Accelerating expansion or inhomogeneity? Part 2: Mimicking acceleration with the energy function in the Lema\^ıtre-Tolman model

Abstract

This is a continuation of the paper published in {\it Phys. Rev.} {\bf D89}, 023520 (2014). It is investigated here how the luminosity distance -- redshift relation $D_L(z)$ of the $\Lambda$CDM model is duplicated in the Lema\^ıtre -- Tolman (L--T) model with $\Lambda =0$, constant bang-time function $t_B$ and the energy function $E(r)$ mimicking accelerated expansion on the observer's past light cone ($r$ is a uniquely defined comoving radial coordinate). Numerical experiments show that $E>0$ necessarily. The functions $z(r)$ and $E(r)$ are numerically calculated from the initial point at the observer's position; then backward from the initial point at the apparent horizon (AH). Reconciling the results of the two calculations allows one to determine the values of $E/r^2$ at $r=0$ and at the AH. The problems connected with continuing the calculation through the AH are discussed in detail and solved. Then $z(r)$ and $E(r)$ are continued beyond the AH, up to the numerical crash that signals the contact of the light cone with the Big Bang. Similarly, the light cone of the L--T model is calculated by proceeding from the two initial points, and compared with the $\Lambda$CDM light cone. The model constructed here contains shell crossings, but they can be removed by matching the L--T region to a Friedmann background, without causing any conflict with the type Ia supernovae observations. The mechanism of imitating the accelerated expansion by the $E(r)$ function is explained in a descriptive way.

18 pages, 14 figures. Penultimate paragraph of Sec. XIII and Refs. 22, 23 added in response to referee's comments. Typos corrected. This text matches the published version

References in corpus (2)

• Solving the Inverse Problem with Inhomogeneous Universes
• The Mass of the Cosmos

Cited by in corpus (3)

• Blueshifts in the Lema\^ıtre -- Tolman models
• Mimicking acceleration in the constant-bang-time Lema\^ıtre -- Tolman model: Shell crossings, density distributions and light cones
• Thermodynamic perfect fluid spheres admitting an orthogonal flat synchronization