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 |
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.