On the Feasibility of Deriving Pseudo-Redshifts of Gamma-ray Bursts from Two Phenomenological Correlations

Abstract: Accurate knowledge of gamma-ray burst (GRB) redshifts is essential for studying their intrinsic properties and exploring their potential application in cosmology. Currently, only a small fraction of GRBs have independent redshift measurements, primarily due to the need of rapid follow-up optical/IR spectroscopic observations. For this reason, many have utilized phenomenological correlations to derive pseudo-redshifts of GRBs with no redshift measurement. In this work, we explore the feasibility of analytically deriving pseudo-redshifts directly from the Amati and Yonetoku relations. We simulate populations of GRBs that (i) fall perfectly on the phenomenological correlation track, and (ii) include intrinsic scatter matching observations. Our findings indicate that, in the case of the Amati relation , the mathematical formulation is ill-behaved so that it yields two solutions within a reasonable redshift range $z \in [0.1, 10] $. When realistic scatter is included, it may result in no solution, or the redshift error range is excessively large. In the case of the Yonetoku relation, while it can result in a unique solution in most cases, the large systematic errors of the redshift calls for attention, especially when attempting to use pseudo redshifts to study GRB population properties.