A New Consistency Test for the $\Lambda$CDM Model using Radial and Transverse BAO Measurements

We present a calibration-free consistency test of spatially flat $\Lambda$CDM based on baryon acoustic oscillation (BAO) distance measurements. The method forms ratios of BAO distances including the Hubble distance, the comoving angular diameter distance, and the volume-averaged distance, so that the sound horizon scale cancels, and then maps each observed ratio to an effective flat-$\Lambda$CDM matter density parameter, ${\Omega}_{\rm M}^{\Lambda}$, defined as the value of $\Omega_{\rm M}$ that reproduces the measured ratio within $\Lambda$CDM. Flat $\Lambda$CDM predicts that ${\Omega}_{\rm M}^{\Lambda}$ should be independent of redshift and of the particular ratio used. For ratios involving the integrated distances, we associate them with well-defined effective line-of-sight redshift intervals based on the integral mean value theorem. We apply the test to BAO measurements from the Dark Energy Spectroscopic Instrument (DESI) Data Release~1 and Data Release~2, propagating the full published BAO covariance matrices into all derived ratios and ${\Omega}_{\rm M}^{\Lambda}$ constraints. Within current uncertainties, the inferred ${\Omega}_{\rm M}^{\Lambda}$ values are broadly consistent with a redshift-independent constant, providing an internal consistency check of flat $\Lambda$CDM that can be strengthened straightforwardly as BAO measurements improve.