Dark energy: the absolute electric potential of the universe
aa r X i v : . [ phy s i c s . g e n - ph ] M a y Essay written for the Gravity Research Foundation 2009Awards for Essays on Gravitation
Dark energy: the absolute electric potentialof the universe
Jose Beltr´an Jim´enez ∗ and Antonio L. Maroto † Departamento de F´ısica Te´oricaUniversidad Complutense de Madrid28040 Madrid, SpainMarch 30th, 2009
ABSTRACT
Is there an absolute cosmic electric potential? The recent discovery of theaccelerated expansion of the universe could be indicating that this is cer-tainly the case. In this essay we show that the consistency of the covariantand gauge invariant theory of electromagnetism is truly questionable whenconsidered on cosmological scales. Out of the four components of the elec-tromagnetic field, Maxwell’s theory only contains two physical degrees offreedom. However, in the presence of gravity, one of the ”unphysical” statescannot be consistently eliminated, thus becoming real. This third polariza-tion state is completely decoupled from charged matter, but can be excitedgravitationally thus breaking gauge invariance. On large scales the new statecan be seen as a homogeneous cosmic electric potential, whose energy densitybehaves as a cosmological constant. ∗ jobeltra@fis.ucm.es † maroto@fis.ucm.es 1he recent discovery of the accelerated expansion of the universe, and thedifficulties found in the context of General Relativity (GR) and the StandardModel (SM) of elementary particles to properly account for this effect, hasled to consider the possibility that physics on large scales could differ fromour well-known small scale laws.In this context, models in which the description of the own gravitationalinteraction are modified on large scales with respect to GR have been ex-tensively considered in recent years. Here we will concentrate in the otherlong-range interaction of nature, and explore the possibility that our stan-dard theory of (quantum) electromagnetism, being valid on small scales,could give rise to unexpected effects on cosmological scales. As a matter offact, this possibility is perfectly compatible with current experimental limitswhich have tested electromagnetism only for wavelengths roughly below theSolar System scales (1.3 A.U. [1]).In this essay we will discuss one of the most striking consequences ofelectromagnetism in the cosmological context, which is the possibility thatthe universe at large scales not only sets a privileged reference frame, butcould also determine an absolute electric potential. Indeed, it is well knownthat the presence of matter and radiation in the universe implies that, onlarge scales, the universe as a whole has associated a privileged referenceframe. That frame is nothing but the cosmic center of mass frame [2] of thedifferent components (baryonic and dark matter, radiation and dark energy).In the case in which all such components are at rest with respect to eachother, the frame can be identified with that of the observers who see anisotropic cosmic microwave background. Thus we can say that, althoughLorentz symmetry is locally a good symmetry of space-time, it is broken onlarge scales by the matter/energy content of the universe. But, what about2he rest of gauge symmetries and, in particular, that of electromagnetism? Isit also possible that, although on small scales we see electromagnetic gaugesymmetry as an exact symmetry of nature, the actual situation is that it isbroken by the content of the universe on large scales? Does it make senseto talk about a privileged electromagnetic gauge? We will argue that darkenergy, responsible for the accelerated expansion of the universe, could benothing but the energy density associated to such absolute electric potential[3, 4].Let us start by briefly reviewing the standard covariant electromagneticquantization in Minkowski space-time [5]. The starting point is the action: S = Z d x − F µν F µν + ξ ∂ µ A µ ) + A