5.4 Models with varying constants

The models that can be constructed are numerous and cannot all be reviewed here. Thus, we focus on the string dilaton model in Section 5.4.1 and then discuss the chameleon mechanism in Section 5.4.2 and the Bekenstein framework in Section 5.4.3.

5.4.1 String dilaton and Runaway dilaton models

Damour and Polyakov [135Jump To The Next Citation Point, 136Jump To The Next Citation Point] argued that the effective action for the massless modes taking into account the full string loop expansion should be of the form

∫ ∘ ---[ { [ ] } k S = d4x − ˆg M 2s Bg(Φ )ˆR + 4B Φ(Φ ) ˆ□ Φ − (∇ˆΦ )2 − BF (Φ)--ˆF 2 ] 4 ¯ˆ ˆ ˆ − Bψ(Φ )ψD ∕ψ + ... (194 )
in the string frame, Ms being the string mass scale. The functions Bi are not known but can be expanded (from the genus expansion of string theory) in the limit Φ → − ∞ as
−2Φ (i) (i)2Φ (i) 4Φ Bi(Φ ) = e + c0 + c1 e + c2 e + ... (195 )
where the first term is the tree level term. It follows that these functions can exhibit a local maximum. After a conformal transformation (−3∕4 1∕2ˆ gμν = CBg ˆgμν,ψ = (CBg ) B ψ ψ), the action in Einstein frame takes the form
∫ [ ] -d4x--√ --- 2 k- 2 ¯ S = 16πG − g R − 2(∇ϕ ) − 4BF (ϕ)F − ψD ∕ψ + ... (196 )
where the field ϕ is defined as
[ ( ) ] ∫ 3 B ′g 2 BΦ′ B ′Φ ϕ ≡ -- --- + 2 ---+ 2--- dΦ. 4 Bg BΦ Bg
It follows that the Yang–Mills coupling behaves as −2 gYM = kBF (ϕ). This also implies that the QCD mass scale is given by
ΛQCD ∼ Ms (CBg )−1∕2e−8π2kBF∕b (197 )
where b depends on the matter content. It follows that the mass of any hadron, proportional to ΛQCD in first approximation, depends on the dilaton, m (B ,B ,...) A g F.

If, as allowed by the ansatz (195View Equation), mA (ϕ) has a minimum ϕm then the scalar field will be driven toward this minimum during the cosmological evolution. However, if the various coupling functions have different minima then the minima of mA (ϕ) will depend on the particle A. To avoid violation of the equivalence principle at an unacceptable level, it is thus necessary to assume that all the minima coincide in ϕ = ϕm, which can be implemented by setting Bi = B. This can be realized by assuming that ϕm is a special point in field space, for instance it could be associated to the fixed point of a Z2 symmetry of the T- or S-duality [129].

Expanding lnB around its maximum ϕm as lnB ∝ − κ(ϕ − ϕm )2∕2, Damour and Polyakov [135Jump To The Next Citation Point, 136Jump To The Next Citation Point] constrained the set of parameters (κ,ϕ0 − ϕm ) using the different observational bounds. This toy model allows one to address the unsolved problem of the dilaton stabilization, to study all the experimental bounds together and to relate them in a quantitative manner (e.g., by deriving a link between equivalence-principle violations and time-variation of αEM). This model was compared to astrophysical data in [306] to conclude that |Δϕ | < 3.4κ10 −6.

An important feature of this model lies in the fact that at lowest order the masses of all nuclei are proportional to ΛQCD so that at this level of approximation, the coupling is universal and the theory reduces to a scalar-tensor theory and there will be no violation of the universality of free fall. It follows that the deviation from general relativity are characterized by the PPN parameters

