Redshift-Space Distortion

A map drawn in the wrong coordinate

To map the three-dimensional positions of millions of galaxies, a redshift survey does something deceptively simple: it points at a galaxy, measures its angular position on the sky (two coordinates) and its redshift (the third), and converts the redshift into a distance using the Hubble law. Angles are clean. The third coordinate is not. The redshift we measure is not the pure cosmological expansion — it carries an extra Doppler shift from the galaxy's own motion through space, its peculiar velocity. When we treat the full redshift as if it were all Hubble flow, we place the galaxy at the wrong distance. The map we build is therefore drawn in redshift space, not real space, and it is systematically warped along the line of sight.

The crucial point is that the warp is purely radial. Peculiar velocities only shift the line-of-sight coordinate; the two angular coordinates are untouched. So clustering measured parallel to the line of sight looks different from clustering measured perpendicular to it. That anisotropy is the redshift-space distortion, and far from being noise to remove, it is signal: the strength and shape of the distortion encode how fast galaxies are falling toward one another, which is to say how fast cosmic structure is growing. RSD turns an annoyance — we can't measure true distances — into one of the few direct probes of the growth of structure, and through it, a test of gravity itself.

Two distortions, opposite signs

RSD has two faces that act on different scales and push the clustering pattern in opposite directions.

Kaiser squashing (large, linear scales). Consider a large overdensity — a forming supercluster or a great wall. Surrounding galaxies are being pulled toward it by gravity in a coherent, organized flow. Galaxies on the near side of the structure (between us and it) are falling away from us, so their peculiar velocity adds a small positive redshift, pushing them farther away in our map. Galaxies on the far side are falling toward us, gaining a small negative redshift that pulls them closer in our map. Both effects move the two sides toward the structure's center: the whole pattern gets compressed along the line of sight. Structures that are spherical in real space appear squashed, like a beach ball stepped on from the radial direction. This is the Kaiser effect (Nick Kaiser, 1987), and because the infall speed is set by the growth rate, the amount of squashing measures how fast structure grows.

Fingers of God (small, nonlinear scales). Now zoom into a single virialized galaxy cluster. Its member galaxies are not flowing coherently — they are buzzing around the cluster center on randomized orbits with velocity dispersions of 500–1000 km/s. Those random radial velocities add a random spread to each galaxy's redshift, scattering the members along the line of sight by tens of h⁻¹ Mpc even though the cluster is only ~1–2 h⁻¹ Mpc across. A compact, roughly round cluster gets stretched into a long, thin radial spike that points directly back at the observer. Every cluster does this, so the map fills with spikes radiating from Earth — the Fingers of God. This is the opposite anisotropy to Kaiser: it elongates rather than compresses, and it dominates on small scales.

From velocity field to anisotropic clustering

The line-of-sight shift for a galaxy is

s = r + (1 + z) v_pec,∥ / H(z)

where r is the true comoving distance, s the apparent redshift-space distance, v_pec,∥ the peculiar velocity component along the line of sight, and H(z) the Hubble parameter at the galaxy's redshift. A peculiar velocity of 300 km/s at z ≈ 0.5, where H ≈ 125 km/s/Mpc (comoving), produces a displacement of order a few h⁻¹ Mpc — comparable to the separations at which we measure clustering, which is why the distortion is so visible.

On linear scales the peculiar velocity field is tied to the density field by the continuity equation: regions that are overdense are pulling matter in, and the divergence of the velocity is proportional to the density contrast times the growth rate, ∇·v ∝ −f δ. Working through the Jacobian of the real-space-to-redshift-space map (Kaiser's derivation) gives the redshift-space power spectrum on linear scales:

P_s(k, μ) = (1 + β μ²)² P_r(k)

  μ = cos(angle to line of sight)
  β = f / b
  f = dln D / dln a   (linear growth rate)
  b = linear galaxy bias

The factor (1 + βμ²)² is the Kaiser term. It is largest for μ = 1 (purely radial modes, along the line of sight) and reduces to 1 for μ = 0 (transverse modes). That angular dependence is the squashing, written in Fourier space. Because the strength is set by β = f/b, and the overall clustering amplitude is set by b·σ₈, the bias cancels in the product the survey can actually constrain, leaving the combination fσ₈.

