#FLRW metric

Cosmologists routinely compress the large-scale geometry of the universe into one line of mathematics.

Written without preparation, that line looks less like an explanation than a security fence:

$$ ds^2=-c^2dt^2+a^2(t)\left[\frac{dr^2}{1-kr^2}+r^2\left(d\theta^2+\sin^2\theta\,d\phi^2\right)\right]. $$

This is the Friedmann–Lemaître–Robertson–Walker metric, usually shortened to the FLRW metric. It is the mathematical framework behind most modern statements about the expansion, age, curvature, and observable history of the universe.

But the equation does not need to remain a ceremonial inscription that everyone respectfully walks past. Each part has a clear job. Taken piece by piece, it expresses a surprisingly economical idea:

If the universe is homogeneous and isotropic on sufficiently large scales, then its geometry can change with time while keeping the same general spatial form everywhere.

The FLRW metric is the measuring rule for that kind of universe.

Begin by Smoothing the Universe

At familiar scales, the universe is plainly not homogeneous. Earth is denser than the space surrounding it. The Milky Way has stars, gas clouds, a central black hole, and large regions containing very little matter. Galaxies collect into groups and clusters, with enormous voids between them.

Cosmology does not deny any of this. Instead, it asks what the universe looks like after we step back far enough that individual galaxies become local texture rather than the main pattern.