27 May 1999

Ž .Physics Letters B 455 1999 96–103

Primordial magnetic fields via spontaneous breaking of Lorentzinvariance

O. Bertolami, D.F. MotaDepartamento de Fısica, Instituto Superior Tecnico, AÕ. RoÕisco Pais, 1096 Lisboa Codex, Portugal´ ´

Received 30 November 1998; received in revised form 31 March 1999Editor: L. Alvarez-Gaume

Abstract

Spontaneous breaking of Lorentz invariance compatible with observational limits may realistically take place in thecontext of string theories, possibly endowing the photon with a mass. In this process the conformal symmetry of the

Ž .electromagnetic action is broken allowing for the possibility of generating large scale ;Mpc magnetic fields withininflationary scenarios. We show that for reheating temperatures safe from the point of view of the gravitino and moduliproblem, T Q109 GeV for m f1 TeV, the strength of the generated seed fields is, in our mechanism, consistent withRH 3r2

amplification by the galactic dynamo processes and can be even as large as to explain the observed galactic magnetic fieldsthrough the collapse of protogalactic clouds. q 1999 Elsevier Science B.V. All rights reserved.

PACS: 98.80 Cq; 98.62 En

1. Introduction

Coherent magnetic fields are observed over awide range of scales in the Universe from the Earth,Solar System and stars to galaxies and clusters of

w xgalaxies 1 .Magnetic fields play an important role in a variety

of astrophysical processes. For instance, the galacticfield affects the dynamics of the galaxy as it confinescosmic rays, influences the dynamics of compact

w xstars and the process of star formation 2 . Largescale magnetic fields are also quite important inquasars, active galactic nuclei and in intercluster gasor rich clusters of galaxies.

The current estimates for the magnetic field of theMilky Way and nearby galaxies is B;10y6 G,

which are supposed to be coherent over length scalesw xcomparable to the size of the galaxies themselves 1 .

The origin of the astrophysical mechanisms responsi-ble for these galactic fields is however poorly under-stood.

Possibly the most plausible explanation for theobserved galactic magnetic field involves some sort

w xof dynamo effect 2,3 . In this mechanism, turbu-lence generated by the differential rotation of thegalaxy enhances exponentially, via non-linear pro-cesses, a seed magnetic field up to some saturationvalue that corresponds to equipartition between ki-netic and magnetic energy. A galactic dynamo mech-anism along these lines can enhance a seed magneticfield by a factor of several orders of magnitude.Indeed, if one assumes that the galactic dynamo has

0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 99 00418-9

( )O. Bertolami, D.F. MotarPhysics Letters B 455 1999 96–103 97

operated during about 10 Gyr, then a seed magneticfield could be amplified by a factor of e30 corre-sponding to about 30 dynamical timescales or com-plete revolutions since the galaxy has formed. Hencethe observed galactic magnetic fields at present may,via the dynamo amplification, have its origin in aseed magnetic field of about B;10y19 G.

Naturally, the galactic magnetic fields can emergedirectly from the compression of a primordial mag-netic field, in the collapse of the protogalactic cloud.In this case, it is required a seed magnetic field ofB;10y9 G over a comoving scale l; Mpc, thecomoving size of a region which condenses to forma galaxy. Since the Universe through most of itshistory has behaved as a good conductor it impliesthat the evolution of any primeval cosmic magnetic

w xfield will conserve magnetic flux 4,5 . Therefore,the ratio denoted by r, of the energy density of a

B2

magnetic field r s relative to the energy densityB 8p

of the cosmic microwave background radiation rgp 2 4s T remains approximately constant and pro-15

vides a invariant measure of magnetic field strength.At present, for galaxies r;1, so pregalactic mag-

netic fields of about rf10y34 are required if oneinvokes dynamo amplification processes, and rf10y8 if the observed galactic magnetic fields arecreated from the compression of the primordial mag-netic field in the collapse of the protogalactic cloudwithout the help of dynamo type processes.

