Late-time Inhomogeneity and Acceleration Without Dark Energy

The inhomogeneous distribution of matter in the non-linear regime of galaxies, clusters of galaxies and voids is described by an exact, spherically symmetric inhomogeneous solution of Einstein’s gravitational field equations, corresponding to an under-dense void. The solution becomes the homogeneous and isotropic Einstein-de Sitter solution for a red shift $z>10-20$, which describes the matter dominated CMB data with small inhomogeneities $\delta \rho \mathrm{/}\rho \sim {10}^{-5}$. A spatial volume averaging of physical quantities is introduced and the averaged time evolution expansion parameter $\theta$ in the Raychoudhuri equation can give rise in the late-time universe to a volume averaged deceleration parameter $<q>$ that is negative for a positive matter density. This allows for a region of accelerated expansion which does not require a negative pressure dark energy or a cosmological constant. A negative deceleration parameter can be derived by this volume averaging procedure from the Lema^{i}tre-Tolman-Bondi open void solution, which describes the late-time non-linear regime associated with galaxies and under-dense voids and solves the ``coincidence’’ problem.