Imitating accelerated expansion of the Universe by matter inhomogeneities: corrections of some misunderstandings

A number of misunderstandings about modeling the apparent accelerated expansion of the Universe and about the ‘weak singularity’ are clarified: (1) Of the five definitions of the deceleration parameter given by Hirata and Seljak (HS), only q ₁ is a correct invariant measure of acceleration/deceleration of expansion. The q ₃ and q ₄ are unrelated to acceleration in an inhomogeneous model. (2) The averaging over directions involved in the definition of q ₄ does not correspond to what is done in observational astronomy. (3) HS’s equation (38) connecting q ₄ to the flow invariants gives self-contradictory results when applied at the centre of symmetry of the Lemaître–Tolman (L–T) model. The intermediate equation (31) that determines q _3' is correct, but approximate, so it cannot be used for determining the sign of the deceleration parameter. Even so, at the centre of symmetry of the L–T model, it puts no limitation on the sign of q _3'(0). (4) The ‘weak singularity’ of Vanderveld et al. is a conical profile of mass density at the centre—a perfectly acceptable configuration. (5) The so-called ‘critical point’ in the equations of the ‘inverse problem’ for a central observer in an L–T model is a manifestation of the apparent horizon (AH)—a common property of the past light cones in zero-lambda L–T models, perfectly manageable if the equations are correctly integrated.