Weak gravitational lensing effects on cosmological parameters and dark
  energy from gamma-ray bursts
  Gamma-ray bursts (GRBs) are attractive objects for constraining the nature of
dark energy in a way complementary to other cosmological probes, especially at
high redshifts. However, the apparent magnitude of distant GRBs can be
distorted by the gravitational lensing from the density fluctuations along the
line of sight. We investigate the gravitational lensing effect on the
cosmological parameters and dark energy equation of state from GRBs. We first
calibrated the GRB luminosity relations without assuming any cosmological
models. The luminosity distances of low-redshift GRBs were calibrated with the
cosmography method using a latest type Ia supernova (SNe Ia) sample. We find
that the gravitational lensing has non-negligible effects on the determination
of cosmological parameters and dark energy. The gravitational lensing shifts
the best-fit constraints on cosmological parameters and dark energy. Because
high-redshift GRBs are more likely to be reduced, the most probable value of
the observed matter density $\Omega_M$ is slightly lower than its actual value.
In the $\Lambda$CDM model, we find that the matter density parameter $\Omega_M$
will shift from 0.30 to 0.26 after including the gravitational lensing effect.
The gravitational lensing also affects the dark energy equation of state by
shifting it to a more negative value. We constrain the dark energy equation of
state out to redshift $z\sim 8$ using GRBs for the first time, and find that
the equation of state deviates from $\Lambda$CDM at the $1\sigma$ confidence
level, but agrees with $w=-1$ at the $2\sigma$ confidence level.
