Self-Assessment Quiz — Chapter 24

Test your understanding of the core concepts before moving on. Try to answer each question before checking the solutions at the end.

Q1. (Multiple Choice) At what temperature does the neutron-to-proton ratio freeze out?

(a) $T \sim 10^{12}\,\text{K}$ ($kT \sim 100\,\text{MeV}$) (b) $T \sim 10^{10}\,\text{K}$ ($kT \sim 1\,\text{MeV}$) (c) $T \sim 10^{9}\,\text{K}$ ($kT \sim 0.1\,\text{MeV}$) (d) $T \sim 10^{8}\,\text{K}$ ($kT \sim 0.01\,\text{MeV}$)

Q2. (Multiple Choice) The baryon-to-photon ratio $\eta$ is approximately:

(a) $6 \times 10^{-4}$ (b) $6 \times 10^{-7}$ (c) $6 \times 10^{-10}$ (d) $6 \times 10^{-13}$

Q3. (True/False) The neutron-to-proton ratio at the onset of nucleosynthesis ($t \approx 3\,\text{min}$) is approximately $n/p = 1/7$.

Q4. (Short Answer) What is the "deuterium bottleneck"? Why does nucleosynthesis not begin immediately after the weak freeze-out, even though deuterium formation is exothermic?

Q5. (Multiple Choice) The primordial helium-4 mass fraction $Y_p$ is approximately:

(a) 0.05 (b) 0.12 (c) 0.25 (d) 0.50

Q6. (True/False) The primordial deuterium abundance D/H depends strongly on the baryon-to-photon ratio $\eta$, scaling approximately as $\eta^{-1.6}$.

Q7. (Short Answer) Why does the BBN reaction network not produce significant amounts of elements heavier than lithium-7? Name the two critical mass numbers where no stable nuclei exist.

Q8. (Multiple Choice) The "lithium problem" refers to the fact that:

(a) The BBN prediction for ${}^7\text{Li}/\text{H}$ is $\sim 3\times$ lower than observed (b) The BBN prediction for ${}^7\text{Li}/\text{H}$ is $\sim 3\times$ higher than observed (c) Lithium is not produced in BBN at all, but is observed in old stars (d) The BBN prediction agrees with observations but conflicts with the CMB

Q9. (True/False) The neutron-proton mass difference $Q = 1.293\,\text{MeV}$ plays a critical role in determining the freeze-out $n/p$ ratio and hence $Y_p$.

Q10. (Short Answer) The BBN prediction for D/H agrees with observations to within about 1%. What type of astrophysical objects are used to measure the primordial deuterium abundance, and why are they considered primordial?

Q11. (Multiple Choice) Adding a fourth light neutrino species would:

(a) Decrease $Y_p$ by slowing the expansion rate (b) Increase $Y_p$ by speeding up the expansion rate and causing earlier freeze-out (c) Have no effect on $Y_p$ because neutrinos do not participate in nuclear reactions (d) Decrease $Y_p$ by increasing the neutron decay rate

Q12. (True/False) In the standard BBN scenario, the triple-alpha process ($3\alpha \to {}^{12}\text{C}$) produces significant amounts of carbon.

Q13. (Short Answer) What is the Spite plateau, and why is it important for the lithium problem?

Q14. (Multiple Choice) The baryon density parameter from BBN and from the CMB agree to within:

(a) A factor of 2 (b) 10% (c) 1% (d) 0.01%

Q15. (True/False) The deuterium bottleneck temperature $kT_D \approx 0.07\,\text{MeV}$ is much lower than the deuterium binding energy $B_d = 2.224\,\text{MeV}$ because of the enormous photon-to-baryon ratio.

Q16. (Short Answer) Name two fundamental quantities (other than $\eta$) whose values significantly affect the predicted primordial helium abundance $Y_p$. Briefly explain the physical mechanism for each.

Q17. (Multiple Choice) The dominant source of theoretical uncertainty in the BBN prediction for $Y_p$ is:

(a) The baryon-to-photon ratio $\eta$ (b) The neutron lifetime $\tau_n$ (c) The $d(p,\gamma){}^3\text{He}$ reaction rate (d) The number of neutrino species

Q18. (Short Answer) BBN and the CMB both constrain the baryon density $\Omega_b h^2$. Why is the agreement between these two independent measurements so significant? What epoch does each probe?

Solutions

Q1. (b) $T \sim 10^{10}\,\text{K}$ ($kT \sim 1\,\text{MeV}$). This is when the weak interaction rate drops below the Hubble expansion rate.

Q2. (c) $6 \times 10^{-10}$. There are approximately 1.6 billion photons for every baryon.

Q3. True. The ratio freezes out at $\sim 1/6$ and then decreases to $\sim 1/7$ due to free neutron decay during the $\sim 3\,\text{min}$ delay imposed by the deuterium bottleneck.

