  We propose to use HII galaxies (HIIG) to trace the redshift-distance
relation, by means of their $L(\mathrm{H}\beta) - \sigma$ correlation, in an
attempt to constrain the dark energy equation of state parameter solution
space, as an alternative to the cosmological use of type Ia supernovae.
  For a sample of 128 local compact HIIG with high equivalent widths of their
Balmer emission lines we obtained ionised gas velocity dispersion from high
S/N, high-dispersion spectroscopy (Subaru-HDS and ESO VLT-UVES) and integrated
H$\beta$ fluxes from low dispersion wide aperture spectrophotometry. We find
that the $L(\mathrm{H}\beta) - \sigma$ relation is strong and stable against
restrictions in the sample. The size of the starforming region is an important
second parameter, while adding the emission line equivalent width or the
continuum colour and metallicity, produces the solution with the smallest rms
scatter.
  We have used the $L(\mathrm{H}\beta) - \sigma$ relation from a local sample
of HIIG and a local calibration or `anchor', given by giant HII regions in
nearby galaxies which have accurate distance measurements determined via
primary indicators, to obtain a value of $H_0$. Using our best sample of 69
HIIG and 23 Giant HII regions in 9 galaxies we obtain $H_{0}=74.3 \pm 3.1$
(statistical)$\pm$ 2.9 (systematic) km s$^{-1}$ Mpc$^{-1}$, in excellent
agreement with, and independently confirming, the most recent SNa Ia based
results.
  Using a local sample (107 sources) and a sample of 21 high redshift HIIG, 6
of them with medium-dispersion spectroscopy (ESO VLT-XShooter) and 17 taken
from the literature, we have obtained constraints on the planes $H_0 -
\Omega_m$, $\Omega_m - w_0$ and $w_0 - w_1$ (CPL model).