A number of proposals have been put forward toexplain the way a primordial magnetic field could be

Ž w x .generated see 6 for a review . In many of theseproposals, changes in the nature of the electromag-

w xnetic interaction at the period of inflation 4,7 areinvolved together with the process of structure for-

w xmation 4,8 . Other proposals invoke collision ofw xphase transition bubbles 9 , fluctuating Higgs field

w x w xgradients 10 , superconducting cosmic strings 11and non-minimal coupling between electromag-netism and gravity via the Schuster-Blacket relationw x12 . In this work we shall describe a mechanism toproduce a seed magnetic field that is based on apossible violation of Lorentz invariance in solutionsof string field theory and that uses inflation foramplification of quantum fluctuations of the electro-magnetic field. We show that these fluctuations arecompatible with both galactic dynamo mechanismsand protogalactic cloud collapse scenario.

Invoking a period of inflation to explain the cre-ation of seed magnetic fields is a quite attractivesuggestion as inflation provides the means of gener-ating large-scale phenomena from microphysics thatoperates on subhorizon scales. More concretely, in-flation, through de Sitter-space-produced quantumfluctuations, provides the means of exciting the elec-tromagnetic field allowing for an increase of themagnetic flux before the Universe gets filled with ahighly conducting plasma. Furthermore, via a mech-anism akin to the superadiabatic amplification, long-wavelength modes, for which lR Hy1, are duringinflation and reheating enhanced. It is certainly quiteinteresting that inflation can play, for generatingprimordial magnetic fields, the same crucial role itplays in solving the problems of initial conditions ofthe cosmological standard model.

w xHowever, as pointed out by several authors 4,7 ,it is not possible to produce the required seed mag-netic fields from a conformally invariant theory as it

Ž .happens with the usual U 1 gauge theory. Thereason being that, in a conformally invariant theory,

Ž .y2 Ž .the magnetic field decreases as a t , where a t isthe scale factor of the Robertson-Walker metric, andduring inflation, the total energy density in the Uni-verse is constant, so the magnetic field energy den-sity is strongly suppressed, yielding rs10y104ly4 ,Mpc

which is far too low for a seed field candidate. Itthen follows that conformal invariance of electro-magnetism must be broken.

In the context of string theories, conformal invari-ance may be broken actually due to the possibility of

w xspontaneous breaking of the Lorentz invariance 13Žthis breaking can also lead to the breaking of CPT

w x.symmetry 14 . This possibility arises explicitly fromsolutions of string field theory, at least for the opentype I bosonic string, as interactions are cubic in thestring field and these give origin in the static fieldtheory potential to cubic interaction terms of the typeSSS, STT and TTT , where S and T denote scalar andtensor fields. The way Lorentz invariance may bebroken can be seen, for instance, from the staticpotential involving the tachyon and a generic vector

w xfield as can be explicitly computed 13 :

w 23 mV w , A , ... sy qagw qbgwV V q . . . ,Ž . Xm m2a

1Ž .

( )O. Bertolami, D.F. MotarPhysics Letters B 455 1999 96–10398

a and b being order one constants and g the on-shellthree-tachyon coupling.

The vacuum of this model is clearly unstable andthis instability gives rise to a mass-square term for

² : ² :the vector field that is proportional to w . If w isnegative, then the Lorentz symmetry is sponta-neously broken as the vector field can acquire itself anon-vanishing vacuum expectation values 1. Thismechanism can give rise to vacuum expectation val-ues to tensor fields inducing for the fields that do notacquire vacuum expectation values, such as the pho-

² : Žton, to mass-square terms proportional to T thisw x.possibility has been briefly discussed in 16 . Hence,

one should expect from this mechanism, terms for² : m ² : m nthe photon such as T A A , T A A and som mn

on. Naturally, these terms break explicitly the con-formal invariance of the electromagnetic action.

