Inhomogeneity and the Observed Light Cone

Cosmology is inferred from events on an observer’s past light cone, not from a photograph of a constant-time spatial slice. Redshift, direction, flux, shape, and source counts are measured in an observer’s local frame; distances, density contrasts, and velocity fields are reconstructed only after a spacetime model, source model, calibration, and selection function have been supplied. This chapter builds that map and then separates it from the distinct question of whether inhomogeneity changes an averaged background dynamics.

Helpful background. Use Einstein equations for local dynamics, relativistic matter for source physics, the cosmology overview for the homogeneous baseline, geodesics and Jacobi fields for light propagation, and probability and inference for catalog likelihoods.

From spacetime to a catalog

The reusable chain is

$$\begin{gathered} \text{metric, matter, and sources} \longrightarrow \text{null geodesic and Jacobi map}\\ \longrightarrow \{z,\hat{\boldsymbol n},F,\text{shape}\} \longrightarrow \text{calibration and selection}\\ \longrightarrow \text{catalog statistic and likelihood}. \end{gathered}$$

Each arrow is physical. The source and observer four-velocities determine redshift. The observer tetrad determines sky direction. Geodesic deviation determines the angular-diameter distance, convergence, and shear. The detector and source population determine which photons become catalog entries. A likelihood then compares those entries with a forward model. Coordinate position, coordinate density, and a spatially averaged scale factor can enter intermediate calculations, but none is itself a detector reading.

The order matters because perturbations displace the emission event associated with fixed observed redshift. Evaluating a field at the unperturbed coordinate event and then appending a lensing correction misses radial, temporal, volume, and observer contributions. The correct organizing variables are what the observer holds fixed: measured redshift, measured angle, a flux or size threshold, and the survey definition.

The operational observables

For a photon tangent $k^a$ and four-velocity $u^a$, the measured frequency is $\omega=-u_a k^a$, so

$$1+z=\frac{(u_a k^a)_{\rm s}}{(u_a k^a)_{\rm o}}.$$

Screen-projected Jacobi fields map an observed angular separation to a source-plane separation. Their determinant gives $D_A^2$ up to orientation; before a caustic one chooses the positive-area branch. Under metric null propagation with photon-number conservation, distance duality gives $D_L=(1+z)^2D_A$ Ellis and van Elst 1999, §§4.5.3 and 4.8, Eqs. (124)–(125) and (138)–(141). These statements define an observational layer that is independent of any particular perturbation gauge.

A catalog adds measures and cuts. A number-count observable is $dN/(dz\,d\Omega)$ for a stated tracer and selection, not a coordinate density contrast Challinor and Lewis 2011, §§II–V. Weak-lensing estimators average noisy galaxy shapes and respond to reduced shear, magnification, blending, and point-spread-function calibration Bartelmann and Schneider 2001, §§4.2 and 6.3–6.4. Peculiar-velocity inference combines redshift with an independent distance indicator in a declared frame; Gaussian distance-modulus scatter must not be silently treated as Gaussian velocity scatter. The same geometry therefore supports several probes, but each has a different measurement model.

Route through the chapter

The six pages follow the arrows from local measurement to interpretation.

Page	Central question	Output to carry forward
Observables on the Past Light Cone	What can one observer operationally measure from a null bundle?	Tetrad redshift and direction, Jacobi map, distances, flux, and count measure
Distance–Redshift Relations with Perturbations	How is distance evaluated at fixed observed redshift in a perturbed spacetime?	Event shift, Doppler and potential terms, integrated effects, convergence, and low-redshift checks
Relativistic Galaxy Number Counts	How do density, redshift displacement, volume distortion, lensing, and selection form one observed scalar?	Gauge-consistent count contrast with evolution and magnification bias
Weak Lensing and Magnification	How does the Jacobi map become a calibrated shear likelihood?	Convergence, shear, reduced shear, tomographic kernels, selection response, and systematics
Peculiar Velocities and Kinematic Probes	How can coherent motion be separated from distance scatter and survey geometry?	Log-distance forward model, velocity covariance, bulk-flow windows, and calibration tests
Averaging, Backreaction, and Model Dependence	When do averages alter domain evolution, and what would connect that change to observations?	Buchert scalar equations, Green–Wald hypotheses, closure limits, and a matched light-cone test

The first two pages establish the geometric dictionary. Number counts, lensing, and velocities then apply it to three different catalog constructions. The final page deliberately changes the question from photon observables to averaged dynamics and makes the additional bridge back to data explicit.

