A blind spot in transverse BAO calibration

Abstract: Transverse baryon acoustic oscillation (BAO) measurements are increasingly used for cosmological inference, and carry a calibration that no such inference can constrain. A constant error in the transverse BAO scale is exactly degenerate with the combination $r_{\rm d} h$: it leaves the goodness of fit unchanged and the recovered parameters plausible, and is therefore invisible to any analysis that uses these measurements alone. The radial BAO sector removes this degeneracy, supplying $D_{\rm M}/r_{\rm d}$ by integration without reference to $H_0$ or to any model for the expansion rate. In flat Friedmann--Lemaître--Robertson--Walker (FLRW) geometry the relation between the two sectors is an identity, so the sound horizon and the dark-energy equation of state cancel as well. An integrated form of this identity reduces the consistency test to a straight line, whose slope measures a relative transverse calibration $\varepsilon$. A departure from $\varepsilon = 1$ cannot be produced by any dark-energy model, nor by spatial curvature: it indicates an inconsistency in the measurement chain rather than in the cosmology. We apply the test to the two main SDSS transverse BAO compilations, which give $\varepsilon = 1.073 \pm 0.021$ and $1.021 \pm 0.029$. The first differs from unity at $3.8σ$ using the published independent errors, while the second is consistent with unity. The two compilations themselves differ by $(5.4 \pm 1.4)\%$, or $3.9σ$, and the offset is constant in redshift. The test provides a direct diagnostic for current and future angular BAO measurements.