Exercises — Chapter 24

Thermal History and Freeze-Out

Problem 24.1 ⭐ The temperature-time relation in the radiation-dominated era is $T \approx 1.5 \times 10^{10}\,\text{K} \cdot (t/\text{s})^{-1/2}$ (for $g_* = 10.75$).

(a) At what time does the universe reach $T = 10^{10}\,\text{K}$? $T = 10^9\,\text{K}$? $T = 10^8\,\text{K}$?

(b) What is the characteristic timescale for temperature to decrease by a factor of 2? By a factor of 10? Express in terms of the current temperature.

(c) How long does the "first three minutes" actually last, from $t = 1\,\text{s}$ to $t = 20\,\text{min}$? What is the temperature range spanned?

Problem 24.2 ⭐ The effective number of relativistic degrees of freedom at the BBN epoch is $g_* = 10.75$.

(a) Verify this by counting the contributions from photons ($g = 2$), electrons and positrons ($g = 2 \times 2 = 4$), and three neutrino species plus antineutrinos ($g = 3 \times 2 = 6$), with the $7/8$ factor for fermions.

(b) After $e^+e^-$ annihilation ($T \ll 0.5\,\text{MeV}$), what is $g_*$ if only photons and (decoupled) neutrinos remain? (Remember that neutrinos are now at a lower temperature: $T_\nu = (4/11)^{1/3} T_\gamma$.)

(c) How does the time-temperature relation change when $g_*$ changes? Specifically, by what factor does the coefficient in $T \propto t^{-1/2}$ change between the two cases?

Problem 24.3 ⭐⭐ Derive the equilibrium neutron-to-proton ratio.

(a) Starting from the Boltzmann distribution, show that in thermal equilibrium the ratio of neutron to proton number densities is:

$$\frac{n_n}{n_p} = \left(\frac{m_n}{m_p}\right)^{3/2} \exp\left(-\frac{(m_n - m_p)c^2}{kT}\right)$$

(b) Argue that $(m_n/m_p)^{3/2} \approx 1$ to excellent approximation. What is the numerical value?

(c) Plot $n/p$ vs. $kT$ from $kT = 5\,\text{MeV}$ to $kT = 0.1\,\text{MeV}$. At what temperature is $n/p = 1/2$? $n/p = 1/6$?

(d) If the neutron-proton mass difference were $Q = 2.0\,\text{MeV}$ instead of $1.293\,\text{MeV}$, what would $n/p$ be at freeze-out ($kT_f = 0.8\,\text{MeV}$)?

Problem 24.4 ⭐⭐ The weak interaction rate governing $n \leftrightarrow p$ scales as $\Gamma_w \approx 0.40\,\text{s}^{-1} (T/10^{10}\,\text{K})^5$, and the Hubble rate is $H \approx 0.68\,\text{s}^{-1} (T/10^{10}\,\text{K})^2$.

(a) Find the freeze-out temperature by setting $\Gamma_w = H$.

(b) At the freeze-out temperature, what is $\Gamma_w/H$? Is this consistent with calling it "freeze-out"?

(c) Calculate the ratio $\Gamma_w/H$ at $T = 3 \times 10^{10}\,\text{K}$ and at $T = 3 \times 10^9\,\text{K}$. By what factor does the ratio change over this temperature range?

(d) The freeze-out is not instantaneous. Estimate the "freeze-out width" — the temperature range over which $\Gamma_w/H$ drops from 10 to 0.1.

Problem 24.5 ⭐⭐ Free neutron decay modifies the $n/p$ ratio between freeze-out and nucleosynthesis.

(a) Show that if the initial $n/p$ ratio is $(n/p)_f$, then after time $t$ the ratio evolves as:

$$\frac{n}{p}(t) = \frac{(n/p)_f \, e^{-t/\tau_n}}{1 + (n/p)_f(1 - e^{-t/\tau_n})}$$

(Hint: $n_n(t) = n_n(0) e^{-t/\tau_n}$, and every neutron that decays becomes a proton.)

(b) Using $(n/p)_f = 1/6$ and $\tau_n = 879.4\,\text{s}$, calculate $n/p$ at $t = 60\,\text{s}$, $120\,\text{s}$, $180\,\text{s}$, and $300\,\text{s}$.

(c) If the neutron lifetime were $\tau_n = 600\,\text{s}$ (hypothetical), what would $n/p$ be at $t = 180\,\text{s}$? What would $Y_p$ be?

(d) If the neutron lifetime were $\tau_n = 1200\,\text{s}$, what would $n/p$ and $Y_p$ be?

