What expansion equations describe

Equations connect a model’s contents to its geometry

A scale-factor curve can be chosen to illustrate definitions, as in the preceding lesson. Predicting a physical history requires more: a gravitational theory, component properties and conditions that select a solution. Those inputs determine which curves the model permits.

The Friedmann description follows from general relativity after imposing a homogeneous and isotropic background geometry and suitable fluid descriptions. Homogeneous means the background has the same properties at different locations on a cosmic-time slice; isotropic means it has no preferred direction.

This is a smooth-background approximation, not a claim that stars and galaxies are individually uniform. Local structures and perturbations need further equations. Nor is a Newtonian expanding ball an exact derivation of all relativistic cosmic geometry.

A perfect-fluid description uses an energy density ε and an isotropic pressure P in the local fluid frame. An equation of state specifies their relationship for a component. Pressureless nonrelativistic matter has P approximately zero; radiation has P = ε/3 in this ideal description.

Three related equations answer different questions

A Friedmann constraint relates H² to the contents and spatial curvature at an epoch. An acceleration equation describes ä/a and shows how density and pressure influence the change of the scale-factor slope. A continuity equation expresses the local fluid energy balance during expansion.

For a spatially flat model with no cosmological constant, the constraint is H² = 8πGε/(3c²). Here G is the gravitational constant, c is light speed and ε is total fluid energy density in J/m³. H² has units s⁻². Inserting mass density in kg/m³ into this energy-density formula would be a unit error.

Under the same conditions, ä/a = −4πG(ε + 3P)/(3c²). Pressure has the same units as energy density: one pascal is one joule per cubic metre. Positive radiation pressure therefore contributes to the decelerating term rather than being omitted simply because photons have no rest mass.

The continuity relation is ε̇ + 3H(ε + P) = 0 for the non-exchanging total fluid account. It is consistent with the preceding equations, rather than a freely independent extra rule that can be chosen inconsistently. For separately conserved components, each has its own corresponding relation.

Choose the contents before calculating H

Set the reference scale factor to a = 1 and choose a positive reference rate H_ref. Define ε_ref = 3c²H_ref²/(8πG), which has energy-density units. Supply reference densities ε_r,ref = ε_ref/4 for radiation and ε_m,ref = 3ε_ref/4 for pressureless matter.

Assume spatial flatness, no cosmological constant and no energy exchange. The separate scalings give (H/H_ref)² = (1/4)a⁻⁴ + (3/4)a⁻³. These reference weights are chosen for the example, not fitted fractions of our present Universe.

At a = 1/2, the radiation contribution to the squared-rate ratio is (1/4) × 16 = 4 and the matter contribution is (3/4) × 8 = 6. Their sum is 10. Selecting the expanding branch gives H/H_ref = √10 ≈ 3.162.

At a = 2, the contributions are 1/64 and 3/32, whose sum is 7/64. The expanding rate ratio is √7/8 ≈ 0.3307. The component densities and H change together according to the supplied model; a reference fraction cannot simply be held fixed as a evolves.

A square does not select the branch

A positive value for H² allows both positive and negative H. The density constraint alone therefore does not decide whether the background is expanding or contracting. The branch and initial conditions are part of specifying the solution.

The acceleration equation adds another distinction. In this matter-plus-radiation example, ε + 3P = ε_m + 2ε_r is positive, so ä is negative. On the selected expanding branch, a can increase while its positive slope decreases. Decelerating expansion is not the same as contraction.

To obtain a time history, one must evolve the components and solve ȧ = aH with consistent conditions. Specifying one density and one squared rate at an epoch is not a complete record of the past. Observations then test the history and the assumptions that generated it.

The worked calculation cancels the dimensional constants by taking ratios. On a first reading, its separate contributions and square root carry the main reasoning. Derivations from the gravitational field equations are a later mathematical return path.

An optional fuller equation needs a convention

With dimensionless a, write a spatial-curvature coefficient K with units m⁻². One convention is K = k/R_ref², where k is −1, 0 or +1 and R_ref is a fixed length. The constraint becomes H² = 8πGε/(3c²) + Λc²/3 − Kc²/a².

Here Λ is an explicit cosmological constant with units m⁻². Both Λc² and Kc²/a² have inverse-time-squared units. A dimensionless sign k alone cannot replace K in this SI expression without the required length scale. Some sources instead give a length-valued scale factor; their symbols cannot be mixed with this convention.

If Λ is written explicitly, ε and P in the fluid terms exclude its contribution. Alternatively, it can be included as a component with ε_Λ = Λc⁴/(8πG) and P_Λ = −ε_Λ, in which case the separate Λ term is removed. Counting it both ways would count the same contribution twice.

The acceleration equation with explicit Λ adds Λc²/3 to its right-hand side. Changing the contents or their pressure behaviour changes the possible history. A spatial-curvature label alone does not establish an unconditional future fate without specifying the remaining physical model.

Worked reasoning

Try the reasoning

1. Does H² = 10H_ref² alone select positive H?

   Show the answer and reasoning

   No.

   Both signs have the same square. Expansion is an additional branch condition.

2. May kilograms per cubic metre be substituted directly for ε in H² = 8πGε/(3c²)?

   Show the answer and reasoning

   No. ε in this formula is energy density in J/m³.

   A mass-density form uses a different placement of c². Consistent definitions and units are essential.

3. If the Λ term is explicit in the constraint, should ε also include ε_Λ?

   Show the answer and reasoning

   No.

   Use the explicit term or the equivalent fluid contribution consistently; including both double-counts it.

What to carry forward

• Expansion equations require stated gravity, geometry, contents, pressure behaviour and conditions.
• A constraint on H², an acceleration law and a continuity relation play different roles.
• The expanding branch, density units, curvature convention and treatment of Λ must be explicit.