Redshift-Space Distortions

A map drawn with a corrupted ruler

Astronomers cannot measure the distance to a faraway galaxy directly. What a spectrograph delivers is a redshift — the fractional stretch in the wavelength of a known atomic line. We convert that redshift into a distance by assuming the galaxy is carried outward purely by the smooth, uniform expansion of the universe: the Hubble flow. Multiply the redshift by the speed of light and divide by the Hubble constant, and you have a radial coordinate. Do this for a million galaxies and you have a three-dimensional map of the cosmos.

But the ruler is corrupted. No galaxy moves with the pure Hubble flow. Each one also has a peculiar velocity — a real, physical motion driven by the gravity of everything around it: a few hundred kilometres per second of infall toward the nearest supercluster, plus whatever orbital motion it inherits from sitting inside a group or cluster. That extra velocity Doppler-shifts the light, and the redshift cannot tell the two contributions apart. The galaxy gets placed at the wrong distance — but only along the line of sight, because only the radial component of the velocity affects the redshift. The transverse position on the sky stays perfectly correct.

The result is a map that is distorted in one direction only. Structures get squashed or stretched along the line of sight while their across-the-sky dimensions are untouched. That anisotropy is not noise to be removed. It is a direct fingerprint of the velocity field — and the velocity field is gravity in action. Reading the distortion backwards tells you how fast structure is growing. That is the surprising payoff of redshift-space distortions: a mapping error, properly understood, becomes one of cosmology's sharpest tools.

The mechanism: real space versus redshift space

Define the true comoving position of a galaxy in real space as r. The position we actually reconstruct, in redshift space, is s. They differ only along the line-of-sight unit vector r̂ by the displacement that the peculiar velocity introduces:

s = r + (v · r̂ / aH) r̂

Here v is the peculiar velocity, a the scale factor, and H the Hubble rate at the galaxy's epoch. The factor aH converts a velocity into a comoving-distance displacement. The observed redshift obeys, to first order,

c z_obs ≈ c z_cosmo + v_los   →   Δs_los = v_los / H(z)

where v_los is the line-of-sight peculiar velocity. The displacement is purely radial, which is why the distortion is anisotropic. On large scales the velocity field is set by linear gravitational infall and is coherent: neighbouring galaxies fall in concert. On small scales, inside collapsed halos, the velocities are random and virialized. These two regimes produce the two opposite distortions.

The Kaiser effect — squashing on large scales

Consider a mild overdensity — a forming supercluster — in the linear regime. Gravity pulls the surrounding galaxies inward. Galaxies on the far side of the overdensity (more distant than its centre) are falling toward us, which reduces their redshift and brings them apparently closer. Galaxies on the near side are falling away from us, increasing their redshift and pushing them apparently farther. Both edges move toward the centre in redshift space. The structure looks flattened — pancaked — perpendicular to the line of sight.

Nick Kaiser worked this out in linear perturbation theory in 1987. In Fourier space the redshift-space power spectrum is amplified relative to the real-space one by an angle-dependent factor:

P_s(k, μ) = (1 + β μ²)² P_r(k)
β = f / b
f = d ln D / d ln a ≈ Ω_m(z)^γ,   γ ≈ 0.55 (GR)

where μ = cos θ is the cosine of the angle between the wavevector k and the line of sight, b is the linear galaxy bias (how strongly galaxies trace the underlying matter), f is the linear growth rate, and D(a) is the linear growth factor. The boost is largest for modes pointing along the line of sight (μ = 1) and absent for transverse modes (μ = 0). Crucially, the amplitude of the squashing is controlled by f — the speed at which structure grows. Faster growth means faster infall means stronger squashing. This is the link that turns a geometric distortion into a measurement of gravity.

Fingers of God — stretching on small scales

Now zoom into a virialized galaxy cluster. The member galaxies are not falling coherently; they are buzzing around the cluster centre on random orbits with a velocity dispersion of 700–1500 km/s. A cluster only a couple of Mpc across in real space gets its galaxies scattered over tens of Mpc in redshift space, because each random radial velocity displaces the inferred distance by v/H. The compact knot is stretched into a thin, elongated spike — and because the elongation is always along the line of sight, in a 3D map every cluster points like a finger straight back at the observer.

These "Fingers of God" are a strongly nonlinear effect and the bane of large-scale-structure analysis: they leak power from small scales up to larger scales and contaminate the clean Kaiser signal. Analyses model them with a small-scale velocity-dispersion damping term, classically a Lorentzian or Gaussian in kμσ_v:

P_s(k, μ) = (1 + β μ²)² P_r(k) × 1 / [1 + (k μ σ_v / H)²]   (dispersion model)

The net picture is memorable: on large scales the cosmic web is squashed toward the observer; on small scales individual clusters are stretched away into radial fingers. Squashing and stretching, on opposite scales, both produced by the same velocity field.

