The Friedmann Equation and Critical Density

Does the universe expand forever? The equation that answers it was found by Alexander Friedmann in 1922 from general relativity, but its heart can be derived with nothing more than orbital energy. It links the expansion rate to the density of the universe.

Why it works

Picture a sphere of radius $R = ar$ cut out of a uniform universe of density $\rho$. By the shell theorem, a galaxy on its surface feels only the mass inside, $M = \tfrac43\pi R^3\rho$. Its energy per unit mass is constant: $$\tfrac12\dot R^2 - \frac{GM}{R} = \text{const}.$$ Divide by $\tfrac12R^2$ and use $H = \dot R/R = \dot a/a$: $$H^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2},$$ where the constant has been written as $-kc^2/a^2$. General relativity gives the same equation and says what $k$ means: the curvature of space. The universe is flat ($k = 0$) when $\rho$ equals the critical density $\rho_c = 3H^2/(8\pi G)$.

Key idea

The Friedmann equation $H^2 = \tfrac{8\pi G}{3}\rho - kc^2/a^2$. With $\Omega = \rho/\rho_c$: $\Omega > 1$ means positive curvature, $\Omega < 1$ negative, $\Omega = 1$ flat. Measurements (Planck satellite and others) find $\Omega$ equal to 1 within about 0.2%: about 31% matter and 69% dark energy.

The scale factor over time for different mixes of matter and dark energy.

Worked examples

Worked example: The critical density today

Find $\rho_c$ for $H_0 = 70$ km/s/Mpc.

$H_0 = 70/3.09 \times 10^{19} = 2.27 \times 10^{-18}$ s⁻¹.
$\rho_c = 3(2.27 \times 10^{-18})^2/(8\pi \times 6.67 \times 10^{-11})$.

Answer: About $9.2 \times 10^{-27}$ kg/m³, the mass of about 5.5 hydrogen atoms per cubic meter.

Worked example: The expansion rate in the past

With $\Omega_m = 0.31$ and $\Omega_\Lambda = 0.69$, $H(z) = H_0\sqrt{\Omega_m(1 + z)^3 + \Omega_\Lambda}$. Find $H$ at $z = 1$ for $H_0 = 70$.

$0.31 \times 2^3 + 0.69 = 3.17$, and $\sqrt{3.17} = 1.78$.
$H = 70 \times 1.78$.

Answer: About 125 km/s/Mpc: the universe expanded faster when it was denser.

Worked example: Escape speed for the universe

For the sphere above, compare the expansion speed $v = HR$ with the escape speed $\sqrt{2GM/R}$.

$v^2/v_{\text{esc}}^2 = H^2R^2/(\tfrac83\pi G\rho R^2) = \rho_c/\rho = 1/\Omega$.
With matter alone, $\Omega < 1$ means faster than escape speed.

Answer: $v/v_{\text{esc}} = 1/\sqrt{\Omega}$: a matter-only universe with $\Omega < 1$ expands forever, and one with $\Omega > 1$ recollapses. Dark energy changes this story (next topic).

Common mistake

Thinking a "flat universe" is shaped like a pancake. Flat describes the geometry of space everywhere, in three dimensions: parallel lines stay parallel and the angles of a giant triangle add to 180°. A closed universe ($k > 0$) is like the surface of a sphere, with triangles adding to more than 180°; an open one like a saddle. None of them has an edge. Astronomers test it with triangles billions of light-years across, using the size of hot spots in the microwave background.

The Friedmann Equation and Critical Density

Worked examples

Practice

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