Large-Scale Magnetic Fields, Dark Energy and QCD
PPreprint typeset in JHEP style - HYPER VERSION
Large-Scale Magnetic Fields, Dark Energy and QCD.
Federico R. Urban and Ariel R. Zhitnitsky
Department of Physics & Astronomy, University of British Columbia, Vancouver, B.C.V6T 1Z1, Canada
Abstract:
Cosmological magnetic fields are being observed with ever increasing corre-lation lengths, possibly reaching the size of superclusters, therefore disfavouring the con-ventional picture of generation through primordial seeds later amplified by galaxy-bounddynamo mechanisms. In this paper we put forward a fundamentally different approach thatlinks such large-scale magnetic fields to the cosmological vacuum energy. In our scenariothe dark energy is due to the Veneziano ghost (which solves the U (1) A problem in QCD).The Veneziano ghost couples through the triangle anomaly to the electromagnetic fieldwith a constant which is unambiguously fixed in the standard model. While this interac-tion does not produce any physical effects in Minkowski space, it triggers the generationof a magnetic field in an expanding universe at every epoch. The induced energy of themagnetic field is thus proportional to cosmological vacuum energy: ρ EM (cid:39) B (cid:39) ( α π ) ρ DE , ρ DE hence acting as a source for the magnetic energy ρ EM . The corresponding numericalestimate leads to a magnitude in the nG range. There are two unique and distinctivepredictions of our proposal: an uninterrupted active generation of Hubble size correlatedmagnetic fields throughout the evolution of the universe; the presence of parity violationon the enormous scales 1 /H , which apparently has been already observed in CMB. Thesepredictions are entirely rooted into the standard model of particle physics. Keywords: . a r X i v : . [ a s t r o - ph . C O ] A ug ontents
1. Introduction 12. Cosmological Magnetic Fields: Observations vs Theoretical Models 3 µ G fields 4
3. Coupling Dark Energy with Magnetism 4
4. Dark Energy and Large Scale Magnetic Field, or Two Sides of the SameCoin 10 B B
5. Results and conclusion 16
1. Introduction
The origin of cosmological magnetic fields, that is, magnetic fields which permeate thelargest structures found in the universe such as galaxies, clusters, and so on, still enjoysthe status of “yet to be solved” mistery. Such cosmologically correlated magnetic fields havehistorically been discovered at ever increasing distances (see the comprehensive reviews [1,2, 3, 4, 5, 6], mostly due to the fact that experimentalists have come up with new methodsand new devices capable of observing their effects in the ever farer and bigger structuresin universe.The models and mechanisms that aim at explaining the origin of the large-scale mag-netic fields can be roughly filed under two wide categories, astrophysical and cosmologi-cal ones. The astrophysical mechanisms mostly rely on what is typically termed battery ,see [2, 6] for a review. Such mechanisms generally provide protogalactic magnetic fieldsbut they are not likely to be correlated much beyond galactic sizes.On the other hand, cosmological models for the origin of cosmological magnetic fieldstypically involve some sort of (possibly first order) phase transition, such as the electroweak– 1 –ransition or the QCD confinement one, in order to generate Hubble scale seed fields whichthen are amplified through the mechanism known as dynamo [5, 6]. The typical correlationlength of such seeds is typically many orders of magnitude smaller than needed today, unlessa process called inverse cascade takes place (see for instance [7, 8, 9]). An inverse cascade,in brief, spreads highly energetic short wavelength modes to larger correlation lengths atsmaller amplitudes. This effect is expected to take place whenever a helical field evolves in aturbulent plasma, as the early universe mostly is, see again [2, 5, 6] for details. Dependingon the architectures used to model such phenomena, one expects to obtain correlationlengths that vary between a few pc to the very optimistic 100 kpc, but a very weak fieldintensity.In the case of an inflationary universe there is also the opportunity of a super-horizonmagnetogenesis due to the peculiar expansion character of an inflating universe [10], whichhowever demands a beyond the standard model (SM) coupling for electromagnetism forthe latter is Weyl invariant and is not amplified by gravitational interactions (unless oneresorts to the conformal anomaly, see [11]–but that again is inefficient as long as we stickto SM physics).In all cases, the dynamo is a necessary step in order to revitalise the weakened magneticfields. The dynamo must be very efficient for most of the models pushed forward, and, moreimportantly, should be operating at very large-scales (beyond galaxies). All these scenarioshowever incur in a series of difficulties when faced with observations; these problems becomeprofoundly more difficult as observations reach further in intensity and distance, as we arguebelow.The structure of our presentation is as follows. In the following section 2 we discusshow observations compare with most theoretical paradigms on the generation of large scalemagnetic fields. We elaborate on a number of problems which appear to be beyond theabilities of such models. Moving on to section 3 we offer an alternative approach to resolvethe mystery of large scale magnetic field based on the anomalous coupling between darkenergy and magnetism. In subsection 3.1 we review the dynamics of the Veneziano ghost(which solves a fundamental problem in QCD and plays the rˆole of source for the DEin our framework) in an expanding universe, and sketch how the observed vacuum energyarises, and how the auxiliary conditions on the physical Hilbert space (similar to the Gupta-Bleuler condition in QED) keep the theory unitary. In subsection 3.2 we derive the couplingbetween the Veneziano ghost and electromagnetism via the conventional triangle anomaly.We shall argue that this coupling, despite being of the same order of magnitude of g π γγ (which describes the π → γγ decay), still does not lead to any physical effects in Minkowskispace. By contrast, in an expanding universe when the Veneziano sources the appearanceof the cosmological vacuum energy, it does generate large scale magnetic fields which wecould be observing now. In section 4 we outline the arguments leading to our quantitativeestimates on such fields, with particular emphasis on the radical differences between theapproach pursued here and the far more common generation through primordial seeds.Lastly, the final section 5 wraps up the main ideas of the paper with a concise summarywith conclusions. – 2 – . Cosmological Magnetic Fields: Observations vs Theoretical Models Large-scale magnetic fields have been first discovered in our Milky Way, and subsequentlyin a number of other galaxies of different sizes and shapes, with characteristic intensity ofaround a few µ G. The correlation lengths of such fields range between a kpc and 30 kpc.However, this is not the end of the story, as magnetic fields of very similar strengths havebeen observed in clusters of galaxies, where they appear to be correlated over larger dis-tances reaching the Mpc region. It is important to notice that such fields are not associatedwith individual galaxies, as they are observed in the intergalactic medium as well [12, 13](see also the recent papers [14, 15]). This poses a serious challenge to astrophysically basedmechanisms, for then for some sort of conspiracy a series of unrelated galaxy-bound fieldswould all choose to align in one specific direction.As for cosmological magnetogenesis, already at this point one may notice that even themost optimistic and efficient inverse cascades are not going to be able to stretch primordialseeds to such wide lengths, as far as the seeds come from the electroweak phase transition,or even in the case of the QCD one. In the latter case the best result one can hopeto obtain is around 30 kpc (see for instance [16, 17]) using Son’s estimates for inversecascade parameters [9], which are actually already known to be over-optimistic, see [18].Moreover, recent analyses of the development of magnetic fields in cosmological turbulentplasma show that more realistic parameters for the inverse cascade lead to even smallercorrelations [19, 20].The most recent hints towards a possible magnetisation of gigantic supercluster struc-tures [21, 22], whose size can easily be two orders of magnitude beyond the clusters withinit, although not yet fully conclusive, would represent a further theoretical challenge inmodelling magnetogenesis, pushing the correlation lengths further away up to fractions ofGpc. It is clear that in this case all of the mechanisms mentioned above have very littlechance of being successful: such correlation lengths are simply beyond their capabilities.