PPN 2 2 2 2 PPN 1-2 dfA- 1- 3 3 2 γ − 1 ≃ − 2fA = − 2 βsκ Δ ϕ0, β − 1 ≃ 2fA dϕ = 2βsκ Δϕ 0
[ ] ∂ ln ΛQCD (ϕ) Ms 1 dlnB dln B fA = ----∂-ϕ------= − ln m---+ 2- -dϕ---≡ − βs-dϕ---= βsκΔ ϕ0 (198 ) A
with Δ ϕ0 = ϕ0 − ϕm and βs ∼ 40 [135Jump To The Next Citation Point]. The variation of the gravitational constant is, from Equation (167View Equation), simply
[ ] G˙ Ms 1 dlnB -- = 2fA ˙ϕ0 = − 2 ln ----+ -- -----ϕ˙0. G mA 2 dϕ
The value of ˙ϕ = H ϕ′ 0 0 0 is obtained from the Klein–Gordon equation (168View Equation) and is typically given by ′ ϕ 0 = − Z βsκH0 Δ ϕ0 were Z is a number that depends on the equation of state of the fluid dominating the matter content of the universe in the last e-fold and the cosmological parameters so that
| G˙|| ˙ 2 2 2 G || = 2fAϕ0 = − 2ZH0 βsκ Δϕ 0. (199 ) 0
The factor Z is model-dependent and another way to estimate ϕ˙0 is to use the Friedmann equations, which imply that ∘ ---------3---- ˙ϕ0 = H0 1 + q0 − 3Ωm0 where q is the deceleration parameter.

When one considers the effect of the quark masses and binding energies, various composition-dependent effects appear. First, the fine-structure constant scales as −1 αEM ≃ B so that

|| α˙| = κ Δ ϕ0 ˙ϕ0 = − ZH0 βsκ2 Δ ϕ2. (200 ) α |0 0
The second effect is, as pointed out earlier, a violation of the universality of free fall. In full generality, we expect that
[ ] ∑ m mA (ϕ ) = N ΛQCD (ϕ) 1 + 𝜖qA ---q--+ 𝜖EAMαEM . (201 ) q ΛQCD
Using an expansion of the form (17View Equation), it was concluded that
[ ( ) ( ) ( ) ] 2 2 B D E ηAB = κ (ϕ0 − ϕm) CB Δ --- + CD Δ --- + CE Δ --- (202 ) M M M
with B = N + Z, D = N − Z and E = Z(Z − 1)∕(N + Z)1∕3 and where the value of the parameters Ci are model-dependent.

It follows from this model that:

This model was extended [133] to the case where the coupling functions have a smooth finite limit for infinite value of the bare string coupling, so that −ϕ Bi = Ci + 𝒪 (e), folling [229]. The dilaton runs away toward its attractor at infinity during a stage of inflation. The late time dynamics of the scalar field is similar as in quintessence models, so that the model can also explain the late time acceleration of the cosmic expansion. The amplitude of residual dilaton interaction is related to the amplitude of the primordial density fluctuations and it induces a variation of the fundamental constants, provided it couples to dark matter or dark energy. It is concluded that, in this framework, the largest allowed variation of αEM is of order 2 × 10–6, which is reached for a violation of the universality of free fall of order 10–12 and it was established that

∘ --------------- α˙ || 3 ∘ ------ -EM-|| ∼ ±10 −16 1 + q0 − -Ωm0 1012η yr−1, (203 ) αEM 0 2
where the first square-root arises from the computation of ˙ϕ0. The formalism was also used to discuss the time variation of αEM and μ [97].

The coupling of the dilaton to the standard model fields was further investigated in [122Jump To The Next Citation Point, 121Jump To The Next Citation Point]. Assuming that the heavy quarks and weak gauge bosons have been integrated out and that the dilaton theory has been matched to the light fields below the scale of the heavy quarks, the coupling of the dilaton has been parameterized by 5 parameters: de and dg for the couplings to the electromagnetic and gluonic field-strength terms, and dme, dmu and dmd for the couplings to the fermionic mass terms so that the interaction Lagrangian is reduces to a linear coupling (e.g., ∝ deϕF 2 for the coupling to electromagnetism etc.) It follows that Δ αEM ∕αEM = deκ ϕ for the fine structure constant, Δ ΛQCD ∕ ΛQCD = ddκϕ for the strong sector and Δmi ∕mi = dmi κϕ for the masses of the fermions. These parameters can be constrained by the test of the equivalence principle in the solar system [see Section 6.3].

