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Cosmological magnetic fields from inflation in extended electromagnetism Jose Beltrán Jiménez and Antonio L. Maroto Departamento de F´ısica Teórica, Universidad Complutense de Madrid, 28040, Madrid, Spain. (Dated: January 11, 2011) In this work we consider an extended electromagnetic theory in which the scalar state which is usually eliminated by means of the Lorenz condition is allowed to propagate. This state has been shown to generate asmall cosmological constant in thecontextof standard inflationary cosmology. Hereweshow that theusualLorenzgauge-breaking term now playstherole of an effectiveelectro- magneticcurrent. Suchacurrentisgeneratedduringinflationfromquantumfluctuationsandgives rise to a stochastic effective charge density distribution. Due to the high electric conductivity of 1 the cosmic plasma after inflation, the electric charge density generates currents which give rise to 1 both vorticity and magnetic fields on sub-Hubble scales. Present upper limits on vorticity coming 0 from temperature anisotropies of the CMB are translated into lower limits on the present value of 2 cosmicmagneticfields. Wefindthat,foranearlyscaleinvariantvorticityspectrum,magneticfields n Bλ > 10−12 G are typically generated with coherence lengths ranging from sub-galactic scales up a tothepresentHubbleradius. Thosefieldscouldactasseedsforagalactic dynamoorevenaccount J for observations just by collapse and differential rotation of theprotogalactic cloud. 0 1 PACSnumbers: 98.80.-k,98.62.En ] O Traditionally it has been argued that due to the elec- not explain the claimed extragalactic detection, and, in C tric neutrality of the universe on large scales, the only principle,onlyproductionduringinflationcouldaccount relevant interaction in cosmology should be gravitation. for observations. Generation of magnetic fields during . h However, the behaviour of electromagnetic fields on as- inflation requires the breaking of the conformal trivial- p trophysicalandcosmologicalscalesis still farfromclear, ity of standard electromagnetism in a Robertson-Walker - o the most evident example being the unknown origin of background. For that reason, modified electromagnetic r magnetic fields observed in galaxies and galaxy clusters. theories, including non-minimal curvature couplings or t s Magnetic fields with large coherence lengths (around couplings to extra fields such as inflaton or dilaton have [a 10 kpc or even larger), and strengths around 10−6 G [1] been studied in the literature [6]. havebeenmeasuredingalaxiesofalltypesandingalaxy Recently, the possibility of producing a small cosmo- 2 logical constant during inflation in the context of an ex- clusters located at very different redshifts. Also, recent v tendedelectromagneticmodelhasbeenconsideredin[9– works[2]showevidencefortheexistenceofstrongextra- 0 galacticmagneticfieldsabove3 10 16Gwithcoherence 11]. The proposed theory involves a modification of the 6 − × non-transverseelectromagneticsector,whichbreakscon- 9 lengths much larger than the cluster scales and eventu- formal triviality but respects the ordinary (transverse) 3 ally reaching the present Hubble radius. . photonsdynamics. Unlikepreviousmodels,themodified 0 Two different types of scenarios have been considered equationsremainlinear,withoutpotentialterms,dimen- 1 for the generation of such fields. On one hand, the sional parameters or explicit curvature couplings. The 0 primordial field hypothesis, i.e. the existence of relic 1 aim of this work will be to explore the possibility that magnetic fields from the early universe