Redshifting of cosmological black bodies in BSBM varying-alpha theories
aa r X i v : . [ g r- q c ] N ov Redshifting of cosmological black bodies in BSBM varying-alpha theories
John D. Barrow and Jo˜ao Magueijo
2, 3 DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd., Cambridge, CB3 0WA, U.K. Theoretical Physics, Blackett Laboratory, Imperial College, London, SW7 2BZ, U.K. Dipartimento di Fisica, Universit`a La Sapienza and Sez. Roma1 INFN, P. le A. Moro 2, 00185 Rome, Italy (Dated: September 28, 2018)We analyse the behaviour of black-body radiation in theories of electromagnetism which allowthe electron charge and the fine structure constant to vary in space and time. We show that suchtheories can be expressed as relativistic generalizations of a conventional dielectric. By making theappropriate definition of the vector potential and associated gauge transformations, we can identifythe equivalent of the electric and displacement fields, E and D , as well as the magnetic B and H fields. We study the impact of such dielectrics on the propagation of light in the so-called “BSBM”theory. We examine the form of simple cosmological solutions and conclude that no changes arecreated to the standard cosmological evolution of the temperature and energy-density of black-bodyradiation. Nonetheless the matter evolution changes and the behaviour of the entropy per baryonis modified, and the ratios of different dark matter components may be changed too. I. INTRODUCTION
There has been considerable observational and theo-retical interest in the possibility that some dimensionlessatomic constants of Nature, like the fine structure con-stant, α, or the proton-electron mass ratio, µ , might bevarying very slowly in space and time [1]. A range ofhigh-precision astronomical instruments have opened upnew ways to find very small changes in the values of phys-ical “constants” that would be undetectable in currentterrestrial experiments [2]. Often, observational data isused simply to deduce the value of a constant at someredshift z > , and compare it with the value measuredhere and now in the laboratory at z = 0: no actual the-ory of the constant’s variation is used to connect the twovalues or to include other possible consequences of vary-ing constants on the cosmological history. By contrast,theories which promote constants to become variables ina self-consistent way do so by making them into space-time variable (scalar) fields. These fields must then grav-itate and satisfy the constraints imposed by energy andmomentum conservation. These requirements determinethe generalisations of Einstein’s general relativity whichincorporate varying constants.In the well-studied case of varying G, if we ignoredthis dictate and simply compared values of G at fixed z, then the conclusions drawn would be unreliable becausethe time-variation of G couples to the evolution of thecosmological scale factor at zero-order. A self-consistentBrans-Dicke (BD) scalar-tensor theory of gravity [3] hasa solution for a time-variation of the form G ∝ t − n anda cosmological scale factor evolution a ( t ) ∝ t (2 − n ) / ina flat, dust-filled BD universe [4] (and its Newtoniananalogue has the same solution [5]). In the case of avarying α, a corresponding self-consistent scalar theoryfor its space-time variation is provided by the theory ofBekenstein [6], Sandvik, Barrow and Magueijo (BSBM)[7–10]. Exact, approximate and asymptotic solutions ofthis theory and its extensions can be found and used tofit observational data from quasar spectra and elsewhere. Analogous theories have also been created to study self-consistently the variation of µ [11]. Unlike the situationwith varying- G, variations of atomic “constants”, like α and µ, do not have significant effects on the evolution ofthe cosmological scale factor; for example during a