Cosmological standard ruler

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.

The epoch at which momentum exchange with photons ceases to drag baryons efficiently. The sound horizon at this epoch calibrates the late-time baryon acoustic oscillation ruler. It is related to, but not identical with, photon last scattering.

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 conversions between observed separations and comoving distances at one cosmological redshift. For a small intrinsically isotropic feature of comoving size $r$, the radial and transverse relations areHere $D_M$ is the transverse comoving distance, $H$ the Hubble parameter, and the angle $\Delta \theta$ is in radians. Eliminating the unknown ruler gives the dimensionless combinationwhere $D_A$ is the angular diameter distance. In spatially flat geometry $D_M=\chi$; the earlier expression $H\chi$ additionally uses units with $c=1$. For nonzero cosmological spatial curvature, the radial comoving distance cannot simply replace $D_M$.^[ref]

This ratio is useful because an absolute ruler calibration cancels. Its inference still assumes statistical isotropy of the selected feature. Peculiar velocities produce redshift-space distortions, so an observed anisotropy must be modeled jointly with these effects before being interpreted as an incorrect distance conversion.