In conjunction with the observations of supercluster magnetic fields there are new pieces ofevidence that show how µ G magnetic fields were present at much earlier epochs than pre-viously thought [23, 24]. Given this empirical picture, one immediately sees the problem inall mechanism which require a very efficient dynamo amplification be brought to successfulcompletion, for if strong magnetic fields are observed at redshifts as high as z ∼ µ G fields everywhere, for that implies roughlythe same effective number of turns for all of them.
Cosmological seeds suffer from another problem besides those just mentioned, as it hasrecently been emphasised in [19]. This can be schematised as follows. Primordial magnetic– 3 –elds trigger the appearance of gravitational waves which would be around during theformation of the light elements (Big Bang Nucleosynthesis - BBN). Such component istightly constrained by the beautiful concordance between theory and observations (a reviewcan be found in [25]), and puts problematic limits on the maximal intensity of primordialmagnetic fields. In particular, for non-helical seeds it has been show that neither inflation-born, nor electroweak-born, and not even QCD-born ones would be able to satisfy suchlimits and provide enough power for a dynamo [26, 27]. In the helical case these limitsare mitigated, but still on the verge of being excluded, for they demand the most efficientamplification [19].One further issue plaguing inflation-dawned magnetic gemmae is the fact that theback-reaction of the field on the inflaton itself is, in many practical realisations, of non-negligible importance [28]. Indeed, such back-reaction would make inflation unable toreproduce the observed spectrum of temperature fluctuations as observed in the cosmicmicrowave background (CMB), unless the seeds are smaller that what is required to feedgalactic dynamos [28].Finally, if the dynamo amplification of primordial seeds serves as the main field gener-ation mechanism, it is hard to imagine that low mass dwarf galaxies could produce againjust the same µ G fields, because of their relative lower rotation rates. µ G fields
Briefly, let us list a few scattered ideas supporting an ever existing and all-pervading fairlystrong magnetic field.Coles [29] and Kim et al. [30] have argued that sufficiently strong magnetic fields shouldbe around during most of the structure formation epoch of the universe, for they would helpsolving some of the discrepancies between ΛCDM simulations and the observed structures.The very simple observation that magnetic fields of intensity spanning at most oneorder of magnitude in the microgauss are found in completely different environments, fromgalaxies to intracluster medium, seems to be telling us that such fields are allegedly inde-pendent on the matter density they are found to be immersed in. Indeed, Kronberg [1],corroborating this suggestion with a number of other pieces of evidence, had proposed thatgalaxies and clusters have formed in a µ G environmental field, rather than the other wayaround.Let us close this overview by mentioning the few models which account for magneticfields in the formation of very peculiar structures such as filaments, which work better thantheir magnetic-less counterparts [31, 32, 33], and the correlation between star-burst ratesand the intensity of a pervading strong magnetic field within the galaxy, which holds to agood accuracy and may signal how stars are given birth in strongly magnetised surround-ings [34].
3. Coupling Dark Energy with Magnetism
On a completely different side, na¨ıvely unrelated to large scale magnetic fields, the lastdecade or so has seen the scientific community steadily realise and acknowledge the exis-– 4 –ence of the so-called dark energy component which appears to be accounting for over 70%of the total energy density of the universe [35, 36, 37]. In this unclustered energy densitythe entire cosmos is immersed, and its origin is very often thought of as the most intricateproblem of modern cosmology and particle physics.In this work, we propose that the two problems are intertwined at their hearts, andsuggest that the same mechanism which explains the fundamental origin of the cosmologi-cal vacuum energy should also be able to encompass the generation of cosmologically sizedmagnetic fields with intensity in the µ G range. More specifically, it has been recently beenshown [38, 39] how dark energy can be explained entirely within the SM, more precisely,within QCD, without the need for any new field or symmetry. In the proposal [38, 39], theinformation about the vacuum dark energy is carried by the so-called Veneziano ghost [40],whose properties in the expanding universe will be reviewed shortly. The basic idea is thatthe Veneziano ghost, which is a known, unphysical, degree of freedom in QCD, gives risenon a non-zero vacuum energy (“ghost condensation” similar to “gluon condensation” inQCD) if the universe is expanding. This effect of the Veneziano ghost is in many respectsakin to the known phenomenon of particle emission in a time-dependent gravitational back-ground, with the important difference that there is no emission of “real”, i.e., asymptoticstates here . Rather, the effect should be interpreted as the injection of extra energy (incomparison with Minkowski space) into ghost waves when the universe slowly expands.The average momentum resulting from this pumping is obviously zero as momentum isstill good quantum number in the expanding universe. Overall, this effect is clearly verytiny as it is proportional to H . What is also important is that the extra energy is storedin the form of ghost waves with momenta ω k (cid:39) k (cid:39) H , as only this much energy can belent by the expanding background, higher frequencies being exponentially suppressed [38].The arguments presented above imply that the typical wavelengths λ k associated with thisenergy density are of the order of the Hubble parameter, λ k ∼ /k ∼ /H ∼
10 Gyr, andthe corresponding excitations do not clump on scales smaller than this, in contrast withall other types of matter.The very same ghost field couples via the triangle anomaly to electromagnetism witha constant which is unambiguously fixed in the SM. While the interaction of the ghostwith electromagnetic field is sufficiently strong, it does not produce any physical effects inMinkowski space as a result of the auxiliary conditions on the physical Hilbert space (similarto the Gupta-Bleuler condition in QED) that are necessary to keep the theory unitary.However, the induced extra time-dependent energy due to the Veneziano ghost in theexpanding universe automatically leads to the generation of the physical electromagneticfield. What is important is that the typical momentum k EM of the generated EM field willbe of the same order of magnitude of a typical momentum of the the ghost k . Consequently,typical frequencies of the generated EM field will also be the same order, ω EMk (cid:39) k EM (cid:39) H . One should remark here that our preference in using the approach of Veneziano in describing the QCD-related vacuum energy is a matter of convenience. In principle, the same physics is also (hidden) in Witten’sapproach to the resolution of the U (1) problem [41]. However, the corresponding technique is much moreinvolved when the system is promoted to time dependent backgrounds, see subsection 3.1 below, and [38]for a detailed discussion. – 5 – .1 Dark energy and the Veneziano ghost It has been suggested recently that the solution of cosmological vacuum energy puzzle maynot require any new field beyond the SM [38, 39]. The idea is based on the philosophy thatgravitation can not be a truly fundamental interaction, but rather it must be consideredas a low-energy effective quantum field theory (QFT) [42]. The first application of thisproposal was the computation of the cosmological constant in a spacetime with non-trivialtopological structure when a Casimir type effect emerges. It was shown that the cosmo-logical constant does not vanish if our universe is enclosed in a large but finite manifoldwith typical size L (cid:39) /H , where H is the Hubble parameter. The cosmological vacuumenergy density ρ Λ in this framework is expressed in terms of QCD parameters for N f = 2light flavours as follows [38, 39]: ρ Λ (cid:39) N f | m q (cid:104) ¯ qq (cid:105)| m η (cid:48) L ∼ (4 . · − eV) . (3.1)This estimate should be compared with the observational value ρ Λ ≈ (2 . · − eV) [37].The