Cosmological variation of the fine structure constant $\alpha$ due to the evolution of a spatially homogeneous ultralight scalar field $\left(m\sim H_0\right)$ during the matter and $\Lambda$ dominated eras is analyzed. Agreement of $\Delta \alpha /\alpha$ with the value suggested by recent observations of quasar absorption lines is obtained by adjusting a single parameter, the coupling of the scalar field to matter. Asymptotically $\alpha \mathrm{}\left(t\right)\mathrm{}$ in this model goes to a constant value $\overline{\alpha }\approx {\alpha }_0$ in the early radiation and the late $\Lambda$ dominated eras. The coupling of the scalar field to (nonrelativistic) matter drives $\alpha$ slightly away from $\overline{\alpha }$ in the epochs when the density of matter is important. Simultaneous agreement with the more restrictive bounds on the variation $|\Delta \alpha /\alpha |$ from the Oklo natural fission reactor and from meteorite samples can be achieved if the mass of the scalar field is on the order of 0.5–0.6 $H_{\Lambda },$ where $H_{\Lambda }={\Omega }_{\Lambda }^{1/2}{H}_0.$ Depending on the scalar field mass, $\alpha$ may be slightly smaller or larger than ${\alpha }_0$ at the times of big bang nucleosynthesis, the emission of the cosmic microwave background, the formation of early solar system meteorites, and the Oklo reactor. The effects on the evolution of $\alpha$ due to nonzero mass for the scalar field are emphasized. An order of magnitude improvement in the laboratory technique could lead to a detection of $(\dot{\alpha }/\alpha {)}_0.$

• Received 7 May 2003

DOI:https://doi.org/10.1103/PhysRevD.68.043513