Problem 24.6 ⭐⭐ Sensitivity of $Y_p$ to fundamental parameters.

(a) Using $Y_p = 2(n/p)/[1 + (n/p)]$ with $n/p = \exp(-Q/kT_f) \cdot \exp(-t_D/\tau_n)$, calculate $\partial Y_p / \partial Q$ at the standard values. Express your answer in units of MeV$^{-1}$.

(b) Calculate $\partial Y_p / \partial \tau_n$ at the standard values. Express in s$^{-1}$. How large a change in $\tau_n$ would produce a 1% change in $Y_p$?

(c) The number of neutrino species affects $Y_p$ through $\Delta Y_p \approx 0.013 \Delta N_\nu$. If the observed $Y_p = 0.245 \pm 0.004$, what range of $N_\nu$ is consistent at $2\sigma$? Does this exclude a fourth light neutrino species?

The Deuterium Bottleneck

Problem 24.7 ⭐ The photon number density at temperature $T$ is $n_\gamma = (2\zeta(3)/\pi^2)(kT)^3$ in natural units, or numerically $n_\gamma = 2.03 \times 10^7 (T/\text{K})^3\,\text{cm}^{-3}$.

(a) At $T = 10^{10}\,\text{K}$, calculate $n_\gamma$. What is the baryon number density $n_b = \eta n_\gamma$?

(b) What is the baryon mass density $\rho_b = m_N n_b$ at this temperature? Express in g/cm$^3$.

(c) At $T = 10^9\,\text{K}$ (the onset of nucleosynthesis), calculate $n_b$ and $\rho_b$. Compare to the density of air at sea level ($\rho_{\text{air}} \approx 1.2 \times 10^{-3}\,\text{g/cm}^3$).

Problem 24.8 ⭐⭐ The nuclear Saha equation for deuterium equilibrium.

(a) Starting from Eq. (24.8), show that the deuterium-to-baryon ratio can be written as:

$$\frac{n_d}{n_b} = C \cdot \eta \cdot T_9^{3/2} \exp(25.82/T_9)$$

where $T_9 = T/10^9\,\text{K}$ and $C$ is a numerical constant. Find $C$.

(b) Plot $n_d/n_b$ vs. $T_9$ from $T_9 = 10$ to $T_9 = 0.5$ for $\eta = 6 \times 10^{-10}$.

(c) At what temperature does $n_d/n_b = 1$? (This defines the deuterium bottleneck temperature.) Solve numerically or graphically.

(d) Repeat for $\eta = 10^{-9}$ and $\eta = 10^{-10}$. How does the bottleneck temperature depend on $\eta$?

Problem 24.9 ⭐⭐⭐ The deuterium bottleneck temperature from the Saha equation.

(a) Starting from $n_d/n_b = 1$, show that the bottleneck temperature satisfies:

$$\frac{B_d}{kT_D} = \ln\left(\frac{1}{\eta}\right) + \frac{3}{2}\ln\left(\frac{m_N c^2}{kT_D}\right) + \text{const.}$$

Derive the constant.

(b) Solve this transcendental equation iteratively for $\eta = 6.14 \times 10^{-10}$. Start with $kT_D^{(0)} = B_d/\ln(1/\eta)$ and iterate.

(c) Show that the bottleneck temperature depends only logarithmically on $\eta$: $kT_D \propto 1/\ln(1/\eta)$. If $\eta$ increases by a factor of 10, by how much does $kT_D$ change?

The Reaction Network and Abundances

Problem 24.10 ⭐ The helium-4 mass fraction from neutron counting.

(a) If the $n/p$ ratio at the start of nucleosynthesis is $n/p = 1/7$, and all neutrons are incorporated into ${}^4\text{He}$, show that $Y_p = 2(n/p)/(1+n/p) = 1/4$.

(b) If 1% of neutrons remain in deuterium and 0.5% in ${}^3\text{He}$, what is the corrected $Y_p$?

(c) In a sample of primordial matter (no stellar processing), what fraction of atoms are helium? What fraction of the mass is helium?

Problem 24.11 ⭐ Mass gaps and the limits of BBN.

(a) Explain why the absence of stable nuclei at $A = 5$ stops the reaction ${}^4\text{He} + n \to {}^5\text{He}$ from building heavier elements. What is the half-life of ${}^5\text{He}$?

(b) ${}^8\text{Be}$ is unbound by only 92 keV. Calculate its lifetime against decay to two alpha particles using a simple estimate: $\tau \sim \hbar/\Gamma$ with $\Gamma \sim 92\,\text{keV}$. Compare to the BBN timescale ($\sim 100\,\text{s}$).