cold-dark-matter era α ( t ) ∝ ln( t/t ) and a ( t ) ∝ t / [ln( t/t )] λ where λ ≃ × − and t are constants [10].Despite this general pattern, it has been claimed thatvariations of α and µ are able to create discernible (loga-rithmic) differences in the temperature-redshift evolutionfor massless particles in BSBM theories [13–15]. Thisis one aspect of the original formulation of a varying α theory by BSBM that will be the focus of this pa-per. In the original formulation it appears that the evo-lution of a black body distribution of equilibrium pho-tons can pick up logarithmic corrections to the stan-dard cosmological evolution of the standard temperature-redshift relation, T γ ∝ a − ∝ (1 + z ), changing it to T γ ∝ α / a − ∝ α / (1 + z ). If true, this would have po-tentially observable consequences – including slow evolu-tion of the photon entropy per baryon and of the ratioof the neutrino to photon temperatures, and deviationfrom linearity in the relation between radiation tempera-ture and redshift z . All have observational consequences.In this paper we analyse this feature of BSBM theoriesin detail and show that in BSBM, with the appropriatechoice of generalisation from Maxwell’s theory, the evo-lution of black-body radiation follows the standard cos-mological trajectory followed by theories with constant α . The same conclusions will hold mutatis mutandis fortheories of this type with varying µ [11].In order to investigate how a black body reacts to theBSBM field, ψ , and whether it might self-consistentlydrive variations in the electron charge e ≡ e exp(2 ψ )(and hence in α ≡ e / ℏ c ), we develop an analogy witha dielectric medium [12] in Section II. We find that theBSBM field ψ behaves like a relativistic generalizationof a dielectric or insulator. It is linear ( D and H areproportional to E and B ) and the proportionality con-stants ǫ and µ − are isotropic and frequency indepen-dent. Unlike in standard media, ǫ and µ − obey a rel-ativistic Klein-Gordon equation that is sourced by theEM lagrangian. We find that ǫ = 1 /µ , so the medium isnon-dispersive. In Section III we examine the propertiesof photons in ’ free’ flight through such a dielectric andfind no frequency shift or photon production. Hence wedo not expect a non-interacting black body to be affectedby changes in α created by the space-time variation of ψ .The same should hold true for a coupled thermalised sys-tem. Therefore there will be no new observable effects onthe redshift history of the cosmic microwave backgroundradiation. II. BSBM VARYING-ALPHA AS ARELATIVISTIC DIELECTRIC EFFECT
BSBM [6–8] was built upon the principles of relativis-tic Lorentz invariance and the gauge principle. The ψ field it employs to carry variations in α , which obeys arelativistic Klein-Gordon equation, can therefore neverbe identical to a dielectric medium in conventional elec-trodynamics, which is usually a non-relativistic material(an insulator). However, the ψ field may be regarded asa relativistic generalization of a conventional dielectric.Although this creates some important differences, nev-ertheless with regard to the effects on electromagneticradiation much of the formalism is similar, as we nowshow.In setting up BSBM there is an “ambiguity” in thedefinition of electric and magnetic fields similar to thatfound for insulators, where one can use E or D for theelectric field, and B or H for the magnetic field. In realityboth concepts play a role, with E and B convenient forwriting the homogeneous Maxwell equations, and D and H better suited for writing the inhomogeneous equations,even when there are no sources.A first decision fork appears in the literature in thedefinition of the vector potential (and the expression ofgauge transformations). One can use either A µ (as in[6]), or a µ (as in [7]), with the two related by: a µ = e ψ A µ , (1)where ψ = ln ˜ ǫ = ln ee = 12 ln α. (2)The last expression links ψ to the fine structure “con-stant”, α . Here ˜ ǫ corresponds to the “ ǫ ” used in [6],which we stress is not the relative permittivity of the“medium”( ǫ , in our notation here), as we shall see.Gauge transformations can be performed as a µ → a µ + ∂ µ Λ (3)or as A µ → A µ + ∂ µ Λ˜ ǫ . (4) This fork propagates into the definition of gauge-invariant field tensors, with [6] led to the natural defi-nition: F µν = e − ψ (cid:2) ∂ µ ( e ψ A ν ) − ∂ ν ( e ψ A µ ) (cid:3) , (5)and [7] to f µν = ∂ µ a ν − ∂ ν a µ . (6)The two are related by F µν = e − ψ f µν . (7)The electromagnetic action, from which the non-homogeneous Maxwell’s equations are derived, can bewritten in the two forms: S EM = − Z d x F = − Z d x e − ψ f . (8)In order to study which quantities play the role of E and B we examine the non-homogeneous Maxwell equa-tions. These are best written in terms of f µν , in the formof the integrability condition: ǫ αβµν ∂ β f µν = 0 . (9)This is obviously a necessary condition for (6), but notethat the same argument cannot be made directly for F µν (derivatives of ψ would appear in the corresponding con-dition in terms of F µν ; cf. (5) and (6)). Thus, in orderto parallel the usual theory of electrodynamics in mediawe should associate E and B (appearing in the inhomo-geneous Maxwell equations) with f µν , with entries in theusual places. With this identification we obtain the stan-dard inhomogeneous Maxwell equations: ∇ · B = 0 , (10) ∇ ∧ E + ∂ B ∂t = 0 . (11)This was already noted in [6] (however, the wrong iden-tification was made in ref.[16], cf. their Eq.(25)).In order to find the equivalent of D and H , we considerinstead the inhomogeneous Maxwell equations. In theabsence of currents these can be written in the two forms: ∂ µ ( e − ψ f µν ) = ∂ µ ( e − ψ F µν ) = 0 , (12)and we see that neither of them leads to the equivalentstandard expression for dielectric media (in both casesextra terms in the derivatives of ψ appear). Therefore,we should define the alternative tensor: F µν = e − ψ F µν = e − ψ f µν , (13)in terms of which we have ∂ µ F µν = 0 . (14)We should then define D and H from the appropriateentries in F µν , so as to get: ∇ · D = 0 , (15) ∇ ∧ H − ∂ D ∂t = 0 . (16)With these identifications BSBM becomes equivalent toelectromagnetism in dielectric media with only smalladaptations. We have: D = ǫ E = e − ψ E , (17) H = µ − B = e − ψ B , (18)and so ǫ = 1 µ = e − ψ . (19) D and H are proportional to E and B , and the propor-tionality constants ǫ and µ − are isotropic and frequency-independent. However, they do not depend locally onthe EM field (as is the case for standard media). Rather,they obey a relativistic Klein-Gordon equation. Since ǫ = 1 /µ , the medium is non-dispersive, as we shall ex-plicitly prove in the next Section. Our discussion hasbeen for Minkowski spacetime, but replacing derivativeswith covariant derivatives gives the usual generalizationto curved space-time . III. ELECTROMAGNETIC WAVES: PHOTONSIN FREE FLIGHT
Maxwell’s equations may be combined in the usual wayto produce electromagnetic wave equations. As in thecase of conventional dielectrics, these are most symmet-rically written in terms of E and H . For simplicity, as-suming that ǫ = ǫ ( t ) and µ = µ ( t ) only, they are: − ∇ E + ∂∂t µ ∂∂t ǫ E = 0 , (20) −∇ H + ∂∂t ǫ ∂∂t µ H = 0 . (21)Generalizations for space-dependent fields could be ex-pressed, but will not be required in this paper. So far wehave not assumed that n = 1. Several of these points were noted before in refs.[6, 17], althoughour analogy of a relativistic dielectric medium was not used there.For instance, in [6] (using the notation ˜ ǫ for the “ ǫ ” used therein),we should have ǫ = 1 / ˜ ǫ , as recognised in the Section IIIC of [6]. A. WKB solution
