The Hubble parameter and its present value

H measures a fraction per unit time

Suppose a representative background separation is growing. Its absolute increase per unit time and the fraction of its current length gained per unit time are different rates. The Hubble parameter describes the fractional one.

For a dimensionless scale factor a and cosmic time t, H(t) = ȧ(t)/a(t). A dot means the instantaneous change per unit time. Because a has no units, both ȧ and H have inverse-time units, but dividing by a gives them different meanings.

For a comoving separation χ that stays fixed, D = aχ and Ḋ = HD. At one cosmic time in the ideal homogeneous background, the same H relates different comoving pairs. This does not turn the relation into a local force that makes all bound objects grow.

The subscript zero in H₀ conventionally labels the present cosmic epoch: H₀ = H(t₀). Calling it the Hubble constant does not say that H was identical at every past epoch. Conversely, H need not change in every imaginable model; some ideal histories have constant H.

Speed per distance is still inverse time

The customary unit km/s/Mpc divides a speed by a distance. Converting the megaparsec to kilometres leaves inverse seconds. One parsec is exactly (648,000/π) astronomical units, with the astronomical unit exactly 149,597,870,700 m; the commonly printed decimal parsec conversion is rounded.

For the chosen exercise coefficient H★ = 60 km/s/Mpc, using 1 Mpc ≈ 3.08567758149 × 10¹⁹ km gives H★ ≈ 1.94447 × 10⁻¹⁸ s⁻¹. This is a unit conversion of the supplied number, not an independent measurement or a current value of H₀.

Its reciprocal is a characteristic time about 5.14280 × 10¹⁷ s. If a Julian year is defined as 365.25 days of 86,400 s each, that is about 16.30 billion Julian years. A different year convention must be stated if it is used.

The reciprocal’s time units do not, by themselves, make it the age of the Universe. An age depends on how the scale factor evolved before the reference epoch. The last lesson of this chapter compares physical model histories with the same reference rate and different inferred ages.

Use one smooth history to keep the rates separate

Choose a positive time unit T and define elapsed stage u = (t − t_ref)/T. On 0 ≤ u ≤ 10, supply the kinematic history a(u) = 1 + 0.08u + 0.002u². This is a chosen curve for comparing rates, not a fit to cosmic data or a history derived from specified matter contents.

Its slope is ȧ = (0.08 + 0.004u)/T. The absolute slope increases from 0.08/T to 0.12/T across the interval. Its second time derivative is 0.004/T², which is positive: the scale factor’s growth accelerates.

The fractional rate is H = (0.08 + 0.004u)/(Ta). At u = 0, 5 and 10, a is 1, 1.45 and 2. The corresponding HT values are 0.08, about 0.06897 and 0.06. H falls throughout this interval even though the absolute slope rises.

The table uses ȧT and HT so the listed rate combinations are dimensionless. If T were chosen as 1 Gyr, for example, 0.08/T would be 0.08 Gyr⁻¹. No value of T or identification of t_ref with today is required for the comparison.

An optional calculation checks the whole interval

The finite comparisons are enough for a first reading. With derivative notation, Ḣ = ä/a − H². Its units are inverse time squared. This identity shows why the sign of Ḣ alone does not determine the sign of ä.

For the supplied curve, ḢT² = [0.004a − (0.08 + 0.004u)²]/a². Its numerator simplifies to −0.0024 − 0.00032u − 0.000008u², which is negative for every u in the chosen interval. Meanwhile äT² stays positive at 0.004.

These equations check a statement about the entire interval rather than just three table samples. They do not determine which cosmological contents could generate the curve. That additional dynamical question requires gravitational field equations, component properties and initial conditions.

The mathematical preparation reference to functions is a return path if slopes and derivatives are unfamiliar. Retaining the table’s increasing ȧT and decreasing HT already preserves the physical distinction.

Name the epoch as well as the estimator

A present-rate estimate uses observations and a method that connects them to H₀. Other observations can constrain expansion at earlier epochs or constrain a model’s whole history. An observed source at a redshift does not directly display the present simultaneous separation rate of a remote pair.

Comparing reported values therefore requires the data, calibration, physical model and uncertainty conventions introduced in the preceding chapter. One should not identify every numerical rate with the same target or use the word constant to suppress the epoch label.

The supplied curve provides a useful check on language: faster expansion could mean increasing separation, increasing absolute slope or increasing fractional rate. State which quantity is intended. H(t) gives one precise fractional-rate definition; H₀ names its present-epoch value.

Worked reasoning

Try the reasoning

1. Does the subscript zero in H₀ assert the same H at every cosmic time?

   Show the answer and reasoning

   No. It labels the present epoch.

   A cosmological history supplies H(t); constant H is a special possible behaviour rather than the meaning of the subscript.

2. Does the converted H★⁻¹ ≈ 16.30 Gyr establish a measured cosmic age?

   Show the answer and reasoning

   No.

   H★ was chosen, and an age additionally requires an expansion history and its domain.

3. Can ä be positive while Ḣ is negative in the supplied curve?

   Show the answer and reasoning

   Yes.

   The absolute slope grows while division by the increasing a gives a decreasing fractional rate. Both the table and derivative identity check this.

What to carry forward

• H = ȧ/a is a fractional expansion rate with inverse-time dimensions.
• H₀ labels the present epoch, while H(t) describes a history.
• A rate, an acceleration and an inferred age are different quantities whose relationship requires a stated history and model.