Cosmological constraints on $\Lambda(\alpha)$CDM models with
  time-varying fine structure constant
  We study the $\Lambda(\alpha)$CDM models with $\Lambda(\alpha)$ being a
function of the time-varying fine structure constant $\alpha$. We give a close
look at the constraints on two specific $\Lambda(\alpha)$CDM models with one
and two model parameters, respectively, based on the cosmological observational
measurements along with 313 data points for the time-varying $\alpha$. We find
that the model parameters are constrained to be around $10^{-4}$, which are
similar to the results discussed previously but more accurately.
