Observables on the Past Light Cone

A cosmological observation is not the value of a perturbation at a chosen coordinate point. It is a comparison made by an observer: a photon frequency against a clock, a ray direction against a tetrad, a flux against detector area and proper time, or a count in bins of measured redshift and solid angle. The central construction is therefore a map from emission events and null propagation to quantities on the observer’s past light cone. This page gives that map in an arbitrary metric spacetime, within geometric optics and before any weak-field expansion.

Required background. Geodesics and Jacobi Fields supplies the affine null rays and screen-space deviation map used below.

Helpful background. Einstein Equations connects optical tidal focusing to matter, Relativistic Matter and Compact Objects supplies the source four-current, Relativistic Cosmology supplies the FLRW limit, and Probability and Statistical Inference clarifies how a selection function turns physical sources into a catalog.

Frequencies and directions in an observer’s tetrad

Let a future-directed photon momentum $p^\mu$ connect a source event $S$ to an observation event $O$. An observer with future-directed unit velocity $u^\mu$ measures

$$\omega=-u_\mu p^\mu, \qquad p^\mu=\omega\bigl(u^\mu-n^\mu\bigr), \qquad u_\mu n^\mu=0, \qquad n_\mu n^\mu=1,$$

where $n^\mu$ points on the observer’s sky from the observer toward the source. Its three components on the observer’s spatial tetrad form a unit vector and hence specify two sky angles. An orthonormal screen basis $e_A{}^\mu$, $A=1,2$, is perpendicular to this direction and describes changes in the observed angle; the direction itself has zero screen components. The spectroscopic redshift is the scalar ratio

$$1+z=\frac{\omega_S}{\omega_O} =\frac{(-u_\mu p^\mu)_S}{(-u_\mu p^\mu)_O}.$$

The same relation says that two neighboring wave crests emitted a proper-time interval $d\tau_S$ apart arrive with $d\tau_O=(1+z)d\tau_S$. Thus redshift and arrival-time dilation are one observable statement, not independent coordinate effects. These definitions require a specified source and observer velocity; a null curve alone does not determine a redshift. The arbitrary-spacetime frequency ratio, with emission frequency in the numerator, is given in Challinor and Lewis 2011, § II, eq. (1).

The Jacobi map and distance observables

Neighboring rays are compared in the two-dimensional screen. Choose an affine parameter $\lambda$ increasing toward the past, with $\lambda(O)=0$ and tangent $k^\mu=-p^\mu$. Parallel-transport the screen basis along the ray. If $\delta\theta^B$ is the angular separation at $O$, the physical transverse separation at $\lambda$ is

$$\xi^A(\lambda)=\mathcal D^A{}_B(\lambda)\,\delta\theta^B, \qquad \frac{D^2\mathcal D^A{}_B}{d\lambda^2} =\mathcal T^A{}_C\mathcal D^C{}_B,$$

with vertex data and optical tidal matrix

$$\mathcal D(0)=0, \qquad \frac{d\mathcal D}{d\lambda}(0)=\omega_O\mathbf 1, \qquad \mathcal T_{AB} =R_{\mu\nu\rho\sigma}e_A{}^\mu k^\nu k^\rho e_B{}^\sigma.$$

The initial derivative makes a nearby bundle’s size equal to its physical depth times its opening angle. With the site’s curvature convention, $\operatorname{tr}\mathcal T=-R_{\mu\nu}k^\mu k^\nu$: positive null Ricci curvature focuses the beam, providing a useful sign check. The positive angular-diameter distance and the signed image parity are

$$D_A=\sqrt{\bigl\lvert\det\mathcal D\bigr\rvert}, \qquad \operatorname{parity}=\operatorname{sgn}(\det\mathcal D).$$

A vanishing determinant marks a caustic; replacing it by an unsigned scalar too early erases parity and can merge distinct images. The matrix reciprocity relation remains meaningful through such parity changes. For a resolved image in a metric spacetime, geometric optics gives the reciprocity factor, and photon conservation then gives luminosity distance:

$$D_L=(1+z)^2D_A, \qquad F_O=\frac{L_S}{4\pi D_L^2}.$$

Absorption, photon conversion, unresolved image aggregation, or finite-beam effects change the flux model; they do not change the local definition of $D_A$. The Jacobi, parity, and luminosity relations are stated together in Perlick 2010, §§ 2.3–2.4, printed pp. 16–22, eqs. (34)–(35) and (39)–(48), PDF.

