**1. Introduction**

The ΛCDM model is one of the most accredited models, which implies an accelerated expansion phase [1,2]. Although it represents the favored paradigm, it is affected by great challenges: the fine-tuning, the coincidence [3,4], and the Dark Energy nature's problems.

More importantly, the *H*<sup>0</sup> tension represents a big challenge for modern cosmology. Indeed, the 4.4 up to 6.2 *σ* discrepancy, depending on the sample used [5–7], between the local value of *H*<sup>0</sup> obtained with Cepheids observations and SNe Ia, *<sup>H</sup>*<sup>0</sup> <sup>=</sup> 74.03 <sup>±</sup> 1.42 km s−<sup>1</sup> Mpc−<sup>1</sup> [8,9], and the Planck data of Cosmic Microwave background radiation (CMB), *<sup>H</sup>*<sup>0</sup> <sup>=</sup> 67.4 <sup>±</sup> 0.5 km s−<sup>1</sup> Mpc−<sup>1</sup> from the Planck Collaboration [10] requires further investigation. From now on, *H*<sup>0</sup> will be in the units km s−<sup>1</sup> Mpc−<sup>1</sup> .

We stress that other probes report values of *H*<sup>0</sup> ≈ 72 ± 2, similar to the value obtained with the SNe Ia. Surely, to solve the Hubble tension it is necessary to use probes that are standard candles. SNe Ia, considered one of the best standard candles, are observed only up to a low redshift range: the farthest so far discovered is at *z* = 2.26 [11].

It is important for studying the evolution of the cosmological parameters to investigate probes at high redshift. One of the best candidates in this regard is represented by the Gamma-ray Bursts (GRBs).

GRBs are observed up to cosmological redshifts (the actual record is of *z* = 9.4 [12]) and surpassed even the quasars (the most distant being at *z* = 7.64 [13]). Due to their detectability at high redshift, GRBs allow extending the current Hubble diagram to new redshift ranges [14–18].

Indeed, it is important to stress that once we have established if the Hubble constant undergoes redshift evolution, the Pantheon sample can safely be combined with other probes. Surely the advantage of the use of the SNe Ia is that their emission mechanism is pretty clear, namely they originate from the thermonuclear explosion of carbon–oxygen white dwarfs (C/O WDs).

For GRBs, more investigation about their progenitor mechanism is needed. We here stress that this work can be also preparatory to the work of future application of GRBs as cosmological tools together with SNe Ia and Baryon Acoustic Oscillations (BAOs) through well-established correlations among the prompt variables, such as: the Amati relation [19], which connects the peak in the *νF<sup>ν</sup>* spectrum to the isotropic energy in the prompt emission (*Eiso*), the Yonetoku relation [20,21] between *Epeak* and the peak luminosity of the prompt emission, *Lpeak*, the Liang and Zhang relation [22] between *Eiso*, the rest-frame break time of the GRB *t* 0 *b* and the peak energy spectrum in the rest frame *E* 0 *p* , the Ghirlanda relation (*Epeak* − *Ejet* = *Eiso* × (1 − *cosθ*)) [23], and the prompt-afterglow relations for the GRBs with the plateau emission investigated in [24–38], which have as common emission mechanism most likely the magnetar model, where a neutron star with an intense magnetic field undergoes a fast-spinning down [39–43].

A feasibility study shows that GRBs can give relevant constraints on the cosmological parameters [17,44]. We here give a list of examples of other probes used for measuring the Hubble constant tension. One of them is the use of data from time-delay measurements and strong lens systems [45,46]. On the contrary, additional probes carry similar values of *H*<sup>0</sup> to the ones of Planck, based on the Cosmic Chronometers (CC) (*H*<sup>0</sup> = 67.06 ± 1.68) in [8]. Besides, there is a series of independent probes, such as quasars [47], the Tip of the Red-Giant Branch (TRGB) calibration through SNe Ia [48], and also GRBs [14,15,17,18,49,50], which bring estimates of *H*<sup>0</sup> ranging between the values obtained with local measurements (SNe Ia and Cepheids) and Baryon Acoustic Oscillations (BAO)+CMB. Ref. [51] discuss possible

reasons behind the *H*<sup>0</sup> tension in the Pantheon sample: selection biases of parameters of the SNe Ia, unknown systematics, internal inconsistencies in the Planck data, or alternative theoretical interpretations compared to the standard cosmological model. Furthermore, the use of type 1 Active Galactic Nuclei (AGN) represents another promising cosmological probe given the peculiarity of their spectral emission [52].

