Relativistic Corrections to Large-Scale Structure

On short scales, galaxy clustering is well approximated by biased density plus the Kaiser velocity gradient. Near the Hubble scale, that approximation stops being an observable by itself: a survey selects sources by measured flux, redshift, and angle on the past light cone, so Doppler, gravitational, lensing, time-delay, evolution, and observer terms must accompany the local density. This page identifies their hierarchy, constructs a gauge-safe bias term, and separates analytic necessity from the still-unsettled question of detection.

Helpful background. The Einstein Equation supplies Einstein Equations and their constraint structure; Relativistic Matter and Compact Objects supplies conserved source currents; Relativistic Cosmology supplies the perturbation variables; Computational Relativity supplies gauge and mock-validation tests; and Probability and Statistical Inference supplies selection, covariance, and likelihood reasoning.

A catalog is a light-cone observable

Work to first order in scalar perturbations of spatially flat FLRW, with $a_0=1$, Fourier modes $e^{i\mathbf k\cdot\mathbf x}$, and Newtonian-gauge metric

$$ds^2=a^2\left[-(1+2\Phi)d\eta^2 +(1-2\Psi)\delta_{ij}dx^i dx^j\right], \qquad \mathcal H=\frac{a'}a.$$

Here $\Phi$ is the lapse potential and $\Psi$ the spatial-curvature potential. Let $\boldsymbol n$ point from observer to source, $\chi=\eta_0-\eta$, and $v_r=\boldsymbol n\mathbin{\cdot}\boldsymbol v$. The quantity actually estimated is the fractional count per observed $dz\,d\Omega$ and flux cut. Schematically, but with each physical class shown,

$$\begin{aligned} \Delta_{\rm obs}={}&b\,\delta_m^{\rm com} -\frac{1}{\mathcal H}\partial_\chi v_r +(5s-2)\kappa +A(z)v_r \\ &+(3-b_e)\mathcal HV_m+B(z)\Phi+C(z)\Psi +D(z)\frac{\Psi'}{\mathcal H} +\int_0^\chi d\tilde\chi\, \left[E(z,\tilde\chi)(\Phi'+\Psi') +F(z,\tilde\chi)(\Phi+\Psi)\right] +\Delta_O . \end{aligned}$$

The first line contains intrinsic clustering, Kaiser redshift-space distortion, magnification, and local Doppler terms. The second contains the conversion from comoving tracer density to Newtonian-gauge density, local potentials, integrated redshift and time-delay terms, and the observer monopole/dipole required to define the frame. The coefficients $A$ through $F$ are fixed by background geometry, the evolution bias, and the magnification slope $s$; they are not arbitrary relativistic nuisance amplitudes. The next chapter’s complete number-count derivation gives every coefficient and endpoint term. The present page asks when this full object can be replaced by a smaller one.

Yoo derives the observed-redshift surface, radial and angular displacements, and volume distortion in Yoo 2010, §§II–IV (Open PDF). Bonvin and Durrer obtain the same observable through an independent gauge-invariant construction in Bonvin and Durrer 2011, §§II–IV (Open PDF), while Challinor and Lewis begin from the exact source current and Jacobi map before specializing to linear theory Challinor and Lewis 2011, §§II–V (Open PDF). Agreement requires translating the metric—not merely exchanging the letters $\Phi$ and $\Psi$.

Bias must be attached to a physical slicing

A raw coordinate density is gauge dependent. Define $V_m=\delta q_m/\rho_m$, so under the passive time shift $\tilde\eta=\eta+T$ used in this volume,

$$\widetilde V_m=V_m+T, \qquad \widetilde\delta_m=\delta_m+3\mathcal H T.$$

The matter density on comoving slices is therefore

$$\delta_m^{\rm com}=\delta_m-3\mathcal H V_m.$$

For a tracer with physical mean density $\bar n_g$, define the evolution bias

$$b_e=\frac{d\ln(a^3\bar n_g)}{d\ln a}.$$

Since $\bar n_g'/\bar n_g=(b_e-3)\mathcal H$, evaluate the tracer and matter fields on the same matter-comoving slicing:

$$\delta_g^{(m\text{-}{\rm com})} =\delta_g-(3-b_e)\mathcal H V_m, \qquad \delta_m^{\rm com}=\delta_m-3\mathcal H V_m.$$

Both combinations are invariant under $T$, and the leading local statement is then meaningful as $\delta_g^{(m\text{-}{\rm com})}=b\,\delta_m^{\rm com}$. For locally formed, freely falling tracers sharing the regular adiabatic large-scale flow, there is no leading velocity bias, $V_g=V_m$ as $k\to0$. The equivalence principle fixes their gravitational acceleration, but does not by itself exclude an independently specified initial relative-velocity mode. If a higher-derivative or formation model retains $V_g-V_m\ne0$, the tracer density on its own comoving slicing is instead

$$\delta_g^{(g\text{-}{\rm com})} =b\,\delta_m^{\rm com} -(3-b_e)\mathcal H(V_g-V_m),$$

so the slicing difference cannot be hidden inside $b$. Writing $\delta_g=b\delta_m$ in an unspecified gauge creates apparent horizon-scale terms that are coordinate artifacts. The same discipline is needed for selection: $b_e$ and $s$ are properties of the declared population and observing pipeline, not universal galaxy constants.

