9.2.1. Bézier Polynomials and Cosmographic Series

The first method utilizes GRB data, calibrated through the above Bézier polynomials, to extract cosmological constraints by means of cosmographic model-independent series and heal de facto the circularity problem without postulating the model a priori [216]. This method has been implemented to Amati, Ghirlanda, Yonetoku, and Combo GRB correlations in conjunction with SNe Ia and BAO data sets to get more stable and narrow constraints. We considered the most recent approaches to cosmography, comparing among them Taylor expansions with *z* and *y*<sup>2</sup> series, and Padé polynomials. Two hierarchies have been considered: *hierarchy 1*, up to *j*0, and *hierarchy 2*, up to *s*0.

Reasonable results have been found for both hierarchies through several MCMC fits showing possible matching with the standard paradigm (see Tables 3–6). Moreover, we only partially alleviated the tension on local *H*<sup>0</sup> measurements as hierarchy 2 is considered. Taylor outcomes are quite stable within each hierarchy, as portrayed by the results in Table 3, and work well with Amati, Ghirlanda, and Yonetoku correlations in the sense that the corresponding numerical outcomes are consistent within 1–*σ* with previous findings. Again, this suggests a spatially flat ΛCDM paradigm as a statistically favored model, with mass density parameter Ω*<sup>m</sup>* = 2(1 + *q*0)/3 ∼ 0.3 for Combo correlation, whereas the other correlations seem to indicate smaller values.

The auxiliary *y*<sup>2</sup> variable is not stable enough compared to Taylor expansions. It significantly enlarges *h*0, see, e.g., Table 4, and the overall results are however quite nonpredictive at the level of hierarchy 1. Moreover, Padé fits seem to improve the quality of Taylor expansion hierarchy 1, as expected by construction (see Table 5). This is particularly evident for Combo and Yonetoku correlations, while, for Amati and Ghirlanda correlations, it is not. It is worth noticing that, to go further, jerk term implies ≥(3,1), leading to higher orders than *P*3,1, quite unconstrained at higher redshift domains.

Quite surprisingly, our findings summarized in Tables 3–6 show that the ΛCDM model is not fully confirmed using GRBs. Although this can be an indication that more refined analyses are necessary, as GRBs are involved, simple indications seem to be against a genuine cosmological constant [122] and may be interpreted either with a barotropic dark energy contribution or with the need of non-zero spatial curvature [216]. Nevertheless, at this stage, our findings are in line with recent claims on tensions with the ΛCDM model [195,219,225].


**Table 3.** Cosmographic best fits and 1–*σ* (2–*σ*) errors from Taylor expansions labeled as *hierarchy 1* (*h*0, *q*0, *j*0) and *hierarchy 2* (*h*0, *q*0, *j*0, *s*0). Letters A, G, Y, and C indicate Amati, Ghirlanda, Yonetoku, and Combo correlations, respectively.

**Table 4.** Cosmographic best fits and 1–*σ* (2–*σ*) errors from expansions with *y*<sup>2</sup> labeled as *hierarchy 1* (*h*0, *q*0, *j*0) and *hierarchy 2* (*h*0, *q*0, *j*0, *s*0). Letters A, G, Y, and C indicate Amati, Ghirlanda, Yonetoku, and Combo correlations, respectively.


**Table 5.** Cosmographic best fits and 1–*σ* (2–*σ*) errors from Padé expansions labeled as *hierarchy 1*. Letters A, G, Y, and C indicate Amati, Ghirlanda, Yonetoku, and Combo correlations, respectively.


**Table 6.** *χ* <sup>2</sup> values of the cosmographic fits performed over the considered approximants. For each GRB correlation, the number of degrees of freedom (DoF) and the considered hierarchy are reported. Correlations are sorted for increasing values of the ratio *χ* <sup>2</sup>/DoF with respect to the Taylor hierarchy 1 expansion.