A WKB expansion may now be sought, with ans¨atze: E = e E E ( t ) exp (cid:20) i (cid:18) k · x − Z ωdt (cid:19)(cid:21) , (22) H = e H H ( t ) exp (cid:20) i (cid:18) k · x − Z ωdt (cid:19)(cid:21) . (23)To order ω , this translates into the dispersion relations: ω = n k , (24)with refractive index n = ǫµ . (25)To first order in ω we get the equations:˙ µµ + 2 ˙ ǫǫ + 2 ˙ E E = 0 , (26)˙ ǫǫ + 2 ˙ µµ + 2 ˙ H H = 0 , (27)so that E ∝ ǫ √ µ , (28) H ∝ µ √ ǫ , (29)and also D ∝ √ µ , (30) B ∝ √ ǫ . (31)Specializing to BSBM, we can assume (19), and thus n = 1 , (32)so that BSBM does not induce modified dispersion rela-tions or a variation in the speed of light. This was notedbefore in [18], and is hardly surprising. It is built intothe theory, since ψ may be regarded as an “insulatingaether” which does not break Lorentz invariance. There-fore, it is expected that the central property of light—theconstancy of its speed—is left undisturbed, since it un-derpins the Lorentz invariance.The absence of dispersion in the free propagation ofelectromagnetic waves implies that in BSBM there is noextra shift in the frequency of photons. In addition, wecan derive the scalings for the amplitudes: E ∝ e ψ ∝ B , (33) H ∝ e − ψ ∝ D . (34)These are consistent with E = B and the orthogonalityof E and B . B. Energy density
The stress-energy tensor in BSBM can be obtained tak-ing variations of (8) with respect to the metric. The ex-pressions thus obtained match the well-know results inthe theory of electrodynamics in media. For example,the energy density (i.e. T ) is given by: e ρ = 12 ( ED + BH ) = e − ψ E + B ) . (35)For a wave packet, averaging over many wavelengths andperiods, this becomes: e ρ = 14 e − ψ ( E + B ) ω = 12 e − ψ E ω , (36)and so e ρ does not depend on ψ , since E ∝ B ∝ e ψ , aswe saw in our WKB solution.Therefore, in BSBM the energy of a wave packet doesnot change because of a varying e (or α ). Combining thiswith the absence of dispersion we can conclude that thereis no photon production, since the density of photonswith frequency ω is N = ρ/ ( ~ ω ). The Stefan-Boltzmannlaw for a decoupled black body is therefore also preserved,with the temperature receiving no new effects due to avarying α (other than those due to the gravitational effectof ψ in the Friedman equation, which, as we stated, arenegligible).We note that if we want ψ to be absent from the ex-pressions of the electromagnetic stress-energy tensor interms of the fields, then, as evident from (35), we shouldwork with electric and magnetic fields which are “geo-metric averages” of E and D , and of B and H . In fact,these are the fields which make up Bekenstein’s F µν (see[6]), and this is the tensor which enters without explicitcoupling to ψ in the Lagrangian and the stress-energytensor (although of course ψ is hidden inside the defini-tion of F µν in terms of the 4-potential). We argue thatthis is the correct definition for the energy for these theo-ries, based on the results in this Section, as well as thosefound in the next two. IV. COSMOLOGICAL EQUATIONS
The previous argument can be reinforced by looking atthe cosmological equations. With a standard Lagrangianfor ψ : L ψ = − ω B ∂ µ ψ∂ µ ψ (37)we are led to a driven Klein-Gordon equation for theevolution of ψ : ∇ ψ = 2 ω B e L EM . (38)where e L EM = e − ψ ( E − B ) / ψ into theirdefinition). Under the assumption of homogeneity andisotropy, this Klein-Gordon becomes the ODE:¨ ψ + 3 ˙ aa ˙ ψ = − ω B e L EM . (39)and this equation can be interpreted as an energy balanceequation, with the driving terms representing energy ex-change between ψ and other forms of matter. Indeedhomogeneity and isotropy imply that ψ must behave likea perfect fluid, and computing the stress-energy tensorreveals: p ψ = ρ ψ = ω B ˙ ψ . (40)Equation (39) is then equivalent to:˙ ρ ψ + 3 ˙ aa ( p ψ + ρ ψ ) = − ψ e