Big Bang Nucleosynthesis; electron-photon ratio

Thread starter ergospherical
Start date May 18, 2024
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Electron Photon

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In the context of Big Bang Nucleosynthesis (BBN), the baryon number density, denoted as ##n_b##, should account for all baryons, including neutrons, protons, hydrogen, and helium. The discussion highlights that while helium contains both neutrons and protons, the ratio of helium to hydrogen is approximately 1/16. It is important to consider the contributions of free neutrons and protons in the baryon density calculation, especially given the hint to not overlook the terms related to the ratio of neutrons to protons (##n_n/n_p##). The non-relativistic and relativistic expressions for electron and photon number densities are also mentioned as part of the calculations. Overall, the focus is on accurately determining the baryon number density in the context of BBN.

May 18, 2024

ergospherical

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Homework Statement: See the image below!

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I'm asking mainly about part (c). Within the context of BBN, I'm a little unsure how you account for different baryons (i.e. does ##n_b## include neutrons, protons, hydrogen and helium, given that helium itself contains both neutrons and protons?)

For completeness, for part (b) I would just use the non-relativistic number density expression for electrons (given that ##T < m_e##) and the relativistic one for photons:\begin{align*}
n_{e} &= 2\left( \frac{m_e T}{2\pi} \right)^{3/2} e^{-m_e/T} \\
n_{\gamma} &= \frac{2\zeta(3)}{\pi^2} T^3
\end{align*}and take the ratio.

So coming back to (c), we have derived elsewhere that ##n_{\mathrm{He}}/n_{\mathrm{H}} \sim 1/16##. What should I write for the baryon number ##n_b##? At this point I would have thought almost all neutrons be inside helium nuclei, but the question hints not to ignore terms of order ##n_n/n_p##.

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