Abstract
We explore the possibility that the reported time variation of the fine structure constant α is due to a coupling between electromagnetism and gravitation. We predict such a coupling from a very simple effective theory of physical interactions, under the form of an improved version of the Kaluza-Klein theory. We show that it precisely leads to a variation of the effective fine structure constant with cosmic conditions, and thus with cosmic time. The comparison with the recent data from distant quasars absorption line spectra gives a good agreement; moreover, this may reconcile the claimed results on α with the upper limit from the Oklo naturel Uranium fission reactor.
Introduction
Recent publications [1, 2] report observations of distant quasars absorption lines, which may indicate a time variation of the fine structure constant α. Different kinds of explications have been proposed (see in conclusion), which all involve new physics [3, 4] . Since many theoretical arguments suggest that our present theories of physical interactions are not the ultimate ones, this possibility deserves serious attention.
Introduction
An ultimate theory would include additional fields and coupling which remain presently unknown. At the effective level, and in particular in astrophysical conditions, those can be manifest as a soft dependence of the "constants" of the interactions with some parameters. For instance, the effective theory considered here predicts a variation of the effective gravitatonal constant G with respect to electromagnetic conditions, and a variation of the effective α with respect to gravitational conditions, and thus with the cosmic time (seen as a parameter expressing the variation of the cosmic gravitational potential, through the Friedmann -Lemaître equations). In this paper, we calculate the expected variation of α, and compare it with astronomical observations. In a companion paper ( [5] ), we have shown that the predicted dependence of G ef f with the value of the geomagnetic field, in the same framework, may explain the discordant terrestrial measurements of the gravitational constant.
Introduction
One of the simplest effective theories that it is possible to build (beside Brans-Dicke type theories) results from the compactification of the Kaluza-Klein (KK) one. As we show here (see also [6] ), this leads to replace the gravitational constant G and the fine structure constant α by effective values, which vary with the scalar field Φ introduced in the theory. On the other hand, different authors [7, 8] have pointed out that a pure KK action leads to instability of the theory, because of the bad sign for the kinetic term of Φ [9] . The same authors suggested the presence of an additional field to cure this problem. In a previous study [10] , we applied an argument initially from Landau and Lifshitz [11] , to study this instability, and we proposed a minimal hypothesis for stabilization: the addition of an external field ψ. Thus we consider this modified KK theory (hereafter KKψ) as the simplest candidates for an effective theory, a prototype to explore the possibility that the observational results are due to a coupling between gravitation and electromagnetism (hereafter GE coupling).
Introduction
Seen in our 4-dimensional space-time, the KKψ Lagrangian leads to a theory of gravitation and electromagnetism, with the additional fields Φ (internal KK field) and ψ (external stabilizing field). The latter induce a GE coupling, which appears precisely as a dependence of the (effective) constants G ef f and α ef f with respect to other fields. This paper explores the cosmic evolution of α ef f generated by that of matter and gravitation (spacetime curvature).
Introduction
In section 2, we calculate the variations predicted by the KKψ theory: section 2.1 recalls the definition of the effective coupling constants, and gives the effective MaxwellEinstein equations in the context of the compactified KK theory; section 2.2 introduces the KKψ Lagrangian and the resulting equations (Maxwell-Einstein and scalar fields evolutions); section 2.3 considers the cosmological solution (weak fields limit, in the matterdominated epoch). In section 3, we compare our calculations of the cosmological evolution of the fine structure constant with the avalaible data from distant quasars absorption lines. Also, we discuss the consistency of our results especially with respect to the Oklo phenomenon. In section 4, the similarities and differences with other work are emphasized.
