*Article* **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**

**Maria Giovanna Dainotti 1,2,3,\* , Biagio De Simone 4,5 , Tiziano Schiavone 6,7 , Giovanni Montani 8,9 , Enrico Rinaldi 10,11,12 , Gaetano Lambiase 4,5 , Malgorzata Bogdan13,14 and Sahil Ugale <sup>15</sup>**


**Abstract:** The difference from 4 to 6 *σ* in the Hubble constant (*H*0) between the values observed with the local (Cepheids and Supernovae Ia, SNe Ia) and the high-z probes (Cosmic Microwave Background 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*) = H0/(1 + *z*) *η* , where H<sup>0</sup> is *H*0(*z* = 0) and *η* is the evolutionary parameter. Here, we investigate if this evolution still holds by using the SNe Ia gathered in the Pantheon sample and the Baryon Acoustic Oscillations. We assume *H*<sup>0</sup> = 70 km s−<sup>1</sup> Mpc−<sup>1</sup> as the local value and divide the Pantheon into three bins ordered in increasing values of redshift. Similar to our previous analysis but varying two cosmological parameters contemporaneously (*H*0, Ω0*<sup>m</sup>* in the ΛCDM model and *H*0, *w<sup>a</sup>* in the *w*0*wa*CDM model), for each bin we implement a Markov-Chain Monte Carlo analysis (MCMC) obtaining the value of *H*<sup>0</sup> assuming Gaussian priors to restrict the parameters spaces to values we expect from our prior knowledge of the current cosmological models and to avoid phantom Dark Energy models with *w* < −1. Subsequently, the values of *<sup>H</sup>*<sup>0</sup> are fitted with the model *<sup>f</sup>*(*z*). Our results show that a decreasing trend with *<sup>η</sup>* <sup>∼</sup> <sup>10</sup>−<sup>2</sup> is still visible in this sample. The *η* coefficient reaches zero in 2.0 *σ* for the ΛCDM model up to 5.8 *σ* for *w*0*wa*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 analysis is meant to further cast light on the evolution of *H*<sup>0</sup> and it does not specifically focus on constraining the other parameters. This work is also a preparatory to understand how the combined probes still show an evolution of the *H*<sup>0</sup> by redshift and what is the current

**Citation:** Dainotti, M.G.; De Simone, B.; Schiavone, T.; Montani, G.; Rinaldi, E.; Lambiase, G.; Bogdan, M.; Ugale, S. 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. *Galaxies* **2022**, *10*, 24. https:// doi.org/10.3390/galaxies10010024

Academic Editor: Elena Moretti and Francesco Longo

Received: 5 November 2021 Accepted: 25 January 2022 Published: 29 January 2022

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status of simulations on GRB cosmology to obtain the uncertainties on the Ω0*<sup>m</sup>* comparable with the ones achieved through SNe Ia.

**Keywords:** supernovae; Ia; cosmology; Hubble; tension; ΛCDM; evolution; modified; gravity; theories