L EM . (41)Each cosmic component i contributes a term proportionalto e L EM i to the right hand side of (41). This shouldbe balanced by a counter-term with opposite sign in theright hand side of the the conservation equation for i : e ρ i + 3 ˙ aa ( e ρ i + e p i ) = 2 ˙ ψ e L EM i . (42)Let us first examine the case of pure radiation. Sinceno driving term exists for pure radiation, the field ψ doesnot lose or gain energy (other than redshifting like 1 /a due to its own pressure) and likewise the energy for ra-diation should be defined so that it does not depart fromthe usual 1 /a law. Thus for radiation it is clear that afull set of equations is: (cid:18) ˙ aa (cid:19) = 13 ( e ρ r + ρ ψ ) (43)˙ e ρ r + 4 ˙ aa e ρ r = 0 (44)˙ ρ ψ + 6 ˙ aa ρ ψ = 0 (45)This lends further support to the choice of e ρ as a suitabledefinition for the energy in electromagnetism with vary-ing alpha, as discussed at the end of last section. Thefield ψ behaves like a non-interacting fluid if only radi-ation is present. Energy should therefore be defined forradiation (and more generally for electromagnetism) sothat it is also non-interacting in this case.For other components (including the dark matter) weneed equations of state relating their energy density withtheir EM Lagrangian content, and this is not always easyto infer. One possibility is to define parameters: ζ i = e L EMi e ρ i . (46)For radiation ζ r = 0, but ζ m = 0 for baryonic as well asfor some types of dark matter. The statement that ζ i isa constant is part of the model (and we stress that such amodel is not the model employed for matter in [7]). Darkmatter candidates with ζ m < (cid:18) ˙ aa (cid:19) = 13 ( e ρ m + e ρ r + ρ ψ ) (47)˙ e ρ r + 4 ˙ aa e ρ r = 0 (48)˙ e ρ m + 3 ˙ aa e ρ m = 2 ˙ ψζ m e ρ m (49)˙ ρ ψ + 6 ˙ aa ρ ψ = − ψζ m e ρ m (50)where ˙ ψ on the right-hand side of the last two equationscan be written in terms of ρ ψ via (40), to form a closedsystem. V. SOME EXACT SOLUTIONS ANDPHYSICAL IMPLICATIONS
We now derive some exact solutions with pure mat-ter and scalar kinetic energy plus matter content (pureradiation is trivial) and discuss the physical implications.
A. Matter-dominated model If ζ m is indeed a constant we find at once that:˜ ρ m = M e ζ m ψ a , (51)with M constant. At late times the cosmological equa-tions with zero radiation density have approximate late-time solutions of the same form as the earlier BSBM anal-ysis [8], now with: a ( t ) = t / (52) e ζ m ψ = ω B M ζ m ln( t )We observe that in the Friedmann equation this leads to˜ ρ m = M e ζ m ψ a = ω B ζ m t ln( t ) ρ ψ = ω B ˙ ψ ω B ζ m t ln ( t )and so the ˜ ρ m term dominates increasingly at large t. During any dark-energy dominated era (say driven by asimple cosmological constant Λ) the expansion dynamicswill approach the de Sitter evolution a ( t ) ∝ exp[ t p Λ / ψ will tend to a constant value, ψ ∞ , exponentiallyrapidly ( ψ → ψ ∞ + O ( t exp[ − t √ t → ∞ . Hencein this era the matter density will quickly approach thestandard evolution with ˜ ρ m ∝ a − . B. Matter with scalar kinetic energy
If eq. 51 holds still then the Friedmann equation (47)becomes:3 (cid:18) ˙ aa (cid:19) = e ρ m + ρ ψ = M e ζ m ψ a + 12 ω B ˙ ψ (53)and the equation for ψ is:¨ ψ + 3 ˙ aa ˙ ψ = − ζ m M e ζ m ψ a . There is a special exact solution of these equations with: a = t n (54) ψ = A + B ln( t )where 3 n = 2(1 + Bζ m ) , (55)3 n = M e ζ m A + 12 ω B B ,B (1 + 2 Bζ m ) = − ζ m M e ζ m A , determine the constants A, B and n in terms of M, ζ m and ω B . Note that in this solution all the terms in theFriedmann eqn fall as t − , specifically:˜ ρ m ∝ e ζ m ψ a ∝ ω B ˙ ψ ∝ t − . However, exact solutions of this simple scaling form areunstable in BSBM [14] and are expected to be so herealso, approaching the form 52 at late times when thereis no dark energy term to drive ψ to a constant value. If n = 1 / e ρ r ∝ a − ∝ t − to eq. (53) and stillobtain an exact solution of this scaling form. C. Physical implications
