  The Baryon Acoustic Oscillations feature (BAO) imprinted in the clustering
correlation function is known to furnish us cosmic distance determinations that
are independent of the cosmological-background model and the primordial
perturbation parameters. These measurements can be accomplished rigorously by
means of the Purely Geometric BAO methods. To date two different Purely
Geometric BAO approaches have been proposed. The first exploits the
linear-point standard ruler. The second, called correlation-function
model-fitting, exploits the sound-horizon standard ruler. A key difference
between them is that, when estimated from clustering data, the linear point
makes use of a cosmological-model-independent procedure to extract the ratio of
the ruler to the cosmic distance, while the correlation-function model-fitting
relies on a phenomenological cosmological model for the correlation function.
Nevertheless the two rulers need to be precisely defined independently of any
specific observable. We define the linear point and sound horizon and we
characterize and compare the two rulers' cosmological-parameter dependence. We
find that they are both geometrical within the required accuracy, and they have
the same parameter dependence for a wide range of parameter values. We estimate
the rulers' best-fit values and errors given the cosmological constraints
obtained by the Planck Satellite team from the CMB measurements. We do this for
three different cosmological models encompassed by the Purely Geometric BAO
methods. In each case we find that the relative errors of the two rulers
coincide 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 toward
a sensitivity of the rulers to different cosmological models when they are
estimated from the CMB.
