Primordial magnetic fields and CMB anisotropies

0.27 MB·English

by  Kandaswamy Subramanian

Astron.Nachr./ANxxx(xxxx)x,xxx–xxx
Primordial magnetic fields and CMB anisotropies
KANDASWAMY SUBRAMANIAN
IUCAA,PostBag4,PuneUniversityCampus,Ganeshkhind,Pune411007,India
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J Abstract. Possible signatures of primordial magnetic fields on the Cosmic Microwave Background (CMB) temperature
andpolarizationanisotropiesarereviewed.Thesignalsthatcouldbesearchedforincludeexcesstemperatureanisotropies
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particularlyatsmallangularscalesbelowtheSilkdampingscale,B-modepolarization,andnon-Gaussianstatistics.Afield
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atafewnGlevelproducestemperatureanisotropiesatthe5µKlevel,andB-modepolarizationanisotropies10timessmaller,
1 andisthereforepotentiallydetectableviatheCMBanisotropies. Anevensmallerfield,withB0 < 0.1nG,couldleadto
v structureformationathighredshiftz >15,andhencenaturallyexplainanearlyre-ionizationoftheUniverse.
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Keywords:cosmicmicrowavebackground–cosmology:theory–magneticfields
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(cid:13)c0000WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
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1. Introduction fluxfreezing,duringgalaxyand cluster formationmayease
/
h the problemsof the dynamo.It is then worth considering if
p
Magnetic fields are ubiquitous in astrophysical systems but onecandetectorconstrainsuchprimordialfields.
-
o theiroriginisstillamystery.Onepossibilityisthattheyarise Here we review possible probes of primordial mag-
r duetothedynamoamplificationofsmallseedfields.Thedy- netic fields, which use CMB temperature and polarization
t
s namo paradigm has been extensively studied (cf. Branden- anisotropies.Indeedtherearealsoanumberofpuzzlingfea-
a
: burg&Subramanian2005forarecentreview),butpotential tures associated with current CMB observations, which are
v problemsremaintobe surmounted.These involvetheques- not fully understood.These include (a) the excess power in
i
X tionofwhethermeanfielddynamocoefficientsforthegalac- temperatureanisotropiesdetectedbytheCosmicBackground
r ticdynamoarecatastrophicallyquenchedornot,andwhether Imager(CBI)experimentonsmallangularscales(Readhead
a fields generated by the fluctuation dynamo in clusters have etal.2004),(b)theobservationsbytheWilkinsonMicrowave
a sufficient degree of coherence to explain the observations AnisotropyProbe(WMAP)satelliteforanunexpectedlyhigh
(Subramanian,Shukurov&Haugen2006;Shukurov,Subra- redshiftofre-ionization(Kogutetal.2003),(c)thelowCMB
manian&Haugen2006;Schekochihinetal.2005). quadrupole anisotropy seen compared to theoretical expec-
Aninterestingalternativeisthattheobservedlarge-scale tations (cf. Spergelet al. 2003), and (d) various asymmetry
magnetic fields are a relic from the early Universe, aris- andalignmenteffects(cf.deOliveira-Costaetal.2004).Such
ing perhaps during inflation or some other phase transition featuresalso encourageustokeepopenpossibilitiesofnew
(Turner&Widrow1988;Ratra1992;andreviewsbyWidrow physicaleffects,perhapshavingtodowithprimordialfields!
