To pursue and cast light into a quantitative form of the idea of a
history of cosmic star formation and metal enrichment - not of any
particular type of galaxy but of the Universe as a whole - it is useful
to start by generalizing the standard equations of galaxy evolution
(Tinsley 1980)
over all galaxies and intergalactic gas in the
Universe. In a representative cosmological comoving volume with density
ρ_{∗} in long-lived stars and stellar remnants (white
dwarfs, neutron stars, black holes) and gas density
ρ_{g} and in which new stars are formed at the rate
ψ, the equations of cosmic chemical evolution can be written as

(1) |

Here, *Z* is the metallicity in the gas and newly born stars,
*R* is the "return fraction" or the mass fraction of each
generation of stars that is put back into the interstellar medium (ISM)
and intergalactic medium (IGM), and *y* is the net metal yield or
the mass of new heavy elements created and ejected into the ISM/IGM by
each generation of stars per unit mass locked into stars. The above
equations govern the formation, destruction, and distribution of heavy
elements as they cycle through stars and are ultimately dispersed into
the ISM/IGM. By treating all galaxies as a single stellar system and all
baryons in the ISM/IGM as its gas reservoir, their solution enables the
mean trends of galaxy populations to be calculated with the fewest
number of free parameters. The equations state that, for every new mass
element locked forever into long-lived stars and stellar remnants,
Δρ_{∗}, the metallicity of the ISM/IGM
increases as Δ*Z* = *y*
Δρ_{∗} / ρ_{g}, whereas the mass of
heavy elements in the ISM/IGM changes as
Δ(*Z*ρ_{g}) = (*y* - *Z*)
Δρ_{∗}. The latter expression is a consequence of
metals being released into the gas from mass loss during
post-main-sequence stellar evolution as well as being removed from the
ISM/IGM when new stars condense out. However, compared with the source
term, the metal sink term can be neglected at early epochs when
*Z* ≪ *y*.

At redshift *z*, Equation 1 can be integrated to give the following:

The total mass density of long-lived stars and stellar remnants accumulated from earlier episodes of star formation,

(2)

where

*H*(*z'*) =*H*_{0}[Ω_{M}(1 +*z'*)^{3}+ Ω_{Λ}]^{1/2}is the Hubble parameter in a flat cosmology.- The total mass density of gas,
(3)

where ρ

_{g,∞}is the comoving density of gas at some suitable high redshift where there are no stars or heavy elements. The total mass density of heavy elements in the ISM/IGM,

(4)

where the term <

*Z*_{∗}> ρ_{∗}is the total metal content of stars and remnants at that redshift. Note that the instantaneous total metal ejection rate,*E*_{Z}, is the sum of a recycle term and a creation term (Maeder 1992),(5)

where the first term is the amount of heavy elements initially lost from the ISM when stars formed that are now being re-released, and the second represents the new metals synthesized by stars and released during mass loss.

For a given universal stellar IMF, the quantities *R* and *y*
can be derived using the following formulas:

(6) (7) |

where *m* is mass of a star, *w*_{m} is its remnant
mass, ϕ(*m*) is the IMF [normalized so that
∫_{ml}^{mu}
*m*ϕ(*m*)d*m* = 1], and *y*_{m} is the
stellar yield, i.e., the fraction of mass *m* that is converted to
metals and ejected. In this review, the term
"yield" generally indicates the net yield *y* of a stellar
population as defined in Equation 7; instead we
explicitly speak of "stellar yields" to indicate the
*y*_{m} resulting from nucleosynthesis calculations.
The above equations have been written under the simplifying assumptions
of "instantaneous recycling"
(where the release and mixing of the products of nucleosynthesis by all
stars more massive than *m*_{0} occur on
a timescale that is much shorter than the Hubble time, whereas stars
with *m* < *m*_{0} live forever), "one zone"
(where the heavy elements are well mixed at all times within the volume
under consideration), "closed box" (flows of gas in and out the chosen
volume are negligible), and "constant IMF and metal yield."

