What the Universe Is Made Of

Section 2 of 7

What Sets the Expansion Rate?

Part 1: What Sets the Expansion Rate?

In the last reading we measured the expansion rate $H_0$ and inverted it for a rough age. But what governs that rate, and how it changes? The answer is the Friedmann equation:

Read it as a sentence, left to right. $H^2$ is the expansion rate squared — how fast space stretches. The first term on the right, $(8\pi G/3)\rho$ , is the gravity of everything the universe contains: $\rho$ is the total energy density, summing radiation, matter, and dark energy. The second term, $-kc^2/a^2$ , is the shape of space — its curvature. More stuff means faster expansion; the curvature sign decides whether space is open, closed, or flat.

Friedmann equation

The equation governing the cosmic expansion rate, $H^2 = (8\pi G/3)\rho - kc^2/a^2$ . It ties the expansion rate $H = \dot a/a$ to the total energy density $\rho$ and the spatial curvature $k$ . Read, not solved, in this course.

We will not solve this differential equation — that is a graduate exercise. We will read it, exactly as we read the structure equations for a star. And the most useful read comes from a single move.

Reading the Math · The Whole Universe

All the way back in Module 3 we made a promise: an equation is a story you read, not a computation you perform under pressure. We read hydrostatic equilibrium and got the pressure inside a star. Here that same disposition reaches its summit. Divide the Friedmann equation by $H^2$ :

1 = \frac{8\pi G\,\rho}{3H^2} - \frac{kc^2}{a^2 H^2}.

Define the critical density $\rho_\text{crit} = 3H^2/8\pi G$ and the density parameters $\Omega_i = \rho_i/\rho_\text{crit}$ . The equation becomes

1 = \Omega_r + \Omega_m + \Omega_\Lambda + \Omega_k.

You did not solve anything. You divided by $H^2$ and read off the budget of the universe. This is the read we have been training for since the first module.

Quick check

In the Friedmann equation, which term would you increase to make the universe expand faster at a given moment, and which term carries the shape of space?