Accelerating expansion or inhomogeneity? II. Mimicking acceleration with the energy function in the Lemaître-Tolman model

This is a continuation of the paper published in Phys. Rev. D 89, 023520 (2014). Here we investigate how the luminosity distance–redshift relation $D_L(z)$ of the $\mathrm{\Lambda }$CDM model is duplicated in the Lemaître-Tolman (L-T) model with $\mathrm{\Lambda }=0$, 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 $\mathrm{\Lambda }\mathrm{CDM}$ 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.

Figure 1

Typical graphs of the function $z(r)$ for two ranges of $k>0$. The cross marks the point of coordinates $(r,z)=(r,z{)}_{\mathrm{AH}}$, given by (2.33) and (2.31). The straight lines are the tangents to $z(r)$ at $r=0$, found by solving (4.11). Left panel: $k=10^{-3}$. Right panel: $k=10^{-16}$.

Figure 2

Graph of $z(r)$ for $0\le r\le r_{\mathrm{AH}}$. The cross marks the point of coordinates $(r,z)=(r,z{)}_{\mathrm{AH}}$, given by (2.33) and (2.31). The dotted straight lines are the tangents to $z(r)$ at $r=0$ [given by (9.2)] and at $r=r_{\mathrm{AH}}$ [given by (10.2)].

Figure 3

Left panel: Close-up view of the vicinity of $r=0$ in Fig. 2. The horizontal axis goes from $r=0$ to $r=0.01$, the tics on it are separated by $\mathrm{\Delta }r=0.002$. Right panel: Close-up view of the vicinity of $r=r_{\mathrm{AH}}$ in Fig. 2. The cross marks the point $(r,z)=(r_{\mathrm{AH}},z_{\mathrm{AH}})$, given by (2.33) and (2.31). The straight line is the theoretical tangent to $z(r)$ at $r=r_{\mathrm{AH}}$ given by (10.2). This mismatch is the best accuracy achieved in FORTRAN 90 at double precision. The leftmost tic on the horizontal axis is at $r=0.3085$, the rightmost one is at $r=0.3115$, the tics are separated by $\mathrm{\Delta }r=0.0005$.

Figure 4

Main panel: Graph of the function $E(r)$ for $0\le r\le r_{\mathrm{AH}}$. An instability is seen near $r=r_{\mathrm{AH}}$—see Fig. 5. Inset: Close-up view of the vicinity of $r=0$. There are no instabilities in this range. The horizontal axis goes from $r=0$ to $r=10^{-4}$, the tics are separated by $\mathrm{\Delta }r=2\times 10^{-5}$.

Figure 5

Graph of the function $E(r)$ in the vicinity of $r_{\mathrm{AH}}$. The straight line nearly coincides with $E(r)$ for $0.1771<E<0.179$. The larger cross marks the point of coordinates $(r_{\mathrm{AH}},{\tilde{E}}_{\mathrm{AH}})$, where ${\tilde{E}}_{\mathrm{AH}}$ is given by (9.7). The leftmost tic on the horizontal axis is at $r=0.304$, the rightmost one is at $r=0.311$, the tics are separated by $\mathrm{\Delta }r=0.001$. See text for more explanation.

Figure 6

Left panel: Close-up view of the vicinity of $r=r_{\mathrm{AH}}$ on the curve $z(r)$ obtained by integrating (7.9) backward and forward from the initial point at $r=r_{\mathrm{AH}}$ (for information on the forward part see Sec. 11). The lower line in the left half is the tangent to $z(r)$ at $r_{\mathrm{AH}}$ given by (10.2). Right panel: A magnified view of the vicinity of $r=r_{\mathrm{AH}}$. The leftmost tic on the horizontal axis is at $r=0.310535$, the rightmost one is at $r=0.31055$, the tics are separated by $\mathrm{\Delta }r=5\times 10^{-6}$. The errors in $r$ show up at the level of $10^{-6}$ in the backward-integrated segment; in the forward-integrated segment they are a few times smaller. The cross marks the point $(r_{\mathrm{AH}},z_{\mathrm{AH}})$.

Figure 7

A comparison of the two $E(r)$ curves. Left panel: In a neighborhood of $r=0$ the backward-integrated $E(r)$ is, at this scale and at all smaller scales, indistinguishable from the forward-integrated one. The left margin of the figure is at $r=0$, the right one at $r=0.001$, the tics on the horizontal axis are separated by $\mathrm{\Delta }r=0.0002$. Right panel: Around $r=0.15$, the two curves differ by $\mathrm{\Delta }E=5\times 10^{-6}$. The backward-integrated $E(r)$ is the upper curve. The left margin of the figure is at $r=0.14998$, the right margin is at $r=0.15002$, the tics on the horizontal axis are separated by $\mathrm{\Delta }r=5\times 10^{-6}$.

