Dynamics
Scalings in Each Era
| Radiation domination | Matter domination | $\Lambda$ domination | |
|---|---|---|---|
| $a(t)$ | $t^{1/2}$ | $t^{2/3}$ | $\exp(H_0 t)$ |
| $\eta(a)$ | $\propto a$ | $\propto a^{1/2}$ | $-1/(aH)$ |
| $H(a)$ | $\propto a^{-2}$ | $\propto a^{-3/2}$ | const. |
In Λ domination $\eta = \int dt/a$ asymptotes to a finite future conformal time; the form above is the leading $a \to \infty$ behavior.
Formula Sheet
$$ \begin{aligned} \frac{dz}{dt} &= -H(z)\,(1+z) \\[4pt] \frac{d}{d\eta} &= a^2 H \,\frac{d}{da} \\[4pt] \frac{H^2}{H_0^2} &= \Omega_{r,0}\,a^{-4} + \Omega_{m,0}\,a^{-3} + \Omega_{k,0}\,a^{-2} + \Omega_{\Lambda,0} \end{aligned} $$Critical density today: $\rho_{\rm crit} = 3 H_0^2 / (8\pi G) \approx 1.878\times 10^{-29}\,h^2\;{\rm g\,cm^{-3}}$.
CMB temperature scaling: $T(z) = T_0\,(1+z)$ with $T_0 = 2.7255\,{\rm K}$.