µ J µ ! (1)which is not invariant under general gauge transformations, but only underresidual ones, given by: A µ → A µ + ∂ µ θ , with ✷ θ = 0. The equations ofmotion obtained from this action read: ∂ ν F µν + ξ∂ µ ( ∂ ν A ν ) = J µ (2)In order to recover ordinary Maxwell’s equation, the Lorenz condition ∂ µ A µ =0 must be imposed so that the ξ term disappears. At the classical level thiscan be achieved by means of appropriate boundary conditions on the field.Indeed, taking the four-divergence of the above equation, we find: ✷ ( ∂ ν A ν ) = 0 (3)where we have made use of current conservation. This means that the field ∂ ν A ν evolves as a free scalar field, so that if it vanishes for large | t | , it willvanish for all time. At the quantum level, the Lorenz condition cannot beimposed as an operator identity, but only in the weak sense ∂ ν A ν (+) | φ i = 0,3here (+) denotes the positive frequency part of the operator and | φ i is aphysical state. This condition is equivalent to imposing that the physicalstates contain the same number of temporal and longitudinal photons, sothat their energy densities, having opposite signs, cancel each other. Thuswe see that the Lorenz condition seems to be essential in order to recoverstandard Maxwell’s equations and get rid of the negative energy states.Now we move to an expanding universe. The curved space-time versionof action (1) reads: S = Z d x √ g − F µν F µν + ξ ∇ µ A µ ) + A µ J µ ! (4)and the modified Maxwell’s equations are: ∇ ν F µν + ξ ∇ µ ( ∇ ν A ν ) = J µ (5)Taking again the four divergence, we get: ✷ ( ∇ ν A ν ) = 0 (6)We see that once again ∇ ν A ν behaves as a scalar field which is decoupled fromthe conserved electromagnetic currents, but it is non-conformally coupled togravity. This means that, unlike the flat space-time case, this field can beexcited from quantum vacuum fluctuations by the expanding backgroundin a completely analogous way to the inflaton fluctuations during inflation.Thus this poses the question of the validity of the Lorenz condition at alltimes.In order to illustrate this effect, we will present a toy example. Let usconsider quantization in the absence of currents, in a spatially flat expandingbackground, whose metric is written in conformal time as: ds = a ( η ) ( dη − d~x ) (7)4or the scale factor we assume the following form: a ( η ) = 2 + tanh( η/η )where η is constant. This metric is asympotically flat in the remote past andfar future. Let us prepare our system in an initial state | φ i belonging to thephysical Hilbert space, i.e. satisfying ∂ ν A ν (+) in | φ i = 0 in the initial flat region.We solve the coupled system of equations (5) for the corresponding Fouriermodes A µ~k . Because of the expansion of the universe, the positive frequencymodes in the in region with a given temporal or longitudinal polarization λ will become a linear superposition of positive and negative frequency modesin the out region and with different polarizations λ ′ [6]. H k L Figure 1:
Occupation numbers for temporal (continuous line) and longitudinal (dashedline) photons in the out region vs. k in η − units. Thus, the system will end up in a final state which no longer satisfiesthe weak Lorenz condition, i.e. in the out region ∂ ν A ν (+) out | φ i 6 = 0. This isshown in Fig. 1, where we have computed the final number of temporal andlongitudinal photons starting from an initial vacuum state with n in ( k ) = n in k ( k ) = 0. We see that in the final region n out ( k ) = n out k ( k ). Notice that the5ailure comes essentially from large scales ( kη ≪ kη ≫ A µ = A (1) µ + A (2) µ + A ( s ) µ + ∂ µ θ (8)where A ( i ) µ with i = 1 , A ( s ) µ is the new scalar state, which is the mode that would have beeneliminated if we had imposed the Lorenz condition and, finally, ∂ µ θ is a puregauge mode which can be eliminated. In order to quantize the free theory,we perform the mode expansion for the three physical states: A µ = Z d ~k X λ =1 , ,s (cid:18) a λ ( k ) A ( λ ) µk + a † λ ( k ) A ( λ ) µk (cid:19) (9)In fact, the three modes can be chosen to have positive normalization and,for ξ = 1 /
