Testing the Copernican Principle with Hubble Parameter

Using the longitudinal expression of Hubble expansion rate for the general Lema\^itre-Tolman-Bondi (LTB) metric as a function of cosmic time, we examine the scale on which the Copernican Principle holds in the context of a void model. By way of performing parameter estimation on the CGBH void model, we show that the Hubble parameter data favors a void with characteristic radius of 2 ~ 3 Gpc. This brings the void model closer, but not yet enough, to harmony with observational indications given by the background kinetic Sunyaev-Zel'dovich effect and the normalization of near-infrared galaxy luminosity function. However, the test of such void models may ultimately lie in the future detection of the discrepancy between longitudinal and transverse expansion rates, a touchstone of inhomogeneous models. With the proliferation of observational Hubble parameter data and future large-scale structure observation, a definitive test could be performed on the question of cosmic homogeneity. Particularly, the spherical LTB void models have been ruled out, but more general non-spherical inhomogeneities still need to be tested by observation. In this paper, we utilise a spherical void model to provide guidelines into how observational tests may be done with more general models in the future.