@Veropalumbo].

Moreover, to probe the early time evolution of the dark energy it is imperative to investigate the highest redshift available cosmological tracer, that is, the quasars [@McDonald; @White; @Bassett]. The first BAO analyses with quasars, at $2.1 \le z \le 3.5$, used the data release 9 (DR9) of the Baryon Oscillation Spectroscopic Survey, part of the Sloan Digital Sky Survey (SDSS) [@SDSS], to measure the BAO features in the three-dimensional correlation function in the Lyman-$\alpha$ flux fluctuations resulting in constraints on the angular diameter distance, $D_A$, and on the Hubble expansion rate, $H$, at the redshifts $z = 2.3$ [@Busca] and $z = 2.4$ [@Slosar]. More recent studies of the DR11 [@Delubac] and DR12 [@Bautista] quasar catalogues achieved more precise measurements of these quantities at $z = 2.34$ and $z=2.33$, respectively (see also [@Ata]).

A common feature in these analyses is the assumption of a fiducial cosmological model to compute the comoving distance between pairs of astrophysical sources, and then calculate the 2PCF in order to analyze the excess probability of finding pairs at a given separation distance. In this case, the sample contains both the transversal and the radial BAO modes, and the traditional way to quantify the 