Completing magnetic field generation from gravitationally coupledelectrodynamics with the curvaton mechanism

Kerstin E. Kunze*

Departamento de Fı́sica Fundamental and IUFFyM, Universidad de Salamanca,Plaza de la Merced s/n, 37008 Salamanca, Spain

(Received 30 October 2012; published 5 March 2013)

Primordial magnetic fields of cosmologically interesting field strengths can be generated from gravita-

tionally coupled electrodynamics during inflation. As the cosmological constraints require this to be

power law inflation it is not possible to generate at the same time the curvature perturbation from inflation.

Therefore here a completion is considered whereby the large scale magnetic field is generated during

inflation and the primordial curvature mode in a subsequent era from a curvaton field. It is found that

constraints on the model to obtain strong magnetic fields and those to suppress the amplitude of the

curvature perturbation generated during inflation can be simultaneously satisfied for magnetic seed fields

Bs * 10�30 G.

DOI: 10.1103/PhysRevD.87.063505 PACS numbers: 98.80.Cq, 98.62.En

I. INTRODUCTION

Observations indicate the existence of large scalemagnetic fields in the Universe. Evidence for magneticfields at galactic and cluster scales has been obtainedfor decades with field strengths of order 10�6 G [1].Moreover, using observations of TeV blazars there havebeen claims of truly cosmologically magnetic fields, per-vading space with a lower bound on the field strength in therange 10�18 G to 10�15 G depending on the model [2–5].However, in Ref. [6] possible uncertainties are discussedconcluding that the data are compatible with a zero inter-galactic magnetic field hypothesis.

There are a multitude of proposals to explain the exis-tence of cosmological magnetic fields (for recent reviews,see, e.g., Ref. [7]). Placing the generation mechanism inthe very early Universe during inflation by significantlyamplifying perturbations in the electromagnetic fieldrequires us to break conformal invariance in flat back-grounds [8] which however is not the case in open geome-tries [9] (see, however, Ref. [10]).

The conformal invariance of Maxwell’s equationsin four dimensions is broken in models in which theelectromagnetic field is gravitationally coupled. Couplingsbetween curvature terms and the Maxwell tensor in theLagrangian of the form RF��F

��, R��F��F�

� and

R����F��F�� are present when describing, e.g., the propa-

gation of a photon in a curved background [11]. In Fourierspace the resulting mode equation for the gauge potentialof the electromagnetic field in an expanding backgroundincludes a term which during inflation can cause sufficientamplification on superhorizon scales so as to generate astrong enough magnetic field in order to seed the galacticdynamo. This was first proposed in Ref. [8] and investi-gated in more detail in Ref. [12].

In Ref. [12] the spectrum of the resulting nonhelicalmagnetic field is calculated assuming that the inflationarystage is directly matched to the standard radiation domi-nated era at some conformal time � ¼ �1, so that the scalefactor has the form

að�Þ ¼

8>>><>>>:a1

���1

��

� < �1

a1

���2�1��1

�� � �1

(1.1)

and the line element has the form ds2 ¼ a2ð�Þð�d�2 þdx2 þ dy2 þ dz2Þ. In the following a1 � 1. For � < �1, deSitter inflation takes place for � ¼ �1 and power lawinflation for �1<�<�1.On large scales the magnetic field simply redshifts with

the expansion as a�2. However, magnetohydrodynamicaland turbulence processes in the expanding Universe canlead to important changes in the shape and amplitude of themagnetic field spectrum. In particular, for helical magneticfields power on small scales can be shifted to larger scalesvia inverse cascade [13–17].In Ref. [12] the fractional magnetic field energy

density, r ¼ �B

��on a galactic scale of order 1 Mpc has

been found as

rð!GÞ ¼ 10�79þ52�½�ð�Þ�2�1

2� �

�2�2

4

����H1

MP

��þ1

2;

(1.2)

where � � j�þ 32 j, 2 ¼ 10��7

7��10 and H1 is the Hubble

parameter at the beginning of the radiation dominatedera, at � ¼ �1 which determines the reheat temperature.It was found that a magnetic field strength B> 10�20 Gcorresponding to r > 10�37 can be reached for �<�2:8

and H1

MP< 10�18, depending on � (cf. Fig. 1). The con-

straint on � prevents the observed curvature perturbation*kkunze@usal.es

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from being generated during inflation since the resultingspectral index is too small. In slow roll inflation power lawinflation [18] is realized by an exponential potential of theinflaton of the form (e.g., Ref. [19])

VðÞ ¼ Vi exp

244

ffiffiffiffi�

p

s ð�iÞMP

35; (1.3)

where the index i indicates initial values and p determinesthe scale factor in cosmic time dt ¼ ad�, that is a� tp.

