**2. SNe Ia Cosmology**

SNe Ia are characterized by an intrinsic luminosity that is almost uniform. Because of this, SNe Ia are considered reliable *standard candles*. We compare the theoretical distance moduli *µth* with the observed distance moduli *µobs* of SNe Ia belonging to the PS. The theoretical distance moduli are defined through the luminosity distance *dL*(*z*) which we need to define based on the cosmological model of interest. We here show the CPL parametrization which describes the *w* parameter as a function of redshift (*w*(*z*) = *w*<sup>0</sup> + *w<sup>a</sup>* × *z*/(1 + *z*)) in the *w*0*wa*CDM model. In the usual assumptions *w*<sup>0</sup> ∼ −1 and *w<sup>a</sup>* ∼ 0, and *dL*(*z*) is defined as the following [285]:

$$d\_{\rm L}(z, H\_{0\cdots}) = \frac{c(1+z)}{H\_0} \int\_0^z \frac{dz^\*}{\sqrt{\Omega\_{0m} \left(1+z^\*\right)^3 + \Omega\_{0\Lambda} \left(1+z^\*\right)^3 \left(w\_0 + w\_d + 1\right)} e^{-3w\_d \frac{z^\*}{1+z^\*}}},\tag{1}$$

where Ω0<sup>Λ</sup> is the Dark Energy component, *c* is the speed of light, and *z* is the redshift. We stress that in this context the relativistic components are ignored. Moreover, since in the present universe the radiation density parameter <sup>Ω</sup>0*<sup>r</sup>* <sup>≈</sup> <sup>10</sup>−<sup>5</sup> , this contribution can be neglected. If we substitute *w<sup>a</sup>* = 0, *w*<sup>0</sup> = −1 in Equation (1) the luminosity distance expression for ΛCDM model is automatically retrieved. According to the distance luminosity expression, the theoretical distance modulus can be written in the following form:

$$
\mu\_{th} = 5 \log\_{10} d\_L(z, H\_{0\prime} \dots) + 25,\tag{2}
$$

which is usually expressed in Megaparsec (Mpc). The observed distance modulus, *µobs* = *m*0 *<sup>B</sup>* − *M*, taken from PS contains the apparent magnitude in the B-band corrected for statistical and systematic effects (*m*0 *B* ) and the absolute in the B-band for a fiducial SN Ia with a null value of stretch and color corrections (*M*). Considering the color and stretch population models for SNe Ia, in our approach we average the distance moduli given by the [286] (G2010) and [287] (C2011) models. We here remind the reader that *H*<sup>0</sup> and *M* are degenerate parameters: in the PS release, *M* = −19.35 such that *H*<sup>0</sup> = 70.0.

Ref. [51] obtain information on *H*<sup>0</sup> by comparing *µobs* in [284] <sup>1</sup> with *µth* for each SN. Moreover, they fix Ω0*<sup>m</sup>* to a fiducial value to better constrain the *H*<sup>0</sup> parameter. Furthermore, according to [288], we consider the correction of the luminosity distance keeping into account the peculiar velocities of the host galaxies which contain the SNe Ia. To perform our analysis, we define the *χ* 2 for SNe:

$$
\chi\_{\rm SN}^2 = \Delta \boldsymbol{\mu}^T \cdot \mathcal{C}^{-1} \cdot \Delta \boldsymbol{\mu}. \tag{3}
$$

Here ∆*µ* = *µobs* − *µth*, and C denotes the 1048 × 1048 *covariance matrix*, given by [284]. As for the *µobs* values of G2010 and C2011, the systematic uncertainty matrices of the two models have been averaged. After building the C total matrix from Equation (16) in [51], we slice the PS in redshift bins, and then we divide C into submatrices considering the order in redshift. More in detail, starting from the 1048 SNe Ia redshift-ordering, we divide the SNe Ia into 3 equally populated bins made up of ≈349 SNe Ia. Concerning only *Dstat*, it is trivial to build its submatrices considering that the statistical matrix is diagonal. Hence, a single matrix element is related to a given SN of the PS. On the other hand, if the non-diagonal matrix *Csys* is included, a customized code will be used<sup>2</sup> to build the submatrices. Our code was developed to select only the total covariance matrix elements related to SNe Ia having redshift within the considered bin.

The choice of three bins is justified by the high number of SNe Ia (around hundreds of SNe per bin) that can still constitute statistically illustrative subsamples of the PS and that can properly consider the contribution of systematic uncertainties. Subsequently to the bins division, we focus on the optimal values of *H*<sup>0</sup> to minimize the *χ* 2 in Equation (3). *H*<sup>0</sup> is regarded as a nuisance parameter, which is free to vary, to better analyze a possible redshift function of *H*0. We follow the assumptions on the fiducial value of *M* = −19.35: while in [51] *M* was estimated assuming a local (*z* = 0) value of *H*<sup>0</sup> = 73.5, we here consider the conventional *H*<sup>0</sup> value of the PS release, namely *H*<sup>0</sup> = 70.0 for three bins. Our choice of a starting value of *H*<sup>0</sup> = 70 is dictated by the presence in the current literature of more than 50 papers that are using the PS in combination with other probes to estimate the value of *H*0, see [172,198,204,205,288–307,307–339]. Thus, if an evolutionary effect is present, it is necessary to investigate to which extent this can affect current and future results largely based on the PS sample. Conversely, we fix Ω0*<sup>m</sup>* = 0.298 ± 0.022 according to [284] for a standard flat ΛCDM model. More specifically, after the minimization of *χ* 2 , we extract the *H*<sup>0</sup> value in each redshift bin, via the *Cobaya* code [340]. To this end, we execute an MCMC using the D'Agostini method to obtain the confidence intervals for *H*<sup>0</sup> at the 68% and 95% levels, in three bins.