2002;Giovannini2005a).Itiswellknownthatscalar(density
or potential) perturbations and gravitational waves (or ten-
sor perturbations) can be generated during inflation. Could 2. MagneticfieldevolutionintheearlyUniverse
magnetic field perturbations could also be generated? In-
deedmechanismsinvolving,say,stringtheory,forthegener- Primordialmagnetogenesisscenariosgenerallyleadtofields
ationofprimordialfieldshavebeenreviewedinthismeeting which are Gaussian random, characterized by a spectrum
(Gasperini 2006 and references therein). These generically M(k) (see below). This spectrum is normalized by giving
involvethebreakingoftheconformalinvarianceoftheelec- thefieldstrengthB0,atsomefiducialscaleandasmeasured
tromagnetic action, and the predicted amplitudes are rather atthepresentepoch,assumingitdecreaseswithexpansionas
modeldependent.Nevertheless,ifaprimordialmagneticfield B = B0/a2(t), wherea(t)is theexpansionfactor.We take
withapresent-daystrengthofB ∼ 10−9 Gandcoherenton B0 to be a free parametersince the predictionsforits value
Mpcscalesisgenerated,itcanstronglyinfluenceGalaxyfor- from magnetogenesistheories are highly modeland param-
mation. An even weaker field, sheared and amplified due to eter sensitive. Note that the magnetic and radiation energy
3 CMBSIGNALSFROMTANGLEDMAGNETICFIELDS
densitiesbothscalewithexpansionas1/a4.Sowecanchar- theCMBleadtothe’primary’anisotropiesintheCMBtem-
acterizethemagneticfieldeffectbytheratioB02/(8πργ0) ∼ peratureasseenatpresentepoch.Furthermore,variationsin
10−7B−29whereργ0isthepresent-dayenergydensityinradi- interveninggravitationaland scattering effects, which influ-
ation,andB−9 =B0/(10−9G).Magneticstressesarethere- encetheCMBphotonsastheycometousfromthelastscat-
fore small comparedto the radiation pressure for nG fields. teringsurface,can leadto additionalsecondaryanisotropies
The scalar, vector and tensor parts of the perturbed stress (cf.Subramanian2005forareview).TheCMBtemperature
tensor associated with magnetic fields lead to correspond- anisotropies∆T(θ,φ)/T areexpandedinsphericalharmon-
ingmetricperturbations.Furtherthecompressiblepartofthe ics,
Lorentzforceleadstocompressible(scalar)fluidvelocityand ∆T
(θ,φ)= a Y (θ,φ); a∗ =(−1)ma , (1)
associateddensityperturbations,whileitsvorticalpartleads T lm lm lm l−m
to vortical(vector)fluidvelocityperturbation.Themagneti- Xlm
and expressed in terms of their angular power spectrum C
callyinducedcompressiblefluidperturbations,fornGfields, l
are highly subdominantcompared to those due to the infla- wherethe ensemble averagehalma∗l′m′i = Clδll′δmm′. The
meansquaretemperaturefractionalanisotropyisgivenby
tionaryscalarmodes.Moreimportantin ourcontextwillbe
the Alfve´n mode driven by the rotational componentof the h(∆T)2i 2l+1 l(l+1)C
l
= C ≈ dlnl
Lorentzforce,especiallysincetheydecaywithoutsuchmag- T2 l 4π Z 2π
Xl
netic driving.We recall below some featuresof their evolu-
with the last approximate equality valid for large l. So
tion(Jedamzik,Katalinic&Olinto1998(JKO);Subramanian
(l(l + 1)C)/2π measures the power in the temperature
&Barrow1998(SB98a)). l
anisotropies per logarithmic interval in l space. (This com-
TheAlfve´nmodeoscillatesnegligiblyonMpcScalesby
bination is used because scale-invariant potential perturba-
recombination,withthephaseofitsoscillationχ=kV η ∼
A tions generate anisotropies, which at large scales (small l)
10−2B−9(k/0.2hMpc−1)(η/η∗). Here the Alfve´n velocity have a nearly constant l(l + 1)C). A convenient charac-
is VA ∼ 3.8×10−4cB−9, k the comovingwavenumber,η terization of the scale-dependent tlemperature anisotropy is
theconformaltimewithη itsvalueatrecombination.Unlike
∗ ∆T(l) = T[l(l+1)C )/2π]1/2.ThisisplottedinFigure1,
thecompressionalmode,whichgetsstronglydampedbelow l
asadashed-triple-dottedlineforastandardΛCDMmodel.
the Silk scale L due to radiative viscosity (Silk 1967), the
S Primordialmagneticfieldsinduceavarietyofadditional
Alfve´n mode behaves like an over-damped oscillator. Note
signals on the CMB. An uniform field would for exam-
thatforanover-dampedoscillatorthereisonenormalmode
ple select out a special direction, lead to anisotropic expan-
which isstronglydampedandanotheronewherethe veloc-
sion around this direction, hence leading to a quadrupole
ity starts from zero and freezes at the terminal velocity till
anisotropy.ThedegreeofisotropyoftheCMBthenimplies
the damping becomes weak at a latter epoch. The net re-
alimitofseveralnGonsuchafield(Barrow,Ferreira&Silk
sultisthattheAlfve´nmodesurvivesSilkdampingforscales
1997).Comparablelimitsmayobtain,atleastfortheuniform
bigger than L ∼ (V /c)L ≪ L , the canonical Silk
A A S S component,fromupperlimitstotheIGMFaradayrotationof
damping scale (JKO; SB98a). The resulting baryon veloc-
high redshift quasars (cf. Blasi et al. 1999). For inhomoge-
ity is potentiallydetectable,since dueto the Dopplereffect,
neous,tangledprimordialfields,thespatialinhomogeneities
CMBtemperatureanisotropies∆T/T ∼V/c∼f χV /c∼
d A aroundthesurfaceoflastscattering,duetomagneticallyin-
10−3fdχ(B−9/3) are induced. Here fd < 1 takes into ac- duced perturbations(scalar, vector and tensor), lead to both
countpossible dampingeffectsdue the thickness of the last
largeand small angularscale anisotropiesin the CMB tem-
scattering surface. One sees that for nG fields, significant
peratureandpolarization.