In these two string-inspired scenarios, the amplitude of the variation of the constants is related to the one of the density fluctuations during inflation and the cosmological evolution.

5.4.2 The Chameleon mechanism

A central property of the least coupling principle, that is at the heart of the former models, is that all coupling functions have the same minimum so that the effective potential entering the Klein–Gordon equation for the dilaton has a well-defined minimum.

It was realized [287] that if the dilaton has a coupling 2 A (ϕ) to matter while evolving in a potential V (ϕ) the source of the Klein–Gordon equation (168View Equation) has a an effective potential

Veff = V (ϕ) + A2(ϕ )ρ.
In the case where V is a decreasing function of ϕ, e.g., a runaway potential, and the coupling is an increasing function, e.g., A2 = exp β ϕ∕MP, the effective potential has a minimum the value of which depends on the matter local density ρ (see also [186]). Thus, the field is expected to be massive on Earth where the density is high and light in space in the solar system. It follows that the experiment on the universality of free fall in space may detect violations of the universality of free fall larger than the bounds derived by laboratory experiments [288, 498]. It follows (1) that the constraints on the time variation of the constants today can be relaxed if such a mechanism is at work and (2) that is the constants depend on the local value of the chameleon field, their value will be environment dependent and will be different on Earth and in space.

The cosmological variation of αEM in such model was investigated in [70, 71]. Models based on the Lagrangian (209View Equation) and exhibiting the chameleon mechanism were investigated in [398Jump To The Next Citation Point].

The possible shift in the value of μ in the Milky Way (see Section 6.1.3) was related [323Jump To The Next Citation Point, 324Jump To The Next Citation Point, 322Jump To The Next Citation Point] to the model of [398] to conclude that such a shift was compatible with this model.

5.4.3 Bekenstein and related models

Bekenstein [39Jump To The Next Citation Point, 40Jump To The Next Citation Point] introduced a theoretical framework in which only the electromagnetic sector was modified by the introduction of a dimensionless scalar field 𝜖 so that all electric charges vary in unison α ei = e0i𝜖(x) so that only αEM is assumed to possibly vary.

To avoid the arbitrariness in the definition of 𝜖, which can be rescaled by a constant factor while e0i is inversely rescales, it was postulated that the dynamics of 𝜖 be invariant under global rescaling so that its action should be of the form

∫ μν S𝜖 = − ℏc- g--∂-μ𝜖∂ν𝜖√ −-gd4x, (204 ) 2l2 𝜖2
l being a constant length scale. Then, 𝜖 is assumed to enter all electromagnetic interaction via eiA μ → e0i𝜖Aμ where Aμ is the usual electromagnetic potential and the gauge invariance is then preserved only if 𝜖A μ → 𝜖A μ + λ,μ for any scalar function λ. It follows that the the action for the electromagnetic sector is the standard Maxwell action
∫ -1-- μν √ --- 4 S𝜖 = − 16π F Fμν − gd x, (205 )
for the generalized Faraday tensor
F μν = 1-[(𝜖A ν),μ − (𝜖A μ),ν.] (206 ) 𝜖
To finish the gravitational sector is assumed to be described by the standard Einstein–Hilbert action. Finally, the matter action for point particles of mass m takes the form S = ∑ ∫ [− mc2 + (e∕c)uμA ]γ −1δ3(xi − xi(τ))d4x m μ where γ is the Lorentz factor and τ the proper time. Note that the Maxwell equation becomes
( ) ∇ μ 𝜖−1F μν = 4πjν, (207 )
which is the same as for electromagnetism in a material medium with dielectric constant −2 𝜖 and permeability 𝜖2 (this was the original description proposed by Fierz [195] and Lichnerowicz [332]; see also Dicke [152]).

It was proposed [445] to rewrite this theory by introducing the two fields

aμ ≡ 𝜖Aμ, ψ ≡ ln 𝜖
so that the theory takes the form
∫ ∫ ∫ c3 √ --- 4 1 − 2ψ μν √ --- 4 1 2√ --- 4 S = 16-πg R − gd x − 16-π e f fμν − gd x − 8π-κ2 (∂μψ) − gd x (208 )
with κ = l∕(4π ℏc) and fμν the Faraday tensor associated with aμ. The model was further extended to include a potential for ψ [32] and to include the electroweak theory [461Jump To The Next Citation Point].

