  The duration of more than one thousand gamma-ray bursts (GRBs) has been
measured by Swift satellite. Besides the redshift distribution of GRBs, the
burst duration could be another significant property of GRBs that can be
analyzed. In this project, First, we find the detection rate of Swift/BAT for a
cosmological model with the $ \omega CDM$ model, then by performing a
Monte-Carlo simulation, we find the "long" GRB duration histogram to compare
the cosmological model with the $ \omega CDM$ model and the $\Lambda CDM$
model. The $\chi^2$ minimization method is employed to determine the optimal
value of $\alpha$ in the dark energy equation of state, $P= c \omega(z) \rho$,
with $\omega(z)=1+\alpha z$ as $\alpha=0.08 ^{+ 0.04}_{-0.02}$. We showed that
the $\omega CDM$ model is statistically favored over the $\Lambda CDM$ model.
Other studies by measuring the expansion rate of the Universe directly from the
SNIa data by Pantheon and DES-SN3YR samples and the combinations of SNIa, CMB,
and BAO data confirmed that the value of $\alpha$ in the dark energy equation
of state is not zero \citep{Dark_energy_Abbott, 2011Beutler, 2014Anderson,
2015Ross, 2016PlanckCollaboration, Alam2017, Mazumdar2021}.