deviation of the cosmological constant from zero is entirely due to the large but finitesize L of the manifold, and, as we have anticipated, should be understood as a Casimireffect in QCD . This proposal has a very simple and analytically trackable analogue in the2d Schwinger model [43], and could be tested in upcoming CMB experiments [44].The result (3.1) is a direct consequence of the existence of a very special degree offreedom in QCD, that is, the Veneziano ghost [40]. This field is unphysical in Minkowskispace (it belongs to the unphysical projection of the Hilbert space) in a way akin to the twounphysical polarisations of the electromagnetic four-potential in conventional QED. Theeffective Lagrangian for this field was known already in the ’80 [45, 46], but it has beenrecently reworked in a very convenient form for studying its curved spacetime propertiesas [38] L = 12 D µ ˆ φD µ ˆ φ + 12 D µ φ D µ φ − D µ φ D µ φ − m η (cid:48) ˆ φ + N f m q |(cid:104) ¯ qq (cid:105)| cos (cid:34) ˆ φ + φ − φ f η (cid:48) (cid:35) , (3.2)where the covariant derivative D µ is defined as D µ = ∂ µ + Γ µ so that, for instance D µ V ν = ∂ µ V ν + Γ νµλ V λ . The fields appearing in this Lagrangian areˆ φ = physical η (cid:48) , φ = ghost , φ = its partner . (3.3)It is important to realise that the ghost field φ is always paired up with φ in each andevery gauge invariant matrix element, as explained in [38]. The condition that enforcesthis statement is the Gupta-Bleuler-like condition on the physical Hilbert space H phys forconfined QCD, and reads like ( φ − φ ) (+) |H phys (cid:105) = 0 , (3.4)where the (+) stands for the positive frequency Fourier components of the quantised fields.In Minkowski space one can ignore the unphysical ghost field φ and its partner φ in– 6 –omputing all S matrix elements precisely in the same way as one always ignores the twounphysical photon polarisations when the Gupta-Bleuler condition in QED are imposed.However, the requirement (3.4) could not be globally satisfied in a general backgroundas explained in details in [38]. This is due to the fact that the Poincar´e group is no longera symmetry of a general curved spacetime (including the FLRW universe) and, therefore,it would be not possible to separate positive frequency modes from negative frequencyones in the entire spacetime, in contrast with what happens in Minkowski space where thevector ∂/∂t is a constant a Killing vector, orthogonal to the t = const hypersurface, andthe corresponding eigenmodes are eigenfunctions of this Killing vector. The Minkowskiseparation is maintained throughout the whole space as a consequence of Poincar´e invari-ance. Therefore, all physical effects related to the ghost dynamics are proportional to thedeviation from Minkowski spacetime geometry, i.e., to the rate of expansion H . This isthe very reason for the Veneziano ghost to exhibit non-trivial dynamics in an expandinguniverse; gravitational interaction however intervene and change this picture, allowing forthe appearance of a non-trivial energy density in the time-dependent background. We referthe interested reader to the original paper [38] for the details.One more comment concerns the appearance of the scale L (cid:39) /H in the energydensity (3.1). As it has been extensively explained in the previous letter [39], and also inthe longer paper [38], the potential felt by the ghost and its partner is a Casimir-like energywhich is a result of a subtraction procedure that compares the values of the vacuum energyin Minkowski space with that in a general compact manifold (such as a torus of size L ). Thisprescription aims at extracting the physical and measurable portion of the vacuum potentialenergy of the ghost field, by taking such difference between the vacuum energy in compactcurved space and that in infinite Minkowski space. Essentially, this is our definition of thevacuum energy when the “renormalised energy density” is proportional to the departurefrom Minkowski spacetime geometry and remains finite. The basic motivation for thisdefinition is the observation (3.4) that in Minkowski infinite space-time the energy due tothe ghost identically vanishes. Technically, it implies that the Lagrangian itself (3.2) doesnot have any small parameters such as 1 /L . However, the vacuum energy thereby definedexhibits a Casimir-like effect. Notice that the correction (which was computed exploitingthe topological susceptibility of QCD when the ghost is present) is linear in the inversesize of the manifold, not exponentially suppressed, as one would normally expect in theconfined phase of QCD where all physical degrees of freedom are massive.The ghost we are and will be working with here, and whose effects are central for ourdiscussion, was postulated by Veneziano in the context of the U (1) problem. However,the same problem had been tackled from a different perspective (although in the samelarge- N c context) by Witten in [41]. In his approach the ghost field does not ever enterthe system, and makes us wonder whether its consequences are physical or just artificious.Without going into the details (see [38] for a technical explanation), it will be enough hereto mention that the curved space effects we have directly computed with the help of, andattributable to, the ghost, are not going to disappear if we follow Witten’s approach. Therelevant physics will be hidden in the contact term which will depend in a highly non-trivialway on the properties of the spacetime (such as curvature) once the apt renormalisation– 7 –rocedure in the expanding background is performed . Our next step is to include the EM field into the low energy effective Lagrangian (3.2).First, in order to do so, we need to know the interaction of the η (cid:48) field with photons. Afterthat, we can recover the interaction of the ghost field φ and its companion φ with theelectromagnetic fields because φ and φ always accompany the physical η (cid:48) in a unique waysimilar to the interaction term (3.2).The interaction of the η (cid:48) field with photons is a textbook example crafted on the almostidentical well known anomalous term describing the π → γγ decay, see, e.g., [47]. Theonly difference is that η (cid:48) is an isotopical singlet state while π is an isotopical triplet. Theinteraction term is L η (cid:48) γγ = α π N c (cid:88) Q i η (cid:48) f η (cid:48) F µν ˜ F µν . (3.5)where α is the fine-structure constant, f η (cid:48) is the decay constant for the η (cid:48) , N c is thenumber of colours, and the Q i s are the light quarks electric charges. Finally, F µν is theusual electromagnetic field strength (in curved space), and ˜ F µν = (cid:15) µνρσ F µν / (cid:15) µνρσ = (cid:15) µνρσM / √− g with the Minkowski antisymmetric tensor following from (cid:15) M = +1, and g = det g µν the determinant of the metric tensor.One should remark here that such kind of anomalous interaction has been studiedin great details in particle physics as well as in cosmology. In particular, the axion (orany other pseudoscalar particle) has exactly the same structure and has been thoroughlyanalysed in cosmological contexts, see e.g., [48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58].We are not really interested in η (cid:48) physics as the heavy η (cid:48) meson of course is notexcited in our universe. We are interested in the interaction of the ghost field φ andits companion φ with the electromagnetic fields. The corresponding terms have neverbeen discussed in the literature because they do not appear in any gauge invariant matrixelement in Minkowski space as a consequence of the auxiliary condition (3.4). In fact,these unphysical degrees of freedom can be completely integrated out in that case, suchthat they even disappear at the Lagrangian level (this is exactly the procedure that wasadopted in the original paper [40], see also [38] in the context of the present work). Incurved space these fields can not be swept under the carpet as they carry relevant physicalconsequences: we must explicitly deal with them.In order to derive the interaction term between the fields φ , φ and A µ one shouldrepeat the steps described in [38]. We can explicitly check that the physical η (cid:48) alwaysenters the Lagrangian in the combination ( η (cid:48) + φ − φ ) /f η (cid:48) . Hence, the interaction termwe are interested in has the structure L ( φ − φ ) γγ = α π N c (cid:88) Q i (cid:18) η (cid:48) + φ − φ f η (cid:48) (cid:19) F µν ˜ F µν . (3.6) The required procedure, for the