To capture the small-scale Fingers of God, the linear formula is multiplied by a damping factor that represents the random velocity smearing — commonly a Lorentzian or Gaussian in k·μ·σ_v:

P_s(k, μ) = (1 + β μ²)² P_r(k) · D(k μ σ_v)

  D_Lorentzian = 1 / [1 + (k μ σ_v)² / 2]

where σ_v is the pairwise velocity dispersion (~3–6 h⁻¹ Mpc in velocity units). The damping kills radial power on small scales — exactly the suppression the fingers produce.

Reading it off: the multipole decomposition

Because the anisotropy depends only on μ, the natural way to compress P_s(k, μ) is to expand it in Legendre polynomials L_ℓ(μ) and keep the leading multipoles:

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

  Monopole   P_0 = (1 + 2β/3 + β²/5) P_r(k)
  Quadrupole P_2 = (4β/3 + 4β²/7) P_r(k)
  Hexadecapole P_4 = (8β²/35) P_r(k)

The monopole is the angle-averaged clustering — the Kaiser term simply boosts its amplitude. The quadrupole is the workhorse: it is identically zero with no peculiar velocities, becomes negative when Kaiser squashing dominates (large scales), and turns positive when Fingers of God take over (small scales). The ratio P_2/P_0 on large scales is a clean function of β alone:

P_2 / P_0 = (4β/3 + 4β²/7) / (1 + 2β/3 + β²/5)

Measure that ratio, solve for β, multiply by the clustering amplitude bσ₈, and you have fσ₈ with the bias divided out. The hexadecapole adds leverage and helps separate RSD from the geometric Alcock-Paczyński distortion, which also makes clustering anisotropic but for a different reason (an assumed cosmology that warps distances).

Worked example: fσ₈ from a quadrupole-to-monopole ratio

Suppose a survey at z = 0.5 measures, on large linear scales, a quadrupole-to-monopole ratio P_2/P_0 ≈ 0.45. Invert the relation above for β. Plugging numbers, β ≈ 0.45 solves the ratio (the function P_2/P_0 rises from 0 toward ~0.8 as β goes from 0 to 1). Now suppose the galaxy clustering amplitude gives bσ₈ ≈ 1.2. Then

f σ₈ = β · (b σ₈) = 0.45 × 1.2 / b ...
     = (f/b) · (b σ₈)
     = f σ₈
     ≈ 0.45 × ( b σ₈ ) with the b cancelling
     ⇒ f σ₈ ≈ 0.46

That result, fσ₈(0.5) ≈ 0.46, can be compared directly to the ΛCDM + General Relativity prediction. For Ω_m = 0.31, the matter density at z = 0.5 is Ω_m(0.5) = Ω_m(1+z)³ / E²(z) ≈ 0.62, so the growth rate is

f(0.5) ≈ Ω_m(0.5)^0.55 ≈ 0.62^0.55 ≈ 0.76

and with σ₈(0.5) ≈ 0.60 (the present-day σ₈ ≈ 0.81 scaled back by the growth factor D(0.5)/D(0) ≈ 0.74), the GR prediction is fσ₈(0.5) ≈ 0.76 × 0.60 ≈ 0.46. The measurement and prediction agree — which is precisely the test: had modified gravity boosted f by 15%, the measured fσ₈ would have come out near 0.53, well outside a 3%-precision error bar.

Variants, regimes and complications

• Linear vs. nonlinear RSD. The clean Kaiser formula holds only on large scales (k ≲ 0.1 h/Mpc). On smaller scales, perturbation theory (TNS, EFT-of-LSS, Gaussian streaming with a velocity dispersion) is needed, and Fingers of God must be modeled, not just damped away.
• Plane-parallel vs. wide-angle. The (1 + βμ²)² result assumes a single line-of-sight direction (plane-parallel approximation). For wide survey footprints the line of sight varies across the volume, adding wide-angle corrections to the multipoles.
• RSD vs. Alcock-Paczyński. Both make clustering anisotropic. AP comes from assuming a wrong cosmology when converting redshifts/angles to distances; it stretches the BAO ruler differently along and across the line of sight. Modern analyses fit RSD and AP jointly, which is why the hexadecapole matters.
• Velocity-field RSD (pairwise velocities). Beyond clustering, peculiar velocities can be measured directly from distance indicators (Tully-Fisher, fundamental plane, supernovae), giving an independent handle on the same velocity field at low z.
• Growth index γ. Writing f = Ω_m(z)^γ turns RSD into a single-number test: γ ≈ 0.55 is the GR value, while f(R) and DGP gravity predict γ ≈ 0.42 and γ ≈ 0.68 respectively.

