**3. The Contribution of BAOs**

The environment of relativistic plasma in the early universe was crossed by the sound waves that were generated by cosmological perturbations. At redshift *z<sup>d</sup>* ∼ 1059.3, which marks the ending of the drag period [341], the recombination of electrons and protons into a neutral gas interrupted the propagation of the sound waves while the photons were able to propagate further [342]. In the period between the formation of the perturbations and the recombination, the different modes produced a sequence of peaks and minima in the anisotropy power spectrum. Given the huge fraction of baryons in the universe, it is expected by cosmological models that the oscillations may affect also the distribution of baryons in the late universe. As a consequence, the BAOs manifest as a local maximum in the correlation function of the galaxies distribution in correspondence of the comoving sound horizon scale at the given redshift *z<sup>d</sup>* , namely *rs*(*z<sup>d</sup>* ): this is associated with the stopping of the propagation of the acoustic waves.

To use the BAOs data for cosmology, we first need to define the following variables:

$$D\_V(z) = \left[\frac{czd\_L^2(z)}{(1+z)^2H(z)}\right]^{1/3}, \qquad d\_z(z) = \frac{r\_s(z\_d)}{D\_V(z)}.\tag{4}$$

The value of the redshift *z<sup>d</sup>* , which corresponds to the drag era ending and marks the decoupling of the photons, allows estimating the sound horizon scale:

$$(r\_d \cdot h)\_{fid} = 104.57 \,\text{Mpc}, \qquad r\_s(z\_d) = \frac{(r\_d \cdot h)\_{fid}}{h},\tag{5}$$

where we use the adimensional ratio *h* = *H*0/100(km s−<sup>1</sup> Mpc−<sup>1</sup> ). To estimate *r<sup>s</sup>* , the following approximated formula [343] can be applied:

$$r\_s \approx \frac{55.154 \cdot e^{-72.3(\omega\_V + 0.0006)^2}}{\omega\_{0m}^{0.25351} \omega\_b^{0.12807}} \text{ Mpc},\tag{6}$$

where *ω<sup>i</sup>* = Ω*<sup>i</sup>* · *h* 2 , and *i* = *m*, *ν*, *b* represent matter, neutrino and baryons. We here assume *ω<sup>ν</sup>* = 0.00064 [344] and *ω<sup>b</sup>* = 0.02237 [10]. Given these quantities, we define the *χ* 2 for BAOs as follows:

$$
\chi^2\_{BAO} = \Delta d^T \cdot \mathcal{M}^{-1} \cdot \Delta d,\tag{7}
$$

where ∆*d* = *d obs z* (*zi*) − *d theo z* (*zi*) and M is the covariance matrix for the BAO *d obs z* (*zi*) values. In this binned analysis, a subset of the 26 BAO observations set available in [341] will be employed.