with comoving : large-scale cosmic magnetic fields could be generated in v strengths around 10 10 10 12 G which permeated the i protogalactic medium− an−d w−ere amplified to the present this extended theory. X Letusstartbywritingthegeneralizedelectromagnetic valuesbycollapseanddifferentialrotation. Ontheother r action which includes, apart from the coupling to the a hand, we have the dynamo mechanism, in which much conserved current Jµ, a gauge-breakingterm [9, 10]: weaker fields, around 10 19 G [3], could have been am- − plified by the galactic rotation. However, it is known 1 ξ S = d4x√g F Fµν + ( Aµ)2+A Jµ . (1) that the latter scenario has certain limitations since the Z (cid:20)−4 µν 2 ∇µ µ (cid:21) timescales for dynamo amplification may be too long to Because ofthe presence of the gauge-breakingterm, this explain the observed fields in young objects [4]. action does not respect the invariance under arbitrary Bothscenariosrequirepreexistingseedfieldstobeam- gauge transformation, but it still preserves a residual plified and proposals for their generation include: astro- gauge symmetry given by A A + ∂ θ provided µ µ µ physical mechanisms [5], production during inflation [6], (cid:3)θ =0. → in phase transitions [7], and others [8]. Although some The corresponding modified Maxwell equations read: of those mechanisms could seed a galactic dynamo, the generation of the stronger seeds required in the primor- νFµν +ξ µ( νAν)=Jµ. (2) ∇ ∇ ∇ dial field hypothesis is much more problematic. In any Taking the 4-divergence of these equations we obtain: case, according to [2], astrophysical processes, generation in phase transitions or during recombination could (cid:3)( Aν)=0 (3) ν ∇ 2 where we have used the fact that the electromagnetic rent. We will show that it can be generated during current is covariantly conserved. inflation and we will compute its corresponding power Thus we see that due to the presence of the ξ-term, spectrum. This implies that the universe will acquire a thefreetheorycontainsthreepropagatingphysicalfields, non-vanishinggaussianstochasticdistributionofeffective whichcorrespondtothetwoordinarytransversephotons electric charge with zero mean but a non-vanishing dis- andathirdscalarstaterelatedto Aν (inprinciple,we persion. Thisinvacuum,couldalsobeseenasthegener- ν ∇ should include a fourth polarization for A , however it ation of a stochastic background of longitudinal electric µ can be seen to correspondto the pure gauge mode ∂ θ). waves. Due to the high electric conductivity after in- µ Notice that inthe ordinaryapproachto electrodynamics flation, an electrically charged universe has been shown [12], the same action (1) is considered, but Aν is im- to lead necessarily to the generationof vorticity and the ν ∇ posed to be zero (Lorenz condition), so that we are left presenceofmagneticfieldsoncosmologicalscales[13,14]. only with the two transverse polarizations. However in Notice that even though conductivity will be in general the modified approachwe will follow, we allow this state high after reheating also in ordinary electromagnetism, to propagate. Despite the fact that the extended theory itis thepresenceofanon-vanishingeffectivechargeden- is not gauge invariant, the transverse photons dynamics sity the crucial ingredient leading to the generation of is not affected and remains gauge invariant. This im- cosmological magnetic fields. Finally, we will show that plies that ordinary QED phenomenology is recovered in the existing upper limits on vorticity coming from CMB Minkowski space-time. On the other hand, the fact that anisotropiesimposealower