Annu. Rev. Astron. Astrophys. 1994. 32: 531-590
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5.1. Background Light Constraints

An effect that constrains the number of Population III objects over every mass range between 0.1 and 105 Msun is the generation of background light during stellar main-sequence phase. The fact that the observed background radiation density over all wavebands cannot exceed OmegaRO approx 10-4 in units of the critical density (and it is smaller in most bands) permits a general constraint on Omega*(M). One can obtain more precise constraints by using information about the waveband in which the radiation is expected to reside (Peebles & Partridge 1967. Thorstensen & Partridge 1975, Carr et al 1984, McDowell 1986, Negroponte 1986) but an integrated background light limit has the virtue of generality.

For stars larger than 0.8 Msun, which have already burnt their nuclear fuel, one just compares their total light production to OmegaRO to obtain a constraint on Omega*(M). Since 7 MeV per baryon is released in burning hydrogen to helium, the background light density generated should be OmegaR = 0.007 Omega*fb(1 + z*)-1 in units of the critical density, where z* is the redshift at which the stars burn their fuel (the minimum of the formation redshift zf and the redshift zMS at which the age of the Universe equals their main-sequence time) and fb is the fraction of the star's mass burnt into helium. By using the known dependence of fb and zMS on M, one can predict the value of OmegaR as a function of Omega*, M, and zf. Since the observed background density over all wavebands does not exceed 10-4, this implies a constraint on Omega* as function of M and zf. For stars with M < 0.8 Msun, which are still burning, one compares the product of the luminosity L(M) and the age of the Universe to OmegaRO. The resulting constraints on Omega*(M, z*) are shown in Figure 2; these are somewhat stronger than indicated by Carr et al (1984) because the limits on OmegaRO have improved (See Section 6.1). Peebles & Partridge (1967) used this argument to preclude stars in the mass range 0.3-2.5 Msun from having the critical density.

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