Luminosity distance-redshift relation for the LTB solution near the center

TL;DR

This paper derives the luminosity distance-redshift relation for the LTB dust solution near the center, providing a unified explicit form and showing it matches the homogeneous case up to second order under regularity conditions.

Contribution

The authors introduce a new explicit and unified form of the LTB solution, enabling the computation of the luminosity distance function up to third order in redshift.

Findings

The luminosity distance function matches that of a homogeneous universe up to second order.

The new LTB solution form simplifies calculations of $D_L(z)$ near the center.

The solution is regular at the center, ensuring physical plausibility.

Abstract

Motivated by the inverse problem for the Lemaitre-Tolman-Bondi dust solution, in which problem the luminosity distance function $D_L(z)$ is taken as an input to select a specific model, we compute the function $D_L(z)$ of the LTB solution up to the third order of $z$. To perform the otherwise cumbersome computation, we introduce a new convenient form of the LTB solution, in which the solution is explicit and unified. With this form of the LTB solution we obtain the luminosity distance function with the full generality. We in particular find that the function exactly coincides with that of a homogeneous and isotropic dust solution up to second order, if we demand that the solution be regular at the center.