Large scale magnetic fields from gravitationally coupled electrodynamics

Kerstin E. Kunze*

Departamento de Fı́sica Fundamental, Universidad de Salamanca, Plaza de la Merced s/n, E-37008 Salamanca, Spain(Received 11 November 2009; published 16 February 2010)

The generation of primordial magnetic seed fields during inflation is studied in a theory derived from

the one-loop vacuum polarization effective action of the photon in a curved background. This includes

terms which couple the curvature to the Maxwell tensor. The resulting magnetic field strength is estimated

in a model where the inflationary phase is directly matched to the standard radiation dominated era. The

allowed parameter region is analyzed and compared with the bounds necessary to seed the galactic

magnetic field. It is found that magnetic fields of cosmologically interesting field strengths can be

generated.

DOI: 10.1103/PhysRevD.81.043526 PACS numbers: 98.80.Cq, 98.62.En

I. INTRODUCTION

Magnetic fields are found to be associated with nearly allstructures in the Universe. They are observed on stellar upto possible supercluster scales [1]. In some processes suchas star formation they can play an important role. Equally,the physics of cosmic rays indicate the existence of a largescale galactic magnetic field. Furthermore, ultra high en-ergy cosmic rays, which are most likely of extra galacticorigin, could be used to study the properties of galactic andextra galactic magnetic fields [2].

As to the origin of the observed magnetic fields thereseem to be generally speaking two broad classes of mecha-nisms [1]. On the one hand there are battery-type mecha-nisms which work on the basis of charge separation whichleads to a current and finally induces a magnetic field. Onthe other hand there are dynamo-type mechanisms whichamplify an initial seed magnetic field. The former have acoherence scale of the order of the domain associated withthe battery mechanism, which in a cosmological context isalways smaller than the horizon at the epoch of creation.The latter involves magnetic fields whose correlationlength is of the order of the region of interest, which, inthe case of a galactic magnetic field, would be a protogalactic scale of the order of 1 Mpc today which requires amechanism to create magnetic fields with rather largecoherence scales. A natural mechanism for this is providedby the amplification of perturbations of the electromag-netic field during inflation. Since quantum perturbationsare stretched beyond the horizon during inflation and uponleaving the causal domain becoming classical, there is noproblem with the coherence length. However, in general, ina background geometry with a flat spatial section theresulting field strength of the primordial magnetic fieldafter inflation in standard electrodynamics is far too smallto, for example, serve as a seed field for a potential galacticdynamo explaining the observed galactic magnetic field ofthe order of 10�6 G. Starting with [3] this motivated an

intensive study of alternative models of some kind ofcoupling of linear electrodynamics to either other fieldsin the theory, such as scalar fields [4] or gravity [3,5]including extra dimensions [6]. Other models break explic-itly Lorentz invariance considering a non zero photon mass[3,7]. Recently there has also been interest in magneticfield generation within models of nonlinear electrodynam-ics which naturally occur when quantum corrections andself-couplings of the electromagnetic field are taken intoaccount [8]. Furthermore, there are models within thestandard model or its supersymmetric extensions [9]. Inthe case of models with curved spatial sections it wasshown that even in the minimally coupled model of linearelectrodynamics strong enough magnetic fields can becreated [10].Here we are returning to a model where the electromag-

netic field is gravitationally coupled which first has beenproposed in the context of the generation of primordialmagnetic fields during inflation in [3]. In particular themodel under consideration derives from the one-loop ef-fective action of vacuum polarization in QED in a gravi-tational background [11]. As was shown in [11] this leadsto birefringence of the electromagnetic wave where thephotons have velocities depending on their polarizationwhich can exceed the speed of light. This last observationleads to an interesting causal structure of these space-times[12].

