The slope of the IMF controls the rate of luminosity evolution of a SSP, as shown by Figure 2.6 for the bolometric light. The flatter the IMF the more rapid the luminosity declines past an event of star formation. On the contrary, the steeper the IMF the slower such decline, as the light from many low-mass stars compensates for the progressive death of the rarer, more massive and brighter stars. Having identified and studied passively evolving elliptical galaxies all the way to z~ 2 and even beyond, one expects that their M / L ratio must systematically decrease with increasing redshift, and do so by an amount that depends on the slope of the IMF. This test is particularly effective if undertaken for cluster ellipticals, as clusters provide fairly numerous samples of ellipticals at well defined redshifts. Besides the IMF, the rate of luminosity (M / L) evolution also depends on the age of a SSP, being much faster at young ages than at late epochs. Thus, the rate of M / L evolution of elliptical galaxies with redshift must depend on both the IMF slope and the luminosity-weighted age of their stellar populations, or, equivalently on their formation redshift.
We know that the bulk of stars in local massive ellipticals are very old, and for an age of ~ 12 Gyr the light of such galaxies comes from a narrow range around the turnoff mass MTO at ~ 1 M. Their progenitors at z~ 1.5 must be younger by the corresponding lookback time, that is, ~ 9 Gyr younger, and from the MTO-age relation (Eq. (2.2)) we see that the bulk of light of such progenitors has to come from stars of mass around MTO 1.4 M. Thus, the evolution of the M / L ratio of old stellar populations from redshift zero all the way to redshift ~ 1.5 is controlled by the IMF slope in the narrow interval between ~ 1 and ~ 1.4 M. The IMF slope below ~ 1 M has no influence on the luminosity evolution, and that above ~ 1.4 M was in control of the luminosity evolution at redshifts beyond ~ 1.5. Therefore, the evolution of the M / L ratio of elliptical galaxies from z = 0 to ~ 1 allows us to measure the slope of the IMF just near M ~ 1 M.
Figure 8.6 shows the evolution with redshift of the M* / LB ratio of solar composition SSPs, for various IMF slopes and different formation redshifts. Also plotted is the average M* / LB ratio of cluster ellipticals from the literature, from local clusters at z~ 0 all the way to clusters at z ~ 1.3. A Salpeter slope (s = 2.35) fits the data for a formation redshift zF between ~ 2 and ~ 3, which is in pretty good agreement with both the formation redshift derived from age-dating local ellipticals in various ways, and with the observed rapid disappearance of quenched galaxies beyond z ~ 2.
Figure 8.6. The differential redshift evolution (with respect to the value at z = 0) of the M* / LB mass-to-light ratio of solar composition SSPs, for various choices of the IMF slope between ~ 1 and ~ 1.4 M, and for various assumed formation redshifts zF, as indicated. The data points refer to the M* / LB ratio of elliptical galaxies in clusters at various redshifts, from z ~ 0 up to z 1.3. (Updated from Renzini, A. (2005) The Initial Mass Function 50 Years later, (ed. E. Corbelli et al. , Ap. Sp. Sci. Library, 327, 221).
Assuming that the IMF at the formation redshift of ellipticals was like line b in Figure 8.3, then with s = 1.3 at M = 1 M this IMF would require a formation redshift well beyond 3 in order to fit the data. Line d instead, with s = 0.8 at M = 1 M would fail to match the data even assuming zF = ∞, as shown in Figure 8.6. One can conclude that the evolution of the M / L ratio of cluster elliptical galaxies up to redshift ~ 1.3 does not favor any significant departure from the Salpeter value s = 2.35 of the slope of the IMF in the vicinity of M ~ 1 M, all the way to a formation redshift beyond ~ 2.