Dark energy, non-minimal couplings and the origin of cosmic magnetic fields
aa r X i v : . [ a s t r o - ph . C O ] O c t Dark energy, non-minimal couplings and the origin of cosmic magnetic fields
Jose Beltr´an Jim´enez and Antonio L. Maroto
Departamento de F´ısica Te´orica, Universidad Complutense de Madrid, 28040, Madrid, Spain. (Dated: November 20, 2018)In this work we consider the most general electromagnetic theory in curved space-time leadingto linear second order differential equations, including non-minimal couplings to the space-timecurvature. We assume the presence of a temporal electromagnetic background whose energy densityplays the role of dark energy, as has been recently suggested. Imposing the consistency of the theoryin the weak-field limit, we show that it reduces to standard electromagnetism in the presence of aneffective electromagnetic current which is generated by the momentum density of the matter/energydistribution, even for neutral sources. This implies that in the presence of dark energy, the motion oflarge-scale structures generates magnetic fields. Estimates of the present amplitude of the generatedseed fields for typical spiral galaxies could reach 10 − G without any amplification. In the case ofcompact rotating objects, the theory predicts their magnetic moments to be related to their angularmomenta in the way suggested by the so called Schuster-Blackett conjecture.
PACS numbers: 95.36.+x,98.80.-k,98.62.En
I. INTRODUCTION
The origin of the magnetic fields observed in galax-ies and galaxy clusters with large coherence lengths andstrengths around 10 − G still remains an open prob-lem in astrophysics [1] (recent works [2] also show evi-dence for the existence of extragalactic magnetic fieldswith strengths above 3 × − G). Two different typesof mechanisms have been proposed for the generation ofsuch fields. On one hand, we have the primordial fieldhypothesis, i.e. relic fields from the early universe withcomoving strengths around 10 − − − G are ampli-fied to the present values in the protogalactic collapse.On the other, much weaker fields around 10 − G at de-coupling time could have been amplified by the galacticrotation through a dynamo mechanism. In both cases,preexisting seed fields are required. In fact, there arealso several proposals for the generation of fields whichcould seed a galactic dynamo. They include astrophysicalmechanisms [3], production during inflation [4], in phasetransitions [5], by spontaneous breaking of Lorentz in-variance [6] or by metric perturbations [7]. Nevertheless,it has been argued that the timescales for dynamo ampli-fication may be too long to explain the observed fields inyoung objects [1]. In addition, the origin of the strongerlarge-scale seeds in the primordial approach is even moreproblematic.A very interesting framework for magnetic field gen-eration is the possibility that the standard electromag-netic theory could be modified in the presence of gravity.Thus in [4], couplings of the electromagnetic field to thespace-time curvature were proposed as a way of produc-ing magnetic fields during inflation. In this paper we willconsider a generalized electromagnetic action in curvedspace-time, including also non-minimal couplings. Thecrucial difference with respect to previous works is thatwe allow for the presence of a homogeneous temporalelectromagnetic background potential. This is motivatedby the fact that, as has been recently shown [8], the pres- ence of temporal electromagnetic potentials on cosmo-logical scales could play the role of dark energy. Indeed,this type of fields can be amplified from quantum fluctu-ations during inflation in a completely analogous way tometric perturbations. The initial amplitude of the fieldbeing given by h A i / ∼ H I , where H I is the value ofthe Hubble parameter during inflation. The field is thenshown to grow linearly in time in the matter and radi-ation eras, the corresponding energy density on cosmo-logical scales behaving as a cosmological constant. Inter-estingly, the predicted value of the cosmological constantagrees with observations provided inflation took place atthe electroweak scale. In such a case, the present valueof the background field would be ¯ A ≃ . M P . Herewe show that the non-minimal coupling of the tempo-ral background to the space-time curvature implies thatthe energy-momentum density of any matter/energy dis-tribution generates an effective electromagnetic current,even for neutral sources. This allows to establish a nat-ural link between dark energy and the origin of cosmicmagnetic fields. II. GENERALIZED ELECTROMAGNETISM