II. ESTIMATING THE MAGNETIC FIELDSTRENGTH

In [11] it was shown that to lowest order the propagationof a photon in a gravitational background is described bythe Lagrangian [3,11]

L ¼ � 1

4F��F

�� � 1

4m2e

½bRF��F�� þ cR��F

��F��

þ dR����F��F�� þ fðr�F

��Þðr�F��Þ� (2.1)

The expansion parameter is basically the square of theCompton wave length of the electron which enters due to*kkunze@usal.es

PHYSICAL REVIEW D 81, 043526 (2010)

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the fact that vacuum polarization effectively gives thephoton a non zero ‘‘size’’ due to electron positron paircreation. The parameters b, c, d, and f are assumed to befree parameters here, which, however, have been calcu-lated in [11] in the weak gravitational field limit. It wasfound that the coefficient of the last term is of the order e2

and can therefore be neglected. Moreover, unless f is to bechosen much larger than the other coefficients it is ex-pected to be negligible in general, as can be appreciatedfrom the equations of motion,

r�F�� þ 1

m2e

r�

�bRF�� þ c

2ðR�

�F�� � R��F��Þ

þ dR����F��

�þ f

2m2e

ðr�r�r�F�� þ R��r�F��Þ ¼ 0:

(2.2)

Thus the new type of term that the last term contributesinvolves higher derivatives which are suppressed by afactor �1=�2, where � is a typical scale. Therefore, inthe following this term will neglected and f � 0.

The Maxwell tensor is expressed in terms of the gaugepotential A�, that is F�� ¼ @�A� � @�A� and the

Coulomb gauge is used A0 ¼ 0, @iAi ¼ 0. Furthermore,the background cosmology is described by the line ele-ment,

ds2 ¼ a2ð�Þð�d�2 þ dx2 þ dy2 þ dz2Þ; (2.3)

where að�Þ is the scale factor describing a model, in whichthe inflationary stage is directly matched at � ¼ �1 to thestandard radiation dominated era,

að�Þ ¼� a1ð ��1

Þ� � < �1

a1ð��2�1

��1Þ � � �1:

(2.4)

In the following a1 � 1. For � ¼ �1 de Sitter inflation isrealized and for �1<�<�1 the model has a stage ofpower law inflation for �< �1. The exponent � is relatedto the equation of state of matter, defined by, p ¼ , by

� ¼ 23þ1 . Note that ¼ 2��

3� takes values between�1 and

� 13 . The Fourier expansion of the gauge potential is given

by

Ajð�; ~xÞ ¼Z d3k

ð2�Þ3=2X2�¼1

�ð�Þ~kj½að�Þ~k

Akð�Þei ~k� ~x

þ að�Þy~kA�kð�Þe�i ~k� ~x�; (2.5)

where the sum is over the two polarization states and �ð�Þ~kj

are the polarization vectors satisfying ~�ð�Þ~k� ~k ¼ 0.

Furthermore, the amplitude Ak satisfies the same modeequation for both polarization states. Thus the index ð�Þis suppressed in Ak.

In Fourier space in the background model (2.3) Eq. (2.2)results in

F1ð�Þ €Ak þ F2ð�Þ _Ak þ F3ð�Þk2Ak ¼ 0; (2.6)

where a dot indicates dd� and

F1ð�Þ ¼ 1þ �1

m2e�

21

��

�1

��2ð�þ1Þ;

�1 ¼ �½6bð�� 1Þ þ cð�� 2Þ � 2d�;F2ð�Þ ¼ �2

�31m

2e

��

�1

��2��3;

�2 ¼ �2ð�þ 1Þ�1;

F3ð�Þ ¼ 1þ �3

�21m

2e

��

�1

��2ð�þ1Þ;

�3 ¼ �½6bð�� 1Þ þ cð2�� 1Þ þ 2d��:

(2.7)

The standard quantization procedure requires the corre-sponding action of the field to be diagonal. This can beachieved by using the canonical field � defined by

