*8.3. The Use of Bézier Polynomials*

The left term of Equation (30) can be approximated by means of particular choices, such as using model-independent Bézier parametric curves. They are constructed to be stable at the lower degrees of control points. They can be rotated and translated by performing the operations on the points and assuming a degree *n*. They formally are defined as

$$H\_n(z) = \sum\_{d=0}^n \beta\_d h\_n^d(z) \quad , \quad h\_n^d(z) \equiv \frac{n!(z/z\_\mathbf{m})^d}{d!(n-d)!} \left(1 - \frac{z}{z\_\mathbf{m}}\right)^{n-d} \tag{59}$$

where we notice the linear combination of Bernstein basis polynomials *h d n* (*z*). Assuming the coefficients *β<sup>d</sup>* to be positive in the range of 0 ≤ *z*/*z*<sup>m</sup> ≤ 1, where *z*<sup>m</sup> is the maximum *z* of OHD, we soon can classify those polynomials by means of the exponent *n*.

In particular, besides the constant case, *n* = 0, both linear growth that happens for *n* = 1 and oscillatory regimes, say *n* > 2, work well. This implies that a suitable choice is *n* = 2. In this case, we have

$$H\_2(z) = \beta\_0 \left(1 - \frac{z}{z\_{\rm m}}\right)^2 + 2\beta\_1 \left(\frac{z}{z\_{\rm m}}\right) \left(1 - \frac{z}{z\_{\rm m}}\right) + \beta\_2 \left(\frac{z}{z\_{\rm m}}\right)^2. \tag{60}$$

The comparison between *H*2(*z*) and the OHD data points give *β*<sup>0</sup> = 67.76 ± 3.68, *<sup>β</sup>*<sup>1</sup> <sup>=</sup> 103.3 <sup>±</sup> 11.1, and *<sup>β</sup>*<sup>2</sup> <sup>=</sup> 208.4 <sup>±</sup> 14.3, all in units of km s−<sup>1</sup> Mpc−<sup>1</sup> .

After having approximated *H*(*z*) with Equation (60), for spatially flat cosmology, Ω*<sup>k</sup>* = 0, the calibrating luminosity distance becomes

$$d\_{\rm cal}(z) \simeq (1+z) \int\_0^z \frac{dz'}{H\_2(z')}\,. \tag{61}$$

Once the luminosity distance is written, it is possible to calibrate *E*iso, *Eγ*, *L*p, and *L*<sup>0</sup> for the correlations that we intend to test. For *E*<sup>p</sup> − *E*iso, Ghirlanda, Yonetoku, and Combo correlations, we report in Table 1 the corresponding numerical outcomes related to the calibration process.

**Table 1.** For brevity, we report in this table only a few calibrated correlations. In particular, in the columns, we prompt four correlations, i.e., Amati, Ghirlanda, Yonetoku, and Combo, with the data set number points and the corresponding last update year. On the right, we display the calibrated best fit parameters. The statistical method behind these calibrations is reported in Section 9.2.


Once calibrated, the corresponding distance moduli from Equation (24) are computed for each correlation.

#### 8.3.1. Simultaneous Fits

Another relevant strategy is based on the idea to constrain the cosmological parameters *together with* the luminosity correlation [171,172]. In particular, the real distance modulus can be computed as

$$
\mu\_{\rm fit} = \frac{\sum\_{i} \mu\_{i} / \sigma\_{\mu\_{i}}^{2}}{\sum\_{i} \sigma\_{\mu\_{i}}^{-2}},
\tag{62}
$$

where the sum is over a given number of different correlations. In particular, *µ<sup>i</sup>* is the best estimated distance modulus and the subscript *i*-th refers to the correlation, with *σµ<sup>i</sup>* the error bars. The uncertainty of the distance modulus for each burst is *σµ*fit = (∑*<sup>i</sup> σ* −2 *µi* ) −1/2 .

