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Recent evidence based independently on spectral line strengths and dynamicalmodelling point towards a non-universal stellar Initial Mass Function (IMF),probably implying an excess of low-mass stars in elliptical galaxies with ahigh velocity dispersion. Here we show that a time-independent bottom-heavy IMFis compatible neither with the observed metal-rich populations found in giantellipticals nor with the number of stellar remnants observed within thesesystems. We suggest a two-stage formation scenario involving a time-dependentIMF to reconcile these observational constraints. In this model, an earlystrong star-bursting stage with a top-heavy IMF is followed by a more prolongedstage with a bottom-heavy IMF. Such model is physically motivated by the factthat a sustained high star formation will bring the interstellar medium to astate of pressure, temperature and turbulence that can drastically alter thefragmentation of the gaseous component into small clumps, promoting theformation of low-mass stars. This toy model is in good agreement with thedifferent observational constrains on massive elliptical galaxies, such as age,metallicity, alpha-enhancement, M/L, or the mass fraction of the stellarcomponent in low-mass stars.

The (galaxy-wide) IMF in giant elliptical galaxies: From top to bottom