� ¼ F1=21 Ak; (2.8)

which satisfies the mode equation,

�00 þ P� ¼ 0; (2.9)

where a new dimensionless variable z � �k� has beendefined and 0 � d

dz . Moreover,

P ¼ 1

4

�1z�4��6

½1þ �2z�2ð�þ1Þ�2 þ

1

2

�3z�2��4

1þ �2z�2ð�þ1Þ

þ 1þ �4z�2ð�þ1Þ

1þ �2z�2ð�þ1Þ ; (2.10)

and

�1 � �22�

20; �2 � �1�0; �3 � ð2�þ 3Þ�2�0;

�4 � �3�0; where �0 ��me

H1

��2�k

k1

�2ð�þ1Þ

: (2.11)

In deriving these expressions the maximally amplified(comoving) wave number k1 has been defined by k1 �1

j�1j . H1 is the value of the Hubble parameter at the begin-

ning of the radiation dominated stage at �1. It is related tok1 by k1 �H1. The spectrum of the resulting magneticfield is determined by calculating the Bogoliubov coeffi-cients which connect the ‘‘in’’ and ‘‘out’’ vacua. Therelevant solutions of the mode equation (2.9) are those onsuperhorizon scales, z � 1, in which case Eq. (2.9) re-duces to

�00 þ ð 1z�2 þ 2Þ� ¼ 0; (2.12)

where

1 ¼ �ð�þ 1Þð�þ 2Þ;

2 ¼ 6bð�� 1Þ þ cð2�� 1Þ þ 2d�

6bð�� 1Þ þ cð�� 2Þ � 2d:

(2.13)

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For � ¼ �1, which describes de Sitter inflation, 1 ¼ 0and 2 ¼ 1 the solution is a plane wave which was alsonoted in [3,11]. Furthermore, � ¼ �2 also gives 1 ¼ 0,but 2 ¼ 18bþ5cþ4d

18bþ4cþ2d . Solving Eq. (2.12) during power law

inflation, �<�1 and � � �2, results in the followingsolution in terms of the Hankel function of the second kind,

Hð2Þ� ðxÞ:

�ðIÞ ¼ffiffiffiffiffi�

2k

r ffiffiffiz

pHð2Þ

� ð ffiffiffiffiffi 2

pzÞ; where � ¼

���������þ 3

2

��������;(2.14)

which gives the correctly normalized incoming wave func-tion for � ! �1 for 2 > 0. During the radiation domi-nated stage the additional curvature terms can be neglectedand thus the mode equation (2.12) simplifies to that of afree harmonic oscillator which is solved by the superposi-tion of plane waves,

�ðRDÞ ¼ 1ffiffiffik

p ðcþe�iðz�z1Þ þ c�eiðz�z1ÞÞ; (2.15)

where z1 � kj�1j and c are the Bogoliubov coefficients.Since the aim here is to calculate the magnetic field energyspectrum at the galactic scale which reenters during theradiation era it is enough to only consider the radiationdominated stage. Matching the solutions of the gaugepotential and its first derivatives at � ¼ �1 on superhor-izon scales determines cþ and c�. In particular, using thesmall argument limit of the Hankel functions [13] jc�j2 isfound to be, for � � � 3

2 ,

jc�j2 ’ ½�ð�Þ�28��1

�1

2� �

�2�me

H1

�2� 2

4

����k

k1

��1�2�;

(2.16)

where it was used that in the approximation used here,F1ð�1Þ ’ �1ðme

H1Þ�2. In the case � ¼ � 3

2 the limiting be-

havior of the mode function on superhorizon scales leads toa divergent factor ln2ð ffiffiffiffiffi

2

pkk1Þ in jc�j2. Therefore, this case

will not be considered further. Including both polarizationstates the total spectral energy density of the photons isgiven by (cf., e.g., [14])

ð!Þ � d

d logk’ 2

�k

a

�4 jc�j2

�2: (2.17)

