Problems for Chapter 13

Builds on: Sections 13.1 to 13.8
Reading time: 50 min
Exercises: 0
Olympiad problems: 2

The past problems in the earlier sections of this chapter each used one or two ideas at a time. The two in this section join them up. The first, from IOAA 2018, follows the Universe back from today to the first second, using the Friedmann equation, the way each kind of energy thins out, the background radiation and a little dimensional analysis. It is long, at 75 points, but every part is short and most parts use one result of the chapter. The second is a thought experiment in which light bounces off a mirror far from the Sun while the space between them grows.

13.9.1From textbook to olympiad

Part (i) of the first problem below asks for the redshift at which matter and radiation had equal densities, in terms of $Ω_{m0}$ and $h$ . The method of Section S2 takes us from the chapter to the solution in these steps.

We start by drawing it. On a graph of the logarithm of density against the logarithm of the scale factor, matter is a straight line of slope $- 3$ and radiation a steeper line of slope $- 4$ , as in Figure 13.3. Going back in time, the radiation line climbs faster and crosses the matter line. The crossing is the moment we want.
We name the physics. Matter thins out as $(1 + z)^{3}$ and radiation as $(1 + z)^{4}$ ((13.7)), so the ratio of radiation to matter grows as $1 + z$ .
We work backwards from the target. At equality $Ω_{m0} (1 + z)^{3} = Ω_{r0} (1 + z)^{4}$ , so $1 + z_{eq} = Ω_{m0} / Ω_{r0}$ . Everything now rests on finding $Ω_{r0}$ .
We use every given. The problem gives the energy density of blackbody radiation, the temperature of the background today and the extra 68 per cent from neutrinos. The first two give the density of photons, the critical density turns it into $Ω_{γ 0}$ , and the neutrinos multiply it by 1.68. Because the answer is asked for in terms of $h$ , we keep $H_{0} = 100 h$ km/s/Mpc as a symbol.
We check the answer. With $Ω_{m0} = 0.3$ and $h = 0.7$ the result is a redshift of about 3500. This is close to the 3570 found in Section 13.4, whose model has a slightly smaller density of radiation, and to the crossing in Figure 13.3.

Olympiad problemsPast IOAA problems, restated2 problems

2018The thermal history of the Universe75 points

Adapted from IOAA 2018 (Beijing), theory problem 11. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

The Friedmann equation for a uniform universe can be written

{(\frac{\dot{a}}{a})}^{2} = \frac{8 π G}{3} (ρ_{m} + ρ_{r}) + \frac{Λ c^{2}}{3} - \frac{k c^{2}}{a^{2}},

where $a$ is the scale factor, $ρ_{m}$ the density of matter, ordinary and dark, $ρ_{r}$ the density of radiation, $Λ$ the cosmological constant and $k$ the curvature of space. The Hubble parameter is $H = \dot{a} / a$ , and a subscript 0 marks a value today. Take $H_{0} = 2.19 \times 10^{- 18}$ s⁻¹, the value the official solution uses, and $T_{0} = 2.73$ K for the background radiation today.

(a) What are the dimensions of $H$ ? Define a timescale for the expansion from it, the Hubble time, and find its value today. (5 points)

(b) The critical density is the density of matter that a flat universe with no radiation and no dark energy needs for its expansion. Express it in terms of $H$ and $G$ , and find its value today. (5 points)

(c) With the density parameters $Ω_{i} = ρ_{i} / ρ_{c}$ , the Friedmann equation becomes $Ω_{m} + Ω_{r} + Ω_{Λ} + Ω_{k} = 1$ . Find $Ω_{Λ}$ and $Ω_{k}$ in terms of $H$ , $c$ , $Λ$ , $k$ and $a$ . (6 points)

(d) Each component also obeys the fluid equation $\dot{ρ} + 3 (\dot{a} / a) (ρ + p / c^{2}) = 0$ , where $p$ is its pressure. The pressure of radiation, photons and neutrinos alike, is a third of its energy density. Show that $ρ_{r} \propto (1 + z)^{4}$ . If $\dot{ρ} / ρ = n \dot{a} / a$ , then $ρ \propto a^{n}$ . (7 points)

