatedepoch,
the transverse vector and tensor parts of the magnetic field
withtheenergydensityofthefieldafirst-orderperturbation
to the standard Friedmann-Lemaˆıtre-Robertson-Walker ho- stress-energytensorτij(k).ThisisdonethroughΠ(ijV)(k) =
mogeneouscosmologicalspacetimemodel.Neglectingfluid (P (kˆ)kˆ +P (kˆ)kˆ )kˆ τ (k)(forvectorperturbations)and
ib j jb i a ab
back-reactionontothemagneticfield,thespatialandtempo- Π(T)(k)=[P (kˆ)P (kˆ)−1P (kˆ)P (kˆ)]τ (k)(forten-
ral dependence of the field separates, B(t,x) = B(x)/a2; soirjperturbatioinas);fojrbdetails2seeijMackaebtal.(2a0b02).
hereaisthecosmologicalscalefactor.Asaphenomenolog- ForvectorperturbationstheΠ(V) tensorisrelatedtothe
icalnormalizationofthemagneticfield,wesmooththefield ij
vector (divergenceless and transverse) part of the Lorentz
onacomovinglengthλwithaGaussiansmoothingkernel∝
forceL(V)(k) = k Π (k) = P (kˆ)k τ (k). Forthe nor-
exp[−x2/λ2]toobtainthesmoothedmagneticfieldwithav- i j ij ib a ab
eragevalueofsquaredmagneticfieldBλ2 ≡hB(x)·B(x)i|λ malized Lorentz force vector, Πi ≡ L(iV)/k, the general
andmagnetichelicityHλ2 ≡λ|hB(x)·[∇×B(x)]i|λ. spectrum of hΠ⋆i(k)Πj(k′)i in wavenumber space is simi-
lartoEq.(2),andhastwoparts,symmetricandhelical.Both
We also assume that the primordial plasma is a perfect
contain contributions from P (K), and so magnetic helic-
conductor on all scales larger than the Silk damping wave- H
ity affects the vector magnetic source term, Kahniashvili &
length λ (the thickness of the last scattering surface) set
S
Ratra(2005).
by photon and neutrino diffusion. We model magnetic field
Thetensormagneticsourcetermisobtainedthroughthe
dampingbyanultravioletcut-offwavenumberk =2π/λ ,
D D
Subramanian&Barrow(1998), source two-point function hΠ(T)⋆(k)Π(T)(k′)i. This is de-
ij lm
k nB+5 termined by the magnetic field two-point function, and like
D ≈ the vector magnetic source term has symmetric and helical
(cid:18)Mpc−1(cid:19)
parts(Caprinietal.2004;Kahniashvilietal.2005).Thehe-
B −2 k nB+3 lical(parity-odd)pieceresultsincircularpolarizationofthe
≈2.9×104 λ λ h. (1)
(cid:18)10−9G(cid:19) (cid:18)Mpc−1(cid:19) induced gravitational waves. For a maximally helical mag-
netic field with P (k) ≃ P (k), the polarizationdegree is
Here n is the spectral index of the symmetric part of the H B
B
highenoughtoallowustoconsiderthepossibilityoftesting
magnetic field powerspectrum(see Eq. (3) below),h is the
Hubble constant in units of 100 km sec−1 Mpc −1, k = formagnetichelicitythroughameasurementofthepolariza-
λ
tionofrelicgravitationalwaves(thismightbepossiblewith
2π/λ is the smoothing wavenumber, and λ ≪ λ . This
D S
futuregravitationalwavesdetectors);see Kahniashviliet al.
assumes that magnetic field damping is due to the damping
(2005).
ofAlfve´nwavesfromphotonviscosity.
In both cases (vector and tensor perturbations) the con-
AssumingthatthestochasticmagneticfieldisGaussianly
tribution of magnetic field helicity to the symmetric part of
distributed, and accounting for the possible helicity of the
themagneticsourceisnegative.Itis clearthatthemagnetic
field, the magnetic field spectrum in wavenumber space is,
sourcetermsvanishonscalessmallerthanthecutoffscaleλ
Pogosianetal.(2002), D
becauseofmagneticfielddamping.Dependingonspectralin-
hB⋆(k)B (k′)i=
i j dexesn and n the magneticsource termsare dominated
B H
=(2π)3δ(3)(k−k′)[Pij(kˆ)PB(k)+iǫijlkˆlPH(k)]. (2) eitherbysmallwavenumber,∝ k2nB+3 fornB < −3/2,or
Here Pij(kˆ) ≡ δij −kˆikˆj is the transverse plane projector by the high frequency cut-off and so ∝ kD2nB+3 for nB >
withunitwavenumbercomponentskˆ = k /k,ǫ isthean- −3/2.
i i ijl
tisymmetricsymbol,andδ(3)(k−k′)istheDiracdeltafunc-
tion. P (k) and P (k) are the symmetric and helical parts 3. CMBanisotropies
B H
ofthemagneticfieldpowerspectrum,assumedtobesimple
powerlawsonlargescales, ForourcomputationsweusetheformalismgiveninMacket
2π2λ3B2 al.(2002),extendingittoaccountformagneticfieldhelicity.