Q4. The deuterium bottleneck is the delay between weak freeze-out ($t \sim 1\,\text{s}$) and the onset of nucleosynthesis ($t \sim 3\,\text{min}$) caused by the photodissociation of deuterium. Although deuterium formation ($p + n \to d + \gamma$) is exothermic, the enormous number of photons per baryon ($\eta^{-1} \sim 10^9$) means that the high-energy tail of the Planck distribution contains enough energetic photons to destroy deuterium until the temperature drops to $kT \sim 0.07\,\text{MeV}$ — roughly 30 times lower than the deuterium binding energy.

Q5. (c) 0.25. This follows from the neutron-to-proton ratio of $\sim 1/7$ at the start of nucleosynthesis.

Q6. True. D/H $\propto \eta^{-1.6}$, making deuterium the most sensitive BBN "baryometer."

Q7. There are no stable nuclei with mass number $A = 5$ (${}^5\text{He}$ and ${}^5\text{Li}$ are unbound) or $A = 8$ (${}^8\text{Be}$ decays in $\sim 10^{-16}\,\text{s}$). These mass gaps prevent the reaction network from building nuclei beyond ${}^7\text{Li}/{}^7\text{Be}$, because there is no way to add a nucleon to ${}^4\text{He}$ and produce a stable product.

Q8. (b) The BBN prediction for ${}^7\text{Li}/\text{H}$ is approximately $5 \times 10^{-10}$, while observations of old, metal-poor halo stars give $\sim 1.6 \times 10^{-10}$ — a factor of $\sim 3$ discrepancy.

Q9. True. The equilibrium $n/p$ ratio at freeze-out is $\exp(-Q/kT_f)$, so $Q$ directly determines how many neutrons are available for helium synthesis.

Q10. Primordial deuterium is measured in high-redshift ($z \sim 2$–$4$), low-metallicity gas clouds observed as absorption systems in quasar spectra (damped Lyman-$\alpha$ systems). These clouds have undergone minimal stellar processing, so their D/H ratio is close to the primordial value. Additionally, since deuterium is only destroyed (never produced) in stars, any observed D/H is a lower limit on the primordial value.

Q11. (b) A fourth neutrino species increases $g_*$, which increases the expansion rate $H \propto \sqrt{g_*}$. Faster expansion means the weak interactions freeze out earlier (at a higher temperature), giving a larger $n/p$ ratio and hence a higher $Y_p$.

Q12. False. The triple-alpha process requires high density ($\rho \sim 10^5\,\text{g/cm}^3$) because its rate scales as $\rho^2$. The BBN baryon density is only $\sim 10^{-5}\,\text{g/cm}^3$, making the triple-alpha rate completely negligible.

Q13. The Spite plateau is the observation (by Monique and Francois Spite, 1982) that old, metal-poor halo stars with $[\text{Fe/H}] < -1.5$ all show approximately the same lithium abundance, $A(\text{Li}) \approx 2.2$, independent of temperature and metallicity. This uniformity suggests a primordial origin. However, the plateau value (${}^7\text{Li}/\text{H} \approx 1.6 \times 10^{-10}$) is a factor of $\sim 3$ lower than the BBN prediction, creating the lithium problem.

Q14. (c) 1%. BBN gives $\Omega_b h^2 = 0.0224 \pm 0.0007$ (from deuterium), while the Planck CMB gives $\Omega_b h^2 = 0.02242 \pm 0.00014$. These agree to better than 1%.

Q15. True. The bottleneck temperature is $kT_D \approx B_d / \ln(1/\eta)$, and $\ln(1/\eta) \approx 21$. The enormous photon-to-baryon ratio means even the tiny high-energy tail of the photon distribution is sufficient to destroy deuterium until $T$ drops far below $B_d/k$.

Q16. Two examples: (1) The neutron-proton mass difference $Q = 1.293\,\text{MeV}$: a larger $Q$ shifts the equilibrium toward protons, reducing the freeze-out $n/p$ ratio and hence $Y_p$. (2) The number of neutrino species $N_\nu$: more species increase $g_*$ and the expansion rate, causing earlier freeze-out at higher $T$, larger $n/p$, and higher $Y_p$. Other valid answers: the neutron lifetime $\tau_n$ or the Fermi constant $G_F$.

Q17. (b) The neutron lifetime $\tau_n$. The current uncertainty $\Delta\tau_n \approx 0.6\,\text{s}$ translates to $\Delta Y_p \approx 0.0001$, which is the dominant source of theoretical uncertainty.

Q18. BBN probes nuclear reactions at $t \sim 3\,\text{min}$ ($z \sim 10^9$), while the CMB probes photon-baryon acoustic oscillations at $t \sim 380{,}000\,\text{yr}$ ($z \sim 1100$). These are completely independent physical processes at completely different epochs, separated by a factor of $\sim 10^5$ in time and $\sim 10^6$ in temperature. Their agreement to $\sim 1\%$ on $\Omega_b h^2$ is powerful evidence that the standard hot Big Bang model with known particle physics is correct over this enormous range.