Optimizing the $L$-$\sigma$ Relation of HII Galaxies for Improving Cosmological Application

The basic premise of using HII starburst galaxies (HIIGs) as cosmic "standard candels" is that there is a significant correlation between the H$\beta$ luminosity ($L$) and the velocity dispersion ($\sigma$) of the ionized gas from HIIGs measurements, which can be called as the empirical $L$ - $\sigma$ relation. However, the scaling $L$ - $\sigma$ relation well-calibrated with the lower-redshift HIIGs is unfitted for the higher-redshift HIIGs. To solve this problem, we explore new relational expression for the $L$ - $\sigma$ relation which should be suitable for both lower-redshift and higher-redshift HIIGs. After reconstructing the Hubble diagram with the Gaussian process (GP) method from the Pantheon+ supernovae Ia sample, we examine and compare six different revised formulas of $L$ - $\sigma$ relation. Furthermore, we use the Bayesian evidence to compare the revised $L$ - $\sigma$ relations with the analysis of a joint sample of 36 giant extragalactic HII regions (GEHRs) and 145 HIIGs. It turns out that the redshift-dependent bilinear correction and the quadratic $\sigma$ based correction are significantly better than the others. Moreover, a quadratic $\sigma$ based correction is the most supported one. It suggests that the appropriate corrections to the $L$ - $\sigma$ relation should be considered when the HIIGs are used as a kind of cosmological probes.
Comment: 12 pages, 4 figures