On astrophysical explanations due to cosmological inhomogeneities for the observational acceleration

We review various cosmological models with a local underdense region (local void) and the averaged models with the backreaction of inhomogeneities, which have been proposed to explain (without assuming a positive cosmological constant) the observed accelerating behaviors appearing in the magnitude-redshift relation of SNIa. To clarify their reality, we consider their consistency with the other observational studies such as CMB temperature anisotropy, baryon acoustic oscillation, kinematic Sunyaev-Zeldovich effect, and so on. It is found as a result that many inhomogeneous models seem to be ruled out and only models with the parametrs in the narrow range remain to be examined, and that, unless we assume very high amplitudes of perturbations or gravitational energies, the averaged models cannot have the accelerated expansion and the fitted effective Lambda has not the value necessary for the observed acceleration.