(c) In stellar interiors, the triple-alpha process bridges the $A = 8$ gap. Its rate scales as $\rho^2$. If the triple-alpha rate is adequate at $\rho = 10^5\,\text{g/cm}^3$ (helium-burning core), by what factor is it suppressed at $\rho = 10^{-5}\,\text{g/cm}^3$ (BBN)? Is this consistent with negligible triple-alpha production during BBN?

Problem 24.12 ⭐⭐ The BBN reaction flow.

(a) Draw a schematic diagram showing the main reaction pathways during BBN. Use boxes for the eight species ($n$, $p$, $d$, ${}^3\text{H}$, ${}^3\text{He}$, ${}^4\text{He}$, ${}^7\text{Li}$, ${}^7\text{Be}$) and arrows for the reactions.

(b) Identify the "fast" reactions (those with large cross sections that proceed quickly) and the "slow" reactions (bottlenecks). Which reaction limits the overall rate of helium production?

(c) In the reaction $d + d \to {}^3\text{He} + n$ (R2), the $Q$-value is 3.269 MeV. What is the kinetic energy of the outgoing neutron in the center-of-mass frame? (Assume the deuterium nuclei are approximately at rest — valid when $kT \ll Q$.)

Problem 24.13 ⭐⭐ Lithium-7 production channels.

(a) At low $\eta$ ($\eta < 3 \times 10^{-10}$), the direct production channel ${}^3\text{H} + {}^4\text{He} \to {}^7\text{Li} + \gamma$ dominates. At high $\eta$ ($\eta > 3 \times 10^{-10}$), the indirect channel via ${}^7\text{Be}$ dominates. Explain qualitatively why the ${}^7\text{Be}$ channel becomes more important at higher $\eta$.

(b) The destruction reaction ${}^7\text{Li} + p \to {}^4\text{He} + {}^4\text{He}$ has a large cross section (it proceeds through a broad resonance in ${}^8\text{Be}$). Why does this destruction channel not eliminate all ${}^7\text{Li}$?

(c) The net ${}^7\text{Li}$ abundance curve (as a function of $\eta$) has a minimum near $\eta \approx 3 \times 10^{-10}$. Sketch this curve and label the dominant production mechanism on each side of the minimum.

Problem 24.14 ⭐⭐⭐ Write the full rate equations for the minimal BBN network.

(a) Write the ODE for $dY_d/dt$ including all reactions involving deuterium (R1–R7). Be careful with stoichiometric coefficients — the $d + d$ reactions consume two deuterium nuclei.

(b) Write the ODE for $dY_{{}^4\text{He}}/dt$ including all reactions that produce ${}^4\text{He}$.

(c) Verify conservation: show that $\sum_i A_i (dY_i/dt) = 0$ (baryon number conservation) when you sum over all eight species.

(d) Show that the system is stiff by estimating the ratio of the fastest to slowest reaction rates at $T_9 = 1$.

BBN as a Cosmological Probe

Problem 24.15 ⭐ The baryon density of the universe.

(a) Using $\eta = 6.14 \times 10^{-10}$ and the present CMB temperature $T_0 = 2.725\,\text{K}$, calculate the present baryon number density $n_{b,0}$.

(b) Calculate the present baryon mass density $\rho_{b,0} = m_N n_{b,0}$. Express in g/cm$^3$ and in protons per cubic meter.

(c) Calculate $\Omega_b = \rho_{b,0}/\rho_c$ where $\rho_c = 3H_0^2/(8\pi G) = 1.88 \times 10^{-29} h^2\,\text{g/cm}^3$ is the critical density. Verify that $\Omega_b h^2 \approx 0.022$.

(d) What fraction of the total matter density is baryonic? (Use $\Omega_m h^2 = 0.143$.)

Problem 24.16 ⭐⭐ The Schramm plot.

(a) The deuterium abundance scales as D/H $\propto \eta^{-1.6}$ for $\eta$ near the concordance value. If the observed D/H is measured with 1% uncertainty, what is the corresponding uncertainty in $\eta$?

(b) $Y_p$ depends on $\eta$ as $\partial Y_p / \partial \ln\eta \approx 0.012$. If $Y_p$ is measured with an uncertainty of 0.004, what is the corresponding uncertainty in $\eta$?

(c) Compare the constraining power of deuterium and helium-4 on $\eta$. Which is the better "baryometer" and by what factor?

Problem 24.17 ⭐⭐ Constraining the number of neutrino species.

