Reading cosmic time from an expansion history

A rate at one epoch does not fix the past

The Hubble time 1/H_ref converts a positive reference expansion rate into a time scale. Calling that reciprocal the age assumes more than the rate itself supplies. Age depends on how the expansion rate changed over the interval being counted.

Compare two ideal models governed by general relativity. Both are spatially flat, expanding and have no cosmological constant. One contains only separately conserved pressureless matter; the other contains only separately conserved radiation. These are separate mathematical histories, not successive pieces of a fitted history of our Universe.

Normalize each scale factor to a = 1 at its reference time t_ref and give both the same positive H_ref there. Thus both have the same scale and the same instantaneous slope ȧ = H_ref at that reference epoch, despite having different contents.

Define dimensionless lookback τ = H_ref(t_ref − t). Moving toward the past increases τ. It is a convenient comparison coordinate, not an observed redshift or an age measured in seconds until H_ref is supplied.

Two contents give two histories

The ideal flat matter-only solution is a = (1 − 3τ/2)²ᐟ³ for 0 ≤ τ < 2/3 in this past-directed comparison. Its formal a → 0 endpoint occurs at τ = 2/3. The radiation-only solution is a = √(1 − 2τ) for 0 ≤ τ < 1/2, with its formal a → 0 endpoint at τ = 1/2.

Both curves have a = 1 and da/dτ = −1 at τ = 0. They therefore share the same tangent at the reference epoch. Farther back, their different density scalings and dynamics produce different curves and different elapsed times to their formal endpoints.

At τ = 0.2, the matter curve gives a ≈ 0.7884 and the radiation curve gives a ≈ 0.7746. At τ = 0.4, the values are about 0.5429 and 0.4472. A shared present slope does not make the whole earlier curve identical.

At τ = 0.5, radiation reaches its limiting a = 0 endpoint while the matter curve still has a ≈ 0.3969. For τ > 0.5, the supplied radiation expression has no real positive-scale-factor value. Do not extend the plotted curve along zero or label that interval another radiation epoch.

The same Hubble time gives different ideal ages

Measured from the formal a → 0 endpoint to the reference time, the matter-only model age is 2/(3H_ref). The radiation-only model age is 1/(2H_ref). Their ratio is 4/3, although both models were assigned the same H_ref.

For a numerical illustration only, choose H_ref = 60 km/s/Mpc. Using consistent length and time units, its reciprocal is about 16.30 Julian gigayears. The matter-only age is then about 10.86 Gyr and the radiation-only age about 8.15 Gyr.

Those are ages of the two supplied ideal models. The chosen reference rate is not presented as a current measurement, and neither result is an estimate of the age of our actual Universe. A realistic inference requires a calibrated model including the relevant components and observational constraints.

The example demonstrates non-uniqueness from one rate, rather than a competition between two observational fits. The different pressure and density histories change the relation between the reference rate and elapsed time.

An age calculation adds time along a specified history

From H = ȧ/a, an expanding branch with H(a) > 0 gives dt = da/[aH(a)]. The elapsed time between an earlier positive a_i and the reference a = 1 is therefore t_ref − t_i = ∫ from a_i to 1 of da/[aH(a)].

The numerator da describes a small scale-factor interval. Dividing by aH(a), the scale-factor slope, converts it into the time taken to cross that interval. The integral adds those times, so the result depends on the function H(a) throughout the range.

For the ideal matter-only model, H(a) = H_ref a⁻³ᐟ². Substitution gives elapsed time 2(1 − a_i³ᐟ²)/(3H_ref). For radiation only, H(a) = H_ref a⁻², giving (1 − a_i²)/(2H_ref). Their limits as a_i approaches zero reproduce the two formal model ages.

A first reading can use the table and the two age factors without doing calculus. On a later return, the integral shows precisely what additional information beyond H_ref is required. A mixed-component history needs its consistent combined H(a), rather than arbitrary joins between these two ideal formulas.

A formal endpoint is a limit of the stated description

The a → 0 endpoints belong to extrapolations of classical ideal solutions. They do not verify a complete physical account of the origin, establish that quantum effects can be ignored there or describe a tested state at exactly a = 0.

A calculated time between two specified positive scale factors can be meaningful within an appropriate model even when its extrapolation to an extreme endpoint is uncertain. Separate the interval supported by the model from the additional assumptions used in a formal age.

An epoch label such as radiation domination is also an approximation about relative contributions over an interval. It does not say every other component vanished or that a single-component solution is exact through a transition.

When reporting an age, name the reference epoch, rate units, contents, gravitational and geometric assumptions, integration limits and observational status. These details turn a bare number into an interpretable statement about a history.

Worked reasoning

Try the reasoning

1. If two histories share a = 1 and H_ref at the reference epoch, must they have identical earlier scale factors?

   Show the answer and reasoning

   No.

   The reference rate fixes the instantaneous slope there; different contents and dynamics can produce different past curves.

2. What should a plot show for the supplied radiation-only history at τ = 0.6?

   Show the answer and reasoning

   No real positive-scale-factor branch from this formula; the stated past domain has ended.

   Its formal a → 0 endpoint is τ = 0.5. Continuing it along a = 0 would invent an extension.

3. Do the illustrative values 10.86 and 8.15 Gyr establish the measured age of our Universe?

   Show the answer and reasoning

   No.

   They follow from separate ideal single-component models and a chosen reference rate, rather than a calibrated observational inference.

What to carry forward

• A reciprocal rate is a time scale; an age depends on the assumed expansion history.
• Two ideal histories can share a reference rate yet have different formal ages.
• State the integration limits and physical validity of a history before interpreting its endpoint.