Rolling tachyon field models are among the candidates suggested as explanations for the recent acceleration of the Universe .
In these models the field is expected to interact with gauge fields and lead to variations of the fine-structure constant \alpha .
Here we take advantage of recent observational progress and use a combination of background cosmological observations of Type Ia supernovas and astrophysical and local measurements of \alpha to improve constraints on this class of models .
We show that the constraints on \alpha imply that the field dynamics must be extremely slow , leading to a constraint of the present-day dark energy equation of state ( 1 + w _ { 0 } ) < 2.4 \times 10 ^ { -7 } at the 99.7 \% confidence level .
Therefore current and forthcoming standard background cosmology observational probes can ’ t distinguish this class of models from a cosmological constant , while detections of \alpha variations could possibly do so since they would have a characteristic redshift dependence .