temperature anisotropies with ∆T/T ∼ 10−6 can result,
For magneticallyinducedscalar perturbations,the ques-
even for f χ ∼ 10−3. Detailed computations bear out this
d tionoftheinitialconditionshasonlybeenanalyzedinsome
simpleestimateandshowthatthesignalpeaksonarcminute
detailrecentlyinapaperbyGiovannini(2004),anddetailed
scales,l ∼103(correspondingtotheanglesubtendedbythe
predictionsoftheCMBanisotropies,forgeneralinitialcondi-
Silkscaleatrecombination),andbecauseL ≪ L ,itis
A Silk tions,areyettobeworkedout.(Forsomeapproximatetreat-
significantevenbelowtheSilkscale.
ments of the magnetosonic scalar modes see Adams et al.
Afterrecombination,whenradiationdecouplesfromthe
1996; Koh & Lee 2000; Yamazaki, Ichiki & Kajino 2005).
baryons, the cosmic pressure drops by a large factor of or-
Further, scalar modes are dominated by the standard infla-
der the photon to baryon ratio, n /n ≫ 1. The surviving
γ b tionaryscalar perturbationandarealsodampedbyradiative
tangled magnetic fields can now drive strong compressible
viscositybelowtheSilkscale.
motions and seed density fluctuations (Wassermann 1978;
Much more work has been done on the vector modes
SB98a; Sethi 2003), which could well be very important
(Subramanian & Barrow 1998 (SB98b), 2002 (SB02); Se-
in forming the first structures and leading to an early re-
shadri & Subramanian 2001 (SS01); Mack, Kahniashvilli
ionizationoftheUniverse(Sethi&Subramanian2005).
& Kosowsky 2002; Subramanian, Seshadri & Barrow 2003
(SSB03);Lewis2004).Theytypicallylead toa temperature
3. CMBsignalsfrom tangled magneticfields anisotropy∆T ∼5µK(B−9/3)2atl ∼1000andabove(see
below). A comparable signal arises at large angular scales,
CMBanisotropiesingeneralarise intwoways.Firstly,spa- l < 100, due to gravitational wave perturbations (tensors).
tial inhomogeneitiesaroundthe surface of last scattering of All modes lead to much smaller polarization signals. The
3.1 Vectormodes 3 CMBSIGNALSFROMTANGLEDMAGNETICFIELDS
tensor and vector components in particular also lead to B- Here v(k,η) is the magnitude of the rotational component
type polarization,which can help distinguishmagnetic field of the fluid velocity vi in Fourier space, and η0 the present
inducedsignalsfromthoseduetoinflationaryscalarmodes. value of η. The ’visibility function’ g(η0,η) determines the
In addition, the presence of tangled magnetic fields in probability that a photon reaches us at epoch η0 if it was
the intergalactic medium can cause Faraday rotation of the last scattered at the epoch η. So it weighs the contribu-
polarizedcomponentofthe CMB, leadingto the generation tion at any conformaltime η by the probabilityof last scat-
of new B-type signals from the inflationary E-mode signal. tering from that epoch. We have shown as a solid line in
Theirdampinginthepre-recombinationeracanleadtospec- Fig. 3 the visibility function for a standard ΛCDM model.
tral distortions of the CMB (Jedamzik, Katalinic & Olinto The spherical Bessel function of order l, the jl(z) term,
2000),whiletheirdampinginthepost-recombinationeracan projectsvariationsin space, at the conformaltime η around
changetheionizationandthermalhistoryoftheUniverse.A the last scattering epoch, to angular (or l) anisotropies at
potentially important consequence which we discuss below thepresentepoch.ThesesphericalBesselfunctionsgenerally
isthemagneticfieldinducedstructureformation,whichmay peakaroundk(η0−η)≈l.Themultipoleslarethenprobing
be relevantto explain the early re-ionizationimplied by the generallyspatialscaleswithwavenumberk ∼ l/(η0−η)at
WMAPdata.Wediscusssomeoftheseeffectsfurtherbelow, aroundlastscattering.
