A More Accurate Parameterization based on cosmic Age (MAPAge)
  Recently, several statistically significant tensions between different
cosmological datasets have raised doubts about the standard Lambda cold dark
matter ($\Lambda$CDM) model. A recent letter~\citet{Huang:2020mub} suggests to
use "Parameterization based on cosmic Age" (PAge) to approximate a broad class
of beyond-$\Lambda$CDM models, with a typical accuracy $\sim 1\%$ in angular
diameter distances at $z\lesssim 10$. In this work, we extend PAge to a More
Accurate Parameterization based on cosmic Age (MAPAge) by adding a new degree
of freedom $\eta_2$. The parameter $\eta_2$ describes the difference between
physically motivated models and their phenomenological PAge approximations. The
accuracy of MAPAge, typically of order $10^{-3}$ in angular diameter distances
at $z\lesssim 10$, is significantly better than PAge. We compare PAge and
MAPAge with current observational data and forecast data. The conjecture
in~\citet{Huang:2020mub}, that PAge approximation is sufficiently good for
current observations, is quantitatively confirmed in this work. We also show
that the extension from PAge to MAPAge is important for future observations,
which typically requires sub-percent accuracy in theoretical predictions.
