The tests of the constancy of fundamental constants take all their importance in the realm of the
tests of the equivalence principle [540]. Einstein general relativity is based on two independent
hypotheses, which can conveniently be described by decomposing the action of the theory as
.
The equivalence principle has strong implication for the functional form of . This principle
includes three hypotheses:
In its weak form (that is for all interactions but gravity), it is satisfied by any metric theory of gravity
and general relativity is conjectured to satisfy it in its strong form (that is for all interactions including
gravity). We refer to [540] for a detailed description of these principles. The weak equivalence principle can
be mathematically implemented by assuming that all matter fields are minimally coupled to a single metric
tensor
. This metric defines the length and times measured by laboratory clocks and rods so that it
can be called the physical metric. This implies that the action for any matter field,
say, can be written
as
As an example, the action of a point-particle reads
with Note that in the Newtonian limit where
is the Newtonian potential. It
follows that, in the slow velocity limit, the geodesic equation reduces to
The assumption of a metric coupling is actually well tested in the solar system:
Large improvements are expected thanks to existence of two dedicated space mission projects: Microscope [493] and STEP [355].
We can conclude that the hypothesis of metric coupling is extremely well-tested in the solar system.
The second building block of general relativity is related to the dynamics of the gravitational sector, assumed to be dictated by the Einstein–Hilbert action
This defines the dynamics of a massless spin-2 fieldThe variation of the total action with respect to the metric yields the Einstein equations
where The dynamics of general relativity can be tested in the solar system by using the parameterized
post-Newtonian formalism (PPN). Its is a general formalism that introduces 10 phenomenological
parameters to describe any possible deviation from general relativity at the first post-Newtonian
order [540, 541
] (see also [60] for a review on higher orders). The formalism assumes that gravity is
described by a metric and that it does not involve any characteristic scale. In its simplest form, it reduces to
the two Eddington parameters entering the metric of the Schwartzschild metric in isotropic
coordinates
These two phenomenological parameters are constrained (1) by the shift of the Mercury perihelion [457],
which implies that
, (2) the Lunar laser ranging experiments [543
], which
implies that
and (3) by the deflection of electromagnetic signals,
which are all controlled by
. For instance the very long baseline interferometry [459] implies that
, while the measurement of the time delay variation to the Cassini spacecraft [53]
sets
.
The PPN formalism does not allow to test finite range effects that could be caused, e.g., by a massive degree of freedom. In that case one expects a Yukawa-type deviation from the Newton potential,
General relativity is also tested with pulsars [125, 189
] and in the strong field regime [425]. For more
details we refer to [129
, 495, 540
, 541]. Needless to say that any extension of general relativity
has to pass these constraints. However, deviations from general relativity can be larger in the
past, as we shall see, which makes cosmology an interesting physical system to extend these
constraints.
As the previous description shows, the constancy of the fundamental constants and the universality are two
pillars of the equivalence principle. Dicke [152] realized that they are actually not independent and that if
the coupling constants are spatially dependent then this will induce a violation of the universality of free
fall.
The connection lies in the fact that the mass of any composite body, starting, e.g., from nuclei, includes
the mass of the elementary particles that constitute it (this means that it will depend on the Yukawa
couplings and on the Higgs sector parameters) but also a contribution, , arising from the
binding energies of the different interactions (i.e., strong, weak and electromagnetic) but also gravitational
for massive bodies. Thus, the mass of any body is a complicated function of all the constants,
.
It follows that the action for a point particle is no more given by Equation (2) but by
This anomalous acceleration is generated by the change in the (electromagnetic, gravitational, …)
binding energies [152, 246, 386
] but also in the Yukawa couplings and in the Higgs sector parameters so
that the
-dependencies are a priori composition-dependent. As a consequence, any variation of the
fundamental constants will entail a violation of the universality of free fall: the total mass of the body being
space dependent, an anomalous force appears if energy is to be conserved. The variation of the constants,
deviation from general relativity and violation of the weak equivalence principle are in general expected
together.
On the other hand, the composition dependence of and thus of
can be used to optimize the
choice of materials for the experiments testing the equivalence principle [118
, 120
, 122
] but also to
distinguish between several models if data from the universality of free fall and atomic clocks are
combined [143
].
From a theoretical point of view, the computation of will requires the determination of the
coefficients
. This can be achieved in two steps by first relating the new degrees of freedom of the
theory to the variation of the fundamental constants and then relating them to the variation
of the masses. As we shall see in Section 5, the first issue is very model dependent while the
second is especially difficult, particularly when one wants to understand the effect of the quark
mass, since it is related to the intricate structure of QCD and its role in low energy nuclear
reactions.
As an example, the mass of a nuclei of charge and atomic number
can be expressed
as
For macroscopic bodies, the mass has also a negative contribution
from the gravitational binding energy. As a conclusion, from (17Note that varying coupling constants can also be associated with violations of local Lorentz invariance and CPT symmetry [298, 52, 242].
Most constraints on the time variation of the fundamental constants will not be local and related to physical systems at various epochs of the evolution of the universe. It follows that the comparison of different constraints requires a full cosmological model.
