Cosmology with gamma-ray bursts: I. The Hubble diagram through the
  calibrated $E_{\rm p,i}$ - $E_{\rm iso}$ correlation
  Gamma-ray bursts are the most energetic explosions in the Universe. They are
detectable up to very high redshifts, therefore can be used to study the
expansion rate of the Universe and to investigate the observational properties
of dark energy, provided that empirical correlations between spectral and
intensity properties are appropriately calibrated. We used the type Ia
supernova luminosity distances to calibrate the correlation between the peak
photon energy, $E_{p, i}$, and the isotropic equivalent radiated energy, $
E_{iso}$ in GRBs. With this correlation, we tested the reliability of applying
GRBs to measure cosmological parameters and to obtain indications on the basic
properties and evolution of dark energy. Using 162 GRBs with measured redshifts
and spectra, we applied a local regression technique to calibrate the $E_{p,
i}$-$E_{iso}$ correlation against the type Ia SN data to build a calibrated GRB
Hubble diagram. We tested the possible redshift dependence of the correlation
and its effect on the Hubble diagram. Finally, we used the GRB Hubble diagram
to investigate the dark energy EOS. For this, we focused on the so-called
Chevalier-Polarski-Linder (CPL) parametrization of the dark energy EOS and
implemented the Markov chain Monte Carlo (MCMC) method to efficiently sample
the space of cosmological parameters. Our analysis shows once more that the
$E_{p, i}$-$E_{iso}$ correlation has no significant redshift dependence.
Therefore the high-redshift GRBs can be used as a cosmological tool to
determine the basic cosmological parameters and to test different models of
dark energy in the redshift region ($z\geqslant 3$), which is unexplored by the
SNIa and baryonic acoustic oscillations data. Our updated calibrated Hubble
diagram of GRBs provides some marginal indication (at $1\sigma$ level) of an
evolving dark energy EOS.
