The First Three Minutes

Show explicit units, and run a sanity check on every result. Worked solutions are released after the homework due date.

Useful values:

Quantity	Value
Matter density parameter	$\Omega_m \approx 0.31$
Radiation density parameter	$\Omega_r \approx 9 \times 10^{-5}$
CMB temperature today	$T_0 = 2.73~\mathrm{K}$
Temperature at recombination	$T_\text{rec} \approx 3000~\mathrm{K}$
Frozen neutron-to-proton ratio	$n/p \approx 1/7$

Useful relations: densities dilute as $\rho_m \propto a^{-3}$ , $\rho_r \propto a^{-4}$ , $\rho_\Lambda \approx \text{const}$ ; redshift $1+z = 1/a$ (with $a_0 = 1$ ); CMB temperature $T \propto 1/a \propto (1+z)$ ; helium mass fraction $Y = \dfrac{2(n/p)}{1 + (n/p)}$ .

Conceptual

Problem

⭐⭐ Put cosmic history in order. Five epochs/events, scrambled: matter–radiation equality; inflation; recombination (the CMB); Big Bang nucleosynthesis; dark-energy domination.

(a) List them in chronological order, earliest first.
(b) For each transition, name in a few words what changes.
(c) Does BBN come before or after the CMB? State which, and roughly how long after the Big Bang each occurred.

Problem

⭐⭐ “Before the Big Bang.” A student says: “Inflation happened before the Big Bang — first inflation, then a gap, then the Big Bang created everything.”

(a) Identify what is wrong with treating inflation and the Big Bang as separate events.
(b) What does “the Big Bang” actually name?
(c) Where does inflation sit in that picture?

Problem

⭐ The freeze-out window. Big Bang nucleosynthesis lasted only a few minutes.

(a) Why couldn’t helium form much earlier than three minutes? (Think about deuterium.)
(b) Why couldn’t it continue indefinitely?
(c) In one phrase, what is a “freeze-out”?

Calculation

Problem

⭐⭐ When matter took over. Matter and radiation dilute as $\rho_m \propto a^{-3}$ and $\rho_r \propto a^{-4}$ . Today $\Omega_m \approx 0.31$ , $\Omega_r \approx 9\times10^{-5}$ (take $a_0 = 1$ ).

(a) Set $\Omega_m\,a^{-3} = \Omega_r\,a^{-4}$ and solve for the scale factor $a_\text{eq}$ at matter–radiation equality.
(b) Convert to a redshift, $z_\text{eq} = 1/a_\text{eq} - 1$ .
(c) Which component dominated before equality — and what does that say about the era in which BBN occurred?

Problem

⭐⭐ The CMB’s long stretch. The cosmic microwave background was released at recombination, $T_\text{rec} \approx 3000~\mathrm{K}$ ; today $T_0 = 2.73~\mathrm{K}$ . The CMB temperature scales as $T \propto 1/a \propto (1+z)$ .

(a) Find the stretch factor $1+z$ from the temperature ratio.
(b) What redshift is the surface of last scattering?
(c) Compare to $z_\text{eq}$ from the previous problem: was recombination before or after matter–radiation equality?

Problem

⭐⭐ A quarter helium — and a counterfactual. The helium mass fraction is $Y = \dfrac{2(n/p)}{1 + (n/p)}$ , with $n/p$ the frozen neutron-to-proton ratio.

(a) Evaluate $Y$ for the actual $n/p = 1/7$ .
(b) Suppose neutrons had instead frozen out at $n/p = 1/5$ . Recompute $Y$ .
(c) The observed primordial helium is about $25\%$ . What does the agreement tell you, and why is $Y$ so sensitive to early-universe physics?

Synthesis

Problem

⭐⭐⭐ The periodic table as cosmic history. Explain why hydrogen, helium, carbon, oxygen, iron, and the heaviest elements (e.g., gold) point to different astrophysical environments.

(a) Assign each to its cosmic foundry: the Big Bang, stellar interiors, or extreme/explosive events.
(b) Why can’t stars alone account for all the elements?
(c) Read one everyday object (a glass of water, a gold ring) as a multi-chapter cosmic archive.

Problem

⭐⭐ The age of an atom. Consider the atoms in your own body.

(a) Which of your atoms are (mostly) relics of the first three minutes?
(b) Which were forged inside stars and stellar deaths?
(c) In what sense are you made of both the Big Bang and dead stars?

Problem

⭐⭐⭐ From a parallax angle to the Big Bang. The course capstone. Start from a single parallax measurement and trace the observe → model → infer method outward to the first three minutes.

(a) At each milestone, name the observable measured and the hidden quantity inferred: (i) parallax; (ii) a star’s spectrum / HR position; (iii) galaxy rotation speeds; (iv) galaxy redshifts; (v) the Friedmann budget; (vi) the cosmic helium fraction.
(b) Mark where each of the three course-long threads ends: the distance ladder, the origin of the elements, and “reading the equation.”
(c) In one or two sentences, state what unifies all six measurements into a single method.