non-abelian and strongly interacting theory under consideration is notknown yet. – 8 –or our future discussions we safely neglect the massive physical η (cid:48) field, and keep only theghost field φ and its companion φ along with the EM field, L = − F µν F µν + 12 D µ φ D µ φ − D µ φ D µ φ (3.7) − α π N c (cid:88) Q i (cid:18) φ − φ f η (cid:48) (cid:19) (cid:126)E · (cid:126)B + N f m q |(cid:104) ¯ qq (cid:105)| cos (cid:18) φ − φ f η (cid:48) (cid:19) , where the electric and magnetic fields are the usual Minkowski ones (not rescaled by thescale factor of the universe a ( t )). We claim that the expression (3.7) is the exact lowenergy Lagrangian describing the interaction of the ghost field φ and its companion φ with electromagnetism in the gravitational background defined by the Γ µ . In Minkowskispace the expectation value (cid:104)H phys | ( φ − φ ) |H phys (cid:105) = 0 , (3.8)vanishes, implying that φ and φ are decoupled from QED, as they should in order topreserve the unitarity of the system.However, as we have pointed out before [38], the constraint (3.4) can not be globallymaintained in the entire space in a general curved background. Thus, the ghost field andits partner do couple to electromagnetism, and consequently we do expect some physicaleffects to occur as a result of this interaction. Notice that we are not claiming that the ghostfield becomes a propagating degree of freedom, or becomes an asymptotic state. Rather,we propose the description in terms of the ghost as it is a convenient way to account forthe physics hidden in the non-trivial boundary conditions, see [38] once again. In thecontext of this discussion, a very illuminating example is that of 2d Rindler spacetimeand the associated Unruh effect, see [59], where all different approaches are workable andcomparable to give the same result. One should emphasise that the Veneziano ghost isvery different from all other types of ghosts, including the conventional Fadeev-Popovfields. The peculiar feature of the Veneziano ghost resides in its close connection withthe topological properties of the theory, and expresses the necessity to sum over differenttopological sectors in QCD [59]. This uniqueness manifests itself, in particular, in thespectrum of the Veneziano ghost: while conventional ghosts may have arbitrary largefrequencies, and essentially, are introduced only to cancel unphysical polarisations of gaugefields for arbitrary large ω , typical frequencies of the Veneziano ghosts are order of thehorizon scale, ω ∼ H .The most immediate consequence of the interaction term (3.6) is that the magnetic fieldwhich will be generated this coupling will have a typical Fourier mode k EM (cid:39) k of the sameorder of magnitude of the ghost mode. On the other hand, gravity can only lend energiesof order ω k (cid:39) k (cid:39) H , higher frequencies being exponentially suppressed. Consequently, atypical EM mode will be around ω EMk (cid:39) k EM (cid:39) H . Moreover, this interaction is active atevery epoch, and we shall see in the forthcoming section how (3.7) naturally explains the µ G intensity apparently observed in galaxies and clusters, and maybe needed from the verybeginning of structure formation. To conclude, let us notice once again that everything wehave been discussing so far is part of the SM, and the coupling constants are all known.– 9 –owever, all our conclusions will apply to most (pseudo)scalar models of dark energy, aslong as they are augmented with a coupling with the same structure as (3.6); in this sensethe results we will discuss in the upcoming section are general, and widely applicable todark energy theories.
4. Dark Energy and Large Scale Magnetic Field, or Two Sides of the SameCoin
The Lagrangian density (3.7) explicitly contains a coupling between the ghost and the P -odd operator (cid:126)E · (cid:126)B made of electromagnetic fields. The structure of this term is identicalto the textbook example describing the anomalous π → γγ decay. Let us explore theconsequences of this interaction in our specific circumstances. While the structure of theghost-photon interaction is a photocopy of that of the pion-photon one, there is a funda-mental difference between the two: π is a massive physical particle which can decay totwo photons, whereas the ghost is massless and unphysical, and can not decay into twophotons. The ghost field in an expanding universe should be treated as the large correlatedclassical field which emerges from a non-zero expectation value (cid:104)H phys | ( φ − φ ) |H phys (cid:105) (cid:54) = 0as explained in [38]. The Fourier expansion of these classical fields φ and φ is saturatedby very low frequencies ω k (cid:39) k (cid:39) H , while higher frequency modes ω k (cid:29) H are stronglysuppressed as a result of the relative suppression of the so-called Bogolubov coefficients [38]. B For future convenience we introduce the dimensionless coupling constant which appears inour basic expression (3.7) β ≡ α π N c (cid:88) Q i . (4.1)In nature β (cid:28)
1, but we still want to study the rate of energy transfer from darkenergy (which is represented by the ghost field) to the electromagnetic energy as a functionof β . Hence, we treat β as a free parameter in this section. If the fields are interactingsufficiently strongly ( β (cid:29)
1) and the potential minimum is reached sufficiently rapidly,one can estimate the expectation value (cid:104) (cid:126)E · (cid:126)B (cid:105) in terms of the external background field (cid:104) φ − φ (cid:105) , which is treated as a source. By following this procedure we arrive to (cid:104) (cid:126)E · (cid:126)B (cid:105) = 1 β N f m q |(cid:104) ¯ qq (cid:105)|(cid:104) φ − φ f η (cid:48) (cid:105) (cid:39) β Λ H , β (cid:29) . (4.2)As expected, in Minkowski space there will be no generation of EM field as (cid:104) φ − φ (cid:105) = 0according to (3.4), (3.8). In an expanding universe (cid:104) φ − φ (cid:105) is proportional to the deviationfrom Minkowski space, and is expected to be around H . In this case the magnetic field isproduced as an outcome of the energy flow from the ghost to the EM field. At the timeat which each field mode is born, our equations are symmetric under the permutation of (cid:126)E and (cid:126)B . Therefore, one could estimate the absolute value of | (cid:126)B | as , (cid:104) (cid:126)E · (cid:126)B (cid:105) (cid:39) (cid:104) (cid:126)B (cid:105) . The exact statement is [ (cid:104) (cid:126)E (cid:105) + (cid:104) (cid:126)B (cid:105) ] ≥ (cid:104) (cid:126)E · (cid:126)B (cid:105) but the electric field will be screened soon after it isgenerated, and it can then be neglected in the evolution. – 10 –f course, the evolution of the electric field and magnetic fields are drastically differentas electric charges do exist in nature while magnetic ones do not. However, we expectthat (cid:104) (cid:126)E · (cid:126)B (cid:105) (cid:39) (cid:104) (cid:126)B (cid:105) is a reasonable approximation for the absolute value of | (cid:126)B | at itsbirth. Consequently, a simple estimate of the intensity of the magnetic field based on theassumption that the system (ghost + EM) can reach its minimum energy configurationsufficiently quickly (fast equilibration) can be presented as (cid:104) (cid:126)B (cid:105) (cid:39) (cid:104) (cid:126)E · (cid:126)B (cid:105) (cid:39) Λ Hβ (cid:39) ρ DE β , β (cid:29) , (4.3)where ρ DE (cid:39) Λ H in our framework. For large coupling constant β this estimate istotally justified. Indeed, eq. (4.2) corresponds to the minimum of the potential energyof the system , and it becomes a precise statement for β (cid:29) /H .However, in nature β (cid:28)
1. Let us see what is happening with our formula (4.3) whenwe start to decrease β . This equation tells us that we generate magnetic energy which(parametrically!) starts to exceed the energy of the source when β ∼
1. Clearly this cannot be true, and the loophole in the above line of arguments lies in the assumption ofequilibrium. The minimisation procedure described above works only when the reactionsinvolved in transferring energy from one source (the ghost) to a recipient (the electromag-netic field) are efficient enough to be in equilibrium. The lowest energy of the systemsimply can not be achieved when the coupling between the two components is weak.Thus, in a slowly expanding universe one needs to have a handle on the rate of energytransfer at each epoch, and compare it with the Hubble time, just as it is typically donein studying the thermal history of particle species in the early universe. The differencehere is that this rate depends itself on the rate of expansion H due to the fact that theeffective coupling constant is proportional to (cid:104) φ − φ (cid:105) (cid:39) H as formula (4.2) states. Aswe will see shortly, in one Hubble time only a very small fraction of the ghost’s potentialenergy is transferred to the magnetic field at β (cid:28)
