Low red-shift formula for the luminosity distance in a LTB model with cosmological constant

TL;DR

This paper derives a low red-shift Taylor expansion for luminosity distance in a spherically symmetric inhomogeneous universe with a cosmological constant, aiding understanding of local inhomogeneity effects.

Contribution

It provides a general low red-shift formula for luminosity distance in LTB models with a cosmological constant, without assuming specific inhomogeneity profiles.

Findings

Formulas accurately match numerical calculations

Applicable to various inhomogeneity profiles

Useful for studying local universe effects

Abstract

We calculate the low red-shift Taylor expansion for the luminosity distance for an observer at the center of a spherically symmetric matter inhomogeneity with a non vanishing cosmological constant. We then test the accuracy of the formulas comparing them to the numerical calculation for different cases for both the luminosity distance and the radial coordinate. The formulas can be used as a starting point to understand the general non linear effects of a local inhomogeneity in presence of a cosmological constant, without making any special assumption about the inhomogeneity profile.