Since the electric field decays rapidly due to the highconductivity of the radiation dominated universe, the spec-tral energy density (2.17) gives a measure of the magneticfield energy density, B, which results in the commonlyused ratio of magnetic field over background radiation

energy density r � B

¼ �B

�[3], where the density parame-

ter of radiation is given by � ¼ ðH1

H Þ2ða1a Þ4, for � � �2,

� 32 , �1,

r ’ 2½�ð�Þ�23�2�1

�1

2� �

�2�me

MP

�2� 2

4

����k

k1

�3�2�

; (2.18)

where MP is the Planck mass. An expression similar to(2.18) is obtained when the magnetic field energy density iscalculated using the two point function of the magnetic

field, hBið ~kÞB�j ð ~k0Þi. The form of the magnetic field spec-

trum (2.18) imposes the constraint � 32 which implies the

range for� given by�3<�<�1 taking into account theconstraint from power law inflation.An initial magnetic field with strength Bs � 10�20 G

could seed the galactic magnetic field assuming that inaddition there is a galactic dynamo operating [15]. Thisleads to r� 10�37 at a galactic scale of order 1 Mpc. Theformer estimate does not take into account the presence ofa cosmological constant which reduces the minimal mag-netic field strength to Bs � 10�30 G and r� 10�57 [16]. Inorder to directly seed the magnetic field, r has to be of theorder of 10�8. In the following, r [cf. Eq. (2.18)] is calcu-lated at the galactic scale !G ¼ 10�14 Hz, correspondingto a length scale of 1 Mpc. Furthermore the physicalfrequency corresponding to the maximally amplified

wave number, k1, today, is given by !1ð�0Þ ’ 6�1011ðH1

MPÞ1=2 Hz [6]. The expression for r [cf. Eq. (2.18)]

depends on the parameters b, c, and d in addition to theparameters characterizing inflation � and the Hubble pa-rameter at the beginning of the radiation dominated stageH1

MP. The corresponding reheat temperature T1 is given by

T1

MP�

ffiffiffiffiffiH1

MP

q. An upper limit on r can be derived by using the

constraint

�1

�me

H1

��2> 1; (2.19)

consistent with the approximation used to deriveEq. (2.12). This yields to, for � � �2, � 3

2 , �1, at !G ¼10�14 Hz

rmaxð!GÞ ¼ 10�79þ52�½�ð�Þ�2�1

2� �

�2� 2

4

���

��H1

MP

��þð1=2Þ

: (2.20)

For simplicity assuming that the parameters determiningthe curvature terms are all of the same order, so that b�c� d, the maximal possible ratio rmax is independent of

these parameters, since 2 in this case is given by 2 ¼10��77��10 . In Fig. 1 (left) the contour lines for log10rmax are

shown in this case as a function of � and log10ðH1

MPÞ. The

constraint on �1 [cf. Eq. (2.19)] implies in the case b�c� d a constraint on the parameter b,

b > bmin � 10�45

�ð7�� 10Þ�H1

MP

��2: (2.21)

logbmin is shown in Fig. 1 (right).

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043526-3

In [17] a strong bound on primordial magnetic fieldscreated before nucleosynthesis has been derived. Thebound is due to the conversion of magnetic field energyinto gravitational wave energy. Translating the bound onthe magnetic field strength given in [17] into a bound onthe magnetic field to background radiation energy densityratio r results in

rGW ’ 2� 10�61þ52�N GWh20

�H1

MP

���ð3=2Þ

; (2.22)

at the galactic scale used here, � ¼ 1 Mpc, and

N GW ¼ 2ð5=2Þ���

�5

2� �

�: (2.23)

Therefore, the requirement rmax rGW implies an upper

limit on H1

MP, namely,

log 10

�H1

MP

�max

� 9þ 1

2log10

�2ð7=2Þ��h20�ð52 � �Þ�2ð�Þð12 � �Þ2

� 2

4

���;

(2.24)

so that the allowed range is given by ðH1

MPÞ ðH1

MPÞmax. This is

plotted in Fig. 2 for small values of �. As can be appre-ciated from Fig. 2 the maximally allowed value for

log10ðH1

MPÞ is always much bigger than 1 for �<�2 which

is the relevant range here. This can always be satisfiedsince it is assumed that H1 <MP.

