Cosmology · Galaxies as tracers of cosmic structure

Redshift-space patterns and peculiar velocities

Map five labelled tracers into redshift-inferred radial coordinates, keeping coordinate distortion separate from physical motion or a dynamical solution.

Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.

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After reading, you can

Separate real spatial offsets from radial coordinates inferred using redshift.
Convert a supplied peculiar velocity into a radial coordinate displacement with units.
Recognise how assigned coherent and residual velocities produce different radial patterns.

Peculiar velocity enters a redshift measurement

A galaxy’s measured redshift is influenced by background expansion and by its motion relative to that background, among other effects requiring a suitable observational model. Converting redshift directly into distance can therefore distort inferred spatial patterns.

In a small region with a common distant line of sight, a first-order nonrelativistic mapping can be written s_∥ = x_∥ + v_∥/(aH) and s_⊥ = x_⊥. Here x and s are comoving position offsets, v_∥ is proper peculiar velocity, a is the dimensionless scale factor and H is the background expansion rate.

Positive radial velocity points away from the observer in this convention. The quotient v_∥/(aH) has comoving length units. A change in s is a change in an inferred coordinate, not a physical jump by the object.

The approximation does not supply the velocities themselves. Their origin, evolution and relation to gravity require additional dynamics and statistical modelling. The following examples assign velocities explicitly.

Keep the same labelled objects in every comparison

Choose one reference epoch with a = 1 and H_* = 100 km/s/Mpc. H_* is a teaching input, not a measured present Hubble constant. The mapped radial displacement in Mpc is then the supplied velocity in km/s divided by 100.

Use five tracers A–E with real radial offsets −2, −1, 0, 1 and 2 Mpc relative to a distant common centre. Their transverse offsets are −0.4, 0.6, 0, −0.6 and 0.4 Mpc respectively.

Negative offsets mean the near side of that reference centre, not negative absolute distance from the observer. The distant-observer approximation lets the same radial direction serve all five objects.

Compare two alternative velocity assignments for these same real positions. They are separate coordinate accounts at one epoch, not successive frames of a simulation or tracked changes over time.

Inward coherent motion compresses one supplied radial pattern

For the coherent assignment choose v_∥ = −50(x_∥/Mpc) km/s. The velocities are 100, 50, 0, −50 and −100 km/s. Each displacement opposes half of its real offset, giving s_∥ = x_∥/2.

The alternative residual assignment is 200, −100, −200, −100 and 200 km/s. Its mapped radial offsets are 0, −2, −2, 0 and 4 Mpc. Transverse offsets stay fixed in both assignments.

A radial coordinate coincidence does not mean two objects collide or disappear. B and C have equal s_∥ in the residual account but retain their own identities and different transverse offsets.

Coherent infall can compress radial patterns, and motions within collapsed systems can produce elongation in real analyses. These supplied velocities illustrate coordinate effects; they are not a solved infall field, a virial velocity distribution or an observed cluster.

Five supplied tracers at a = 1 and chosen H_* = 100 km/s/Mpc; the two velocity accounts are alternatives
Tracer	Real x_∥ / Mpc	x_⊥ / Mpc	Coherent v_∥ / km/s	Coherent s_∥ / Mpc	Alternative v_∥ / km/s	Alternative s_∥ / Mpc
A	−2	−0.4	100	−1	200	0
B	−1	0.6	50	−0.5	−100	−2
C	0	0	0	0	−200	−2
D	1	−0.6	−50	0.5	−100	0
E	2	0.4	−100	1	200	4

A covariance can change the inferred spread

All three lists of radial offsets have zero mean. Using denominator five, the real radial variance is 2 Mpc² and the coherently compressed variance is 0.5 Mpc². Their root-mean-square spreads are about 1.414214 and 0.707107 Mpc.

For the alternative assignment, velocity variance is 28,000 (km/s)² and its finite covariance with real radial position is zero. The inferred variance becomes 2 + 28,000/100² = 4.8 Mpc², with RMS about 2.190890 Mpc.

More generally for these finite centred averages, Var(s_∥) = Var(x_∥) + Var(v_∥)/(aH)² + 2Cov(x_∥,v_∥)/(aH). The coherent assignment has a negative covariance, so simply adding a positive velocity-variance term would miss the compression.