Let us consider the most general expression for theelectromagnetic action in the presence of gravity, includ-ing all the possible terms leading to linear second orderdifferential equations: S = Z d x √− g (cid:20) − F µν F µν + λ ∇ µ A µ ) + σR µν A µ A ν + ωRA µ A µ ] . (1)Notice that this expression does not contain any dimen-sional parameter or potential term. The minimal casewith σ = ω = 0 was studied in detail in [8] and the pos-sibility of generating cosmic magnetic fields in this casehas been considered recently in [9]. In this action, the λ parameter can be fixed by choosing a normalizationof the non-transverse modes and σ and ω are arbitrarydimensionless constants. In order to fix them, we willconsider the weak-field limit of the theory. Thus, thespace-time metric can be written as a small perturba-tion around Minkowski space-time, g µν = η µν + h µν andthe electromagnetic potential reads A µ = ¯ A µ + a µ with¯ A µ = ¯ A δ µ and ¯ A constant. The background electro-magnetic field is determined by the corresponding cosmo-logical value and therefore it could evolve on cosmologicaltimescales. However, for local experiments it is a goodapproximation to assume it constant (in agreement withthe flat space-time background). Notice that the electricand magnetic fields associated to ¯ A µ identically vanish.The corresponding Maxwell equations obtained from (3)read to first order: ∂ ν F µν + λ∂ µ ( ∇ ν A ν ) (1) = J µg . (2)where F µν = ∂ µ a ν − ∂ ν a µ , ( ∇ ν A ν ) (1) denotes the con-tribution to first order and the non-minimal terms giverise to an effective current given also to first order by: J µg = 2( σ R µν (1) + ω R (1) η µν ) ¯ A ν . Imposing this effectivecurrent to be conserved, i.e. ∂ µ J µg = 0, we obtain σ = − ω , i.e. the non-minimal coupling must involvethe conserved Einstein tensor G µν = R µν − Rg µν . No-tice also that conservation implies that taking the diver-gence of (2) we get (cid:3) ( ∇ µ A µ ) (1) = 0, i.e. to first orderit is possible to impose the Lorenz condition ∇ µ A µ = 0at the classical level as in ordinary electromagnetism, sothat the λ term disappears. Thus, for weak gravitationalfields we recover ordinary Maxwell electromagnetism, theonly difference is the appearance of a gravitationally-generated electromagnetic current. Notice that this cur-rent is only present provided the background electromag-netic potential is non-vanishing and in the presence ofspace-time curvature.According to the previous discussion, the form of theaction will be given by: S = Z d x √− g (cid:20) − F µν F µν + λ ∇ µ A µ ) + σG µν A µ A ν (cid:21) (3) III. CONSISTENCY AND STABILITY
This theory is a particular case of the more generalclass of vector-tensor theories [10]. These theories usuallygive rise to modifications of the gravitational interactionat small (Solar System) scales which are encoded in thecorresponding PPN parameters. For this particular case,the PPN parameters are: γ − ≃ − πGσA ⊙ , β − ≃− πGσA ⊙ , α ≃ − πGσA ⊙ , α ≃ − πGσA ⊙ , where A ⊙ is the background amplitude at Solar System scalesand we have assumed | σ | ≪
1, keeping only the leadingorder in the expansion. The most stringent constrainton the PPN parameters is | γ − | ≤ . × − , which imposes the corresponding limit on | σA ⊙ | . If we assumethat the amplitude of the electromagnetic field at SolarSystem scales resembles the cosmological value A ⊙ ≃ ¯ A ≃ . M P , we obtain the constraint | σ | < ∼ − .Let us now study the stability of the theory by ana-lyzing the behavior of the inhomogeneous perturbationsaround the Minkowski background. As usual, we shall de-compose both the electromagnetic perturbation a µ andthe metric perturbation h µν in Fourier modes and sepa-rate them into scalar, vector and tensor contributions (wefollow the same procedure as in [11]). The correspondingpropagation speeds for the perturbations are: c s = 1 (4) c v = 1 − πGσ (1 − σ ) ¯ A − πGσ ¯ A ≃ πGσ ¯ A (5) c t = 1 + 8 πGσ ¯ A − πGσ ¯ A ≃ πGσ ¯ A (6)where we have expanded for | σ | ≪
