Kaiser Effect (Cosmology)

The Kaiser effect is a linear-regime cosmological redshift-space distortion (RSD) derived by British cosmologist Nick Kaiser in 1987 that describes the apparent flattening ("squashing") of large-scale galaxy clustering along the observer's line of sight and the resulting angular anisotropy of the three-dimensional galaxy power spectrum P_s(k). In spectroscopic galaxy redshift surveys, the radial comoving distance to each galaxy is inferred from its observed spectroscopic redshift z_obs, which sums the cosmological Hubble expansion with the Doppler shift from the galaxy's line-of-sight peculiar velocity v_∥. On large scales (r ≳ 10–30 h⁻¹ Mpc) where density perturbations remain in the linear regime, galaxies fall coherently inward toward overdense superclusters and stream outward from underdense cosmic voids. Because foreground galaxies infalling toward a cluster center have positive peculiar velocities away from the observer while background galaxies infalling toward the same center have negative peculiar velocities toward the observer, their redshift-space positions are compressed together along the line of sight—providing modern cosmology's primary observational probe of the linear growth rate of structure f(Ω_m) and tests of general relativity on cosmological scales.

In a Friedmann–Lemaître–Robertson–Walker (FLRW) universe with scale factor a(t) = (1 + z)⁻¹ and Hubble parameter H(z), a galaxy located at true comoving position r possessing peculiar velocity v(r) exhibits an apparent comoving position s in redshift space shifted along the unit line-of-sight vector n̂:

s = r + [v_∥(r) / (aH)] n̂, where v_∥(r) = v(r) · n̂

Because the total number of galaxies in a comoving volume element is conserved between real space and redshift space, the apparent galaxy number density n_s(s) and true number density n_r(r) satisfy the continuity condition [1 + δ_g,s(s)] d³s = [1 + δ_g,r(r)] d³r. In the plane-parallel (distant-observer) approximation along line-of-sight coordinate z = r · n̂, the Jacobian determinant of the coordinate transformation is |∂s/∂r| = 1 + (aH)⁻¹ ∂v_∥/∂z. Expanding to first linear order in the perturbations (|δ_g,r| ≪ 1 and |(aH)⁻¹ ∂v_∥/∂z| ≪ 1) yields the fundamental real-to-redshift-space overdensity mapping:

δ_g,s(r) = δ_g,r(r) − (1 / aH) (∂v_∥ / ∂z)

In the linear regime of cosmological perturbation theory (the same regime that seeds halo collapse in the Press–Schechter formalism), the linearized continuity equation for collisionless dark matter relates the velocity divergence θ = ∇ · v to the time derivative of the linear growth factor D(a):

∇ · v = −aH f δ_m(r), where f ≡ d ln D / d ln a is the dimensionless linear growth rate

Taking the Fourier transform v(k) = i aH f (k / k²) δ_m(k) and defining μ ≡ k** · n̂ / k = cos θ** as the cosine of the angle between the wavevector k and the line of sight, the radial velocity gradient in Fourier space evaluates to −(aH)⁻¹ ∂v_∥/∂z → f μ² δ_m(k).

Because luminous galaxies (whose abundance follows the Schechter luminosity function) form preferentially inside high-density peaks of the underlying dark-matter distribution, their real-space overdensity on large linear scales is related to the matter overdensity by the linear galaxy bias b, such that δ_g,r(k) = b δ_m(k). Substituting δ_m(k) = δ_g,r(k) / b into the Fourier-transformed redshift-space relation yields Kaiser's 1987 linear amplification formula:

δ_g,s(k) = (1 + β μ²) δ_g,r(k) = (b + f μ²) δ_m(k)

where β ≡ f(Ω_m) / b is the dimensionless redshift-space distortion parameter. Squaring the Fourier amplitude ensemble average produces the anisotropic redshift-space galaxy power spectrum:

P_s(k, μ) = (1 + β μ²)² P_g,r(k) = (b + f μ²)² P_m,r(k)

Notice the directional dependence on μ = cos θ:

• Transverse Modes (μ = 0, k** ⊥ n̂):** Modes perpendicular to the line of sight are completely unaffected by peculiar velocities, giving P_s(k, 0) = b² P_m,r(k).
• Line-of-Sight Modes (μ = ±1, k** ∥ n̂):** Modes parallel to the line of sight are amplified by the coherent infall factor (1 + β)², giving P_s(k, 1) = (1 + β)² b² P_m,r(k).

