The Scale Factor, the Friedmann Equation and the Age of the Universe
Expansion is written as one function of time multiplying every distance, and the equation governing it comes from applying general relativity to a uniform universe. What the universe contains fixes how that function behaves, so the measured contents predict an age and a fate.
What a learner can do afterwards
- Writes redshift as a ratio of scale factors and explains why that is not a Doppler shift through space
- States how matter, radiation and a cosmological constant each dilute as the universe expands, and which dominates when
- Estimates an age from an expansion rate and says why the simplest estimate is not the measured answer
1 · Read
Cosmic expansion is written as one function of time, the scale factor, multiplying every distance. Light stretched while it travelled records this growth as redshift: one plus z equals the scale factor now divided by the scale factor back then. Nothing moves through space. Space itself stretches the waves, and the expansion has no centre.
Suppose light left a galaxy when the scale factor was half its present value. One plus z equals one divided by one half, which is two, so z equals one. The same rule runs the contents: if the scale factor doubles, matter thins eightfold with the volume, radiation thins sixteenfold because each photon also reddens, and dark energy stays constant.
The Friedmann equation governs the scale factor by tying the expansion rate to what the universe contains. More density means stronger braking on the expansion, while dark energy pushes outward. Because each component dilutes differently, each era has its own ruler: radiation ruled the hot start, matter built the galaxies, and dark energy drives today.
To estimate the age, run the expansion backwards. The naive guess of one over the Hubble rate lands near fourteen billion years. The measured answer differs because early matter slowed the expansion and late dark energy speeds it. Dating the oldest stars cross checks the result, and the microwave background plus the light elements confirm the hot dense start.
The scale factor stretches all distances, its contents set its pace, and the naive age needs their correction.
More to read
29.1 The Age of the Universe
The age of the universe comes from running the expansion backwards, corrected by what the universe contains. The naive estimate, 1 over the Hubble constant, lands near 14 billion years. Matter and dark energy bend that simple number: early matter slowed the expansion, late dark energy speeds it. Dating the oldest stars cross checks the answer.
29.2 A Model of the Universe
The standard model describes a homogeneous expanding universe governed by the Friedmann equation, which ties the expansion rate to the density of each component. Matter, radiation and dark energy dilute at different rates as the scale factor grows. That difference decides which component rules each era. Geometry plus contents predicts the whole history.
29.5 What Is the Universe Really Made Of?
Ordinary matter is a small slice: dark matter outweighs it several times over, and dark energy outweighs both. Each dilutes differently with expansion, so the budget also sets the timeline. Radiation ruled the hot start, matter built the galaxies, dark energy drives today.
26.5 The Expanding Universe
Distant galaxies recede with redshifts proportional to distance: the Hubble law. The redshift is not motion through space but stretching of space itself, recorded as 1 + z equals the ratio of scale factors. Expansion has no centre and needs nothing to expand into. The microwave background and element abundances confirm the hot dense start.
Explore more
2 · Watch
Take it off screen
Where it sits
Learn first
This opens up
Nothing builds on it yet.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.