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 EBL, and constrain the cosmic history of star birth in galaxies. One of the most serious uncertainties in this calculation is 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 (Gould et al. 1996, hereafter GBF; Gould et al. 1997). For m > 1 this mass distribution agrees well with a Salpeter function, log (m) = const - 2.35 log m. A shallow mass function (relative to the Salpeter slope) below 1 M has been measured in the Galactic bulge as well (Zoccali et al. 1999). 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 (see Elmegreen 1998; Massey 1998, and references therein). In the following we will use a ``universal'' IMF (shown in Figure 3) 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. ⁽²⁾

Figure 3. Stellar initial mass functions, (m), multiplied by m2. Solid line: Salpeter IMF, (m) m-2.35 at high masses, matched to a GBF function at m 1. Dotted line: WD-progenitor dominated IMF in galaxy halos, (m) e-(/ m)3 m-5, with = 2.4 (see text for details). Dot-dashed line: Same for = 4. All IMFs have been normalized to m (m)dm = 1

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 Buzzoni 1995). 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 will be 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, _s, the bolometric emissivity at time t is given by the convolution integral

(5)

Therefore the total background light observed at Earth (t = t_H), generated by a stellar population with a formation epoch t_F, is

(6)

where the factor (1 + z) at the denominator is lost to cosmic expansion when converting from observed to radiated (comoving) luminosity density.

Figure 4. Left: Synthetic (based on an update of Bruzual & Charlot's 1993 libraries) 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 (Z = Z) and mass density g+s h2 = 0.0018, 0.0013, and 0.0008, as a function of zF. Solid curves: EdS universe with h = 0.5 (tH = 13 Gyr). Dashed curves: -dominated universe with M = 0.3, = 0.7, and h = 0.65 (tH = 14.5 Gyr).

To set a lower limit to the present-day mass density, _g+s, 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 _g+s h² = 0.0008, 0.0013, 0.0018, corresponding to 4, 7, and 9 percent of the nucleosynthetic baryon density, _b h² = 0.0193 ± 0.0014 (Burles & Tytler 1998). Two main results are worth stressing here: (1) the time evolution of the luminosity radiated by a simple stellar population (eq. 4) makes the dependence of the observed EBL from z_F much shallower than the (1 + z_F)^-1 lost to cosmic expansion (eq. 6), 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 _g+s h² > 0.0013 I₅₀ for an EdS universe with h = 0.5, hence a mean mass-to-blue light ratio today of < M / L_B >_g+s > 3.5 I₅₀ (the total blue luminosity density at the present-epoch is _B = 2 x 10⁸ h L Mpc^-3, Ellis et al. 1996). The dependence of these estimates on the cosmological model (through eq. 6) 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. ⁽³⁾

A visible mass density at the level of the above lower limit, while able to explain the measured sky brightness, requires (most of the stars) that give origin to the observed light to have formed at very low redshifts (z_F 0.5), a scenario which appears to be ruled out by the observed evolution of the UV luminosity density (Madau 1999). For illustrative purposes, it is interesting to consider instead a model where the star formation rate per unit comoving volume stays approximately constant with cosmic time. In an EdS cosmology with h = 0.5, one derives from equations (4), (5), and (6)

(7)

The observed EBL therefore implies a ``fiducial'' mean star formation density of < s > = 0.034 I50 M yr-1 Mpc-3 (or a factor of 1.6 higher in the case of a Salpeter IMF down to 0.1 M). Any value much greater than this over a sizeable fraction of the Hubble time will generate an EBL intensity well in excess of 50 n W m-2 sr-1. Ignoring for the moment the recycling of returned gas into new stars, the visible mass density at the present epoch is simply g+s = 0tH s (t)dt = 4.4 x 108 I50 M Mpc-3, corresponding to g+s h2 = 0.0016 I50 (< M / LB >g+s = 4.4 I50).

Perhaps a more realistic scenario is one where the star formation density evolves as

(8)

This 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 (Madau 1999), and produce a total EBL of about the right magnitude (I₅₀ 1). Since about half of the present-day stars are formed at z > 1.3 in this model and their contribution to the EBL is redshifted away, the resulting visible mass density is _g+s h² = 0.0031 I₅₀ (< M / L_B >_g+s = 8.6 I₅₀), almost twice as large as in the _s = const approximation.

We conclude that, depending on the star formation history and for the assumed IMF, the observed EBL requires between 7% and 16% of the nucleosynthetic baryon density to be 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 (FHP), corresponding to a mean stellar mass-to-blue light ratio of < M / L_B >_s = 3.4^+1.7_-1.3 for h = 0.5 (roughly 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 (8), 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_Z 1.5 Z, ⁽⁴⁾ the predicted mean metallicity at the present epoch is y_{Z g+s} / _b 0.25 Z, in good agreement with the values inferred from cluster abundances (Renzini 1997).

³ An asymptotic mass fraction of stars returned as gas, R = (m - m_f) (m)dm x [ m (m)dm]^-1 0.5, can be obtained by using the semiempirical initial (m)-final (m_f) mass relation of Weidemann (1987) for stars with 1 < m < 10, and by assuming that stars with m > 10 return all but a 1.4 M remnant. Back

⁴ Here we have taken y_Z mp_zm (m)dm x [ m (m)dm]^-1, the stellar yields p_zm of Tsujimoto et al. (1995), and a GBF+Salpeter IMF. Back.