Fingers of God: How Virial Velocities Smear Galaxy Clusters Along the Line of Sight

What the effect is and its physical basis

The Fingers of God (FoG) effect is an apparent, radial stretching of virialized structures — galaxy clusters and groups — in maps built from galaxy redshifts. It is a member of the broader family called redshift-space distortions (RSD).

The root cause is simple. An observed redshift z encodes two contributions: the cosmological redshift from the expanding universe (the Hubble flow, which tracks true distance) and a Doppler shift from the galaxy's own peculiar velocity — its motion relative to the smooth expansion. When we convert redshift to distance, we cannot separate the two, so peculiar velocities masquerade as distance offsets.

Inside a massive cluster, galaxies orbit in the deep gravitational potential well with a line-of-sight velocity dispersion σ_v of order 500–1,500 km/s. These random motions scatter each galaxy's inferred radial position by ±(σ_v / H₀) — tens of Mpc — even though the galaxies physically span only a few Mpc. The result: a real, roughly spherical cluster is mapped as a thin radial spike pointing straight at us.

The mechanism, step by step

Start with the redshift-space position s of a galaxy versus its real-space position r:

• s = r + (v·r̂ / (a·H)) r̂, where v·r̂ is the line-of-sight peculiar velocity, a the scale factor, and H the Hubble rate. Only the radial component is displaced — sky angles are unchanged.
• The peculiar velocities in a cluster are set by the virial theorem: for a self-gravitating, relaxed system, 2K + U = 0, giving a characteristic dispersion σ_v² ≈ GM / (5R) for a system of mass M and radius R.
• Because these virial velocities are large (comparable to or exceeding the Hubble-flow spread across the cluster) and incoherent — pointing every which way — they smear positions symmetrically along the radial axis.

The finger's length in redshift space is roughly ΔS ≈ 2σ_v / H₀. Its orientation is always radial because only the line-of-sight velocity component alters the redshift; transverse motions leave no imprint. That radial pointing — toward the observer — is why the streaks earned the tongue-in-cheek name.

Key quantities and a worked example

Take the Coma cluster (Abell 1656), the textbook case. Fritz Zwicky measured its line-of-sight velocity dispersion at σ_v ≈ 1,000 km/s back in 1933, and modern surveys confirm it.

• Finger length: ΔS ≈ 2σ_v / H₀ = 2 × 1,000 / 70 ≈ 29 Mpc (using H₀ ≈ 70 km/s/Mpc). Physically Coma is only ~6 Mpc across — so the redshift map stretches it roughly fivefold.
• Virial mass: plugging σ_v ≈ 1,000 km/s and R ≈ 1.5–3 Mpc into M ≈ 5·σ_v²·R / G gives M ≈ (1–4)×10¹⁵ M_sun — the same calculation that first revealed dark matter.
• Distance: Coma sits at ~100 Mpc (cz ≈ 7,000 km/s), so its ~1,000 km/s finger is a ~15% radial distortion of its own distance.

For poorer groups, σ_v drops to 100–300 km/s and the fingers shrink to a few Mpc; for the richest clusters σ_v can top 1,500 km/s, producing fingers longer than 40 Mpc.

How it's observed and where it appears

Fingers of God are visible by eye in essentially every wide-field galaxy redshift survey. The 1986 CfA "Slice of the Universe" (de Lapparent, Geller & Huchra) famously showed the Coma cluster as a prominent finger stabbing toward the origin, sitting atop the filament later dubbed the Great Wall. The 2dF Galaxy Redshift Survey and the Sloan Digital Sky Survey (SDSS) show hundreds of them.

Quantitatively, FoG appear as a damping of small-scale power along the line of sight in the galaxy two-point correlation function ξ(σ, π) or power spectrum P(k, μ):

• In ξ(σ, π), where σ is transverse and π is radial separation, the contours are stretched along π — the statistical fingerprint of FoG.
• In P(k), FoG multiply the linear (Kaiser) spectrum by a damping factor, often modeled as a Gaussian exp(−k²μ²σ_v²) or a Lorentzian 1/(1 + k²μ²σ_v²/2), where μ is the cosine of the angle to the line of sight.

Analysts routinely compress the fingers using cluster-finding algorithms (e.g., Huchra & Geller 1982) before measuring large-scale structure statistics.

Relation to the Kaiser effect and other regimes

FoG have a large-scale mirror image: the Kaiser effect (Nick Kaiser, 1987). On linear scales of tens of Mpc, galaxies stream coherently toward overdense regions (gravitational infall). This coherent inflow does the opposite of FoG — it squashes or flattens structures along the line of sight rather than stretching them.

• The Kaiser boost multiplies the real-space power spectrum by (1 + β μ²)², where β = f / b, f ≈ Ω_m(z)^0.55 is the linear growth rate of structure and b is galaxy bias. Measuring β from RSD is a premier probe of gravity and dark energy.
• FoG, being non-linear and small-scale, act as contamination for the Kaiser signal: their damping term must be modeled and marginalized over (the "Gaussian streaming model") to extract fσ₈ cleanly.

Distinguish FoG from unrelated effects: the Alcock–Paczyński distortion is a geometric anisotropy from assuming the wrong cosmology; gravitational redshift adds a tiny (~km/s) asymmetric shift. Only FoG stem from incoherent virial motions.

Significance, history, and open questions

The term "Fingers of God" traces to J. C. Jackson's 1972 paper on the clustering of QSO absorption redshifts; the physics of redshift-space elongation was elaborated by Tully & Fisher (1978) and others as redshift surveys matured. The effect is now foundational to interpreting the cosmic web.

Why it matters:

• Cosmology: FoG must be modeled to recover the growth rate fσ₈ from surveys like BOSS, eBOSS, DESI and Euclid — a key test of General Relativity on cosmic scales.
• Cluster physics: the very velocity dispersions that create fingers deliver virial masses, the historic evidence for dark matter (Zwicky 1933).

Open problems center on the non-linear regime: the velocity distribution is not perfectly Gaussian (infalling satellites add skew and non-virialized "caustic" structure), scale-dependent bias couples to the damping, and the mapping is not one-to-one where fast galaxies cross in redshift space ("shell crossing"). Getting σ_v modeling wrong biases fσ₈ by up to ~10%, which now exceeds the statistical error budget of flagship surveys — so FoG modeling is an active frontier, not a solved problem.