Observational constraints on the breaking of theLorentz invariance arise from measurements of thequadrupole splitting time dependence of nuclear Zee-man levels along Earth’s orbit, the so-called

w xHughes-Drever experiment 17,18 , and have beenw xperformed over the years 19,20 , the most recent

y21 w xone indicating that d-3=10 21 . Bounds onthe violation of momentum conservation and exis-tence of a preferred reference frame can be alsoextracted from limits on the parametrized post-New-tonian parameter a obtained from the pulse period3

w x w xof pulsars 22 and millisecond pulsars 23 . Thisparameter vanishes identically in general relativity

< < y20and the most recent limit a -2.2=10 ob-3w xtained from binary pulsar systems 24 implies the

Lorentz symmetry is unbroken up to this level. Theselimits indicate that if the Lorentz invariance is bro-ken then its violation is suppressed by powers ofenergy over the string scale. Similar conclusions canbe drawn for putative violations of the CPT symme-

w xtry 14 .In order to relate the theoretical possibility of

spontaneous breaking of Lorentz invariance to theobservational limits discussed above we parametrize

1 Actually, it is known that negative quadratic mass statescorresponding to tachyonic solutions are admitted in the represen-tation space of massive vector particles implying in a violation of

w xthe rotational sector of Lorentz invariance 15 .

the vacuum expectation values of the Lorentz tensorsin the following way:

2 lE2² :T sm , 2Ž .L ž /MS

where m is a light mass scale when compared toL

string typical energy scale, M , presumably M fS S

M ; E is the temperature of the Universe in a givenP

period and 2 l is a positive integer. Given that theexpansion of the Universe is adiabatic, we shallfurther replace in what follows the temperature of

Žthe Universe by the inverse of the scale factor theproportionality constant is absorbed in the yet un-

.specified light mass scale, m . Notice thatLŽ .parametrization Eq. 2 used here is somewhat differ-

w xent than the ones used in previous work 14,25,26 .

2. Generation of seed magnetic fields

We are going to consider spatially flatFriedmann-Robertson-Walker cosmologies, wherethe stress tensor is described by a perfect fluid withan equation of state psgr. The metric in theconformal time, h, is given by

2g sa h diag y1,1,1,1 , 3Ž . Ž . Ž .mn

Ž .where a h is the scale factor.The present value of the Hubble parameter is

written as H s100 h km sy1 Mpcy1 and the pre-0 0

sent Hubble radius is R s1026 hy1 m, where 0.4F0 0

h F1. We shall assume the Universe has gone0

through a period of exponential inflation at a scaleM and whose associated energy density is givenGUT

by r 'M 4 . The details of this de Sitter phase areI GUTw xnot relevant and will play, as discussed in 4 , no role

in our mechanism for generating a primordial seedmagnetic field. From the Friedmann equation, HdS

8p G 1r2 8p 1r2 2MGU Ts r s , where M is theŽ . Ž .I P3 3M P

Planck mass.From our discussion on the breaking of Lorentz

invariance we consider for simplicity only a single² : mterm, namely T A A , from which implies them

following Lagrangian density for the photon:1 mn 2 y2 l mLLsy F F qM a A A , 4Ž .mn L m4

2 m2Lwhere M ' .L2 lMp

( )O. Bertolami, D.F. MotarPhysics Letters B 455 1999 96–103 99

The field strength tensor, F , is given bymn

0 yE yE yEx y z

E 0 B yBx z y2F sa h ,Ž .mn E yB 0 By z x� 0E B yB 0z y x

and it satisfies the Bianchi identity

E F qE F qE F s0 5Ž .m lk l km k ml

as well as the equation of motion for the photon field

= mF qM 2 ay2 lA s0 . 6Ž .mn L n

Ž . Ž .Eq. 5 and Eq. 6 can be then explicitly writtenas

1 E2a Bq==Es0 , 7Ž .2 Eha

1 E n A2a Ey==By s0 , 8Ž .2 2 2Eha h a

where n'yh 2M 2 ay2 lq2 and A is the vector po-L

tential.Ž . Ž .Taking the curl of Eq. 8 and using Eq. 7 we

obtain the wave equation for the magnetic field:

1 E 2 n2 2a By= Bq Bs0 . 9Ž .2 2 2a Eh h

The corresponding equation for the Fourier com-ponents of B is given by

n2F qk F q F s0 , 10Ž .k k k2h

where the dots denote derivatives according to theconformal time and

F h 'a2 d3 x ei k . xB x ,h , 11Ž . Ž . Ž .Hk

F being a measure of the magnetic flux associatedk

with the comoving scale l;ky1. It follows that theenergy density of the magnetic field is given by

