Lookback Time and Conformal Distance

An observed redshift labels both an emission time and a radial null path, but the two associated integrals are not the same. The lookback time weights each redshift interval by $1/(1+z)$, whereas the conformal, or line-of-sight comoving, distance does not. This page derives both quantities from the scale factor, checks their low-redshift and single-component limits, and separates them from transverse distance measures. Instrument calibration and current distance-ladder results are deliberately left to observational analyses.

Required background. Affine and Variational Geodesics supplies the Geodesics and Jacobi Fields capability; only its affine null-ray and observer conventions are needed for this radial calculation.

Helpful background. The Einstein Equation supplies the Einstein Equations from which an expansion history may be found; Relativistic Matter and Compact Objects supplies the Relativistic Matter and Compact Objects models that determine $H(z)$; Relativistic Cosmology orients the earlier FLRW, redshift, and Friedmann results used here; and The Wave Equation and Characteristics supplies the Waves, Green Functions, and Geometric Optics interpretation of a radial light ray as a null characteristic.

From redshift to time

Use a dimensionless scale factor normalized by $a(t_0)=1$, set $c=1$, and write the FLRW line element in radial-geodesic form,

$$ds^2=-dt^2+a^2(t) \left[d\chi^2+S_K^2(\chi)\,d\Omega^2\right].$$

Here $K$ is the time-independent, dimensionful curvature of the reference spatial metric, $\chi$ has dimensions of length, and

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

For comoving emission and observation,

$$1+z=\frac{1}{a(t_{\rm e})}.$$

Differentiate this relation along the background history:

$$\frac{dz}{dt} =-\frac{\dot a}{a^2} =-(1+z)H(z), \qquad \boxed{\frac{dt}{dz} =-\frac{1}{(1+z)H(z)}}.$$

The elapsed proper time on the comoving observer congruence is therefore

$$\boxed{ t_{\rm L}(z) \equiv t_0-t_{\rm e} =\int_0^z \frac{dz'}{(1+z')H(z')} }.$$

This is a duration, not a spatial distance. Hogg gives the same redshift and lookback-time dictionary in Hogg 2000, §§3 and 10, printed pp. 2–3 and 8, eqs. (12) and (30) (Open PDF).

The radial null integral

Define conformal time by $d\eta=dt/a(t)$. A radial null ray obeys

$$0=-d\eta^2+d\chi^2,$$

so an incoming ray satisfies $d\chi=-d\eta$. If the observer is at $\chi=0$, the source’s radial coordinate is

$$\chi(z) =\eta_0-\eta_{\rm e} =\int_{t_{\rm e}}^{t_0}\frac{dt}{a(t)}.$$

Using $a^{-1}=1+z$ and the time–redshift derivative above makes the extra factor cancel:

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

Thus $\chi$ is also the line-of-sight comoving distance $D_C$. The curvature constant $K$ does not enter this radial integral explicitly. A chosen dynamical model may correlate $K$ with the function $H(z)$; the explicit curvature mapping enters only when one forms the transverse comoving distance.

The redshift forms of both integrals assume a monotonic branch on which $H\ne0$. At a smooth turnaround or bounce, redshift ceases to be a good global time coordinate: split the history into monotonic branches or integrate in $t$ directly. The time-domain definitions themselves remain regular whenever the geometry does.

$$D_M=S_K(\chi).$$

At observation the proper radial separation on the $t=t_0$ slice is $a_0\chi=\chi$; on the emission slice it is $a_{\rm e}\chi=\chi/(1+z)$. Neither slice-dependent separation is the length of the photon’s null worldline. The line-of-sight and transverse definitions, including their curvature distinction, are given in Hogg 2000, §§4–5, printed pp. 4–5, eqs. (15)–(16) (Open PDF).

The pair of functions contains a useful kinematic consistency relation. If both were known as smooth functions of the same cosmological redshift, then

$$H(z)=\left(\frac{d\chi}{dz}\right)^{-1} =\left[(1+z)\frac{dt_{\rm L}}{dz}\right]^{-1}, \qquad \frac{d\chi}{dt_{\rm L}}=1+z.$$

No Einstein equation was used to obtain these identities; dynamics enters only through the supplied function $H(z)$. In practice, clocks and distance indicators have different calibration, selection, and population assumptions, so this relation is a theoretical cross-check rather than a claim that either function can be read directly from raw data.

