Big Bang nucleosynthesis

Adapted from IOAA 2025 (Mumbai), theory problem 10. We have restated it in our own words and kept the original data. The parts that need the problem's own graphs, 2b and 6, are left out, so the points add up to 27 of the original 35. The solution was last checked on 21 October 2026.

In the radiation era the scale factor grows as $a\propto t^{1/2}$. Until $t_{\text{wk}}=1.70$ s, when $k{T}_{\text{wk}}=800$ keV, reactions keep neutrons and protons in balance, with the number of each in proportion to $m^{3/2}\exp (-m{c}^2/kT)$. After that, free neutrons decay into protons with a half-life of 610.4 s. Use $m_{\text{n}}{c}^2=939.565$ MeV and $m_{\text{p}}{c}^2=938.272$ MeV.

(1) Find the share of neutrons, $X_{\text{n,wk}}=N_{\text{n}}/(N_{\text{n}}+N_{\text{p}})$, at $t_{\text{wk}}$. [4 points]

(2) The photons keep a blackbody spectrum. Find the power $\beta$ in $T\propto a^{\beta }$. [2 points]

(3) After $t_{\text{wk}}$ the formation of deuterium follows the simplified Saha equation of (13.16), with $\eta =6.1\times {10}^{-10}$ and $B_{\text{D}}=2224.6$ keV. Find the temperature $k{T}_{\text{nuc}}$ at which $N_{\text{D}}=N_{\text{n}}$, and the time $t_{\text{nuc}}$ at which it is reached, assuming that then all free neutrons go into helium at once. [10 points]

(4) Find $X_{\text{n,nuc}}$ just before $t_{\text{nuc}}$. [5 points]

(5) Estimate the helium fraction by mass, $Y_{\text{prim}}$, taking $m_{\text{p}}=m_{\text{n}}$ and $m_{\text{He}}=4m_{\text{n}}$. [3 points]

(6) If $\eta$ were smaller, would $N_{\text{D}}/N_{\text{n}}$ at a given temperature, $T_{\text{nuc}}$, $t_{\text{nuc}}$, $X_{\text{n,nuc}}$ and $Y_{\text{prim}}$ be larger or smaller? [3 points]

Answer(1) 0.166. (2) $\beta =-1$. (3) $k{T}_{\text{nuc}}=66.0$ keV at $t_{\text{nuc}}=250$ s. (4) 0.125. (5) 0.250. (6) The ratio is smaller, $T_{\text{nuc}}$ lower, $t_{\text{nuc}}$ later, and $X_{\text{n,nuc}}$ and $Y_{\text{prim}}$ smaller.