PB(k) ≡ PB0knB = λ (λk)nB, To compute CMB temperature and polarization anisotropy
Γ(n /2+3/2)
B
powerspectraweusethetotalangularmomentummethodof
2π2λ3H2
P (k) ≡ P knH = λ (λk)nH, (3) Hu & White (1997). Our analytical approximations are de-
H H0
Γ(nH/2+2) rivenin(Caprinietal.2004;Kahniashvili&Ratra2005).
and vanishing on small scales when k > k . Here Γ is
D
the EulerGamma function.These powerspectraare generi-
3.1. Parity-evenCMBfluctuations
callyconstrainedbyP (k)≥|P (k)|,Caprinietal.(2004),
B H
whichimpliesn > n .Inaddition,finitenessofthemag- Cosmological magnetic field vector and tensor mode con-
H B
netic field energy density requiresn > −3 (to preventan tributions to CMB fluctuations at large angular scales (with
B
4 CONCLUSION
multipole number l < 100) are of the same order of mag- CΘB(T).Theratiobetweentemperature–B-polarizationsig-
l
nitude, Mack et al. (2002), while for small angular scales nalsfromvectorandtensormodesisindependentoftheam-
(wherel>100)CMBfluctuationsaredominatedbythevec- plitudesoftheaveragemagneticfield(B )andaveragemag-
λ
tormodecontributionbecauseofgravitationalwavedamping netichelicity(H ).
λ
(Lewis2004). For small angular scales (l > 100) where the tensor
Thecompleteparity-evenCMBfluctuationpowerspectra modesignalvanishes,foramaximallyhelicalmagneticfield
maybeexpressedas, with n ≃ n , due to the suppression factor L /η
H B γ,dec 0
CXX′ =CXX′ −CXX′, (4) (where Lγ,dec is the photon mean free path at decoupling
l (S)l (A)l andη conformaltimetoday)thetemperature-E-polarization
0
where X is either Θ, E, or B, which represent respec- cross-correlation power spectrum, CΘE, is smaller than
l
tively the temperature, E-polarization, and B-polarization thetemperature-B-polarizationcross-correlationpowerspec-
anisotropies,andCXX′ aretheantisymmetricpowerspectra trum, CΘB, but both are ∝ l2, if n + n > −5. The
(A)l l B H
inducedbymagnetichelicity.Theminussignreflectstheneg- samesuppressionfactormakesCΘB smallerthanCΘΘ.For
l l
ativecontributionofmagnetichelicitytothetotalparity-even an arbitrary helical field CΘB/CΘE depends on the ratio
l l
CMB fluctuation power spectra (from terms proportionalto (P /P )knH−nB and order unity prefactors that depend
H0 B0 D
d3pP (p)P (|k−p|)).Forlargeangularscalesthisresult onn andn .Adependenceonlappearsonlyifn +n <
H H B H B H
hRoldsforboth(vectorandtensor)modes,whileforl > 100 −5, when the ratio, CΘB/CΘE decreases as ∝ lnB+nH+5,
l l
itappliesonlyforvectorperturbations.Thefractionaldiffer- Kahniashvili&Ratra(2005).
ences κXX′ ≡ 1 − CXX′/CXX′, where 0 < κXX′ < 1,
l (A)l (S)l l
can be used to characterize the reduction of the parity-even
3.2.2. E andB-polarizationcross-correlations
CMBfluctuationpowerspectraamplitudesasaconsequence
of non-zero magnetic helicity. The ratio CXX′/CXX′ may For a tensor mode signal at large angular scales (l <
(A)l (S)l
beexpressedintermsofP /P andspectralindexesn 100),E-andB-polarizationcross-correlationCEB isofthe
0H 0B H l
and n (Caprini et al. 2004; Kahniashvili & Ratra 2005). same order of magnitude as the tensor mode temperature–
B
Whilethereductionofparity-evenpowerspectraamplitudes B-polarizationanisotropycross-correlationspectrum,CΘB,
l
are significant for a maximally helical field, the most inter- Caprini et al. (2004). The situation is different for a vector
esting effects from magnetic helicity are the generation of mode which survives up to small angular scales (e.g., Sub-
parity-odd CMB fluctuations, such as the cross-correlations ramanian & Barrow 1998; Mack et al. 2002; Lewis 2004;
between temperature and B-polarization, and between E- Giovannini 2005). In this case, the E- and B-polarization
andB-polarizations. anisotropy cross-correlation power spectrum has a suppres-
sion factor of kL implyingthat CEB ≪ CΘB. This is
γ,dec l l
consistentwiththeresultofHuandWhite(1997).