Counts as a light-cone volume measurement

Let $J^\mu=n_su_s^\mu$ be the physical source-number current, with $n_s$ measured in the source rest frame. A ray bundle subtending $d\Omega_O$ has transverse area $D_A^2d\Omega_O$ at the source. During affine thickness $d\lambda$ it sweeps a rest-frame depth $\lvert p\cdot u_s\rvert d\lambda$. Therefore, before applying detection completeness or flux cuts,

$$\frac{dN}{dz\,d\Omega_O} =D_A^2\,\bigl\lvert p_\mu J^\mu\bigr\rvert \left\lvert\frac{d\lambda}{dz}\right\rvert.$$

This expression applies to each branch on which redshift is a regular coordinate along the ray. If several source events have the same observed redshift and direction, their contributions must be summed; image counts also require an explicit rule for multiply imaged sources. The expression already contains radial redshift distortion, angular focusing, source evolution, and the conversion from proper density to observed volume. A catalog adds a selection probability $S(L,\mathrm{color},z,\boldsymbol n,\ldots)$ and integrates the luminosity function above the direction-dependent flux threshold. Coordinate density by itself is never the observed count. The nonperturbative current-times-Jacobian result is derived in Challinor and Lewis 2011, § II, eqs. (1)–(5).

Gauge invariance and the FLRW check

Every quantity above is either a tetrad contraction at a physical endpoint, a screen Jacobi map, or a count in observed bins. A coordinate gauge transformation changes the labels assigned to the source event and redistributes terms among velocity, metric, and displacement perturbations; it cannot change the complete observable. This is why truncating the calculation to a coordinate density or to lensing alone generally leaves gauge-dependent pieces.

In unperturbed FLRW with $a_O=1$, comoving radial distance $\chi(z)=\int_0^z dz'/H(z')$, and curvature function $S_K(\chi)$, the construction reduces to

$$1+z=\frac{1}{a_S}, \qquad D_A=a_S S_K(\chi), \qquad D_L=(1+z)S_K(\chi),$$

and, for comoving source density $\bar n_c(z)$,

$$\frac{d\bar N}{dz\,d\Omega} =\bar n_c(z)\frac{S_K^2(\chi)}{H(z)}.$$

These limits check the affine normalization, the two powers of transverse distance, and the four powers of $1+z$ relating flux to area distance. They also expose the boundary of the chapter: spatial averages on a time slice do not directly supply any of these light-cone quantities.

Check your understanding

Show why an achromatic opacity with optical depth $\tau(z)$ violates the luminosity-distance relation inferred from flux but leaves geometric reciprocity intact.

Solution

Geometric reciprocity compares two Jacobi maps for the same null bundle, so it depends on the metric, endpoint normalization, and geometric-optics propagation, not on photon number. Opacity multiplies the received flux by $e^{-\tau}$: $F_O=L_S e^{-\tau}/(4\pi D_L^2)$. If one interprets this attenuated flux using a transparent model, the inferred distance is $D_L^{\mathrm{flux}}=D_L e^{\tau/2}$. Meanwhile $D_A$ and the corrected geometric luminosity distance still obey their reciprocity relation. Hence the failed step is photon conservation in the flux law, not the Jacobi-map theorem.

References

• Challinor, Anthony, and Antony Lewis. 2011. “The Linear Power Spectrum of Observed Source Number Counts.” Physical Review D 84, 043516. DOI.
• Perlick, Volker. 2004; updated 2010. “Gravitational Lensing from a Spacetime Perspective.” Living Reviews in Relativity 7, 9. DOI. Open PDF.