Thus p ¼ ��þ1 . The slow roll parameters are given by � ¼

2 ¼ 1

p . Ending inflation by bubble formation puts the con-

straint p < 10 [20,21]. p as a function of � is monotoni-cally growing for �< 0, approaching 1 for � ! �1 andat � ¼ �2:8, p ¼ 1:56, so that the upper limit on p doesnot put any additional constraint on the model at hand. Thespectral index of the curvature perturbation created duringinflation, ns ¼ 1þ 2 � 6�, is in the range�1 and�0:29for �<�2:8. The power spectrum of the curvature per-turbation evaluated at the horizon crossing is given by [19]

P � ¼ 1

�M2P

�2��3

2�ð�Þ�ð32Þ

�2��� 1

2

��2�þ1�H2

�

�k¼aH

;

(1.4)

where � � 32 þ 1

p�1 . The amplitude of the curvature per-

turbation generated during inflation should only contributenegligibly to the final curvature perturbation. Starting withthe slow roll equations

3H _þ V ¼ 0 (1.5)

H2 ¼ 8�

3M2P

V (1.6)

the evolution of the inflaton is found to be

�i

MP

¼ � 1

2

ffiffiffiffip

�

rð�þ 1Þ ln

��

�i

�: (1.7)

Thus the potential is given by

VðÞM4

P

¼ 3�2

8�

��1M�1

P

��2��

�1

��2ð�þ1Þ: (1.8)

This allows us to calculate Hk � HðkÞ ¼ Hjk¼aH. Using

that aH ¼ �� and a1 ¼ að�1Þ ¼ 1 it follows that�

Hk

MP

�2 ¼

�H1

MP

��2��k

MP

�2ð�þ1Þ

: (1.9)

Finally, using ��;0 ¼ ðH1

H0Þ2ða1a0Þ4,�

Hk

MP

�2 ¼ ð5:24� 10�58�

�14

�;0Þ2ð�þ1Þ�

kp

Mpc�1

�2ð�þ1Þ

��H0

MP

��ð�þ1Þ�H1

MP

�1��

; (1.10)

where kp is going to be chosen to be the pivot wave number

of WMAP 7 today; kp ¼ 0:002 Mpc�1 [22]. Thus the

amplitude of the curvature spectrum at kp is determined by

P� ¼ p

�

�2��3

2�ð�Þ�ð32Þ

�2��� 1

2

��2�þ1

� ð5:24� 10�58��1

4

�;0Þ2ð�þ1Þ�

kp

Mpc�1

�2ð�þ1Þ

��H0

MP

��ð�þ1Þ�H1

MP

�1��

: (1.11)

In Fig. 1 a contour plot of log 10P� is shown. As can be

appreciated from Fig. 1, satisfying the constraints to gen-erate a magnetic field with Bs � 10�20 G corresponding tor� 10�37 (that is, �<�2:8Þ leads to curvature perturba-tions which are in general, too large, not only for thecurvaton mechansim to work, but even larger than theobserved amplitude P � ¼ 2:43� 10�9 [22]. However, it

is possible to generate a magnetic field with Bs � 10�30 Gcorresponding to r� 10�57 which is the lower limitrequired to seed the galactic dynamo when the nonvanish-ing cosmological constant is taken into account [23].In order for the curvature perturbation created during

inflation to be subdominant � and H1

MPhave to be in the

32

0

32 64 96 128 160

logr : 57

logr : 37

12

9

40 35 30 25 20 156

5

4

3

2

LogH1

MP

log10P

FIG. 1 (color online). Contour plot of log 10P� shown for

kp ¼ 0:002 Mpc�1. The contour lines corresponding to

P� ¼ 10�9 (thick, black dashed line) and P

� ¼ 10�12

(thick, blue, long-dashed line) are marked. The red dotted-dashed and green dotted lines show the maximal value of thelogarithm of the ratio of magnetic to background radiationenergy density of the model [12], corresponding to r ¼ 10�37

(red dotted-dashed) and r ¼ 10�57 (green dotted).

KERSTIN E. KUNZE PHYSICAL REVIEW D 87, 063505 (2013)

063505-2

ranges�4:8 � � � �3:6 and 10�40 � H1

MP� 10�34 which

correspond to reheat temperatures between 0.1 and100 GeV.

