Varying-$\alpha $ Theories and Solutions to the Cosmological Problems

If the fine structure constant $\alpha =e^2/(\hbar c)$ were to change, then a number of interpretations would be possible, attributing this change either to variations in the electron charge, the dielectric constant of the vacuum, the speed of light, or Planck's constant. All these variations should be operationally equivalent and can be related by changes of standard units. We show how the varying speed of light cosmology recently proposed can be rephrased as a dielectric vacuum theory, similar to the one proposed by Bekenstein. The cosmological problems will therefore also be solved in such a theory.