< <2FkŽ .r k A .B4a

For modes well outside the horizon, al4Hy1

< < Ž .or kh <1, solutions of Eq. 10 are given in termsof the conformal time:

< < m "F Ah , 12Ž .k

where1 'm s 1" 1y4n . 13Ž ." 2

In order to estimate n we consider the differentphases of evolution of the Universe. By requiringthat n is not a growing function of conformal time, itfollows that n has to be either a constant or that 2 lis negative, which is excluded by our assumption Eq.Ž .2 . Hence for different phases of evolution of theUniverse:

Ž . Ž . Ž .I Inflationary de Sitter dS phase, where a h1Ay , it follows that ls0 and

hHdS

M 2dS

nsy , 14Ž .2HdS

where we denote the light mass M by the index ofL

the relevant phase of evolution of the Universe.Ž . Ž .II Phase of Reheating RH and Matter Domina-

1 2 3 2Ž . Ž .tion MD , where a h A H R h , yields from the0 04

condition n is a constant that 2 ls3 and

4M 2MD

nsy . 15Ž .2 3H R0 0

Ž . Ž .III Phase of Radiation Domination RD , whereŽ . 2a h AH R h, from which follows that ls2 and0 0

M 2RD

nsy . 16Ž .2 4H R0 0

It is clear that in this case last case n<1.We mention that, we could have obtained the

Ž .same results comparing Eq. 10 to the correspond-ing equation of the Fourier modes of a scalar field,coupled non-minimally to gravity through the term1 2 aj Rf , where Rs6 is the Ricci scalar. The2

3a

relevant Lagrangian density is1 1m 2LL sy E fE fy j Rf . 17Ž .f m2 2

In terms of the k-th Fourier component of thecombination wsaf, the equation of motion of anon-minimally coupled scalar field can be written asw x4

nf2w qk w q w s0 , 18Ž .¨ k k k2h

a 2< <¨2 wkŽ . Ž .where n sh 6jy1 and as before r k A .f fa4a

Notice that the correspondence between w andk

the components of F implies, for instance, that ifk1

js then the components of A behave like am6

( )O. Bertolami, D.F. MotarPhysics Letters B 455 1999 96–103100

scalar field conformally coupled and if js0 thenthe components of A behave like a scalar fieldm

minimally coupled. Furthermore, a correspondencebetween w and F , requires that nsn implyingk k f

in the following condition:

M 2L y2 lq3aq a s0 . 19Ž .¨

6jy1Ž .It is easy to show that for different phases of

evolution of the Universe one can obtain, via Eq.Ž .19 , essentially the same results for n as the oneswe have obtained from requiring that n is constant.

Hence for each phase of evolution of the Universewe find from the corresponding values of l andassociated conditions to M the following behaviourL

< <for F :kŽ .I de Sitter Phase

< < ym "F Aa , 20Ž .k

where

21 2 MdSm s 1" 1q 21Ž .)" dS ž /2 HdS

and

r Aay2 m " dSy4 . 22Ž .BdS

Notice that the most relevant exponent is given bym , as it corresponds to the fastest growing solu-ydS

tion for F .kŽ .II Phase of Reheating and Matter Domination

1m "2< <F Aa , 23Ž .k

where

21 MRHm s 1" 1q16 , 24Ž .)" RH 2 32 H Rž /0 0

and therefore

r Aam " MDy4 . 25Ž .B

The relevant exponent here is m , as it corre-qRD

sponds to the fastest growing solution for F .kŽ .III Phase of Radiation Domination

< < m "F Aa , 26Ž .k

where

m s1, 0 , 27Ž ." RD

from which follows that

r Aay2 or r Aay4 . 28Ž .B B

As expected, for m one obtains that r Aay4.yRD B

These results enable us to estimate strength of theprimordial magnetic field. Assuming the Universehas gone through a period of inflation at scale MGUT

and that fluctuations of the electromagnetic fieldhave come out from the horizon when the Universehad gone through about 55 e-foldings of inflation,

w xthen in terms of r 4 :Ž .4 qyp r3MŽ . GUTy2 pq225rf 7=10 =Ž . ž /MP

Ž . y8 qr32 2 qyp r3T TRH ) y2Ž pq2.= = =l ,Mpcž / ž /M MP P

29Ž .where T is the temperature at which plasma effects

)

become dominant, that is, the temperature when theUniverse first becomes a good conductor and thatcan be estimated from details of the reheating pro-

