Expansion means changing cosmic separations

Ask what is farther apart, and at which times

Imagine three labelled markers A, B and C following an ideal smooth cosmic flow. At a first cosmic time, A and B are 10 megaparsecs apart and B and C are 20 megaparsecs apart along a straight line. At a later time the same separations are 15 and 30 Mpc. Both have grown by the factor 1.5.

The comparison follows the same markers, rather than selecting a new, more distant pair in the later picture. It also compares distances defined at a common cosmic time in each picture. We call these proper separations on the chosen cosmic-time slices. They describe the background geometry; they are not automatically the distance numbers delivered by an observation of old light.

A megaparsec, Mpc, is a length: one million parsecs. It is not an age or a speed. A large separation can be quoted in these units without saying that a photon took that number of million years to cross it. Light propagation depends on the geometry along its path.

Use stable labels while the distance scale changes

A comoving coordinate is a label attached to the ideal flow. A marker that follows that flow retains its label even as the proper separation from another such marker changes. The dimensionless scale factor a(t) records the common change in length scale with cosmic time t.

For a flat-background straight-line example, write D(t) = a(t)χ. Here D is the proper separation of a specified pair, and χ is its fixed comoving separation. In this lesson χ has length units, such as Mpc; a has no units. Other coordinate conventions put a separate reference length beside a dimensionless comoving coordinate, so the convention must be read before comparing formulas.

Choose a = 1 at the first picture. The three marker labels can be 0, 10 and 30 Mpc. At a = 1.5 their proper positions relative to A are 0, 15 and 45 Mpc. The coordinate labels did not change. The physical ruler assigned to a fixed coordinate interval did.

Normalizing a to one is a choice of reference, not a measurement that the total Universe has a unit radius. If we multiply all a values by two and divide the comoving lengths by two, every D is unchanged. The ratios of a between epochs carry the relative expansion in this description.

No marker has to be the centre of expansion

Now describe the same two snapshots relative to B. At the first epoch, A lies 10 Mpc to its left and C lies 20 Mpc to its right. At the second epoch those distances are 15 and 30 Mpc. B sees the same fractional growth in both directions. Choosing B as origin has not changed the model.

The increase in A-to-C separation is 15 Mpc, whereas the increase in A-to-B is 5 Mpc. The larger absolute increase comes from the larger starting separation; the fractional increase is the same. This is a pattern of changing distances throughout the ideal background, not evidence that marker A pushed the others away.

The three-marker picture is only a finite one-dimensional sample of the assumed background. Its page boundary and its leftmost marker are drawing choices. Neither establishes a physical edge of space. Extending the same geometric idea into three spatial dimensions requires no surrounding material container.

Real objects need not remain on the ideal flow

A real galaxy can move relative to the smooth background flow because of nearby structure. This is peculiar motion. Its comoving position then changes, so its total separation change need not be given by a fixed χ multiplied by a alone. Some nearby objects can approach each other while the large-scale background expands.

The same warning applies inside a bound structure. The separation of two stars in an orbit is governed by their local dynamics; it is not required to grow by the scale-factor ratio. The markers in the toy define comoving trajectories. They are not a claim that every galaxy centre, planet or atom must follow one.

Separating background expansion from peculiar motion is part of a model. An observed spectral shift can include both, along with other effects discussed in the preceding chapter. A redshift is not a label that independently certifies pure background motion.

The two snapshots are a model comparison, not two direct cosmic photographs

We receive signals along light paths. Distant light was emitted earlier, so a telescope image mixes emission times rather than showing one simultaneous cosmic-time slice. Reconstructing a separation at a chosen epoch uses a model and a distance definition. The earlier lesson on lookback time explains why those distinctions cannot be dropped.

For light exchanged by ideal comoving source and receiver in the homogeneous background, the cosmological wavelength ratio is 1 + z = a at reception divided by a at emission. It connects two events. It does not by itself provide the pair's comoving separation or the elapsed interval between them.

Our two scale-factor values were supplied as a teaching model. They were not derived from three real observed galaxies, and no actual expansion rate has been measured here. A physical cosmology must predict an expansion history and confront distance, redshift and other observations with their uncertainties.

That makes the next question precise: how quickly does a change, and is its absolute change per unit time the same thing as its fractional change? The next lesson introduces the expansion rate while retaining the time-slice and comoving assumptions used here.

Worked reasoning

Try the reasoning

1. In the three-marker example, does A-to-C grow by a larger fraction than A-to-B?

   Show the answer and reasoning

   No. Both grow by 50 percent.

   The absolute increases are 15 and 5 Mpc because their starting separations differ. The later/earlier ratio is 1.5 for each.

2. Replace a by 2a and every comoving length χ by χ/2. Does the predicted proper separation change?

   Show the answer and reasoning

   No.

   The product (2a)(χ/2) is still aχ. The coordinate normalization is not an independent physical size measurement.

3. Does an approaching nearby galaxy, by itself, rule out an expanding homogeneous background?

   Show the answer and reasoning

   No.

   Its peculiar motion and local environment can make its separation decrease. The ideal comoving assumption must be tested rather than assigned to every real object.

What to carry forward

• Expansion compares changing separations of specified ideal comoving markers on stated cosmic-time slices.
• A dimensionless scale factor multiplies the chosen comoving length; its normalization is conventional.
• The same fractional growth can be described from any marker in the homogeneous background.
• Local dynamics and the light-cone nature of observations must be added before interpreting real measurements.