  To determine whether or not H II starburst galaxies (H IIG) are
standardizable candles, we study the correlation between the H$\beta$
luminosity ($L$) and the velocity dispersion ($\sigma$) of the ionized gas from
H IIG measurements by simultaneously constraining the $L-\sigma$ relation
parameters and the cosmological model parameters. We investigate six flat and
nonflat relativistic dark energy cosmological models. We find that low-redshift
and high-redshift H IIG data subsets are standardizable but obey different
$L-\sigma$ relations. Current H IIG data are too sparse and too non-uniformly
distributed in redshift to allow for a determination of why the samples follow
different relations, but it could be caused by the high-redshift sample
containing relatively fewer intrinsically dimmer sources (Malmquist bias) or it
could be a consequence of H IIG evolution. Until this issue is better
understood, H IIG data cosmological constraints must be treated with caution.