focusing in more detail on vector modes and post recombi- Weassumethatb0isaGaussianrandomfield.ItsFourier
nation effects, where we have been directly involved in the components satisfy < bi(k)b∗j(q) >= δk,qPij(k)M(k),
computationofthemagneticsignals. where the magnetic power spectrum is normalized using
a top hat filter in k-space, and taking k3M(k)/(2π2) =
(B2/2)(n+3)(k/k )3+n withn > −3.We generallynor-
0 G
3.1. Vectormodes malizethefieldatk = 1hMpc−1.Thespectrumiscut-off
G
atk ,determinedbydissipativeprocesses.
On galactic scales and above, the induced velocity due to c
An analytic estimate of the temperature anisotropy, for
the Lorentz forces is generally so small that it does not
lead to any appreciable distortion of the initial field (JKO; kLS <1isthen(SB98b;SB02)
SB98a). Hence, the magnetic field simply redshiftsaway as B−9 2 l l
B(x,t) = b0(x)/a2. TheLorentzforceassociatedwith the ∆TB(l)≈5.8µK(cid:18) 3 (cid:19) (cid:18)500(cid:19)I(R ), (4)
∗
tangled field FL = F/(4πa5), with F = (∇×b0)×b0, whereasforscalessmallerthanSilkscaleswithkL >1,
pushes the fluid to create rotational velocity perturbations. S
2 −3/2
ThesecanbeestimatedbyusingtheNavier-Stokesequation ∆T (l)≈13.0µK B−9 l I( l ). (5)
forthebaryon-photonfluidintheexpandingUniverse, B (cid:18) 3 (cid:19) (cid:18)2000(cid:19) R
∗
4 ∂v ρ da k2η P Fˆ HereI(k)isamodecouplingintegral
ρ +ρ i + b + v = ij j. (2)
(cid:18)3 γ b(cid:19) ∂t (cid:20) a dt a2 (cid:21) i 4πa5 I2(k) = 8(n+3)( k )6+2n; n<−3/2
3 k
Here, ρ is the photon density, ρ the baryon density, and G
γ b 28(n+3)2 k k
η = (4/15)ργlγ the shear viscosity coefficient associated = ( )3( c )3+2n; n>−3/2.
withthedampingduetophotons,wherelγisthephotonmean 15(3+2n) kG kG
free path. The projection tensor, P (k) = [δ −k k /k2] Note that for n < −3/2, I is independent of k . For a
ij ij i j c
projects Fˆ, the Fourier component of F onto its transverse nearlyscale-invariantspectrum,say with n = −2.9,we get
componentsperpendiculartok. ∆T(l) ∼ 4.7µK(l/1000)1.1 for scales larger than the Silk
One can solve Eq. 2 in two asymptotic limits, which al- scale, and ∆T(l) ∼ 5.6µK(l/2000)−1.4 for scales smaller
lowssemi-analyticestimatesofthesignals.Forscaleslarger than LS but largerthan Lγ. Largersignals will be expected
thantheSilkscale,kL < 1,theradiativeviscousdamping forsteeperspectra,n>−2.9atthehigherlend.
S
can be neglected to get vi = 3PijFˆjD(η)/(16πρ0), where One can also do a similar calculation for the expected
D(η) = η/(1+R ). Since |Fˆ | ∼ kV2, we get for large CMB polarizationanisotropy (SS01;SSB03). Note that po-
∗ ij A
larizationoftheCMBarisesduetoThomsonscatteringofra-
scales v/c ∼ χV /c, as in our earlier simple estimate. For
A
diationfromfreeelectronsandissourcedbythequadrupole
scales smaller than the Silk scale, kL > 1, one assumes a
S
componentoftheCMBanisotropy.Forvectorperturbations,
terminalvelocityapproximationtobalanceviscousdamping
theB-typecontributiondominatesthepolarizationanisotropy
with thedrivingduetotheLorentzforce.Thisthenleadsto
D(η) ∼ 5/ck2L , and v/c ∼ (5/kL )V2/c2, where L is (HW97), unlike for inflationary scalar modes. Further, as
γ γ A γ
mentioned earlier, the vector mode signals can also be im-
theco-movingphotonmeanfreepath.
portantbelowtheSilkdampingscale(l≥103).
