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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.

Yes. A radial baryon acoustic oscillation measurement and a transverse baryon acoustic oscillation measurement of the same ruler at one redshift obeyTheir ratio cancels both the unknown ruler length and the overall Hubble scale:This is the Alcock-Paczyński parameter. At a specified redshift its dependence on ${\Omega }_{m,0}$ permits a one-redshift determination within the assumed flat Lambda-CDM model.