A scaling relation in [C II]-detected galaxies and its likely application in cosmology

We identify and investigate a possible correlation between the $\rm{[CII]} 158{\mu}m$ luminosity and linewidth in the $\rm{[CII]}$-detected galaxies. Observationally, the strength of the $\rm{[CII]} 158{\mu}m$ emission line is usually stronger than that of the CO emission line and this $\rm{[CII]}$ line has been used as another tracer of the galactic characteristics. Moreover, many $\rm{[CII]}$-detected galaxies are identified in $z > 4$. Motivated by previous studies of the CO luminosity - FWHM correlation relation (LFR) and the available new $\rm{[CII]}$ measurements, we compile samples of the $\rm{[CII]}$-detected galaxies in the literature and perform the linear regression analysis. The $\rm{[CII]}$ LFR is confirmed at a robust level. We also demonstrate the possible application of the $\rm{[CII]}$ LFR by utilizing it on the distance measurement of the high-$z$ galaxy. As a result, we extend the cosmic spatial scale beyond the redshift $z$ of $7$. With the outcome of the distance measurement, we constrain the cosmology parameters in the Chevallier-Polarski-Linder model, which considers the evolution of dark energy. Consequently, the uncertainties of the $\textit{w}_{0}$ and $\textit{w}_{a}$ are reduced significantly when the measured distance data of the $\rm{[CII]}$-detected galaxies are included in the cosmological parameter constraint, exemplifying the potential of using the $\rm{[CII]}$-detected galaxies as a tracer to constrain the cosmological parameters.