To date, a wide range of different solutions to the Hubble constant tension has been provided by several groups [53–86]. Concerning the observational solutions, we here detail a series of proposals [87–130]. In [131], the simulations of data taken from the anomalously fast-colliding El Gordo galaxy clusters allow discussing the probability of observing such a scenario in a ΛCDM framework. Ref. [132] perform a re-calibration of Cepheids in NGC 5584, thus obtaining a relation between the periods of Cepheids and their amplitude ratios (tighter than the one obtained in SH0ES [9]) which could be useful to better estimate the value of *H*0. In [133], the UV and X-ray data coming from quasars are used to constrain *H*<sup>0</sup> in the Finslerian cosmology. Ref. [123] demonstrate that the Planetary Nebula Luminosity Function (PNLF) can be extended beyond the Cepheid distances, thus promoting it to be an additive probe for constraining *H*0. In [134], the analysis of Pan-STARRS telescope SNe Ia data provides a value of *H*<sup>0</sup> which lies between the SH0ES and Planck values.

Ref. [135] investigated how the *H*<sup>0</sup> measurements can depend on the choice of different probes (SNe, BAO, Cepheid, CC, etc.), showing also that through the set of filters on cosmological models, such as fiducial values for cosmological parameters (*w* = −1, with *w* parameter for the equation of state, or Ω*<sup>k</sup>* = 0, namely the curvature parameter set to zero), the tension can be alleviated.

Ref. [136] extended a Hubble diagram up to redshift *z* ∼ 8 combining galaxies and high-redshift quasars to test the late-time cosmic expansion history, giving a constraint on the upper-value of *H*<sup>0</sup> which is only marginally consistent with the results obtained by the Cepheids.

Ref. [137] further tests the *w*CDM (with varying parameters of the equation of state), and oCDM models (with varying curvature) through the merging of BAOs, SNe Ia, CC, GRBs, and quasars data, after the analysis of the standard ΛCDM model.

In [138], the combination of strongly lensed quasars and SNe Ia led the authors to conclude that the solution to the tension should be found outside of the Friedmann– Lemaitre–Robertson–Walker metric.

Refs. [125,139] detect in the Stochastic Gravitational Wave Background a new method to alleviate the tension, while [140,141] focuses on the gravitational-wave signals from compact star mergers as probes that can give constraints on the *H*<sup>0</sup> value.

Ref. [142] combine the SNe Ia and the VLT-KMOS HII galaxies data to put new constraints on the cosmokinetic parameters. The proposed solutions deal also with models that are alternative to the standard ΛCDM or, in other cases, that can extend it.

Ref. [143] constrain the Brans–Dicke (BD) theory through CMB and BAOs. The TRGB method, combined with SNe Ia, gives a value of *H*<sup>0</sup> compatible with the one from CMB [144].

Ref. [145] obtain an 8%-precise value of *H*<sup>0</sup> through the Fast Radio Bursts (FRB).

In [146], the Cepheids calibration parameters are allowed to vary, thus leading to an estimated value of *H*<sup>0</sup> which is compatible with the CMB one. The possibility that the Solar system's proper motion may induce a bias in the measurement of *H*<sup>0</sup> has been subject to study in [147], finding out that there is no degeneracy between the cosmological parameters and the parameters of the Solar system motion.

Ref. [148] measure *H*<sup>0</sup> through the galaxies parallax having as reference the CMB rest-frame, being this parallax caused by the peculiar motions.

Ref. [149] verified through the measurements on GRBs and quasars that the Hubble constant has a bigger value in the sky directions aligned with the CMB dipole polarization, suggesting that a detachment from the FLRW should be considered.

Refs. [150–153] investigate how the dark sirens producing gravitational waves could help to probe *H*0. Despite being a promising method, the incompleteness of galaxy catalogs may hinder the outcome of this method, thus [154] proposes a pixelated-sky approach to overcome the issue of event redshifts which are missing but may be retrieved through the galaxies present on the line of sight.

A review of the most promising emerging probes to measure the Hubble constant can be found in [155].

