Maria Giovanna Dainotti, Aleksander Łukasz Lenart, Biagio De Simone, William Giarè, Eleonora Di Valentino, Dieter H. Hartmann, Nissim Fraija, Kazunari Iwasaki, Gaetano Lambiase

Testing the $Λ$CDM model requires cosmological probes spanning the wide redshift interval between Type Ia Supernovae (SNe Ia, $z\lesssim2.9$) and the Cosmic Microwave Background (CMB, $z\approx1100$). Gamma-Ray Bursts (GRBs), observed up to redshift $z=9.2$, offer the opportunity to explore this regime. Here, we investigate how many GRBs are needed to become a useful cosmological probe capable of independently testing deviations from $Λ$CDM suggested by the recent DESI BAO observations. We develop forecasts based on the two-dimensional X-ray and optical Dainotti relations, between the luminosity at the end of the plateau phase and its rest-frame duration. Using simulated GRB samples constructed from the observed population, we evaluate the constraining power of GRBs on cosmological parameters within the $w$CDM and $w_0w_a$CDM models, both independently and in combination with CMB observations. Our results show that GRB samples containing several tens to hundreds of well-characterized plateau can already approach the precision currently achieved by CMB measurements on the Dark Energy (DE) equation-of-state parameter $w$. Particularly, a sample of $\sim66$ optical GRBs can reach a precision $σ_w \approx 0.47$, comparable to that obtained from Planck within the $w$CDM framework. Such sample sizes are already attainable through Machine Learning techniques that double the number of GRBs using inferred redshifts. These forecasts indicate that future GRB observations, when combined with next-generation transient missions and improved statistical techniques, will provide an independent high-redshift probe of cosmic expansion and will play an important role in testing the robustness of potential Dynamical DE signals suggested by other cosmological datasets.