When radiation and matter were equal

Adapted from IOAA 2016 (Bhubaneswar), 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.

The energy density of radiation in the Universe grows into the past as $(1+z{)}^4$, and that of matter as $(1+z{)}^3$. Today the density parameters of radiation and matter are ${\mathrm{\Omega }}_{\text{r},0}={10}^{-4}$ and ${\mathrm{\Omega }}_{\text{m},0}=0.3$.

(a) Find the redshift $z_{\text{e}}$ at which radiation and matter were equally dense. [3 points]

(b) The radiation has a blackbody spectrum, now at 2.732 K. Estimate its temperature $T_{\text{e}}$ at $z_{\text{e}}$. [4 points]

(c) Estimate the energy of a typical photon of that radiation, in eV, at $z_{\text{e}}$. [3 points]

Answer(a) $z_{\text{e}}\approx 3000$. (b) $T_{\text{e}}\approx 8200$ K. (c) A typical photon had about 3.5 eV, the energy at the peak of the spectrum, about $5kT$. The official solution also accepts $kT=0.71$ eV or $2.7kT=1.9$ eV, and $2.7kT$ is the mean energy of a photon.