Checks that should survive every formulation

Several tests catch errors before a detailed likelihood does.

Observer check. A pure change of coordinates must not change a tetrad-measured redshift, angle, or final catalog scalar. Dropping observer terms can leave a spurious monopole or dipole even when the source terms look gauge invariant.

Homogeneous limit. Setting perturbations and peculiar velocities to zero must recover the FLRW distance–redshift relation, isotropic beam map, homogeneous count measure, and vanishing shear. At low redshift, the velocity contribution scales inversely with the background recession speed and can dominate distance residuals.

Area and parity check. The determinant of the Jacobi map controls area, while its trace-free part controls shear. Beyond caustics, signed parity and physical area cannot be interchanged without stating the image-counting rule.

Selection check. Changing a flux, size, morphology, redshift, or quality threshold can change the sample response even if the underlying spacetime is fixed. Magnification bias, evolution bias, photometric-redshift tails, and distance-indicator cuts belong in the forward model.

Reproduction check. A numerical constraint is not reproduced merely because its formula is quoted. One needs the named data product, covariance, cuts, nuisance priors, software or transparent calculation, and an independently recomputed statistic. Where those ingredients are absent, the relevant leaf page states the boundary instead of presenting an attributed result as a rerun.

Spatial averages and light-cone fits

Let $a_{\mathcal D}$ describe the proper volume of a chosen comoving spatial domain. It is a meaningful dynamical variable once the matter model, foliation, and domain are fixed Buchert 2000, §2.2, Eqs. (9a)–(13b). It is nevertheless not the quantity plotted on a Hubble diagram. An observer receives photons emitted at different times and locations; their fluxes and redshifts depend on the metric along each null path, and the catalog depends on which sources were selected.

Consequently, neither $\ddot a_{\mathcal D}\gt0$ nor a small effective stress tensor alone determines the sign or magnitude of a fitted acceleration. A defensible comparison evolves one inhomogeneous spacetime, computes both its spatial averages and its synthetic light-cone observables, applies the survey selection, and fits the same comparison model to those observables. Agreement can then be reported with its domain, observer, and approximation; it cannot be assumed from notation.

Review the chapter: identify the missing maps

A calculation supplies a gauge-fixed matter contrast $\delta(\eta,\boldsymbol x)$ on constant-$\eta$ slices and a positive acceleration $\ddot a_{\mathcal D}$ for one spatial domain. A survey instead reports galaxy counts binned by observed $(z,\hat{\boldsymbol n})$ and supernova fluxes. What is still required before either theoretical quantity can be compared with the catalog?

Solution

The coordinate contrast must first be evaluated at the perturbed emission event selected by observed redshift. Source velocity and metric perturbations shift that event and distort radial and angular volume; lensing, evolution bias, magnification bias, observer terms, tracer bias, and the actual selection then combine into the observed count contrast. Gauge dependence must cancel in the complete scalar.

The domain acceleration requires a different bridge. One must propagate null rays through the same inhomogeneous spacetime, compute redshifts and fluxes in chosen observer frames, apply the supernova selection and calibration, and fit those synthetic data with the survey likelihood. A positive $\ddot a_{\mathcal D}$ by itself supplies none of those maps, so it neither proves nor disproves a positive fitted acceleration.

References

• Bartelmann, Matthias, and Peter Schneider. 2001. “Weak Gravitational Lensing.” Physics Reports 340: 291–472. DOI. Open PDF.
• Buchert, Thomas. 2000. “On Average Properties of Inhomogeneous Fluids in General Relativity. I: Dust Cosmologies.” General Relativity and Gravitation 32: 105–125. DOI. Open PDF.
• Challinor, Anthony, and Antony Lewis. 2011. “The Linear Power Spectrum of Observed Source Number Counts.” Physical Review D 84: 043516. DOI. Open PDF.
• Ellis, George F. R., and Henk van Elst. 1999. “Cosmological Models.” In Theoretical and Observational Cosmology, 1–116. Dordrecht: Springer. DOI. Open PDF.