Regarding the words in bold, why is that the case that if the early Universe had not been dominated by the radiation of a fireball, all of the superdense hydrogen would have been fused into helium?

At temperatures of say 100 miliion degress Kelvin, Fusion [thermonuclear] occurs.....like hyrdogen combining into helium in some stellar objects....

But the big bang temperatures were on the order of 10^{30} or 10^{35}, anyway, a LOT more decimal places, a LOT hotter, and so nuclear attraction could not overcome the exhorbitant heat to fuse neclei...that is, radiation blasted stuff apart.

I'm trying to find Peeble's original paper, but I'm guessing that in a matter dominated universe, the universe cools less quickly in response to expansion, which leaves the universe hot enough to fuse all of the protons to helium. In a radiation-dominated universe, the temperature goes down very quickly, and so it freezes the proton/neutron ratio.

This is a guess, and I've been trying to do scaling numbers and I haven't been able to get them to work.

George, I think Twofish's guess is right. Radiation cools faster than matter, in an expanding volume.

So in an expanding world that is mostly radiation, cooling is quick and there is only a brief window of time when helium can be fused.

But in an expanding world that is mostly matter particles, cooling takes longer (requires more expansion) --- perhaps with a longer window there would be a danger that all the H would be fused into He. Bummer!

A partial explanation for the quicker cooling goes like this: with matter particles if you double the scale the volume goes up by 8, so the density goes down to 1/8.
(same number of particles, just in 8 times bigger volume)

But with photons of light, not only are they spread out by the expansion but they are also REDSHIFTED, SO EACH PHOTON HAS ONLY HALF THE ENERGY. So doubling the scale cuts the density of energy down to 1/16 what it was. There are only 1/8 as many photons per unit volume and each one is only worth half as much energy.

When people estimate the amount of expansion (and cooling) after one second, two seconds, etc. they get that the temperature was only right for making helium between 3 and 20 minutes. So there was only a 17 minute window.

See if you can get this toy to work. It's not perfect by a long shot but it does seem to convert from seconds of expansion time into what temperature it was. Just type 180 into the seconds box (leave the power of ten box blank) and then click anywhere else on the spreadsheet.

It is a spreadsheet relating time to temperature. Put in 180 seconds and see if you can get the temperature. It should come out around 1.4 billion Kelvin.
Hotter (i.e. earlier) than that, a necessary intermediate product (deuterium) would not stay together, so helium could not form. The deuterium would keep getting knocked apart. This is called "the deuterium bottleneck".

Then put in 20 minutes, i.e. 1200 seconds and see if you can get the temperature. It should come out around 0.5 billion Kelvin. Colder (i.e. later) than that helium would not form because the deuteriums weren't being banged together hard enough to make them fuse together.

"Radiation domination" is just a technical codeword for how much cooling to expect from a given amount of expansion. You have to know whether to expect energy density to go down to 1/8 or to 1/16 when you double the distance scale.

It's nearly 1 AM here. Maybe I'll say this better tomorrow, or someone else will. Got to sleep now.

For both radiation and matter solutions of ρ(t), with curvature = 0, ρ [itex]\propto[/itex] 1/t^{2} (although the radiation value of ρ is 9/8 of ρ for a matter-only solution). For radiation, a [itex]\propto[/itex] t^{1/2}, whereas for matter only, a [itex]\propto[/itex] t^{2/3}. For both solutions, the temperature scales as 1/a. Therefore it's easy to run the numbers, although you need to plug in a matter:energy ratio at a selected time.
Assume there was lots of matter (as ionized hydrogen) compared to energy at a time when T= 10^{9} K. At this temp (kT~100 keV), the maximum possible energy:matter ratio would have been ~ 10,000:1 because the proton and electron thermal energy would have radiated photons, and the particles also had thermal energy. In the real universe, the matter:energy ratio at the time when T = 10^{9} K was more like 1:10^{5}. This is much lower matter density than the hypothetical high ratio of 10,000:1. The high ratio (matter-dominated) solution would therefore yield much more fusion, because the hydrogen density would have been much higher at the time the temperature passed a billion K.