This article describes the various experimental bounds on the variation of the fundamental constants of nature. After a discussion of the role of fundamental constants, their definition and link with metrology, it reviews the various constraints on the variation of the fine-structure constant, the gravitational, weak- and strong-interaction couplings and the electron-to-proton mass ratio. The review aims (1) to provide the basics of each measurement, (2) to show as clearly as possible why it constrains a given constant, and (3) to point out the underlying hypotheses. Such an investigation is of importance in comparing the different results and in understanding the recent claims of the detection of a variation of the fine-structure constant and of the electron-to-proton mass ratio in quasar absorption spectra. The theoretical models leading to the prediction of such variation are also reviewed, including Kaluza-Klein theories, string theories, and other alternative theories. Cosmological implications of these results are also discussed. The links with the tests of general relativity are emphasized.

Fundamental constants are a cornerstone of our physical laws. Any constant varying in space and/or time would reflect the existence of an almost massless field that couples to matter. This will induce a violation of the universality of free fall. Thus, it is of utmost importance for our understanding of gravity and of the domain of validity of general relativity to test for their constancy. We detail the relations between the constants, the tests of the local position invariance and of the universality of free fall. We then review the main experimental and observational constraints that have been obtained from atomic clocks, the Oklo phenomenon, solar system observations, meteorite dating, quasar absorption spectra, stellar physics, pulsar timing, the cosmic microwave background and big bang nucleosynthesis. At each step we describe the basics of each system, its dependence with respect to the constants, the known systematic effects and the most recent constraints that have been obtained. We then describe the main theoretical frameworks in which the low-energy constants may actually be varying and we focus on the unification mechanisms and the relations between the variation of different constants. To finish, we discuss the more speculative possibility of understanding their numerical values and the apparent fine-tuning that they confront us with.

Unresolved sources of gravitational waves are at the origin of a stochastic gravitational wave background. While the computation of its mean density as a function of frequency in a homogeneous and isotropic universe is standard lore, the computation of its anisotropies requires understanding the coarse graining from local systems, to galactic scales and then to cosmology. An expression of the gravitational wave energy density valid in any general spacetime is derived. It is then specialized to a perturbed Friedmann-Lemaître spacetime in order to determine the angular power spectrum of this stochastic background as well as its correlation with other cosmological probes, such as the galaxy number counts and weak lensing. Our result for the angular power spectrum also provides an expression for the variance of the gravitational wave background.

We present the time drift of the cosmological redshift in a general spherically symmetric spacetime. We demonstrate that its observation would allow us to test the Copernican principle and so determine if our Universe is radially inhomogeneous, an important issue in our understanding of dark energy. In particular, when combined with distance data, this extra observable allows one to fully reconstruct the geometry of a spacetime describing a spherically symmetric underdense region around us, purely from background observations.

The effect of variations of the fundamental nuclear parameters on big-bang nucleosynthesis are modeled and discussed in detail taking into account the interrelations between the fundamental parameters arising in unified theories. Considering only ${}^4\mathrm{He}$, strong constraints on the variation of the neutron lifetime, neutron-proton mass difference are set. These constraints are then translated into constraints on the time variation of the Yukawa couplings and the fine structure constant. Furthermore, we show that a variation of the deuterium binding energy is able to reconcile the ${}^7\mathrm{Li}$ abundance deduced from the WMAP analysis with its spectroscopically determined value while maintaining concordance with D and ${}^4\mathrm{He}$. The analysis is strongly based on the potential model of Flambaum and Shuryak that relates the binding energy of the deuteron with the nucleon and $\sigma$ and $\omega$ mesons masses; however, we show that an alternative approach that consists of a pion mass dependence necessarily leads to equivalent conclusions.

This article provides a general study of the Hamiltonian stability and the hyperbolicity of vector field models involving both a general function of the Faraday tensor and its dual, f(F2,FF\texttildelow{}), as well as a Proca potential for the vector field, V(A2). In particular it is demonstrated that theories involving only f(F2) do not satisfy the hyperbolicity conditions. It is then shown that in this class of models, the cosmological dynamics always dilutes the vector field. In the case of a nonminimal coupling to gravity, it is established that theories involving Rf(A2) or Rf(F2) are generically pathologic. To finish, we exhibit a model where the vector field is not diluted during the cosmological evolution, because of a nonminimal vector field-curvature coupling which maintains second-order field equations. The relevance of such models for cosmology is discussed.