Dark matter against dark energy at the time of the background

Adapted from IOAA 2008 (Bandung), 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 now has a temperature of 2.73 K, and it set out at $z=1100$. Today the densities of dark energy, dark matter and ordinary matter are $7.1\times {10}^{-30}$, $2.4\times {10}^{-30}$ and $0.5\times {10}^{-30}$ g/cm³. Assuming that dark energy is the energy of the vacuum, what was the ratio of the density of dark matter to that of dark energy when the background radiation set out?

AnswerDark matter was about $4.5\times {10}^8$ times as dense as dark energy. The official solution gives $4.8\times {10}^8$ because it divides by $6.7\times {10}^{-30}$ g/cm³ in place of the given $7.1\times {10}^{-30}$.