Figure 8

The $E(r)$ curve in the neighborhood of $r=r_{\mathrm{AH}}$, marked by the vertical stroke in both panels. Left panel: The curve that bends up is $E(r)$ integrated forward from $r=0$. The other line is $E(r)$ integrated backward and forward from $r=r_{\mathrm{AH}}$. The leftmost tic on the horizontal axis is at $r=0.309$, the rightmost one is at 0.3115, the tics are separated by $\mathrm{\Delta }r=5\times 10^{-4}$. Right panel: The neighborhood of $r=r_{\mathrm{AH}}$ magnified $\approx 100$ times. The leftmost tic on the horizontal axis is at $r=0.310525$, the rightmost one is at $r=0.31056$, the tics are separated by $\mathrm{\Delta }r=5\times 10^{-6}$.

Figure 9

Main panel: The continuous curve is the graph of $z(r)$ for $z\in [0,10]$. The dotted curve is $z(r)$ for the $\mathrm{\Lambda }\mathrm{CDM}$ model—see text for explanations. The straight line is the tangent to $z(r)$ at $r=r_{\mathrm{AH}}$, the vertical stroke marks $r=r_{\mathrm{AH}}$. Inset: The graph of $z(r)$ for $z\in [0,1100]$. The curve at right is for the L-T model, the curve at left is for the $\mathrm{\Lambda }\mathrm{CDM}$ model.

Figure 10

The function $E(r)$ extended into the range $r>r_{\mathrm{AH}}$. The vertical stroke is at $r=r_{\mathrm{AH}}$. Since $E(r)$ becomes decreasing at $r=r_{\mathrm{sc}}$ given by (11.3), there are shell crossings in the region $r>r_{\mathrm{sc}}$.

Figure 11

The light cone profile obtained by integrating (12.1) from $r=0$ to $r=r_{\mathrm{AH}}$ and then integrating (12.2) beyond $r_{\mathrm{AH}}$ with the initial values of $r=r_{\mathrm{AH}}$ and $t=t_{\mathrm{AH}}$ given by (2.33) and (10.3). The upper horizontal line is the $t=t_B$ given by (9.4). The cross marks the point $(r_{\mathrm{AH}},t_{\mathrm{AH}})$. The dotted lines are the $\mathrm{\Lambda }\mathrm{CDM}$ light cone and the $\mathrm{\Lambda }\mathrm{CDM}$ bang time. Inset: Close-up view of the neighborhood of $r=r_{\mathrm{AH}}$. The gap in the $t(r)$ curve is $\delta r\approx 0.0005$. The tics on the horizontal axis are separated by $\mathrm{\Delta }r=0.0005$, the leftmost one is at $r=0.309$, the rightmost one is at $r=0.312$.

Figure 12

Close-up views of key segments of $t(r)$ integrated in the two ways described in the text. The cross marks the point $(r_{\mathrm{AH}},t_{\mathrm{AH}})$. Upper-left panel: The neighborhood of $(r,t)=(r_{\mathrm{AH}},t_{\mathrm{AH}})$. The right end of $t(r)$ integrated forward from $r=0$ is seen at left. Upper-right panel: The neighborhood of $(r,t)=(r_{\mathrm{AH}},t_{\mathrm{AH}})$ magnified 100 times with respect to the left panel. The leftmost tic on the horizontal axis is at $r=0.310535$, the rightmost one is at $r=0.31055$, the tics are separated by $\mathrm{\Delta }r=5\times 10^{-6}$. Lower-left panel: The neighborhood of $r=0$. The forward-integrated $t(r)$ is the lower curve. The two curves differ by $\mathrm{\Delta }{t}_1\approx 1.9\times 10^{-5}\mathrm{NTU}\approx 1.86\times 10^5\mathrm{y}$. Lower-right panel: The end point of $t(r)$ misses the BB time by $\mathrm{\Delta }{t}_2\approx 10^{-6}\mathrm{NTU}\approx 9.8\times 10^4\mathrm{y}$. The left end of the horizontal axis is at $r=1.02$, the right end is at $r=1.05$, the tics are separated by $\mathrm{\Delta }r=0.005$.

Figure 13

Comparison of the function $R(t(r),r)$ along the past light cone calculated from (2.26) with the same function calculated by substituting $t(r)$, the solution of (12.2), into $R(t,r)$. At the scale of the upper panel, the two functions seem to coincide. The differences between them are listed in Table 1. The lower panel shows the neighborhood of $r=r_{\mathrm{AH}}$—see the text for an explanation. The inset in the upper panel shows the numerical instability in $F_l$ at the AH described in the text. The tics on the horizontal axis are separated by $\mathrm{\Delta }r=5\times 10^{-6}$, the depth of the dip is $\mathrm{\Delta }{F}_l=5\times 10^{-9}$.

Figure 14

Graph of the function $|k_F(r)|\equiv |-k+\mathcal{F}|$. It is decreasing all the way to that $r$, at which the past light cone reaches the BB. Thus, shells of matter closer to the observer evolve by the Friedmann equation corresponding to larger $|k_F|$. See text for the interpretation. The vertical stroke is at $r=r_{\mathrm{AH}}$.