Abstract
Here we present a full description of the integrated galaxy-wide initial mass function (IGIMF) theory in terms of the optimal sampling and compare it with available observations. Optimal sampling is the method we use to discretize the IMF into stellar masses deterministically. Evidence has been indicating that nature may be closer to deterministic sampling as observations suggest a smaller scatter of various relevant observables than random sampling would give, which may result from a high level of self-regulation during the star formation process. The variation of the IGIMFs under various assumptions are documented. The results of the IGIMF theory are consistent with the empirical relation between the total mass of a star cluster and the mass of its most massive star, and the empirical relation between a galaxy's star formation rate (SFR) and the mass of its most massive cluster. Particularly, we note a natural agreement with the empirical relation between the IMF's power-law index and a galaxy's SFR. The IGIMF also results in a relation between the galaxy's SFR and the mass of its most massive star such that, if there were no binaries, galaxies with SFR $<10^{-4}$ M$_\odot$/yr should host no Type II supernova events. In addition, a specific list of initial stellar masses can be useful in numerical simulations of stellar systems. For the first time, we show optimally-sampled galaxy-wide IMFs (OSGIMF) which mimics the IGIMF with an additional serrated feature. Finally, A Python module, GalIMF, is provided allowing the calculation of the IGIMF and OSGIMF in dependence on the galaxy-wide SFR and metallicity.
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