Relativistic Galaxy Number Counts

A redshift survey records sources per observed solid angle, redshift, and flux threshold. Its fluctuation is therefore not merely a galaxy density evaluated on a coordinate slice: the radial redshift Jacobian, transverse lensing map, source evolution, flux selection, velocities, gravitational potentials, and observer frame all change the catalog. At linear order these pieces combine into one gauge-invariant observed count. The Newtonian density-plus-Kaiser approximation is a controlled small-scale limit, not the definition of the observable.

Required background. Geodesics and Jacobi Fields supplies the null and screen maps used to convert source current into observed volume.

Helpful background. Einstein Equations controls the potential constraints, Relativistic Matter and Compact Objects supplies conserved source currents, Relativistic Cosmology supplies perturbation variables, and Probability and Statistical Inference supplies selection and likelihood concepts.

Counts are density times an observed Jacobian

For source current $J^\mu=n_su_s^\mu$, the exact count along a ray is

$$\frac{dN}{dz\,d\Omega_O} =\bigl\lvert\det\mathcal D_O\bigr\rvert\, \bigl\lvert p_\mu J^\mu\bigr\rvert \left\lvert\frac{d\lambda}{dz}\right\rvert,$$

where $\mathcal D_O$ is the observer-vertex Jacobi map. Three factors are visible: physical source density, transverse area per observed solid angle, and affine depth per observed redshift. A magnitude cut also perturbs which intrinsic luminosities enter because flux depends on the same Jacobi determinant. This nonperturbative construction and its Lorentz transformation under a change of observer are given in Challinor and Lewis 2011, § II, eqs. (1)–(5).

Work now to first order in scalar perturbations of flat FLRW, in conformal Newtonian gauge,

$$ds^2=a^2\left[-(1+2\Phi)d\eta^2+(1-2\Psi)d\boldsymbol x^2\right],$$

with $a_O=1$, $\boldsymbol n$ pointing from observer to source, $\chi=\eta_O-\eta$, and $\mathcal H=a'/a$; primes denote conformal-time derivatives. Sources follow the pressureless geodesic Euler equation, $\boldsymbol v'+\mathcal H\boldsymbol v=-\boldsymbol\nabla\Phi$, and photons propagate transparently. Thus $\Phi$ is the lapse potential and $\Psi$ the spatial-curvature potential. Let $\bar n_s(a,L\gt L_*)$ be the physical mean source density above intrinsic luminosity $L_*$. Define the magnification slope and evolution bias by

$$s(z)=\frac{\partial\log_{10}\bar N(z,m\lt m_*)}{\partial m_*}, \qquad b_e(z)=\left. \frac{\partial\ln\!\left[a^3\bar n_s(a,L\gt L_*)\right]}{\partial\ln a} \right|_{L_*}.$$

Here $\bar N$ in the definition of $s$ is the mean count per redshift and solid angle, and its derivative is at fixed $z$. Evaluate $b_e$ at the background luminosity threshold corresponding to $m_*$. Its derivative holds that intrinsic threshold fixed: differentiating the observed redshift histogram also differentiates volume and flux selection and does not give this evolution bias.

The intrinsic fluctuation $\delta_N$ below is the Newtonian-gauge density fluctuation of this luminosity-selected population. For linear local bias and no velocity bias,

$$\delta_N=b\,\delta_m^{\mathrm{com}}+(3-b_e)\mathcal H V_m, \qquad \boldsymbol v=\boldsymbol\nabla V_m.$$

The velocity-potential term converts the physical comoving-slice bias prescription into Newtonian gauge; simply replacing $\delta_N$ by $b\delta_m^{\mathrm{com}}$ loses a horizon-scale term. The transformation is derived on Relativistic Effects in Large-Scale Structure.

The complete linear scalar count

Let $v_r=\boldsymbol n\cdot\boldsymbol v$ and introduce

$$I=\int_{\eta}^{\eta_O}(\Phi'+\Psi')\,d\tilde\eta, \qquad T=\int_{\eta}^{\eta_O}(\Phi+\Psi)\,d\tilde\eta.$$

Define the background coefficient

$$\mathcal A(z)=\frac{2-5s}{\mathcal H\chi} +5s-b_e+\frac{\mathcal H'}{\mathcal H^2}.$$

After subtracting the angular monopole, the magnitude-limited fluctuation can be written with the observer dipole retained:

$$\begin{aligned} \Delta_N={}&\delta_N -\frac{1}{\mathcal H}\partial_\chi v_r +(5s-2)\kappa +(2-5s)\frac{T}{\chi} \\ &+\mathcal A\left(\Phi+I-v_r\right) \\ &+\frac{\Psi'}{\mathcal H} +\Phi+(5s-2)\Psi +\Delta_O. \end{aligned}$$

The angular mean of the right-hand side is understood to be removed. In the fixed observation-event convention used here, the surviving local term is

$$\Delta_O=\left[\mathcal A+2-5s\right] \boldsymbol n\cdot\boldsymbol v_O.$$

The observer-potential term $-\mathcal A\Phi_O$ and any observer-clock lapse contribute only to the removed monopole at this order. The displayed dipole vanishes after a consistent boost to the frame with $\boldsymbol v_O=0$; it also has no full-sky multipoles with $\ell\geq2$. The first line contains intrinsic clustering, Kaiser redshift-space distortion, lensing area plus flux selection, and radial time delay. The second converts the redshift perturbation into a background volume and population shift. The third contains local and evolving potentials. This translates Challinor and Lewis 2011, §§ III–IV, eqs. (28)–(31) and (36)–(38), PDF: their lapse $\psi$ is site $\Phi$, and their spatial potential $\phi$ is site $\Psi$.

