A scale factor and a changing expansion rate

The same length increase can be a different fraction

Compare two separate hypothetical growth scenarios. In the first, a model separation grows from 10 to 11 Mpc in a short interval. Its increase is 1 Mpc, or 10 percent of its starting length. In the second scenario, a separation grows from 20 to 21 Mpc over an equal interval. Its increase is also 1 Mpc, but only 5 percent. These are separate scenarios rather than two comoving pairs in one homogeneous background. Absolute change per unit time and fractional change per unit time answer different questions.

The dimensionless scale factor a tells us relative lengths in the smooth background. Its slope with respect to cosmic time t is written ȧ, pronounced a-dot. A dot denotes a time derivative: the instantaneous rate of change obtained as the comparison interval becomes very short. It has units of inverse time because a itself has no units.

The Hubble parameter is H(t) = ȧ(t)/a(t). Dividing by the current scale factor turns the absolute slope into a fractional expansion rate. H also has inverse-time units. For a fixed comoving separation χ, D = aχ and Ḋ = ȧχ = HD. Ḋ is the change in the specified simultaneous proper separation, not a local relative speed measured by one nearby observer.

The derivative notation gives a compact exact statement for these smooth model functions. On a first reading, follow the finite changes and table below. A slope at one instant should not be replaced by a long-interval average without checking how much the history varies.

A steady absolute slope gives a falling fractional rate

Choose an illustrative time scale T = 10 Gyr, where a Gyr is one billion Julian years. Let t = 0 mark a chosen reference epoch and examine 0 ≤ t ≤ 20 Gyr. This is a synthetic interval for comparing functions, not a measured future history of our Universe.

Model A is a(t) = 1 + t/T. Since t and T have the same units, t/T and a are dimensionless. At t = 0, 10 and 20 Gyr, a is 1, 2 and 3. It increases by one in each 10-Gyr interval, so ȧ = 1/T = 0.1 Gyr⁻¹ throughout.

Its fractional expansion rate is H(t) = 1/(T + t). At the same three times, H is 0.1, 0.05 and about 0.0333 Gyr⁻¹. The expansion continues while H decreases. The same increment becomes a smaller fraction of an already larger scale factor.

For fixed χ = 10 Mpc, this history gives D = 10, 20 and 30 Mpc at those epochs and a constant Ḋ = 1 Mpc/Gyr. A falling H has not meant that this pair's separation decreases or that its absolute separation rate slows. Keeping the denominator in view resolves the apparent contradiction.

A decreasing H does not decide the sign of acceleration

Expansion means ȧ > 0 in this background description. Accelerated expansion means ä > 0, where ä is the time derivative of ȧ. It describes an increasing absolute slope of a, with units of inverse time squared. Model A has ä = 0: its scale factor increases linearly rather than accelerating.

Now choose Model B, a(t) = (1 + t/T)² on the same interval. It has a = 1, 4 and 9 at t = 0, 10 and 20 Gyr. Its absolute increases over successive 10-Gyr intervals are 3 and 5. Its slope is ȧ = (2/T)(1 + t/T), and ä = 2/T² > 0.

Nevertheless H for Model B is 2/(T + t), which falls from 0.2 to 0.1 to about 0.0667 Gyr⁻¹ at the three times. An accelerating scale factor can therefore have a decreasing Hubble parameter. Saying only that the expansion rate is going down is ambiguous unless the quantity is named.

These two functions demonstrate the definitions and their logical differences. They have not been derived from particular matter contents or fitted to observations. Later dynamics lessons ask which histories follow from a specified relativistic cosmological model. Do not read a dark-energy measurement into either chosen curve.

Why is H often quoted in speed per distance?

In the relation Ḋ = HD, a separation rate can be written in km/s and a separation in Mpc. Then H is quoted in km/s/Mpc. This is still inverse time: dividing a speed by a length leaves a reciprocal time once the two length units are converted consistently.

For a conversion example, choose H = 0.05 Gyr⁻¹. Using a Julian year of 31,557,600 seconds and 1 Mpc ≈ 3.08568 × 10¹⁹ km gives H ≈ 1.5844 × 10⁻¹⁸ s⁻¹, or about 48.9 km/s/Mpc. These are equivalent units for one chosen rate; they are not three independent measurements or a current estimate of H₀.

The subscript zero in H₀ conventionally means the Hubble parameter at the present cosmic epoch. Calling it the Hubble constant does not say that H has stayed constant over cosmic history. Within the ideal homogeneous background at one time, the same H relates separations of comoving pairs; its value can change between times.

Recession defined from cosmic-time separation should also be distinguished from a special-relativistic local velocity. The first relation does not impose a universal conversion z = HD/c for arbitrarily distant observed sources. Redshift and observed distances involve the expansion history and the relevant propagation geometry.

A present slope is not a cosmic stopwatch

The reciprocal H⁻¹ is a time scale. Its units are appropriate for comparing expansion with other processes. Its units alone do not make it the age of the Universe. Determining an elapsed cosmic time requires the previous history, not only the current fractional rate.

The same caution holds for estimating a future doubling time. Multiplying today's rate by a long interval assumes a behavior that must be stated. If H were exactly constant, the fractional change would accumulate differently from Model A, where ȧ is constant instead. A single number does not choose between those histories.

Observations constrain a history through multiple kinds of evidence and a physical model. Supernova distances, characteristic scales and other methods appear later in the approved learning path. At this stage, the important calculation is knowing which quantity a proposed observation or statement is meant to constrain.

Worked reasoning

Try the reasoning

1. Model A has constant positive ȧ and falling H. Is its scale factor contracting, decelerating or growing linearly?

   Show the answer and reasoning

   Growing linearly over the specified interval.

   ȧ remains positive and ä is zero. The decreasing H comes from dividing by a larger a.

2. In Model B, H decreases while ä is positive. Is that mathematically possible?

   Show the answer and reasoning

   Yes; the supplied quadratic history explicitly has both properties.

   H = 2/(T + t) decreases while ä = 2/T² stays positive. A fractional rate and the derivative of an absolute slope are different quantities.

3. Does quoting H⁻¹ = 20 Gyr establish a cosmic age of 20 Gyr?

   Show the answer and reasoning

   No.

   It supplies a time scale at the quoted epoch. Cosmic age is an elapsed interval inferred from the earlier expansion history and the model's applicability.

What to carry forward

• a is dimensionless; ȧ is its instantaneous slope; H = ȧ/a is the fractional expansion rate.
• Expansion requires positive ȧ; accelerated expansion requires positive ä.
• Both steady and accelerating absolute growth can occur with a falling H.
• Units and epoch labels clarify a measured rate; they do not supply the full cosmic history.