We perform unbiased tests for the clumpiness of the 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 the 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 the clumpiness parameter \eta which quantifies the fraction of homogeneously distributed matter within a given light cone .
At a 1 \sigma confidence level , the constraints are \Omega _ { m } = 0.34 \pm 0.02 and \eta = 1.00 ^ { +0.00 } _ { -0.02 } from the joint analysis .
The results suggest that the Universe full of Friedman-Lemaître-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 sufficiently accurate form to describe the effects of local homogeneity on the expanding Universe .