Annu. Rev. Astron. Astrophys. 2001. 39:
249-307
Copyright © 2001 by . All rights reserved |

**4.2. Element Production and the Cosmic Star Formation Rate**

The total EBL intensity is a measure of the bolometric energy output in
the universe. It can be expressed as the integral of the comoving
luminosity density
(*z*) over redshift
*z*
(Dwek et al. 1998):

(5) |

where (Longair 1998; Chapter 7)

(6) |

_{M}
_{M} /
_{c} is
the present mass density of the universe normalized to the critical
density, and
_{}
/
3*H*^{2}_{0} is the dimensionless cosmological
constant.

If we make the assumption that most of the EBL energy is produced by
fusion of hydrogen, then the comoving luminosity density
can be directly related to
the cosmic star formation and element production rates. Fusion of
hydrogen into heavier elements liberates 0.7% of the rest mass
energy. The mass fraction of baryons in the form of hydrogen is denoted
by *X*, and the decrement in this fraction due to nucleosynthesis
in stars by
*X*. The
luminosity density can therefore be expressed in terms of
_{X}, the comoving mass production rate of
helium and heavier elements:

(7) |

The relation between and
the comoving star formation rate,
_{*}, depends on the stellar initial mass
function (IMF) and is therefore not as robust as the relation between
and the metal production
rate. It is given by the convolution

(8) |

where *L*_{b}(*t*) is the bolometric luminosity
per unit mass of a stellar population of age *t*. For a constant
star formation rate and a Salpeter IMF, *n*(*M*)
*M*^{-2.35} for 0.1
*M*_{} <
*M* < 120
*M*_{}, where
*n*(*M*) is the number of stars formed per unit mass *M*,
the relation is given by

(9) |

for populations of ages 10^{8} to 10^{10} years. If we
assume a constant star formation rate over cosmic history, <
_{*}
>, then Equations 5 and 9 imply

(10) |

similar to the result of
Madau & Pozzetti
(2000).
In the local universe, the comoving star formation rate is only about
0.01 *M*_{}
yr^{-1} Mpc^{-3}
(Madau et al. 1998).
Thus, the observed intensity of the EBL suggests an average cosmic star formation rate that is 10-40 times higher than the rate at the present epoch.

To get a simple estimate of the amount of metals produced in the
universe, we first assume that all metals were produced in a burst of
star formation at redshift *z*_{e} and that all of
the stellar energy was instantaneously released. The expression for
*I* then simplifies to
(Peebles 1993,
Pagel 1997)

(11) |

where _{X} is the comoving mass density of consumed
hydrogen atoms. We can write
*X*
_{X}
/ _{b}
= (_{X}
/ _{b}),
where
_{b} and
_{b} are,
respectively, the baryonic mass density and its value as a fraction of
the critical mass density. The mass fraction of processed baryonic
material can then be expressed in terms of the EBL intensity as

(12) |

The value of
_{b} derived
from Big Bang nucleosynthesis calculations is
_{b}h^{2} = 0.0192 ± 0.0018
(Olive et al. 2000).
For an emission redshift of *z*_{e} = 1, we get

(13) |

Hence, to get the observed EBL (*I*_{100} = 0.45-1.7) with
an instantaneous energy release at *z*_{e} = 1, one
needs to convert 2%-6% of the nucleosynthetic baryon mass density into
helium and heavier elements. The range of
*X* values is
comparable to the solar value,

where *Y*
0.04 is the
difference between the solar (*Y* = 0.28) and the primordial
(*Y* = 0.24) ^{4}He mass fraction, and
*Z*_{} = 0.02
is the solar metallicity. Hence, the processing of the local ISM has
been very similar to that of the average baryonic matter in the
universe.

The instantaneous injection of energy is a considerably oversimplified
description of reality. Even if all stars were formed instantaneously at
some redshift *z*_{e}, their energy output would be
spread over some time interval (Equation 8) that is at least as long as
their main sequence lifetime
(Madau & Pozzetti
2000).
The EBL intensity will consequently fall off less steeply with
increasing *z*_{e} than the (1 +
*z*_{e})^{-1} cosmic expansion factor,
lowering the fraction of processed baryonic mass density needed to
produce the observed EBL intensity.

