Downsizing revised: Star formation timescales for elliptical galaxies
  with an environment-dependent IMF and number of SNIa
  Previous studies of the stellar mean metallicity and [Mg/Fe] values of
massive elliptical (E)~galaxies suggest that their stars were formed in a very
short timescale which cannot be reconciled with estimates from stellar
population synthesis (SPS) studies and with hierarchical-assembly. Applying the
previously developed chemical evolution code, GalIMF, which allows an
environment-dependent stellar initial mass function (IMF) to be applied in the
integrated galaxy initial mass function (IGIMF) theory instead of an invariant
canonical IMF, the star formation timescales (SFT) of E galaxies are
re-evaluated. The code's uniqueness lies in it allowing the galaxy-wide IMF and
associated chemical enrichment to evolve as the physical conditions in the
galaxy change. The calculated SFTs become consistent with the independent SPS
results if the number of type Ia supernovae (SNIa) per unit stellar mass
increases for more massive E~galaxies. This is a natural outcome of galaxies
with higher star-formation rates producing more massive star clusters, spawning
a larger number of SNIa progenitors per star. The calculations show E~galaxies
with a stellar mass $\approx 10^{9.5} M_\odot$ to have had the longest mean
SFTs of $\approx2\,$Gyr. The bulk of more massive E~galaxies were formed faster
(SFT$\,\approx 1\,$Gyr) leading to domination by M~dwarf stars and larger
dynamical mass-to-light ratios as observed, while lower-mass galaxies tend to
lose their gas supply more easily due to their shallower potential and
therefore also have similarly-short mean SFTs. This work achieves, for the
first time, consistency of the SFTs for early-type galaxies between
chemical-enrichment and SPS modelling and leads to an improved understanding of
how the star formation environment may affect the total number of SNIa per unit
stellar mass formed.