(a) The standard BBN prediction with $N_\nu = 3$ gives $Y_p = 0.247$. Using $\Delta Y_p = 0.013 \Delta N_\nu$, calculate $Y_p$ for $N_\nu = 2, 3, 4, 5$.

(b) If the observed $Y_p = 0.245 \pm 0.004$ (at $1\sigma$), what is the allowed range of $N_\nu$ at the $2\sigma$ level?

(c) At the $Z$ resonance, LEP measured $N_\nu = 2.984 \pm 0.008$ from the invisible width of the $Z$ boson. Compare the precision of the LEP and BBN constraints. Why is the BBN constraint still valuable despite being less precise?

(d) Could a sterile neutrino with mass $m_s \sim 1\,\text{eV}$ affect BBN? Under what conditions?

Problem 24.18 ⭐⭐⭐ Constraining the variation of fundamental constants.

(a) The weak interaction rate scales as $\Gamma_w \propto G_F^2$. If $G_F$ at the BBN epoch were 5% larger than today, how would the freeze-out temperature change? (Express as a fractional change $\Delta T_f / T_f$.)

(b) What would be the resulting change in $n/p$ at freeze-out and in $Y_p$?

(c) The Coulomb barrier penetration in the Gamow peak depends on the fine-structure constant $\alpha$. If $\alpha$ were 2% larger at the BBN epoch, qualitatively describe how the reaction rates and primordial abundances would change.

Computational Problems

Problem 24.19 ⭐⭐ 🖥️ Implement the simple freeze-out calculation.

(a) Numerically integrate the rate equation:

$$\frac{d(n/p)}{dt} = -\Gamma_{n \to p}(T) \frac{n}{p} + \Gamma_{p \to n}(T) - \frac{n/p}{\tau_n}$$

from $T = 10^{11}\,\text{K}$ to $T = 10^9\,\text{K}$, using the weak rates and the time-temperature relation. Plot $n/p$ vs. $T$.

(b) Compare your numerical result to the instantaneous freeze-out approximation. At what temperature do they diverge by more than 10%?

Problem 24.20 ⭐⭐ 🖥️ Explore the dependence of BBN abundances on $\eta$.

Using the bbn_network.py code from the chapter project:

(a) Run the code for $\eta = 10^{-10}$, $3 \times 10^{-10}$, $6 \times 10^{-10}$, $10^{-9}$, and $3 \times 10^{-9}$.

(b) Plot D/H, $Y_p$, and ${}^7\text{Li}/\text{H}$ vs. $\eta$ on a log-log plot. Reproduce the qualitative features of the Schramm plot.

(c) Verify that D/H $\propto \eta^{-1.6}$ near the concordance value.

(d) Identify the $\eta$ value at the minimum of the ${}^7\text{Li}$ curve.

Problem 24.21 ⭐⭐⭐ 🖥️ Sensitivity to nuclear reaction rates.

Using the bbn_network.py code:

(a) Increase the $d(p,\gamma){}^3\text{He}$ rate by 10%. How does the predicted D/H change?

(b) Increase the ${}^3\text{He}({}^4\text{He},\gamma){}^7\text{Be}$ rate by 20%. How does ${}^7\text{Li}/\text{H}$ change?

(c) Which single reaction rate has the largest effect on D/H? On ${}^7\text{Li}/\text{H}$?

(d) Can any reasonable change in nuclear reaction rates ($\leq 2\sigma$ from measured values) resolve the lithium problem?

Observations and the Lithium Problem

Problem 24.22 ⭐ Deuterium as a one-way tracer.

(a) Explain why deuterium is destroyed but never produced in significant quantities in stars. (Hint: What happens to deuterium at stellar interior temperatures?)

(b) If a gas cloud has D/H $= 2.5 \times 10^{-5}$, is this a lower limit or an upper limit on the primordial value? Why?

(c) The D/H ratio in the local interstellar medium is $\approx 1.5 \times 10^{-5}$, lower than the primordial value. What fraction of the primordial deuterium has been destroyed by stellar processing?

Problem 24.23 ⭐⭐ Measuring primordial helium.

(a) In an H II region with both hydrogen and helium fully ionized, the helium mass fraction $Y$ is related to the He/H number ratio by $Y = 4y/(1 + 4y)$ where $y = n_{\text{He}}/n_{\text{H}}$. Derive this relation.

(b) If the observed He/H number ratio in a metal-poor H II region is $y = 0.082 \pm 0.003$, calculate $Y$ and its uncertainty.

(c) The primordial value $Y_p$ is obtained by extrapolating $Y$ vs. metallicity ($Z$) to $Z = 0$. If $Y = 0.256 \pm 0.003$ at $Z = 0.004$ (a metal-poor H II region) and the slope $dY/dZ \approx 2$, estimate $Y_p$.

