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

in the string frame, being the string mass scale. The functions are not known but can be expanded (from the genus expansion of string theory) in the limit as where the first term is the tree level term. It follows that these functions can exhibit a local maximum. After a conformal transformation (), the action in Einstein frame takes the form where the field is defined asIf, as allowed by the ansatz (195), has a minimum 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 will depend on the particle . To avoid violation of the equivalence principle at an unacceptable level, it is thus necessary to assume that all the minima coincide in , which can be implemented by setting . This can be realized by assuming that is a special point in field space, for instance it could be associated to the fixed point of a symmetry of the - or -duality [129].

Expanding around its maximum as , Damour and Polyakov [135, 136] constrained the set of parameters 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 ). This model was compared to astrophysical data in [306] to conclude that .

An important feature of this model lies in the fact that at lowest order the masses of all nuclei are proportional to 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

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

The second effect is, as pointed out earlier, a violation of the universality of free fall. In full generality, we expect that Using an expansion of the form (17), it was concluded that with , and and where the value of the parameters are model-dependent.It follows from this model that:

- The PPN parameters, the time variation of and today and the violation of the university of free-fall all scale as .
- The field is driven toward during the cosmological evolution, a point at which the scalar field decouples from the matter field. The mechanism is usually called the least coupling principle.
- Once the dynamics for the scalar field is solved, can be related to at the end of inflation. Interestingly, this quantity can be expressed in terms of amplitude of the density contrast at the end of inflation, that is to the energy scale of inflation.
- The numerical estimations [135] indicate that showing that in such a class of models, the constraint on implies , which is a better constraint that the one obtained directly.

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 ), 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 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

The coupling of the dilaton to the standard model fields was further investigated in [122, 121]. 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: and for the couplings to the electromagnetic and gluonic field-strength terms, and , and for the couplings to the fermionic mass terms so that the interaction Lagrangian is reduces to a linear coupling (e.g., for the coupling to electromagnetism etc.) It follows that for the fine structure constant, for the strong sector and 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.

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 to matter while evolving in a potential the source of the Klein–Gordon equation (168) has a an effective potential

The cosmological variation of in such model was investigated in [70, 71]. Models based on the Lagrangian (209) and exhibiting the chameleon mechanism were investigated in [398].

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

Bekenstein [39, 40] introduced a theoretical framework in which only the electromagnetic sector was modified by the introduction of a dimensionless scalar field so that all electric charges vary in unison ) so that only is assumed to possibly vary.

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

being a constant length scale. Then, is assumed to enter all electromagnetic interaction via where is the usual electromagnetic potential and the gauge invariance is then preserved only if for any scalar function . It follows that the the action for the electromagnetic sector is the standard Maxwell action for the generalized Faraday tensor 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 takes the form where is the Lorentz factor and the proper time. Note that the Maxwell equation becomes which is the same as for electromagnetism in a material medium with dielectric constant and permeability (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

As discussed previously, this class of models predict a violation of the universality of free fall and, from Equation (14), it is expected that the anomalous acceleration is given by .

From the confrontation of the local and cosmological constraints on the variation of Bekenstein [39] concluded, given his assumptions on the couplings, that “is 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 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 would couple more strongly to dark matter than to baryon. For instance, supersymmetrizing Bekenstein model, will get a coupling to the kinetic term of the gaugino of the form . Assuming that the gaugino is a large fraction of the stable lightest supersymmetric particle, the coupling to dark matter would then be of order 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

where the sum is over proton [] and neutron []. The functions can be expanded (since one focuses on small variations of the fine-structure constant and thus of ) as . It follows that so that . This framework extends the analysis of [39] to a 4-dimensional parameter space (). It contains the Bekenstein model (, , ), a Jordan–Brans–Dicke model (, , ), a string-like model (, , ) so that ) and a supersymmetrized Bekenstein model (, , so that ). In all the models, the universality of free fall sets a strong constraint on (with ) and the authors showed that a small set of models may be compatible with a variation of from quasar data while being compatiblewith the equivalence principle tests. A similar analysis [347] concluded that such models can reproduce the variation of 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 would decrease the black-hole entropy but with half the rate expected from the earlier analysis [139, 339].

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

- Models involving a late time phase transition in the electromagnetic sector [87, 10];
- Braneworld models [336, 8, 73, 331, 403] or extra-dimensions [477];
- Model with pseudo-scalar couplings [203];
- Growing neutrino models [9, 533] in which the neutrino masses are a function of a scalar field, that is also responsible for the late time acceleration of the universe. In these models the neutrinos freeze the evolution of the scalar field when they become non-relativistic while its evolution is similar as in quintessence when the neutrinos are ultra-relativistic;
- Models based on discrete quantum gravity [223] or on loop quantum gravity in which the Barbero–Immirzi parameter controls the minimum eigenvalue of the area operator and could be promoted to a field, leading to a classical coupling of Einstein’s gravity with a scalar-field stress-energy tensor [354, 483]
- “varying speed of light” models for which we refer to the review [341] and our previous analysis [183] for a critical view;
- Quintessence models with a non-minimal coupling of the quintessence field [20, 11, 96, 112, 162, 217, 315, 314, 389, 347, 404, 531] [see discussion Section 2.2.3];
- Mass varying neutrinos [9, 533];
- Holographic dark energy models with non-minimal couplings [235]

Living Rev. Relativity 14, (2011), 2
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