Radiation domination and matter domination

Both components remain in the model

Radiation and pressureless matter dilute at different rates in an expanding background when each is separately conserved. Their relative contributions therefore change continuously, even if neither component disappears.

Calling an interval radiation dominated or matter dominated identifies an approximation about the leading contribution. It does not by itself set the smaller component to exactly zero or make the change between them instantaneous.

The earlier density-scaling lesson compared their ratios. Here one shared general-relativistic background connects both densities to a continuous Hubble rate and an elapsed time, rather than joining unrelated pure-component histories.

The reference equality stage below is chosen for normalization. No measured equality redshift, present component density, physical date or cosmic origin state is assigned.

Use one density convention and one equality rate

Choose a spatially flat general-relativistic background containing only separately conserved pressureless matter and radiation, with no cosmological constant or energy exchange. Let q = a/a_eq range from 1/4 to 4.

At q = 1 each component has the same positive energy density ε_eq in J/m³. Thus ε_r = ε_eq q⁻⁴ and ε_m = ε_eq q⁻³. The total at equality is 2ε_eq, not ε_eq.

Let H_eq > 0 be the actual expanding Hubble rate at that equality stage. The flat Friedmann equation then gives (H/H_eq)² = (q⁻⁴ + q⁻³)/2. The factor of 2 is required because H_eq includes both components.

The corresponding fractions are f_r = 1/(1 + q) and f_m = q/(1 + q). They sum to 1. Assume fixed radiation entropy degrees of freedom so that the radiation temperature ratio is T/T_eq = 1/q over this interval.

A continuous ratio describes the change of dominance

At q = 1/4, radiation contributes 80 percent and matter 20 percent of the two-component energy total. At equality each contributes half. At q = 4 the fractions have exchanged, with matter contributing 80 percent.

For q = 4, the component densities in equality units are ε_r/ε_eq = 1/256 and ε_m/ε_eq = 1/64. Both have decreased substantially; matter’s larger relative contribution does not mean its absolute density increased.

The rate H/H_eq decreases continuously from about 12.64911 at q = 1/4 to 1 at equality and 0.0988212 at q = 4. Equality creates no jump in the supplied density laws or the rate derived from them.

When a fraction is quoted, name its denominator. The matter fraction of this two-component energy total is not the same as a critical-density ratio in a more general model with curvature or additional components.

A mixed history needs its own time integral

Because dq/dt = qH, the dimensionless elapsed time from q = 1/4 is τ = H_eq[t(q) − t(1/4)] = ∫ from 1/4 to q of dq′/[q′(H(q′)/H_eq)]. The table uses this one shared mixed rate throughout.

An optional analytic check is τ = F(q) − F(1/4), where F(q) = (2√2/3)(q − 2)√(1 + q). Differentiation gives F′(q) = √2 q/√(1 + q), equal to the integrand obtained from the supplied H.

The elapsed time to equality is about 0.511329 H_eq⁻¹; to q = 4 it is about 6.061032 H_eq⁻¹. These are intervals measured from the declared starting stage, not the Universe’s age.

Switching abruptly between exact pure-radiation and pure-matter power-law solutions at q = 1 would not give this mixed solution exactly. Pure-era approximations need a quantified domain where the omitted contribution is small.

Normalization alone does not supply an observed calendar

The finite interval avoids extrapolating this model to a limiting zero scale factor. Its useful result is the internally consistent change of fractions, rate and elapsed time within a stated domain.

An actual equality chronology requires measured or inferred component values, appropriate radiation contents and a larger model. Additional energy components or changes in species thermodynamics can alter the temperature and expansion accounts.

The phrase “radiation era” can hide these assumptions if used as an exact description of every earlier event. State whether it is a dominant-component approximation, a two-component calculation or a more complete thermal model.

A careful reconstruction combines the background evolution with relevant interaction and observation models. Fractions, temperatures and ages should be compared only after their normalizations and domains match.

Worked reasoning

Try the reasoning

1. At equality, may H_eq be computed from ε_eq as though it were the total energy density?

   Show the answer and reasoning

   No; the total is 2ε_eq.

   Each of the two components has energy density ε_eq at q = 1.

2. Does choosing q = 1 for equality assign a measured date or redshift?

   Show the answer and reasoning

   No.

   The example supplies a relative scale normalization without present component measurements or an empirical calendar.

3. Does an abrupt switch between exact pure-era solutions reproduce this mixed elapsed time exactly?

   Show the answer and reasoning

   No.

   The mixed H(q) contains both components continuously and must be integrated as such.

What to carry forward

• Dominance describes a relative contribution, while the smaller component can remain important.
• A shared mixed model keeps density, rate and time normalizations consistent.
• The finite elapsed interval is neither a measured equality date nor an origin-to-present age.