Exploring the "$L$--$\sigma$" relation of HII galaxies and giant extragalactic HII regions acting as standard candles

Cosmological applications of HII galaxies (HIIGx) and giant extragalactic HII regions (GEHR) to construct the Hubble diagram at higher redshifts require knowledge of the "$L$--$\sigma$" relation of the standard candles used. In this paper, we study the properties of a large sample of 156 sources (25 high-$z$ HII galaxies, 107 local HII galaxies, and 24 giant extragalactic HII regions) compiled by Terlevich et al.(2015). Using the the cosmological distances reconstructed through two new cosmology-independent methods, we investigate the correlation between the H$\beta$ emission-line luminosity $L$ and ionized-gas velocity dispersion $\sigma$. The method is based on non-parametric reconstruction using the measurements of Hubble parameters from cosmic clocks, as well as the simulated data of gravitational waves from the third-generation gravitational wave detector (the Einstein Telescope, ET), which can be considered as standard sirens. Assuming the emission-line luminosity versus ionized gas velocity dispersion relation, $\log L ($H$\beta) = \alpha \log \sigma($H$\beta)+\kappa$, we find the full sample provides a tight constraint on the correlation parameters. However, similar analysis done on three different sub-samples seems to support the scheme of treating HII galaxies and giant extragalactic HII regions with distinct strategies. Using the corrected "$L$--$\sigma$" relation for the HII observational sample beyond the current reach of Type Ia supernovae, we obtain a value of the matter density parameter, $\Omega_{m}=0.314\pm0.054$ (calibrated with standard clocks) and $\Omega_{m}=0.311\pm0.049$ (calibrated with standard sirens), in the spatially flat $\Lambda$CDM cosmology.