3.2. Parity-oddCMBfluctuations
Magnetic helicity induces parity-odd cross correlations be-
4. Conclusion
tween the E- and B-polarization anisotropies, as well
as between temperature and B-polarization anisotropies,
Ihavediscussedhowcosmologicalmagnetichelicityaffects
(Pogosian et al. 2002; Caprini et al. 2004; Kahniashvili &
CMB fluctuations. Even for a cosmological magnetic field
Ratra2005).Suchoff-diagonalparity-oddcrosscorrelations
with maximal helicity such effects may be detectable only
occur also in the case of an homogeneous magnetic field
if the current magnetic field amplitude is at least 10−10 or
from the Faraday rotationeffect, Scoccola et al. (2004),but 10−9 G on Mpc scales. A non-helicalseed field of this am-
not in the case of Faraday rotation in a stochastic magnetic
plitudecanbegeneratedduringinflation(Ratra1992;Bamba
field,evenonewithnon-zerohelicity,Kosowskyetal.(2005).
& Yokoyama 2004). A cosmological magnetic field gener-
Faraday rotation measurementsused to measure a magnetic
ates a B-polarization signal via induced vector and/or ten-
field amplitude cannot be used to detect magnetic helicity
sor modes, so detection of such a signal may indicate the
(Ensslin&Vogt2003;Campanellietal. 2004;Kosowskyet
presence of a cosmological magnetic field. However, it has
al. 2005).A possible way of detectingmagnetic helicity di-
to be emphasized that a B-polarization anisotropy signal
rectly fromCMB fluctuationdatais todetecttheparity-odd
can also arise in other ways, such as from primordial ten-
CMB fluctuation cross-correlations or/and to detect the ef-
sor perturbations, gravitational lensing, or Faraday rotation
fectsmagnetichelicityhasonparity-evenCMBfluctuations,
of the CMB anisotropy polarization plane; for a review see
Kahniashvili&Ratra(2005).
Subramanian (2004). The peak position of B-polarization
anisotropypowerspectrum,l2CBB,mayhelptoidentifythe
l
3.2.1. Temperature-B-polarizationcross-correlations B-polarizationsource.Forexample,cosmological-magnetic-
field-induced tensor perturbations only contribute on large
Atlargeangularscales(l<100)wherethecontributionfrom angularscalesl <100,whileB-polarizationanisotropyfrom
thetensormodeissignificant,fornB+nH >−2thevector gravitationallensinghasapeakamplitudel2CBB ∼10−14at
mode CΘB(V) and the tensor mode CΘB(T) have the same l∼1000Challinor&Lewis(2005).TheFaradlayrotationB-
l l
l dependence ∝ l2. For all other values of spectral indexes polarizationanisotropysignalfroma field with B = 10−9
λ
n andn ,thegrowthrate(withl)ofCΘB(V)isfasterthan G (at λ = 1 Mpc) and spectral index n = −2 peaks
B H l B
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22222.....55555eeeee-----1111155555 Challinor,A.,Lewis,A.:2005,astro-ph/0502425
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amplitude on the right side of the plot correspond to mag-
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neticfieldpowerspectralindicesn = 2,1,0,−1,and−2.
B 123004
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l
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B H
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Vishniac,E.,Cho,J.:2001,Astrophys.J.550,752
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andprocessesintheearlyuniverse(Cornwall1997;Banerjee
&Jedamzik2004).Tomeasureprimordialmagnetichelicity
through these effects very accurate CMB fluctuation data is
required.
Acknowledgements. Theauthorthankstheorganizersoftheconfer-
enceTheOriginandEvolutionofCosmicMagnetism forhospital-
ity,andtheSKAprojectforpartialsupporttoattendtheconference.
The author acknowledges her collaborators C. Caprini, R. Durrer,
G. Gogoberidze, A. Kosowsky, G.Lavrelashvili, A.Mack, and B.
Ratra, and thanks A. Brandenburg, A. Dolgov, M. Giovannini, D.
Grasso,K.Jedamzik,andT.Vachaspatifordiscussions.Thiswork
issupportedbyDOEEPSCoRgrantDE-FG02-00ER45824.
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