II. GENERATING THE CURVATUREPERTURBATION

In the curvaton model the curvature perturbation isgenerated after inflation has ended by the curvaton field� which has been subdominant during inflation [24]. Thespectral index of the final curvature perturbation is givenby [20]

ns ¼ 1þ 2 �� � 2�; (2.1)

where

�� � �M2P

V

@2V

@�2(2.2)

and �M2P is the reduced Planck mass M2

P=ð8�Þ. Assumingthe simplest model [24,25] the potential during inflation isdefined to be

Vð;�Þ ¼ Vi exp

�4

ffiffiffiffi�

p

s ð�iÞMP

�þ 1

2m2

��2; (2.3)

where the contribution from the curvaton has to besubleading to the one of the inflaton. Equation (2.1)requires �� to be of the order of � for a nearly scaleinvariant spectrum. Thus

�� ’ �M2Pm

2�

Vi exph4

ffiffiffi�p

q ðk�iÞMP

i ’ m2�

3H2k

; (2.4)

calculated at the time of the first horizon crossing duringinflation of the mode k ¼ aH, so that the mass of thecurvaton is determined by�

m�

MP

�¼

�3

p� 3

2ð1� nsÞ

�12

�Hk

MP

�: (2.5)

This is shown in Fig. 2 for the region allowed by the

constraints that P� � 10�12 and r > 10�57 for P � ¼

2:43� 10�9 and 1� ns ¼ 0:037 [22].The evolution of the background curvaton is

determined by

€�þ 3H _�þ V� ¼ 0: (2.6)

It is assumed [20] that jV��j � H2 which in the modelat hand results in m2

�t2 � 1 since p is Oð1Þ which

implies that during and after inflation, before the curvatonbecomes effectively massive, � stays approximately con-stant, � ’ �.

In the subsequent analysis we follow [25]. The end ofinflation takes place at � ¼ �1 when the Hubble parameterhas the valueH1. During the following radiation dominatedera it evolves as H2 � 1=a4 so that

H2 ¼ H21

�a1a

�4: (2.7)

The curvaton becomes massive at m2� ¼ H2 so that

�a1

amass

�4 ¼ m2

�

H21

: (2.8)

To prevent an additional stage of inflation driven by thecurvaton, thus requiring the Universe to be radiation domi-nated at �mass, imposes �radð�massÞ 1

2m2��

2, where � isthe value of the curvaton during inflation when the modesof the observable perturbations leave the horizon, assum-ing that, during the late stages of slow roll inflation, thechange in � can be neglected. Then together with the valueof �rad at the beginning of the radiation dominated era,�radð�1Þ ¼ 3 �M2

PH21 , it follows that

�2 � 3M2P

4�; (2.9)

which is the same constraint as in the chaotic inflationmodel of Ref. [25].In Ref. [24] two separate cases have been considered. If

the curvaton decays during the radiation dominated era,that is until its decay it stays subdominant, the resultingcurvature perturbation is given by [24]

P � ¼S2decay

16

H2k

�2�2(2.10)

8

7

6

40 39 38 37

5.0

4.8

4.6

4.4

4.2

40 39 38 37

LogH1

MP

log10 m MP

FIG. 2 (color online). Contour plot of log 10ðm�

MPÞ shown for

kp ¼ 0:002 Mpc�1 in the region allowed by the constraints

P� � 10�12 and r > 10�57 for P � ¼ 2:43 � 10�9 and

1� ns ¼ 0:037 [22].

COMPLETING MAGNETIC FIELD GENERATION FROM . . . PHYSICAL REVIEW D 87, 063505 (2013)

063505-3

using that �k ¼ � is approximately constant during in-flation and S � ��

�rad. Moreover, here �� ¼ 1

2m2��

2. As in

Ref. [25] it is found that at the time of decay,

Sdecay ¼ �26 �M2

P

adecay

amass

(2.11)

and defining the decay constant of the curvaton ��,

together with �� ¼ Hdecay results in adecay=amass ¼ðm�=��Þ1=2, implying

Sdecay ’ �26 �M2

P

�m�

��

�12; (2.12)

so that finally, the spectrum of the curvature perturbation,for Sdecay < 1, is found to be

P � ¼ 1

9

�Hk

MP

�2�m�

MP

���MP

�2���

MP

��1: (2.13)

Moreover, for the perturbations to be GaussianHk=� � 1 [24]. This together with Sdecay < 1 determines

the range of possible values of �=Hk to be

1<�Hk

<1

4�P12

�

: (2.14)

30.6

25.5

20.4

40 39 38 37

5.0

4.8

4.6

4.4

4.2

40 39 38 37

LogH1

MP

log10 MP , Hk 10

25.5

20.4

15.3

40 39 38 37

5.0

4.8

4.6

4.4

4.2

40 39 38 37

LogH1

MP

log10 MP , Hk 1 32 P

87

6

5

40 39 38 37

5.0

4.8

4.6

4.4

4.2

40 39 38 37

LogH1

MP

log10 MP , Hk 10

5

4

3

40 39 38 37

5.0

4.8

4.6

4.4

4.2

40 39 38 37

LogH1

MP

log10 MP , Hk 1 32 P

FIG. 3 (color online). Upper panel: Contour plot of log 10ð��

MPÞ. Lower panel: Contour plot of the corresponding values of log 10ð�

MPÞ.