One can check the selection coefficients without memorizing the final formula. At fixed measured redshift, changing the area distance changes the luminosity threshold by $\delta\ln L_*=2\delta D_A/\bar D_A$. Since $\partial\ln\bar n_s/\partial\ln L_*=-5s/2$, flux selection adds $-5s\,\delta D_A/\bar D_A$ to the unselected count. Combining this with the geometric area term $2\delta D_A/\bar D_A$ gives $(2-5s)$ times the distance fluctuation. The displacement $\delta\chi=-\delta\eta+T$ therefore supplies $(2-5s)T/\chi$, while $-\kappa$ supplies $(5s-2)\kappa$.

For $k\gg\mathcal H$ in a distant, plane-parallel patch, the density and radial velocity-gradient terms dominate if selection varies slowly and lensing and other integrated terms are negligible for the chosen statistic:

$$\Delta_N\longrightarrow b\,\delta_m^{\mathrm{com}} -\frac{1}{\mathcal H}\partial_\chi v_r.$$

Local relativistic terms scale roughly as $\mathcal H/k$ or $(\mathcal H/k)^2$, with low-redshift coefficients such as $1/(\mathcal H\chi)$ capable of enhancing Doppler terms. Lensing is a line-of-sight projection and can matter even on small angular scales or for narrow bins. Wide angles, very large scales, cross-bin correlations, and multi-tracer differences invalidate a blanket omission.

Gauge, number-conservation, and observer checks

Under an infinitesimal time-slicing change, coordinate density acquires a background-evolution term; the inferred source time, radial volume, and velocity-potential terms change at the same order. Their sum in $\Delta_N$ cancels. The three independent derivations assigned to this page use different variable packages but agree that observed redshift and angle, not an arbitrary coordinate position, define the comparison. Yoo constructs the observed power spectrum from the perturbed redshift hypersurface Yoo 2010, §§ II–IV, especially eqs. (1), (11), and (36), while Bonvin and Durrer derive the redshift-space density and angular spectra in generic scalar gauge variables Bonvin and Durrer 2011, §§ II–IV, especially eqs. (2), (8), and (31).

Two further checks are decisive. First, $\kappa$ is an angular divergence, so its full-sky integral vanishes at first order when the mapping is single-valued; lensing redistributes sources but does not create them. Second, an observer boost changes the measured redshift and solid angle together. Removing a kinematic dipole while retaining only one of those changes produces a frame-dependent residual. A realistic mask mixes monopole and dipole power into higher multipoles, so subtraction and window convolution must be performed in the same pipeline.

Results and limitations through 2026-08-19

The analytic existence of relativistic projection terms is settled at linear order under the assumptions above. Their detectability in a particular survey remains unresolved because the signal depends on scale range, tracer population, $s(z)$, $b_e(z)$, window geometry, foregrounds, and covariance. The assigned sources are derivations and forecasts, not an observational detection or null test.

Formulation	Evidence or result	Scope or limitation
Yoo 2010, §§ II–IV	Gauge-invariant redshift-surface construction and forecasted observed power spectrum	Linear scalar theory and survey forecasts; no end-to-end calibrated detection or null result
Bonvin and Durrer 2011, §§ II–IV	Independent redshift, volume, lensing, and angular-spectrum derivation	Idealized linear kernels; bias, mask, selection, and covariance remain survey-specific
Challinor and Lewis 2011, §§ II–V	Exact current–Jacobi starting point, magnitude selection, linear transfer implementation, and forecasts	Requires calibrated $s$ and $b_e$ plus observer and window treatment; not observational evidence by itself
Survey-template boundary at the cutoff	Newtonian, partial Doppler/lensing/wide-angle, full relativistic, and multi-tracer templates are all used in different regimes	No universal verdict follows without common mock injections, actual windows, nuisance calibration, and full covariance

The credible competing positions are therefore conditional. A Newtonian template is adequate on restricted subhorizon scales; partial templates can capture a targeted Doppler, lensing, or wide-angle signal; a full gauge-invariant transfer treatment is required for horizon-scale interpretation; and multi-tracer estimators may reduce cosmic variance but amplify sensitivity to mismatched evolution and magnification biases. None makes the other formulations universally wrong outside its claimed regime.

Spatial averages on constant-time hypersurfaces do not directly equal these light-cone observables. Relating an averaged cosmology to a catalog requires a separate null-propagation, source-selection, and fitting calculation rather than substituting an averaged density into $\Delta_N$.

Review is triggered by a survey detection or null result with calibrated evolution and magnification bias and full window covariance, or by a public mock challenge that injects the complete observable and demonstrates unbiased recovery across Newtonian, partial, and relativistic templates. A corrected derivation of observer or selection terms would also require reassessment.

Formulation diagnostic

For a sample with $s=0.4$, the coefficient $(5s-2)$ vanishes. Does this prove that lensing has no effect on the observed catalog?

Solution

No. At this particular count slope, the first-order dilution of sources per solid angle cancels the first-order promotion of fainter sources across the flux threshold, so the explicit convergence term in this ideal magnitude-limited number count vanishes. Lensing can still affect source selection, shapes, higher-order reduced shear and magnification, cross-correlations, and any sample for which the effective slope varies with redshift or observing conditions. The cancellation is a formulation- and population-specific diagnostic, not a statement that the light bundle is unlensed.

References

• Bonvin, Camille, and Ruth Durrer. 2011. “What Galaxy Surveys Really Measure.” Physical Review D 84, 063505. DOI.
• Challinor, Anthony, and Antony Lewis. 2011. “The Linear Power Spectrum of Observed Source Number Counts.” Physical Review D 84, 043516. DOI.
• Yoo, Jaiyul. 2010. “General Relativistic Description of the Observed Galaxy Power Spectrum.” Physical Review D 82, 083508. DOI.