In reality, the cosmic star formation rate (CSFR) evolves with
redshift. The EBL intensity can be used to test the validity of proposed
star formation histories. For example, the CSFR determined from
UV-optical observations of galaxies by
Madau et al. (1998)
implies a total extragalactic background in the 0.1 to 1000
*µ*m range of ~ 30 nW m^{-2} sr^{-1}
(Dwek et al. 1998).
This increases to about 47 nW m^{-2} sr^{-1} for the
CSFR of
Steidel et al. (1999).
Even the larger of these values is only marginally consistent with the
allowed range of integrated EBL.

In general, the rate of star formation in individual galaxies can be
inferred from a number of diagnostics based on the spectral energy
distribution and energy output from young massive stars
(Kennicutt 1998).
Combining these diagnostics into a comprehensive picture of the global
star formation history requires knowledge of the redshift of the
galaxies being studied, and verification that no significant energy
release has been overlooked by the use of a limited number of
diagnostics. Hence, UV-optical observations of galaxies will a priori
provide only a partial description of the true CSFR. Indeed, the
discovery of dust-enshrouded luminous infrared galaxies in SCUBA 850
*µ*m surveys and follow-up studies of the redshift
distribution of these sources show that a significant fraction of the
star formation activity at high redshift is taking place behind a veil
of dust hidden from UV and optical observers
(Hughes et al. 1998,
Barger et al. 1998,
1999b,
2000,
Lilly et al. 1999).
Additional evidence suggesting that a significant fraction of the star
formation occurred at redshifts *z* > 1-2 was provided by the
iron abundance determinations in galaxy clusters with the *Advanced
Satellite for Cosmology and Astrophysics (ASCA)*
(Mushotzky &
Lowenstein 1997,
Renzini 1997),
and by radio observations of galaxies
(Haarsma et al. 2000)
(Section 4.3). Revised estimates of the CSFR
as a function of redshift that take into account some of the more recent
developments are presented by
Blain et al. (1999c),
Haarsma et al. (2000).

Since a significant fraction of the energy released by stars is
ultimately emitted at far-infrared wavelengths, it is interesting to
examine what constraints can be set on the CSFR from the CIB detections
alone. Equation 14 (Section 5) shows the
relation between the EBL spectrum and the spectral luminosity density,
_{}( ,
*z*), which, integrated over frequency, is proportional to the
CSFR. The formal determination of the CSFR from the EBL is an inversion
problem.
Gispert et al. (2000)
examined the possibility of determining the CSFR solely from the CIB
measurements longward of 140 *µ*m. To simplify the problem,
they assumed that the intrinsic spectral luminosity,
*L*_{}(), of the galaxies that dominate
the CIB does not evolve with time, so that
_{}( , *z*) can be written as the
product
_{}( ,
*z*) = *L*_{}()
(*z*), where
(*z*) is the
comoving number density of galaxies. Gispert et al. used Monte Carlo
simulations to explore a range of possible solutions for
(*z*) for various
spectral shapes and intensities of
*L*_{}().
The method provided only weak constraints on the bolometric luminosity
density (*z*) at
*z* < 1 because of the lack of CIB data at wavelengths shorter
than 140 *µ*m. Their CSFR in the redshift range *z*
1-4 was found to be
consistent with that inferred from galaxy observations
(Steidel et al. 1999,
Haarsma et al. 2000).

Gispert et al. (2000)
concluded that the far-infrared region of the CIB strongly constrains
the CSFR in the redshift range *z*
1-4. However,
Dwek et al. (1998)
showed that with different assumptions on the spectra of the CIB sources
and the stellar initial mass function, one can obtain the observed CIB
spectrum at wavelengths above 240 *µ*m for two distinctly
different star formation histories [see
Section 5.2.2 and
Figure 8]. The fact that different
star formation histories provide equally good fits to the far-infrared
CIB shows that only deep surveys that resolve the EBL into its
individual sources can provide the definitive star formation history.

Galaxy counts obtained by the *IRAS, ISO*, and SCUBA observations
have already provided useful constraints on the redshift evolution of
the CSFR. The IRAS 60 *µ*m counts suggest strong luminosity
evolution
(1 +
*z*)^{3.2±0.5} up to *z*
0.2
(Bertin et al. 1997).
The ISO counts extended this evolutionary trend to *z*
1
(Puget et al. 1999,
Altieri et al. 1999,
Elbaz et al. 1999).
The SCUBA observations of dust enshrouded sources at *z*
2 suggest that the CSFR at
these redshifts should be moderately declining as (1 +
*z*)^{-1.1}, or at most be constant. Otherwise, the
integrated source light would exceed the submillimeter background
(Dwek et al. 1998,
Blain et al. 1999a,
Takeuchi et al. 2001).