*5.3. BAO Measurements*

The BAO measurements are due to overdensity of baryonic matter due to acoustic waves. These waves propagate in the early universe [119,120] and represent the *standard ruler* for cosmological length scale. This signature, in the large-scale clustering of galaxies, constrains cosmological parameters by detection of a peak in the correlation function [121], by defining the *A* parameter as follows:

$$A = \frac{\sqrt{\Omega\_m}}{z\_1} \left[ \frac{z\_1}{E(\mathbf{x}, z\_1)} \frac{1}{|\Omega\_k|} \text{sinc}^2 \left( \sqrt{|\Omega\_k|} \int\_0^{z\_1} \frac{dz}{E(\mathbf{x}, z)} \right) \right]^{\frac{1}{5}} \tag{27}$$

where **x** is the set of cosmological density parameters, *E*(**x**, *z*) = *H*(**x**, *z*)/*H*0, and sinn(*x*) = sinh(*x*) for the curvature parameter Ω*<sup>k</sup>* > 0, sinn(*x*) = *x* for Ω*<sup>k</sup>* = 0, and sinn(*x*) = sin(*x*) for Ω*<sup>k</sup>* < 0. The *A* parameter has been measured from the SDSS data and reads to be *A* = 0.469(0.95/0.98) <sup>−</sup>0.35 <sup>±</sup> 0.017, with *<sup>z</sup>*<sup>1</sup> <sup>=</sup> 0.35, so the *<sup>χ</sup>* 2 in terms of *A* reads *χ* 2 BAO = (*A* − 0.469) <sup>2</sup>/0.017<sup>2</sup> . The BAO corresponding angular distance measures can be defined by means of

$$d\_{\mathbf{z}}(\mathbf{x}, z) \equiv r\_{\mathbf{s}}(z\_{\mathbf{d}}) \left[ \frac{cz}{H(\mathbf{x}, z)} \right]^{-1/3} \left[ \frac{d\_{\mathbf{L}}(\mathbf{x}, z)}{1 + z} \right]^{-2/3}. \tag{28}$$

The corresponding *χ* 2 is given by

$$\chi^2\_{\rm BAO} = \sum\_{i=1}^{N\_{\rm BAO}} \left[ \frac{d\_{\rm z}^{\rm th}(\mathbf{x}\_i z\_i) - d\_{\rm z,i}^{\rm obs}}{\sigma\_{d\_{\rm z,i}}} \right]^2 \,. \tag{29}$$

It is clear that BAO measures are slightly model-dependent as they depend on the comoving sound horizon *r*s(*z*d). In particular, in Equation (28), the sound horizon depends upon the baryon drag redshift *z*d. This quantity requires calibration that typically is performed with CMB data, adopting a given background model that commonly is the ΛCDM scenario. Very often, the best expected values are given by *z*<sup>d</sup> = 1059.62 ± 0.31 and *r*s(*z*d) = 147.41 ± 0.30 [122].