Our current cosmological model is known as the CDM (see [409
] for a detailed description, and
Table 4 for the typical value of the cosmological parameters). It is important to recall that its construction
relies on 4 main hypotheses: (H1) a theory of gravity; (H2) a description of the matter components
contained in the Universe and their non-gravitational interactions; (H3) symmetry hypothesis; and (H4) a
hypothesis on the global structure, i.e., the topology, of the Universe. These hypotheses are not on the same
footing since H1 and H2 refer to the physical theories. However, these hypotheses are not sufficient to solve
the field equations and we must make an assumption on the symmetries (H3) of the solutions
describing our Universe on large scales while H4 is an assumption on some global properties of these
cosmological solutions, with same local geometry. But the last two hypothesis are unavoidable
because the knowledge of the fundamental theories is not sufficient to construct a cosmological
model [504
].
The CDM model assumes that gravity is described by general relativity (H1), that the Universe
contains the fields of the standard model of particle physics plus some dark matter and a cosmological
constant, the latter two having no physical explanation at the moment. It also deeply involves the
Copernican principle as a symmetry hypothesis (H3), without which the Einstein equations usually cannot
been solved, and assumes most often that the spatial sections are simply connected (H4). H2 and H3 imply
that the description of the standard matter reduces to a mixture of a pressureless and a radiation perfect
fluids. This model is compatible with all astronomical data, which roughly indicates that
,
and
. Thus, cosmology roughly imposes that
, that is
.
Hence, the analysis of the cosmological dynamics of the universe and of its large scale structures requires the introduction of a new constant, the cosmological constant, associated with a recent acceleration of the cosmic expansion, that can be introduced by modifying the Einstein–Hilbert action to
Classically, this value is no problem but it was pointed out that at the quantum level, the vacuum
energy should scale as , where
is some energy scale of high-energy physics. In such a case, there
is a discrepancy of 60 – 120 order of magnitude between the cosmological conclusions and the theoretical
expectation. This is the cosmological constant problem [528
].
Parameter |
Symbol | Value |
Reduced Hubble constant |
![]() |
0.73(3) |
Baryon-to-photon ratio |
![]() |
6.12(19) × 10–10 |
Photon density |
![]() |
2.471 × 10–5 |
Dark matter density |
![]() |
0.105(8) |
Cosmological constant |
![]() |
0.73(3) |
Spatial curvature |
![]() |
0.011(12) |
Scalar modes amplitude |
![]() |
(2.0 ± 0.2) × 10–5 |
Scalar spectral index |
![]() |
0.958(16) |
Neutrino density |
![]() |
(0.0005 – 0.023) |
Dark energy equation of state |
![]() |
–0.97(7) |
Scalar running spectral index |
![]() |
–0.05 ± 0.03 |
Tensor-to-scalar ratio |
T/S | < 0.36 |
Tensor spectral index |
![]() |
< 0.001 |
Tensor running spectral index |
![]() |
? |
Baryon density |
![]() |
0.0223(7) |
Two approaches to solve this problem have been considered. Either one accepts such a constant and
such a fine-tuning and tries to explain it on anthropic ground. Or, in the same spirit as Dirac,
one interprets it as an indication that our set of cosmological hypotheses have to be extended,
by either abandoning the Copernican principle [508] or by modifying the local physical laws
(either gravity or the matter sector). The way to introduce such new physical degrees of freedom
were classified in [502]. In that latter approach, the tests of the constancy of the fundamental
constants are central, since they can reveal the coupling of this new degree of freedom to the
standard matter fields. Note, however, that the cosmological data still favor a pure cosmological
constant.
Among all the proposals quintessence involves a scalar field rolling down a runaway potential hence
acting as a fluid with an effective equation of state in the range if the field is
minimally coupled. It was proposed that the quintessence field is also the dilaton [229
, 434, 499
].
The same scalar field then drives the time variation of the cosmological constant and of the
gravitational constant and it has the property to also have tracking solutions [499]. Such models do
not solve the cosmological constant problem but only relieve the coincidence problem. One of
the underlying motivation to replace the cosmological constant by a scalar field comes from
superstring models in which any dimensionful parameter is expressed in terms of the string mass
scale and the vacuum expectation value of a scalar field. However, the requirement of slow roll
(mandatory to have a negative pressure) and the fact that the quintessence field dominates
today imply, if the minimum of the potential is zero, that it is very light, roughly of order
[81
].
Such a light field can lead to observable violations of the universality of free fall if it is non-universally
coupled to the matter fields. Carroll [81] considered the effect of the coupling of this very light quintessence
field to ordinary matter via a coupling to the electromagnetic field as . Chiba and Kohri [96
] also
argued that an ultra-light quintessence field induces a time variation of the coupling constant if it is
coupled to ordinary matter and studied a coupling of the form
, as, e.g., expected from
Kaluza–Klein theories (see below). This was generalized to quintessence models with a couplings of the
form
[11
, 112
, 162
, 315
, 314
, 347
, 404
, 531
] and then to models of runaway
dilaton [133
, 132] inspired by string theory (see Section 5.4.1). The evolution of the scalar field drives both
the acceleration of the universe at late time and the variation of the constants. As pointed
in [96
, 166
, 532
] such models are extremely constrained from the bound on the universality of free-fall (see
Section 6.3).
We have two means of investigation:
In conclusion, cosmology seems to require a new constant. It also provides a link between the microphysics and cosmology, as foreseen by Dirac. The tests of fundamental constants can discriminate between various explanations of the acceleration of the universe. When a model is specified, cosmology also allows to set stringer constraints since it relates observables that cannot be compared otherwise.
http://www.livingreviews.org/lrr-2011-2 |
Living Rev. Relativity 14, (2011), 2
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