The key numbers

Quantity	Typical value	Note
Coherent infall velocity	~300 km/s	Drives the Kaiser squashing
Cluster velocity dispersion	700–1500 km/s	Coma ≈ 1000 km/s; drives Fingers of God
Local Group vs CMB frame	≈ 630 km/s	Our own peculiar velocity (Milky Way ≈ 550 km/s)
Growth rate f at z = 0	≈ 0.53	From Ω_m(0)^0.55 with Ω_m ≈ 0.31
Growth rate f at z = 1	≈ 0.87	Approaches 1 at high z (matter-dominated)
fσ₈ at z ≈ 0.5	≈ 0.46	BOSS/eBOSS/DESI measurement
σ₈ (Planck 2018)	0.811 ± 0.006	rms fluctuation on 8 h⁻¹ Mpc
Linear bias of LRGs	b ≈ 1.8–2.2	Luminous red galaxies trace peaks
Distortion scale at z = 0.5	~3 h⁻¹ Mpc per 300 km/s	v/H(z) displacement

The single most important number is fσ₈. RSD do not measure f and the galaxy bias b separately, and they do not measure f and σ₈ separately. The Kaiser amplitude is ∝ f · (matter fluctuation amplitude), and because galaxy clustering measures bσ₈, the bias cancels in the right combination, leaving the bias-independent product fσ₈. That is what surveys report, and it is what cosmological models must predict.

How it is measured: multipoles of the two-point statistic

To extract the anisotropy you measure clustering as a function of both the separation (or wavenumber) and the angle μ to the line of sight, then project onto Legendre polynomials ℒ_ℓ(μ):

P_ℓ(k) = (2ℓ + 1)/2  ∫₋₁¹  P_s(k, μ) ℒ_ℓ(μ) dμ

• Monopole (ℓ = 0): the angle-averaged power. Carries the isotropic clustering and the baryon-acoustic-oscillation ruler, but no direct velocity information.
• Quadrupole (ℓ = 2): the leading anisotropy. In linear theory P₂/P₀ ∝ β, so the quadrupole-to-monopole ratio directly measures β = f/b. Kaiser squashing gives the quadrupole a characteristic negative sign on large scales.
• Hexadecapole (ℓ = 4): a higher-order check that helps break degeneracies between the Kaiser term and the Fingers-of-God damping.

Equivalently, in configuration space one measures the correlation function ξ(s_⊥, s_∥) on a grid of transverse and line-of-sight separations and sees the contours literally squashed on large scales and elongated (the finger) at small s_⊥. Modern pipelines fit the full multipole shape with one-loop perturbation-theory or effective-field-theory models that include nonlinear bias, the Kaiser term, and a streaming model for the small-scale velocity dispersion.

Worked example: how far does 300 km/s move a galaxy?

Take a galaxy at redshift z = 0.5 with a line-of-sight peculiar velocity of 300 km/s, in a standard flat ΛCDM universe with H₀ = 70 km/s/Mpc and Ω_m = 0.31. First evaluate the Hubble rate at that redshift:

H(z) = H₀ √[Ω_m (1+z)³ + Ω_Λ]
     = 70 × √[0.31 × 1.5³ + 0.69]
     = 70 × √[0.31 × 3.375 + 0.69]
     = 70 × √[1.046 + 0.69]
     = 70 × √1.736
     ≈ 70 × 1.318
     ≈ 92.2 km/s/Mpc

The comoving displacement the redshift error introduces is the velocity divided by this Hubble rate (in physical units the factor is H(z) directly for the apparent shift in redshift-inferred distance):

Δs = v_los / H(z) = 300 / 92.2 ≈ 3.3 Mpc

So a perfectly ordinary 300 km/s infall mislocates the galaxy by about 3 Mpc — a few percent of the scale on which superclusters and the BAO feature live. Multiply this small shift across millions of galaxies all falling coherently toward the same overdensities and the squashing becomes a clean statistical signal in the quadrupole. For a cluster galaxy with a 1000 km/s random velocity the displacement is ≈ 11 Mpc — large enough to smear the whole cluster into the dramatic radial finger.

Now translate the amplitude into physics. At z = 0.5 in ΛCDM, Ω_m(z) = Ω_m (1+z)³ / [H(z)/H₀]² = 0.31 × 3.375 / 1.736 ≈ 0.60, so the growth rate is f ≈ 0.60^0.55 ≈ 0.75. With σ₈(z = 0.5) ≈ 0.62 (the z = 0 value of 0.81 grown down by the linear factor), the predicted fσ₈ ≈ 0.46 — in line with what BOSS measured at that redshift. Measure fσ₈ a few percent low or high, and you are constraining whether gravity grows structure exactly as General Relativity predicts.

