Cosmological distance measure

A cosmological distance measure converts an observed redshift, angle, flux, or travel time into a distance in an expanding spacetime. Different measures answer different operational questions and therefore need not agree numerically.

The angular diameter distance is defined by $d_A=x/\theta$ for a small transverse physical size $x$ subtending angle $\theta$. In a spatially flat Friedmann-Lemaitre-Robertson-Walker metric,where $\chi$ is the comoving radial distance.

In a standard expanding cosmology, $d_A(z)$ initially increases from zero but eventually decreases because the factor $(1+z{)}^{-1}$ dominates the bounded comoving distance. Beyond its maximum, a fixed physical object subtends a larger angle at a larger redshift.

A cosmological standard ruler is a feature with a fixed known or redshift-independent physical or comoving size. Comparing its observed angular and radial extents constrains the expansion history even when its absolute size is treated as a nuisance parameter.

Baryon acoustic oscillations are the relic spatial scale of sound waves in the pre-recombination photon-baryon fluid. The scale appears both as a preferred separation in galaxy clustering and as oscillatory structure in the cosmological density power spectrum.

For a comoving BAO scale ${\chi }_d$ perpendicular to the line of sight in a spatially flat universe,Its redshift dependence constrains the shape of the distance--redshift relation.

For a small comoving BAO separation parallel to the line of sight,in units with $c=1$. Combining this with a transverse measurement eliminates the unknown ruler size.

The matter power spectrum contains low-amplitude oscillatory features because baryons preserve their pre-recombination acoustic phase and later transfer part of it gravitationally to the combined matter distribution.

The Alcock-Paczyński test compares radial and transverse distance conversion using a feature that is statistically isotropic in comoving space.

The Alcock-Paczyński parameter compares radial and transverse distance conversion at one redshift. An intrinsically isotropic comoving feature obeysso its unknown absolute size cancels.