The IMF from Low to High Redshift
  From time to time, and quite more frequently in recent years, claims appear favoring a variable Initial Mass Function (IMF), one way or another, either in time or space. In this chapter we add our two pennies of wisdom, illustrating how the IMF affects various properties of galaxies and galaxy clusters. We start by showing that even relatively small variations of the IMF slope have large effects on the demography of stellar populations, moving the bulk of the stellar mass from one end to the other of the distribution. We then point out how the slope of the IMF in different mass ranges controls specific major properties of galaxies and clusters. The slope of the IMF below ~1 solar mass controls the M/L ratio of local ellipticals, whereas the slope between ~1 and ~1.4 solar masses controls the evolution with redshift of such ratio, hence of the fundamental plane of elliptical galaxies. Similarly, the slope between ~1 and ~40 solar masses drives the ratio of the global metal mass in clusters of galaxies to their total luminosity. While we believe that it is perfectly legitimate to entertain the notion that the IMF may not be universal, our message is that when proposing IMF variations to ease a specific problem then one should not neglect to explore the full consequences of the invoked variations.   This paper is integrally reproduced from Chapter 8 of the book by L. Greggio and A. Renzini: Stellar Populations. A User Guide from Low to High Redshift (2011, Wiley-VHC Verlag-GmbH & Co., ISBN 9783527409181), whose index is also appended. 
Laura Greggio and Alvio Renzini
astro-ph.GA astro-ph.CO