A Note on the Inverse Problem with LTB Universes

The inverse problem with Lema\^itre-Tolman-Bondi (LTB) universe models is discussed. The LTB solution for the Einstein equations describes the spherically symmetric dust-filled spacetime. The LTB solution has two physical functional degrees of freedom of the radial coordinate. The inverse problem is constructing an LTB model requiring that the LTB model be consistent with selected important observational data. In this paper, we assume that the observer is at the center and consider the distance-redshift relation $\da$ and the redshift-space mass density $\mu$ as the selected important observational data. We give $\da$ and $\mu$ as functions of the redshift $z$. Then, we explicitly show that, for general functional forms of $\da(z)$ and $\mu(z)$, the regular solution does not necessarily exist in the whole redshift domain. We also show that the condition for the existence of the regular solution %in terms of $\da(z)$ and $\mu(z)$ is satisfied by the distance-redshift relation and the redshift-space mass density in $\Lambda$CDM models. Deriving regular differential equations for the inverse problem with the distance-redshift relation and the redshift-space mass density in $\Lambda$CDM models, we numerically solve them for the case $(\Omega_{\rm M0},\Omega_{\Lambda0})=(0.3,0.7)$. A set of analytic fitting functions for the resultant LTB universe model is given. How to solve the inverse problem with the simultaneous big-bang and a given function $\da(z)$ for the distance-redshift relation is provided in the Appendix.