**9. Recent Developments of Cosmology with Gamma-Ray Bursts**

#### *9.1. Numerical Results Using Correlations*

Observational data indicate that the cosmological expansion is currently accelerating. They also indicate that, in the recent past, the expansion was decelerated. The standard spatially-flat ΛCDM model [176–178] is the simplest model consistent with these observations [122,179–181]. Here, a cosmological constant Λ dominates the current energy budget and fuels the currently-accelerating cosmological expansion. In this model, above a redshift *z* ≈ 0.75, non-relativistic cold dark matter and baryons dominate over Λ and powered the then-decelerating cosmological expansion. While the observations are consistent with dark energy being time- and space-independent, they do not rule out slowly-evolving and weakly spatially-inhomogeneous dynamical dark energy or spatial flatness.

Significant constraints on cosmological parameters come from the CMB anisotropy data—that primarily probe the *z* ∼ 1100 part of redshift space—as well as from BAO observations—the highest of which reach to *z* ∼ 2.3—and other lower-redshift SNeIa and OHD measurements. Observational data in the intermediate redshift range, between *z* ∼ 2.3 and ∼ 1100, are not as constraining as the lower and higher redshift data, but hold

significant promise. Intermediate redshift observations include those of HII starburst galaxies that reach *z* ∼ 2.4 [182–188], quasar angular sizes that reach *z* ∼ 2.7 [186,189–193], quasar X-ray, and UV fluxes that reach *z* ∼ 7.5 [193–201], as well as GRBs that have now been detected to *z* = 9.4 [202]. Observed correlations between GRB photometric and spectroscopic properties that can be related to an intrinsic burst physical property would allow GRBs to be used as valuable standard candles that reach high *z* and probe a largely unexplored region of cosmological redshift space (see, e.g., [136,140,142,145,151,203–206], and references therein), similar to how SNeIa are used as standard candles [118] at *z* < 2.3. However, as stressed many times in this review, this is still a challenge for GRBs.

After it was established that GRBs were at cosmological distances, many attempts have been made to use burst correlations to constrain cosmological parameters. The first GRB Hubble diagram of a small sample of 9 bursts, obtained in Ref. [207] from the *L*iso–*V* correlation [131], led to a current non-relativistic matter energy density parameter limit of Ω*m*<sup>0</sup> < 0.35 at the 1*σ* confidence level (for the flat ΛCDM model). Soon after, using the Ghirlanda correlation, in Ref. [208], with a sample of 12 bursts, it has been found Ω*m*<sup>0</sup> = 0.35 ± 0.15 for the flat ΛCDM model, and in Ref. [171], with 14 GRBs as well as SNeIa, it has been inferred that Ω*m*<sup>0</sup> = 0.37 ± 0.10 and a cosmological constant energy density parameter Ω<sup>Λ</sup> = 0.87 ± 0.23, in the non-flat ΛCDM model, and Ω*m*<sup>0</sup> = 0.29 ± 0.04 in the flat model. Similar constraints were obtained in Ref. [135], using the *E*p–*E*iso–*t*<sup>b</sup> correlation: 0.13 < Ω*m*<sup>0</sup> < 0.49 and 0.50 < Ω<sup>Λ</sup> < 0.85 at 1*σ* confidence level in the flat ΛCDM model.

More recently, many contrasting results have been reported in the literature. In Ref. [209], cosmological parameter constraints, within the ΛCDM model and dynamical dark energy models from two different GRB data sets, were found to be different from the two data sets and also relatively broad. Similarly, in Ref. [210], it has been shown that at that time GRB data could not significantly constrain cosmological parameters. In addition, in Refs. [211,212], it has been shown that most GRB correlations have large scatter and/or their parameters differ somewhat significantly between low- and high*z* GRB data sets. From the calibration of the Ghirlanda correlation, by using a SNeIa distance-redshift relation—through the (3, 2) Padé approximant—in Ref. [211], it has been obtained <sup>Ω</sup>*m*<sup>0</sup> <sup>=</sup> 0.302 <sup>±</sup> 0.142 within the flat <sup>Λ</sup>CDM model.<sup>26</sup> Based on a cosmographic approach, an updated *E*p–*E*iso correlation with 162 GRBs has been used to get cosmological constraints. In Ref. [12], GRBs were calibrated with SNeIa, resulting in Ω*m*<sup>0</sup> = 0.25+0.29 −0.12 within the flat ΛCDM model, whereas, in Ref. [214], a cosmographic expansion, up to the fifth order, involving SNeIa is used to calibrate the *E*p–*E*iso correlation for GRBs, which are then used in conjunction with OHD and BAO measurements to constrain cosmographic parameters, resulting in a 1*σ* deviation from the ΛCDM cosmological model.