Recall now that the main-sequence timescale is shorter than 0.6 Gyr (the
age of the Universe at *z* = 8.5) for stars more massive than 2.5
*M*_{⊙}, whereas stars less massive than 0.9
*M*_{⊙} never evolve off the main sequence. [Stellar
evolutionary models by
Schaller et al. (1992)
show that, for
*m* < 7 *M*_{⊙}, solar-metallicity stars have
longer lifetimes than their metal-poor counterparts, whereas the
opposite is true for *m* > 9 *M*_{⊙}.] So over
the redshift range of interest here, the instantaneous recycling
approximation may break down in the limited mass range
0.9 < *m* < 2.5 *M*_{⊙}. For illustrative
purposes, in the following we adopt the initial-final mass values for
white dwarfs tabulated by
Weidemann (2000),
which can be fit to few-percent accuracy over the interval 1
*M*_{⊙} < *m* < 7 *M*_{⊙}
as *w*_{m} = 0.444 + 0.084*m*. We also assume that all
stars with 8
*M*_{⊙} < *m* < *m*_{BH} = 40
*M*_{⊙} return all but a *w*_{m}=1.4
*M*_{⊙} remnant, and stars above
*m*_{BH} collapse to black holes without ejecting
material into space, i.e., *w*_{m} = *m*. Few stars form
with masses above 40 *M*_{⊙}, so the impact of the
latter simplifying assumption on chemical evolution is minimal. Thus,
taking *m*_{0} = 1 *M*_{⊙} as the dividing
stellar mass for instantaneous recycling and a
Salpeter (1955)
IMF with ϕ(*m*) ∝ *m*^{-2.35} in the range
*m*_{l} = 0.1
*M*_{⊙} < *m* < *m*_{u} = 100
*M*_{⊙}, one derives a return fraction of
*R* = 0.27. Under the same assumptions, a
Chabrier (2003)
IMF,

(8) |

(with *m*_{c} = 0.08 *M*_{⊙} and
σ = 0.69) is more weighted toward short-lived massive stars and
yields a larger return fraction, *R* = 0.41. In the instantaneous
recycling approximation, the fraction of "dark" stellar remnants
formed in each generation is

(9) |

The two IMFs produce a dark remnant mass fraction of *D* = 0.12 and
*D* = 0.19, respectively.
The stellar nucleosynthetic yields depend on metallicity, rotation, and
the mass limit for black hole formation *m*_{BH}.
By integrating over the IMF the subsolar metallicity stellar yields
(where the effect of mass loss is negligible) tabulated by
Maeder (1992)
from 10 *M*_{⊙} to
*m*_{BH} = 40, *M*_{⊙}, we obtain
*y* = 0.016 for a Salpeter and *y* = 0.032 for a Chabrier
IMF. When integrated to *m*_{BH} = 60
*M*_{⊙}, the same tabulation implies *y* = 0.023
(with *R* = 0.29) and *y* = 0.048 (with *R* = 0.44) for a
Salpeter and Chabrier IMF, respectively.
Notice that some of the uncertainties associated with the IMF and the
mass cutoff *m*_{BH} become smaller when computing the term
*y*(1 - *R*) in the equations (Equation 1). For massive stars
at solar metallicities, stellar winds eject large amount
of helium and carbon into the ISM before these are processed into
heavier elements, but the effect on the integrated metal yields (10-40
*M*_{⊙}) is weak
(Maeder 1992).
Total stellar yields (including the wind and pre-SN contributions)
obtained from rotating stellar models at solar metallicity
have been presented by
Hirschi et al. (2005).
Over the same range 10-40 *M*_{⊙}, in this case we
derive *y* = 0.019 for Salpeter and *y* = 0.038
for Chabrier. For comparison, the zero-metallicity stellar yields of
Chieffi & Limongi
(2004)
imply *y* = 0.015 for Salpeter and *y* = 0.030 for Chabrier.

Although disfavored by many observations, a Salpeter IMF in the mass
range 0.1-100 *M*_{⊙} is used as a reference
throughout the rest of this
review. Similarly, for consistency with prior work, we assume the
canonical metallicity scale where solar metallicity is
*Z*_{⊙} = 0.02, rather than the revised value
*Z*_{⊙} = 0.014 of
Asplund et al. (2009).