1 122 3�Ž . Ž . 4 w xcess T smin T M ; T M 4 ; for TF) RH GUT RH P

T , r necessarily evolves as Aay4. For the reheat-) B

ing temperature we assume either a poor or a quite� 9 4efficient reheating, T s 10 GeV;M , seeRH GUT

however the discussion below. Finally,

21 2 MdSp'm s 1y 1q)ydS ž /2 HdS

and

21 MLq'm s 1q 1q16)qRH 2 32 H Rž /0 0

are the fastest growing solutions for F in the dek

Sitter and Reheating phases, respectively.In order to obtain numerical estimates for r we

have to compute M . At the de Sitter phase we haveL

that M sm . As we have seen m is a lightdS dS L

energy scale when compared to M and M , theP GUT

energy scale of the de Sitter phase. Thus in order toestimate M at the de Sitter phase we introduce aL

parameter, x , such that m sx M and x<1.dS GUT

( )O. Bertolami, D.F. MotarPhysics Letters B 455 1999 96–103 101

Table 1r 15BValues of r s at 1 Mpc for M s10 GeVGUTrg

Ž . Ž .x p q T GeV T GeV log rRH )

y4 9 125=10 y1.67 1 10 10 y37y4 15 155=10 y1.67 1 10 10 y31y4 9 126=10 y2.08 1 10 10 y21y4 15 156=10 y1.08 1 10 10 y13

At the matter domination phase, we have to im-T

lgpose that the mass term M sm , T be-MD MD gž /MP

ing the temperature of the cosmic background radia-tion at about the recombination time, is consistentwith the present-day limits of experiments and obser-vations to the photon mass. Thus, at the matterdomination phase, we have to satisfy the condition,

T y36gM Fm which implies that 3r F3=102MD g ž /MPw x 4GeV 27 , following that m F7.8=10 GeV. AMD

more stringent bound on m could be obtainedMD

from the limit m F1.7=10y42 h GeV arising fromg 0

the absence of rotation in the polarization of light ofw xdistant galaxies due to the Faraday effect 28 .

In Tables 1–3 we present our estimates for theratio r for different values of M . One can seeGUT

that we obtain values that are in the range 10y37 -r-10y5. For M s1015,1016 GeV one sees that aGUT

poor reheating and the lower values for x tend torender r too low even for an amplification viadynamo processes. On the other hand, for M sGUT

1017 GeV a quite efficient reheating leads for x)5=10y2 to rather large primordial magnetic fields.Actually, primordial fields greater than 3.4 =

y9 Ž 0 .1r2 210 V h G , where V is the density0 50 0

parameter at present and h is the present value of50

the Hubble parameter in units of 50 km sy1 Mpcy1,are ruled out as they lead to more anisotropies thanthe ones observed in the microwave background

w xradiation 29 . Hence, our results indicate that mod-els with M s1017 GeV, x)5=10y2 and TGUT RH

sM should be disregarded.GUT

2 This constraint is much more stringent than the ones arisingfrom nucleosynthesis that imply, at the end of nucleosynthesiswhen T f0.01 MeV, that B -2=109 G and that r F0.28 rg P B n

w x30 .

