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

The Lema\^ıtre -- Tolman model with $Λ= 0$ and constant bang time that imitates the luminosity distance -- redshift relation of the $Λ$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 $t \approx 1064 T$ 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 $Λ$CDM last scattering epoch.