Mimicking acceleration in the constant-bang-time Lema\^ıtre -- Tolman model: Shell crossings, density distributions and light cones

Abstract

The Lema\^ıtre -- Tolman model with $\Lambda =0$ and constant bang time that imitates the luminosity distance -- redshift relation of the $\Lambda$CDM model using the energy function $E$ alone contains shell crossings. In this paper, the location in spacetime and the consequences of existence of the shell-crossing set (SCS) are investigated. The SCS would come into view of the central observer only at $t\approx 1064T$ to the future from now, where $T$ is the present age of the Universe, but would not leave any recognizable trace in her observations. Light rays emitted near to the SCS are blueshifted at the initial points, but the blueshift is finite, and is overcompensated by later-induced redshifts if the observer is sufficiently far. The local blueshifts cause that $z$ along a light ray is not a monotonic function of the comoving radial coordinate $r$. As a consequence, the angular diameter distance $D_A$ and the luminosity distance $D_L$ from the central observer fail to be functions of $z$; the relations $D_A(z)$ and $D_L(z)$ are multiple-valued in a vicinity of the SCS. The following quantities are calculated and displayed: (1) The distribution of mass density on a few characteristic hypersurfaces of constant time; some of them intersect the SCS. (2) The distribution of density along the past light cone of the present central observer. (3) A few light cones intersecting the SCS at characteristic instants. (4) The redshift profiles along several light cones. (5) The extremum-redshift hypersurface. (6) The $D_A(z)$ and $D_L(z)$ relations. (7) The last scattering time and its comparison with the $\Lambda$CDM last scattering epoch.

arXiv comment13 pages, 13 figures. New secs. X and XI added, secs. I, II, V, IX and the old XI(now XIII) re-edited to accommodate the new material, Figs. 3, 8 and old 11 (now 10) redrawn, old Fig. 10 removed. The extensions were inspired by the referee. New text matches that accepted for Phys. Rev. D

References in corpus (14)

• Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant
• Measurements of Omega and Lambda from 42 High-Redshift Supernovae
• Inhomogeneous cosmological models: exact solutions and their applications
• Is dark energy the only solution to the apparent acceleration of the present universe?
• Mimicking Dark Energy with Lemaitre-Tolman-Bondi Models: Weak Central Singularities and Critical Points
• 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
• Imitating accelerated expansion of the Universe by matter inhomogeneities - corrections of some misunderstandings
• CMB observations in LTB universes: Part I: Matching peak positions in the CMB spectrum
• CMB observations in LTB universes: Part II: the kSZ effect in an LTB universe
• Accelerating expansion or inhomogeneity? A comparison of the $\Lambda$CDM and Lema\^ıtre - Tolman models
• Radiation in the process of the formation of voids
• Accelerating expansion or inhomogeneity? Part 2: Mimicking acceleration with the energy function in the Lema\^ıtre-Tolman model

Cited by in corpus (4)

• Blueshifts in the Lema\^ıtre -- Tolman models
• Towards the geometry of the universe from data
• Inverse Construction of the $\Lambda$LTB Model from a Distance-redshift Relation
• Cosmological blueshifting may explain the gamma ray bursts