3.2 How uncertain is the ionization epoch?

Even if we knew exactly what the initial fluctuations were, and when the first bound systems on each scale formed, the above-mentioned uncertainties would render the ionization redshift is uncertain by at least a factor of 2. This can be easily seen as follows:

Ionization breakthrough requires at least 1 photon for each ionized baryon in the IGM (one photon per baryon is obviously needed; extra photons are needed to balance recombinations, which are more important in clumps and filaments than in underdense regions). An OB star produces 10⁴ - 10⁵ ionizing photons for each constituent baryon, so (again in very round numbers) 10^-3 of the baryons must condense into stars with a standard IMF to supply the requisite UV. Photoionization will be discussed in Madau's contribution to this conference. Earlier references include Ciardi and Ferrara (1997), Gnedin and Ostriker (1998), Madau, Haardt and Rees (1999) and Gnedin (1999).

We can then contrast two cases:
(A) If the star formation were efficient, in the sense that all the baryons that `went non-linear', and fell into a CDM clump larger than the Jeans mass, turned into stars, then the rare 3- peaks on mass-scales 10⁸ M would suffice.

On the other hand:
(B) Star formation could plausibly be so inefficient that less than 1 percent of the baryons in a pregalaxy condense into stars, the others being expelled by stellar winds, supernovae, etc., In this case, production of the necessary UV would have to await the collapse of more typical peaks (1.5-, for instance).

A 1.5- peak has an initial amplitude only half that of a 3- peak, and would therefore collapse at a value of (1 + z) that was lower by a factor of 2. For plausible values of the fluctuation amplitude this could change z_i from 15 (scenario A) to 7 (scenario B). There are of course other complications, stemming from the possibility that most UV photons may be reabsorbed locally; moreover in Scenario B the formation of sufficient OB stars might have to await the build-up of larger systems, with deeper potential wells, in which stars could form more efficiently.

The above examples have assumed a `standard' IMF, and there is actually further uncertainty. If the Population III IMF were biased towards low-mass stars, the situation resembles inefficient star formation in that a large fraction of the baryons (not just the rare 3- peaks) would have to collapse non-linearly before enough UV had been generated to ionize the IGM. By the time this happened, a substantial fraction of the baryons could have condensed into low mass stars. This population could even contribute to the MACHO lensing events (see section 6).