How matter and radiation densities evolve

Use energy per volume consistently

Choose a representative region following a homogeneous expanding background. Define q = a/a_ref, the length scale relative to an arbitrary reference epoch. Its proper volume changes by q³. A reference volume of 1 m³ would become 8 m³ at q = 2.

Write energy density as ε, pronounced epsilon, measured in joules per cubic metre. The same symbol and units will be used for both components below. Mass density in kilograms per cubic metre is another quantity; for the rest-energy contribution of nonrelativistic matter, multiplying that mass density by c² gives its energy density.

Assume a conserved population of nonrelativistic particles whose kinetic and pressure contributions are negligible for this account. Their rest energy in the following region stays approximately fixed while its volume grows. Their energy density therefore changes in proportion to q⁻³.

This approximation concerns the component being modelled. A sufficiently energetic particle population need not behave as pressureless matter. A classification appropriate at one epoch should not automatically be extended across every temperature or species transition.

Photons lose energy as well as number per volume

For conserved freely propagating photons, expansion spreads their number over the larger volume, giving a number-density factor q⁻³. It also stretches their wavelengths by q. Their frequencies and energies therefore receive another factor q⁻¹.

Multiplying the number-density and photon-energy factors gives ε_r ∝ q⁻⁴. The subscript r labels radiation in this comparison. Doubling the scale reduces this radiation energy density by 16, whereas the pressureless matter energy density falls by 8.

The separate rules assume no energy transfer, photon production, absorption or entropy release during the interval. If interactions exchange substantial energy between components, their individual accounts need exchange terms. Applying the separate power laws unchanged across such a transition would misstate the physics.

The total photon energy in the following region falls by q⁻¹, not q⁻⁴: total energy is density times volume. The next conservation lesson explains how this behaviour fits the local fluid balance without assigning the lost photon energy to an invented reservoir.

A supplied ratio identifies a model equality stage

Now supply unequal reference energy densities: ε_m,ref = 8ε_unit for matter and ε_r,ref = 2ε_unit for photons. Here ε_unit is one shared arbitrary energy-density unit. The numbers are hypothetical, and the reference epoch is not identified with today.

The rules give ε_m = 8ε_unit/q³ and ε_r = 2ε_unit/q⁴. Dividing cancels the unit and three inverse powers: ε_r/ε_m = 1/(4q). The relative contribution changes with q even though both components remain present.

At q = 1/4, each density is 512ε_unit, so the two chosen components are equal. At q = 1, matter has four times the photon energy density. At q = 2, matter has eight times the photon energy density. Equality is a stage defined by a ratio, not the disappearance of either component.

The table gives no years or measured equality redshift. Those would require a calibrated physical history and reference conditions. Other components have been omitted from this two-component example, so its equality should not silently be identified with the full cosmic matter–radiation equality inferred in an observational model.

A larger density is a statement about the specified account

Saying matter dominates this energy-density comparison at q = 1 means ε_m exceeds ε_r. It does not say photons are absent or unimportant for every observation. Even a lower-density radiation field can carry information about earlier conditions.

Nor does an energy-density comparison alone describe every gravitational effect. Pressure enters relativistic acceleration equations, while the distribution and motion of components matter for structure. The next lesson introduces those roles in a homogeneous model.

A component’s scaling follows its physical behaviour, not merely its name. Neutrinos, for example, cannot be treated as radiation at every epoch without checking how their energies compare with their rest energies. No neutrino component is included in the table.

When the assumptions change, preserve the accounting questions: What is conserved? How does individual-particle energy change? Which volume is being used? What energy moves between components? These questions indicate why a new rule or coupled evolution is required.

Reference conditions and evolution play different roles

For any positive reference densities obeying these same rules, the ratio is (ε_r,ref/ε_m,ref)/q. Equality occurs at q_eq = ε_r,ref/ε_m,ref. The power-law behaviour tells how the ratio evolves, while the reference ratio tells where equality falls.

Changing the reference epoch consistently changes q and the quoted reference densities without changing the underlying physical history. Changing only one input, while leaving the others inconsistent, creates a different model or a bookkeeping error.

For a first reading, follow the ratio from 1 to 1/2 when q doubles from 1/4 to 1/2. Radiation loses relative share because its individual photons also redshift. The equality calculation then turns that mechanism into a specific, checkable model statement.

Worked reasoning

Try the reasoning

1. If q doubles while the stated assumptions hold, by what factor does ε_r/ε_m change?

   Show the answer and reasoning

   It halves.

   The ratio is proportional to 1/q, reflecting the extra photon-energy redshift factor.

2. When q doubles, does the total photon energy in the following region fall by 16?

   Show the answer and reasoning

   No. It halves in this model.

   Radiation energy density falls by 16 while the following volume grows by 8; their product falls by 2.

3. May separate q⁻³ and q⁻⁴ laws be used unchanged through a substantial energy exchange between the components?

   Show the answer and reasoning

   No.

   The separate conservation assumptions fail, so the individual accounts need the relevant interaction terms.

What to carry forward

• Conserved pressureless rest energy gives ε_m ∝ a⁻³; conserved free photons give ε_r ∝ a⁻⁴.
• The reference ratio sets a chosen component-equality stage; equality does not remove either component.
• Volume, particle energies and exchanges must be specified before applying a density scaling.