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The universe, with large-scale homogeneity, is locally inhomogeneous,clustering into stars, galaxies and larger structures. Such property isdescribed by the smoothness parameter $\alpha$ which is defined as theproportion of matter in the form of intergalactic medium. If we takeconsideration of the inhomogeneities in small scale, there should bemodifications of the cosmological distances compared to a homogenous model.Dyer and Roeder developed a second-order ordinary differential equation (D-Requation) that describes the angular diameter distance-redshift relation forinhomogeneous cosmological models. Furthermore, we may obtain the D-R equationfor observational $H(z)$ data (OHD). The density-parameter $\Omega_{\rm M}$,the state of dark energy $\omega$, and the smoothness-parameter $\alpha$ areconstrained by a set of OHD in a spatially flat $\Lambda$CDM universe as wellas a spatially flat XCDM universe. By using of $\chi^2$ minimization method weget $\alpha=0.81^{+0.19}_{-0.20}$ and $\Omega_{\rm M}=0.32^{+0.12}_{-0.06}$ at$1\sigma$ confidence level. If we assume a Gaussian prior of $\Omega_{\rmM}=0.26\pm0.1$, we get $\alpha=0.93^{+0.07}_{-0.19}$ and $\Omega_{\rmM}=0.31^{+0.06}_{-0.05}$. For XCDM model, $\alpha$ is constrained to$\alpha\geq0.80$ but $\omega$ is weakly constrained around -1, where $\omega$describes the equation of the state of the dark energy ($p_{\rmX}=\omega\rho_{\rm X}$). We conclude that OHD constrains the smoothnessparameter more effectively than the data of SNe Ia and compact radio sources.

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