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 . 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 beThe 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].
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] |
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
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
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 toThis 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 (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].
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
This work is licensed under a Creative Commons License. E-mail us: |