1. Notice that thescalar modes propagate at the speed of light irrespectiveof the value of the parameter σ . However, the speedof photons c v would be larger than the ”speed of light” c = 1 which determines the null cones of the Minkowskigeometry. This in principle could give rise to inconsisten-cies with causality in the theory. However, it is knownthat in scenarios with violations of the strong equivalenceprinciple, as the one considered here, superluminal prop-agation can be consistent with causality, provided stablecausality is ensured [12]. For that purpose, if the newlight cone can be written as G µν k µ k ν = 0, then theremust exist a globally defined function f , such that ∇ µ f must be non-vanishing and timelike everywhere with re-spect to ( G − ) µν . In our case, the light cones for vectorscan be written as: (cid:2)(cid:0) πGκ ¯ A (cid:1) η µν − πGκ ¯ A µ ¯ A ν (cid:3) k µ k ν = 0 (7)with κ = σ / (1 − σA ), whereas for tensors: (cid:2)(cid:0) πGσ ¯ A (cid:1) η µν − πGσ ¯ A µ ¯ A ν (cid:3) k µ k ν = 0 (8)Since ¯ A < ∼ M P and | σ | ≪
1, in both cases, the effectivemetric ( G − ) µν is a small perturbation with respect toMinkowski. This implies that we can use the time coor-dinate t as the globally defined function f . Thus, we seethat, for small σ , the theory does not exhibit classicalinstabilities or causality inconsistencies.In order to study the presence of quantum instabilities(ghosts), we analyze the positiveness of the energy den-sity of the three types of perturbations considered before.Thus, we define the energy for the modes as [11, 13]: ρ = (cid:28) T (2)00 − πG G (2)00 (cid:29) (9)where T (2) µν and G (2) µν are the energy-momentum tensor ofthe vector field and the Einstein’s tensor calculated up toquadratic terms in the perturbations and h· · · i denotesan average over spatial regions. Although the calcula-tion has been performed in the longitudinal gauge, both,mode frequencies and energies, do not depend on thegauge choice.For scalar modes we find that the energy density van-ishes identically if we impose the Lorenz condition, as inordinary electromagnetism (see [8] for expanding back-grounds). For vector and tensor modes, the energy den-sities are: ρ v = 2 k − πGσ (cid:2) πGσ (2 σ −
1) ¯ A (cid:3) ¯ A (1 − πGσ ¯ A ) | ~C | ≃ k (1 − πGσ ¯ A ) | ~C | (10) ρ t = k − πGσ (2 + 8 πGσ ¯ A ) ¯ A − πGσ ¯ A (cid:0) | C ⊕ | + | C ⊗ | (cid:1) ≃ k (1 − πGσ ¯ A ) (cid:0) | C ⊕ | + | C ⊗ | (cid:1) (11)where ~C is the amplitude of the Fourier mode for thevector modes and C ⊕ , ⊗ are the amplitudes of the twopolarizations of the gravitational waves. From these ex-pressions we see that the theory is also free from quantuminstabilities for small | σ | . IV. COSMOLOGICAL EVOLUTION
In the following we shall show that, due to the small-ness of the parameter σ , the cosmological evolution ofthe homogeneous mode becomes modified in a negligi-ble way by the presence of the coupling to the Einsteintensor. This ensures that the inflationary generation andevolution discussed in [8] is also a good description in thenon-minimal case. We shall consider an electromagneticfield of the form A µ = ( A ( t ) , , , A z ( t )) in a FLRWmetric ds = dt − a ( t ) d~x . In this case, the equationsof motion read:¨ A + 3 H ˙ A + 3 (cid:16) ˙ H − σ λ H (cid:17) A = 0 (12)¨ A z + H ˙ A z + σ (4 ˙ H + 6 H ) A z = 0 (13)where σ λ = σλ . In a de-Sitter inflationary era with H = H I constant, the growing mode solutions behave for small σ as: A ( t ) ∝ exp(2 σ λ H I t ) , A z ( t ) ∝ exp( − σH I t ) (14)During the radiation and matter dominated epochs inwhich H = p/t with p = 1 / p = 2 / A ( t ) ∝ t σ λ / , A z ( t ) ∝ t / σ (15)in the radiation era, and A ( t ) ∝ t σ λ / , A z ( t ) ∝ t / (16)in the matter era. We see that the the only effect ofthe non-minimal coupling is a slight modification in the power exponents. Finally, in a universe dominated by theelectric potential A ( t ), we have a power law expansion ofthe form a ( t ) ∝ t − λ σ . For small σ , we have an acceleratedexpansion which corresponds to a quasi de Sitter phasewith slow-roll parameter ǫ = − σλ . Notice that, in thelimit σ →
0, we also recover the pure de Sitter solutionfound in the minimal case.