limitontheamplitudeofthe the theorycontainsanadditionalpolarizationcouldsug- produced magnetic fields. gest the possibility that such a mode is a ghost and the The power spectrum of super-Hubble fluctuations of theory would be quantum-mechanically unstable. How- Aν produced during an inflationary phase character- ν ∇ ever,as shownin[10], thanks to the residualgaugesym- ized by a slow-rollparameter ǫ, can be written as [9]: metry of the theory it is possible to eliminate the ghost state, so that the new mode has positive norm (details P (k)= 9Hk40 k −4ǫ (5) ontheoreticalandexperimentalaspectsofthetheorycan ∇A 16π2 (cid:18)k0(cid:19) be found in [9–11] and references therein). As seen from (3) the new state is completely decou- where Hk0 is the Hubble parameter when the k0 mode pled from the conserved currents, although it is non- left the horizon, and we have fixed ξ = 1/3. As shown conformally coupled to gravity. This means that this in [9], this value of ξ corresponds to canonical normal- state cannot be excited from electromagnetic currents. ization of commutation relations for creation and anni- However, it could be produced from quantum fluctua- hilation operators for states built out of the standard tions in a curved space-time, in a similar way as infla- Bunch-Davies vacuum. Of course, in curved space-time ton fluctuations during inflation. Moreover, due to the it would be possible to choose a different normalization well-known fact that a massless scalar field gets frozen condition,butaslongasitisanaturalchoice,wedonot on super-Hubble scales for a Robertson-Walker universe expect deviations from ξ being of order unity and, thus, (ds2 = a2(η)(dη2 d~x2)), we get Aν const. on the results obtained in this work will remain essentially ν scales larger than−the Hubble radiu∇s, givi∼ng rise to a unchanged. The pivot point will be chosen as k0 H0 ≃ cosmological constant-like term in the action (1). This withH0theHubbleparametertoday. Thecorresponding constanthasbeen shownto agreewith observationspro- field variance will read: videdinflationtookplaceattheelectroweakscale[9,10]. On the other hand, for sub-Hubble scales, we have that ( Aµ)2 = H0 dk9Hk40 k −4ǫ 9Hk40 H0 4ǫ νAν a−1ei(kη−~kx~). h ∇µ i Zkc k 16π2 (cid:18)k0(cid:19) ≃ 64π2ǫ(cid:18)kc (cid:19) ∇It is∼interesting to note that the ξ-term can be seen, (6) at the equations of motion level, as a conserved current where k is the infrared cutoff which is usually set by actingasasourceoftheusualMaxwellfield. Toseethis, c we can write ξ µ( Aν) Jµ which, according to thecomovingHubble radiusatthe beginningofinflation (3), satisfies t−he∇cons∇erνvation≡eq∇ua·Ation Jµ = 0 and (see[15]andreferencesthereinforproblemswithinfrared we can express (2) as: ∇µ ∇·A divergences during inflation). The above expression, for super-Hubble modes today, can be identified with the ∇νFµν =JTµ (4) cosmologicalconstantscaleMΛ ≃2×10−3 eVand,thus: wmietahnsJTµth=at,Jwµh+ileJ∇tµh·Ae naenwds∇caµlJaTrµm=od0e. cPahnyosinclayllyb,etehxis- MΛ4 ≃ 694Hπk420ǫ(cid:18)Hk0(cid:19)4ǫ. (7) c cited gravitationally,once it is producedit will generally behave as a source of electromagnetic fields. Therefore, Since ǫ is positive, we see that, in general, H < M . k0 Λ the modified theory is described by ordinary Maxwell Notice that Aν is constant on super-Hubble∼scales ν ∇ equations with an additional ”external” current. and starts decaying as 1/a once the mode reenters the In the following we will study the phenomenological Hubble radius. Thus, today, a mode k will have been consequences of the presence of this new effective cur- suppressed by a factor a (k) (we are