We further investigate the dark energy model based on the Finsler geometry inspired osculating Barthel–Kropina cosmology. The Barthel–Kropina cosmological approach is based on the introduction of a Barthel connection in an osculating Finsler geometry, with the connection having the property that it is the Levi-Civita connection of a Riemannian metric. From the generalized Friedmann equations of the Barthel–Kropina model, obtained by assuming that the background Riemannian metric is of the Friedmann–Lemaitre–Robertson–Walker type, an effective geometric dark energy component can be generated, with the effective, geometric type pressure, satisfying a linear barotropic type equation of state. The cosmological tests, and comparisons with observational data of this dark energy model are considered in detail. To constrain the Barthel–Kropina model parameters, and the parameter of the equation of state, we use 57 Hubble data points, and the Pantheon Supernovae Type Ia data sample. The st statistical analysis is performed by using Markov Chain Monte Carlo (MCMC) simulations. A detailed comparison with the standard $$\Lambda $$Λ CDM model is also performed, with the Akaike information criterion (AIC), and the Bayesian information criterion (BIC) used as the two model selection tools. The statefinder diagnostics consisting of jerk and snap parameters, and theOm(z) diagnostics are also considered for the comparative study of the Barthel–Kropina and $$\Lambda $$Λ CDM cosmologies. Our results indicate that the Barthel–Kropina dark energy model gives a good description of the observational data, and thus it can be considered a viable alternative of the $$\Lambda $$Λ CDM model.

Delta-gravity (DG) is a gravitational model based on an extension of general relativity given by a new symmetry called . In this model, new matter fields are added to the original matter fields, motivated by the additional symmetry. We call them matter fields. This model predicts an accelerating universe without the need to introduce a cosmological constant. In this work, we study the scalar cosmic microwave background (CMB) temperature (TT) power spectrum predicted by DG using an analytical hydrodynamic approach. To fit the Planck satellite’s data with the DG model, we used a Markov Chain Monte Carlo analysis. We also include a study about the compatibility between Type Ia supernovae (SNe Ia) and CMB observations in the DG context. Finally, we obtain the scalar CMB TT power spectrum and the fitted parameters needed to explain both SN Ia data and CMB measurements. The results are in reasonable agreement with both observations considering the analytical approximation. We also discuss whether the Hubble constant and the accelerating universe are in concordance with the observational evidence in the DG context.

A gravitational field model based on two symmetric tensors, $g_{\mu \nu }$ and ${\tilde{g}}_{\mu \nu }$ , is studied, using a Markov Chain Monte Carlo (MCMC) analysis with the most updated catalog of SN-Ia. In this model, new matter fields are added to the original matter fields, motivated by an additional symmetry ( $\tilde{\delta }$ symmetry). We call them $\tilde{\delta }$ matter fields. This theory predicts an accelerating Universe without the need to introduce a cosmological constant $\mathsf{\Lambda }$ by hand in the equations. We obtained a very good fit to the SN-Ia Data, and with this, we found the two free parameters of the theory called C and $L_2$ . With these values, we have fixed all the degrees of freedom in the model. The last $H_0$ local value measurement is in tension with the CMB Data from Planck. Based on an absolute magnitude $M_V=-19.23$ for the SN, Delta Gravity finds $H_0$ to be $74.47\pm 1.63$ km/(s Mpc). This value is in concordance with the last measurement of the $H_0$ local value, $73.83\pm 1.48$ km/(s Mpc).

We introduce a modification of the Press–Schechter formalism aimed to derive general mass functions for primordial black holes (PBHs). In this case, we start from primordial power spectra (PPS) which include a monochromatic spike, typical of ultra slow-roll inflation models. We consider the PBH formation as being associated to the amplitude of the spike on top of the linear energy density fluctuations coming from a PPS with a blue index. By modelling the spike with a lognormal function, we study the properties of the resulting mass function spikes, and compare these to the underlying extended mass distributions. When the spike is at PBH masses, which are much lower than the exponential cut-off of the extended distribution, very little mass density is held by the PBHs within the spike, and it is not ideal to apply the Press–Schechter formalism in this case as the resulting characteristic overdensity is too different from the threshold for collapse. It is more appropriate to do so when the spike mass is similar to, or larger than the cut-off mass. Additionally, it can hold a similar mass density as the extended part. Such particular mass functions also contain large numbers of small PBHs, especially if stable PBH relics are considered, and they can provide ${\sim}1000\, {\rm M}_\odot$ seeds for the supermassive black holes at the centres of present-day galaxies. The constraints on the fraction of dark matter in PBHs for monochromatic mass functions are somewhat relaxed when there is an additional underlying extended distribution of masses.