The updated luminosity correlations of gamma-ray bursts and cosmological
  implications
  Several interesting luminosity correlations among gamma-ray burst (GRB)
variables have been recently discussed extensively. In this paper, we derive
the six luminosity correlations ($\tlag-L$, $V-L$, $\epkk-L$, $\epkk-E_\gamma$,
$\trt-L$, $\epkk-E_{\gamma, \mathrm{iso}}$) from the light curves and spectra
of the latest 116 long GRBs, including the time lag ($\tlag$) between low and
high photon energy light curves, the variability ($V$) of the light curve, the
peak energy of the spectrum ($\epkk$), and the minimum rise time ($\trt$) of
the peaks. We find that the intrinsic scatter of the $V-L$ correlation is too
large and there seems no inherent correlation between the two parameters using
the latest GRB data. The other five correlations indeed exist when the sample
is enlarged. The $\epkk-E_\gamma$ correlation has a significantly lower
intrinsic scatter compared to the other correlations. We divide the full data
into four redshift bins when testing possible evolution of the correlations
with redshift. We find no statistically significant evidence for the redshift
evolution of the luminosity correlations. To avoid the circularity problem when
constraining the cosmological parameters, we simultaneously minimize $\chi^2$
with respect to both correlation parameters $a$, $b$ and the cosmological
parameters using the maximum likelihood method. For the flat $\Lambda$CDM, the
best fit is $\omm=0.31^{+0.13}_{-0.10}$. We also constrain the possible
evolution of the equation of state (EOS) of the dark energy using the GRBs
together with the Union2 compilation of SNe Ia and the $H(z)$ data. The result
is consistent with the cosmological constant at $2 \sigma$ confidence level and
mainly due to the GRB data, the dark energy EOS shows slight deviation from -1
at $z \geq0.5$ as was persistently presented with many previous data sets.
