Galaxy Bias and Redshift-Space Distortions

A redshift survey does not observe the matter density directly. It selects galaxies or quasars whose abundance responds to long-wavelength fields, then places them in three dimensions using angles, redshifts, and a fiducial distance law. Tracer bias changes the amplitude and scale dependence; line-of-sight velocities make the clustering anisotropic; and the survey window mixes the theoretical modes. This page derives the linear control limit, then shows how a versioned full-shape likelihood turns that limit into a conditional inference rather than a direct measurement of gravity.

Helpful background. The Einstein Equation supplies Einstein Equations and their weak-field limit; Relativistic Matter and Compact Objects supplies fluid and kinetic closures; Relativistic Cosmology supplies the expanding background; Computational Relativity supplies convergence and benchmark practice; and Probability and Statistical Inference supplies covariance, priors, and likelihood reasoning.

Bias is an operator expansion

Work first on scales much smaller than the Hubble radius, in the tracer rest frame, where the retained large-scale fields admit a perturbative expansion. Galaxies and their host halos have already formed nonlinearly on smaller scales; that unresolved formation physics enters the bias coefficients and stochastic terms. A galaxy sample is defined by its selection as well as its astrophysics. Its Eulerian density contrast therefore has the schematic renormalized expansion

$$\delta_g= b_1\delta +\frac{b_2}{2}[\delta^2] +b_{K^2}[K_{ij}K^{ij}] +b_{\nabla^2\delta}\frac{\nabla^2\delta}{k_M^2} +\epsilon+\cdots,$$

where

$$K_{ij}=\left(\frac{\partial_i\partial_j}{\nabla^2} -\frac{\delta_{ij}}{3}\right)\delta .$$

The brackets denote renormalized composite operators, not merely squares of a chosen smoothed map. The coefficients depend on tracer definition and epoch; $k_M^{-1}$ represents a formation scale; and the stochastic field $\epsilon$ has zero mean but nonzero auto-correlations. Tidal, higher-derivative, and stochastic terms are required by symmetry and scale separation even when a local linear relation looks adequate over a narrow range. Desjacques, Jeong, and Schmidt construct this expansion and its finite-scale and stochastic extensions in Desjacques, Jeong, and Schmidt 2018, §§2.5–2.10, printed pp. 32–60 (Open PDF).

At the lowest order, $\delta_g=b_1\delta+\epsilon$. This is a controlled large-scale limit, not the assertion that every galaxy is a deterministic multiple of the matter field. Assembly history, baryon physics, sample selection, and halo occupation enter through coefficients, additional operators, stochasticity, or an explicitly different model.

Peculiar velocities produce the Kaiser anisotropy

In the distant-observer approximation, let $\hat{\mathbf n}$ be a fixed line of sight and $v_\parallel=\mathbf v\mathbin{\cdot}\hat{\mathbf n}$. The redshift-space comoving coordinate is

$$\mathbf s=\mathbf x+ \frac{v_\parallel}{aH}\hat{\mathbf n}.$$

Number conservation under this map gives, to first order,

$$\delta_g^s=\delta_g- \frac{1}{aH}\partial_\parallel v_\parallel.$$

For the growing pressureless mode, $\nabla\mathbin{\cdot}\mathbf v=-aHf\delta$, with $f=d\ln D/d\ln a$. In Fourier space, with $\mu=\hat{\mathbf k}\mathbin{\cdot}\hat{\mathbf n}$,

$$\delta_g^s(\mathbf k) =\left(b_1+f\mu^2\right)\delta(\mathbf k)+\epsilon(\mathbf k),$$

and the deterministic Kaiser spectrum is

$$P_g^s(k,\mu)=\left(b_1+f\mu^2\right)^2P_m(k).$$

Using

$$P_\ell(k)=\frac{2\ell+1}{2} \int_{-1}^{1}d\mu\,P_g^s(k,\mu){\cal L}_\ell(\mu),$$

the nonzero linear multipoles are

$$\begin{aligned} P_0&=\left(b_1^2+\frac23b_1f+\frac15f^2\right)P_m,\\ P_2&=\left(\frac43b_1f+\frac47f^2\right)P_m,\\ P_4&=\frac{8}{35}f^2P_m. \end{aligned}$$