Effective coupling constants 2.1 The Kaluza-Klein theory
The original KK theory, after dimensional reduction, leads to the effective action in the Jordan-Fierz frame (e.g., see [6, 12] )
Effective coupling constants 2.1 The Kaluza-Klein theory
where A α is the potential 4-vector of the electromagnetic field, F αβ = ∂ α A β − ∂ β A α is the electromagnetic field strength tensor, Φ the (internal) scalar field related byĝ 44 = − Φ 2 to the fifteenth degree of freedom,ĝ 44 , of the 5-dimensional metric and G the (true) gravitational constant. We emphasize that this is not a theory with minimally coupled scalar field. According to Lichnerowicz [6] , the quantity G ef f := G Φ of the Einstein-Hilbert term, and the factor ε 0ef f = ε 0 Φ 3 of the Maxwell term in (1), should be interpreted respectively as the effective gravitational "constant" and the effective vacuum dielectric permittivity. These terms depend on the local (for terrestrial experiments) or global (at cosmological scale) value of Φ, assumed to be positive defined. Clearly, the previous considerations lead to an effective fine structure constant
Effective coupling constants 2.1 The Kaluza-Klein theory
to be compared to the true fine structure constant α := e 2 /4πε 0h c. It is worth noticing that this does not involve any variation of the electric charge, unlike the earlier suggestion of Bekenstein [13] . Also, the velocity of light remains constant since the value of the effective vacuum magnetic permeability, µ 0ef f = µ 0 Φ −3 , insures ε 0ef f µ 0ef f = ε 0 µ 0 (see [6] ). Applying the least action principle to the action (1) yields
Effective coupling constants 2.1 The Kaluza-Klein theory
• the generalized Einstein-Maxwell equations with the additional source term
Effective coupling constants 2.1 The Kaluza-Klein theory
in addition to the electromagnetic stress-energy tensor T (EM ) αβ
Effective coupling constants 2.1 The Kaluza-Klein theory
. They identify to the usual expressions, where G and ε 0 are replaced respectively by G ef f and ε 0ef f .
Effective coupling constants 2.1 The Kaluza-Klein theory
• the KK scalar field equation
Stabilizing the Kaluza-Klein action
To stabilize the KK action ( [10] ), the simplest possibility is to introduce an additional matter field: a real bulk scalar field minimally coupled to gravity. After dimensional reduction, this field appears as a scalar field ψ in spacetime, with the effective action (in the Jordan-Fierz frame)
Stabilizing the Kaluza-Klein action
where U = U (ψ) denotes the self-interaction potential of ψ and J its source term. The latter includes contributions from the matter and from Φ, both proportional to the trace of their energy-momentum tensor, viz.
Stabilizing the Kaluza-Klein action
and that of the (traceless) electromagnetic field, ε 0 f (ψ , Φ) F αβ F αβ . Generally speaking these coupling functions are temperature dependent, with magnitude decreasing as the temperature increases (this prevents from any significant modification of the big bang nucleosynthesis). The necessity to recover the usual physics whenever the ψ-field is not excited requires g(v , 1) = f (v , 1) = 0 and U (v) = 0. We have written v the vacuum expectation value (VEV) of ψ, such that Applying the variational principle to (5) yields
Stabilizing the Kaluza-Klein action
and
Stabilizing the Kaluza-Klein action
The effective Einstein equations are unchanged, apart from a new source term: the effective energy momentum tensor of ψ,
Stabilizing the Kaluza-Klein action
Since we know that, for the effects examined here, the effective values are close to the usual one, we linearize the two scalar fields around their respective VEVs. Hence, equations (6, 7) above reduce to
Stabilizing the Kaluza-Klein action
and
Cosmological solutions
For cosmology, we assume spatially constant values of the fields and we follow their evolutions with respect to the cosmic time, t. Hence, ψ = ψ(t) and Φ = Φ(t). Besides, F αβ F αβ vanishes for the pure EM cosmic background radiation. Thus, the cosmological equations reduce to the effective Friedmann equation (with a cosmological constant Λ)
Cosmological solutions
and, for the scalar fields,ψ
Cosmological solutions
The dot denotes the derivative with respect to the cosmic time, H =ȧ/a is the expansion rate (Hubble parameter), a = a(t) the scale factor, k is the spatial curvature parameter and P the pressure; we have set β ψ = ∂g ∂ψ (v , 1) and
Cosmological solutions
. The smallness of the observed effects implies | β Φ v | ≪ 1, | β ψ v | ≪ 1, whereas the consistency of the model implies |ψ | ≪ H and |Φ | ≪ H (all confirmed by the numerical calculations below). Hence, the small excitations of the scalar fields do not modify significantly the variation of the scale factor with respect to the cosmic time. As this is suggested by observations, we assume zero spatial curvature. Let us emphasize that equation (14) implies that the extrema of Φ are necessary maxima during the matter or matter-Λ dominated era (present era). On account of equation (13), the same conclusion applies to ψ under the condition β ψ v > 0 (choosing β ψ v < 0 would lead to minima of ψ, instead).