Checks and analytic limits

Differentiating the two integrals returns their integrands:

$$\frac{dt_{\rm L}}{dz} =\frac{1}{(1+z)H(z)}, \qquad \frac{d\chi}{dz} =\frac{1}{H(z)}.$$

This is the quickest normalization check. Define the present deceleration parameter by

$$q_0=-\frac{\ddot a_0}{a_0H_0^2},$$

and near the observer write $H(z)=H_0[1+(1+q_0)z+O(z^2)]$. Then

$$\begin{aligned} H_0t_{\rm L}(z) &=z-\frac12(2+q_0)z^2+O(z^3),\\ H_0\chi(z) &=z-\frac12(1+q_0)z^2+O(z^3). \end{aligned}$$

Both begin with the Hubble-law result $z/H_0$, while their quadratic coefficients differ by the redshift factor in the clock integral.

For a flat universe dominated by one constant-$w$ component, define $n=3(1+w)/2$, so $H(z)=H_0(1+z)^n$. For $n\ne0$ and $n\ne1$,

$$\begin{aligned} H_0t_{\rm L}(z) &=\frac{1-(1+z)^{-n}}{n},\\ H_0\chi(z) &=\frac{(1+z)^{1-n}-1}{1-n}. \end{aligned}$$

The exceptional cases are continuous limits: $n=0$ gives $H_0t_{\rm L}=\ln(1+z)$ and $H_0\chi=z$, while $n=1$ gives $H_0\chi=\ln(1+z)$. Matter domination, $n=3/2$, yields

$$H_0t_{\rm L} =\frac23\left[1-(1+z)^{-3/2}\right], \qquad H_0\chi =2\left[1-(1+z)^{-1/2}\right].$$

Radiation domination, $n=2$, instead gives $H_0t_{\rm L}=[1-(1+z)^{-2}]/2$ and $H_0\chi=z/(1+z)$. Both expressions acquire dimensions of time or length from $H_0^{-1}$, and direct differentiation recovers the original integrands. The finite limit $\chi(\infty)=H_0^{-1}$ is the comoving particle horizon of that idealized radiation history, whereas the lookback limit $t_{\rm L}(\infty)=(2H_0)^{-1}$ is its present cosmic age.

These limits also expose a causal distinction: $\chi(z\to\infty)$ is finite when $n\gt1$, but a finite integral must still be interpreted as a particle-horizon statement only after the lower endpoint and spacetime history are specified. Tong develops the conformal-time and radial-null integrals in Tong 2019, §1.1.4, printed pp. 15–19 (Official PDF).

Common pitfalls

Dropping the factor $1+z$. It belongs in $dt/dz$ and hence in lookback time. It cancels from $d\eta=dt/a$ and therefore does not appear in $\chi(z)$.

Calling every distance “comoving distance.” $D_C=\chi$ measures the radial null integral, while $D_M=S_K(\chi)$ converts it into transverse separation. They agree only when $K=0$.

Treating a coordinate distance as a null-path length. A null curve has zero spacetime interval. Proper distances such as $a(t)\chi$ live on chosen spacelike slices.

Exercises

In a flat matter-only model, evaluate $t_{\rm L}$ and $\chi$ for a source at $z=3$, and verify both answers by differentiating the general matter-era expressions.

Solution

For $z=3$, $(1+z)^{-1/2}=1/2$ and $(1+z)^{-3/2}=1/8$. Therefore

$$t_{\rm L}(3) =\frac{2}{3H_0}\left(1-\frac18\right) =\frac{7}{12H_0}, \qquad \chi(3) =\frac{2}{H_0}\left(1-\frac12\right) =\frac{1}{H_0}.$$

The source is therefore one Hubble length away in line-of-sight comoving distance, while the elapsed comoving-observer time is $7/12$ of a Hubble time. These are different observables even in this simplest background; the agreement of their leading small-redshift limits does not make them interchangeable.

Differentiation gives

$$\frac{d t_{\rm L}}{dz} =\frac{1}{H_0(1+z)^{5/2}}, \qquad \frac{d\chi}{dz} =\frac{1}{H_0(1+z)^{3/2}},$$

which are exactly $[(1+z)H(z)]^{-1}$ and $H^{-1}(z)$ for $H=H_0(1+z)^{3/2}$. Their ratio is $1+z$, providing an additional check on the two powers and on the time–redshift conversion.

References

• Hogg, David W. “Distance Measures in Cosmology.” arXiv:astro-ph/9905116v4, 16 December 2000. Open PDF.
• Tong, David. Cosmology. University of Cambridge Part II Mathematical Tripos lecture notes, Michaelmas Term 2019. Official PDF.