Table 2r 16BValues of r s at 1 Mpc for M s10 GeVGUTrg

Ž . Ž .x p q T GeV T GeV log rRH )

y3 9 125=10 y1.67 1 10 2.3=10 y35y3 16 165=10 y1.67 1 10 10 y27y3 9 126=10 y2.08 1 10 2.3=10 y18y3 16 166=10 y1.08 1 10 10 y9

However, an important issue is that in supersym-metric theories the reheating temperature is severelyconstrained in order to avoid that gravitinos andmoduli are not copiously regenerated in the post-in-flationary epoch. This is indeed a difficulty as asonce regenerated beyond a certain density these par-ticles dominate the energy density of the Universeor, if decay, have undesirable effects on nucleosyn-thesis and lead to distortions of the microwave back-

Žground. The relevant bounds are the following seew x .31 and references therein :

T F 2y6 =109 GeVŽ .RH

for m s 1y10 TeV . 30Ž . Ž .3r2

15 Ž 16 .Therefore it follows that for M s10 10GUTy4 Ž y3 .GeV, only for x)5=10 x)5=10 the

generated seed magnetic fields are large enough. ForM s1017 GeV, the generated seed fields, al-GUT

though rather small, can be still compatible withobservations, for the chosen x values, via the dy-namo amplification process.

Finally, we mention that from the comparisonwith a non-minimally coupled scalar field, our re-sults allow inferring, for example that for M sGUT

1017 GeV, at the de Sitter phase jsy0.02 andjsy0.04 for xs5=10y2 and xs6=10y2 , re-spectively. This allows us to conclude that the growthin magnetic flux is analogous to the phenomenon ofsuperadiabatic amplification, as the components of

Table 3r 17BValues of r s at 1 Mpc for M s10 GeVGUTrg

Ž . Ž .x p q T GeV T GeV log rRH )

y2 9 135=10 y1.67 1 10 10 y33y2 17 175=10 y1.67 1 10 10 y24y2 9 136=10 y2.08 1 10 10 y16y2 17 176=10 y1.08 1 10 10 y5

( )O. Bertolami, D.F. MotarPhysics Letters B 455 1999 96–103102

A behave like the scalar field with a negativem

mass-square term. For the matter domination phase1we find that js , actually the expected result.6

3. Conclusions

Galactic dynamo or protogalactic cloud collapsecan explain the observed magnetic fields of galaxiesat present provided a primordial seed magnetic fieldis generated prior the Universe turns a good conduc-tor and the magnetic flux turns into a conservedquantity.

A particularly interesting proposal is to invokeinflation for explaining the origin of the primordialmagnetic field. This is a quite good idea since infla-tion naturally relates microphysics with macro-physics as it allows for amplification of quantumfluctuations of fields, and in particular of the electro-magnetic field, stretching fluctuations beyond thehorizon prior the Universe becomes a good conduc-tor and locks the growth of electromagnetic flux.However, this scenario requires the breaking of theconformal invariance as otherwise the r will bediluted to quite small values r-10y104ly4 at MpcMpc

w xscales. In 4 it was suggested, in order to break theconformal symmetry of electromagnetism, to intro-duce in a somewhat ad hoc way terms such asRA A m, R A mAn, etc., into the electromagneticm mn

action. In this work we have pointed out that theexistence of string field theory solutions, in thecontext of which Lorentz invariance can be sponta-neously broken, leads for the photon to terms like² : m ² : m nT A A T A A , hence breaking conformalm mn

symmetry. We then showed that this allows inflationto generate rather large seed magnetic fields.

Furthermore, we have demonstrated that thestrength of the magnetic field produced by our mech-anism is sensitive to the values of a light mass, m ,LŽ Ž ..cf. Eq. 2 , M and the reheating temperatureGUT

T . Our results indicate that for rather diverseRH

values of these parameters we can obtain values forr that are consistent with amplification via galacticdynamo or collapse of protogalactic clouds. ForM s1015, 1016 GeV we find that poor reheatingGUT

and xs5=10y4 and xs5=10y3 tend to giverise to too low r values. For M s1017 GeV,GUT

very efficient reheating leads for xs6=10y2 to

rather large primordial magnetic fields, these beingactually incompatible with upper limits derived fromthe study of the microwave background anisotropies.Since in supersymmetric theories the reheating tem-perature is strongly constrained not to be greater than

9 Ž .about 10 GeV for OO TeV gravitino and modulimasses, our results indicate that for M sGUT

1015, 1016 GeV only by choosing x)5=10y4 andx)5=10y3, respectively, the generated seed fieldsare large enough for accounting the observations.There is actually no such a limitation for M sGUT

1017 GeV.

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