Cosmology distance methods

Cosmology has multiple distance definitions because the universe expands while light travels. The main trick: always ask what the distance is being used for.

Comoving distance

The line-of-sight comoving distance is the distance in coordinates that expand with the universe:

$$\chi (z)=c{\int }_0^z\frac{d{z}^{\prime }}{H(z^{\prime })}.$$

In scale factor form, using $a=1/(1+z)$:

$$\chi (a)=c{\int }_a^1\frac{d{a}^{\prime }}{a^{\prime 2}H(a^{\prime })}.$$

Comoving distance is useful for large-scale structure because objects at rest in the Hubble flow keep roughly fixed comoving coordinates.

Transverse comoving distance

For a flat universe, the transverse comoving distance is just:

$$D_M=\chi .$$

With curvature, $D_M$ is the curvature-corrected version of $\chi$ using the usual $S_k(\chi )$ function.

Angular-diameter distance

Angular-diameter distance relates physical transverse size to observed angular size:

$$\theta =\frac{L}{d_A},$$

so

$$d_A(z)=\frac{D_M(z)}{1+z}.$$

See Angular diameter distance.

Luminosity distance

Luminosity distance is defined through flux and luminosity:

$$F=\frac{L}{4\pi d_L^2}.$$

In an expanding universe:

$$d_L(z)=(1+z)D_M(z).$$

The extra redshift factors come from photon energy loss and time dilation of photon arrival rates.

Etherington reciprocity relation

Combining the above:

$$d_L=(1+z{)}^2{d}_A.$$

This is a classic exam sanity check. If your answer violates this without a stated exotic assumption, something is probably off.

Quick use guide

• Standard ruler / angular size: use $d_A$.
• Standard candle / flux: use $d_L$.
• Large-scale coordinate separation: use $\chi$ or $D_M$.
• Age/lookback-time problems: integrate $dt=da/(aH)$ instead of distance.

Related

• Angular diameter distance
• Age of Universe
• Cosmology MOC