History and the surveys that measure it

• 1970s — Fingers of God. As the first redshift surveys (the CfA survey and predecessors) built 3D maps, rich clusters appeared as elongated radial spikes. The effect was understood as virial velocity scatter; the evocative name traces to work by Jim Peebles and contemporaries.
• 1987 — Kaiser's linear theory. Nick Kaiser's paper "Clustering in real space and redshift space" derived the (1 + βμ²)² amplification and showed that the large-scale anisotropy measures β = f/b. This is the theoretical foundation of all modern RSD cosmology.
• 2001 — 2dFGRS. The 2dF Galaxy Redshift Survey (~221,000 redshifts) yielded the first competitive β ≈ 0.43 (Peacock et al.), an early growth-rate constraint from the squashing.
• 2009–2014 — SDSS-III BOSS. The Baryon Oscillation Spectroscopic Survey mapped ~1.5 million luminous galaxies, delivering fσ₈ measurements at the few-percent level across z ≈ 0.2–0.7, jointly with the BAO standard ruler.
• 2014–2019 — SDSS-IV eBOSS. Extended BOSS to quasars and emission-line galaxies, pushing fσ₈ measurements out to z ≈ 1.5 and combining 20+ years of SDSS data into a consistent growth history.
• 2024 — DESI. The Dark Energy Spectroscopic Instrument's first-year results measured BAO and RSD from tens of millions of redshifts, the most precise growth-rate map to date, with the full 5-year survey targeting ~40 million galaxies and quasars.
• 2023+ — Euclid and Vera Rubin. ESA's Euclid satellite (launched July 2023) and the Vera C. Rubin Observatory's LSST extend RSD growth measurements to higher redshift and larger volumes, sharpening the test of gravity.

Why RSD test gravity itself

Cosmology has two independent things to measure: the expansion history (how fast the universe grows in size, set by H(z) and probed by supernovae, BAO, the CMB) and the growth history (how fast lumps grow inside that expanding background). General Relativity ties them together: given an expansion history, GR predicts exactly how fast structure must grow, with f ≈ Ω_m(z)^0.55. Many alternatives — f(R) gravity, DGP braneworlds, and other modified-gravity models invented to explain cosmic acceleration without dark energy — can be tuned to reproduce the same expansion history while predicting a different growth rate.

RSD measure growth directly and independently of expansion. If the measured fσ₈ tracks the GR prediction across redshift, modified gravity is squeezed; if it deviates, you have evidence for new physics. This is why fσ₈ is one of the headline numbers of every modern spectroscopic survey, and why the mild tension between some low-redshift fσ₈ measurements and the value extrapolated from Planck's CMB (related to the broader "S₈ tension") is taken seriously.

RSD versus other large-scale-structure probes

Probe	Measures	Direction sensitivity	Main systematic
RSD (this article)	Growth rate fσ₈	Anisotropy along line of sight	Nonlinear velocities, Fingers of God, bias
BAO	Standard ruler → H(z), D_A(z)	Isotropic + Alcock-Paczynski	Nonlinear smearing of the peak
Weak lensing	σ₈, Ω_m (matter amplitude)	Transverse shear of background galaxies	Shape measurement, photo-z, intrinsic alignment
Galaxy clustering (real-space)	bσ₈, shape of P(k)	Isotropic	Galaxy bias degeneracy
Peculiar-velocity surveys	fσ₈ at very low z	Direct radial velocities	Distance-indicator scatter

RSD are uniquely sensitive to velocities rather than positions. The Alcock-Paczynski test exploits a related anisotropy — if you assume the wrong cosmology, spherical features look elliptical — and modern analyses fit RSD and Alcock-Paczynski jointly, since both produce line-of-sight-versus-transverse distortions that must be disentangled.

Common misconceptions and subtleties

• "Redshift-space distortions are an instrumental error to be removed." They are a real, physical, information-rich signal. You do not subtract them; you model them and read out the growth rate.
• "It's the same as gravitational redshift or cosmological redshift." No. RSD come from the Doppler shift of peculiar velocities, not from the expansion of space or from light climbing out of a potential well. It is the line-of-sight component of a galaxy's real motion.
• "The Kaiser formula is exact." It is linear theory, valid on large scales (k ≲ 0.1 h/Mpc). On smaller scales nonlinear evolution, scale-dependent bias, and the Fingers-of-God dispersion all break it, which is why modern fits use perturbation-theory or effective-field-theory models with nuisance parameters.
• "Squashing and stretching cancel out." They act on different scales — squashing on tens of Mpc, stretching inside clusters of a few Mpc — and have opposite signs in the quadrupole, so they are separable rather than cancelling.
• "You measure f directly." You measure the degenerate combination fσ₈, because the clustering amplitude and the velocity amplitude both scale with the matter fluctuation amplitude. Breaking that degeneracy requires external information (e.g. weak lensing or the CMB) on σ₈ or the bias.
• "The plane-parallel approximation is always fine." It assumes one fixed line of sight across the survey. For very wide-angle surveys (DESI, Euclid) wide-angle corrections matter and are included explicitly.