  The presented study gives a comprehensive overview of the theory and the
evidence for a systematically varying stellar initial mass function (IMF). Then
we focus on the impact of this paradigm change, that is, from the universal
invariant IMF to a variable IMF, on galaxy chemical evolution (GCE) studies.
For this aim, we developed the first GCE code, GalIMF, that is able to
incorporate the empirically calibrated environment-dependent IMF variation
theory, the integrated galactic initial mass function (IGIMF) theory. In this
theory, the galaxy-wide IMF is calculated by summing all the IMFs in all
embedded star clusters which formed throughout the galaxy in 10 Myr time
epochs. The GalIMF code recalculates the galaxy-wide IMF at each time step
because the integrated galaxy-wide IMF depends on the galactic star formation
rate and metallicity. The resulting galaxy-wide IMF and metal abundance evolve
with time. Using this code, we examine the chemical evolution of early-type
galaxies (ETGs) from dwarf to the most massive. We find that the introduction
of the non-canonical IMF affects the best estimation of the galaxy properties
such as their mass, star formation history, and star formation efficiency.
Moreover, we are able to provide an independent estimation on the stellar
formation timescale of galaxies, the type Ia supernova production efficiency,
and constrain the IMF variation law of the low-mass stars. This work provides
to the community the publicly available GalIMF code with improved constraints
on the IMF variation and has, for the first time, resolved the discrepancy
between the galaxy formation timescales obtained from stellar population
synthesis and chemical enrichment studies.
