Constraining the dark energy and smoothness parameter with type Ia
  Supernovae and Gamma-Ray Bursts
  The existence of inhomogeneities in the observed Universe modifies the
distance-redshift relations thereby affecting the results of cosmological tests
in comparison to the ones derived assuming spatially uniform models. By
modeling the inhomogeneities through a Zeldovich-Kantowski-Dyer-Roeder (ZKDR)
approach which is phenomenologically characterized by a smoothness parameter
$\alpha$, we rediscuss the constraints on the cosmic parameters based on
Supernovae type Ia and Gamma-Ray Bursts (GRBs) data. The present analysis is
restricted to a flat $\Lambda$CDM model with the reasonable assumption that
$\Lambda$ does not clump. A $\chi^{2}$-analysis using 557 SNe Ia data from the
Union2 Compilation Data (Amanullah {\it et al.} 2010) constrains the pair of
parameters ($\Omega_m, \alpha$) to $\Omega_m=0.27_{-0.03}^{+0.08}$($2\sigma$)
and $\alpha \geq 0.25$. A similar analysis based only on 59 Hymnium GRBs (Wei
2010) constrains the matter density parameter to be $\Omega_m=
0.35^{+0.62}_{-0.24}$ ($2\sigma$) while all values for the smoothness parameter
are allowed. By performing a joint analysis, it is found that $\Omega_m =
0.27^{+0.06}_{-0.03}$ and $\alpha \geq 0.52$. As a general result, although
considering that current GRB data alone cannot constrain the smoothness
$\alpha$ parameter our analysis provides an interesting cosmological probe for
dark energy even in the presence of inhomogeneities.