The angular dependence helps separate $b_1$ from growth, but observations usually constrain amplitude combinations such as $b_1\sigma_8$ and $f\sigma_8$, not $f$ in isolation. Here $\sigma_8(z)$ is the linear matter RMS smoothed with a spherical top hat of comoving radius $8h^{-1}\,\mathrm{Mpc}$, with $h=H_0/(100\,\mathrm{km\,s^{-1}\,Mpc^{-1}})$; it is computed from the windowed matter spectrum. The derivation assumes one resolved large-scale velocity field, no large-scale velocity bias, a fixed line of sight, and linear dynamics. Virial velocities generate Fingers of God; wide-angle and horizon-scale terms are deferred to Relativistic Corrections to Large-Scale Structure. The exact nonlinear redshift mapping and its selection boundary are reviewed by Desjacques, Jeong, and Schmidt in Desjacques, Jeong, and Schmidt 2018, §9.3, printed pp. 208–217 (Open PDF).

Distance conversion, windows, and nuisance models

A catalog is built using fiducial distances. Define

$$q_\parallel=\frac{D_H}{D_H^{\rm fid}}, \qquad q_\perp=\frac{D_M}{D_M^{\rm fid}}, \qquad D_H=\frac{c}{H}.$$

The true wavevector corresponding to $(k_{\rm fid},\mu_{\rm fid})$ obeys

$$k=k_{\rm fid} \left[\frac{\mu_{\rm fid}^2}{q_\parallel^2} +\frac{1-\mu_{\rm fid}^2}{q_\perp^2}\right]^{1/2}, \qquad \mu=\frac{\mu_{\rm fid}/q_\parallel} {\left[\mu_{\rm fid}^2/q_\parallel^2 +(1-\mu_{\rm fid}^2)/q_\perp^2\right]^{1/2}}.$$

The model spectrum also carries the Jacobian $(q_\perp^2q_\parallel)^{-1}$. These Alcock–Paczynski distortions can mimic some RSD anisotropy, so distance and growth parameters must be fit jointly. DESI 2024 V, §4.2, printed pp. 13–17, eqs. (4.1)–(4.12) (Open PDF) gives the exact dictionary used in the example below.

A production model replaces the tree-level spectrum by a perturbative or emulated biased-tracer spectrum, adds counterterms and stochastic terms, and convolves it with the survey window. Short-scale velocities may be represented by a perturbative counterterm or by a phenomenological Gaussian or Lorentzian damping factor. These are competing regime choices, not interchangeable notations. The likelihood must declare bias and counterterm bases, nuisance priors, scale cuts, covariance corrections, window normalization, and whether the covariance depends on the parameters.

A pinned DESI DR1 example

The data example is the official DESI DR1 Full Shape and BAO clustering products v1.0, not a DR2 galaxy full-shape release. The parent analysis used more than 4.7 million unique BGS, LRG, ELG, and quasar redshifts in six bins over about $7{,}500\,\mathrm{deg}^2$. Catalog-level BAO, RSD, and large-scale blinding remained in place while the baseline model, priors, fitting range, multipoles, covariance, and systematic budget were fixed. The unblinding checks compared photometric regions, analytic and mock covariances, configuration and Fourier space, and reconstructed BAO. DESI 2024 V, §§2–6, printed pp. 4–45 and Tables 7–8, PDF documents those choices.

The release contract also includes catalog v1.5, matched randoms, veto masks, angular and redshift completeness, fiber-assignment mitigation, redshift failure and imaging weights, distance conversion, and estimator/window normalization. Its covariance program used 1,000 EZmocks, while 25 Abacus cut-sky mocks supplied independent pipeline checks. The sample construction and correction chain is specified in DESI 2024 II, §§2–8, printed pp. 4–38; §§10–11, pp. 40–50 (Open PDF).

