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I'm reading through Dodelson chapter on BBN. I'm trying to follow the examples, but having trouble with the basics. First, the proton to neutron ratio is quoted as: $$\frac{n_p}{n_n}=e^{\frac{Q}{T}}$$ Where Q = 1.293 MeV and T is the temperature of the soup. Then they go on to say that at $T_{FO}$ (The freeze-out temperature), of $1 MeV$, the ratio of protons to neutrons is roughly 6. This doesn't make sense as $e^{\frac{1.293}{1}}=3.64$ protons to every neutron.

What am I getting wrong?

Quark Soup
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  • That works for a freeze-out temperature of roughly 0.72 MeV. Generally, temperatures are only known to the order of magnitude (we definitely don't know that the freeze-out temperature was exactly 1 MeV). – probably_someone Jun 08 '20 at 22:15
  • I don't understand. The difference between 0.72 and 1.0 is pretty significant. It's the difference between 3 protons per neutron and 6 at freeze out which would completely alter the composition of light elements. I've seen this in nearly every paper and textbook I've found. Why are they so certain of the ratio when they're apparently very uncertain of the temperature? – Quark Soup Jun 09 '20 at 00:24
  • Indeed, the Wikipedia article on BBN seems to confirm my assessment: "These reactions continued until the decreasing temperature and density caused the reactions to become too slow, which occurred at about T = 0.7 MeV (time around 1 second) and is called the freeze out temperature." – probably_someone Jun 09 '20 at 00:29
  • Anyway, "roughly 6" isn't a whole lot of precision either. I suspect that "roughly 6" and "roughly 1 MeV" weren't meant to be compatible. My point was mainly that it's generally easier to place bounds on the proton-to-neutron ratio (by counting the number of protons and neutrons existing today and extrapolating backward) than it is to determine the temperature of the universe precisely in a period where it was completely opaque (so we can't actually see light emitted from this period of time, or rather, it's hidden behind the horizon of last scattering). – probably_someone Jun 09 '20 at 00:32

1 Answers1

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The ratio you quote is in thermal equilibrium. The whole point of the discussion about freezeout is that the rate $n+\nu\leftrightarrow p+e$ becomes too slow for thermal equilibrium to be maintained.

We can compute the neutron to proton ratio by solving a simple rate equation for $np$ conversion. If the rate is large compared to the expansion rate of the universe the solution is the thermal equilibrium abundance. As the rate drops neutrons start to decay and cannot be replenished sufficiently quickly. We typically defined freezeout by the condition that the expansion rate equals the $np$ conversion rate. Numerically this comes out to be $T\simeq $ 1 MeV. Integrating the rate equation gives the $n/p$ ratio at that temperature. This exercise can be found in standard text books, including Dodelson.

Thomas
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  • Thanks, but I'm not interested in the problem. I've seen it repeated at least a dozen times. I want the solution, with numbers. As I pointed out in the previous post, there's a huge difference between $\simeq 1 MeV$ and 0.72 when it comes to the light element production. So far, I've just seen hand waving. – Quark Soup Jun 09 '20 at 10:13
  • @GluonSoup Well, if you are ultimately interested in element formation then yo have to do the same thing: Integrate rate equations for $pn\to d\gamma$ etc forward in time. There is no shortcut, element formation does not take place in NSE (nuclear statistical equilibrium). For this excercise the whole concept of an $np$ freezeout temperature, and the corresponding equilibrium abundance are irrelevant. They are just numbers that are used to get a rough idea what is going on. The actual numbers always have to come from rate equations. – Thomas Jun 11 '20 at 15:34
  • Thank you. That's what I'm trying to do, but there are several steps involved in what is a very complex calculation. Step One: calculate the p/n ratio. Do you see my issue now? It'd be handy to have some reference where I can check my work before I go on to Step 2. – Quark Soup Jun 11 '20 at 23:54