V. EFFECTIVE ELECTROMAGNETICCURRENT: GRAVITATIONAL MAGNETISM
Let us now consider the possible effects of the neweffective electromagnetic current J µg = 2 σG µ ¯ A in (2).Using Einstein equations to relate G µν to the matter con-tent, we obtain: J µg = 16 πGσT µ ¯ A (17)so that the effective electromagnetic current is essentiallydetermined by the four-momentum density. Moreover,if we assume T µν = ( ρ + p ) u µ u ν − pη µν at first order,we can see that the energy density of any perfect fluidhas an associated electric charge density given, for smallvelocities, by: ρ g = J g = 16 πGσρ ¯ A (18)and the three-momentum density generates an electriccurrent density given by ~J g = 16 πGσ ( ρ + p ) ~v ¯ A (19)This theory effectively realizes the old conjecture bySchuster, Einstein and Blackett [14] of gravitational mag-netism, i.e. neutral mass currents generating electromag-netic fields. Early attempts to encompass this conjecturein a gravitational theory can be found in [15].In the case of a particle of mass m at rest, (18) intro-duces a small contribution to the active electric charge(the source of the electromagnetic field), given by ∆ q =16 πGσm ¯ A ≃ σ ( m/M P ), but does not modify the pas-sive electric charge (that determining the coupling to theelectromagnetic field). In fact, this would give differentactive charges to electrons and protons due to their massdifference and, in addition, would provide the neutronwith a non-vanishing active electric charge. However, theeffect is very small in both cases ∆ q ≃ σ − e where e = 0 .
303 is the electron charge in Heaviside-Lorentzunits. Present limits on the electron-proton charge asym-metry and neutron charge are both of the order 10 − e [16], implying | σ | < ∼ − which is less stringent thanthe PPN limit discussed before. Notice also that pho-tons would acquire a non-vanishing active electric charge.However tight existing limits imposed by deflection of ra-dio pulsar emission by galactic magnetic fields [17] onlyapply to passive charge which is not modified gravita-tionally.On the other hand, for any compact object, even in thecase it is neutral, the effective electric current will gener-ate an intrinsic magnetic moment ~m = R ~r × ~J g ( ~r ) d ~r given by: ~m = β √ G ~L (20)with ~L the corresponding angular momentum and β aconstant parameter whose value is: β = 16 π √ Gσ ¯ A (21)Notice that relation (20) resembles the Schuster-Blackettlaw, which is an empirical relation between the magneticmoments and the angular momenta found in a wide rangeof astrophysical objects from planets, to galaxies, includ-ing those related to the presence of rotating neutron starssuch as GRB or magnetars [18]. Let us mention that theobservational evidence on this relation is still not conclu-sive. From observations, the β parameter is found to bein the range 0.001 to 0.1.Imposing the PPN limits on the σ parameter, we find β < ∼ − , which is just below the observed range. Thusfor a typical spiral galaxy, a direct calculation provides: B ∼ σ − G, i.e. according to the PPN limits, the fieldstrength could reach 10 − G without amplification.However, notice that even in the case in which thisgeneration mechanism took place, the determination ofthe actual amplitudes of magnetic fields in astrophysicalobjects would require to take into account the full mag-netohydrodynamical evolution. Therefore, we do not ex-pect the gravitationally generated magnetic field to nec-essarily agree with observations. However this mecha-nism could help seeding standard amplification mecha-nisms such as dynamo with appropriate fields correlatedto the object angular momentum.It is also interesting to evaluate the maximum mag-netic field that could be generated in a Blackett-like ex-periment in laboratory [19]. Thus for a rotating neutralsphere of M = 500 kg, radius R = 0 . ω = 100 Hz, the field amplitude would be B ∼ σ − T < ∼ − T, which is just below thefundamental sensitivity limit of SQUID or SERF magne-tometers [20].
VI. DISCUSSION
Notice that in this scenario, it is the non-vanishingRicci curvature what generates electromagnetic fields inthe presence of dark energy. Notice, however, that theonly requirement for the main results of the present workis the presence of a cosmological electric potential. Thisimplies that magnetic fields would be associated to thepresence of a non-vanishing energy-momentum distribu-tion. In other words, the effect would be absent in vac-uum even for curved backgrounds.Another interesting consequence of the presence ofnon-minimal couplings in the electromagnetic action (3)is the fact that they play the role of an effective massterm for the electromagnetic field during inflation. Thisnaturally provides an infrared cutoff in the calculation ofthe field dispersion from quantum fluctuations [21].As shown before, fields of strengths up to 10 − G couldbe generated on galactic scales in this theory, which couldseed a galactic dynamo or even play the role of ”primor-dial” seeds and account for the observed magnetic fieldsin galaxies and clusters just by adiabatic compression inthe collapse of the protogalactic cloud. Therefore, a de-tailed study of the magnetohydrodynamical evolution inthe presence of the gravitationally-induced current willhelp establishing the importance of dark energy in theorigin of cosmic magnetic fields.
