Accelerating expansion or inhomogeneity? A comparison of the $\Lambda$CDM and Lema\^{\i}tre - Tolman models

It is shown how certain observations interpreted in the background of the Friedmann model with $\Lambda < 0 = k$ (the $\Lambda$CDM model) can be re-interpreted using the $\Lambda = 0$ Lema\^{\i}tre - Tolman (L-T) model so as to do away with the "dark energy". The purpose of the paper is to clarify the underlying geometrical relations by doing the calculations as much as possible analytically or by very simple numerical programs. In the first part of the paper (fictitious) observations of the distribution of expansion velocity along the past light cone of the observer are considered. It is shown that the whole past light cone of the $\Lambda$CDM observer can be reproduced in the L-T model with $\Lambda = 0 = E$. This is a geometric exercise that has the advantage of being free of numerical complications. In the second part, the luminosity distance - redshift relation of the $\Lambda$CDM model is duplicated using the L-T model with $-k = 2E/r^2 =$ constant $> 0$. The value of $k$ and the function $t_B(r)$ are determined by the $\Lambda$CDM parameters. General properties of this L-T model are described. Difficulties of carrying the numerical calculations through the apparent horizon are presented in detail and mostly solved. The second model is a counterexample to the claim that an L-T model mimicking $\Lambda$CDM must contain a void around the center - it has a peak of density at $R = 0$.