Radiation era
Before the recombination, the content of the universe is well described by the equation of state P = 1 3 ρ c 2 (matter negligible, no spatial curvature): putting H = 1/2t, a = a(t 0 ) (t/t 0 ) 1/2 , ρ = ρ(t 0 ) (t 0 /t) 2 , we obtain
Radiation era
where t 0 is the present time. The solutions of equations (15) and (16) take the forms
Radiation era
and
Radiation era
where
Radiation era
and t d denotes the epoch of matter-radiation decoupling. Now, requiring that both ψ and Φ be bounded at any time past the big bang involves δψ(t d ) = 0 and δΦ(t d ) = 0. Hence, both scalar fields remain constant and equal to their respective VEV, during the radiation era. As a consequence, the effective fine structure constant identifies to the true fine structure constant during the radiation era.
A model without cosmological constant
After recombination, the matter and the cosmological constant play their role. We explore two different cosmological models: first, in this section, an Einstein -de Sitter model, with no cosmological constant and the critical density (no spatial curvature, no pressure). In the next section we explore a more realistic model with a cosmological constant. Putting H = 2/3t, a = a(t 0 ) (t/t 0 ) 2/3 , ρ = ρ(t 0 ) (t 0 /t) 2 , we obtain
A model without cosmological constant
The solutions of equations (19) and (20) take the forms
A model without cosmological constant
and
A model without cosmological constant
where t 0 denotes the present epoch in the cosmic time, and we have set δψ 0 = ψ(t 0 ) − v and δΦ 0 = Φ(t 0 ) − 1. On account of the results obtained previously at the epoch of matter-radiation decoupling, on gets in addition the following constraints
A model without cosmological constant
and
Model with cosmological constant

Present era
The present cosmological data seem to favor a model where Ω Λ = 2Ω m ≈ 2/3, that we explore now (still no spatial curvature and no pressure). We write the solution as
Present era
, and x = 3 2 √ λ H 0 t. Solving equations (13) and (14), we obtain
Present era
and
Present era
The parameters δψ Λ and δΦ Λ have been introduced in such a way that, formally, they reduce respectively to δψ 0 = ψ(t 0 ) − v and δΦ 0 = Φ(t 0 ) − 1 for Λ = 0. Further, because of the matching conditions at t d , relations (25) and (26) are respectively subject to the constraints
Present era
and
Present era
where we have set
Present era
Clearly, because of the constraints (23), (24), (27) and (28), only two parameters (the coupling constants β Φ v and β ψ v) are left free to fit the data.
The future universe (cosmological constant era)
In the future of the universe, the matter density and pressure become quite negligible with respect to the Λ term: P = ρ = 0, still no spatial curvature. Putting a ∝ exp (− Λ/3 ct) assuming Λ > 0, we obtain
The future universe (cosmological constant era)
The solutions of equations (30) and (31) are respectively of the same form as (17) and (18). Hence, each scalar field tends to a constant equal to its VEV, during the cosmological constant era. As a consequence, here again the effective fine structure constant will approach asymptotically the true fine structure constant.
Comparison with the observational data
We compare our prediction (2) of the time variation of α ef f , for the two different cosmological models, with the observational data. A quasar of redshift z = a(t 0 ) a(te) − 1 emits photons at t e , that we receive at t 0 on Earth. Defining α z = α ef f (t e ), α 0 = α ef f (t 0 ) and ∆α z = α z − α 0 , a least-squares fit to the observational data ( figure 1 of [1] ) gives,
Comparison with the observational data
• for λ = 0: β Φ v ≃ 8.014 10 −7 and | β ψ v | ≃ 0.0793, with χ 2 = 0.948 per degree of freedom (dof).
Comparison with the observational data
• for λ = 0.7: β Φ v ≃ 3.393 10 −6 and | β ψ v | ≃ 0.0317, with χ 2 = 0.779 per dof. Figure 1 shows, in the same plot, the observational results [1] and our theoretical prediction (2) of ∆α z /α 0 versus the redshift, for our two models λ = 0 and λ = 0.7 (assuming no spatial curvature).
Comparison with the observational data
The consistency of the recent observation from the distant quasars absorption line spectra with the constraints from the Oklo uranium deposit have been discussed in [1] . At the corresponding redshift, our best fits imply (assuming a spatially flat universe):
Comparison with the observational data
Including the Oklo point in the fit modify the χ 2 as indicated in the figure caption, and we conclude that our two best fits are consistent with the Oklo bounds. This gives an averaged decreasing rate approximately equal to − 10 −17 per year, consistent with the recent analyses [14] on account of the remark made in [1] and in [15] for the case of a non-linear time-evolution in ∆α z /α 0 . Note that this implies also a non trivial cosmic evolution for G ef f which yieldsĠ ef f /G ef f ≃ − 1.6 10 −17 per year at present, consistent with all the current bounds.
