Constraining the dark energy and smoothness-parameter with supernovae

The presence of inhomogeneities modifies the cosmic distances through the gravitational lensing effect, and, indirectly, must affect the main cosmological tests. Assuming that the dark energy is a smooth component, the simplest way to account for the influence of clustering is to suppose that the average evolution of the expanding Universe is governed by the total matter-energy density whereas the focusing of light is only affected by a fraction of the total matter density quantified by the $\alpha$ Dyer-Roeder parameter. By using two different samples of SNe type Ia data, the $\Omega_m$ and $\alpha$ parameters are constrained by applying the Zeldovich-Kantowski-Dyer-Roeder (ZKDR) luminosity distance redshift relation for a flat ($\Lambda$CDM) model. A $\chi^{2}$-analysis using the 115 SNe Ia data of Astier {\it et al.} sample (2006) constrains the density parameter to be $\Omega_m=0.26_{-0.07}^{+0.17}$($2\sigma$) while the $\alpha$ parameter is weakly limited (all the values $\in [0,1]$ are allowed even at 1$\sigma$). However, a similar analysis based the 182 SNe Ia data of Riess {\it et al.} (2007) constrains the pair of parameters to be $\Omega_m= 0.33^{+0.09}_{-0.07}$ and $\alpha\geq 0.42$ ($2\sigma$). Basically, this occurs because the Riess {\it et al.} sample extends to appreciably higher redshifts. As a general result, even considering the existence of inhomogeneities as described by the smoothness $\alpha$ parameter, the Einstein-de Sitter model is ruled out by the two samples with a high degree of statistical confidence ($11.5\sigma$ and $9.9\sigma$, respectively). The inhomogeneous Hubble-Sandage diagram discussed here highlight the necessity of the dark energy, and a transition deceleration/accelerating phase at $z\sim 0.5$ is also required.