The Friedmann equation says the expansion rate is set by the density. But different ingredients dilute at different rates as the universe grows, so the mix changes over time, and each ingredient has its turn in charge.
Why it works
Matter (ordinary and dark): the mass in a region stays fixed while its volume grows as $a^3$, so $\rho_m \propto a^{-3}$. Radiation (photons, and neutrinos early on): the number per volume also falls as $a^{-3}$, but each photon's wavelength stretches as $a$, so its energy falls as $a^{-1}$: $\rho_r \propto a^{-4}$. Dark energy, in the simplest model a cosmological constant, has the same density at every time: $\rho_\Lambda$ = const. Today $\Omega_r \approx 9 \times 10^{-5}$, $\Omega_m \approx 0.31$ and $\Omega_\Lambda \approx 0.69$. Going back in time ($a$ smaller, $z$ larger) radiation grows fastest, so it ruled the earliest era; then matter; and now dark energy.
Key idea
$\rho_r \propto (1 + z)^4$, $\rho_m \propto (1 + z)^3$, $\rho_\Lambda$ constant. Radiation and matter were equal at $1 + z = \Omega_m/\Omega_r \approx 3400$ (about 50,000 years after the Big Bang); matter and dark energy were equal at $z \approx 0.3$, a few billion years ago. In the radiation era $a \propto t^{1/2}$, in the matter era $a \propto t^{2/3}$, and with dark energy in charge the expansion accelerates.
Worked examples
Worked example: When did matter take over?
With $\Omega_m = 0.31$ and $\Omega_r = 9.0 \times 10^{-5}$, find the redshift of matter–radiation equality.
- $\Omega_r(1 + z)^4 = \Omega_m(1 + z)^3$ gives $1 + z = \Omega_m/\Omega_r$.
- $1 + z = 0.31/9.0 \times 10^{-5} = 3440$.
Answer: $z \approx 3400$.
Worked example: When did dark energy take over?
With $\Omega_m = 0.31$ and $\Omega_\Lambda = 0.69$, find when the two densities were equal, and when the expansion began to speed up (when $\Omega_\Lambda = \tfrac12\Omega_m(1 + z)^3$).
- Equal: $(1 + z)^3 = 0.69/0.31 = 2.23$, so $z = 0.31$.
- Acceleration: $(1 + z)^3 = 2(0.69)/0.31 = 4.45$, so $z = 0.65$.
Answer: The expansion started speeding up at $z \approx 0.65$, roughly 6 billion years ago; dark energy outweighed matter from $z \approx 0.3$.
Worked example: Growth in the matter era
During the matter era, $a \propto t^{2/3}$. While the age of the universe grows by a factor of 8, how much do distances grow?
- $a_2/a_1 = 8^{2/3}$.
Answer: 4 times.
Common mistake
Thinking the mix of matter and radiation has always been what it is today. Radiation is now a tiny 0.009% of the total, but because its density grows as $(1 + z)^4$ going back, it outweighed matter before $z \approx 3400$. Dark energy, now 69%, was negligible in the early universe. Check: the ratio $\rho_r/\rho_m = (\Omega_r/\Omega_m)(1 + z)$ was about 0.3 even at $z = 1100$, when the microwave background was released.