9.2.2. Bézier Polynomials and ΛCDM and *ω*CDM Cosmological Models

In a second method summarized here, the circularity problem affecting GRBs is again overcome by using Bézier polynomials to calibrate, in this case, the Amati correlation alone [123]. Unlike the previous method, GRB data are utilized now in conjuction with the SNe Ia JLA data set alone [226] and employed to explicitly constrain two different cosmological scenarios: the concordance ΛCDM model and the *ω*CDM model, with the dark energy equation of the state parameter free to vary [123].

In the Monte Carlo integration, through the Metropolis–Hastings algorithm, *H*<sup>0</sup> has been fixed to the best-fit value obtained from the model-independent analysis over OHD data, i.e., *H*<sup>0</sup> = 67.76 km s−<sup>1</sup> Mpc−<sup>1</sup> . The results for Ω*<sup>m</sup>* and *w* (see Table 7) agree with previous findings making use of GRBs. The statistical performance of the models under study has been evaluated through the Akaike information criterion (AIC) criterion [227]

$$\text{AIC} \equiv \text{ } 2p - 2 \ln \mathcal{L}\_{\text{max}}$$

where *p* is the number of free parameters in the model and L*max* is the maximum probability function calculated at the best-fit point, and the deviance information criterion (DIC) criterion [228]

$$\text{DIC} \equiv 2p\_{eff} - 2\ln \mathcal{L}\_{\text{max}} \ \mu$$

where *pe f f* = h−2 lnLi + 2 lnL*max* is the number of parameters that a data set can effectively constrain28. The best model is the one that minimizes the AIC and DIC values. Unlike the AIC criterion, the DIC statistics do not penalize for the total number of free parameters of the model, but only for those which are constrained by the data [229]. Differently from the previous approach, we found that the ΛCDM model is preferred with respect to the minimal *ω*CDM extension (see Table 7) and then conclude that no modifications of the standard paradigm are expected as intermediate redshifts are involved. However, future efforts dedicated to the use of our new technique to fix refined constraints over dynamical dark energy models are encouraged in order to fix the apparent dichotomy in the results of the two described methods.

**Table 7.** 95% confidence level results of the MCMC analysis for the SN+GRB data. The AIC and DIC differences are intended with respect to the ΛCDM model.


#### *9.3. The Role of Spatial Curvature*

An updated sample of GRBs has been developed in 2020, in which the Combo relation extracts bounds on the spatial curvature with no other probes [141], differently from previous attempts that commonly assume a spatially flat background, with the inclusion of SNe Ia and BAO.

The way in which the Combo relation is calibrated is without an OHD data set, but rather invokes two step methods. In this picture, we assume [140]


GRB sub-samples with the same *z* are chosen among those ones providing well constrained best-fit parameters.

Considering that in each sub-sample the GRB luminosity distances *d*<sup>L</sup> are quite the same, we employ the rest-frame 0.3–10 keV energy flux *F*0. This improves the dependence on the model and enables one to render our procedure cosmology-independent. In the

specific example, reported here, seven sub-samples have been suggested and the best fits reported in Table 8, showing no evident trends with *z* within the errors. This technique is used for Combo relation since previous results suggest its advantages in the fitting strategies described above29. One can perform a simultaneous fit of the above sub-samples with the same *k*<sup>1</sup> and *σ*<sup>k</sup> implying *k*<sup>1</sup> = 0.90 ± 0.13 and *σ*<sup>k</sup> = 0.28 ± 0.03.

**Table 8.** Best-fit parameters of the seven sub-samples at average redshift h*z*i: the slope *q*1,z, the normalization log *F*0,z, and the extrascatter *σ*q,z are shown.


The calibration of *k*<sup>0</sup> is performed by means of the nearest couple of GRBs of the employed sample with the same redshift. In particular, it is possible to take GRB 130702A at *z* = 0.145 and GRB 161219B at *z* = 0.1475. Then, *µ* obs C can be replaced via its average distance modulus h*µ*SNIai = 39.21 ± 0.24, determined by SNe Ia with the same *z* as the above two GRBs, considering the bound over *k*<sup>1</sup> and the values of *F*0, *E*p, *τ*, and *α* for the two GRBs adopted throughout the computation. The computed value is *k*<sup>0</sup> = 49.54 ± 0.21.

At this stage, comparing between GRB distance moduli *µ* obs C , with uncertainties *σµ*obs C , with theoretical expectations, it is possible to get constraints over background cosmologies. In particular, for a non-flat ΛCDM model, i.e., the simplest scenario to work with, we write

$$d\_{\mathcal{L}} = \frac{c}{H\_0} \frac{(1+z)}{\sqrt{|\Omega\_{\mathbb{R}}|}} \text{sinc}\left(\int\_0^z \frac{\sqrt{|\Omega\_{\mathbb{R}}|} dz'}{\sqrt{\Omega\_m (1+z)^3 + \Omega\_\Lambda + \Omega\_k (1+z)^2}}\right),\tag{72}$$

and we compute numerical bounds once *H*<sup>0</sup> is marginalized<sup>30</sup> as in Table 9.

In particular, slightly larger estimations on matter density are obtained from GRBs, i.e., Ω*<sup>m</sup>* = 0.32+0.05 <sup>−</sup>0.05 with *<sup>H</sup>*<sup>0</sup> of Ref. [122] and the opposite, i.e., <sup>Ω</sup>*<sup>m</sup>* <sup>=</sup> 0.22+0.04 <sup>−</sup>0.03, for the *<sup>H</sup>*<sup>0</sup> of Ref. [222]. Analogous results, i.e., compatible with the flat case, are computed using the non-flat ΛCDM model.

**Table 9.** Best-fit parameters with 1–*σ* uncertainties for the various cosmological cases discussed in this work. The last column lists the values of the *χ* 2 . *H*<sup>0</sup> is fixed to the values given by Ref. [122,222], respectively the Planck and Riess expectation values.

