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
If, 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
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 ), 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
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
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:
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Living Rev. Relativity 14, (2011), 2
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