Cosmic Distances and Causal Scales

Cosmological observations arrive on a past light cone, not on a spatial slice. Turning redshift, angle, flux, and source counts into geometry therefore requires a chain of constructions: integrate the radial null path, map it through spatial curvature, attach the appropriate endpoint redshift factors, and only then add source populations and selection. This chapter builds that chain and keeps it separate from a second question—whether the entire past or future of an observer is connected by causal curves.

Helpful background. The Einstein Equation supplies the Einstein Equations behind a physical expansion history; Relativistic Matter and Compact Objects supplies the Relativistic Matter and Compact Objects used as sources and tracers; Affine and Variational Geodesics supplies Geodesics and Jacobi Fields for rays and beams; Relativistic Cosmology orients the background conventions; and The Wave Equation and Characteristics supplies Waves, Green Functions, and Geometric Optics for light and signal propagation.

Check your preparation

Before starting, try two short checks. Can you combine $1+z=a^{-1}$ with a radial null path to explain why one redshift integral contains $1/(1+z)$ and another does not? Can you distinguish an area measured at the source from a solid angle measured at the observer? If the first is unclear, review the preceding redshift and conformal-time pages; if the second is unclear, review the linked geodesic treatment’s discussion of Jacobi fields.

Six routes from a light cone to an observable

The pages form a constructive sequence, with a separate causal branch.

1. Lookback Time and Conformal Distance derives the time and radial-distance integrals and states when redshift fails as a global clock.
2. Angular-Diameter and Luminosity Distances converts the radial result into transverse area and bolometric flux distances.
3. Etherington Distance Duality proves reciprocity for a general metric null beam and isolates the additional photon-conservation assumption.
4. Comoving Volume and Number Counts constructs the FLRW volume element and separates geometry from population evolution and selection.
5. Particle Horizons, Event Horizons, and Hubble Radius distinguishes global causal integrals from instantaneous expansion and sound scales.
6. Standard Candles, Standard Rulers, and Cosmography compares calibration contracts and tests covariance and series truncation with pinned public releases.

For the geometric route, read pages 1–4 in order and use page 6 as the observational stress test. For the causal route, read pages 1–2 and then page 5; page 3 is the deeper route if lensing, caustics, or reciprocity matter.

The distance dictionary

Use a dimensionless scale factor with $a(t_0)=1$ and dimensionful reference curvature $K$. Analytic derivations set $c=1$; the formulas below restore $c$ to display units. The line-of-sight comoving distance is

$$D_C(z)=\chi(z) =c\int_0^z\frac{dz'}{H(z')}.$$

Curvature acts on this radial coordinate through

$$D_M=S_K(\chi), \qquad S_K(\chi)= \begin{cases} \sin(\sqrt K\,\chi)/\sqrt K, & K\gt0,\\ \chi, & K=0,\\ \sinh(\sqrt{-K}\,\chi)/\sqrt{-K}, & K\lt0. \end{cases}$$

The observer and source endpoints then determine the familiar distances:

$$D_A=\frac{D_M}{1+z}, \qquad D_G=D_M, \qquad D_L=(1+z)D_M=(1+z)^2D_A.$$

Here $D_A$ converts observer solid angle to source area, while the galaxy-area distance $D_G$ reverses those endpoints. The last equality requires metric geometric optics plus photon-number conservation for bolometric flux; reciprocity alone establishes $D_G=(1+z)D_A$. Hogg assembles the FLRW dictionary in Hogg 2000, §§4–9, printed pp. 4–8 (Open PDF), while the general beam theorem is developed in Perlick 2004, updated preprint §§2.3–2.4, printed pp. 16–21, eqs. (30)–(48) (Open PDF).

The same geometry gives the background shell volume

$$\frac{dV_c}{dz\,d\Omega} =\frac{cD_M^2(z)}{H(z)} =\frac{c(1+z)^2D_A^2(z)}{H(z)}.$$

Observed counts are not this Jacobian alone: they also require a comoving population density, source evolution, survey selection, and measurement likelihood.

Causal integrals are different objects

At time $t$, a particle horizon and a future event horizon use opposite ends of the conformal-time domain:

$$\chi_{\rm PH}(t) =c\int_{t_i}^{t}\frac{dt'}{a(t')}, \qquad \chi_{\rm EH}(t) =c\int_t^{t_f}\frac{dt'}{a(t')}.$$

Their physical radii are $a\chi_{\mathrm{PH}}$ and $a\chi_{\mathrm{EH}}$. The first asks how far signals could have arrived since a specified past endpoint; the second exists only when the specified future integral converges. By contrast, $c/|H|$ is an instantaneous expansion length and can diverge at a bounce even when the causal integrals remain finite. A sound horizon replaces $c$ in the past integral by the medium’s sound speed. Tong gives the integral definitions and conformal diagrams in Tong 2019, §1.1.4, printed pp. 15–19, eqs. (1.24)–(1.26) (PDF).

Scope and observational boundary

Pages 1–5 construct background quantities and theorem-level relations. They do not infer an expansion history from a catalog, and they do not fold weak lensing, peculiar velocities, or relativistic light-cone perturbations into an observed volume. Page 6 provides a bounded comparison of candles, rulers, and cosmography, including exact release pins and covariance checks, but leaves full survey likelihoods and current parameter verdicts to the later inference chapter. Instrument calibration and current distance-ladder results remain outside this chapter.

Review the chapter

In a flat, matter-only expanding model, $H(z)=H_0(1+z)^{3/2}$. Derive $D_C$, $D_A$, $D_L$, and the comoving volume element. Then decide whether the equality of a distance with $c/H$ at one redshift would establish a causal horizon.

Solution

Direct integration gives

$$D_C=D_M =\frac{2c}{H_0} \left[1-(1+z)^{-1/2}\right].$$

Therefore

$$D_A=\frac{2c}{H_0} \left[(1+z)^{-1}-(1+z)^{-3/2}\right],$$ $$D_L=\frac{2c}{H_0} \left[(1+z)-(1+z)^{1/2}\right],$$

and

$$\frac{dV_c}{dz\,d\Omega} =\frac{4c^3}{H_0^3} \frac{\left[1-(1+z)^{-1/2}\right]^2}{(1+z)^{3/2}}.$$

All four results have the correct low-redshift limits. An equality with $c/H$ is only a numerical relation between a path-integrated distance and an instantaneous scale. A particle or event horizon is established by the relevant conformal-time integral and its endpoint, not by Hubble-radius crossing.

References

• Hogg, David W. “Distance Measures in Cosmology.” arXiv:astro-ph/9905116v4, December 2000. Open PDF.
• Perlick, Volker. “Gravitational Lensing from a Spacetime Perspective.” Living Reviews in Relativity 7, 9 (2004), updated arXiv:1010.3416v1 (2010). DOI. Open PDF.
• Tong, David. Cosmology. University of Cambridge Part II Mathematical Tripos lecture notes, Michaelmas 2019. Official PDF.