Acknowledgments:
We would like to thank Prof. MisaoSasaki for useful comments. This work has been sup-ported by MICINN (Spain) project numbers FIS 2008-01323 and FPA 2008-00592, CAM/UCM 910309, MECgrant BES-2006-12059 and MICINN Consolider-IngenioMULTIDARK CSD2009-00064. J.B. also wishes tothank support from the Norwegian Council under theYGGDRASIL project no 195761/V11. [1] L. M. Widrow, Rev. Mod. Phys. (2002) 775;R. M. Kulsrud and E. G. Zweibel, Rept. Prog. Phys. (2008) 0046091; P. P. Kronberg, Rept. Prog. Phys. (1994) 325.[2] A. Neronov and I. Vovk, Science (2010) 73; F. Tavec-chio, et al., arXiv:1004.1329 [astro-ph.CO]; S. Ando,A. Kusenko, Astrophys. J. (2010) L39.[3] E.R. Harrison, MNRAS (1970) 279; Phys. Rev. Lett. (1973) 18[4] M.S. Turner and L.M. Widrow, Phys. Rev.
D37 : 2743,(1988); K. Bamba and M. Sasaki, JCAP (2007)030. R. Durrer, L. Hollestein and R. Kumar Jain,arXiv:1005.5322 [astro-ph.CO][5] J. M. Quashnock, A. Loeb and D. N. Spergel, Astrophys. J. (1989) L49; T. Vachaspati, Phys. Lett.
B265 (1991) 258[6] O. Bertolami and D. F. Mota, Phys. Lett. B (1999)96[7] A. L. Maroto, Phys. Rev. D (2001) 083006; S. Matar-rese, S. Mollerach, A. Notari and A. Riotto, Phys. Rev.D (2005) 043502; K. Ichiki, K. Takahashi, H. Ohno,H. Hanayama and N. Sugiyama, Science (2006) 827;K. Ichiki, K. Takahashi, N. Sugiyama, H. Hanayama andH. Ohno, arXiv:astro-ph/0701329.[8] J. Beltr´an Jim´enez and A.L. Maroto, JCAP (2009)016; Phys. Lett. B (2010) 175; Int. J. Mod. Phys.D (2009) 2243; J. B. Jimenez, T. S. Koivisto,A. L. Maroto and D. F. Mota, JCAP (2009) 029 [9] J. Beltr´an Jim´enez and A.L. Maroto, arXiv:1010.3960[astro-ph.CO][10] C. Will, Theory and experiment in gravitational physics ,Cambridge University Press, (1993)[11] J. Beltr´an Jimenez and A. L. Maroto, JCAP (2009)025[12] S.W. Hawking and G.F.R. Ellis,
The large scale struc-ture of space-time , Cambridge (1973); G. M. Shore,arXiv:gr-qc/0302116.[13] C. Eling, Phys. Rev. D (2006) 084026[14] A. Schuster, Proc. Lond. Phys. Soc. (1912) 121; A.Einstein, Schw. Naturf. Ges. Verh. 105 Pt. 2, 85 (1924)S. Saunders S and H.R. Brown, Philosophy of Vacuum(Oxford: Clarendon) (1991); P. M. S. Blackett, Nature (1947) 658[15] W. Pauli, Ann. Phys. (Leipzig) , 305 (1933); J. G.Bennett et al., Proc. R. Soc. London A 198, 39 (1949); A. Papapetrou, Philos. Mag. , 399 (1950); G. Luchak,Can. J. Phys. , 470 (1952); A. O. Barut and T. Gornitz,Found. Phys. , 433 (1985).[16] C. Amsler et al. (Particle Data Group), Phys. Lett. B (2008) 1[17] G. Raffelt, Phys. Rev. D (1994) 7729[18] R. Opher and U. F. Wichoski, Phys. Rev. Lett. (1997)787; R. da Silva de Souza, R. Opher, JCAP (2010)022[19] S.-P. Sirag, Nature 278, 535 (1979)[20] Ya. S. Greenberg, Rev. Mod. Phys. (1998) 175[21] D. H. Lyth, JCAP (2007) 016; K. Enqvist,S. Nurmi, D. Podolsky and G. I. Rigopoulos, JCAP (2008) 025; Y. Urakawa and T. Tanaka, Prog. Theor.Phys.122