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Abstract: The inverse problem with Lema\^itre-Tolman-Bondi LTB universe models isdiscussed. The LTB solution for the Einstein equations describes thespherically symmetric dust-filled spacetime. The LTB solution has two physicalfunctional degrees of freedom of the radial coordinate. The inverse problem isconstructing an LTB model requiring that the LTB model be consistent withselected important observational data. In this paper, we assume that theobserver is at the center and consider the distance-redshift relation $\da$ andthe redshift-space mass density $\mu$ as the selected important observationaldata. We give $\da$ and $\mu$ as functions of the redshift $z$. Then, weexplicitly show that, for general functional forms of $\daz$ and $\muz$,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 %interms of $\daz$ and $\muz$ is satisfied by the distance-redshift relationand the redshift-space mass density in $\Lambda$CDM models. Deriving regulardifferential equations for the inverse problem with the distance-redshiftrelation and the redshift-space mass density in $\Lambda$CDM models, wenumerically solve them for the case $\Omega { mM0},\Omega {\Lambda0}=0.3,0.7$. A set of analytic fitting functions for theresultant LTB universe model is given. How to solve the inverse problem withthe simultaneous big-bang and a given function $\daz$ for thedistance-redshift relation is provided in the Appendix.

Autor: Chul-Moon Yoo

Fuente: https://arxiv.org/

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