Finally, the energy density associated with the perturba-tions in the electromagnetic field during inflation will beestimated and compared with the total energy densityduring inflation. During inflation the energy-momentumtensor associated with the Lagrangian (2.1) is given by [18]

T�� ¼ Tð0Þ�� þ 1

m2e

�bTð1Þ

�� þ c

2Tð2Þ�� þ dTð3Þ

��

�; (2.25)

where, neglecting terms involving derivatives of theMaxwell tensor F��, the different contributions are given

by, (for the exact expressions, cf. [18])

Tð0Þ�� ’ F��F�

� � 1

4g��F��F

��;

Tð1Þ�� ’ RTð0Þ

�� þ 1

2R��F��F

��;

Tð2Þ�� ’ �

�1

2g��R��F

��F�� � F��ðR��F�� þ R��F��Þ

� R��F��F��

�;

Tð3Þ�� ’ �

�1

4g��R

����F��F��

� 3

4F��ðF�

�R���� þ F��R����Þ

�: (2.26)

3.0 2.8 2.6 2.4 2.28.8

9.0

9.2

9.4

9.6

9.8

Log

H1

MP

max

b c d

FIG. 2. The maximal value log10ðH1

MPÞmax [cf. Eq. (2.24)] al-

lowed by the bound from gravitational wave production is shownfor the model with b ¼ c ¼ d for h0 ¼ 0:73.

78

65

52

39

2613

40 30 20 103.0

2.8

2.6

2.4

2.2

LogH1

MP

b c d

33

22

11

011

22

40 30 20 103.0

2.8

2.6

2.4

2.2

LogH1

MP

b c d

FIG. 1 (color online). Left: The contour lines for the maximum value of the logarithm of the ratio of magnetic to backgroundradiation energy density is shown, that is log10rmax, for b ¼ c ¼ d. The values of logðH1

MPÞ correspond to reheat temperatures between

0.1 GeVand 1019 GeV. The numbers within the graph refer to the value of log10rmax along the closest contour line. Right: The contourlines for the minimum value of the logarithm of the parameter b, that is log10bmin, are shown in the model b ¼ c ¼ d. The numberswithin the graph refer to the value of log10bmin along the closest contour line.

KERSTIN E. KUNZE PHYSICAL REVIEW D 81, 043526 (2010)

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The energy-momentum tensor of the electromagnetic fieldcan be written in terms of an imperfect fluid with fluidvelocity u� ¼ ða�1; 0; 0; 0Þ. The energy density is definedby ¼ T��u

�u�. Thus for the metric (2.3) the energy

density associated with the perturbations in the electro-magnetic field during inflation is given by

hðemÞið�Þ ’ 2

a4

Zd3kðF4ð�Þj _Akj2 þ F5ð�Þk2jAkj2Þ;

(2.27)

where

F4ð�Þ ¼ 1þ �4

m2e�

21

��

�1

��2ð�þ1Þ;

�4 ¼ �½6bð�� 2Þ þ cð�� 5Þ � 4d�;

F5ð�Þ ¼ 1þ �5

m2e�

21

��

�1

��2ð�þ1Þ;

�5 ¼ �½6b�þ cð2�� 1Þ þ 2d��:

(2.28)

In deriving the average energy density (2.27) the correla-tion function of the amplitude of the gauge potential in kspace was assumed to be of the form

hA~kA�~qi ¼ ð2�Þ3�ð3Þð ~k� ~qÞjAkj2; (2.29)

and �ð�Þ � �ð�0Þ ¼ ���0 was used. During inflation F4 ’ F1

and F5 ’ F1. The main contribution to hðemÞi comes from

the superhorizon modes (e.g. [19]). Thus using the super-horizon solution during inflation (2.14) together with thesmall argument limit of the Hankel function (cf. [13]) theelectromagnetic energy density can be approximated by

hðemÞið�Þ ’ 2

a4�2

Z k�

0dkk2j�ðIÞj2

’ 1

�

½�ð�Þ�23� 2�

� 2

4

���H4

1

�k�k1

�3�2�

��

�1

��4��2��1;

