The First Three Minutes (3) ...

The way in which nucleosynthesis proceeds in the first few minutes of the Universe's history depends critically on the ratio of the number of neutrons to the number of protons.
The Neutron to Proton Ratio
The table below summarizes the calculated evolution of this ratio as a function of temperature in the early Universe.

Neutron to Proton Ratio in the Early Universe
Time (s) T(K) nn/np Protons per
1000 Nucleons*		 Neutrons per
1000 Nucleons*
2.3 x 10-8 1 x 1014 1.000 500 500
2.3 x 10-4 1 x 1012 0.985 504 496
2.3 x 10-2 1 x 1011 0.861 537 463
2.3 1 x 1010 0.223 818 182
6.9 5 x 109 0.221 819 181
37 2.5 x 109 0.212 825 175
231 1 x 109 0.164 859 141
*Rounded to nearest integers

The neutron and proton have almost the same mass, but the neutron is slightly more massive (by about 0.14 percent). This favors conversion of neutrons to protons by weak interactions. At very high temperatures the mass difference doesn't matter much and the ratio of neutrons to protons is about one. However, as the temperature drops neutrons are converted to protons and the ratio begins to favor protons. All the neutrons would be converted to protons if the neutrons and protons remained free for a long enough period (a few hours would be sufficient once the temperature is below about 10 billion K). However, if a neutron is bound up in a stable nucleus like helium-4 or deuterium it no longer is susceptible to being converted to a proton. Therefore, the neutron to proton ratio drops as the temperature drops until deuterium can hold together and the neutrons can be bound up in stable nuclei. Calculations indicate that this happens at a temperature of about a billion degrees, by which time we see from the table that the neutron to proton ratio has fallen to a little more than 16 percent.

The Production of Helium-4
Except for generating very small concentrations of helium-3, lithium-7, and deuterium, the essential result of big bang nucleosynthesis is to convert the initial neutrons and protons to helium and free hydrogen. From the table we may estimate how much of each is produced. For convenience, in the table we have also expressed the neutron to proton ratio in terms of the number of nucleons out of every 1000 that are neutrons and the number that are protons. For example, at a temperature of a billion degrees there are 141 neutrons and 859 protons for every 1000 nucleons in the Universe (corresponding to a neutron/proton ratio of 141 / 859 = 0.164). How many helium-4 nuclei can we make from these 1000 nucleons? That is a matter of simple counting. Each helium-4 consists of 2 neutrons and 2 protons. Since there are 141 neutrons available for each 1000 nucleons, we can make 141 / 2 = 71 helium-4 nuclei (rounded to an integer). To make these helium nuclei requires one proton for every neutron used, so the number of protons left over after all the neutrons are bound into helium nuclei is 859 - 141 = 718 protons.

Summary of Big Bang Nucleosynthesis
We conclude that for every 1000 nucleons the Universe is left after the initial period of big bang nucleosynthesis with 71 helium nuclei, 718 protons, no free neutrons, and a trace of other light nuclei that we shall ignore in this discussion. Since helium-4 is about 4 times as massive as a proton, the mass fractions are (4 x 71) / 1000 = 0.28 for helium-4 and 718 / 1000 = 0.72 for hydrogen. Therefore, these very simple considerations suggest that the baryonic matter of the Universe should be about 28 percent helium-4 by mass, with most of the rest hydrogen. Considering the simplicity of our estimate, that is rather close to the 22-24 percent measured abundance for helium-4. More careful considerations than the ones used here give even better agreement with the observations.
Constraints on Dark Matter
This agreement between theory and observation for light-element abundances also constrains the total amount of mass in the Universe that can be in baryons. That constraint is the basis for our earlier assertion that most of the dark matter dominating the mass of the Universe cannot be ordinary baryonic matter. If enough baryons were present in the Universe to make that true, and our understanding of the big bang is anywhere near correct, the distribution of light element abundances would have to differ substantially from what is observed. The implication is that the matter that we are made of (baryonic matter) is but a small impurity compared to the dominant matter in the universe (nonbaryonic matter). As someone has put it, "not only are we not the center of the Universe, we aren't even made of the right stuff!" Furthermore, we don't yet even know what the right stuff is!

Matter Dominated Universe

The analysis of the main text indicates that for every massive baryon in the present Universe there are several billion photons. However, each baryon carries much more energy that each photon because of their large mass and the Einstein mass-energy relation. When this is accounted for, the baryons of the present Universe are found to have more than a thousand times as much mass-energy as the photons. This, and the large amount of cold dark matter, are the reasons that the present Universe is matter (and dark energy) dominated.

The Baryon to Photon Ratio
The above right figure illustrates a comparison of calculated abundances with observed abundances for the light elements produced mostly in the big bang (D is deuterium). The regions shaded in dark blue are excluded as possibilities by experimental observations. For example, observations indicate with high certainty that the abundance of helium-4 in the Universe can be no more than 24 percent and no less than 22 percent. Therefore, only the part of the helium-4 curve shaded in yellow is consistent with the observed amount of helium-4. Considerations such as these allow us to fix with considerable confidence the quantity on the horizontal axis, which is the ratio of the number of baryons to number of photons in the present Universe. The total number of each kind of particle is not expected to change in the absence of interactions, so this ratio is also characteristic of that at the time when matter and radiation decoupled.
The Constraint from Nucleosynthesis
The only values permitted for the baryon to photon ratio by the observed abundances of the light nuclei included in the plot lie in a band that brackets the four vertical dotted lines. The analysis concludes that there are about 3-4 billion photons for every baryon in the present Universe (but the equivalent mass of these photons as obtained from the Einstein mass-energy relation is at least 10,000 times less than the total mass contained in visible and dark-matter massive particles). In terms of actual number densities, there are about 400 million photons in each cubic meter of the Universe, but only about 1 baryon for every 5 cubic meters of space. Most of these baryons are neutrons and protons. As we shall see, most of the photons are in the cosmic microwave background radiation.
Cosmological Constraint on the Number of Neutrino Families
One of the successes of the hot big bang theory is that the observed abundance of light elements, coupled with the theoretical understanding of big bang nucleosynthesis, tells us something about neutrinos. We saw earlier in the discussion of solar neutrinos in Chapter 18 and dark matter in Chapter 24 that the known neutrinos come in three families. This number of families is favored in the simplest elementary particle theories, but in principle there could be additional families that are not yet discovered. However, the successful predictions of big bang nucleosynthesis require that there be no more than 4 such families total. High-energy particle physics experiments have now found more directly that (with certain technical theoretical assumptions) the number of neutrino families is 3, confirming the limit placed by big bang nucleosynthesis.

This animation illustrates the first three minutes.