Recent results on the measurements of the Hubble parameter and constant through the Third LIGO-Virgo-KAGRA Gravitational-Wave Transient Catalog (GWTC-3) can be found in [156]. An evolving trend for *H*<sup>0</sup> may be naturally predicted in Teleparallelism [157–161], as well as in modified gravity theories [162–167]. Refs. [168–170] study the *f*(*Q*, *T*) models in Teleparallel Gravity through CC and SNe Ia, thus obtaining a value of *H*<sup>0</sup> compatible in 1 *σ* with the SH0ES result. The linear theory of perturbation for the *f*(*Q*, *T*) theory is investigated in [171], allowing the future tests of this model through CMB data.

In [172], the Unimodular Gravity model is constrained with Planck 2018 [10], SH0ES, SNe Ia, and H0LiCOW collaboration [7]. Furthermore, the Axi–Higgs model is tested with CMB, BAO, Weak Lensing data (WL), and SNe [173]: in another paper, it is shown how this model relaxes the Hubble tension [173]. Ref. [174] describe the modified inflationary models considering constant-roll inflation. Ref. [175] give boundaries on the Hubble constant value with the gravitino mass conjecture.

Refs. [176–180] show the role of cosmological second-order perturbations of the flat ΛCDM model in the *H*<sup>0</sup> tension. Ref. [181] discuss how Dark Energy may be generated by quantum fluctuations of an inflating field and how the Hubble tension may be reduced by the spatial correlations induced by this effect. The Dark Energy itself may be subject to evolution, as pointed out in [182]. Ref. [183] show how a modification of the Friedmann equation may naturally explain the inconsistency between the local and the cosmological measurements of the Hubble constant.

Ref. [184] explain how the search for low-frequency gravitational waves (GWs) justifies the Hubble tension's solution through the assumption of neutrino-dark sector interactions. Ref. [185] show how the *R* (∗) *K* anomalies (namely, the discrepancy between the theoretical ratio of the fractions *B* → *K* ∗*µ* <sup>+</sup>*µ* <sup>−</sup>/*B* → *K* ∗ *e* +*e* − for the dilepton invariant mass bins from the Standard Model and the observed one, see [186]) and the *H*<sup>0</sup> tension can be solved by Dirac neutrinos in a two-Higgs-doublet theory.

The introduction of models where the cosmological axio-dilation is present may lead to a solution of the Hubble tension [187].

Refs. [188–191] discuss how the Early Dark Energy models (EDE) can be used to alleviate the *H*<sup>0</sup> tension. Ref. [192] analyze how the phantom Dark Energy models can give a limited reduction of the *H*<sup>0</sup> tension, while [193] explore how the Kaniadakis holographic Dark Energy model alleviates the *H*<sup>0</sup> tension.

In [194], the Viscous Generalized Chaplygin Gas (VGCG) model is used to diminish the Hubble tension. The holographic Dark Energy models are pointed as a possible solution through the study of unparticle cosmology [195].

Ref. [196] test seven cosmological models through the constraints of SNe Ia, BAO, CMB, Planck lensing, and Cosmic Chronometers with the outcome that in the ΛCDM scenario a flat universe is favored.

Ref. [197] discuss how the new physical scenarios before the recombination epoch imply the shift of cosmological parameters and how these shifts are related to the discrepancy between the local and non-local values of *H*0.

Ref. [198] proposes that the *H*<sup>0</sup> tension may be solved if the speed of light is treated as a function of the scale factor (as in [199]), and applies this scenario to SNe Ia data.

Refs. [200,201] discuss the implementation of the alternative Phenomenologically Emergent Dark Energy model (PEDE), which can be also extended to a Generalized Emergent Dark Energy model (GEDE) with the addition of an extra free parameter. This shows the possibility of obtaining the PEDE or the ΛCDM cosmology as sub-cases of the GEDE scenario.

Ref. [202] consider a scenario of modified gravity predicting the increase of the expansion rate in the late-universe, thus proving that in this scenario the Hubble tension reduces significantly.

Ref. [203] study the ΛCDM model constrained, at the early-time universe, by the presence of the early Integrated Sachs–Wolfe (eISW) effect, proving that the early-time models aimed at attenuating the Hubble tension should be able to reproduce the same eISW effect just like the ΛCDM does. The observations of a locally higher value for *H*<sup>0</sup> led to the discussion of local measurements, constraints, and modeling. In this regards, the assumption of a local void [204–206] may produce locally an increased value for *H*0.