TheangularpowerspectrumC ofCMBanisotropiesdue
l
We show in Figs. 1 and 2 the temperatureand polariza-
torotationalvelocityperturbationsisgivenby(Hu&White
tionanisotropyforthe magneticfield inducedvectormodes
1997(HW97))
obtained by evaluating the η and k integrals in Eq. (3) nu-
∞ k2dk l(l+1) merically.WeretaintheanalyticapproximationstoI(k)and
C = 4π
l Z0 2π2 2 vi(k). These curves show the build up of power in temper-
× <|Z0η0dηg(η0,η)v(k,η)jl(kk((ηη00−−ηη)))|2 > (3) afrtoumretaanndgleBd-tmypaegnpeotliacrfiizealtdiosnwdhuicehtsourvvoirvteicSalilkpedratumrbpaintigonast
3 CMBSIGNALSFROMTANGLEDMAGNETICFIELDS 3.2 Tensormodes
Fig.1. ∆T versus l predictions for different cosmological Fig.2. ∆TBB versus l predictions for different cosmologi-
P
models and M(k) ∝ kn, for B−9 = 3. The bold solid line cal models and magnetic power spectrum M(k) ∝ kn, for
is for a canonicalflat, Λ-dominatedmodel,with ΩΛ = 0.7, B−9 = 3. The bold solid line is for a standard flat, Λ-
Ωm =0.3,Ωbh2 =0.02,h=0.7andalmostscale-invariant dominated model, with ΩΛ = 0.73, Ωm = 0.27, Ωbh2 =
spectrumn=−2.9.Thedottedcurve(....)obtainswhenone 0.0224,h = 0.71 and almost scale invariantspectrum n =
changes to Ωm = 1 and ΩΛ = 0 model. The dashed line −2.9. The dashed curve obtains when one changes to n =
is for the Λ-dominated model with a larger baryon density −2.5. The dotted curve gives results for a Ω = 1 and
m
Ωbh2 = 0.03, and a larger n = −2.5. We also show for ΩΛ = 0 model, with n = −2.9. We also show for qualita-
qualitativecomparison(dashed-tripledottedcurve),thetem- tive comparison (dashed-dottedcurve), the B-type polariza-
peratureanisotropyina’standard’ΛCDMmodel,computed tionanisotropyduetogravitationallensing,inthecanonical
using CMBFAST (Seljak & Zaldarriaga 1996) with cosmo- ΛCDM model, computed using CMBFAST (Seljak & Zal-
logical parameters as for the first model described above. darriaga1996).Thesignalduetomagnetictanglesdominate
(AdaptedfromSB02andSSB03) forllargerthanabout1000.Finally,thethinsolidlinegives
the expected galactic foreground contribution estimated by
Prunetetal.(1998),whichisalsosmallerthanthepredicted
highl∼1000−3000.Theeventualslowdeclineisduetothe signals.(AdaptedfromSSB03)
dampingbyphotonviscosity,whichisonlyamilddeclineas
the magneticallysourced vortical mode is over damped.By
contrast, in the absence of magnetic tanglesthere is a sharp 3.2. Tensormodes
cut-off due to Silk damping.Our numericalresults are con-
Tangled magnetic fields also produce anisotropies on large
sistent analytic estimates givenin Eqs. (4) and (5). A scale-
angular scales, or small l, dominated by tensor metric per-
invariant spectrum of tangled fields with B0 = 3 × 10−9
turbations induced by anisotropic magnetic stresses (Dur-
Gauss,producesreducestemperatureanisotropiesatthe5µK
rer, Ferreira & Kahniashvili 2000; Mack, Kahniashvili &
level and B-type polarization anisotropies ∆T ∼ 0.3 −
B Kosowsky2002;Caprini&Durrer2002;Giovannini2005b).
0.4µK between l ∼ 1000−3000. Larger signals result for
Thetensormetricperturbationh obeystheequation
steeperspectrawithn>−3.Notethattheanisotropiesinhot ij
orcoldspotscouldbeseveraltimeslarger,becausethenon- h′′ +2Hh′ −∇2h =−16πGa2δTTT
ij ij ij ij
linear dependence of C on M(k) will imply non-Gaussian
l where H = a′/a, a prime denotes derivative with respect
statisticsfortheanisotropies.
