Supernova Magnitude Evolution and PAge Approximation
  The evidence of environmental dependence of Type Ia supernova luminosity has
inspired recent discussion about whether the late-universe cosmic acceleration
is still supported by supernova data. We adopt the
$\Delta\mathrm{HR}/\Delta\mathrm{age}$ parameter, which describes the
dependence of supernova absolute magnitude on the age of supernova progenitor,
as an additional nuisance parameter. Using the Pantheon supernova data, a lower
bound $\ge 12\,\mathrm{Gyr}$ on the cosmic age, and a Gaussian prior $H_0 =
70\pm 2\,\mathrm{km\,s^{-1}Mpc^{-1}}$ on the Hubble constant, we reconstruct
the cosmic expansion history. Within the flat $\Lambda$ cold dark matter
($\Lambda$CDM) framework, we still find a $5.6\sigma$ detection of cosmic
acceleration. This is because a matter dominated decelerating universe would be
too young to accommodate observed old stars with age $\gtrsim
12\,\mathrm{Gyr}$. A decelerating but non-flat universe is marginally
consistent with the data, however, only in the presence of a negative spatial
curvature $\sim$ two orders of magnitude beyond the current constraint from
cosmic microwave background data. Finally, we propose a more general
Parameterization based on the cosmic Age (PAge), which is {\it not} directly
tied to the dark energy concept and hence is ideal for a null test of the
cosmic acceleration. We find that, for a magnitude evolution rate
$\Delta\mathrm{HR}/\Delta\mathrm{age} \lesssim
0.3\,\mathrm{mag}/5.3\,\mathrm{Gyr}$ (Kang et al. 2020), a spatially flat and
decelerating PAge universe is fully consistent with the supernova data and the
cosmic age bound, and has no tension with the geometric constraint from the
observed CMB acoustic angular scales.
