The bolometric flux of the EBL derived in Section 6.1 is a record of the total energy produced in stellar nucleosynthesis in the universe, and so can be used to constrain estimates of the baryonic mass which has been processed through stars. The relationship between processed mass and background flux depends strongly on the redshift dependence of star formation and on the stellar IMF, but is only weakly dependent on the assumed cosmology for the reasons discussed in Section 4.2 and Section 5.

As an illustrative case, we can obtain a simple estimate of
the total mass processed by stars by assuming that all stars formed in
a single burst at an effective redshift *z*_{e}, and that
all the energy
from that burst was emitted instantaneously. The assumption of
instantaneous emission does not strongly affect the result because
most of the light from a stellar population is emitted by hot,
short-lived stars in the first ~ 10 Myr. The integrated EBL at
*z* = 0 in Equation 1 then simplifies to

(6) |

in which
_{*}
is the bolometric energy density from
stellar nucleosynthesis and (1 + *z*_{e}) compensates for
energy lost to cosmic expansion. In the case of instantaneous formation and
emission,
_{*}
can be expressed in terms of the total
energy released in the nucleosynthesis of He and heavier elements:

(7) |

in which (~
0.0075) is the mean conversion efficiency of energy released in nuclear
reactions and
*Y* and
*Z* are the
mass fractions of ^{4}He and metals. Inverting Equation
6, the total baryonic mass processed through stars
in this model can be derived from a measurement of the bolometric EBL
using the expression:

(8) |

We can bracket a reasonable range for
<*Y* +
*Z*> by
assuming the solar value as a lower limit, and the mass weighted average
of the metal conversion fraction in E/S0 and spirals galaxies as the
upper limit. ^{(3)}
Assuming a 3:2 ratio of E/S0 to Sabc galaxies
(Persic & Salucci 1992),
we find
<*Y* +
*Z*> = 0.25±0.15. For *z*_{e} = 1.5, the total
baryonic mass processed through stars corresponding to
a bolometric EBL of 100±20 nW m^{-2}sr^{-1} is then
_{*} =
0.0030(±0.0019)*h*^{-2} in units of the critical
density, or 0.16(±
0.10)_{B}
for _{B} =
0.019(± 0.001)*h*^{-2}
(Burles & Tytler 1998).
Again, this
calculation assumes a single redshift for star formation with all
energy radiated instantaneously at the redshift of formation.

The true history of star formation is obviously quite different from
this illustrative case. For time-dependent emission and formation,
the bolometric EBL is the integral of the comoving luminosity density
corresponding to realistic age- and redshift-dependent emission (see
Equation 1). For comparison, instantaneous formation at
the same redshift assumed above (*z*_{e} = 1.5) with a
modified Salpeter IMF and time-dependent emission based on SEDs from
Buzzoni (1995)
would imply
_{*} =
0.0037(± 0.0007)*h*^{-2} for our estimate of
the bolometric EBL (for details see MPD98 and
Madau & Pozzetti 1999,
MP99).
The mean of this estimate is about 20% higher than that
from the instantaneous formation and emission model discussed above. The
two models are very similar because the vast majority of energy from a
stellar population is emitted in the first ~ 10 Myrs. The quoted
uncertainty is smaller than for our illustrative model because the
error range reflects only the uncertainty in our estimate of the
bolometric EBL and no uncertainties in the adopted IMF.

For the same IMF and SEDs, a redshift-dependent star formation rate
for 0 < *z* < 4 based on the observed UV luminosity density
and taking dust obscuration into account (see
Steidel *et al.* 1999)
would imply that
almost twice as much mass is processed through stars than in the
instantaneous formation model above (MP99).
Relative to the instantaneous-formation models, the same
bolometric EBL flux corresponds to a larger value of
_{*}
when we consider time-dependent star formation because
more of the emission occurs at higher redshifts, resulting in greater
energy losses to cosmic expansion. For our estimate of
*I*^{bol}_{EBL} and the calculations of MP99
discussed above, we
therefore estimate that total mass fraction processed through stars is
_{*}
= 0.0062(± 0.0012)*h*^{-2} or 0.33±
0.07_{B}.
We adopt this value for the remainder of the paper.

For this value of the total processed mass, we can calculate the
corresponding metal mass which is produced in stellar nucleosynthesis.
To do so requires an estimate of the metal yield - the mass fraction
of metals returned to the ISM relative to the mass remaining in stars
and stellar remnants. Best estimates of the metal yield,
*y*_{Z}, lie
between 0.01 (corresponding to a Scalo IMF) and 0.034 (as observed in
the Galactic bulge)
(Pagel 1987,
1999).
These values incorporate the full
range predicted by IMF models and observations (see
Woosley & Weaver 1995;
Tsujimoto *et al.*
1995;
Pagel & Tautviaisiene
1997;
Pagel 1997).
For *Z*_{} =
0.017
(Grevesse, Noels, & Sauval
1996),
this metal yield range in solar units is
*y*_{Z} = 1.3± 0.7
*Z*_{}. If
the mass fraction remaining in stars and stellar remnants is *f*, then
the predicted metal mass density is given by

(9) |

which gives
_{Z} =
0.0040(± 0.0022)*h*^{-2}
*Z*_{}, or
_{Z} =
0.24(± 0.13)
*Z*_{}
_{B},
for an assumed lock-up fraction of *f* = 0.5.

Note that we have assumed that the full flux of the EBL is due to
stellar nucleosynthesis in the above calculations of
_{*}
and _{Z}.
If 10*h* nW
m^{-2}sr^{-1} of the IR EBL is due to
AGN, as estimated in Section 6.2, then the
energy emitted by stars is
smaller by about about 7%, and the inferred mass fractions are then
smaller by about 7% as well.

^{3} Solar values of
*Y* and
*Z* are 0.04 and 0.02, implying
*DY*/*DZ* = 2. Interstellar absorption measurements of
*DY*/*DZ* in the
solar neighborhood are closer to the range 3-4, implying
*Y* ~
0.07. Helium white dwarfs contribute an
additional 10% of the local stellar mass to the estimate of
*Y*
(Fleming, Liebert & Green
1986),
so that we have
<*Y* +
*Z*> = 0.07 + 0.02 + 0.1 ~ 0.2 as a local estimate for systems
with solar metallicity. This is similar to estimates for other local
spiral galaxies. Estimates for E/S0 galaxies are as high as 0.5
(Pagel 1997).
(Note that the He mass produced in stars is written as
*Y* to
distinguish it from the total He mass, which includes a primordial
component.)
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