The Hubble constant (H ₀) tension between Type Ia supernovae (SNe Ia) and Planck measurements ranges from 4 to 6σ. To investigate this tension, we estimate H ₀ in the ΛCDM and w 0 w a CDM (cold dark matter) models by dividing the Pantheon sample, the largest compilation of SNe Ia, into 3, 4, 20, and 40 bins. We fit the extracted H ₀ values with a function mimicking the redshift evolution: g ( z ) = H 0 ( z ) = H ˜ 0 / ( 1 + z ) α , where α indicates an evolutionary parameter and H ˜ 0 = H 0 at z = 0. We set the absolute magnitude of SNe Ia so that H 0 = 73.5 km s − 1 Mpc − 1 , and we fix fiducial values for Ω 0 m Λ CDM = 0.298 and Ω 0 m w 0 w a CDM = 0.308 . We find that H ₀ evolves with redshift, showing a slowly decreasing trend, with α coefficients consistent with zero only from 1.2 to 2.0σ. Although the α coefficients are compatible with zero in 3σ, this however may affect cosmological results. We measure locally a variation of H 0 ( z = 0 ) − H 0 ( z = 1 ) = 0.4 km s − 1 Mpc − 1 in three and four bins. Extrapolating H 0 ( z ) to z = 1100, the redshift of the last scattering surface, we obtain values of H ₀ compatible in 1σ with Planck measurements independent of the cosmological models and number of bins we investigated. Thus, we have reduced the H ₀ tension in the range from 54% to 72% for both cosmological models. If the decreasing trend of H 0 ( z ) is real, it could be due to astrophysical selection effects or to modified gravity.

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 (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 $f\left(z\right)={\mathcal{H}}_0/{(1+z)}^{\eta }$, 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 Baryon Acoustic Oscillations. We assume $H_0=70{\mathrm{km}\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$ 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 ($H_0$, ${\Omega }_{0m}$ in the $\Lambda$CDM model and $H_0$, $w_a$ in the $w_0{w}_a$CDM model), for each bin we implement a Markov-Chain Monte Carlo analysis (MCMC) obtaining the value of $H_0$ 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 $H_0$ are fitted with the model $f\left(z\right)$. Our results show that a decreasing trend with $\eta \sim {10}^{-2}$ 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, $f\left(R\right)$. This analysis is meant to further cast light on the evolution of $H_0$ 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 $H_0$ 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.

We analyze the propagation of gravitational waves in metric $f(R)$ theories of gravity, on the special setting of flat background geometry (Minkowski spacetime). In particular, adopting a gauge invariant formalism, we clearly establish that the exact number of propagating degrees of freedom is three, consisting of the standard tensorial modes along with an additional massive scalar field. Then, investigating their effects on test masses via the geodesic deviation equation, we show that the additional dynamical degree contained in such extended formulations is actually detectable as a superposition of longitudinal and breathing stresses, which even though in principle correspond to distinct pure polarizations turn out to be never separable in the wave dynamics and cannot be interpreted as a proper independent excitations.

We extend the notion of the Nieh-Yan invariant to generic metric-affine geometries, where both torsion and nonmetricity are taken into account. Notably, we show that the properties of projective invariance and topologicity can be independently accommodated by a suitable choice of the parameters featuring this new Nieh-Yan term. We then consider a special class of modified theories of gravity able to promote the Immirzi parameter to a dynamical scalar field coupled to the Nieh-Yan form, and we discuss in more detail the dynamics of the effective scalar tensor theory stemming from such a revised theoretical framework. We focus, in particular, on cosmological Bianchi I models and we derive classical solutions where the initial singularity is safely removed in favor of a big bounce, which is ultimately driven by the nonminimal coupling with the Immirzi field. These solutions, moreover, turn out to be characterized by finite time singularities, but we show that such critical points do not spoil the geodesic completeness and wave regularity of these spacetimes.

We analyze the dynamics of the mixmaster universe on the basis of a standard Arnowitt-Deser-Misner Hamiltonian approach showing how its asymptotic evolution to the cosmological singularity is isomorphic to a billiard ball on the Lobachevsky plane. The key result of our study consists in the temporary gauge invariance of the billiard ball representation, once provided the use of very general Misner-Chitré-like variables.

The implications of a generalized Uncertainty principle on the Taub cosmological model are investigated. The model is studied in the Arnowitt-Deser-Misner reduction of the dynamics and therefore a time variable is ruled out. Such a variable is quantized in a canonical way and the only physical degree of freedom of the system (related to the universe anisotropy) is quantized by means of a modified Heisenberg algebra. The analysis is performed at both the classical and quantum level. In particular, at quantum level, the motion of wave packets is investigated. The two main results obtained are as follows: (i) The classical singularity is probabilistically suppressed. The universe exhibits a stationary behavior and the probability amplitude is peaked in a determinate region. (ii) The generalized uncertainty principle wave packets provide the right behavior in the establishment of a quasi-isotropic configuration for the universe.