Problem 24.24 ⭐⭐ The Spite plateau.

(a) The lithium abundance in metal-poor halo stars is conventionally expressed as $A(\text{Li}) = \log_{10}(\text{Li/H}) + 12$. If $A(\text{Li}) = 2.2$, what is Li/H?

(b) The BBN prediction is ${}^7\text{Li}/\text{H} = 5.2 \times 10^{-10}$. What is the corresponding $A(\text{Li})$?

(c) What is the discrepancy in dex (logarithmic units)?

(d) If you were told that old, metal-poor stars deplete their surface lithium by a factor of 3 through some stellar mechanism, would this resolve the lithium problem? Explain.

Problem 24.25 ⭐⭐⭐ Proposed solutions to the lithium problem.

(a) Nuclear physics solution: The main production channel for ${}^7\text{Li}$ (at high $\eta$) goes through ${}^7\text{Be}$. The destruction reaction ${}^7\text{Be} + n \to {}^7\text{Li} + p$ competes. If the ${}^7\text{Be}(n,p){}^7\text{Li}$ reaction rate were 50% higher than currently measured, would this help? Why or why not? (Think about what then happens to the ${}^7\text{Li}$.)

(b) Stellar depletion: Lithium is destroyed at temperatures $T > 2.5 \times 10^6\,\text{K}$ by ${}^7\text{Li}(p,\alpha){}^4\text{He}$. In main-sequence halo stars, the base of the convection zone is at $T \sim 2 \times 10^6\,\text{K}$. Explain why the Spite plateau (uniform $A(\text{Li})$ across different stellar temperatures) is problematic for simple convective depletion models.

(c) New physics: A late-decaying particle (mass $\sim 100\,\text{GeV}$, lifetime $\sim 10^3\,\text{s}$) that decays during BBN could inject energetic photons or hadrons. Qualitatively, how could this affect the ${}^7\text{Be}$ abundance?

Advanced and Research Problems

Problem 24.26 ⭐⭐⭐ Derive the correction to $N_{\text{eff}}$ from non-instantaneous neutrino decoupling.

(a) Neutrinos decouple when $\Gamma_{\nu e} \sim G_F^2 T^5$ drops below $H$. Show that the decoupling temperature is $T_{\nu,\text{dec}} \approx 2\,\text{MeV}$.

(b) $e^+e^-$ annihilation ($T \lesssim 0.5\,\text{MeV}$) heats the photon bath. In the instantaneous decoupling limit, the neutrino temperature after $e^+e^-$ annihilation is $T_\nu = (4/11)^{1/3} T_\gamma$. Derive this ratio from entropy conservation.

(c) Since $T_{\nu,\text{dec}} \approx 2\,\text{MeV}$ is not infinitely far above the $e^+e^-$ annihilation temperature $\sim 0.5\,\text{MeV}$, some energy leaks into the neutrino sector. This gives $N_{\text{eff}} = 3.044$ instead of 3.000. The correction $\delta N_{\text{eff}} = 0.044$ corresponds to a fractional increase in neutrino energy density of 0.044/3 = 1.5%. Estimate the resulting shift in $Y_p$.

Problem 24.27 ⭐⭐⭐ Research problem: The deuterium abundance in the local bubble.

Recent measurements suggest that the D/H ratio varies significantly within the local interstellar medium, from $\sim 0.5 \times 10^{-5}$ to $\sim 2.2 \times 10^{-5}$, even over relatively short distances. Research this topic and answer:

(a) What physical mechanism can produce local variations in D/H without invoking different amounts of stellar processing?

(b) How does deuterium depletion onto dust grains affect the gas-phase D/H measurement?

(c) What does the highest local D/H value ($\sim 2.2 \times 10^{-5}$) tell us about the primordial value?

Problem 24.28 ⭐⭐⭐ Research problem: ${}^6\text{Li}$ in metal-poor stars.

Some observers have claimed detections of ${}^6\text{Li}$ (in addition to ${}^7\text{Li}$) in metal-poor halo stars at levels ${}^6\text{Li}/{}^7\text{Li} \sim 0.05$ — far above the BBN prediction of ${}^6\text{Li}/\text{H} \sim 10^{-14}$.

(a) Research the observational status of this claim. Is it confirmed?

(b) What astrophysical processes could produce ${}^6\text{Li}$ (cosmic-ray spallation)?

(c) If the ${}^6\text{Li}$ detection were real and primordial, what would it imply about new physics during BBN?