Observational constraints from SNe Ia and Gamma-Ray Bursts on a clumpy
  universe
  The luminosity distance describing the effect of local inhomogeneities in the
propagation of light proposed by Zeldovich-Kantowski-Dyer-Roeder (ZKDR) is
tested with two probes for two distinct ranges of redshifts: supernovae Ia (SNe
Ia) in 0.015 < z < 1.414 and gamma-ray bursts (GRBs) in 1.547 < z < 3.57. Our
analysis is performed by a Markov Chain Monte Carlo (MCMC) code that allows us
to constrain the matter density parameter \Omega_m as well as the smoothness
parameter $\alpha$ that measures the inhomogeneous-homogeneous rate of the
cosmic fluid in a flat \LambdaCDM model. The obtained best fits are
(\Omega_m=0.285^{+0.019}_{-0.018}, \alpha= 0.856^{+0.106}_{-0.176}) from SNe Ia
and (\Omega_m=0.259^{+0.028}_{-0.028}, \alpha=0.587^{+0.201}_{-0.202}) from
GRBs, while from the joint analysis the best fits are
(\Omega_m=0.284^{+0.021}_{-0.020}, \alpha= 0.685^{+0.164}_{-0.171}) with a
\chi^2_{\rm red}=0.975. The value of the smoothness parameter $\alpha$
indicates a clumped universe however it does not have an impact on the amount
of dark energy (cosmological constant) needed to fit observations. This result
may be an indication that the Dyer-Roeder approximation does not describe in a
precise form the effects of clumpiness in the expansion of the universe.
