Non-parametric study of the evolution of the cosmological equation of
  state with SNeIa, BAO and high redshift GRBs
  We study the dark energy equation of state as a function of redshift in a
non-parametric way, without imposing any {\it a priori} $w(z)$ (ratio of
pressure over energy density) functional form. As a check of the method, we
test our scheme through the use of synthetic data sets produced from different
input cosmological models which have the same relative errors and redshift
distribution as the real data. Using the luminosity-time $L_{X}-T_{a}$
correlation for GRB X-ray afterglows (the Dainotti et al. correlation), we are
able to utilize GRB sample from the {\it Swift} satellite as probes of the
expansion history of the Universe out to $z \approx 10$. Within the assumption
of a flat FLRW universe and combining SNeIa data with BAO constraints, the
resulting maximum likelihood solutions are close to a constant $w=-1$. If one
imposes the restriction of a constant $w$, we obtain $w=-0.99 \pm 0.06$
(consistent with a cosmological constant) with the present day Hubble constant
as $H_{0}=70.0 \pm 0.6$ ${\rm km} \, {\rm s}^{-1} {\rm Mpc}^{-1}$ and density
parameter as $\Omega_{\Lambda 0}=0.723 \pm 0.025$, while non-parametric $w(z)$
solutions give us a probability map which is centred at $H_{0}=70.04 \pm 1$
${\rm km} \, {\rm s}^{-1} {\rm Mpc}^{-1}$ and $\Omega_{\Lambda 0}=0.724 \pm
0.03$. Our chosen GRB data sample with full correlation matrix allows us to
estimate the amount, as well as quality (errors) of data, needed to constrain
$w(z)$ in the redshift range extending an order of magnitude in beyond the
farthest SNeIa measured.
