Exact nonlinear inhomogeneities in $\Lambda$CDM cosmology
  At a time when galaxy surveys and other observations are reaching
unprecedented sky coverage and precision it seems timely to investigate the
effects of general relativistic nonlinear dynamics on the growth of structures
and on observations. Analytic inhomogeneous cosmological models are an
indispensable way of investigating and understanding these effects in a
simplified context.
  In this paper, we develop exact inhomogeneous solutions of general relativity
with pressureless matter (dust, describing cold dark matter) and cosmological
constant $\Lambda$, which can be used to model an arbitrary initial matter
distribution along one line of sight. In particular, we consider the second
class Szekeres models with $\Lambda$ and split their dynamics into a flat
$\Lambda$CDM background and exact nonlinear inhomogeneities, obtaining several
new results. One single metric function $Z$ describes the deviation from the
background. We show that $F$, the time dependent part of $Z$, satisfies the
familiar linear differential equation for $\delta$, the first-order density
perturbation of dust, with the usual growing and decaying modes. In the limit
of small perturbations, $\delta \approx F$ as expected, and the growth of
inhomogeneities links up exactly with standard perturbation theory. We provide
analytic expressions for the exact nonlinear $\delta$ and the growth factor in
our models. For the case of over-densities, we find that, depending on the
initial conditions, the growing mode may or may not lead to a pancake
singularity, analogous to a Zel'dovich pancake. This is in contrast with the
$\Lambda=0$ pure Einstein-de-Sitter background where, at any given point in
comoving (Lagrangian) coordinates pancakes will always occur.
