Testing homogeneity with galaxy number counts : light-cone metric and general low-redshift expansion for a central observer in a matter dominated isotropic universe without cosmological constant

TL;DR

This paper develops a fifth-order analytical formula for redshift spherical shell mass and introduces a gauge-invariant light-cone metric component to test inhomogeneous universe models as alternatives to dark energy.

Contribution

It provides a new analytical framework for describing light-cone geometry in LTB models, enabling tests of inhomogeneity without dark energy.

Findings

Derived a fifth-order formula for redshift spherical shell mass.

Introduced a gauge-invariant quantity G(z) for light-cone metric description.

Complete characterization of light-cone geometry in LTB models.

Abstract

As an alternative to dark energy it has been suggested that we may be at the center of an inhomogeneous isotropic universe described by a Lemaitre-Tolman-Bondi (LTB) solution of Einstein's field equations. In order to test this hypothesis we calculate the general analytical formula to fifth order for the redshift spherical shell mass. Using the same analytical method we write the metric in the light-cone by introducing a gauge invariant quantity $G(z)$ which together with the luminosity distance $D_L(z)$ completely determine the light-cone geometry of a LTB model.