We examine variations of the stellar initial mass function ( IMF ) in extreme environments within the formalism derived by Hennebelle & Chabrier .
We focus on conditions encountered in progenitors of massive early type galaxies and starburst regions .
We show that , when applying the concept of turbulent Jeans mass as the characteristic mass for fragmentation in a turbulent medium , the peak of the IMF in such environments is shifted towards smaller masses , leading to a bottom-heavy IMF , as suggested by various observations .
In very dense and turbulent environments , we predict that the high-mass tail of the IMF can become even steeper than the standard Salpeter IMF , with a limit for the power law exponent \alpha \simeq - 2.7 , in agreement with recent observational determinations .
This steepening is a direct consequence of the high densities and Mach values in such regions but also of the time dependence of the fragmentation process , as incorporated in the Hennebelle-Chabrier theory .
We provide analytical parametrizations of these IMFs in such environments , to be used in galaxy evolution calculations .
We also calculate the star formation rates and the mass-to-light ratios expected under such extreme conditions and show that they agree well with the values inferred in starburst environments and massive high-redshift galaxies .
This reinforces the paradigm of star formation as being a universal process , i.e .
the direct outcome of gravitationally unstable fluctuations in a density field initially generated by large scale shock-dominated turbulence .
This globally enables us to infer the variations of the stellar IMF and related properties for atypical galactic conditions .