tothe conformaltime,andδTTT isthetransverse,traceless
ij
Thesemagneticallyinducedsignalscouldthereforecon- componentoftheenergymomentumtensor(duetothemag-
tribute to the excess power detected by the CBI experi- neticfield).TheresultingCMBanisotropyisthencomputed
ment.Adistinguishingfeature,comparedtosaytheSunyaev- using
Zeldovich (SZ) effect (Zeldovich & Sunyaev 1969) canoni-
1 η0
callyinvokedtoexplaintheCBIexcess,willbeprovidedby (∆T/T)=− h′ ninjdη
theB-typepolarizationsignals.Notethatiftheobservedex- 2Zηi ij
cesspowerintheCBI experimentarisesfromtheSZeffect, where ni is a unit vector along the line of sight, and prime
thisisnotexpectedtobestronglypolarized.Thespectralde-
denotes a conformal time derivative. Using the formalism
pendence of the SZ signal could also help to distinguish it
describedin these papers, we estimate a tensor contribution
from any magnetically induced signals, which are expected at small l < 100 of ∆T ∼ 7(B−9/3)2(l/100)0.1µK, for
tobefrequencyindependent,atleastathighfrequencies.
n = −2.9. (One has to account for neutrino anisotropic
Recently, Lewis (2004) has done a detailed numerical stresscompensation(Lewis2004)afterneutrinodecoupling).
computationofthevectormodesignalandfindsqualitatively Since we have to add this powerto the standardpowerpro-
similarresultstothesemi-analyticalestimatesgivenabove. duced by inflationary scalar perturbations in quadrature, a
5 CONCLUSIONS
tangled field with B−9 ∼ 3 will produce of order a few to
10 percent perturbation to the power in the standard CMB
anisotropy at large angular scales. So if they are indeed de-
tectedatlargel,belowtheSilkdampingscale,onewillalso
havetoconsidertheireffectsseriouslyatlargeangularscales,
especiallyin cosmologicalparameterestimation.Thetensor
mode also contributes to the B-type polarization anisotropy
atlargeangularscales(l < 100orso),with∆T < 0.1µK
B
forB−9 <3.Theproductionofgravitationalwaveshasbeen
usedinanindirectmannerbyCaprini&Durrer(2002)toset
strongupperlimitsonB0forspectrawithn>−2.5orso.
3.3. Faradayrotationduetoprimordialfields
Another interesting effect of primordial fields is the the
Faraday rotation it induces on the polarization of the CMB Fig.3.Visibilityfunctionfordifferentmodels.Thesolidand
(Kosowsky&Loeb1996;Kosowskyetal.2005;Campanelli the dotted curves are for the standard recombination and a
etal.2005).Therotationangleisabout model in which the Universe re-ionizes at z = 17, respec-
tively. The dashed curve corresponds to a decaying turbu-
η0
∆Φ = λ20 (3/e) dη′g(η0,η′)n·B0[n(η′−η0)] lence modelwith B0 = 3×10−9G. The dot-dashedcurve
Z0 correspondstotheambipolardiffusioncase with B0 = 3×
≈ 1.6oB−9(ν/30GHz)−2 10−9Gandn=−2.8.(AdaptedfromSS05)
where λ0 is the wavelength of observation.So this effect is
important only at low frequencies, and here it can lead to
powerspectra,ascomputedbySS05.WhenRisnormalized
thegenerationofB-modepolarizationfromtheFaradayrota-
to the magnetic Jeans scale, λ , it turns out that σ depends
tionoftheinflationaryE-mode.FromtheworkofKosowsky J
onlyonthe ratioR/λ andnotexplicitlyonthe strengthof
et al. (2005) one can estimate a B-mode signal ∆T ∼ J
B thefield.Thisinterestingfeaturearisesbecausedensityfluc-
0.4(B−9/3)(ν/30GHz)−2µK,forn = −2,atl ∼ 104.The tuations are generated by the divergence of F and so are
signalsaresmalleratsmallern.TheFaradayrotationsignal L
canbedistinguishedfromtheB-modepolarizationgenerated ∝ k2B02. Since the magnetic Jeans scale λJ ∝ kJ−1 ∝ B0,
the magneticfield dependencecancelsoutin σ whenscales
bysayvectormodes,orgravitationallensing,becauseoftheir
frequencydependence(ν−2). areexpressedintermsofR/λJ (seeSS05fordetails).Struc-
turescollapsewhenσ >1(forthethesphericaltophatmodel
whenσ =1.68).Forthenearlyscale-freepowerlawmodels,
4. Postrecombination blues eventypicalstructuresatthemagneticJeansscalecollapseat
highredshiftsintherangebetween10to20.
