2024

When the background radiation was infrared

Round: Theory
Points: 10
Level: Bronze
Taught in: Section 13.5

Adapted from IOAA 2024 (Rio de Janeiro), theory problem 5. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

The cosmic microwave background is close to a blackbody spectrum, now at about 3 K.

(a) At what redshift did its spectrum peak at an infrared wavelength of about 0.1 mm? [3 points]

(b) Assuming a flat universe of matter alone with $H_{0} = 70$ km/s/Mpc, how old was the Universe at that redshift? [7 points]

I solved it without the solution

Answer(a) $z \approx 9$ . (b) The Universe was about 0.31 billion years old, 0.29 billion years if $z = 9$ is used. Our actual Universe, with dark energy, was about 0.56 billion years old at that redshift.

This problem also appears among the olympiad problems of Section 13.5, The age and history of the Universe, which teaches the ideas it uses. Equation and figure numbers refer to the course.