How much dark energy

Adapted from IOAA 2023 (Katowice), 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 fluctuations of the cosmic microwave background point to flat space, in which the total density, counting the mass equivalent of every form of energy, equals the critical density ${\rho }_{\text{cr}}=3H_0^2/8\pi G$, which is $9.202\times {10}^{-27}$ kg/m³ for $H_0=70$ km/s/Mpc. The total density of matter, luminous and dark, is estimated at ${\rho }_{\text{m},0}=2.8\times {10}^{-27}$ kg/m³. The standard model makes up the difference with dark energy of constant energy density ${\epsilon }_{\mathrm{\Lambda }}$. Find ${\epsilon }_{\mathrm{\Lambda }}$, and the redshift at which the density of matter, as energy, equalled that of dark energy. Ignore radiation.

Answer${\epsilon }_{\mathrm{\Lambda }}=5.75\times {10}^{-10}$ J/m³, and the two were equal at $z=0.32$.