Light is affected by local inhomogeneities in its propagation, which may alter distances and so cosmological parameter estimation. In the era of precision cosmology, the presence of inhomogeneities may induce systematic errors if not properly account
ed. In this vein, a new interpretation of the conventional Dyer-Roeder (DR) approach by allowing light received from distant sources to travel in regions denser than average is proposed. It is argued that the existence of a distribution of small and moderate cosmic voids (or black regions) implies that its matter content was redistributed to the homogeneous and clustered matter components with the former becoming denser than the cosmic average in the absence of voids. Phenomenologically, this means that the DR smoothness parameter (denoted here by $alpha_E$) can be greater than unity, and, therefore, all previous analyses constraining it should be rediscussed with a free upper limit. Accordingly, by performing a statistical analysis involving 557 type Ia supernovae (SNe Ia) from Union2 compilation data in a flat $Lambda$CDM model we obtain for the extended parameter, $alpha_E=1.26^{+0.68}_{-0.54}$ ($1sigma$). The effects of $alpha_E$ are also analyzed for generic $Lambda$CDM models and flat XCDM cosmologies. For both models, we find that a value of $alpha_E$ greater than unity is able to harmonize SNe Ia and cosmic microwave background observations thereby alleviating the well-known tension between low and high redshift data. Finally, a simple toy model based on the existence of cosmic voids is proposed in order to justify why $alpha_E$ can be greater than unity as required by supernovae data.
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 inhomogeneitie
s 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 $Omega_m=0.27_{-0.03}^{+0.08}$($2sigma$) and $alpha geq 0.25$. A similar analysis based only on 59 Hymnium GRBs (Wei 2010) constrains the matter density parameter to be $Omega_m= 0.35^{+0.62}_{-0.24}$ ($2sigma$) 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.
In this Comment we discuss a recent analysis by Yu et al. [RAA 11, 125 (2011)] about constraints on the smoothness $alpha$ parameter and dark energy models using observational $H(z)$ data. It is argued here that their procedure is conceptually incons
istent with the basic assumptions underlying the adopted Dyer-Roeder approach. In order to properly quantify the influence of the $H(z)$ data on the smoothness $alpha$ parameter, a $chi^2$-test involving a sample of SNe Ia and $H(z)$ data in the context of a flat $Lambda$CDM model is reanalyzed. This result is confronted with an earlier approach discussed by Santos et al. (2008) without $H(z)$ data. In the ($Omega_m, alpha$) plane, it is found that such parameters are now restricted on the intervals $0.66 leq alpha leq 1.0$ and $0.27 leq Omega_m leq 0.37$ within 95.4% confidence level (2$sigma$), and, therefore, fully compatible with the homogeneous case. The basic conclusion is that a joint analysis involving $H(z)$ data can indirectly improve our knowledge about the influence of the inhomogeneities. However, this happens only because the $H(z)$ data provide tighter constraints on the matter density parameter $Omega_m$.
