How many protons in the universe

What is the probability to create a proton with two kinds of quarks (u and d)?

We know that
1. only protons are stable. Free neutrons (udd) will decay to protons (uud) in 15 minutes.
2. baryons (3-quark configurations) may have different angular momentum (J): 1/2 or 3/2. Protons and neutrons have their higher J companions, while uuu and ddd exist only in J=3/2 due to Pauli exclusion principle. All these four are called Delta particles.
3. anti-quarks form anti-matter, which has the same mass as particles of *ordinary* matter but opposite charge and quantum spin.
4. baryons contain virtual quark-antiquark pairs known as sea quarks, along the valence quarks.

To make life easier, let’s forget these complications for a moment and touch a simple 3-kid problem. What is the probability to find a family with 2 girls among all the families having three children?

You may think as follows. The configurations are {GGG, GGB, GBB, BBB}, where G=girl and B=boy. So the probability (P) = 1/4.

Unfortunately this does not sound right. From our everyday experience, 2-girl families are more common than 3-girl ones!

To correctly calculate the probability, we need to list every configuration with exactly the same possibility. “First girl, then boy, last girl” denotes as GBG, and so on. So the configurations are {GGG, GGB, GBG, BGG, BBG, BGB, GBB,BBB}. Counting{GGB,GBG, BGG} and P=3/8! This means there are 3 times chances for a 3-kid family to have 2 girls than 3 girls. Also notice that when a 2-girl family are expecting, the third kid is equally possible for girl or boy.

Back to the proton problem. The universe was a quark soup right after the big bang.  The *initial* particles were 3/8 protons, 3/8 neutrons, thanks to the insight from the 3-kid problem. The other 2/8 two Delta particles decayed in 10^(-24) seconds to protons. Therefore, the neutron-proton ratio was roughly 3/8:5/8=3:5.

Now comes the part that physics is more than math. When the universe was less than 1 second old, the temperature was so high that the protons and neutrons, with electrons and neutrinos, were in thermal equilibrium and could change one to other freely. The neutron-proton ratio was actually 1:1, before temperature dropped below the neutron-proton mass difference, and froze out at about 1:7. As a result, 7/8 of  the matter in the universe is proton!

Bonus: we can figure out that, based on the neutron-proton ratio, 75% quarks are in hydrogen (1 proton) and 25% helium (2 protons & 2 neutrons) when the big bang nucleosynthesis stopped in about 20 minutes.

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