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.

Geometrical distances to nearby calibrators determine the luminosities of indicators that can be observed farther away. Those indicators calibrate still more luminous indicators in turn. A ladder may connect parallax or maser distances to Cepheid variables and then to Type Ia supernovae. Each step propagates its calibration uncertainty to the final distance scale and the inferred Hubble constant.

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-Lemaître-Robertson-Walker metric,where $\chi$ is the comoving radial distance.

In an Einstein-de Sitter universe, $H(z)=H_0(1+z{)}^{3/2}$ and the radial comoving distance is $2H_0^{-1}[1-(1+z{)}^{-1/2}]$. A physical Hubble radius $H(z{)}^{-1}$ has comoving size $(1+z)/H(z)$. Dividing by the radial distance gives the displayed small angle, in radians. A Hubble diameter has twice this angle; the matter-era particle-horizon radius also equals twice the Hubble radius. These distinctions matter when comparing approximate CMB angular scales.

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.

For a flat universe with a single constant equation-of-state parameter $w>-1$, $H(z)=H_0(1+z{)}^{\alpha }$. Its angular diameter distance is $H_0^{-1}[(1+z{)}^{-\alpha }-(1+z{)}^{-1}]/(1-\alpha )$, with logarithmic limit $H_0^{-1}\log (1+z)/(1+z)$ at $\alpha =1$. Differentiation gives the displayed unique maximum, interpreted as $z_m=e-1$ at $\alpha =1$. The distance vanishes at both endpoints $z=0$ and $z\to \infty$. As $w\downarrow -1$, the maximum moves to infinite cosmological 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.