2020

A mirror in an expanding universe

Round: Theory
Points: 10
Level: Silver
Taught in: Section 13.9

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.

This problem also appears among the olympiad problems of Section 13.9, Problems for Chapter 13, which teaches the ideas it uses. Equation and figure numbers refer to the course.