Cosmic rays (CRs) control the thermal, ionization and chemical state of the dense H_2 gas regions that otherwise remain shielded from far-UV and optical stellar radiation propagating through the dusty ISM of galaxies. It is in such CR-dominated regions (CRDRs) rather than Photon-dominated regions (PDRs) of H_2 clouds where the star formation initial conditions are set, making CRs the ultimate star-formation feedback factor in galaxies, able to operate even in their most deeply dust-enshrouded environments. CR-controlled star formation initial conditions naturally set the stage for a near-invariant stellar Initial Mass Function (IMF) in galaxies as long as their average CR energy density U_{CR} permeating their molecular ISM remains within a factor of ~10 of its Galactic value. Nevertheless, in the extreme environments of the compact starbursts found in merging galaxies, where U_{CR}sim(few)x10^{3}U_{CR,Gal}, CRs dramatically alter the initial conditions of star formation. In the resulting extreme CRDRs H_2 cloud fragmentation will produce far fewer low mass (<8 M_{sol}) stars, yielding a top-heavy stellar IMF. This will be a generic feature of CR-controlled star-formation initial conditions, lending a physical base for a bimodal IMF during galaxy formation, with a top-heavy one for compact merger-induced starbursts, and an ordinary IMF preserved for star formation in isolated gas-rich disks. In this scheme the integrated galactic IMFs (IGIMF) are expected to be strong functions of the star formation history of galaxies.
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