- deuterium/hydrogen -- 7 x 10-5 ---> 2 x 10-5
- helium(3)/hydrogen -- 1.5 x 10-5 ---> 4 x 10-5
- lithium/hydrogen -- ~ 10-10
The constraint arises from the fact that the expansion of the early Universe is dominated by the energy contained in particles like photons, electrons, neutrinos, their anti-particles, and so on. These are light particles. The rate at which the early Universe expands is independent of the density of more massive particles such as protons and neutrons (baryons) (the baryons are simply along for the ride).
Nuclear Reaction Rates in the Early Universe
The yield of Big Bang nucleosynthesis, however, depends strongly upon the density of protons and neutrons in the early Universe. Because nuclear reactions involve only the neutrons and protons, it is the density of the protons and neutrons which controls the rate at which the nuclear reactions proeced; the density is crucial for the rate of nuclear reactions.
The reaction rates for the protons and neutrons (and therefore the yields of Big Bang nucleosynthesis) can be changed by making different assumptions about the baryon density in the Universe. The important point is that this twiddling does not alter the rate at which the Universe expands. This is a nice state of affairs as it then makes it very easy to think about how changes in the baryon density of the Universe change the outcome of Big Bang nucleosynthesis. Detailed calculations lead to the following relationships between Big Bang nucleosynthesis yields and Omega.
The number of helium nuclei is not very sensitive to the baryon density of the current Universe, however, species like deuterium, helium(3; 2 protons + 1 neutron), and lithium are quite sensitive to Omega. The observations of these trace constitutents of the Universe are very difficult, but the following are the best results:
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Based on Big Bang nucleosynthesis, Omega << 1, and baryons cannot close the Universe. If Omega ~ 1, then the Universe must be made up of exotic material.