The applicable calibration domain is therefore the coupled angular and redshift completeness, fiber-assignment, redshift-failure, imaging-weight, window-normalization, and fiducial-distance pipeline for these declared samples. None of those corrections calibrates away tracer astrophysics or the nonlinear velocity model; those remain likelihood ingredients.

This page uses the exact LRG combined-footprint bin $0.6\lt z\lt0.8$, $z_{\rm eff}=0.7057956472$, with a $0.05^\circ$ pair cut. Its packaged object combines a rotated pre-reconstruction power-spectrum ShapeFit likelihood with post-reconstruction BAO and includes HOD, window-rotation, and residual photometric systematic covariance. The upstream filename is likelihood_shapefit_spectrum-poles-rotated+bao-recon_syst-rotation-hod-photo_LRG_GCcomb_z0.6-0.8_thetacut0.05.h5; its SHA-256 hash on retrieval, 2026-08-19, was b8e08ca9e4de83ef8f667ce6c9e11a5bd72b9ce6fd7562b032937a5abbf4af0f.

Reproducing the compressed Gaussian

The HDF5 labels are $(q_{\rm iso},q_{\rm AP},d_f,d_m)$: isotropic and anisotropic template dilations, the growth-rate ratio, and the broadband-shape shift. The released mean and marginal standard deviations are

Coordinate	Mean	Marginal standard deviation
$q_{\rm iso}$	$0.9674904744$	$0.0106970043$
$q_{\rm AP}$	$1.0367005807$	$0.0342718007$
$d_f$	$1.0232653079$	$0.1050889211$
$d_m$	$0.0223166549$	$0.0676149519$

The compressed-space window is exactly the $4\times4$ identity, while the released correlation matrix is

$$R=\begin{pmatrix} 1&0.096351&0.232615&-0.235579\\ 0.096351&1&-0.393082&-0.066163\\ 0.232615&-0.393082&1&-0.300596\\ -0.235579&-0.066163&-0.300596&1 \end{pmatrix}.$$

For fixed covariance $C$, the normalized Gaussian is

$$-2\ln L(\mathbf x) =(\mathbf x-\bar{\mathbf x})^{\mathsf T} C^{-1}(\mathbf x-\bar{\mathbf x}) +\ln\det(2\pi C).$$

Using Cholesky solves, the fiducial-template coordinate $\mathbf x_0=(1,1,1,0)$ has quadratic distance $\Delta\chi^2=14.07134467$. Replacing $C$ by its diagonal changes this to $10.54098363$; correlations are consequential. Retaining only $(q_{\rm iso},q_{\rm AP},d_f)$ gives $14.03020090$, whereas the $(d_f,d_m)$ subvector gives $0.22193055$. These are likelihood-algebra checks, not detection significances: the template point is not a parameter-free null, and the compression inherited priors and a fitted nonlinear model.

The calculation used Python 3.12.13, NumPy 2.5.2, and h5py 3.16.0 on arm64 macOS. It verified exact window identity, covariance symmetry to $4.34\times10^{-19}$, positive eigenvalues, and a successful Cholesky factorization. To rerun, download the exact official HDF5 object, verify the hash, read observable/shapefit, window/value, and covariance/value, and solve the displayed quadratic without forming $C^{-1}$ explicitly.

This compressed file cannot vary the underlying $k$ cuts, bias operator basis, or velocity damping. Those checks require the release’s raw multipoles, nontrivial window, covariance, official preparation script, and full-model likelihood. The boundary is part of the reproduction, not a missing numerical detail.

