Mapping luminosity-redshift relationship to Lemaitre-Tolman-Bondi cosmology

We derive a direct general map from the luminosity distance $D_L(z)$ to the inhomogeneous matter distribution $M(r)$ in the Lemaitre-Tolman-Bondi (LTB) cosmology and compute several examples. One of our examples explicitly demonstrates that it is possible to tune the LTB cosmological solution to approximately reproduce the luminosity distance curve of a flat Friedmann-Robertson-Walker universe with a cosmological constant. We also discuss how smooth matter distributions can evolve into naked singularities due to shell crossing when the inhomogeneous “curvature” $E(r)$ is a function which changes sign.

Figure 1

In the graph on the left, the solid curve shows the reconstructed $r(z)$ (in units of $1/H_0$) through the inversion method and the dashed curve shows the $r(z)$ that would be obtained if $M(r)$ of Eq. (36) were given as an input. The breakdown of the inversion method for this model around $z\approx 0.6$ is apparent and is explained in the text. In the graph on the right, the solid curve gives the $M(r)$ as determined through the numerical evaluation of $M_1(z(r))$, and the dashed curve gives $M(r)$ as given by Eq. (36).

Figure 2

$ER/M$ versus photon redshift $z$. For $z=0.6$, $ER/M\sim {10}^{-2}$ and the $dr/dz$ differential equation is unstable. Another way to view the instability is that the luminosity distance function is not sensitive to the geometry for $z\gtrsim 0.6$ in this model.

Figure 3

Using the inversion method, we have computed the energy density $\rho (t,r)$ corresponding to the luminosity distance of an FRW universe with ${\Omega }_M=0.3$ and ${\Omega }_{\Lambda }=0.7$. The solid, dashed, and dotted curves correspond to $\rho (t,r)$ at times $t_{\mathrm{today}}$, $t_{\mathrm{today}}/2$, and $t_{\mathrm{today}}/3$, respectively. With $H_0=70\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$, $t_{\mathrm{today}}$ is $13\times {10}^9$ years. The redshift corresponding to $r\approx 0.5/H_0$ is $z\approx 0.4$. The energy density at $r=0$ is nonzero: $4\pi \rho (t_{\mathrm{today}},0)/(M_{pl}^2{H}_0^2)\sim {10}^{-3}$.

Figure 4

The solid curve corresponds to $(D_L^{\Lambda =0.7}(z)-D_L^{\mathrm{model}}(z))/D_L^{\Lambda =0.7}(z)$ which corresponds to the deviation of the numerically constructed “model” luminosity distance curve from the FRW curve (where the FRW model is one with $\{{\Omega }_{\Lambda }=0.7,{\Omega }_M=0.3\}$). For comparison, we plot the dashed curve, which corresponds to $(D_L^{\Lambda =0.7}(z)-D_L^{\Lambda =0}(z))/D_L^{\Lambda =0.7}(z)$, where $D_L^{\Lambda =0}$ corresponds to a flat matter-dominated FRW model without any cosmological constant. It is clear that the model reproduces the luminosity curve exactly from $z=0$ up to $z=0.4$ (by construction) and there is a less than about 5% deviation from the FRW $\{{\Omega }_{\Lambda }=0.7,{\Omega }_M=0.3\}$ luminosity distance curve until about $z=3$. Furthermore, the deviation error is seen to plateau at large redshifts.

Figure 5

The energy density is plotted as a function of radius for times $t=0.01L$ (solid line) and $t=0.042L$ (dotted line) for the model of Eqs. (45, 46) with $\{r_0=0.7L,\kappa =50,{\rho }_0=3.5\frac{1}{L^2}\}$. As expected, the energy density diverges as $R{,}_r\to 0$ at $r\approx r_0$ near the time $t\gtrsim 0.04L$. Note also that the energy density for $r>r_0$ is larger than $r<r_0$, since one side is overdense while the other side is underdense.

Figure 6

The energy density as a function of radius for three different times. The dotted curve corresponds to $t=0$ (recombination), the dashed curve corresponds to $t=0.1/H_0$, and the solid curve corresponds to $t=0.5/H_0$.