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 $R = 0$.
