Unbiased constraints on the clumpiness of universe from standard candles

We perform unbiased tests for the clumpiness of universe by confronting the Zel'dovich-Kantowski-Dyer-Roeder luminosity distance which describes the effect of local inhomogeneities on the propagation of light with the observational one estimated from measurements of standard candles, i.e., type Ia supernovae (SNe Ia) and gamma-ray bursts (GRBs). Methodologically, we first determine the light-curve fitting parameters which account for distance estimation in SNe Ia observations and luminosity/energy relations which are responsible for distance estimation of GRBs in the global fit to reconstruct the Hubble diagrams in the context of a clumpy universe. Subsequently, these Hubble diagrams allow us to achieve unbiased constraints on the matter density parameter $\Omega_m$ as well as clumpiness parameter $\eta$ which quantifies the fraction of homogeneously distributed matter within a given light cone. At 1$\sigma$ confidence level, the constraints are $\Omega_m=0.34\pm0.02$ and $\eta=1.00^{+0.00}_{-0.02}$ from the joint analysis. The results suggest that the universe full of Friedman-Lema\^{i}tre-Robertson-Walker fluid is favored by observations of standard candles with very high statistical significance. On the other hand, they may also indicate that the Zel'dovich-Kantowski-Dyer-Roeder approximation is a sufficient accurate form to describe the effects of local homogeneity in the expanding universe.