Distance–Redshift Relations with Perturbations

At fixed observed redshift, the source is not located at its background FLRW time or radius. Doppler shifts, endpoint gravitational potentials, integrated evolution of the potentials, Shapiro delay, and transverse focusing all change the distance assigned to the same measured $(z,n^A)$. The result is meaningful only when those effects are combined at the perturbed emission event. Here the calculation is linear in scalar perturbations about spatially flat FLRW, before caustics and with geometric optics and photon conservation.

Required background. Geodesics and Jacobi Fields supplies the null trajectory and screen Jacobi map whose determinant defines angular distance.

Helpful background. Einstein Equations supplies the relation between potentials and matter, Relativistic Matter and Compact Objects supplies source four-velocities, Relativistic Cosmology supplies the FLRW background, and Probability and Statistical Inference is useful when distance scatter enters a likelihood.

Redshift fixes a perturbed emission event

Use conformal Newtonian gauge,

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

with $a_O=1$, lapse potential $\Phi$, and spatial-curvature potential $\Psi$. The unit vector $\boldsymbol n$ points from observer to source, $\boldsymbol v$ is physical peculiar velocity in the orthonormal cosmological frame, primes denote $d/d\eta$, and every perturbation below is evaluated on the unperturbed ray $\boldsymbol x=\boldsymbol n(\eta_O-\eta)$. Initially fix the observation event at coordinate time $\eta_O$ and normalize the background there; an observer proper-time correction is discussed below. Let $\eta_*$ satisfy $1+z=1/a(\eta_*)$ and write $\chi_*=\eta_O-\eta_*$ and $\mathcal H_*=a'(\eta_*)/a(\eta_*)$.

At fixed coordinate emission time, the fractional redshift perturbation is

$$\Delta_z =\Phi_O-\Phi_S +\int_{\eta_O}^{\eta_*}(\Phi'+\Psi')\,d\eta +\boldsymbol n\cdot(\boldsymbol v_S-\boldsymbol v_O).$$

Holding the observed redshift fixed moves the source time by

$$\mathcal H_*\,\delta\eta=\Delta_z.$$

The signs follow directly from $1+z=(-u\cdot p)_S/(-u\cdot p)_O$: motion of the source away from the observer raises the redshift. The endpoint and integrated pieces are not separately gauge-invariant; the event selected by measured $z$ is.

Radial displacement and transverse focusing

The radial coordinate of that event receives both the time shift and the null time-delay correction,

$$\delta\chi =-\delta\eta -\int_{\eta_O}^{\eta_*}(\Phi+\Psi)\,d\eta.$$

The integral is oriented from observation to the earlier emission time. Transverse tidal focusing is summarized by the Born-level convergence

$$\kappa(\boldsymbol n,z) =\frac12\int_0^{\chi_*}d\chi\, \frac{\chi(\chi_*-\chi)}{\chi_*} \nabla_\perp^2(\Phi+\Psi),$$

where $\nabla_\perp$ differentiates with respect to comoving transverse position. A positive overdensity normally gives positive $\kappa$ and a smaller angular-diameter distance. This expression assumes a unique weakly perturbed ray; strong-lensing image multiplicity requires the signed Jacobi map described on Observables on the Past Light Cone.

Distance at fixed observed redshift

Expanding the observer-normalized Jacobi determinant gives

$$\frac{\delta D_A}{\bar D_A} =\frac{\delta\chi}{\chi_*} +\mathcal H_*\delta\eta -\Psi_S-\kappa +\boldsymbol n\cdot\boldsymbol v_O.$$

The last term is the aberration correction between the cosmological tetrad and the actual observer tetrad. Substituting the event and radial shifts makes the physical groups explicit:

$$\frac{\delta D_A}{\bar D_A} =\left(1-\frac{1}{\mathcal H_*\chi_*}\right)\Delta_z -\frac{1}{\chi_*}\int_{\eta_O}^{\eta_*}(\Phi+\Psi)\,d\eta -\Psi_S-\kappa +\boldsymbol n\cdot\boldsymbol v_O.$$