assuming that the in 3 scale factor today is a = 1). This factor will be given scalesasa3w 1 [16],withw the equationofstate param- 0 − by: a (k) = Ω H2/k2 for modes entering the Hub- eter of the dominant component. However, due to the in M 0 ble radius in the matter era, i.e. for k < k with presence of the effective current, we find that vorticity eq k (14Mpc) 1Ω h2 the value of the mode which grows as ~ω a, from radiation era until present. eq − M ≃ | |∝ entered at matter-radiation equality. For k > k we Using (11), it is possible to estimate a lower limit on eq have a (k) = √2Ω (1+z ) 1/2H /k. It is then pos- the present amplitude of the magnetic fields generated. in M eq − 0 sible to compute the corresponding power spectrum for Since we are not assuming any particular mechanism for the effective electric charge density today ρ0 = J0 = the generation of the primordial magnetic and vorticity g A ξ∂ ( Aν). Thus from: ∇· perturbationsintheearlyuniverse,wewillconsiderthem 0 ν − ∇ for simplicity as gaussian stochastic variables such that: ρ(~k)ρ (~h) =(2π)3δ(~k ~h)ρ2(k) (8) h ∗ i − (2π)3 B (~k)B (~h) = P δ(~k ~h)B2(k) we define P (k)= k3 ρ2(k), which is given by: h i j∗ i 2 ij − ρ 2π2 (2π)3 ω (~k)ω (~h) = P δ(~k ~h)ω2(k) (12) 0, k <H h i j∗ i 2 ij − 0  with B2(k) = Bkn, ω2(k) = Ωkm and where P = Pρ(k)= Ω2M1H6π022Hk40 (cid:16)kk0(cid:17)−4ǫ−2, H0 <k<keq (9) eδrijt−ieskîokˆfjBisinatnrdodωu.ceTdhbeecsapuecsetroafltihneditcreasnsnvearnsdalmityapirjreopin- i i  126ΩπM2(H1+02Hzek4q0)(cid:16)kk0(cid:17)−4ǫ, k>keq. pdstruricanticinvitpistleyonabretbchioetmrpaerosyw.learrNgsopeteiaccftetreatrhcraoetmhewinahtgeinnfrgo,tmhwee(1pe1lxa)ps.meWcateccwoonnil--l Therefore the corresponding charge variance will read: be interested in calculating the mean fluctuation of the ρ2 = dkP (k). Notice that for modes entering the h i k ρ magnetic field in a region of size λ using a gaussianwin- Hubble Rradius in the radiation era, the power spectrum dow function: is nearly scale invariant. Also, due to the constancy of νAν on super-Hubble scales, the effective charge den- B2 = 4π dkk3BknW2(kλ) (13) ∇sity power spectrum is negligible on such scales, so that λ (2π)3 Z k we do not expect magnetic field nor vorticity generation where W(kλ) = exp( k2λ2/2). Similar expressions can on those scales. Notice that, on sub-Hubble scales, the − be written for ω and ρ . Thus from (11) it is possible presentamplitudeofthelongitudinalelectricfieldswould λ λ to obtain [14, 19]: be precisely E Aν. L ν ≃∇ For an observer moving with the cosmic plasma with 1 ΩB four-velocity uµ, it is possible to decompose the Fara- ρ2λ = (2π)32π2S(λ,n,m) (14) day tensor in its electric and magnetic parts as: F = µν 2E u + ǫµνρσBρuσ, where Eµ = Fµνu and Bµ = with: [µ ν] √g ν Oǫµhνmρσ’/s(2la√wg)JFµρσuνu.µuInJνthe=inσfiFnµitνeucoinmdpulcietsiviEtyµ l=imi0t,. S(λ,n,m) = ∞ dkk3W2(kλ) (15) Therefore,intha−tcaseνtheonlycontrνibutionwouldcome ZH0 k [ from the magnetic part. Here, Jµ is the current gen- d3p~k p~mpn[1+(~k ~p p~)2] erated in the plasma which is assumed neutral, i.e., × Z | − | − · J uµ =0. Thus, from (4), we get: b µ where H < p < k and ~k p~ > H . For the upper 0 cB 0 | − | ǫµνρσ cutoff of the magnetic power spectrum, we take a con- Fµν;νuµ = √g Bρuσ;νuµ =J∇µ·Auµ (10) ssecravleatwivheicvhaliusegcivoernrebspyoknding t1o01t1heMmpcagn1e[t1i7c]d(iaffumsioorne cB − ≃ that for comoving observersin a Robertson-Walkermet- detailed analysis can be found in [18]). Let us define: ric imply (see also [14]): kmax dk G(λ,n)= k3knW2(kλ) (16) 1 ~ω B~ = ρ0g (11) Zkmin k a2 · a2 where,as before, due to the vanishing ofthe chargeden- sity on super-Hubble