1, thereby mining the foundations of theminimisation procedure. Yet, there is an important lesson to be learnt from this discussion:the dynamics described by the Lagrangian (3.7) does lead to energy transfer from theghost field (which is the DE in our model) into the magnetic field, though numericallyformula (4.3) can not be trusted for the physically relevant case β (cid:28) As we mentioned above, in order to understand the dynamical effects of the couplingbetween the ghost and electromagnetism one should look for the rate at which energy is In particular, a similar procedure of minimisation of the effective potential allows to compute the exactvacuum expectation value for the gluon field (cid:104) α s π G aµν ˜ G µνa (cid:105) = θN f m q (cid:104) ¯ qq (cid:105) which is analogous to (4.2) whenthe colour gluon fields G µνa replace the EM fields, and the so-called θ parameter of QCD replaces theexpectation value of the ghost field (cid:104) φ − φ (cid:105) . The equilibrium for (cid:104) α s π G aµν ˜ G µνa (cid:105) is obviously achieved veryquickly and the relation (cid:104) α s π G aµν ˜ G µνa (cid:105) = θN f m q (cid:104) ¯ qq (cid:105) becomes exact at the minimum of the potential. Infact, one can differentiate this relation with respect to θ one more time to arrive to well-known exact WardIdentity for the topological susceptibility, i (cid:82) dx (cid:104) α s π G aµν ˜ G µνa ( x ) , α s π G aµν ˜ G µνa (0) (cid:105) = m q N f (cid:104) ¯ qq (cid:105) , see, e.g., [60]and references therein. – 11 –ransferred while the universe expands. In order to do so let us write down the Hamiltonianfor the system (3.7) asH = 12 (cid:0) B + E (cid:1) + 12 D µ φ D µ φ − D µ φ D µ φ (4.4)+ α π N c (cid:88) Q i (cid:18) φ − φ f η (cid:48) (cid:19) (cid:126)E · (cid:126)B − N f m q |(cid:104) ¯ qq (cid:105)| cos (cid:18) φ − φ f η (cid:48) (cid:19) . As we discussed earlier in the text, neither the Lagrangian (3.7), nor the Hamiltonian (4.4)contains any small coupling constant in their definitions as all the parameters describing thesystem are known SM parameters. All small effects are proportional to H/ Λ QCD ∼ − and are brought about only at the level of the “renormalisation procedure”, i.e., whenthe subtraction is explicitly performed. If we are computing the energy related to themagnetic field in Minkowski space, this identically vanishes (4.2) as a consequence of theconditions (3.4).The time derivative dH / d t governs the efficiency of the energy flow between ghost andelectromagnetic fields. Explicitly, dH / d t = 0 implies thatdd t (cid:20) (cid:0) B + E (cid:1)(cid:21) (cid:39) − β (cid:32) ˙ φ − ˙ φ f η (cid:48) (cid:33) (cid:126)E · (cid:126)B − β (cid:18) φ − φ f η (cid:48) (cid:19) dd t (cid:104) (cid:126)E · (cid:126)B (cid:105) − dd t H ghost , (4.5)where dH ghost / d t essentially describes the dynamics of the dark energy component, andcan be neglected in this simple evaluation.We want to gain some intuition about the rate of the energy transfer by considering thecase of small β when we know that the minimisation procedure is not justified. Assumingfor simplicity that the Hubble parameter is a constant and the rate of energy transfer isalso a constant, one can estimate the typical time (relaxation period) τ which is requiredfor the system to reach its equilibrium value. Indeed, the left hand side of eq. (4.5) can beapproximated as l.h.s. of eq. (4.5) = dd t (cid:20) (cid:104) B + E (cid:105) (cid:21) (cid:39) ρ EM τ . (4.6)At the same time, the right hand side of (4.5) isr.h.s. of eq. (4.5) (cid:39) − β (cid:104) ˙ φ − ˙ φ f η (cid:48) (cid:105) ρ EM − (cid:104) φ − φ f η (cid:48) (cid:105) ρ EM τ , (4.7)where we have substituted (cid:104) (cid:126)E · (cid:126)B (cid:105) (cid:39) ρ EM and its time derivative with a 1 /τ . The laststep we need to do is simply to plug in the values (cid:104) ˙ φ − ˙ φ (cid:105) /f η (cid:48) (cid:39) (cid:104) φ − φ (cid:105) (cid:39) H (non-zerosolely due to the expanding gravitational background). A more precise numerical estimatefor (cid:104) ˙ φ − ˙ φ (cid:105) will be given below. Comparing (4.6) with (4.7) one can immediately inferthat τ ∼ βH , β (cid:28) , (4.8)which is the main result of this subsection.– 12 – .3 Slow equilibration: more realistic values for B Now we want to see what happens with our estimate (4.3) when β decreases, as it corre-sponds to the physical value (4.8). From what we have said above, it is clear that the timescale which is required to attain equilibrium will be order of the Hubble time τ (cid:39) /H for β ∼ β decreases even further to reach the physically interesting region β (cid:28) β − (cid:29) /H , which really makes no sense. Theappropriate interpretation in this case is that equilibrium will be never achieved with suchsmall coupling constant during one Hubble time. Instead, a very small portion β of theavailable energy can be at most injected into the magnetic field within the same Hubbletime.This is however not the end of the story for small β (cid:28)
1. The point is that, for β ∼
1, each event leading to equilibration transfers an amount of energy of order one.For small β (cid:28) β in comparison withcase β ∼
1, when formula (4.3) can be still marginally trusted.With this interpretation in mind we arrive at our final estimate for the magnetic energythat has flowed from the ghost field ρ EM (cid:39) β (cid:104) ˙ φ − ˙ φ Hf η (cid:48) (cid:105) ρ DE (cid:39) (cid:16) α π N c (cid:88) Q i (cid:17) (cid:104) ˙ φ − ˙ φ Hf η (cid:48) (cid:105) ρ DE , (4.9)where we inserted the extra small parameter β accounting for the two suppression factorsmentioned above, accounting for the fact that only a small fraction of the total availableenergy will effectively exchanged during one Hubble time.One should remark here that the β suppression which enters in (4.9) can be intuitivelyunderstood by considering a system of particles with magnetic moment. In this case, asis known, the interaction between the magnetic moment and an external magnetic fieldis proportional to the coupling constant e . However, the contribution to the energy dueto the induced magnetic moment is proportional to e . In different words, the magneticsusceptibility of the system goes as e rather than linearly with e , in spite of the fact thatthe strength of the interaction itself is ∝ e . Our analysis of the induced EM field follows thefootprints of the analogous induced magnetic argument, where now the rˆole of the couplingconstant e is played by the parameter β , and where the source of the energy is ρ DE ratherthan the external magnetic field. Finally, in our expressions above we assumed that theHubble constant does not depend on time, so that the transfer rate is time-independentitself.A few comments are in order before we proceed with the numerical evaluations. Firstof all, eq. (4.9) has the following parametrical dependence on fine structure constant α and– 13 –ubble constant H : ρ EM (cid:39) (cid:104) (cid:126)B (cid:105) (cid:39) (cid:16) α π (cid:17) H Λ , where ρ DE (cid:39) H Λ . (4.10)In this form it is easy to interpret the appearance of each parameter. Dark energy is aQCD effect related to the mismatch between vacuum energies in infinite Minkowski andcompact or curved spaces, and it is conceivably proportional to the rate of the expansionof the universe. As is known, the EM field does not interact directly with gravity, however,it does interact (through a standard triangle anomaly) with a (pseudo) scalar ghost fieldwhich gives rise to a correlated external source for the EM field in an expanding universe.This explains the extra parameter ( α/ π ) in (4.10). There are other numerical factors oforder one which would also appear in (4.10), which for now are glossed over to simplifythe interpretation. However, we believe that our formula (4.10) has a well understood andphysically motivated behavior in terms of the relevant physical parameters α , H , Λ QCD .We are now in the position to make a more quantitative estimate for the magneticfield as the combination which enters expression (4.9) and which describes the dynamicsof the ghost field, that is, c ∼ (cid:104) ˙ φ − ˙ φ (cid:105) / ( Hf η (cid:48) ), had been previously discussed in [38]and it essentially corresponds to the initial field velocity in the classical potential. Anuncertainty in value of this constant, c , is not related to any physics beyond the SM, butis simply determined by the initial conditions when the dynamics of φ and φ are treatedclassically . For our order of magnitude estimate the specific value of this parameter isunimportant, and, following [38], we will fix it at the value of c ∼ − .With these remarks in mind we can finally work out some numerology. The typicalvalue for the energy density of the magnetic field which is generated by the Venezianoghost, which in turn is responsible for the cosmological dark energy, is B (cid:39) (cid:16) α π N c (cid:88) Q i (cid:17) · c · ρ DE , c ∼ − . (4.11)From this expression it straightforwardly follows that during the last Hubble time of lifeof the universe, an O ( H − ) correlated magnetic field is born with intensity B (cid:39) α π √ c · (2 . · − eV) ∼ nG ,
