Varying-alpha cosmologies with potentials

We generalize the Bekenstein-Sandvik-Barrow-Magueijo model for the variation of the fine-structure ``constant,'' α, to include an exponential or inverse power-law self-potential for the scalar field φ which drives the time variation of α, and consider the dynamics of φ in such models. We find solutions for the evolution of φ or α in matter-, radiation-, and dark-energy-dominated cosmic eras. In general, the evolution of φ 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 α between redshifts z1--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.