The Universe appears locally inhomogeneous below a scale of roughly 100 Mpc. The question some cosmologists are attempting to solve is whether local inhomogeneities have impact on cosmological measurements and the Hubble diagram. Many observables are related to photons paths, which may be directly affected by the matter distribution. Many theoretical attempts were made during the last few decades to develop the necessary average prescription to evaluate the photon propagation on the observer's past light cone based on covariant and gauge-invariant observables [207–210]. Local inhomogeneities and cosmic structure cause scattering and bias effects in the Hubble diagram, which are due to peculiar velocities, selection effects, and gravitational lensing, but also to non-linear relativistic corrections [210–212]. This question was addressed in [213] utilising the N-body simulation of cosmic structure formation through the numerical code *gevolution*. This non-perturbative approach pointed out discrepancies in the luminosity distance between a homogeneous and inhomogeneous scenario, showing, in particular, the presence of non-Gaussian effects at higher redshifts. These studies related to distance indicators will become even more significant considering the large number of the forthcoming surveys designed to the observations on the Large Scale Structure of the Universe in the next decade (for instance, the Euclid survey [214,215] and the Vera C. Rubin Observatory's LSST [216]). The effect of local structures in an inhomogeneous universe should be considered in the locally measured value of *H*<sup>0</sup> [217,218]. The local under-density interpretation was also studied in Milgromian dynamics [131,219,220], but in [221] it is shown how this interpretation does not solve the tension. Ref. [219] study the KBC local void which is in contrast with the ΛCDM, thus proposing the Milgromian dynamics as an alternative to standard cosmology. Milgromian dynamics are studied also in [222] where, through the galactic structures and clusters, it is shown how this model can be consistent at different scales and alleviate the Hubble tension.

Ref. [223] describe the late time approaches and their effect on the Hubble parameter. The bulk viscosity of the universe is also considered the link between the early and late universe values of *H*<sup>0</sup> [224].

Ref. [225] explain how the local measurements over-constrain the cosmological models and propose the graphical analysis of the impact that these constraints have on the *H*<sup>0</sup> estimation through ad hoc triangular plots.

Refs. [220,226,227] describes the effects of inhomogeneities at small scales in the baryon density. Ref. [228] find out that the late time modifications can solve the tension between the *H*<sup>0</sup> SH0ES and CMB values through a parametrization of the comoving distance.

Ref. [229] propose to alleviate the Hubble tension considering an abrupt modification of the effective gravitational constant at redshift *z* ≈ 0.01.

Other proposals are focused on the existence of different approaches.

Refs. [230,231] show how *H*<sup>0</sup> evolves with redshift at local scales.

Ref. [232] discuss how the breakdown of Friedmann–Lemaitre–Robertson–Walker (FLRW [233]) may be a plausible assumption to alleviate the Hubble tension. Ref. [234] investigate the binary neutron stars mergers and, with the analysis of simulated catalogs, show their potential to help to alleviate the *H*<sup>0</sup> tension.

Ref. [235] explain how Gaussian process (GP) and locally weighted scatter plot smoothing are used in conjunction with simulation and extrapolation (LOESS-Simex) methods can reproduce different sets of data with a high level of precision, thus giving new perspectives

on the Hubble tension through the simulation of Cosmic Chronometers, SNe Ia, and BAOs data sets.

Ref. [236] focus on the GP and state the necessity of lower and upper bounds on the hyperparameters to obtain a reliable estimation of *H*0.

On the other hand, Ref. [237] suggested a novel approach to measure *H*<sup>0</sup> based on the distance duality relation, namely a method that connects the luminosity distance of a source to its angular diameter. In this case, data do not require a calibration phase and the relative constraints are not dependent on the underlying cosmological model.

Ref. [238] showed how the tension can be solved with a modified weak-field General Relativity theory, thus defining a local *H*<sup>0</sup> and a global *H*<sup>0</sup> value.

Ref. [239] investigated how a specific Dark Energy model in the generalized Proca theory can alleviate the tension.

In [240], the Horndeski model can describe with significantly good precision the late expansion of the universe thanks to the Hubble parameter data. The same model is considered promising for the solution of the *H*<sup>0</sup> tension in [241].

Ref. [242] described how the transition observed in Tully–Fisher data could imply an evolving gravitational strength and explain the tension.