1G = 1 . · − (eV) . (4.12)As we mentioned previously, formula (4.12) should be treated as an order of magnitudeestimate due to a number of numerical factors which have been consistently neglected whilearriving at our final analytical result (4.9) and its numerical expression (4.12). We should The classical treatment of the system is by far the widest chosen approach in dealing with dark energymatters. However, while in other, perhaps more familiar, cosmological models such as inflation, the passagefrom quantum to classical is justified a posteriori (see for instance the discussion in [10], section 7.4.7,and reference to original works therein), in coping with our quantum fields we do not expect such a “littlemiracle” to happen. In other words, the quantum nature of our fields which appears in their non-trivialdependence on the gravitational background as well as on the global properties of the manifold, is brought inat the level of the renormalisation procedure, and is therefore not describable in a purely classical approach.Nevertheless, as long as we are interested in order of magnitude estimates, it will suffice to confine ourselveswithin the boundaries of such classical framework. – 14 –mphasise that the most important outcome here is not the precise numerical value foundin (4.12), which is likely to change due to the very complicated evolution of the field in itsmost recent history ( z ∼
1) well after its formation. Rather, the main result of this paperis the observation that due to the coupling with DE the EM field will be correlated onenormous scales of order 1 /H . This distinguishes our mechanism from everything whichhas been suggested previously. Another important qualitative consequence is the predictionthat parity will be locally violated on the same scales for which the field does not change,that is, 1 /H as a result of the pseudoscalar nature of the DE field. This distinct andunambigious prediction can be tested in the CMB sky, where apparently parity violationon such scales indeed has been observed, see [61, 62] (we will comment further on thisaspect below). Finally, our field is highly helical with typical wavelengths λ k ∼ /H ;therefore, the induced (cid:126)B is expected to flip sign on the same scale. We notice in passingthat a nG intensity would account for the observed galactic field if it were frozen in thepre-galactic plasma. In spite of its attractiveness however, this possibility should be takenwith a grain of salt; furthermore, the analysis of the evolution of such fields is beyond thescope of the present work.This mechanism operates all the time, and there will be an uninterrupted flow ofmagnetic fields produced at different correlation lengths (proportional to the Hubble pa-rameter). Such fields however, being proportional to the vacuum energy component, willbe of significantly relative weaker amplitude compared to the other components of theuniverse, following the twin evolution of the dark energy. Nevertheless, they could stillbehave as seeds for plasma mechanisms to process them, the outcome of which is beyondthe scope of this paper and shall not be pursued any further. As we have already mentioned, interactions of the form (3.6) have been detailedly exploitedand analysed in the literature, initially in the context of the anomalous axion-photoninteraction, and then extended to general pseudoscalar interactions. What makes this workdepart from all the paper referenced in the bibliography is that we are dealing entirely withSM physics, including the interaction (3.6). We should remark here that this interactiondoes not violate P and CP invariance on the fundamental level, similarly to the π → γ decay. However, on small scales, one can interpret the interaction (3.6) as a P and CP violating coupling similar to the θ term in QCD. Such a violation occurs only locally, onthe correlation scales λ k ∼ /H , while globally one should expect P and CP conservationaccording to the symmetry of the fundamental interactions (3.7).In this paper we have worked out the first application of such interaction in our model,that is, the generation of cosmological magnetic fields. It is easy to see, however, thatthere are several possible effects in addition to what has just been computed, especiallyin connection with possible observables which would be able to confirm or falsify ourframework. We will limit ourselves to a mention for some of these possibilities, the specificdetails of which are beyond the scope of the present work and will be addressed in futurepublications. – 15 – Early universe large scale magnetic fields have a sound impact on the mechanisms ofstructure formation (especially an active source of nanogauss intensity); there existsa vast literature on the subject, although this point is still somewhat overlooked:we refer to the comprehensive reviews [1] and [2] and to section 2.4 where someinteresting consequences of primordial magnetism in the formation of early structuresare investigated. • The presence of a pre-decoupling magnetic field is able to leave significant imprintsin the cosmic microwave background radiation (CMB). The literature on this specifictopic is particularly copious, and we shall mention only a few specific papers. Theeffects of magnetic helicity (particularly relevant for us since in our case the magneticfield is highly helical) have been discussed in [63, 64]. Primordial magnetism is alsoable to de-polarise the CMB (which acquired a small degree of linear polarisationdue to the finite thickness of the last scattering surface), see [65, 66, 67, 68, 69]. Fi-nally, an impressive and complete analysis of the distortions caused by pre-decouplingmagnetic fields (stochastically distributed) to the CMB acoustic peaks has been re-cently undertaken in the series of papers [70, 71, 72, 73, 74]. The novel aspect of ourproposal compared to the models adopted in all the aforementioned efforts is thatour source is active at all times, has a highly non-trivial redshift dependence, and iscapable of producing otherwise forbidden by parity cross-correlation spectra such asthose known as EB and T B on top of the distortions predicted for the usual
T T and EE (and BB ) autocorrelations and the parity even T E cross correlation. Anomalousparity violation in the low multipoles of the CMB has been discussed in [61, 62]. • In addition to the pervading magnetic fields generated through the Veneziano ghost,one could also look for other signatures of the parity-violating coupling (3.6); inthe early expansion history this interaction is able to trigger a sizeable amount ofbirefringence of the CMB [75, 76, 77] (the peculiar effects of the parity violation havebeen considered in [78, 79, 80, 81]); at late times such coupling can be sought afterin its impact to the travelling light signals from the CMB, or any emitting sourcein the universe, to us, the observers, see again [49, 52, 54] and [87, 88, 89]. Thelatter is of prominent relevance due to the recent claims of a preferred orientation ofthe polarisation angles of quasars in optical wavelengths [82, 83, 84] which seems todisappear at radio frequencies [85] (see also [86] for an analysis of the possible biasesaffecting these results): a possible explanation is again to be found in the combinedeffect of a background magnetic field and an anomalous coupling of (3.6) fame whichmixes light and the pseudoscalar field [87].
