The Observed Luminosity Correlations of Gamma-Ray Bursts and Their Applications

Gamma-ray bursts (GRBs) are among the most luminous electromagnetic transients in the universe, providing unique insights into extreme astrophysical processes and serving as promising probes for cosmology. Unlike Type Ia supernovae, which have a unified explosion mechanism, GRBs cannot directly act as standard candles for tracing cosmic evolution at high redshifts due to significant uncertainties in their underlying physical origins. Empirical correlations derived from statistical analyses involving various GRB parameters provide valuable information regarding their intrinsic properties. In this brief review, we describe various correlations among GRB parameters involving the prompt and afterglow phases, discussing possible theoretical interpretations behind them. The scarcity of low-redshift GRBs poses a major obstacle to the application of GRB empirical correlations in cosmology, referred to as the circularity problem. We present various efforts aiming at calibrating GRBs to address this challenge and leveraging established empirical correlations to constrain cosmological parameters. The pivotal role of GRB sample quality in advancing cosmological research is underscored. Some correlations that could potentially be utilized as redshift indicators are also introduced.