Constraining the dark energy and smoothness parameter with supernovae

Abstract

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 $\ensuremath{\alpha}$ Dyer-Roeder parameter. By using two different samples of SNe type Ia data, the ${\ensuremath{\Omega}}_{m}$ and $\ensuremath{\alpha}$ parameters are constrained by applying the Zeldovich-Kantowski-Dyer-Roeder luminosity distance-redshift relation for a flat ($\ensuremath{\Lambda}\mathrm{CDM}$) model. A ${\ensuremath{\chi}}^{2}$-analysis using the 115 SNe Ia data of the Astier et al. sample (2006) constrains the density parameter to be ${\ensuremath{\Omega}}_{m}={0.26}_{\ensuremath{-}0.07}^{+0.17}$ ($2\ensuremath{\sigma}$) while the $\ensuremath{\alpha}$ parameter is weakly limited (all the values $\ensuremath{\in}[0,1]$ are allowed even at $1\ensuremath{\sigma}$). However, a similar analysis based the 182 SNe Ia data of Riess et al. (2007) constrains the pair of parameters to be ${\ensuremath{\Omega}}_{m}={0.33}_{\ensuremath{-}0.07}^{+0.09}$ and $\ensuremath{\alpha}\ensuremath{\ge}0.42$ ($2\ensuremath{\sigma}$). Basically, this occurs because the Riess et al. sample extends to appreciably higher redshifts. As a general result, even considering the existence of inhomogeneities as described by the smoothness $\ensuremath{\alpha}$ parameter, the Einstein-de Sitter model is ruled out by the two samples with a high degree of statistical confidence ($11.5\ensuremath{\sigma}$ and $9.9\ensuremath{\sigma}$, respectively). The inhomogeneous Hubble-Sandage diagram discussed here highlights the necessity of the dark energy, and a transition deceleration/accelerating phase at $z\ensuremath{\sim}0.5$ is also required.