### 2.2 The constancy of constants as a test of general relativity

The previous paragraphs have yet emphasize why testing for the consistency of the constants is a test of fundamental physics since it can reveal the need for new physical degrees of freedom in our theory. We now want to stress the relation of this test with other tests of general relativity and with cosmology.

#### 2.2.1 General relativity

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:

• the universality of free fall,
• the local position invariance,
• the local Lorentz invariance.

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

This metric coupling ensures in particular the validity of the universality of free-fall. Since locally, in the neighborhood of the worldline, there always exists a change of coordinates so that the metric takes a Minkowskian form at lowest order, the gravitational field can be locally “effaced” (up to tidal effects). If we identify this neighborhood to a small lab, this means that any physical properties that can be measured in this lab must be independent of where and when the experiments are carried out. This is indeed the assumption of local position invariance, which implies that the constants must take the same value independent of the spacetime point where they are measured. Thus, testing the constancy of fundamental constants is a direct test of this principle and therefore of the metric coupling. Interestingly, the tests we are discussing in this review allow one to extend them much further than the solar scales and even in the early universe, an important information to check the validity of relativity in cosmology.

As an example, the action of a point-particle reads

with . The equation of motion that one derives from this action is the usual geodesic equation
where , being the proper time; is the covariant derivative associated with the metric and is the 4-acceleration. Any metric theory of gravity will enjoy such a matter Lagrangian and the worldline of any test particle shall be a geodesic of the spacetime with metric , as long as there is no other long range force acting on it (see [190] for a detailed review of motion in alternative theories of gravity).

Note that in the Newtonian limit where is the Newtonian potential. It follows that, in the slow velocity limit, the geodesic equation reduces to

hence defining the Newtonian acceleration . Recall that the proper time of a clock is related to the coordinate time by . Thus, if one exchanges electromagnetic signals between two identical clocks in a stationary situation, the apparent difference between the two clocks rates will be
at lowest order. This is the universality of gravitational redshift.

The assumption of a metric coupling is actually well tested in the solar system:

• First, it implies that all non-gravitational constants are spacetime independent, which have been tested to a very high accuracy in many physical systems and for various fundamental constants; this is the subject of this review.
• Second, the isotropy has been tested from the constraint on the possible quadrupolar shift of nuclear energy levels [99, 304, 422] proving that different matter fields couple to a unique metric tensor at the 10–27 level.
• Third, the universality of free fall can be tested by comparing the accelerations of two test bodies in an external gravitational field. The parameter defined as
can be constrained experimentally, e.g., in the laboratory by comparing the acceleration of a beryllium and a copper mass in the Earth gravitational field [4] to get
Similarly the comparison of Earth-core-like and Moon-mantle-like bodies gave [23]
and experiments with torsion balance using test bodies composed of tellurium an bismuth allowed to set the constraint [450]
The Lunar Laser ranging experiment [543], which compares the relative acceleration of the Earth and Moon in the gravitational field of the Sun, also set the constraints
Note that since the core represents only 1/3 of the mass of the Earth, and since the Earth’s mantle has the same composition as that of the Moon (and thus shall fall in the same way), one loses a factor of three, so that this constraint is actually similar to the one obtained in the lab. Further constraints are summarized in Table 3. The latter constraint also contains some contribution from the gravitational binding energy and thus includes the strong equivalence principle. When the laboratory result of [23] is combined with the LLR results of [542] and [365], one gets a constraints on the strong equivalence principle parameter, respectively

Large improvements are expected thanks to existence of two dedicated space mission projects: Microscope [493] and STEP [355].

Table 3: Summary of the constraints on the violation of the universality of free fall.
 Constraint Body 1 Body 2 Ref. (–1.9 ± 2.5) × 10–12 Be Cu [4] (0.1 ± 2.7 ± 1.7) × 10–13 Earth-like rock Moon-like rock [23] (–1.0 ± 1.4) × 10–13 Earth Moon [543] (0.3 ± 1.8) × 10–13 Te Bi [450] (–0.2 ± 2.8) × 10–12 Be Al [481] (–1.9 ± 2.5) × 10–12 Be Cu [481] (5.1 ± 6.7) × 10–12 Si/Al Cu [481]

• Fourth, the Einstein effect (or gravitational redshift) has been measured at the 2 × 10–4 level [517].

