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The Baryon Acoustic Oscillations feature (BAO) imprinted in the clusteringcorrelation function is known to furnish us cosmic distance determinations thatare independent of the cosmological-background model and the primordialperturbation parameters. These measurements can be accomplished rigorously bymeans of the Purely Geometric BAO methods. To date two different PurelyGeometric BAO approaches have been proposed. The first exploits thelinear-point standard ruler. The second, called correlation-functionmodel-fitting, exploits the sound-horizon standard ruler. A key differencebetween them is that, when estimated from clustering data, the linear pointmakes use of a cosmological-model-independent procedure to extract the ratio ofthe ruler to the cosmic distance, while the correlation-function model-fittingrelies on a phenomenological cosmological model for the correlation function.Nevertheless the two rulers need to be precisely defined independently of anyspecific observable. We define the linear point and sound horizon and wecharacterize and compare the two rulers' cosmological-parameter dependence. Wefind that they are both geometrical within the required accuracy, and they havethe same parameter dependence for a wide range of parameter values. We estimatethe rulers' best-fit values and errors given the cosmological constraintsobtained by the Planck Satellite team from the CMB measurements. We do this forthree different cosmological models encompassed by the Purely Geometric BAOmethods. In each case we find that the relative errors of the two rulerscoincide and they are insensitive to the assumed cosmological model.Interestingly both the linear point and the sound horizon shift by $0.5\sigma$when we do not fix the spatial geometry to be flat in LCDM. This points towarda sensitivity of the rulers to different cosmological models when they areestimated from the CMB.

Linear Point and Sound Horizon as Purely Geometric standard rulers