5. THE STELLAR MASS DENSITY TODAY

With the help of some simple stellar population synthesis tools we can now set a lower limit to the total stellar mass density that produced the observed sky brightness and constrain the cosmic history of star birth in galaxies. One of the most serious uncertainties in this calculation has always been the lower cutoff (usually treated as a free parameter) of the initial mass function (IMF). Observations of M subdwarfs stars with the HST have recently shed some light on this issue, showing that the IMF in the Galactic disk can be represented analytically over the mass range 0.1 <m < 1.6 (here m is in solar units) by log (m) = const - 2.33 log m - 1.82(log m)² ([17], hereafter GBF). For m > 1 this mass distribution agrees well with a Salpeter function. Observations of normal Galactic star-forming regions also show some convergence in the basic form of the IMF at intermediate and high masses, a power-law slope that is consistent with the Salpeter value [13]. In the following I will use a ``universal'' IMF with the GBF form for m < 1, matched to a Salpeter slope for m 1; the mass integral of this function is 0.6 times that obtained extrapolating a Salpeter function down to 0.1 M. ⁽¹⁾

Figure 4. Left: Synthetic [4] bolometric luminosity versus age of a simple stellar population having total mass M = 1 M, metallicity Z = Z (solid line) and Z = 0.2 Z (dotted line), and a GBF + Salpeter IMF (see text for details). Right: EBL observed at Earth from the instantaneous formation at redshift zF of a stellar population having the same IMF, solar metallicity, and mass density * h2 = 0.00175, 0.00125, and 0.00075, as a function of zF. Solid curves: EdS universe with h = 0.5. Dashed curves: -dominated universe with M = 0.3, = 0.7, and h = 0.65.

As shown in Figure 4, the bolometric luminosity as a function of age of a simple stellar population (a single generation of coeval, chemically homogeneous stars having total mass M, solar metallicity, and the above IMF) can be well approximated by

(4)

(cf [6]). Over a timescale of 13 Gyr (the age of the universe for an EdS cosmology with h = 0.5), about 1.3 MeV per stellar baryon are radiated away. This number depends only weakly on the assumed metallicity of stars. In a stellar system with arbitrary star formation rate per comoving cosmological volume, _*, and formation epoch t_F, the comoving bolometric emissivity at time t is given by the convolution integral

(5)

The total background light observed at Earth (t = t_H) is

(6)

where the factor (1 + z) at the denominator is lost to cosmic expansion when converting from observed to radiated (comoving) luminosity density. To set a lower limit to the present-day mass density, _*, of processed gas + stars (in units of the critical density _crit = 2.77 x 10¹¹ h² M Mpc^-3), consider now a scenario where all stars are formed instantaneously at redshift z_F. The background light that would be observed at Earth from such an event is shown in Figure 4 as a function of z_F for _* h² = 0.00075, 0.00125, 0.00175 (corresponding to 9, 6.5, and 4 percent of the nucleosynthetic baryon density, _b h² = 0.0193 ± 0.0014 [5]), and two different cosmologies. A couple of points are worth noting here: (1) the time evolution of the luminosity radiated by a simple stellar population makes the dependence of the observed EBL from z_F much shallower than the (1 + z_F)^-1 lost to cosmic expansion, as the energy output from stars is spread over their respective lifetimes; and (2) in order to generate an EBL at a level of 50 I₅₀ n W m^-2 sr^-1, one requires _* h² > 0.00125 I₅₀ (for an EdS universe with h = 0.5), hence a mean mass-to-blue light ratio today of < M / L_B >_* > 3.5 I₅₀ for a present-day blue luminosity density of _B = 2.0 x 10⁸ h L Mpc^-3 [11]. As shown in Figure 4, the dependence of these estimates on the cosmological model is rather weak. With the adopted IMF, about 30% of this mass will be returned to the interstellar medium in 10⁸ yr after intermediate-mass stars eject their envelopes and massive stars explode as supernovae. This ``return fraction'', R, becomes 50% after about 10 Gyr.

A visible mass density at the level of the above lower limit, 0.00125h^-2 I₅₀, while able to explain the measured sky brightness, requires all the stars that give origin to the observed light to have formed at z_F 0.2, and is, as such, rather implausible. A more realistic scenario appears to be one where the star formation density evolves as

(7)

This model fits reasonably well all measurements of the UV-continuum and H luminosity densities from the present-epoch to z = 4 after an extinction correction of A₁₅₀₀ = 1.2 mag (A₂₈₀₀ = 0.55 mag) is applied to the data [21], and produce a total EBL of the right magnitude (I₅₀ = 1). Since about half of the present-day stars are formed at z > 1.3 (hence their contribution to the EBL is redshifted away), the resulting visible mass density is _* h² = 0.0031 I₅₀ (< M / L_B >_* = 8.6 I₅₀). Note that this estimate ignores the recycling of returned gas into new stars.

The observed EBL therefore requires that between 7% and 16% of the nucleosynthetic baryons are today in the forms of stars, processed gas, and their remnants. According to the most recent census of cosmic baryons, the mass density in stars and their remnants observed today is _s h = 0.00245^+0.00125_-0.00088 [16], corresponding to a mean visible mass-to-blue light ratio of < M / L_B >_s = 3.4^+1.7_-1.3 (h = 0.5) (about 70% of this mass is found in old spheroidal populations). While this is about a factor of 2.5 smaller than the visible mass density predicted by equation (7), efficient recycling of ejected material into new star formation would tend to reduce the apparent discrepancy in the budget. Alternatively, the gas returned by stars may be ejected into the intergalactic medium. With an IMF-averaged yield of returned metals of y 1.5 Z, ⁽²⁾ the predicted mean metallicity at the present epoch is y _* / _b = 0.25 Z, in good agreement with the values inferred from cluster abundances [26].

¹ The bolometric light contributed by stars less massive than 1 M is quite small for a ``typical'' IMF. The use of the GBF mass function at low masses instead of Salpeter leaves then the total radiated luminosity of the stellar population virtually unaffected. Back.
² Here we have taken y mp_zm (m)dm x [ m (m)dm]^-1, the stellar yields p_zm of [33], and a GBF + Salpeter IMF. Back.