The nearby Hubble relation and peculiar motions

Begin with the quantities that were measured

A spectrum supplies a redshift z from the displacement of identifiable spectral features. A distance estimate requires additional observations and calibration. Neither measurement is simply a direct reading of a galaxy’s simultaneous distance from us and the rate at which that distance changes.

In a homogeneous expanding background, a pair of comoving locations has proper separation D at a specified cosmic time and separation rate Ḋ = HD at that same time. H is the fractional expansion rate. This relation describes the chosen background geometry; it is different from a general conversion of an observed redshift into a speed.

For sufficiently nearby objects with small redshift, cz can be used as an approximate velocity proxy, where c is light speed. Nearby distance definitions also approach a common limit. This first-order description is useful only with its approximation and observer frame stated; it is not a universal formula for remote objects.

A galaxy can also move relative to the smooth background flow. That contribution is called peculiar motion. Its radial part affects the redshift along our sightline. Positive radial motion here means away from the observer, while negative means toward the observer.

A small-distance example can have a blueshift

Choose an illustrative background coefficient H★ = 60 km/s/Mpc. The star marks a number supplied for this exercise, not a current measurement of H₀. Use nearby distance estimates D = 2, 10 and 50 Mpc in one consistent observer frame.

The toy relation is cz_proxy ≈ H★D + u, where u is the signed radial peculiar contribution in km/s. It is a first-order nearby, nonrelativistic approximation. Measurement errors, gravitational redshift and higher-order propagation effects are omitted from this chosen comparison.

At 2 Mpc the background term is 60 × 2 = 120 km/s. Choose u = −180 km/s. The proxy becomes 120 − 180 = −60 km/s, corresponding to a small blueshift. The background coefficient is still positive: the toward-us contribution is larger than the background term in this example.

At 10 Mpc, choose u = +180 km/s, giving 600 + 180 = 780 km/s. At 50 Mpc, choose u = −180 km/s, giving 3,000 − 180 = 2,820 km/s. All proxy redshift magnitudes are below 0.01 using c = 299,792.458 km/s. The table is a supplied model, not a fitted galaxy survey.

The same speed is a larger fraction nearby

Compare the magnitude 180 km/s with each background term. At 2 Mpc it is 180/120 = 1.5, or 150 percent. At 10 Mpc it is 180/600 = 0.3, or 30 percent. At 50 Mpc it is 180/3,000 = 0.06, or 6 percent.

This explains why a peculiar contribution can dominate a very nearby redshift while being a smaller fraction of a larger background trend. It does not establish that the peculiar motions of real galaxies all have magnitude 180 km/s, or that every object at the same distance receives the same correction.

Dividing a single object’s proxy by its distance gives −30, 78 or 56.4 km/s/Mpc in the three rows. None equals the supplied H★ = 60. The ratio has mixed the background term with u/D; it has not isolated a background expansion measurement.

Going farther is not an unrestricted cure. The simple cz approximation becomes inadequate as the relevant corrections grow, and distance estimates carry their own calibration and selection uncertainties. A useful analysis balances those effects within a stated measurement and cosmological model.

An expansion inference needs a consistent comparison

Observer motion also contributes to measured redshifts. Before combining observations, an analysis specifies the reference frame and accounts for the corrections appropriate to it. Mixing uncorrected values from different conventions can introduce a pattern that does not belong to the background expansion.

A collection of galaxies can constrain a background trend when the distance calibration, peculiar-motion treatment and other uncertainties are modelled. It is not enough to assume that every contribution disappears merely because more objects were listed. The inferred coefficient remains tied to those choices.

The distinction between same-time separation rate and observed redshift becomes increasingly important at larger distances. Redshift records the propagation history as well as local contributions, and different observational distance definitions no longer agree. Later distance-method lessons develop those comparisons explicitly.

This lesson therefore extends the conceptual relation into a measurement question. A nearby departure from the trend can contain information about local motion; it need not disprove a positive large-scale expansion rate.

Do not apply the background law to every bound object

The background relation concerns the ideal comoving flow. A bound galaxy, star system or laboratory does not have to expand at HD at its own local scale. Its dynamics and peculiar motion require the appropriate local model, as the earlier bound-systems lesson explained.

For a first reading, keep the signed addition at 2 Mpc. A positive background term plus a larger negative peculiar contribution gives a negative nearby proxy. The observation and the background inference can coexist once the quantities and approximation are named.

Worked reasoning

Try the reasoning

1. What fraction of the 50-Mpc background term is a peculiar speed magnitude of 180 km/s in this model?

   Show the answer and reasoning

   0.06, or 6 percent.

   The background term is 3,000 km/s, so 180/3,000 = 0.06.

2. May the single-object value 780/10 = 78 km/s/Mpc be identified with the supplied background coefficient?

   Show the answer and reasoning

   No.

   It includes the chosen radial peculiar contribution divided by distance; the supplied coefficient is 60.

3. Is cz an exact conversion from arbitrary high redshift to a galaxy’s same-time recession rate?

   Show the answer and reasoning

   No.

   The nearby proxy has a limited domain. Distant observations require the relevant distance and expansion-history relations.

What to carry forward

• Nearby redshift proxies combine a background trend with local and observer contributions.
• A fixed peculiar speed is a larger fraction of the trend at smaller D.
• A background expansion inference requires a calibrated, frame-consistent model rather than one galaxy’s cz/D.