The H/k hierarchy explains the Newtonian limit

The Einstein constraints give

$$-k^2\Psi=4\pi Ga^2\rho_m\delta_m^{\rm com}.$$

During ordinary matter growth this and the continuity equation imply the order-of-magnitude hierarchy

$$\Psi\sim\left(\frac{\mathcal H}{k}\right)^2\delta, \qquad v_r\sim\frac{\mathcal H}{k}\delta, \qquad -\frac{1}{\mathcal H}\partial_\chi v_r\sim\delta.$$

Thus density and the Kaiser term are order unity in $\delta$; an undifferentiated Doppler term is down by one power of $\mathcal H/k$; and a local potential is down by two. This counting assumes a regular growing mode, smooth background coefficients, and no scale-dependent enhancement. Lensing and other line-of-sight integrals do not obey a single local suppression rule: a long path and transverse derivatives can compensate part of the naive power counting. Wide-angle effects scale with separation divided by distance and can generate the same odd multipoles used to isolate Doppler terms.

For $k\gg\mathcal H$, a narrow redshift bin, a small plane-parallel patch, and a smooth selection, provided lensing and other integrated contributions are demonstrably negligible for the chosen statistic, the controlled reduction is

$$\Delta_{\rm obs}\longrightarrow b\,\delta_m^{\rm com} -\frac{1}{\mathcal H}\partial_\chi v_r.$$

Near $k\sim\mathcal H$, dropping the remaining terms is not a higher-precision Newtonian approximation; it changes the definition of the modeled observable.

A narrow bin alone does not remove foreground lensing: a distant thin source shell still receives deflections along its entire path. For a count sample with $5s-2=0$, the leading magnification contribution cancels between flux selection and solid-angle dilation; otherwise its importance must be checked using the source distance and angular statistic.

A pure long mode must cancel

An adiabatic mode whose wavelength is much longer than the observed region is locally a change of time coordinate, a dilation, and—at first gradient order—a uniform acceleration. The equivalence principle says a freely falling survey cannot measure that constant potential or acceleration as a local clustering signal. In the complete count, source displacements, redshift perturbations, volume and flux selection, and observer terms cancel the response at orders $k_L^0$ and $k_L^1$. The first locally physical piece is tidal, proportional to $\partial_i\partial_j\Phi_L\sim k_L^2\Phi_L$.

A finite-wavelength super-survey mode can couple to an estimator window or survey boundary, but that is a tidal or higher-order response, not an exception to cancellation of a strictly constant or pure-gradient adiabatic gauge mode. The complete source, observer, selection, and window pipeline must therefore pass the $k_L^0$ and $k_L^1$ null test.

This is an independent implementation test. Inject a constant and a gradient adiabatic mode into a gauge-consistent light-cone mock, transform sources and the observer together, apply the same mask and selection, and demand no horizon-scale signal through first gradient order. If the pipeline keeps a source potential but omits the corresponding observer or radial-volume term, it fails even when its small-scale power spectrum is accurate. The test is stronger than checking the formula in one gauge because it probes the actual estimator and window.

Odd multipoles help, but do not decide the interpretation

For one tracer in the plane-parallel linear limit, density and Kaiser terms produce even multipoles. Cross-correlating two populations with different bias, evolution, and magnification slopes allows Doppler and gravitational terms to generate an antisymmetric correlation and hence a dipole. Multiple tracers can also reduce the shared cosmic-variance contribution. These gains come with new dependencies: a mismatched angular selection, unequal redshift evolution, wide angle, nonlinear velocities, and nonlinear gravitational redshift can all produce or modify odd structure.

There is a second large-scale degeneracy. Local-type primordial non-Gaussianity generates a bias correction with $\Delta b(k)\propto f_{\rm NL}/k^2$, the same scale enhancement as potential terms after the Poisson equation. Redshift dependence, parity, angular structure, multi-tracer ratios, and cross-bin lensing information help, but a claimed detection must fit these branches jointly rather than identifying every $k^{-2}$ residual as either relativity or primordial physics.

Evidence through 2026-08-19

The analytic need for gauge-invariant light-cone terms is established. Their survey detection and the regime in which a partial template is sufficient remain unresolved at the evidence cutoff.

