Competing bounds on the present-day time variation of fundamental   constants
  We compare the sensitivity of a recent bound on time variation of the fine structure constant from optical clocks with bounds on time varying fundamental constants from atomic clocks sensitive to the electron-to-proton mass ratio, from radioactive decay rates in meteorites, and from the Oklo natural reactor. Tests of the Weak Equivalence Principle also lead to comparable bounds on present variations of constants. The "winner in sensitivity" depends on what relations exist between the variations of different couplings in the standard model of particle physics, which may arise from the unification of gauge interactions. WEP tests are currently the most sensitive within unified scenarios. A detection of time variation in atomic clocks would favour dynamical dark energy and put strong constraints on the dynamics of a cosmological scalar field. 
Thomas Dent, Steffen Stern and Christof Wetterich