What the Universe Is Made Of

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

Useful constants (CGS) and values:

Constant	Value
$H_0$	$2.27 \times 10^{-18}~\mathrm{s^{-1}}$ ( $= 70~\mathrm{km\,s^{-1}\,Mpc^{-1}}$ )
$G$	$6.67 \times 10^{-8}~\mathrm{cm^3\,g^{-1}\,s^{-2}}$
$m_H$	$1.67 \times 10^{-24}~\mathrm{g}$
$\rho_\text{crit}$	$9.2 \times 10^{-30}~\mathrm{g\,cm^{-3}}$
Density parameters	$\Omega_b \approx 0.05$ , $\Omega_{dm} \approx 0.26$ , $\Omega_m \approx 0.31$ , $\Omega_\Lambda \approx 0.69$ , $\Omega_r \approx 9\times10^{-5}$

Useful relations: Friedmann $H^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2}$ ; critical density $\rho_\text{crit} = 3H_0^2/8\pi G$ ; budget $1 = \Omega_r + \Omega_m + \Omega_\Lambda + \Omega_k$ with $\Omega_i = \rho_i/\rho_\text{crit}$ ; dilution $\rho_r \propto a^{-4}$ , $\rho_m \propto a^{-3}$ , $\rho_\Lambda \approx \text{const}$ .

Conceptual

Problem

⭐⭐ Reading the equation. The Friedmann equation is $H^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2}$ .

(a) Which term sets the expansion rate from the contents of the universe, and which from the shape of space?
(b) If the universe contained only radiation, reason out (do not solve) what happens to $H$ as $\rho$ falls.
(c) What does $k = 0$ mean physically?

Problem

⭐⭐ “Mostly the stars we see”? A student claims: “The universe is mostly the stars, gas, and galaxies we can see — dark matter and dark energy are small corrections.”

(a) Using $\Omega_b \approx 0.05$ , $\Omega_{dm} \approx 0.26$ , $\Omega_\Lambda \approx 0.69$ , identify the error.
(b) What fraction of the universe is the ordinary matter the periodic table is made of?
(c) Is the cosmic budget consistent with the galaxy-by-galaxy ledger from “Weighing the Invisible”?

Problem

⭐ The CMB as evidence. The cosmic microwave background is a near-perfect $2.73~\mathrm{K}$ glow filling the whole sky.

(a) What was happening in the universe when this light was released (name the era)?
(b) Why is it stronger evidence for a hot dense past than simply “old light”?
(c) Name one cosmic quantity the CMB measures.

Calculation

Problem

⭐⭐ The critical density. Use $H_0 = 2.27\times10^{-18}~\mathrm{s^{-1}}$ , $G = 6.67\times10^{-8}$ (CGS).

(a) Compute $\rho_\text{crit} = 3H_0^2/8\pi G$ in $\mathrm{g\,cm^{-3}}$ , with the unit check.
(b) Convert to hydrogen atoms per cubic meter ( $m_H = 1.67\times10^{-24}~\mathrm{g}$ ; $1~\mathrm{m^3} = 10^6~\mathrm{cm^3}$ ).
(c) Compare to the density of air, $\sim 10^{-3}~\mathrm{g\,cm^{-3}}$ .

Problem

⭐⭐ Closing the budget. Measurements give $\Omega_m = 0.31$ , $\Omega_\Lambda = 0.69$ , $\Omega_r \approx 9\times10^{-5}$ .

(a) Compute $\Omega_\text{total} = \Omega_r + \Omega_m + \Omega_\Lambda$ , then $\Omega_k = 1 - \Omega_\text{total}$ .
(b) What geometry does your $\Omega_k$ imply?
(c) If a future survey found $\Omega_\text{total} = 1.05$ , what shape would space be?

Problem

⭐⭐ Matter versus dark energy, today. Use $\rho_\text{crit} = 9.2\times10^{-30}~\mathrm{g\,cm^{-3}}$ , $\Omega_m = 0.31$ , $\Omega_\Lambda = 0.69$ .

(a) Find the present matter density $\rho_m = \Omega_m \rho_\text{crit}$ and dark-energy density $\rho_\Lambda = \Omega_\Lambda \rho_\text{crit}$ .
(b) Which dominates today?
(c) Matter dilutes as $\rho_m \propto a^{-3}$ while $\rho_\Lambda$ stays constant. Was matter denser than dark energy in the past? Reason it out.

Synthesis

Problem

⭐⭐ Which component runs the show? The densities scale as $\rho_r \propto a^{-4}$ , $\rho_m \propto a^{-3}$ , $\rho_\Lambda \approx \text{const}$ .

(a) Going back toward $a \to 0$ , which component grows fastest and so dominated earliest?
(b) Order the three eras of domination from early to late.
(c) Why does dark energy inevitably win at late times, no matter how small it started?

Problem

⭐⭐⭐ The turning point. In 1998, distant Type Ia supernovae were found to be fainter than expected.

(a) Why does “fainter than expected” mean “farther than expected,” and how does that point to acceleration rather than deceleration?
(b) Explain physically why matter decelerates the expansion but dark energy accelerates it.
(c) Why is it fitting that the same standard candles that calibrate $H_0$ also revealed the acceleration?

Problem

⭐⭐ Shape from density.

(a) State the rule connecting $\Omega_\text{total}$ to the three possible geometries.
(b) The CMB measures $\Omega_\text{total} = 1.00 \pm 0.01$ . What geometry, and how does that make the Friedmann budget sum to exactly 1?
(c) Why is “flat” a statement about three-dimensional space, not a claim that the universe is a flat sheet?