In its youth, a stellar
population generates lots of UV photons, core collapse supernovae and
metals that go to enrich the ISM. In its old age, say ~ 12 Gyr later,
the same stellar population radiates optical-near-IR light from its
~ 1 M
stars, while all more massive stars are dead remnants.
The amount of metals (MX) that are produced by such
populations is proportional to the number of massive stars M > ~
8 M
that
have undergone a core collapse supernova explosion,
whereas the luminosity (e.g., LB) at t
12 Gyr is
proportional to the number of stars with M ~ 1
M
. It
follows that the metal-mass-to-light ratio MX /
LB is a measure of the number ratio of massive to ~
M
stars,
that is, of the IMF slope between ~ 1 and ~ 40
M
.
Clusters of galaxies offer an excellent opportunity to measure both
the light of their dominant stellar populations, and the amount of
metals that such populations have produced in their early days.
Indeed, most of the light of clusters of galaxies comes from ~ 12
Gyr old, massive ellipticals, and the abundance of metals can be
measured both in their stellar populations and in the intracluster
medium (ICM). Iron is the element whose abundance can be most
reliably measured both in cluster galaxies and in the ICM, but its
production is likely to be dominated by Type Ia supernovae whose
progenitors are binary stars. As extensively discussed in Chapter 7,
a large fraction of the total iron production comes from Type Ia
supernovae, and the contribution from CC supernovae is uncertain;
therefore the MFe / LB ratio of
clusters is less useful to set constraints on the
IMF slope between ~ 1 and M > ~ 10
M
. For
this reason, we focus on oxygen and silicon, whose production is indeed
dominated by core collapse supernovae.
Following the notations in Chapter 2, the IMF can be written as:
![]() |
(8.4) |
where a(t, Z) is the relatively slow function of SSP age and metallicity shown in Figure 2.10, multiplied by the bolometric correction shown in Figure 3.1. Thus, the metal-mass-to-light ratio for the generic element "X" can be readily calculated from:
![]() |
(8.5) |
where mX(M) is the mass of the element X which
is produced and ejected by a star of mass M. From stellar
population models one has a(12 Gyr, Z) = 2.22 and 3.12,
respectively, for Z =
Z and
2Z
and
we adopt a(12 Gyr) = 2.5 in Eq. (8.5). Using
the oxygen and silicon yields mO(M) and
mSi(M) from
theoretical nucleosynthesis (cf. Figure 7.5), Equation (8.5)
then gives the MO / LB and
MSi / LB metal mass-to-light
ratios that are reported in Figure 8.7 as a
function of the IMF slope between ~ 1 and ~ 40
M
. As
expected, the
MO / LB and MSi /
LB are extremely sensitive to the
IMF slope. The values observed in local clusters of galaxies, from
X-ray observations of the ICM and assuming stars are near solar on
average, are ~ 0.1 and ~ 0.008
M
/ L
,
respectively
for oxygen and silicon, as documented in Chapter 10. These empirical
values are also displayed in Figure 8.7. A
comparison with the calculated values shows that with the Salpeter IMF slope
(s = 2.35) the standard explosive nucleosynthesis from core collapse
supernovae produces just the right amount of oxygen and silicon to
match the observed MO / LB and
MSi / LB ratios in
clusters of galaxies, having assumed that most of the B-band light
of clusters comes from ~ 12 Gyr old populations. Actually,
silicon may be even somewhat overproduced if one allows a ~ 40%
contribution from Type Ia supernovae (cf. Figure 7.17).
Figure 8.7 also shows that with s =
1.35 such a top heavy
IMF would overproduce oxygen and silicon by more than a factor ~
20. Such a huge variation with
s = 1 is
actually expected, given that for a near Salpeter slope the typical mass
of metal producing stars is ~ 25
M
. By
the same token, the IMF labelled b in
Figure 8.3 would overproduce
metals by a
factor ~ 4 with respect to a Salpeter-slope IMF (lines a
and c ), whereas the IMF labelled d in
Figure 8.3 would do so by a
factor ~ 20. Thus,
under the assumptions that the bulk of the light of local galaxy
clusters comes from ~ 12 Gyr old stars, that current stellar theoretical
nucleosynthesis is basically correct, and that no large systematic
errors affect the reported empirical values of the MO /
LB and MSi
/ LB ratios, then one can exclude a significant
evolution of the IMF
with cosmic time, such as for example, one in which the IMF at z ~ 3
would be represented by line b or d in
Figure 8.3, and by line a
or c at z = 0.
A variable IMF is often invoked as an ad hoc fix to specific discrepancies that may emerge here or there, which however may have other origins. For example, an evolving IMF with redshift has been sometimes invoked to ease a perceived discrepancy between the cosmic evolution of the stellar mass density, and the integral over cosmic time of the star formation rate. In other contexts it has been proposed that the IMF may be different in starbursts as opposed to more steady star formation, or in disks vs. spheroids. Sometimes one appeals to a top-heavy IMF in one context, and then to a bottom-heavy one in another, as if it was possible to have as many IMFs as problems to solve. Honestly, we do not know whether there is one and only one IMF. However, if one subscribes to a different IMF to solve a single problem, then at the same time one should make sure the new IMF does not destroy agreements elsewhere, or conflicts with other astrophysical constraints. While it is perfectly legitimate to contemplate IMF variations from one situation to another, it should be mandatory to explore all consequences of postulated variations, well beyond the specific case one is attempting to fix. This kind of sanitary check is most frequently neglected in the literature appealing to IMF variations. A few examples of such checks have been presented in this chapter.