Ref. [197] explain how the physical models of the pre-recombination era could cause the observed *H*<sup>0</sup> values discrepancy and suggest that if the local *H*<sup>0</sup> measurements are consistent then a scale-invariant Harrison–Zeldovich spectrum should be considered to solve the *H*<sup>0</sup> issue. The Dynamical Dark Energy (DDE) models are the object of study in [243,244]: in the former, the DDE is proposed as an alternative to ΛCDM, while in the latter it is shown how the Chevallier–Polarski–Linder (CPL) parametrization [245,246] is insensitive to Dark Energy at low redshift scales.

Refs. [247,248] propose Dark Radiation as a new surrogate of the Standard Model.

In [249], the scalar field cosmological model is used, together with the parametrization of the equation of state, to obtain *H*<sup>0</sup> and investigate the nature of Dark Energy. The possibility of a scalar field non minimally coupled to gravity as a probable solution to the *H*<sup>0</sup> tension is investigated in [250].

Ref. [251] highlight the advantage of the braneworld models to predict the local higher values of *H*<sup>0</sup> and, contemporaneously, respect the CMB constraints. Another approach is to solve the *H*<sup>0</sup> tension by allowing variations in the fundamental constants [252].

Ref. [253] propose a non-singular Einstein–Cartan cosmological model with a simple parametrization of spacetime torsion to alleviate the tension, while [254] propose a model where the Dark Matter is annihilated to produce Dark Radiation.

Ref. [255] introduce a hidden sector of atomic Dark Matter in a realistic model that avoids the fine-tuning problem. The observed weak effect of primordial magnetic fields can create clustering at small scales for baryons and this could explain the *H*<sup>0</sup> tension [256].

Ref. [257] test the General Relativity at galactic scales through Strong Gravitational Lensing. The Strong Lensing is a promising probe for obtaining new constraints on *H*0, thanks to the next generation DECIGO and B-DECIGO space interferometers [258].

In [259], the cosmological constant Λ is considered a dynamical quantity in the context of the running vacuum models and this assumption could tackle the *H*<sup>0</sup> tension. Ref. [260] show the singlet Majoron model to explain the acceleration of the expansion at later times and prove that this is consistent with large-scale data: this model has been subsequently discussed in other works [261]. The vacuum energy density value is affected by the Hubble tension as well and its measurement may cast more light on this topic [262].

Ref. [263] discuss the outcomes of the Oscillatory Tracker Model with an *H*<sup>0</sup> value that agrees with the CMB measurements. In [264], it is explained how the Generalized Uncertainty Principle and the Extended Uncertainty Principle can modify the Hubble parameter. [265] explore the implication of the Mirror Twin Higgs model and the need for future measurements to alleviate the tension.

The artificial neural networks can be applied to reconstruct the behavior of large scale structure cosmological parameters [266].

Another alternative is given by the gravitational transitions at low redshift which can solve the *H*<sup>0</sup> tension better than the late-time *H*(*z*) smooth deformations [84,229].

Another comparison between the late-time gravitational transition models and other models which predict a smooth deformation of the Hubble parameter can be found in [267].

Ref. [268], as modifications to the ΛCDM model, consider as plausible scenarios or a Dark Matter component with negative pressure or the decay of Dark Energy into Dark Matter.

Ref. [269] does not observe the *H*<sup>0</sup> tension through the Effective Field Theory of Large Scale Structure and the Baryon Oscillation Spectroscopic Survey (BOSS) Correlation Function.

Considering the Dark Matter particles with two new charges, Ref. [270] reproduce a repulsive force which has similar effects to the Λ cosmological constant. Furthermore, the models where interaction between Dark Matter and Dark Energy is present are promising for a solution of the Hubble constant tensions, see [271].

In [272], it is shown how two independent sets of cosmological parameters, the background (geometrical) and the matter density (growth) component parameters, respectively, give consistent results and how the preference for high values of *H*<sup>0</sup> is less significant in their analysis.

Ref. [273] introduce a global parametrization based on the cosmic age which rules out the early-time and the late-time frameworks.

Ref. [274] point out, through the use of non-parametric methods, how the cosmological models may induce biases in the cosmological parameters. In the same way, the statistical analysis of galaxies' redshift value and distance estimations may be affected by biases which could, in turn, affect the estimation of *H*0.