5. Results and conclusion
In this work we have proposed that the origin of the observed cosmological magnetic fields,correlated over scales that stretch from galaxies to superclusters, is tied to the existenceand evolution of the vacuum energy component of the universe. P and CP violation may occur locally, not globally, as explained above. – 16 –ere we have shown how the very same field which provides the dark energy in ourframework [38], together with its Gupta-Bleuler partner, possesses an anomalous coupling(the ordinary triangle anomaly) to the magnetic field and can trigger the generation ofmagnetic fields of the observed intensity with correlation lengths of order of the Hubbleparameter today. This interaction is present at all times starting from the QCD phasetransition, and is continuously producing magnetic fields, whose strength is consequentlylinked to that of the vacuum energy, times a dimensionless coefficient of about ( α/ π ) due to the weakness of the electromagnetic interactions. This combination, α ρ DE / π ,where in our set up ρ DE (cid:39) H Λ , is precisely the magnetic field strength (squared) oneneeds to account for the observational evidence.The list of intricacies most large-scale magnetic field models have to face does not clashwith the scheme we have outlined. Correlation lengths up to the size of the universe ateach epoch are a direct consequence of the properties of the Veneziano ghost. Magneticfields will be generated at all times, even at higher redshifts, though their magnitudes willexperience a standard 1 /a ( t ) suppression due to the expansion of the universe. There isno need for seeds to feed a dynamo with, as the largest scales are generated last, and withalready the correct strength, without the risk of overwhelming BBN with gravitationalwaves.Distinctive predictions and unique features of our proposal are the generation of themagnetic field with magnitude in the nG range at all scales up to the Hubble radius today,and its grounds to be found entirely within familiar SM physics, without the need for anynew fields, unconventional couplings to gravity or any modification of gravity itself. Thismechanism is also unique in predicting parity violation on the largest scales 1 /H , whichapparently is already supported by CMB observations [61, 62]. If these predictions wereto be confirmed by future PLANCK observations, it would be a strong (but still implicit)hint towards the QCD nature of dark energy, together with its fundamental kinship withlarge scale magnetic fields, in the way detailed in the present work.Our final comment concerns the scales involved in the model. Our proposal leads tothe order of magnitude estimate B (cid:39) α π (cid:113) H Λ ∼ nG, which is approximately the valuethat, by simple adiabatic compression, could explain the field observed at all scales, fromgalaxies to superclusters. It is already the second time that this “accidental coincidence”happens, the first one being the dark energy density itself, written as ρ DE (cid:39) H Λ ∼ (10 − eV) . We consider this “coincidence” as encouraging support for the entire framework.The SM coupling of the EM field with the DE field leads to the observed µ G galacticmagnetic field (assuming it were frozen in the pre-galactic plasma). It is difficult to inventanother scheme when these relations hold, see, e.g., [90]. This “coincidental” relationshould be added to our “Fine-tuning without fine-tuning” section from [38] where we clarifyhow the intimidating list of fine-tuning issues which always plagues dark energy models,possesses simple explanations without the need to introduce new fields, which come withnew interactions, new coupling constants, and new symmetries. Instead, all our resultsare based on the paradigm according to which the “physical dark energy” is related tothe deviation of the vacuum energy from infinite Minkowski space similarly to the Casimir– 17 –ffect, while all SM fields couple to the dark energy field (the Veneziano ghost) in the wellunderstood way dictated univocally by the standard model of particle physics.
Acknowledgements
AZ thanks P Naselsky for discussions on the possibilities to test some of the ideas formu-lated in this paper (such as Hubble scale magnetic fields and parity violation on the samescales) in the data being gathered and analysed by the PLANCK collaboration. FU wouldlike to thank S Habib for valuable conversations. This research was supported in part bythe Natural Sciences and Engineering Research Council of Canada.
References [1] P. P. Kronberg, Rept. Prog. Phys. , 325 (1994).[2] D. Grasso and H. R. Rubinstein, Phys. Rept. , 163 (2001) [arXiv:astro-ph/0009061].[3] A. D. Dolgov, arXiv:astro-ph/0306443.[4] M. Giovannini, Int. J. Mod. Phys. D , 391 (2004) [arXiv:astro-ph/0312614].[5] A. Brandenburg and K. Subramanian, Phys. Rept. , 1 (2005) [arXiv:astro-ph/0405052].[6] M. Giovannini, Lect. Notes Phys. , 863 (2008) [arXiv:astro-ph/0612378].[7] A. Brandenburg, K. Enqvist and P. Olesen, Phys. Rev. D , 1291 (1996)[arXiv:astro-ph/9602031].[8] A. Brandenburg, K. Enqvist and P. Olesen, Phys. Lett. B , 395 (1997)[arXiv:hep-ph/9608422].[9] D. T. Son, Phys. Rev. D , 063008 (1999) [arXiv:hep-ph/9803412].[10] A. R. Liddle and D. H. Lyth, “The Primordial Density Perturbation: Cosmology, Inflationand the Origin of Structure,” Cambridge University Press; Revised edition , Cambridge, UK(2009)[11] A. Dolgov, Phys. Rev. D , 2499 (1993) [arXiv:hep-ph/9301280].[12] B. M. Gaensler, R. Beck and L. Feretti, New Astron. Rev. , 1003 (2004)[arXiv:astro-ph/0409100].[13] F. Govoni and L. Feretti, Int. J. Mod. Phys. D , 1549 (2004) [arXiv:astro-ph/0410182].[14] F. Tavecchio, G. Ghisellini, L. Foschini, G. Bonnoli, G. Ghirlanda and P. Coppi,arXiv:1004.1329 [astro-ph.CO].[15] A. Neronov and I. Vovk, Science , 73 (2010) [arXiv:1006.3504 [astro-ph.HE]].[16] M. M. Forbes and A. R. Zhitnitsky, Phys. Rev. Lett. , 5268 (2000) [arXiv:hep-ph/0004051].[17] M. M. Forbes and A. R. Zhitnitsky, arXiv:hep-ph/0102158.[18] R. Banerjee and K. Jedamzik, Phys. Rev. D , 123003 (2004) [arXiv:astro-ph/0410032].[19] C. Caprini, R. Durrer and E. Fenu, arXiv:0906.4976 [astro-ph.CO].[20] C. Caprini, R. Durrer and G. Servant, arXiv:0909.0622 [astro-ph.CO]. – 18 –
