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

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.

💡 Research Summary

The paper addresses the possibility that the observed cosmic acceleration could be explained without invoking dark energy, by placing the observer at the centre of a spherically symmetric but radially inhomogeneous universe described by a Lemaitre‑Tolman‑Bondi (LTB) solution of Einstein’s equations. The authors develop a fully analytical framework that extends previous low‑order treatments to fifth order in redshift, allowing precise predictions for both the redshift‑dependent spherical shell mass, (M(z)), and the geometry of the observer’s past light‑cone.

First, they start from the LTB metric in comoving coordinates ((t,r,\theta,\phi)) with dust as the only source ((\Omega_\Lambda=0)). By expressing the radial coordinate as a function of the observed redshift (z) and the angular diameter distance (D_A(z)), they derive a Taylor expansion of the mass enclosed within a spherical shell at redshift (z). The expansion reads
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