Fundamental constants appear everywhere in the mathematical laws we use to describe the phenomena of Nature. They seem to contain some truth about the properties of the physical world while their real nature seem to evade us.
The question of the constancy of the constants of physics was probably first addressed by Dirac [155, 156] who expressed, in his “Large Numbers hypothesis”, the opinion that very large (or small) dimensionless universal constants cannot be pure mathematical numbers and must not occur in the basic laws of physics. He suggested, on the basis of this numerological principle, that these large numbers should rather be considered as variable parameters characterizing the state of the universe. Dirac formed five dimensionless ratios among which1 and and asked the question of which of these ratios is constant as the universe evolves. Usually, varies as the inverse of the cosmic time while varies also with time if the universe is not described by an Einstein–de Sitter solution (i.e., when a cosmological constant, curvature or radiation are included in the cosmological model). Dirac then noticed that , representing the relative magnitude of electrostatic and gravitational forces between a proton and an electron, was of the same order as representing the age of the universe in atomic units so that his five numbers can be “harmonized” if one assumes that and vary with time and scale as the inverse of the cosmic time.
This argument by Dirac is indeed not a physical theory but it opened many doors in the investigation on physical constants, both on questioning whether they are actually constant and on trying to understand the numerical values we measure.
First, the implementation of Dirac’s phenomenological idea into a field-theory framework was proposed by Jordan , who realized that the constants have to become dynamical fields and proposed a theory where both the gravitational and fine-structure constants can vary ( provides a summary of some earlier attempts to quantify the cosmological implications of Dirac’s argument). Fierz  then realized that in such a case, atomic spectra will be spacetime-dependent so that these theories can be observationally tested. Restricting to the sub-case in which only can vary led to definition of the class of scalar-tensor theories, which were further explored by Brans and Dicke . This kind of theory was further generalized to obtain various functional dependencies for in the formalization of scalar-tensor theories of gravitation (see, e.g., ).
Second, Dicke  pointed out that in fact the density of the universe is determined by its age, this age being related to the time needed to form galaxies, stars, heavy nuclei…. This led him to formulate that the presence of an observer in the universe places constraints on the physical laws that can be observed. In fact, what is meant by observer is the existence of (highly?) organized systems and this principle can be seen as a rephrasing of the question “why is the universe the way it is?” (see ). Carter [82, 83], who actually coined the term “anthropic principle” for it, showed that the numerological coincidences found by Dirac can be derived from physical models of stars and the competition between the weakness of gravity with respect to nuclear fusion. Carr and Rees  then showed how one can scale up from atomic to cosmological scales only by using combinations of , and .
To summarize, Dirac’s insight was to question whether some numerical coincidences between very large numbers, that cannot be themselves explained by the theory in which they appear, was a mere coincidence or whether it can reveal the existence of some new physical laws. This gives three main roads of investigation
While “varying constants” may seem, at first glance, to be an oxymoron, it has to be considered merely as jargon to be understood as “revealing new degrees of freedom, and their coupling to the known fields of our theory”. The tests on the constancy of the fundamental constants are indeed very important tests of fundamental physics and of the laws of Nature we are currently using. Detecting any such variation will indicate the need for new physical degrees of freedom in our theories, that is new physics.
The necessity of theoretical physics on deriving bounds on their variation is, at least, threefold:
Therefore, we shall start in Section 2 by recalling the link between the constants of physics and the theories in which they appear, as well as with metrology. From a theoretical point of view, the constancy of the fundamental constants is deeply linked with the equivalence principle and general relativity. In Section 2 we will recall this relation and in particular the link with the universality of free fall. We will then summarize the various constraints that exist on such variations, mainly for the fine structure constant and for the gravitational constant in Sections 3 and 4, respectively. We will then turn to the theoretical implications in Section 5 in describing some of the arguments backing up the fact that constants are expected to vary, the main frameworks used in the literature and the various ways proposed to explain why they have the values we observe today. We shall finish by a discussion on their spatial variations in Section 6 and by discussing the possibility to understand their numerical values in Section 7.
Various reviews have been written on this topic. We will refer to the review  as FVC and we mention the following later reviews [31, 47, 72, 119, 226, 281, 278, 501, 395, 503, 505] and we refer to  for the numerical values of the constants adopted in this review.
Living Rev. Relativity 14, (2011), 2
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