Variation of $\alpha$ from a Dark Matter Force
  We consider a long range scalar force that mainly couples to dark matter and
unstable Standard Model states, like the muon, with tiny strength. Probing this
type of force would present a challenge to observations. We point out that the
dependence of the induced background scalar field on dark matter number density
can cause the mass of the unstable particles to have spatial and temporal
variations. These variations, in turn, leave an imprint on the value of the
fine structure constant $\alpha$, through threshold corrections, that could be
detected in astronomical and cosmological measurements. Our mechanism can
accommodate the mild preference of the Planck data for such a deviation,
$(\alpha_{_{\rm CMB}}-\alpha_{\rm present})/\alpha_{\rm present} = (-3.6\pm
3.7)\times 10^{-3}$. In this case, the requisite parameters typically imply
that violations of Equivalence Principle may be within reach of future
experiments.
