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.

arXiv comment18 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 (11)

• Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant
• Measurements of Omega and Lambda from 42 High-Redshift Supernovae
• Is dark energy the only solution to the apparent acceleration of the present universe?
• A (giant) void is not mandatory to explain away dark energy with a Lemaitre - Tolman model
• The Discovery of the Most Distant Known Type Ia Supernova at Redshift 1.914
• Solving the Inverse Problem with Inhomogeneous Universes
• Formation of a galaxy with a central black hole in the Lemaitre-Tolman model
• The Mass of the Cosmos
• Accelerating expansion or inhomogeneity? A comparison of the $\Lambda$CDM and Lema\^ıtre - Tolman models
• A Note on the Inverse Problem with LTB Universes
• A new method to determine large scale structure from the luminosity distance

Cited by in corpus (8)

• Can a void mimic the $\Lambda$ in $\Lambda$CDM?
• Blueshifts in the Lema\^ıtre -- Tolman models
• Inverse Construction of the $\Lambda$LTB Model from a Distance-redshift Relation
• Thermodynamic perfect fluid spheres admitting an orthogonal flat synchronization
• Mimicking acceleration in the constant-bang-time Lema\^ıtre -- Tolman model: Shell crossings, density distributions and light cones
• Low red-shift effects of local structure on the Hubble parameter in presence of a cosmological constant
• Reconstructing the metric of the local Universe from number counts observations
• Coordinate independent approach to the calculation of the effects of local structure on the luminosity distance