Observational status

RSD has matured from a marginal detection into a percent-level cosmological probe. The 2dF Galaxy Redshift Survey (Peacock et al. 2001) provided the first robust β measurement; SDSS, WiggleZ and VIPERS followed. The Baryon Oscillation Spectroscopic Survey (BOSS) delivered fσ₈ measurements at the ~6–9% level in bins out to z ≈ 0.6, and its extension eBOSS pushed RSD to z ≈ 1.5 using emission-line galaxies and quasars. The Dark Energy Spectroscopic Instrument (DESI), gathering tens of millions of redshifts, is driving per-bin precision toward 1–3%; Euclid and the Nancy Grace Roman Space Telescope will extend high-precision growth measurements to z ≈ 2.

Stacking all measurements, the inferred growth index sits at γ = 0.55 ± ~0.05 — fully consistent with General Relativity. But there is a persistent undercurrent: several RSD compilations prefer a growth amplitude slightly below the value extrapolated from the Planck CMB, the same direction as the broader σ₈ / S₈ tension seen in weak lensing. Whether that is a statistical fluctuation, an unmodeled systematic, or a genuine sign that structure grew a little slower than ΛCDM predicts is one of the live questions RSD is positioned to answer this decade.

Growth-rate measurements across cosmic time

Survey / sample	Redshift z	Tracer	fσ₈ (approx.)	Precision	Notes
6dFGS + SnIa	0.02	Galaxies + velocities	0.43	~14%	Local universe anchor
SDSS MGS	0.15	Main galaxies	0.49	~16%	Low-z RSD
BOSS LOWZ	0.32	LRGs	0.46	~7%	DR12 consensus
BOSS CMASS	0.57	LRGs	0.46	~6%	Peak of growth signal
VIPERS	0.80	Galaxies	0.47	~12%	Intermediate z
eBOSS QSO	1.48	Quasars	0.46	~10%	Highest-z RSD to date
DESI Y1 (target)	0.1–1.5	BGS/LRG/ELG/QSO	0.40–0.47	~1–3%	Percent-level growth history

The striking feature is that fσ₈ is remarkably flat across redshift — it rises from ~0.43 today to a gentle peak of ~0.47 near z ≈ 0.5 and falls again at higher z. That broad maximum reflects a competition: at higher z there is more matter to drive growth (f is larger), but the fluctuation amplitude σ₈(z) is smaller because structure has had less time to grow. The product peaks where the two trends cross.

Common pitfalls and misconceptions

• Thinking Kaiser and Fingers of God are the same distortion. They have opposite signs: Kaiser compresses along the line of sight (negative quadrupole), Fingers of God elongate along it (positive quadrupole). One dominates on large scales, the other on small.
• Believing RSD gives f directly. Surveys cannot separate galaxy bias from the clustering amplitude, so RSD constrains fσ₈, not f alone. Quoting a bare growth rate requires an external assumption about σ₈ or bias.
• Confusing redshift-space distortion with the Alcock-Paczyński effect. Both make clustering anisotropic, but AP is a geometric distortion from assuming the wrong cosmology in the distance conversion, while RSD is a dynamical distortion from real peculiar velocities. Good analyses fit both at once.
• Assuming the fingers are a physical structure. Fingers of God are not real filaments pointing at us — they are a projection artifact of treating cluster velocity dispersion as distance. The galaxies sit in a compact blob; only the map elongates them.
• Treating RSD as just a nuisance. Early on, peculiar velocities were seen as contamination of the clustering signal. The modern view is the reverse: the anisotropy is the measurement, the most direct cosmological probe of how fast gravity assembles structure.