(2.30)

where k� is the wave number corresponding to the scalewhich becomes superhorizon at the time� during inflation.Hence with k� � ���1 the average energy density asso-ciated with the electromagnetic field is found to be

hðemÞið�ÞM4

P

’ 1

�

½�ð�Þ�23� 2�

� 2

4

����H1

MP

�4��

�1

��4ð�þ1Þ:

(2.31)

During inflation the total energy density is given by

M4P¼ 3

8�

�H1

MP

�2��

�1

��2ð�þ1Þ: (2.32)

Thus the ratio, rðIÞ, between the electromagnetic fieldenergy density and the total energy density is given by

rðIÞ � hðemÞi

’8

3

½�ð�Þ�23�2�

� 2

4

����H1

MP

�2��

�1

��2ð�þ1Þ: (2.33)

Neglecting factors of order one it is found that rðIÞ �=M4

P. Since in the classical domain the total energydensity is less than the Planck energy density it follows

that rðIÞ < 1. Thus the energy density associated with theperturbations in the electromagnetic field is subdominantwith respect to the total energy density during inflation andno backreaction on the inflationary dynamics has to betaken into account.

III. CONCLUSIONS

Primordial magnetic field generation has been investi-gated in a model where the electromagnetic field is coupledto various curvature terms which is motivated by the formof the one-loop effective action of vacuum polarization inQED in a gravitational background [3,11]. Here the result-ing magnetic field spectrum has been calculated explicitlyby matching a stage of power law inflation directly to thestandard radiation dominated phase, solving the modeequation and finding the appropriate Bogoliubov coeffi-cient. The ratio of magnetic field energy density overbackground radiation energy density r calculated at agalactic scale of 1 Mpc has been employed to comparethe model with the minimal required values necessary toseed the galactic dynamic field either directly or rely on themechanism of a galactic dynamo. For simplicity it wasassumed that the coefficients of all additional curvatureterms are of the same order. A constraint on the parameters,which is part of the approximation used to derive the modeequation on superhorizon scale, leads to a maximum valueof the magnetic field strength or equivalently the ratio r

which is a function of � and H1

MP. Here � is the exponent

characterizing inflation and H1 is the value of the Hubbleparameter at the beginning of the radiation dominated era.As can be seen from Fig. 1 there is a region in parameterspace where the resulting magnetic field strength is largerthan Bs > 10�20 G corresponding to r > 10�37, namely,

�<�2:4, H1

MP> 10�18 bounded by the corresponding con-

tour line. De Sitter inflation corresponds to� ¼ �1. In thiscase there is no significant magnetic field generation sincethe mode functions during inflation as well as during theradiation dominated stage are plane waves.Furthermore, it has been checked that the resulting

maximum magnetic field strength satisfies the bound dueto gravitational wave production [17]. Imposing this bound

leads to a maximal value of H1

MPwhich, however, is always

much larger than the maximal value considered here, ascan be seen from Fig. 2.Finally, the energy density associated with the perturba-

tions in the electromagnetic field during inflation has beenestimated and compared with the total energy densityduring inflation. It has been found that during inflationthe energy density in the electromagnetic field is subdo-

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043526-5

minant with respect to the total energy density and thusthere is no back reaction on the dynamics of inflation. In

summary, there is a range of parameters� and H1

MPfor which

magnetic fields are generated that are strong enough toexplain the galactic magnetic field.

ACKNOWLEDGMENTS

Financial support by Spanish Science Ministry GrantsNo. FPA2005-04823, No. FIS2006-05319, andNo. CSD2007-00042 is gratefully acknowledged.

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