In observational analyses of spectroscopic surveys (such as 2dFGRS, SDSS-III BOSS/eBOSS, and the Dark Energy Spectroscopic Instrument [DESI]), P_s(k, μ) is decomposed into orthogonal Legendre polynomials L_ℓ(μ) = {1, (3μ² − 1)/2, (35μ⁴ − 30μ² + 3)/8} for the monopole (ℓ = 0), quadrupole (ℓ = 2), and hexadecapole (ℓ = 4) moments:

P₀(k) = [1 + (2/3)β + (1/5)β²] P_g,r(k) P₂(k) = [(4/3)β + (4/7)β²] P_g,r(k) P₄(k) = (8/35)β² P_g,r(k)

Measuring the ratio of the quadrupole P₂(k) to the monopole P₀(k) breaks the degeneracy between galaxy bias b and the growth rate f, directly determining the combination f(z) σ₈(z), where σ₈(z) is the root-mean-square linear matter fluctuation in 8 h⁻¹ Mpc spheres.

In a two-dimensional plot of the anisotropic galaxy two-point correlation function ξ(σ_⊥, π_∥) as a function of transverse separation σ_⊥ and line-of-sight separation π_∥, the linear Kaiser effect competes against a small-scale non-linear distortion known as the Fingers of God effect (discovered by John C. Jackson in 1972):

• On Large Linear Scales (σ_⊥ > 10 h⁻¹ Mpc): Coherent bulk infall compresses iso-correlation contours along π_∥ into oblate, horizontally flattened ellipses (Kaiser squashing).
• On Small Virialized Scales (σ_⊥ < 2 h⁻¹ Mpc): Inside collapsed galaxy clusters—where galaxies orbit within deep Navarro–Frenk–White profile or Einasto profile potentials governed by the Jeans equations with random velocity dispersions σ_v ≈ 500–1,200 km s⁻¹—large random Doppler shifts scatter galaxies by Δπ_∥ = σ_v / (aH) ≈ 5–12 h⁻¹ Mpc along the line of sight, stretching clusters into radial needles pointing at the observer (Fingers of God). Phenomenological dispersion models combine both effects as P_s(k, μ) = (b + fμ²)² P_m,r(k) F_FoG(k² μ² σ_v²).

Because the expansion history H(z) alone cannot distinguish a cosmological constant (ΛCDM) from modified gravity theories that produce the same background expansion, measuring f(z) via the Kaiser effect tests the gravitational Poisson equation directly. Parametrizing the linear growth rate as f(z) ≈ Ω_m(z)^γ (Linder, 2005), standard general relativity in ΛCDM predicts a growth index γ_GR ≈ 6/11 ≈ 0.545, whereas five-dimensional Dvali–Gabadadze–Porrati (DGP) braneworld gravity predicts γ_DGP ≈ 11/16 ≈ 0.687. Precision Kaiser RSD measurements from SDSS BOSS/eBOSS and DESI across 0.1 < z < 2.1 confirm γ = 0.55 ± 0.04 in agreement with general relativity.

Why does the Kaiser effect squash structures along the line of sight instead of stretching them?

On large scales (>10 Mpc), an overdense supercluster has not yet undergone virialization and orbit crossing; instead, galaxies on the near side are falling away from the observer toward the cluster center (increasing their redshift), while galaxies on the far side are falling toward the observer (decreasing their redshift). In redshift space, near-side and far-side galaxies are therefore shifted closer to the cluster's central redshift, compressing the structure along the line of sight.

What is the difference between the Kaiser effect and the Alcock–Paczyński effect?

Both effects introduce line-of-sight anisotropy into galaxy clustering, but the Kaiser effect is a dynamical distortion caused by peculiar velocities (v_∥), whereas the Alcock–Paczyński effect (1979) is a purely geometric distortion that occurs if an incorrect cosmological model (H(z) and angular diameter distance D_A(z)) is assumed when converting redshifts and angles into comoving coordinates.

Why do cosmologists report f*sigma_8 instead of the growth rate f alone?

In the linear regime, the amplitude of the real-space matter power spectrum scales with σ₈²(z), so the galaxy monopole scales as (bσ₈)² and the quadrupole scales as (bσ₈)(fσ₈). Combining the monopole and quadrupole eliminates the unknown galaxy bias combination bσ₈ and yields a model-independent measurement of the product f(z)σ₈(z).

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