Annu. Rev. Astron. Astrophys. 1994. 32: 531-590 Copyright © 1994 by Annual Reviews. All rights reserved

5.3. Helium Constraints

Although stars return helium to the background Universe in most mass ranges, the associated constraints on the fraction of the Universe going into Population III stars are only weak because of the uncertainties in the primordial helium abundance. However, the helium limit is important in the M > M_c range because there may be no heavy element yield here. Because the pulsational instability leads to mass-shedding of material convected from its core, a VMO is expected to return helium to the background medium during core-hydrogen burning (Bond et al 1983). The net yield depends sensitively on the mass loss fraction _L. If this is very high, the yield will be low because most of the mass will be lost before significant core burning occurs. However, for _L below the critical value (1 - Y_i) / (2 - Y_i), the mass loss is always slower than the shrinkage of the convective core and one can show that the fraction of mass returned as new helium is

(5.2)

Here Y_i is the initial (primordial) helium abundance and the equality sign on the right applies only if _L has the critical value. This does not impose a useful constraint on the number of VMOs if _L is well below the critical value since Y is then very small. However, there is some indication from numerical calculations that hydrogen-shell burning may produce a super-Eddington luminosity which completely ejects the stellar envelope (Woosley & Weaver 1982, Bond et al 1984). This would guarantee the maximal helium production permitted by Equation (5.2) and have profound cosmological implications. If Y_i = 0.23, corresponding to the conventional primordial value, Y = 0.17, so one would substantially overproduce helium if much of the Universe went into VMOs. In this case, only black holes in the mass range above 10⁵ M could be viable candidates for the dark matter. On the other hand, if Y_i = 0, then Y = 0.25, which is tantalizingly close to the standard primordial value. This raises the question of whether the Population III VMOs invoked to produce the dark matter might also generate the helium usually attributed to cosmological nucleosynthesis. Of course, the added attraction of the hot Big Bang model is that it predicts the observed abundances of other light elements. One might conceivably generate these elements by invoking high energy photons from accreting black holes to spallate helium-either within the surrounding accretion tori (Rees 1984, Ramadurai & Rees 1985; Jin 1989, 1990) or in the background Universe (Gnedin & Ostriker 1992, Gnedin et al 1994); however, these models seem somewhat contrived.