Ref. [275]. This consideration holds also for the quadruply lensed quasars which are another method to measure *H*<sup>0</sup> [276].

Ref. [277] use the machine learning techniques to measure time delays in lensed SNe Ia, these being an independent method to measure *H*0.

Additionally, in [231] it is explained how an evolution of *H*<sup>0</sup> with the redshift is to be expected. If a statistical approach on the different *H*<sup>0</sup> values is used instead, together with the assumption of an alternative cosmology, another solution to the tension could be naturally implied [278].

Ref. [279] use data to reconstruct the *f*(*T*) gravity function without assuming any cosmological model: this *f*(*T*) could in turn represent a solution to the *H*<sup>0</sup> tension.

Ref. [124] discuss how the addition of scalar fields with particle physics motivation to the cosmological model which predicts Dark Matter can retrieve the observed abundances of the Big Bang Nucleosynthesis.

In [280], a Dark Matter production mechanism is proposed to alleviate the *H*<sup>0</sup> tension. A general review of the perspectives and proposals concerning the *H*<sup>0</sup> tension can be found in [281–283].

SNe Ia represents a very good example of standard candles. Here we consider also the contribution of geometrical probes, the so-called *standard rulers*: while standard candles show a constant intrinsic luminosity (or obey an intrinsic relation between their luminosity and other physical parameters independent of luminosity), standard rulers are characterized by a typical scale dimension. This property allows estimating their distance according to the apparent angular size. Among the possible standard rulers, the BAOs assume great importance for cosmological purposes.

We here investigate the *H*<sup>0</sup> tension in the Pantheon sample (hereafter PS) from [284] and we add the contribution of BAOs to the cosmological computations to check if the trend of *H*<sup>0</sup> found in [36] is present also with the addition of other probes. We here point out that the current analysis is not meant to constrain Ω0*<sup>m</sup>* or any other cosmological parameters, but it is focused to study the reliability of the trend of *H*<sup>0</sup> as a function of the redshift.

We here point out that this analysis is not meant to constrain Ω0*<sup>m</sup>* or any other cosmological parameters, but it is focused to study the reliability of the trend of *H*<sup>0</sup> as a

function of the redshift. The range of redshift in the PS goes from *z* = 0.01 to *z* = 2.26. We tackle the problem with a redshift binning approach of *H*0, the same used in [51], but here we adopt a starting value of *H*<sup>0</sup> = 70 instead of 73.5: if a trend with redshift exists, it should be independent on the initial value for *H*0. The systematic contributions for the PS are calibrated through a reference cosmological model, where *H*<sup>0</sup> is 70.0 [284]. In the current paper, the aforementioned systematic uncertainties are considered for the analysis. Our approach has a two-fold advantage: on the one hand, it is relatively simple and on the other hand, it avoids the re-estimation of the SNe Ia uncertainties and may be able to highlight a residual dependence on the SNe Ia parameters with redshift.

While a slow varying Einstein constant with the redshift, as it emerges in a modified *f*(*R*) gravity, appears as the most natural explanation for a trend *H*0(*z*), the analysis of Section 7 seems to indicate that such effect is not necessarily related with the Dark Energy contribution of the late universe. Since the Hu–Sawicki gravity lacks of reproducing the correct profile *H*0(*z*) shows that a Dark Energy model in the late Universe may not be enough to explain the observed effect since the scalar mode dynamics can not easily conciliate the Dark Energy contribution with the decreasing trend of *H*0(*z*). Thus, it may be necessary a modified gravity scenario more general than a Dark Energy model in the late Universe.

The current paper is composed as expressed in the following: in Section 2 the ΛCDM and *w*0*wa*CDM models are briefly introduced together with SNe Ia properties; Section 3 describes the use of BAOs as cosmological rulers; Section 4 contains our binned analysis results, after slicing the PS in 3 redshift bins for the aforementioned models, and assuming locally *H*<sup>0</sup> = 70; in Section 5, we investigate, through simulated events, how the GRBs will be contributing to cosmological investigations by 2030; in Section 6 we discuss the results; in Section 7 we test the Hu–Sawicki model through a binning approach; in Section 8 we report an overview on the requirements that a suitable *f*(*R*) model should have to properly describe the observed trend of *H*<sup>0</sup> and in Section 9 our conclusions are reported.