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 field , called the Einstein metric. General relativity then assumes that both metrics coincide, (which is related to the strong equivalence principle), but it is possible to design theories in which this is indeed not the case (see the example of scalar-tensor theories below; Section 5.1.1) so that general relativity is one out of a large family of metric theories.

The variation of the total action with respect to the metric yields the Einstein equations

where is the matter stress-energy tensor. The coefficient is determined by the weak-field limit of the theory that should reproduce the Newtonian predictions.

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

Indeed, general relativity predicts .

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,

that can be probed by “fifth force” experimental searches. characterizes the range of the Yukawa deviation of strength . The constraints on are summarized in [256], which typically shows that on scales ranging from the millimeter to the solar system size.

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.

#### 2.2.2 Varying constants and the universality of free fall

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

where is a list of constants including but also many others and where the index in recalls that the dependency in these constants is a priori different for bodies of different chemical composition. The variation of this action gives the equation of motion
It follows that a test body will not enjoy a geodesic motion. In the Newtonian limit , and at first order in , the equation of motion of a test particle reduces to
so that in the slow velocity (and slow variation) limit it reduces to
where we have introduce the sensitivity of the mass with respect to the variation of the constant by
This simple argument shows that if the constants depend on time then there must exist an anomalous acceleration that will depend on the chemical composition of the body .

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

where and are respectively the strong and electromagnetic contributions to the binding energy. The Bethe–Weizäcker formula allows to estimate the latter as
If we decompose the proton and neutron masses as [230] where is the pure QCD approximation of the nucleon mass (, and being pure numbers), it reduces to
with , the neutron number. For an atom, one would have to add the contribution of the electrons, . This form depends on strong, weak and electromagnetic quantities. The numerical coefficients are given explicitly by [230]
Such estimations were used in the first analysis of the relation between variation of the constant and the universality of free fall [135, 166] but the dependency on the quark mass is still not well understood and we refer to [120, 122, 157, 159, 208] for some attempts to refine this description.

For macroscopic bodies, the mass has also a negative contribution

from the gravitational binding energy. As a conclusion, from (17) and (19), we expect the mass to depend on all the coupling constant, . We shall discuss this issue in more detail in Section 5.3.2.

Note that varying coupling constants can also be associated with violations of local Lorentz invariance and CPT symmetry [298, 52, 242].

#### 2.2.3 Relations with cosmology

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

This constant can equivalently be introduced in the matter action. Note, however, that it is disproportionately small compared to the natural scale fixed by the Planck length

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].

Table 4: Main cosmological parameters in the standard -CDM model. There are 7 main parameters (because ) to which one can add 6 more to include dark energy, neutrinos and gravity waves. Note that often the spatial curvature is set to . (See, e.g. Refs. [296, 409]).
 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:

• The field driving the time variation of the fundamental constants does not explain the acceleration of the universe (either it does not dominate the matter content today or its equation of state is not negative enough). In such a case, the variation of the constants is disconnected from the dark energy problem. Cosmology allows to determine the dynamics of this field during the whole history of the universe and thus to compare local constraints and cosmological constraints. An example is given by scalar-tensor theories (see Section 5.1.1) for which one can compare, e.g., primordial nucleosynthesis to local constraints [134]. However, in such a situation one should take into account the effect of the variation of the constants on the astrophysical observations since it can affect local physical processes and bias, e.g., the luminosity of supernovae and indirectly modify the distance luminosity-redshift relation derived from these observations [33, 435].
• The field driving the time variation of the fundamental constants is also responsible for the acceleration of the universe. It follows that the dynamics of the universe, the level of variation of the constants and the other deviations from general relativity are connected [348] so that the study of the variation of the constants can improve the reconstruction of the equation state of the dark energy [20, 162, 389, 404].

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.