3, the canonical commutation relations are satisfied: h a λ ( ~k ) , a † λ ′ ( ~k ′ ) i = δ λλ ′ δ (3) ( ~k − ~k ′ ) , λ, λ ′ = 1 , , s (10)with positive sign for the three physical states, i.e. there are no negative normstates in the theory, which in turn guarantees that there are no negativeenergy states (ghosts). Moreover, as shown in [4, 7], the theory does notexhibit either local gravity inconsistencies or classical instabilities.As shown in (6), ∇ µ A µ evolves as a minimally coupled scalar field. Thismeans that on sub-Hubble scales ( | kη | ≫ |∇ µ A ( s ) µk | ∝ a − . Thus, on small scales, the modifiedMaxwell’s equations (5) will be physically indistinguishable from the flatspace ones. To summarize, from the previous discussion we see that thetheory is consistent even though we have not imposed the Lorenz condition.But, moreover, on super-Hubble scales ( | kη | ≪ |∇ µ A ( s ) µk | = const. which, as shown in [4], implies that the field contributes as an effectivecosmological constant in (4).In order to determine its value, we will assume that the field is generatedduring inflation from quantum vacuum fluctuations, in a completely analo-gous way to cosmological metric perturbations. Thus, in an inflationary deSitter space-time, it is possible to obtain the corresponding dispersion: h | ( ∇ µ A µ ) | i = Z dkk P A ( k ) (11)where P A ( k ) = 4 πk |∇ µ A ( s ) µk | . In the super-Hubble limit, we obtain for thepower-spectrum: P A ( k ) = 9 H I π , (12)with H I the constant Hubble parameter during inflation. Thus the electro-magnetic energy density on cosmological scales is given by h | ρ A | i ∼ ( H I ) .The measured value of the dark energy density then requires H I ∼ − eV, which corresponds to an inflationary scale of M I ∼ dt = a ( η ) dη ), thehomogeneous temporal component A ( t ) (the electric potential) is constantduring inflation and grows as t during matter and radiation eras. When the7 igure 2: Cosmological evolution of the electric potential A from electroweak-scaleinflation until present. electromagnetic dark energy starts dominating, A ( t ) becomes also constant.The spatial components on super-Hubble scales ~A ( t ) are shown to grow moreslowly than A ( t ) and can be neglected. In Fig. 2 we show the cosmologicalevolution of the electric potential from its initial value generated in inflation( A ( t I ) ∼ − eV) up to its present value A ( t ) ∼ . M P with M P ∼ GeV the Planck mass.In conclusion, we have discussed the possibility that the true theory ofelectromagnetism contains three and not only two physical degrees of free-dom. Although the new scalar state is completely decoupled from the con-served currents, it can be gravitationally amplified during inflation, givingrise to the observed dark energy density. The accelerated expansion of the8niverse would be then the natural consequence of the existence of an abso-lute electric potential in the universe.
Acknowledgments:
This work has been supported by Ministerio de Cienciae Innovaci´on (Spain) project numbers FIS 2008-01323 and FPA 2008-00592,UCM-Santander PR34/07-15875, CAM/UCM 910309 and MEC grant BES-2006-12059.
References [1] A.S. Goldhaber and M.M. Nieto, arXiv:0809.1003 [hep-ph].[2] A. L. Maroto, JCAP , 015 (2006); J. Beltr´an Jim´enez andA. L. Maroto, Phys. Rev. D , 023003 (2007)[3] J. Beltr´an Jim´enez and A. L. Maroto, arXiv:0903.4672 [astro-ph.CO].[4] J. Beltr´an Jim´enez and A.L. Maroto, JCAP : 016 (2009), e-Print:arXiv:0811.0566 [astro-ph].[5] C. Itzykson and J.B. Zuber,
Quantum Field Theory , McGraw-Hill (1980);N.N. Bogoliubov and D.V. Shirkov,
Introduction to the theory of quantizedfields , Interscience Publishers, Inc. (1959).[6] N.D. Birrell and P.C.W. Davies,
Quantum fields in curved space , Cam-bridge (1982).[7] J. Beltr´an Jim´enez and A.L. Maroto,
JCAP0902