Addressing the circularity problem in the $E_\text{p}-E_\text{iso}$ correlation of Gamma-Ray Bursts

Given their huge luminosity and redshift distribution extending up to $z\sim9$, Gamma-Ray Bursts (GRBs) are potentially a very powerful tool for studying the geometry and the accelerated expansion of the universe. We here propose a new model-independent technique to overcome the circularity problem affecting the use of GRBs as distance indicators through the use of $E_{\rm p}$--$E_{\rm iso}$ correlation. We calibrate the $E_{\rm p}$--$E_{\rm iso}$ correlation and find the GRB distance moduli that can be used to constrain dark energy models.} We use observational Hubble data to approximate the cosmic evolution through B\'ezier parametric curve obtained through the linear combination of Bernstein basis polynomials. In so doing, we build up a new data set consisting of 193 GRB distance moduli. We combine this sample with the supernova JLA data set to test the standard $\Lambda$CDM model and its $w$CDM extension. We place observational constraints on the cosmological parameters through Markov Chain Monte Carlo numerical technique. Moreover, we compare the theoretical scenarios by performing the AIC and BIC statistics. For the $\Lambda$CDM model we find $\Omega_m=0.397^{+0.040}_{-0.039}$ at the $2\sigma$ level, while for the $w$CDM model we obtain $\Omega_m=0.34^{+0.13}_{-0.15}$ and $w=-0.86^{+0.36}_{-0.38}$ at the $2\sigma$ level. Our analysis suggests that $\Lambda$CDM model is statistically favoured over the $w$CDM scenario. The results of our numerical analysis are consistent with previous findings involving GRB data. Also, the values of $\Omega_m$ and $w$ for the $w$CDM model are in remarkable agreement with those obtained by the Dark Energy Survey collaboration. No evidence for extension of the $\Lambda$CDM model is found.