Mimicking acceleration in the constant-bang-time Lemaître-Tolman model: Shell crossings, density distributions, and light cones

The Lemaître–Tolman model with $\mathrm{\Lambda }=0$ and constant bang time that imitates the luminosity distance–redshift relation of the $\mathrm{\Lambda }\mathrm{CDM}$ 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 $\mathrm{\Lambda }\mathrm{CDM}$ last scattering epoch.

Figure 1

The profiles $t(r)$ of the past light cone of the present central observer (the decreasing curve) and of the apparent horizon (the increasing curve). The horizontal line is $t=t_B$. The inset shows the AH in the full range of $r$.

Figure 2

The function $E(r)$ in the present model. The irregular segment at the right end is caused by numerical fluctuations in the neighbourhood of the Big Bang. The vertical stroke is at $r=r_{\mathrm{AH}}$. Since $E(r)$ becomes decreasing for $r>r_{\mathrm{sc}}$, given by (5.1), there are shell crossings in that region.

Figure 3

Main panel: see text for explanations. Inset: the neighborhood of $r=r_{\min }$ on the shell-crossing set. The cross marks the point $(r,t)=(r_{\min },t_{\min })$ given by (5.7) and (5.8). The leftmost tic on the $r$ axis is at $r=0.82076$, the rightmost one is at $r=0.82094$, the tics are separated by $\mathrm{\Delta }r=2\times 10^{-5}$.

Figure 4

Main panel: the mass density (2.5) calculated along the hypersurface $t=0$ as a function of $r$. The quantity on the vertical axis is $\kappa \rho$, measured in $(\mathrm{NLU}{)}^{-2}$. The left vertical bar marks $r=r_{\mathrm{AH}}$, the right one marks $r=r_{\mathrm{sc}}$. Inset: enlarged view of the neighbourhood of $r=r_{\mathrm{AH}}$. The leftmost tic on the horizontal axis is at $r=0.31046$, the rightmost tic is at $r=0.31064$, the tics are separated by $\mathrm{\Delta }r=2\times {10}^{-5}$. The discontinuity is a consequence of numerical errors.

Figure 5

Main panel: the mass density (2.5) calculated along the hypersurface $t=t_{\min }$, given by (5.7), as a function of $r$. The units on the vertical axis are the same as in Fig. 4. The vertical bar at the top marks the value $r=r_{\min }$, given by (5.8). Inset: enlarged view of the neighborhood of the $r$ axis. The right vertical bar marks $r=r_{\min }$. What looks like another bar at $r=r_{\mathrm{AH}}$ is a numerical fluctuation.

Figure 6

Main panel: the mass density (2.5) as a function of $r$, calculated along the hypersurface $t=t_2=100\mathrm{NTU}$. The units on the vertical axis are the same as in Fig. 4. The vertical bars mark the values $r=r_{2a}$ and $r=r_{2b}$. Inset: enlarged view of the neighborhood of the $r$ axis. What looks like a vertical bar is a numerical fluctuation at $r=r_{\mathrm{AH}}$.

Figure 7

Main panel: the mass density (2.5) as a function of $r$, calculated along the hypersurface $t=t_3=200\mathrm{NTU}$, which intersects the SCS at the value of $r$ given by (5.12). The units on the vertical axis are the same as in Fig. 4. The vertical bar marks the value $r=r_3$. Inset: enlarged view of the neighborhood of the $r$ axis.

Figure 8

The mass density (2.5) calculated along the light cone of Fig. 1 as a function of $r$. The units of $\kappa \rho$ are the same as in Fig. 4. See text for explanations.

Figure 9

The mass density (2.5) along the light cone of Fig. 1 as a function of $z$. The units of $\kappa \rho$ are the same as in Fig. 4.

Figure 10

Graphs of $z(r)$ along the rays reaching the central observer at various times from $t=0\mathrm{NTU}$ to $t\approx 394.3\mathrm{NTU}$. See text for details. Inset: enlarged view of the right part of curve #4 from the main graph. It clearly shows that $z(r)$ along this ray has a local maximum and a local minimum at $(r,z)$ given by (9.1) and (9.2).

Figure 11

Main panel: the relations $D_A(z)$ (lower curve) and $D_L(z)$ (upper curve) along ray #4 from Fig. 10 (this is the lowest ray in Fig. 3 that intersects the extremum-redshift profile twice). Insets: the upper ends of the curves from the main panel, where the relations $D_L(z)$ (left inset) and $D_A(z)$ (right inset) become triple-valued. The nearly straight segments at the upper edges of the graphs are numerical fluctuations at the edge of the model.

Figure 12

The analogue of Fig. 11 along ray #7 from Fig. 10 (this is the highest ray in Fig. 3). Now the relations $D_L(z)$ and $D_A(z)$ become double-valued close to the shell crossing.

Figure 13

Graph of the function $(-k_F(r))\equiv |-k+\mathcal{F}|$. The vertical stroke is at $r=r_{\mathrm{AH}}$.