We generalize the Bekenstein-Sandvik-Barrow-Magueijo model for the variation of the fine-structure “constant,” $\alpha$, to include an exponential or inverse power-law self-potential for the scalar field $\phi$ which drives the time variation of $\alpha$, and consider the dynamics of $\phi$ in such models. We find solutions for the evolution of $\phi$ or $\alpha$ in matter-, radiation-, and dark-energy-dominated cosmic eras. In general, the evolution of $\phi$ is well determined solely by either the self-potential or the coupling to matter, depending on the model parameters. The results are general and applicable to other models where the evolution of a scalar field is governed by a matter coupling and a self-potential. We find that the existing astronomical data stringently constrains the possible evolution of $\alpha$ between redshifts $z\simeq 1–3.5$ and the present, and this leads to a very strong limit on the allowed deviation of the potential from that of a pure cosmological constant.

• Received 11 August 2008

DOI:https://doi.org/10.1103/PhysRevD.78.083536