AfterrecombinationtheUniversebecamemostlyneutral,re- The mass of these objects does depend on the magnetic
sulting also in a sharp drop in the radiative viscosity. Pri- fieldstrengthsmoothedtotheJeansscale,andlieintherange
mordial magnetic fields can then dissipate their energy into 109M⊙ to 3 × 1010M⊙ for B0 ∼ 10−9G to B0 ∼ 3 ×
theintergalacticmedium(IGM)viaambipolardiffusionand, 10−9G. An even smaller field could have a major impact,
forsmallenoughscales,bygeneratingdecayingMHDturbu- providedthecollapsingstructureshaveamasslargerthanthe
lence. These processes can significantly modifythe thermal thermal Jeans mass. For example, a field as small as B =
and ionization history of the post-recombination Universe. 0.1nGcaninducea106M dwarfgalaxycollapseathighz
⊙
WeshowinFig.3themodifiedvisibilityfunctionduetothe (z >15),causingearlyenoughre-ionizationtoexplaintheT-
gradual re-ionization by ambipolar damping and turbulence EcrosscorrelationpeakobservedbyWMAP.Moredetailed
decayfromtheworkofSethi&Subramanian(2005). workonthisaspectisunderway.
These dissipative processes, for B−9 ∼ 3, can give rise
to Thomsonscatteringopticaldepthsτ >∼ 0.1,althoughnot
intherangeofredshiftsneededtoexplaintherecentWMAP 5. Conclusions
polarization observations (the T-E cross correlation seen at
lowl).However,futureCMBprobeslikePLANCK canpo- We briefly reviewed some of the possible ways one could
tentially detect the modified CMB anisotropy signal from detect/constrain primordial magnetic fields using the CMB
such partial re-ionization(Kaplinghatet al. 2003).This can anisotropies and polarization. This endeavor would be even
be used to detector furtherconstrainsmall-scale primordial morefruitfulif there were a compellingmechanismforpri-
fields. mordialmagnetognesis,whichalso producedstrongenough
Potentially more exciting is the possibility that primor- (B−9 ∼ 1) and coherent enough (ordered on Mpc scales)
dial fields could induce the formation of subgalactic struc- fields.Atthisjuncture,theoreticalpredictionsarehighlypa-
tures for z >∼ 15. We show in Fig. 4 the mass dispersion rameter dependent,and so we have taken a more pragmatic
σ(R,z)fortwomodelswithnearlyscale-freemagneticfield approachofassumingthatsuchafieldcouldbegeneratedin
References References
canpotentiallyleadtotheearlyre-ionizationindicatedbythe
presentWMAPpolarizationdatainaverynaturalmanner.
Acknowledgements. Much of the work reviewed here are in joint
papers with John Barrow, T. R. Seshadri and Shiv Sethi. I thank
themforveryenjoyablecollaborations, DmitriSokoloff foruseful
comments,andtheSKAProjectOfficeforpartialsupport.
References
Adams,J.,Danielsson,U.,Grasso,D.Rubinstein,H.:1996,Physics
LettersB388,253
Barrow,J.D.,Ferreira,P.G.,Silk,J.:1997,PRL78,3610
Blasi,P.,Burles,S.,Olinto,A.:1999,ApJ514,L79
Brandenburg,A.,Subramanian,K.:2005,Phys.Rep.417,1
Campanelli, L., Dolgov, A., Giannotti, M., Villante, F.: 2005,
Fig.4.Themassdispersionσ(R,z)isshownfortwomodels ApJ616,1
Caprini,C.,Durrer,R.:2002,PRD65,23517
withnearlyscalefreemagneticfieldpowerspectra.Thesolid
Durrer,R.,Ferreira,P.G.,Kahniashvili,T.:2000,PRD61,043001
and dashedcurvescorrespondto n = −2.9and n = −2.8,
Gasperini,M.:2006,AN,thisvolume(astro-ph/0509174)
respectively.Differentcurves,fromtoptobottom,correspond
Giovannini,M.:2004,PRD70,123507
to redshiftsz = {10,15,20,25,30},respectively.The hori- Giovannini,M.:2005a,Int.J.Mod.Phys.D13,391
zontallinecorrespondstoσ =1.68.(AdaptedfromSS05) Giovannini,M.:2005b,astro-ph/0508544
Grasso,D.,Rubinstein,H.R.:2001,Phys.Rep.348,161
Hu,W.,White,M.:1997,PRD56,596
the early Universe and asking what it would imply for the Jedamzik,K.,Katalinic´,V.,Olinto,A.V.:1998,PRD57,3264
CMBandstructureformationingeneral. Jedamzik,K.,Katalinic´,V.,Olinto,A.V.:2000,PRL85,700
For a field of B ∼ 3 nG and a nearly scale-invariant Kahniashvilli,T.:2006,AN,thisvolume(astro-ph/0510151).
