Abstract
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
δρ/ρ ∼ 10−5. A
spatial volume averaging of physical quantities is introduced and the averaged time evolution expansion
parameter θ
in the Raychaudhuri equation can give rise in the late-time universe to a volume averaged
deceleration parameter 
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 Lemaître–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.