Discussion and conclusion
In the radiation dominated era, energy is present in the form of radiation only, so that the source terms for the scalar fields (equ. 15 and 16) cancel. Therefore, both ψ and Φ remain close to their respective VEV's. As a consequence, the effective fine structure constant α ef f remains practically constant and close to the true fine structure constant α. It follows both for BBN and at z = 1000 (the epoch of matter-radiation decoupling), that ∆α z /α 0 ≃ − 1.5 10 −7 , much below the present observational bound (model dependent) which gives | ∆α z /α 0 | < 10 −4 − 10 −2 (see [16] ).
Discussion and conclusion
At the onset of the matter-Λ dominated era, the scalar fields will continuously start to vary (Φ increases), though at a lesser extent than the density the ordinary matter. The Hubble friction term introduces a relaxation time τ of the order 1/3H. This provides a natural and sufficient way of driving back α ef f (≃ α/Φ 3 ) to its constant value, α, after a lapse of time of a few τ . As an estimate, let us consider the Einstein-de Sitter cosmology: after 2τ = 2/3H 0 = t 0 , one gets | α ef f − α | max ≃ e −2 | α ef f − α | t=t 0 . Since α ef f ≃ α during the radiation dominated era, it follows | ∆α ef f /α 0 | max ≃ 0.135 | ∆α ef f /α 0 | BBN . Hence, | ∆α ef f /α 0 | max < 1.35 10 −5 − 1.35 10 −3 as observed, on account of the bound on the variation of the fine structure ysampson@example.org. Variations of the effective weak and strong coupling constants are also expected in the higher dimensional theories candidates for unification. The properties of the fundamental interactions are connected to the topological properties of the compactified extradimensions. Such theories involve more than one extradimension in order to encompass all of the gauge groups of the standard model of particle physics. In this framework, the effective constants of the gauge fields would be expressed as functions of additional internal fields Φ 1 , ..., Φ n . The effective electromagnetic (fine structure constant), weak and strong coupling constants would be written, respectively, as α ef f = α F 1 (Φ 1 , ..., Φ n ), α wef f = α w F 2 (Φ 1 , ..., Φ n ) and α sef f = α s F 3 (Φ 1 , ..., Φ n ). The functions F 1 and F 2 are related to each other, because of the electroweak unification; and to F 3 , if an unification scheme is already present. Hence, we expect, at any given time scale (dropping the ef f indexes for clarity):α
Discussion and conclusion
| of the order unity, the three rates |α w αw |, |α s αs | and |α α | should be comparable, both at BBN and at the epoch of the Oklo phenomenon (see [17] ).
Discussion and conclusion
We conclude that our modified Kaluza-Klein type action provides a good effective description of interactions at low energy. The instability problem [10] is cured by the introduction of an additional external bulk scalar field minimally coupled to gravity. It accounts naturally for a cosmological time variation of α, in agreement with recent data. It also reconcile the discordant laboratory measurements of G, by interpreting their differences as due to a coupling with the dipolar magnetic field of the Earth [5] .
Discussion and conclusion
Compared to other explanations, our assumption appears as an economical extension of general relativity. Like in the Barrow-Sandvik-Magueijo (BSM) work [3] , the variation of the effective fine structure constant is related to the coupling of a scalar field to the Maxwell invariant F αβ F αβ . Likewise, the effective fine structure constant remains constant during the radiation era. However, in contrast to BSM who predicts the suppression of the changes in α ef f in the Λ dominated era, in our model the effective constants start to vary at the onset of the matter-cosmological constant dominated era. Further, BSM find that the product G ef f α ef f should approach an asymptotic constant, whereas in this paper it is the quotient G 3 ef f /α ef f that remains approximately constant during the cosmic evolution, since G ef f and α ef f vary respectively as Φ −1 and approximately Φ −3 . Above all, the velocity of light in vacuum remains constant, as well as the electric charges. Moreover, our model derives from a very simple geometrical hypothesis. The first scalar field Φ (the fifteen degree of freedom of the metric) is purely geometric, and thus part of gravity in 5D. It affects directly the fine structure constant and the gravitational constant. In contrary to ψ, it is not minimally coupled to gravity (in 4D). Moreover, any kind of matter (except ψ itself) acts as a source for the external scalar field ψ, and not only the EM field like in the BSM work.
Discussion and conclusion
The present model is limited and intends to be effective only. More precise predictions would result from a fundamental theory, in the same spirit.

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