The thermal history of the Universe

Adapted from IOAA 2018 (Beijing), theory problem 11. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

The Friedmann equation for a uniform universe can be written

where $a$ is the scale factor, ${\rho }_{\text{m}}$ the density of matter, ordinary and dark, ${\rho }_{\text{r}}$ the density of radiation, $\mathrm{\Lambda }$ the cosmological constant and $k$ the curvature of space. The Hubble parameter is $H=\dot{a}/a$, and a subscript 0 marks a value today. Take $H_0=2.19\times {10}^{-18}$ s⁻¹, the value the official solution uses, and $T_0=2.73$ K for the background radiation today.

(a) What are the dimensions of $H$? Define a timescale for the expansion from it, the Hubble time, and find its value today. (5 points)

(b) The critical density is the density of matter that a flat universe with no radiation and no dark energy needs for its expansion. Express it in terms of $H$ and $G$, and find its value today. (5 points)

(c) With the density parameters ${\mathrm{\Omega }}_i={\rho }_i/{\rho }_{\text{c}}$, the Friedmann equation becomes ${\mathrm{\Omega }}_{\text{m}}+{\mathrm{\Omega }}_{\text{r}}+{\mathrm{\Omega }}_{\mathrm{\Lambda }}+{\mathrm{\Omega }}_k=1$. Find ${\mathrm{\Omega }}_{\mathrm{\Lambda }}$ and ${\mathrm{\Omega }}_k$ in terms of $H$, $c$, $\mathrm{\Lambda }$, $k$ and $a$. (6 points)

(d) Each component also obeys the fluid equation $\dot{\rho }+3(\dot{a}/a)(\rho +p/c^2)=0$, where $p$ is its pressure. The pressure of radiation, photons and neutrinos alike, is a third of its energy density. Show that ${\rho }_{\text{r}}\propto (1+z{)}^4$. If $\dot{\rho }/\rho =n\dot{a}/a$, then $\rho \propto a^n$. (7 points)

(e) The density of the cosmological constant does not change. Its pressure is $p=w{\rho }_{\mathrm{\Lambda }}{c}^2$ with $w$ a whole number. Find $w$. (4 points)

(f) The Planck time, before which our physics needs a quantum theory of gravity, is a combination of $\hslash$, $G$ and $c$ with a numerical factor close to 1. Find it by dimensional analysis and estimate its value. (13 points)

(g) The Planck length is $l_{\text{P}}=c{t}_{\text{P}}$. The Planck mass $M_{\text{P}}$ is the mass of a black hole whose Schwarzschild radius $2GM/c^2$ is twice the Planck length. Find $M_{\text{P}}$ and the energy $M_{\text{P}}{c}^2$ in GeV. (7 points)

(h) The energy density of blackbody radiation is ${\epsilon }_{\text{r}}={\pi }^2(k_{\text{B}}T{)}^4/(15{\hslash }^3{c}^3)$. Show that the temperature of the background radiation follows $T/(1+z)=$ constant. (4 points)

(i) Neutrinos add 68 per cent to the energy density of the photons, so ${\mathrm{\Omega }}_{\text{r0}}=1.68{\mathrm{\Omega }}_{\gamma 0}$. Find the redshift of equality of matter and radiation in terms of ${\mathrm{\Omega }}_{\text{m0}}$ and $h=H_0/(100{\text{ km s}}^{-1}{\text{ Mpc}}^{-1})$. (16 points)

(j) Neutrinos stopped interacting with the rest of the matter when $k_{\text{B}}T$ was about 1 MeV, when radiation ruled the expansion. Estimate this time after the Big Bang, using $t=1/(2H)$. (8 points)

Answer(a) 1/time, and 14.47 Gyr. (b) $3H^2/(8\pi G)$, and $8.58\times {10}^{-27}$ kg/m³. (c) ${\mathrm{\Omega }}_{\mathrm{\Lambda }}=\mathrm{\Lambda }{c}^2/(3H^2)$ and ${\mathrm{\Omega }}_k=-k{c}^2/(a^2{H}^2)$. (d) ${\rho }_{\text{r}}\propto (1+z{)}^4$. (e) $w=-1$. (f) $t_{\text{P}}=\sqrt{\hslash G/c^5}=5.4\times {10}^{-44}$ s. (g) $M_{\text{P}}=\sqrt{\hslash c/G}$, with $M_{\text{P}}{c}^2=1.22\times {10}^{19}$ GeV. (h) $T\propto 1+z$. (i) $1+z_{\text{eq}}=2.39\times {10}^4{\mathrm{\Omega }}_{\text{m0}}{h}^2$. (j) About 1.3 s, so about a second after the Big Bang. The official solution has 14.46 Gyr, $8.59\times {10}^{-27}$ kg/m³ and $2.47\times {10}^{-5}$ in part (i), which differ from ours only in the rounding of constants.