Other recent works (involving GRB data only or in conjuction with other probes) also report inconsistencies with the ΛCDM model. In Ref. [215], the *E*p–*E*iso correlation has been used, including also modeling the potential evolution of GRB observables, to conclude that calibrated GRB, SNeIa, and OHD data favor a dynamical dark energy model described by a scalar field with an exponential potential energy density. In Ref. [216], Amati, Ghirlanda, Yonetoku, and Combo correlations have been calibrated in a model-independent way via OHD and jointly analyzed with SNeIa and BAO by using cosmographic methods, such as Taylor expansions, auxiliary variables, and Padé approximations, to conclude that GRB do not favor the flat ΛCDM model but instead favor a mildly evolving dark energy density model. Similarly, in Ref. [217], the *E*p–*E*iso and Combo correlations have been calibrated via OHD actual and machine-learned data, and again, based on a joint analysis with SNeIa and BAO, indications against a genuine cosmological constant have been found. Analogously, in Ref. [218], different combinations of SNe Ia, quasar, and GRB data sets have been used for testing the ΛCDM model and dynamical dark energy parametrizations. It was found that GRB and quasar data sets were inconsistent with the flat ΛCDM model, in agreement with Ref. [219] for similar data. In Ref. [220], strong gravitational lensing data in conjunction with SNe Ia and GRBs have been considered, and it has been found that the best-fit value

of the spatial curvature parameter favored a closed universe, although a flat universe can be accommodated at the 68% confidence level.

On the other hand, some recent efforts have shown that the *E*p–*E*iso and Combo correlations calibrated using better-established cosmological data—such as SNe Ia or OHD measurements—provide cosmological constraints that are consistent with the flat ΛCDM model (see Table 2). In Ref. [123], an updated *E*p–*E*iso correlation with 193 GRBs and a calibration based on an interpolation of the OHD data set have been considered, leading to Ω*m*<sup>0</sup> = 0.397+0.040 <sup>−</sup>0.039 in a flat <sup>Λ</sup>CDM cosmology, though the value of the mass density is higher than the one established by Ref. [122]. In Ref. [221], the *E*p–*E*iso correlation, calibrated with the latest OHD data set, has been jointly fit with CMB, BAO, and SNe Ia data in a search for cosmological parameter constraints within the standard cosmological model, as well as in dynamical dark energy parametrizations, finding no evidence in favor of the alternatives to the ΛCDM model. Finally, by using the Combo correlation with 174 GRBs calibrated in a semi-model independent way, in Ref. [141], it has been found: a) for a flat ΛCDM model Ω*m*<sup>0</sup> = 0.32+0.05 <sup>−</sup>0.05 and <sup>Ω</sup>*m*<sup>0</sup> <sup>=</sup> 0.22+0.04 <sup>−</sup>0.03 for the two values of the Hubble constant *H*<sup>0</sup> of Ref. [122] and Ref. [222], respectively, and b) for a non-flat ΛCDM model Ω*m*<sup>0</sup> = 0.34+0.08 <sup>−</sup>0.07 and <sup>Ω</sup><sup>Λ</sup> <sup>=</sup> 0.91+0.22 <sup>−</sup>0.35 for the *<sup>H</sup>*<sup>0</sup> of Ref. [122], and <sup>Ω</sup>*m*<sup>0</sup> <sup>=</sup> 0.24+0.06 −0.05 and Ω<sup>Λ</sup> = 1.01+0.15 <sup>−</sup>0.25 for the *<sup>H</sup>*<sup>0</sup> of Ref. [222].

**Table 2.** Summary of some recent cosmological constraints obtained by using Amati and Combo correlations, with or without other well established cosmological probes, within the flat and non-flat ΛCDM models. The numbers near the correlations name indicate the size of the GRB sample. For details on the names of the other probes, see the text.


*a* Inferred from the interpolation of the OHD data by using Bézier polynomials. *<sup>b</sup>* Inferred from the interpolation of the OHD data with additional systematic errors [221] by using Bézier polynomials. *<sup>c</sup>* Value from Ref. [222]. *<sup>d</sup>* Value from Ref. [122].

> Again, by examining an uncalibrated *E*p–*E*iso correlation built up from a sample of bright *Fermi*-LAT GRBs [223] and another GRB sample with lower average fluence GRBs [224], in Ref. [175], cosmological parameter constraints have been obtained in a number of cosmological models, concluding that current GRB data are not able to restrictively constrain cosmological parameters, and that cosmological parameter constraints from the more-reliable GRBs are consistent with those resulting from better-established cosmological probes. In Ref. [187], a joint *H*(*z*)+BAO+quasar (QSO)+HII starburst galaxy (HIIG)+GRB fit determined Ω*m*<sup>0</sup> = 0.313 ± 0.013 in the flat ΛCDM model, consistency with a cosmological constant and zero spatial curvature, though mild dark energy dynamics or a little spatial curvature are not ruled out at all.

> Mixing all together the cosmological results summarized above, obtained through GRB data, seem to be mutually inconsistent. This reflects all the efforts made so far to employ GRB as distance indicators are still affected by a certain number of issues, as we outlined previously.

> First of all, we recall that GRB correlations involve a number of observable quantities affected by the so-called circularity problem [166], caused by having to compute the GRB correlations in an a priori assumed background cosmological model, being not fully model-independent [12,136,138,140,151,206,209,214]. However, even uncalibrated GRB

correlations, in principle free from the circularity issue, are not able to put stringent constraints on the cosmological parameters, though consistent with those resulting from better-established cosmological probes. In addition, we recall that all GRB correlations are characterized by large intrinsic dispersions, conceivably caused by unknown large systematic errors<sup>27</sup> [64,143–145] in comparison to the case of better-established probes, such as BAO, OHD, and SNeIa, where many error sources have been better modeled. On the other hand, the influence of possible selection bias and evolution effects are currently debated [154–158]. One may therefore conclude that the large intrinsic dispersions of GRB correlations could be a consequence of yet undiscovered GRB intrinsic properties and/or a yet unidentified sub-class within the population of GRBs, analogously to SN populations.
