Variations of the stellar initial mass function in the progenitors of
  massive early-type galaxies and in extreme starburst environments
  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,
instead of the standard thermal Jeans mass for purely gravitational
fragmentation, 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.
