Redshift Evolution of the HII Galaxy $L$-$\sigma$ Relation: Gaussian Process Analysis and Cosmological Implications

The empirical correlation between the H$\beta$ luminosity ($L$) and the ionized gas velocity dispersion ($\sigma$) in HII starburst galaxies (HIIGs) provides a foundation for using them as cosmological standard candles. A key unresolved issue is whether this $L$-$\sigma$ relation changes with redshift, which would impact its application at high redshifts. We test for possible evolution using cosmology-independent distance estimates up to $z \sim 1.8$, obtained from Gaussian Process regression of the Pantheon+ Type Ia supernovae Hubble diagram. These distances allow us to compare the standard $L$-$\sigma$ relation with three redshift-dependent extensions through Bayesian model comparison. We find that a logarithmic redshift correction is statistically preferred when the intrinsic dispersion of the relation is explicitly modeled, significantly improving the fit to high-$z$ data. However, the evidence for evolution strongly depends on how the likelihood function accounts for this intrinsic dispersion and is weaker if it is ignored. We also show that Malmquist bias significantly affects comparisons between low- and high-$z$ samples, reducing -- though not eliminating -- the statistical preference for redshift evolution after matching luminosity ranges. These results indicate that current HIIG data favor a redshift-dependent modification of the standard $L$-$\sigma$ relation, while highlighting the critical role of selection effects and intrinsic dispersion modeling in establishing HIIGs as precise cosmological probes.