All plots are shown for kp ¼ 0:002 Mpc�1 in the region allowed by the constraints P� � 10�12 and r > 10�57 for different values of

ð�HkÞ. The curvature perturbation is assumed to be given by the best fit parameters of WMAP 7, P � ¼ 2:43� 10�9 and 1� ns ¼ 0:037

[22]. It is assumed that the curvaton decays during the radiation dominated era, so that Sdecay < 1.

KERSTIN E. KUNZE PHYSICAL REVIEW D 87, 063505 (2013)

063505-4

Finally, the decay constant �� is determined by���

MP

�¼ 1

9P �

�m�

MP

���Hk

�2�Hk

MP

�4: (2.15)

The decay constant �� and the value of � is shownfor different values of �

Hkin Fig. 3. As can be seen

��=MP ¼ Hdecay > 10�40 which is the minimal value to

ensure standard primordial nucleosynthesis.In the opposite case, Sdecay > 1, the curvaton dominates

before decay and [24]

P � ’ 1

9

H2k

�2�2: (2.16)

In this case the Gaussianity requirement is alwayssatisfied if the curvature perturbations are of theobserved magnitude. Moreover � is completely deter-mined by Eq. (2.16),�

�MP

�¼ 1

3�P12

�

�Hk

MP

�; (2.17)

which is shown in Fig. 4 for the best fit values ofWMAP 7 [22]. In this case Sdecay > 1 which results in

the constraint [25],

10�40 <��

MP

<

�4�

3

�2��MP

�4�m�

MP

�: (2.18)

This implies ð�MPÞ4ðm�

MPÞ> 10�40 which is satisfied in the

allowed region in parameter space as can be seen fromFigs. 2 and 4.

III. CONCLUSIONS

In Ref. [12] it was shown that cosmologically relevantmagnetic fields can result from gravitationally coupledelectrodynamics motivated by the form of the one-loopeffective action of the vacuum polarization in QED in agravitational background. Perturbations in the electromag-netic field are amplified during power law inflation deter-mined by the exponent � and the value of the Hubbleparameter at the end of inflation H1 which here alsodetermines the reheat temperature. The ratio r of magneticfield energy density over background radiation energydensity has to be larger than 10�37, corresponding toBs � 10�20 G to seed the galactic dynamo in a Universewith no cosmological constant. This value is reduced tor�10�57 and Bs�10�30 G in a Universe with�> 0 [23].In Ref. [12] it was found that there is a region in the

½�; ðH1

MP�-plane for which seed magnetic fields are obtained

with Bs > 10�20 G corresponding to r > 10�37. However,the curvature perturbations generated during inflation forthose values are incompatible with the observed nearly scaleinvariant spectrum. Therefore, here the possibility of com-pleting this model of magnetic field generation with thecurvatonmechanism has been considered, thereby generatingthe curvature perturbations after inflation and hence, impos-ing that the amplitude of the curvature perturbations duringinflation is negligible. This restricts significantly the parame-

ter space in � and ðH1

MPÞ, allowing the creation of primordial

magnetic fields only with amplitude Bs � 10�30 G, how-ever, still satisfying the weaker constraint r > 10�57.We have assumed that power law inflation is realized

within slow roll inflation with an exponential potential.The curvaton is described by a simple quadratic potential.Assuming that the curvature perturbation due to the curva-ton is determined by the best fit values ofWMAP 7 and thatthe contribution of the curvature perturbation due to theinflaton is less than 10�3 of that of the curvaton, theparameter space of the curvaton model has been explored.

The corresponding values in the ½�; ðH1

MP�-plane have been

found for the mass of the curvaton m�, its field valueduring inflation � and the decay constant �� in the twocases where the curvaton decays during radiation domina-tion or during curvaton domination.

ACKNOWLEDGMENTS

I am indebted to David Lyth for suggesting the problemand for very useful discussions. Financial support bySpanish Science Ministry Grants No. FPA2009-10612,No. FIS2009-07238, and No. CSD2007-00042 is gratefullyacknowledged.

5

4

3

2

40 39 38 37

5.0

4.8

4.6

4.4

4.2

40 39 38 37

LogH1

MP

log10 MP

FIG. 4 (color online). Contour plot of log 10ð�MP

Þ shownfor kp ¼ 0:002 Mpc�1 in the region allowed by the

constraints P� � 10�12 and r > 10�57 for P � ¼ 2:43� 10�9

and 1� ns ¼ 0:037 [22]. It is assumed that the curvaton domi-nates before decay, so that Sdecay > 1.

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