Figure 1

Upper figure: The density compared to the density of the EdS model as a function of $v=H_0{a}_{0}r$ for various values of $x=a/a_0$. Middle figure: The functions $k=-\beta (v)/v^2$ and $-\beta (v)=1-W^2(v)$ as a function of $v=H_0{a}_{0}r$. Notice that $W^2$ is always very close to unity (an approximation we will employ later). Bottom figure: The current proper distance as a function of the comoving coordinate $v=H_0{a}_{0}r$ for the LTB model and the EdS model ($H_0^{-1}=5878\mathrm{Mpc}$).Reuse & Permissions

Figure 2

Upper figures: The functions $Y$ and $Z$ as a function of $v=H_0{a}_{0}r$ for the indicated values of $x=a/a_0$. At the initial time of the evolution, $x^{-1}=1100$ and $Y=Z=1$. For the other values of $x$, $Y$ is always smaller than unity and asymptotically approaches $Y=1$ (the EdS solution) for large values of $v$. For small values of $v$ the function $Z$ is smaller than unity, then for larger values of $v$ it becomes larger than unity, and eventually $Z$ approaches unity from above. Lower figures: The functions $dY/dx$ and $dZ/dx$ as a function of $v$ for the indicated values of $x=a/a_0$. The function $dY/dx$ never changes sign, while the function $dZ/dx$ does. Note that for $x^{-1}=1100$, the values of $dY/dx$ and $dZ/dx$ have been multiplied by ${10}^{-3}$.Reuse & Permissions

Figure 3

The luminosity distance as a function of redshift. The curve labeled LTB is the LTB model of this paper, the model labeled $\Lambda \mathrm{CDM}$ is the standard $\Lambda \mathrm{CDM}$ model with ${\Omega }_{\Lambda }=0.7$, the model labeled EdS is the EdS model, and the model labeled Empty is an empty universe model. Rather than $H_0{d}_L(z)$, we show the usual difference in the distance modulus compared to the empty model.Reuse & Permissions

Figure 4

The gauge potentials $\psi (x,v)$ and $\varphi (x,v)$ as function of $v$ for the indicated values of $x$. The potential $\psi (x,v)$ is indicated by solid curves and the potential $\varphi (x,v)$ is indicated by dashed curves. In the lower half of the figure the values of $x$ are indicated only for $\varphi (x,v)$; the values of $x$ for $\psi (x,v)$ are, from top to bottom, $x^{-1}=100$, 10, 3, and 1. In the upper half of the figure, the values of $v$ at the horizon are indicated by a vertical dotted line for $x^{-1}=500$ (left-most line) and $x^{-1}=200$.Reuse & Permissions

Figure 5

The time displacement $\Delta t/t$ and coordinate displacement $\Delta r/r$ as a function of $v$ for the indicated values of $x$. The time displacement $\Delta t$ is negative for $x^{-1}=1100$, 500, and 100. It is positive for $x^{-1}=10$, 3, and 1. For $x^{-1}=1100$, $\Delta r$ is positive, for all other values of $x$, $\Delta r$ is negative. Clearly, $\Delta r=r\Delta r/r$ vanishes in the limit $r\to 0$ as it must.Reuse & Permissions

Figure 6

The peculiar velocity in the Poisson frame as a function of $v=H_0{a}_{0}r$ for the indicated values of $x$. A negative (positive) value corresponds to a radial velocity towards (away from) the center.Reuse & Permissions

Figure 7

The expansion rates as a function of $v$ for the indicated values of $x$.Reuse & Permissions

Figure 8

The functions $({\nabla }^2\beta {)}^2$ evaluated at $v=0$ (dashed curve) and ${\varphi }_0=\varphi (v=0,x)$ (solid curve) as a function of $x^{-1}=a_0/a$.Reuse & Permissions