scales, k is typically given by where ~v = d~x/dη, ~ω = ~ ~v is the fluid vorticity, ρ0 min ∇× g the comoving Hubble horizon at the time the fluctua- is the effective charge density today and the B~ compo- tions are evaluated and k =k ( ) in the magnetic max cB nents scale as Bi 1/a as can be easily obtained from (vorticity) cases respectively. Thus, w∞e finally obtain for ǫµνρσFρσ;ν = 0 to∝the lowest order in v. Thus, as com- the magnetic fluctuation on a scale λ: mented before, the presence of the non-vanishing cosmic effectivechargedensitynecessarilycreatesbothmagnetic 4πρ2G(λ,n)G(λ,m) B2 λ . (17) field and vorticity. In the absence of sources, vorticity λ ≃ ω2S(λ,n,m) λ 4 10-11 with η = η η . We will consider the minimum of 0 rec Λ = 0.1 h-1 Mpc the r.h∗.s. of (1−8) with respect to l which, for m< 1, is − located at l 29 and, for m > 1, at l 1200 which is 10-13 the highest m∼ultipole measured−by WMA∼P. HLBGΛ 10-15 latTedheusseinstgri(n1g7e)nitntuopploewrelirmliimtsitosnovnortthieciptyrecsaenntbveatlruaenos-f the magnetic field created by the effective current. For 10-17 the sake of concreteness, we take H 2 10 6 eV k0 − ≃ × in (9). This value corresponds to a scale of inflation around 100 GeV i.e. in the electroweak range. It sat- 10-19 -4 -2 0 2 isfies Hk0 < MΛ and also the limits on the primordial electromag∼netic fluctuations coming from their imprint n on CMB anisotropies (see [11]). We have evaluated nu- 10-9 Λ = 3000 h-1Mpc merically the integrals appearing in (17) for ǫ 0.01, ≃ although the ǫ-dependence of the bounds is very small. 10-11 InFig. 1we showthe lowerlimits onthe magneticfields generated by this mechanism on scales λ=0.1h 1 Mpc, − LG 10-13 (which is the relevant scale for galaxies and clusters (see HBΛ [17])), and λ = 3000h−1 Mpc. These results show that 10-15 the producedfieldscouldhavestrongamplitudes evenin the largest scales and act as seeds for a galactic dynamo 10-17 or even play the role of primordialfields and account for observationsjustbyamplificationduetothecollapseand 10-19 differential rotation of the protogalactic cloud. -4 -2 0 2 n Itisinterestingtonotethat,sincesuper-Hubblemodes FIG. 1: Lower limits on the magnetic fields generated on of the effective electromagnetic current are not gener- galacticscales(upperpanel)andHubblehorizonscales(lower panel) in termsof themagneticspectral index nfor different ated,weexpectmagneticfieldstobepresentonlyonsub- values of the vorticity spectral index m. Dot-dashed bluefor Hubble scales. This means, that the constraints coming m=0, dashed green for m≃−3 and full red for m≃−5. fromthedissipationofsuper-Hubblemagneticfieldsinto gravitywavesbeforenucleosynthesis[19]donotapply in the present case. In any case, these results show that a moreprecisedeterminationofthemagneticandvorticity Vorticity perturbations generate anisotropies in the spectraoncosmologicalscalescouldhelpestablishingthe CMB temperature at recombination time whose ampli- feasibilityoftheextendedtheoryin(1)forproducingthe tude should be compatible with present observations. observed cosmic magnetic fields. Taking into account the scaling properties of vorticity derived before, such limits on a scale λ can be written today as [14]: Acknowledgments: We would like to thank Ruth Dur- rerandMisaoSasakiforusefulcomments. Thisworkhas l2C /(2π)z2 G(λ,m) beensupportedbyMICINN(Spain)projectnumbersFIS ω2 < l rec (18) λ 8l3(l+1)R(l,m) 2008-01323 and FPA 2008-00592, CAM/UCM 910309, ∼ MEC grant BES-2006-12059 and MICINN Consolider- where l2C /(2π) 10 10 and Ingenio MULTIDARK CSD2009-00064. 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