(e) The density of the cosmological constant does not change. Its pressure is $p = w ρ_{Λ} c^{2}$ with $w$ a whole number. Find $w$ . (4 points)

(f) The Planck time, before which our physics needs a quantum theory of gravity, is a combination of $ℏ$ , $G$ and $c$ with a numerical factor close to 1. Find it by dimensional analysis and estimate its value. (13 points)

(g) The Planck length is $l_{P} = c t_{P}$ . The Planck mass $M_{P}$ is the mass of a black hole whose Schwarzschild radius $2 G M / c^{2}$ is twice the Planck length. Find $M_{P}$ and the energy $M_{P} c^{2}$ in GeV. (7 points)

(h) The energy density of blackbody radiation is $ε_{r} = π^{2} (k_{B} T)^{4} / (15 ℏ^{3} c^{3})$ . Show that the temperature of the background radiation follows $T / (1 + z) =$ constant. (4 points)

(i) Neutrinos add 68 per cent to the energy density of the photons, so $Ω_{r0} = 1.68 Ω_{γ 0}$ . Find the redshift of equality of matter and radiation in terms of $Ω_{m0}$ and $h = H_{0} / (100 {km s}^{- 1} {Mpc}^{- 1})$ . (16 points)

(j) Neutrinos stopped interacting with the rest of the matter when $k_{B} T$ was about 1 MeV, when radiation ruled the expansion. Estimate this time after the Big Bang, using $t = 1 / (2 H)$ . (8 points)

I solved it without the solution

Answer(a) 1/time, and 14.47 Gyr. (b) $3 H^{2} / (8 π G)$ , and $8.58 \times 10^{- 27}$ kg/m³. (c) $Ω_{Λ} = Λ c^{2} / (3 H^{2})$ and $Ω_{k} = - k c^{2} / (a^{2} H^{2})$ . (d) $ρ_{r} \propto (1 + z)^{4}$ . (e) $w = - 1$ . (f) $t_{P} = \sqrt{ℏ G / c^{5}} = 5.4 \times 10^{- 44}$ s. (g) $M_{P} = \sqrt{ℏ c / G}$ , with $M_{P} c^{2} = 1.22 \times 10^{19}$ GeV. (h) $T \propto 1 + z$ . (i) $1 + z_{eq} = 2.39 \times 10^{4} Ω_{m0} h^{2}$ . (j) About 1.3 s, so about a second after the Big Bang. The official solution has 14.46 Gyr, $8.59 \times 10^{- 27}$ kg/m³ and $2.47 \times 10^{- 5}$ in part (i), which differ from ours only in the rounding of constants.

2020A mirror in an expanding universe10 points

Adapted from GeCAA 2020 (Estonia, held online in place of the IOAA), theory problem 3. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

A cosmologist imagines measuring $H_{0}$ in a steady-state universe, in which the distance between two free objects grows as $S = s e^{H_{0} t}$ , where $s$ is the distance at $t = 0$ . A large flat mirror, held facing us by gyroscopes, is placed at a distance $D$ from the Solar System, far from anything that could pull on it. A laser beam is sent from Earth to the mirror, and its reflection comes back after a time $T$ .

(a) Find $H_{0}$ in terms of $D$ , $c$ and $T$ . The approximation $e^{x} \approx 1 + x$ for small $x$ may be used. (7 points)

(b) The mirror is placed near Vega, whose parallax is 0.125″. Estimate how long the experiment takes. (3 points)

I solved it without the solution

Answer(a) $H_{0} = (c T - 2 D) / (D T)$ . (b) About 52 years. The delay caused by the expansion would be only $D H_{0} T / c \approx 3$ s in those 52 years, for $H_{0} = 70$ km/s/Mpc. In our real Universe even that would not be seen, since the space round the Sun and Vega is held together by the Galaxy's gravity and does not expand.