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

With large-scale homogeneity, the universe 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 consider the inhomogeneities over a 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 ΩM, the state of dark energy ω, and the smoothness-parameter α are constrained by a set of OHD in a spatially flat ΛCDM universe as well as a spatially flat XCDM universe. By using a χ2 minimization method, we get α = 0.81+0.19-0.20 and ΩM = 0.32+0.12-0.06 at the 1σ confidence level. If we assume a Gaussian prior of ΩM = 0.26 ± 0.1, we get α = 0.93+0.07-0.19 and ΩM = 0.31+0.06-0.05. For the XCDM model, α is constrained to α ≥ 0.80 but ω is weakly constrained around -1, where ω describes the equation of state of the dark energy (pX = ωρX). We conclude that OHD constrains the smoothness parameter more effectively than the data of SNe Ia and compact radio sources.