Kaplinghat,M.,Chu,M.,Haiman,Z.,Holder,G.P.,Knox,L.,Sko-
spectrum one predicts CMB temperature anisotropies with
rdis,C.:2003,ApJ583,24
a ∆T ∼ 5µK, at l < 100 and l > 1000 and polarization
Kosowsky,A.,Loeb,A.:1996,ApJ469,1
anisotropieswith∆T ∼ 0.4µKatl > 1000.Especiallyin-
P Kosowsky, A, Kahniashvili, T., Lavrelashvili, G., Ratra, B.: 2005,
terestingisthatthevectormodesinducedbyprimordialfields PRD71,043006
can contribute significantly below the Silk scale, where the Lewis,A.:2004,PRD70,043011
conventional scalar modes are exponentially damped. Fur- Mack,A.,Kashniashvili,T.,Kosowsky,A.:2002,PRD65,123004
ther,themagneticallyinducedsignalatsmallangularscales Prunet, S., Sethi, S.K., Bouchet, F.R., Miville-Deschenes, M.A.:
will be dominated by B-mode polarization. There do exist 1998,A&A339,187
Ratra,B.:1992,ApJ391,L1
intriguing results from the CBI experimentfor the presence
Readhead,A.C.S.,etal.:2004,ApJ609,498
of a temperature excess at small angular scales. Some part
Schekochihin, A., Cowley, S.,Kulsrud, R.,Hammett, G.,Sharma,
of this excess could arise due to the influence of primordial
P.:2005,ApJ629,139
fields.Thisexcessisconventionallyexplainedasarisingdue Seljak,U.,Zaldarriaga,M.:1996,ApJ469,437
totheSZeffect,butthepowerindensityfluctuationsonclus- Silk,J.:1968,ApJ151,431
terscales(conventionallymeasuredbyσ8),hastobepushed Seshadri,T.R.,Subramanian,K.:2001,PRL87,101301
tobeintheupperrangeofvalues(σ8 ∼ 1),allowedbycur- Sethi,S.K.:2003,MNRAS342,962
Sethi,S.K.,Subramanian,K.:2005,MNRAS356,778
rentCMBandlarge-scalestructuredata.SincetheSZsignal
Shukurov, A., Subramanian, K., Haugen, N.E.L.: 2006, AN, this
is frequency dependent, a crucial test would be to compare
volume(astro-ph/0512608).
CMBobservationsatdifferentfrequencies.
Spergel,D.N.,etal.:2003,ApJS148,175
Also if B-type polarization at these scales is detected Subramanian,K.:2005,CurrentScience88,1068
this could be a good indicator of the magnetic field effects. Subramanian,K.,Barrow,J.D.:1998,PRD58,83502
Clearly it will be important to make further observationsat Subramanian,K.,Barrow,J.D.:1998,PRL81,3575
smallangularscales,especiallyatdifferentfrequencies.Itis Subramanian,K.,Barrow,J.D.:2002,MNRAS335,L57
Subramanian,K.,Seshadri,T.R.,Barrow,J.D.:2003,MNRAS344,
alsoimportanttostudythestatisticsoftheCMBanisotropies,
L31
since the magnetically induced signals are predicted to be
Subramanian,K.,Shukurov,A.,Haugen,N.E.L.:2006,MNRAS(in
strongly non-Gaussian. If magnetic fields have helicity, this
press)(astro-ph/0505144)
wouldinducefurtherinterestingeffects,whichhavebeenre- deOliveira-Costa,A.,Tegmark,M.,Zaldarriaga,M.,Hamilton,A.:
viewed in this meeting (Kahniashvilli 2006 and references 2004,PRD69,063516
therein). Turner,M.S.,Widrow,L.M.:1988,PRD37,2743
We have also emphasized another interesting conse- Wasserman,I.:1978,ApJ224,337
Widrow,L.M.:2002,Rev.Mod.Phys.74,775
quenceoftheexistenceofprimordialmagneticfields,which
Yamazaki,D.G.,Ichiki,K.,Kajino,T.:2005,ApJ625,L1
indirectly affects the CMB anisotropies. Due to their pres-
Zeldovich,Y.B.,Sunyaev,R.A.:1969,Ap&SS4,301
ence the first collapsed objects of dwarf galaxy masses and
smallercanformathighz > 15,evenforB ∼ 0.1nG.This