At fixed observed $z$, distance duality gives the same fractional perturbation for luminosity distance Perlick 2010, § 2.4, eqs. (45)–(48),

$$\frac{\delta D_L}{\bar D_L} =\frac{\delta D_A}{\bar D_A}, \qquad \bar D_L=(1+z)\chi_*.$$

The source-time, radial, determinant, and observer-angle steps are derived as one package in Challinor and Lewis 2011, § III, eqs. (13)–(20), and § IV, eq. (33). In that source, lowercase $\psi$ is the lapse perturbation and therefore maps to site $\Phi$, while lowercase $\phi$ is the spatial perturbation and maps to site $\Psi$. This is why the endpoint redshift term contains $\Phi_O-\Phi_S$ but the source-plane determinant contains $-\Psi_S$. Quoting only $-\kappa$ or only a Doppler term is not a gauge-complete distance perturbation.

The determinant explains every coefficient. Two transverse directions supply $2\delta\chi/\chi_*$, the scale factor at the shifted emission time supplies $2\mathcal H_*\delta\eta$, the spatial metric supplies $-2\Psi_S$, lensing changes the area by $-2\kappa$, and angular aberration supplies $2\boldsymbol n\cdot\boldsymbol v_O$. Taking the square root converts the area perturbation into the distance perturbation written above. This route checks both the factor of two and the sign of convergence without relying on a remembered formula.

All terms are dimensionless: $\mathcal H_*\chi_*$ is dimensionless, the line-of-sight potential integral is divided by $\chi_*$, and the transverse Laplacian in $\kappa$ is balanced by two powers of comoving length. A constant rescaling of the affine parameter cannot change the result because the observer-frequency normalization and the Jacobi initial derivative rescale together.

The observer event also matters when comparing gauges. If it is specified by a fixed proper-clock reading instead of fixed coordinate $\eta_O$, include its coordinate lapse $\delta\eta_O$: replace $\mathcal H_*\delta\eta=\Delta_z$ by $\mathcal H_*\delta\eta=\Delta_z+\mathcal H_O\delta\eta_O$, and replace $\delta\chi$ by $\delta\eta_O-\delta\eta+\int_{\eta_*}^{\eta_O}(\Phi+\Psi)d\eta$. This adds only an angular monopole at linear order for one observer, but it must be retained if an absolute distance or a constant-potential gauge check is claimed. Monopole-subtracted distance anisotropies are independent of that clock calibration.

Limiting checks and observational meaning

With all perturbations zero, $D_A=a_*\chi_*$ and $D_L=(1+z)\chi_*$. At low redshift, $\mathcal H_*\chi_*\simeq z$, so the leading velocity contribution is

$$\frac{\delta D_L}{\bar D_L} \simeq -\frac{\boldsymbol n\cdot(\boldsymbol v_S-\boldsymbol v_O)}{z},$$

in units with $c=1$. This enhanced $1/z$ behavior explains why peculiar-velocity covariance is essential in nearby standard-candle analyses. At larger distance, the lensing term accumulates along the ray and becomes non-Gaussian beyond linear order. Endpoint monopoles can be absorbed into clock or absolute-distance calibration, and an observer dipole can be reported in a chosen rest frame, but those operations must occur after the full observable has been assembled and the frame stated.

The equation predicts a conditional mean shift for specified perturbations. An actual distance catalog also contains intrinsic luminosity or size scatter, calibration covariance, selection, and lensing dispersion. Those statistical ingredients are not additional metric terms and should enter the measurement model rather than be hidden inside $\delta D_L$.

Check your understanding

Ignore potentials and lensing, set $\boldsymbol v_O=0$, and consider a nearby source with $\boldsymbol n\cdot\boldsymbol v_S=v_r\gt0$. At fixed observed redshift, is its inferred background distance larger or smaller than that of a comoving source?

Solution

The peculiar recession contributes $\Delta_z=v_r$. Keeping the measured redshift fixed therefore reduces the cosmological part of the redshift, moving the background emission event nearer: $\delta\chi=-v_r/\mathcal H_*$. The exact velocity factor is $(1-1/(\mathcal H_*\chi_*))v_r$, whose low-redshift term is $-v_r/z$. Thus the distance is smaller. The finite $+v_r$ term cannot compete with the radial-displacement term when $z$ is small.

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.