The alternative’s zero sample covariance is a deliberately supplied property. It proves neither statistical independence nor Gaussian velocities, and the displayed RMS values have no observational error bars or confidence coverage.

This original finite coordinate exercise uses five persistent labelled test tracers at one reference epoch with a = 1 and chosen H_* = 100 km/s/Mpc. The small-region, distant-observer, first-order nonrelativistic map is s_parallel = x_parallel + v_parallel/(aH), s_perp = x_perp. All displayed spatial coordinates are comoving Mpc offsets from a distant common centre, rather than negative absolute observer distances. Proper peculiar velocity is positive away from the observer. The common radial coordinate increases to the right; all three panels have the same radial range -2 to +4 Mpc and transverse range -0.8 to +0.8 Mpc. Both directions use exactly 80 pixels per Mpc, so the spatial axis units are equal within and across panels. Panel A shows real radial offsets -2,-1,0,1,2 Mpc with transverse offsets -0.4,0.6,0,-0.6,0.4 for A,B,C,D,E respectively. Panel B is the separate coherent assignment v = -50(x_parallel/Mpc) km/s, giving velocities 100,50,0,-50,-100 km/s and inferred offsets -1,-0.5,0,0.5,1 Mpc. Panel C is the alternative assignment 200,-100,-200,-100,200 km/s, giving inferred offsets 0,-2,-2,0,4 Mpc. Transverse offsets and A–E labels remain unchanged throughout. Panels are alternative coordinate accounts at one epoch, not a time sequence or physical jumps. B and C can share a radial coordinate while remaining distinct labelled objects at different transverse offsets. Finite centred averages use denominator five. Real radial variance is 2 Mpc², coherent variance 0.5 Mpc² and alternative variance 4.8 Mpc², with RMS values approximately 1.414214,0.707107,2.190890 Mpc. The alternative velocity variance is 28000 (km/s)² and covariance with real radial position is zero by assignment. The coherent covariance is negative; the full variance map includes that covariance term. No Gaussian or independent velocity law, dynamical infall solution, virial equilibrium, observed cluster, physical collision, measured present Hubble constant or halo-mass inference is supplied. Original Ark teaching model.

Coordinate patterns become a probe through a physical model

Redshift-space clustering can carry information about motion and structure growth because velocities and density are related dynamically. Extracting that information requires a model for the selected tracers and the scales used.

A real calculation also includes redshift uncertainties, survey selection, observer geometry and the distance conversion of a specified background. Wide-angle, nonlinear and relativistic effects can require treatment beyond the small-region mapping used here.

The radial elongation of a pattern alone does not directly give a unique halo mass or a time sequence. Different velocity assignments, selection and distance assumptions can affect the inferred shape.

Always state whether a plotted axis is a real-space coordinate, a redshift-inferred coordinate or a velocity. Preserving that distinction lets a geometrical distortion become a testable part of an inference instead of being mistaken for direct material motion.

Worked reasoning

Convert one supplied radial velocity

At a = 1 and H_* = 100 km/s/Mpc, what displacement does v_∥ = 200 km/s produce?

Use Δs_∥ = v_∥/(aH_*).
The units give (200 km/s)/(100 km/s/Mpc) = 2 Mpc.
The positive sign places the inferred coordinate 2 Mpc farther along the chosen away-from-observer direction.

A positive 2 Mpc inferred radial coordinate displacement, not a physical two-megaparsec jump.

Try the reasoning

Why does the coherent assignment give s_∥ = x_∥/2?
Show the answer and reasoning
Its velocity term is −x_∥/2 at the supplied a and H_*.
Adding this coordinate displacement to x_∥ leaves half the original offset.
Are transverse coordinates changed by the stated radial-only mapping?
Show the answer and reasoning
No.
The chosen plane-parallel first-order construction sets s_⊥ = x_⊥.
If B and C share one inferred radial coordinate, does that establish a physical collision?
Show the answer and reasoning
No.
An inferred radial coincidence is not time evolution, and their transverse offsets and identities remain distinct.

What to carry forward

A peculiar radial velocity changes the coordinate inferred from redshift under a stated distance conversion.
Coherent and residual assignments can compress or broaden radial patterns without moving objects between physical locations in the example.
Dynamical interpretation requires tracer, geometry, selection and uncertainty models beyond the coordinate map.

References

Sources supporting the explanations, physical distinctions and examples in this lesson.