A great advantage is that, as one computes bounds on cosmological parameters, the normalization functions and slopes of each correlations are marginalized. Consequently, we write down the *χ* <sup>2</sup> as

$$\chi^2\_{\rm GRB} = \sum\_{i=1}^{N} \frac{[\mu\_i(z\_{i\prime}H\_{0\prime}\Omega\_{M\prime}\Omega\_{DE}) - \mu\_{\rm fit,i}]^2}{\sigma^2\_{\mu\_{\rm fit,i}}},\tag{63}$$

where *µ*fit,*<sup>i</sup>* and *σµ*fit,*<sup>i</sup>* are the fitted distance modulus and its error, respectively.

#### 8.3.2. Narrow Calibration

Another intriguing technique consists of calibrating standard candles using GRBs in a narrow redshift range, hereafter *δz*. This short interval is placed near a fiducial redshift [173,174] with the great advantage that, in some cases, see e.g., Ref. [173], no low-redshift GRB sample is necessary.

#### *8.4. Fitting Procedures without Calibration*

Constraints on the cosmological parameters can be obtained with an alternative method which completely by-passes the calibration procedure. It consists of taking all best data sets of any GRB correlations, introducing the lowest intrinsic dispersion. The method is described below.

Any correlation between a generic energy/luminosity quantity Y and a GRB observable X has the form

$$
\log \mathcal{Y}^{\text{obs}} = a \log \mathcal{X} + b \,. \tag{64}
$$

The energy/luminosity quantity in general contains the information on the cosmological parameters Ω<sup>i</sup> through *d*L(*z*, Ωi), defined by the theoretical model describing the background cosmology

$$\mathcal{Y}^{\text{th}} = 4\pi d\_{\text{L}}^2(z, \Omega\_{\text{i}}) \mathcal{F}\_{\text{bolo}} \tag{65}$$

where Fbolo may be the rest-frame bolometric fluence *S*bolo(1 + *z*) −1 for Amati-like correlations, or the bolometric observed flux *F*bolo for Yonetoku-like correlations. Please notice that we only focus on these two relations just for giving an example. The same can be reformulated for other correlations, although, for brevity, we do not report other treatments here.

The best cosmological and correlation parameters are then obtained by maximizing the log-likelihood function [152]

$$\ln \mathcal{L} = -\frac{1}{2} \sum\_{i=1}^{N} \left[ \frac{\left( \log \mathcal{Y}\_i^{\text{obs}} - \log \mathcal{Y}\_i^{\text{th}} \right)^2}{\sigma\_i^2} + \ln(2\pi \sigma\_i^2) \right],\tag{66}$$

where *σ* 2 *<sup>i</sup>* = *σ* 2 log Y obs i + *a* 2*σ* 2 log X<sup>i</sup> + *σ* 2 ext. Here, *<sup>σ</sup>*log <sup>Y</sup> obs i is the error in the measured value of log Y obs i , *σ*log <sup>X</sup><sup>i</sup> is the error in log X<sup>i</sup> , and *σ*ext is the intrinsic dispersion of the correlation.

The above treatment avoids calibrating GRB data. This procedure has been applied to an uncalibrated *E*p–*E*iso correlation in Ref. [175] and to an uncalibrated Combo correlation in Ref. [10]. In both cases, the correlations have been built up from samples of GRBs that have lower intrinsic dispersion. As byproducts, the resulting GRB correlations are close for different cosmological models, which can be interpreted with the fact that this procedure is model-independent. However, the application of this method seems to indicate that current GRB data are not able to put stringent constraints on cosmological parameters, though consistent with those resulting from better-established cosmological probes.

However, as hinted by the above results, this method may introduce a possible bias, namely that the GRB correlation may adjust itself to the cosmology that maximizes Equation (66), rather than allowing the derivation of Ω<sup>i</sup> from a cosmology independent calibrated correlation. Moreover, it may be possible that a more exotic cosmological model would lead to best-fit GRB correlations significantly different from simpler models, thus failing in providing a model-independent procedure. Therefore, this method is still the subject of ongoing studies.
