# Is it a coincidence that Big Bang nucleosynthesis lasted about one free neutron half-life?

The free neutron half-life is about 10 minutes. Big Bang nucleosynthesis, which mostly involves protons and neutrons interacting, lasted about 20 minutes.

Question: Is it a coincidence that these are basically the same time scale?

I'd be interested in either a basic physics connection, or a (weak) anthropic argument. I think maybe a complete answer would require considering both angles -- it seems the two quantities are physically related, so having them equal implies some relation between other cosmological and particle-physics parameters which might have additional anthropic implications.

Here's a second attempt to piece things together. You can see my first, very flawed, attempt in the edit history.

According setion II.A of this article, what happens is that before BBN, neutrons and protons are in equilibrium via weak-mediated interactions, so the neturon-proton ratio is $$e^{-\Delta m / T}$$ where $$\Delta m$$ is the neutron-proton mass gap. What happens is that these interactions slow down and "freeze out" at some temperature $$T_f$$ which I don't fully understand, when the fraction is about 1/5. The fraction goes down a little bit after that (to about 1/7) due to free neutron decay, but the effect is qualitatively small. Then BBN starts (the initiation time is delayed by a bottleneck due to deuterium being relatively unstable) but after that nothing really matters -- essentially all the neutrons are fused with protons into helium (and measurable trace amounts of other light elements). The length of BBN turns out to be basically irrelevant.

So if the lifetime of the free neutron were much smaller, then BBN would not take place (and it seems like this doesn't really trigger anthropic effects). If the lifetime of the free neutron were much larger, then nothing would really change. So on this reading of it, it does basically look like a coincidence. There is some relationship, insofar as the ratio of the scale of weak interactions and the neutron-proton mass gap is important to the story and is presumably also important to the free neutron half-life, but the relationship to the time taken by BBN seems tenuous.

• Assuming there was a big bang, there was only one and it's done and gone. They work backwards from what they know and what appears to be relics of the big bang - and cosmological models have a lot wiggle room. – Cinaed Simson Jul 8 '19 at 7:18
• It seems logic that a process capturing free neutrons cannot last longer than the free neutron themselves. – Alchimista Jul 8 '19 at 10:39
• @Alchimista The half-life isn't an absolute ceiling. It's a statistical average. Some of them decay faster than that, some of them decay (much more) slowly. – zibadawa timmy Jul 8 '19 at 13:25
• @zibadawa timmy. There is nothing naive here if you keep a statistical view the vast majority of neutrons would be gone in ten minutes. Indeed it seems that the process lasted a good 20 minutes. After the kinetics was either zero or very low and a survivibg neutron wouldn't find a proton and so on. I didn't want to go in details of bariosynthesis. Butt its duration must be on the same time scale of the lifetime of the reacting components. – Alchimista Jul 9 '19 at 10:33