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Constraints on smoothness parameter and dark energy using observational $H(z)$ data

The universe, with large-scale homogeneity, is locally inhomogeneous, clustering into stars, galaxies and larger structures. Such property is described by the smoothness parameter $α$ which is defined as the proportion of matter in the form of intergalactic medium. If we take consideration of the inhomogeneities in small scale, there should be modifications of the cosmological distances compared to a homogenous model. Dyer and Roeder developed a second-order ordinary differential equation (D-R equation) that describes the angular diameter distance-redshift relation for inhomogeneous cosmological models. Furthermore, we may obtain the D-R equation for observational $H(z)$ data (OHD). The density-parameter $Ω_{\rm M}$, the state of dark energy $ω$, and the smoothness-parameter $α$ are constrained by a set of OHD in a spatially flat $Λ$CDM universe as well as a spatially flat XCDM universe. By using of $χ^2$ minimization method we get $α=0.81^{+0.19}_{-0.20}$ and $Ω_{\rm M}=0.32^{+0.12}_{-0.06}$ at $1σ$ confidence level. If we assume a Gaussian prior of $Ω_{\rm M}=0.26\pm0.1$, we get $α=0.93^{+0.07}_{-0.19}$ and $Ω_{\rm M}=0.31^{+0.06}_{-0.05}$. For XCDM model, $α$ is constrained to $α\geq0.80$ but $ω$ is weakly constrained around -1, where $ω$ describes the equation of the state of the dark energy ($p_{\rm X}=ωρ_{\rm X}$). We conclude that OHD constrains the smoothness parameter more effectively than the data of SNe Ia and compact radio sources.