On the Evolution of the Hubble Constant with the SNe Ia Pantheon Sample and Baryon Acoustic Oscillations: A Feasibility Study for GRB-Cosmology in 2030

The difference from 4 to 6 $\sigma$ in the Hubble constant ($H_0$) between values observed with local (Cepheids and Supernovae Ia, SNe Ia) high-z probes (CMB obtained by Planck data) still challenges astrophysics cosmology community. Previous analysis has shown that there is an evolution scales as $f(z)=\mathcal{H}_{0}/(1+z)^{\eta}$, where $\mathcal{H}_0$ $H_{0}(z=0)$ $\eta$ evolutionary parameter. Here, we investigate if this holds using Ia gathered Pantheon sample BAOs. We assume $H_{0}=70 \textrm{km s}^{-1}\textrm{Mpc}^{-1}$ value divide into 3 bins ordered increasing of redshift. Similar our previous but varying two cosmological parameters contemporaneously ($H_0$, $\Omega_{0m}$ $\Lambda$CDM model $H_0$, $w_a$ $w_{0}w_{a}$CDM model), for each bin implement a MCMC obtaining $H_0$ $[...]$. Subsequently, are fitted $f(z)$. Our results show decreasing trend $\eta \sim10^{-2}$ visible sample. coefficient reaches zero 2.0 up 5.8 model. This trend, not due statistical fluctuations, could be explained through hidden astrophysical bias, such effect stretch evolution, or it requires new theoretical models, possible proposition modified gravity theories, $f(R)$ work also preparatory understand how combined redshift what current status simulations on GRB obtain uncertainties comparable ones achieved Ia.