Formulation or evidence	Result	Scope or limitation
Yoo 2010; Bonvin–Durrer 2011; Challinor–Lewis 2011	Independent linear constructions agree on density, velocity, lensing, potential, evolution, and observer classes	Analytic scalar theory; a coordinate-density or source-only truncation fails gauge and long-mode checks
Alam et al. 2017, §§3–6, pp. 4–11	BOSS CMASS shell asymmetry at $2.7\sigma$ and individual-band dipoles up to $2.8\sigma$; two null tests were consistent with zero	Dominated near $10\,h^{-1}\mathrm{Mpc}$, outside clean linear theory; significance changes by $0.5\sigma$ with choices and depends on nonlinear halo/simulation modeling
Bonvin et al. 2023, §§2–6 and Table 5	Synthetic DESI BGS light cone reaches $18.94\sigma$ for an optimized split at $\bar z=0.25$ and agrees with linear theory above about $30\,h^{-1}\mathrm{Mpc}$	A simulation forecast, not DESI data; split, flux limit, biases, covariance, and nonlinear modeling control the result
Euclid Collaboration 2025, §§2–5	Flagship mock forecasts roughly $6\sigma$ total at small scales for Euclid DR2 and beyond	The large-scale mock dipole is consistent with zero; nonlinear gravitational potential dominates the forecast, which is not an observed Euclid detection
DESI DR1 v1.0 full-shape products	Public galaxy/quasar multipoles, windows, covariance, and standard full-shape likelihood exist	The release targets density, Kaiser/AP, BAO, and nonlinear broadband structure, not a dedicated relativistic odd-multipole estimator; it supplies neither a detection nor a null result for this page

Full relativistic light-cone modeling is required for horizon-scale claims. Newtonian density-plus-Kaiser modeling remains controlled under explicit subhorizon, narrow-angle, and selection cuts. Partial Doppler, lensing, or wide-angle templates can be efficient for a targeted statistic, while multi-tracer estimators can improve sensitivity if their population functions are calibrated. The central parameter regime is the wavenumber range relative to $\mathcal H$, supplemented by angular width, redshift-bin width, and source population. These are regime statements, not mutually exclusive verdicts.

Review is triggered by a public survey detection or null result with calibrated evolution and magnification biases, end-to-end window covariance, and gauge-consistent mock injections that separate relativistic terms from wide-angle, nonlinear gravitational redshift, and scale-dependent primordial bias. A revised observer-term derivation or a mock challenge that changes coverage would also require reassessment.

Formulation diagnostic

A pipeline finds a $k^{-2}$ enhancement in the even galaxy power spectrum but does not model odd multipoles, observer terms, or a second tracer. Why is this not by itself a detection of a relativistic correction?

Solution

Potential contributions do acquire $k^{-2}$ scaling relative to density through the Poisson equation, but local-type primordial non-Gaussianity can produce the same scaling in tracer bias. A source-only relativistic template may also retain a pure-gauge long mode if observer, volume, or selection terms are missing. The pipeline must first pass constant- and gradient-mode cancellation through its actual window. It must then compare jointly fitted relativistic, wide-angle, evolution, and primordial-bias branches, preferably using their different parity, redshift, and tracer dependence. An even-power excess alone identifies neither its physical origin nor a gauge-invariant observable.

References

• Alam, Shadab, et al. 2017. “Relativistic Distortions in the Large-Scale Clustering of SDSS-III BOSS CMASS Galaxies.” Monthly Notices of the Royal Astronomical Society 470: 2822–2833. DOI.
• Bonvin, Camille, and Ruth Durrer. 2011. “What Galaxy Surveys Really Measure.” Physical Review D 84, 063505. DOI.
• Bonvin, Camille, et al. 2023. “A Case Study for Measuring the Relativistic Dipole of a Galaxy Cross-Correlation with the Dark Energy Spectroscopic Instrument.” Monthly Notices of the Royal Astronomical Society 525: 4611–4627. DOI.
• Challinor, Anthony, and Antony Lewis. 2011. “The Linear Power Spectrum of Observed Source Number Counts.” Physical Review D 84, 043516. DOI.
• DESI Collaboration. 2025. “DESI 2024 V: Full-Shape Galaxy Clustering from Galaxies and Quasars.” Journal of Cosmology and Astroparticle Physics 09, 008. DOI.
• Euclid Collaboration. 2025. “Euclid: Relativistic Effects in the Dipole of the Two-Point Correlation Function.” Astronomy & Astrophysics 694, A321. DOI.
• Yoo, Jaiyul. 2010. “General Relativistic Description of the Observed Galaxy Power Spectrum.” Physical Review D 82, 083508. DOI.