21] P. P. Kronberg, Astron. Nachr. , 517 (2006).[22] Y. Xu, P. P. Kronberg, S. Habib and Q. W. Dufton, Astrophys. J. , 19 (2006)[arXiv:astro-ph/0509826].[23] P. P. Kronberg, M. L. Bernet, F. Miniati, S. J. Lilly, M. B. Short and D. M. Higdon,Astrophys. J. , 7079 (2008) [arXiv:0712.0435 [astro-ph]].[24] M. L. Bernet, F. Miniati, S. J. Lilly, P. P. Kronberg and M. Dessauges-Zavadsky,arXiv:0807.3347 [astro-ph].[25] G. Steigman, Ann. Rev. Nucl. Part. Sci. , 463 (2007) [arXiv:0712.1100 [astro-ph]].[26] C. Caprini and R. Durrer, Phys. Rev. D , 023517 (2001) [arXiv:astro-ph/0106244].[27] C. Caprini and R. Durrer, Phys. Rev. D , 063521 (2006) [arXiv:astro-ph/0603476].[28] V. Demozzi, V. Mukhanov and H. Rubinstein, arXiv:0907.1030 [astro-ph.CO].[29] P. Coles, Comments Astroph. , 45 (1992)[30] E. j. Kim, A. Olinto and R. Rosner, Astrophys. J. , 28 (1996) [arXiv:astro-ph/9412070].[31] E. Battaner, E. Florido and J. Jimenez-Vicente, Astron. Astrophys. , 13 (1997)[arXiv:astro-ph/9602097].[32] E. Battaner, E. Florido and J. M. Garcia-Ruiz, Astron. Astrophys. , 8 (1997)[arXiv:astro-ph/9710074].[33] E. Florido and E. Battaner, Astron. Astrophys. , 1 (1997)[34] T. Totani, Astrophys. J. , L69 (1999)[35] A. G. Riess et al. [Supernova Search Team Collaboration], Astron. J. , 1009 (1998)[arXiv:astro-ph/9805201].[36] S. Perlmutter et al. [Supernova Cosmology Project Collaboration], Astrophys. J. , 565(1999) [arXiv:astro-ph/9812133].[37] E. Komatsu et al. [WMAP Collaboration], Astrophys. J. Suppl. , 330 (2009)[arXiv:0803.0547 [astro-ph]].[38] F. R. Urban and A. R. Zhitnitsky, Nucl. Phys. B , 135 (2010) [arXiv:0909.2684[astro-ph.CO]].[39] F. R. Urban and A. R. Zhitnitsky, Phys. Lett. B , 9 (2010) [arXiv:0906.2162 [gr-qc]].[40] G. Veneziano, Nucl. Phys. B , 213 (1979).[41] E. Witten, Nucl. Phys. B , 269 (1979).[42] E. C. Thomas, F. R. Urban and A. R. Zhitnitsky, JHEP , 043 (2009) [arXiv:0904.3779[gr-qc]].[43] F. R. Urban and A. R. Zhitnitsky, Phys. Rev. D , 063001 (2009) [arXiv:0906.2165 [hep-th]].[44] F. R. Urban and A. R. Zhitnitsky, JCAP , 018 (2009) [arXiv:0906.3546 [astro-ph.CO]].[45] P. Di Vecchia and G. Veneziano, Nucl. Phys. B , 253 (1980).[46] C. Rosenzweig, J. Schechter and C. G. Trahern, Phys. Rev. D , 3388 (1980). – 19 –
47] K. Huang, “Quarks, Leptons And Gauge Fields,”
World Scientific , Singapore, Singapore(1982)[48] P. Sikivie, Phys. Rev. Lett. , 1415 (1983) [Erratum-ibid. , 695 (1984)].[49] G. Raffelt and L. Stodolsky, Phys. Rev. D , 1237 (1988).[50] M. S. Turner and L. M. Widrow, Phys. Rev. D , 2743 (1988).[51] S. M. Carroll and G. B. Field, Phys. Rev. D , 3789 (1991).[52] D. Harari and P. Sikivie, Phys. Lett. B , 67 (1992).[53] W. D. Garretson, G. B. Field and S. M. Carroll, Phys. Rev. D , 5346 (1992)[arXiv:hep-ph/9209238].[54] E. D. Carlson and W. D. Garretson, Phys. Lett. B , 431 (1994).[55] R. Brustein and D. H. Oaknin, Phys. Rev. Lett. , 2628 (1999) [arXiv:hep-ph/9809365].[56] R. Brustein and D. H. Oaknin, Phys. Rev. D , 023508 (1999) [arXiv:hep-ph/9901242].[57] S. J. Asztalos, L. J. Rosenberg, K. van Bibber, P. Sikivie and K. Zioutas, Ann. Rev. Nucl.Part. Sci. , 293 (2006).[58] K. van Bibber and L. J. Rosenberg, Phys. Today , 30 (2006).[59] A. R. Zhitnitsky, arXiv:1004.2040 [gr-qc].[60] I. E. Halperin and A. Zhitnitsky, Phys. Rev. Lett. , 4071 (1998) [arXiv:hep-ph/9803301].[61] J. Kim and P. Naselsky, Astrophys. J. , L265 (2010) [arXiv:1001.4613 [Unknown]].[62] J. Kim and P. Naselsky, arXiv:1002.0148 [Unknown].[63] L. Pogosian, T. Vachaspati and S. Winitzki, Phys. Rev. D , 083502 (2002)[arXiv:astro-ph/0112536].[64] T. Kahniashvili and B. Ratra, Phys. Rev. D , 103006 (2005) [arXiv:astro-ph/0503709].[65] A. Kosowsky and A. Loeb, Astrophys. J. , 1 (1996) [arXiv:astro-ph/9601055].[66] J. A. Adams, U. H. Danielsson, D. Grasso and H. Rubinstein, Phys. Lett. B , 253 (1996)[arXiv:astro-ph/9607043].[67] D. D. Harari, J. D. Hayward and M. Zaldarriaga, Phys. Rev. D , 1841 (1997)[arXiv:astro-ph/9608098].[68] M. Giovannini, Phys. Rev. D , 3198 (1997) [arXiv:hep-th/9706201].[69] E. S. Scannapieco and P. G. Ferreira, Phys. Rev. D , 7493 (1997) [arXiv:astro-ph/9707115].[70] M. Giovannini and K. E. Kunze, Phys. Rev. D , 061301 (2008) [arXiv:0712.1977 [astro-ph]].[71] M. Giovannini and K. E. Kunze, Phys. Rev. D , 063003 (2008) [arXiv:0712.3483 [astro-ph]].[72] M. Giovannini and K. E. Kunze, Phys. Rev. D , 123001 (2008) [arXiv:0802.1053 [astro-ph]].[73] M. Giovannini and K. E. Kunze, Phys. Rev. D , 023010 (2008) [arXiv:0804.3380 [astro-ph]].[74] M. Giovannini and K. E. Kunze, Phys. Rev. D , 063007 (2009) [arXiv:0812.2207 [astro-ph]].[75] J. N. Clarke, G. Karl and P. J. S. Watson, Can. J. Phys. , 1561 (1982).[76] M. Giovannini, Phys. Rev. D , 021301 (2005) [arXiv:hep-ph/0410387]. – 20 –
77] M. Giovannini and K. E. Kunze, Phys. Rev. D , 087301 (2009) [arXiv:0812.2804 [astro-ph]].[78] S. M. Carroll, G. B. Field and R. Jackiw, Phys. Rev. D , 1231 (1990).[79] N. F. Lepora, arXiv:gr-qc/9812077.[80] A. Lue, L. M. Wang and M. Kamionkowski, Phys. Rev. Lett. , 1506 (1999)[arXiv:astro-ph/9812088].[81] K. R. S. Balaji, R. H. Brandenberger and D. A. Easson, JCAP , 008 (2003)[arXiv:hep-ph/0310368].[82] D. Hutsemekers, Astron. Astrophys. , 410 (1998)[83] D. Hutsemekers and H. Lamy, Astron. Astrophys. , 381 (2001) [arXiv:astro-ph/0012182].[84] D. Hutsemekers, R. Cabanac, H. Lamy and D. Sluse, Astron. Astrophys. , 915 (2005)[arXiv:astro-ph/0507274].[85] S. A. Joshi, R. A. Battye, I. W. A. Browne, N. Jackson, T. W. B. Muxlow andP. N. Wilkinson, Mon. Not. Roy. Astron. Soc. , 162 (2007) [arXiv:0705.2548 [astro-ph]].[86] R. A. Battye, I. W. A. Browne and N. Jackson, Mon. Not. Roy. Astron. Soc. , 274 (2008)[arXiv:0902.1619 [astro-ph.CO]].[87] S. Das, P. Jain, J. P. Ralston and R. Saha, JCAP , 002 (2005) [arXiv:hep-ph/0408198].[88] S. Das, P. Jain, J. P. Ralston and R. Saha, Pramana , 439 (2008) [arXiv:hep-ph/0410006].[89] A. K. Ganguly, P. Jain and S. Mandal, Phys. Rev. D , 115014 (2009) [arXiv:0810.4380[hep-ph]].[90] S. M. Carroll, Phys. Rev. Lett. , 3067 (1998) [arXiv:astro-ph/9806099]., 3067 (1998) [arXiv:astro-ph/9806099].