In contrast with earlier work we find no variations inthe radiation evolution. However, as we have seen in thissection, the matter evolution is changed (including darkmatter, depending on the value of ζ m ). This leads tonumber of new effects, analogous to those derived basedon variations in the radiation evolution. The main im-plications are as follows: • The density of non-relativistic matter which, likeweakly interacting CDM, does not couple to ψ orto electric charge (i.e. ζ i = 0), will scale with ex-pansion as ρ cdm ∝ a − . Therefore, in the generalasymptotic matter-dominated solution 52 it falls offat a different rate to that of the electrically coupledmatter density, ˜ ρ m , and we will have˜ ρ m ρ cdm ∝ e ζ m ψ ∝ / ln( t ) • The entropy per baryon is no longer constant dur-ing adiabatic expansion. Although e ρ / r /ρ cdm re-mains constant during the expansion (and the ra-tio of the neutrino to photon temperatures will beconstant), the combination e ρ / r / ˜ ρ m ∝ e − ζ m ψ ∝ ln( t ) does not remain constant during the matter-dominated era. Some care must therefore be ex-ercised in comparing constraints on the entropyper baryon at late times with those deduced frompredictions of the primordial deuterium abundanceand observations. VI. CONCLUSION
The absence of dispersion in the free propagation ofelectromagnetic waves implies that in BSBM there isno extra shift in the frequency of photons (other thanthe usual effect of the cosmological expansion). For ablack-body in free flight this means that the temperature-redshift relation T γ ( z ) = T γ (1 + z ) is not modified. Thespectrum remains Planckian when the universe expandsisotropically and homogeneously, and its energy densityis still given by ρ γ ∝ T γ ∝ a − . Thus, nothing changesin the temperature-redshift relation for a black body infree flight in BSBM, as we have shown explicitly in thispaper. This conclusion has been known for a while inthe context of string cosmology and elsewhere [22–26],and in fact is more general than proved here: no energyexchange occurs with a pure radiation system for any“multiplicative” theory (i.e. one of the dilaton type).This conclusion remains also true for a coupled, ther-malized black body. In fact this is implied by theKirchhoff-Clausius law, which states that “the rate atwhich a body emits heat radiation is inversely propor-tional to the square of the speed at which the radia-tion propagates in the medium in which the body is im-mersed” (e.g. [19]). If the medium is non-dispersive, asis the case with BSBM, then the Planck law and Stefan-Boltzmann law, receive no direct corrections due to avarying α . We still have ρ ∝ /a (since p = 1 / ρ ) and ρ ∝ T γ , so T γ ∝ /a remains true. Even though this con- clusion seems supported by copious past literature, it re-mains controversial in some quarters [13, 15, 27] (in partdue to a misunderstanding for which [7] may be partlyresponsible). Clearly there are many possible definitionsfor the energy in a dielectric, but the one responsiblefor a conservation law is the most appropriate. Our sec-tions on cosmological applications should have clarifiedthe matter.At first it might appear we have derived a negativeresult. Since ρ ∝ /a and ρ ∝ T γ we may concludethat T γ ∝ /a remains true. Obviously the field φ maystill affect the function a ( t ), due to its presence in theFriedman equations, but the temperature-redshift rela-tion remains unmodified and astronomical observationsof T γ ( z ) at z > Acknowledgments
We acknowledge STFC for consolidated grant supportand thank Aur´elien Hees, Julien Larena, Olivier Minaz-zoli and Alec Graham for helpful discussions. [1] J-P. Uzan,
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