Baryon acoustic oscillations as standard rulers

Define r d as the comoving distance traveled by sound before the baryon drag epoch. This epoch marks effective release of baryons from photon drag and differs from the photon last-scattering epoch used for the microwave acoustic angle. The integral depends on both the sound speed and expansion rate at early times. Baryon loading lowers the sound speed relative to a photon-only fluid, while extra radiation or altered recombination can change the calibration.

The ruler is commonly computed within a specified early-universe cosmology, with baryon and radiation densities constrained by other observations. A BAO distance expressed in megaparsecs inherits that calibration. Calling the measured feature a standard ruler does not make its absolute length independent of all cosmological physics. Reporting distance divided by r d retains the distinction and allows an alternative early-time model to be tested without pretending that its ruler is unchanged.

For a sufficiently narrow redshift bin in an FLRW background, a transverse comoving separation equals D M times its angle. A radial comoving interval is approximately c times its redshift interval divided by H. Applying these relations to the acoustic feature gives two ratios: transverse comoving distance over r d, and the radial Hubble distance c over H divided by r d. The second is not H itself, so reciprocal conventions must be checked carefully.

Real analyses fit anisotropic correlation functions or power spectra using a fiducial distance-redshift conversion. Their fitted dilation parameters correct transverse and radial scales relative to that fiducial model and its chosen ruler. A spherically averaged measurement instead constrains a combination of radial and transverse distances, losing some anisotropic information. The simple small-separation equations below expose the geometry, but are not a replacement for the survey’s window-convolved likelihood or its broadband nuisance model.

Nonlinear evolution and large-scale motions broaden and shift the acoustic feature. Reconstruction estimates some coherent displacements to sharpen it, using assumptions about bias, growth, and the observed density field. Calibration with simulations and scale cuts checks the remaining error. Smooth galaxy bias generally affects the broadband correlation more strongly than the oscillatory scale, but this useful robustness is not a license to ignore tracer selection, redshift errors, or nonlinear modeling.

A finite angular mask and radial selection function mix Fourier modes. Random catalogs represent the survey selection, while covariance includes sample variance and finite tracer density. Overlapping tracers and redshift bins can therefore be correlated even when their catalog labels differ. DESI’s dated 2024 BAO analysis explicitly reports ruler-relative distances and its modeling framework. Here we use synthetic narrow-bin measurements to isolate the distance conversion, with no claim that these numbers reproduce a particular observed bin.