As discussed previously, this class of models predict a violation of the universality of free fall and, from Equation (14View Equation), it is expected that the anomalous acceleration is given by −1 δa = − M (∂EEM ∕∂𝜖)∇ 𝜖.

From the confrontation of the local and cosmological constraints on the variation of 𝜖 Bekenstein [39Jump To The Next Citation Point] concluded, given his assumptions on the couplings, that αEMis a parameter, not a dynamical variable” (see, however, [40] and then [301]). This problem was recently bypassed by Olive and Pospelov [397] who generalized the model to allow additional coupling of a scalar field − 2 𝜖 = BF (ϕ) to non-baryonic dark matter (as first proposed in [126]) and cosmological constant, arguing that in supersymmetric dark matter, it is natural to expect that ϕ would couple more strongly to dark matter than to baryon. For instance, supersymmetrizing Bekenstein model, ϕ will get a coupling to the kinetic term of the gaugino of the form −1 M ∗ ϕχ¯∂χ. Assuming that the gaugino is a large fraction of the stable lightest supersymmetric particle, the coupling to dark matter would then be of order 103 − 104 times larger. Such a factor could almost reconcile the constraint arising from the test of the universality of free fall with the order of magnitude of the cosmological variation. This generalization of the Bekenstein model relies on an action of the form

∫ ∫ [ ] S = 1M 2 R√ −-gd4x − 1M 2∂ μϕ∂μϕ + 1BF (ϕ)F μνFμν √ −-gd4x (209 ) 2 4 2 ∗ 4 ∫ {∑ 1 1 } √ --- − N¯i[iD ∕ − miBNi (ϕ)]Ni + -χ¯∂χ + M 42B Λ(ϕ )Λ + -M χBχ(ϕ )χT χ − gd4x 2 2
where the sum is over proton [D ∕ = γ μ(∂μ − ie0A μ)] and neutron [D ∕ = γμ∂μ]. The functions B can be expanded (since one focuses on small variations of the fine-structure constant and thus of ϕ) as 2 BX = 1 + ζX ϕ + ξXϕ ∕2. It follows that 2 αEM (ϕ) = e0∕4πBF (ϕ) so that Δ αEM ∕αEM = ζF ϕ + (ξF − 2ζ2F)ϕ2∕2. This framework extends the analysis of [39Jump To The Next Citation Point] to a 4-dimensional parameter space (M ∗,ζF,ζm, ζΛ). It contains the Bekenstein model (ζF = − 2, ζΛ = 0, ζm ∼ 10−4ξF), a Jordan–Brans–Dicke model (ζF = 0, ∘ --------- ζΛ = − 2 2∕2ω + 3, ------- ζm = − 1∕√ 4ω + 6), a string-like model (√ -- ζF = − 2, √ -- ζΛ = 2, √-- ζm = 2 ∕2) so that Δ αEM ∕αEM = 3) and a supersymmetrized Bekenstein model (ζF = − 2, ζχ = − 2, ζm = ζχ so that Δ αEM ∕αEM ∼ 5∕ω). In all the models, the universality of free fall sets a strong constraint on √ -- ζF ∕ ω (with ω ≡ M ∗∕2M 24) and the authors showed that a small set of models may be compatible with a variation of αEM from quasar data while being compatiblewith the equivalence principle tests. A similar analysis [347Jump To The Next Citation Point] concluded that such models can reproduce the variation of αEM from quasars while being compatible with Oklo and meteorite data. Note that under this form, the effective theory is very similar to the one detailed in Section 5.4.2.

This theory was also used [41] to study the spacetime structure around charged black-hole, which corresponds to an extension of dilatonic charged black hole. It was concluded that a cosmological growth of αEM would decrease the black-hole entropy but with half the rate expected from the earlier analysis [139, 339].

5.4.4 Other ideas

Let us mention without details other theoretical models, which can accommodate varying constants:

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