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