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Leveraging the Baryon Acoustic Oscillations (BAO) feature present inclustering 2-point statistics, we aim to measure cosmological distancesindependently of the underlying background cosmological model. However thisinference is complicated by late-time non-linearities that introduce model andtracer dependencies in the clustering correlation function and power spectrum,which must be properly accounted for. With this in mind, we introduce the"Purely-Geometric-BAO," which provides a rigorous tool to measure cosmologicaldistances without assuming a specific background cosmology. We focus on the2-point clustering correlation function monopole, and show how to implementsuch an inference scheme employing two different methodologies: the LinearPoint standard ruler (LP) and correlation-function model-fitting (CF-MF). Forthe first time we demonstrate how, by means of the CF-MF, we can measure veryprecisely the sound-horizon/isotropic-volume-distance ratio,$r_{d}/D_{V}(\bar{z})$, while correctly propagating all the uncertainties.Using synthetic data, we compare the outcomes of the two methodologies, andfind that the LP provides up to $50\%$ more precise measurements than theCF-MF. Finally, we test a procedure widely employed in BAO analyses: fittingthe 2-point function while fixing the cosmological and the non-linear-dampingparameters at fiducial values. We find that this underestimates the distanceerrors by nearly a factor of $2$. We thus recommend that this practice bereconsidered, whether for parameter determination or model selection.

Cosmic distance inference from purely geometric BAO methods: Linear Point standard ruler and Correlation Function Model Fitting