  We report the first results of a programme aimed at studying the properties
of high redshift galaxies with on-going massive and dominant episodes of star
formation (HII galaxies). We use the $L(\mathrm{H}\beta) - \sigma$ distance
estimator based on the correlation between the ionized gas velocity dispersions
and Balmer emission line luminosities of HII galaxies and Giant HII regions to
trace the expansion of the Universe up to $z \sim 2.33$. This approach provides
an independent constraint on the equation of state of dark energy and its
possible evolution with look-back time.
  Here we present high-dispersion (8,000 to 10,000 resolution) spectroscopy of
HII galaxies at redshifts between 0.6 and 2.33, obtained at the VLT using
XShooter. Using six of these HII galaxies we obtain broad constraints on the
plane $\Omega_m - w_0$. The addition of 19 high-z HII galaxies from the
literature improves the constraints and highlights the need for high quality
emission line profiles, fluxes and reddening corrections. The 25 high-z HII
galaxies plus our local compilation of 107 HII galaxies up to $z=0.16$ were
used to impose further constraints. Our results are consistent with recent
studies, although weaker due to the as yet small sample and low quality of the
literature data of high-z HII galaxies.
  We show that much better and competitive constraints can be obtained using a
larger sample of high redshift HII galaxies with high quality data that can be
easily obtained with present facilities like KMOS at the VLT.