Evidence through 2026-08-19

Evidence or result	What it controls	Scope or limitation
Linear bias plus Kaiser multipoles	Exact large-scale angular structure and limiting coefficients	Plane-parallel, single-stream, linear growth; no window or nonlinear velocity field
Renormalized operator expansion in Desjacques–Jeong–Schmidt	Nonlocal, tidal, derivative, stochastic, and selection terms	Coefficients and truncation are tracer- and scale-dependent; the framework does not select a unique production model
DESI DR1 v1.0 full-shape products and mock program	Released multipoles, windows, covariance, ShapeFit/full-model implementations, and coverage/null checks	Conditional on DR1 samples, cuts, priors, fiducial mapping, perturbation models, and systematic construction
DESI 2024 V systematic tests	No detected theoretical shift above $0.1\sigma_{\rm DR1}$ at the stated baseline; total retained systematic budget about $0.46\sigma_{\rm DR1}$	A release-specific null result, not proof that all smaller scales, tracers, or bias/Fingers-of-God descriptions are unbiased
Public DESI status at the cutoff	DR2 galaxy/quasar cosmology products are BAO; DR2 full-shape Lyman-alpha is a distinct forest analysis	Neither product can be relabeled as a public DR2 galaxy/quasar full-shape likelihood

Perturbative local/nonlocal bias, effective-field-theory expansions, halo-model or emulator descriptions, conservative phenomenological scale cuts, and alternative velocity-damping prescriptions can agree in a validated overlap regime yet diverge outside it. The current evidence supports covered, model-conditional growth and distance inference; it does not establish an unconditional verdict about gravity or galaxy formation. The status remains unresolved outside the validated scale range: evidence does not establish that any one bias and Fingers-of-God model is generically correct.

The interpretation could change after a public DESI DR2 galaxy/quasar full-shape release, a corrected DR1 HDF5 object, a covariance or window revision, a blind mock challenge that changes coverage, or a controlled model comparison that shifts the covered growth result.

The February 2026 erratum corrected ShapeFit data vectors and covariances that involve $f\sigma_{s8}$; it did not alter the original figures, tables, results, or conclusions. The pinned operational HDF5 is used here rather than the superseded Appendix A entries. DESI 2024 V Erratum, §1, printed pp. 1–8, eqs. (1.1)–(1.36).

Exercises

Derive the Kaiser monopole, quadrupole, and hexadecapole. Then explain why replacing the released covariance by its diagonal is not a conservative approximation in the DESI reproduction above.

Solution

Expand

$$(b_1+f\mu^2)^2=b_1^2+2b_1f\mu^2+f^2\mu^4$$

and project with ${\cal L}_0=1$, ${\cal L}_2=(3\mu^2-1)/2$, and ${\cal L}_4=(35\mu^4-30\mu^2+3)/8$. Using $\int_{-1}^{1}\mu^{2n}d\mu=2/(2n+1)$ gives

$$P_0=\left(b_1^2+\frac23b_1f+\frac15f^2\right)P_m, \quad P_2=\left(\frac43b_1f+\frac47f^2\right)P_m, \quad P_4=\frac{8}{35}f^2P_m.$$

Diagonalization is not conservative in any general sense: it rotates neither the residual nor the uncertainty ellipsoid into its eigenbasis; it simply deletes measured correlations. Here it lowers the same fiducial-template quadratic from $14.0713$ to $10.5410$, while a different residual direction could move it upward. The full released covariance is therefore part of the observable definition.

References

• DESI Collaboration. “DESI 2024 II: Sample Definitions, Characteristics, and Two-point Clustering Statistics.” Journal of Cosmology and Astroparticle Physics 07 (2025) 017. DOI. Open PDF.
• DESI Collaboration. “DESI 2024 V: Full-Shape Galaxy Clustering from Galaxies and Quasars.” Journal of Cosmology and Astroparticle Physics 09 (2025) 008. DOI. Open PDF, arXiv:2411.12021v5. Erratum: JCAP 02 (2026) E02. DOI.
• DESI Collaboration. “DESI DR1 Full Shape and BAO Clustering Products.” Version 1.0, retrieved 19 August 2026. Official release documentation.
• Desjacques, Vincent, Donghui Jeong, and Fabian Schmidt. “Large-Scale Galaxy Bias.” Physics Reports 733 (2018) 1–193. DOI. Open PDF.