The Einstein-Straus model consists of a Schwarzschild spherical vacuole in a
Friedman-Lema^ tre-Robertson-Walker (FLRW) dust spacetime (with or without ).
It constitutes the most widely accepted model to answer the question of the in
uence
of large scale (cosmological) dynamics on local systems. The conclusion drawn by
the model is that there is no in
uence from the cosmic background, since the spher-
ical vacuole is static. Spherical generalizations to other interior matter models are
commonly used in the construction of lumpy inhomogeneous cosmological models.
On the other hand, the model has proven to be reluctant to admit non-spherical
generalizations. In this review, we summarize the known uniqueness results for
this model. These seem to indicate that the only reasonable and realistic non-
spherical deformations of the Einstein-Straus model require perturbing the FLRW
background. We review results about linear perturbations of the Einstein-Straus
model, where the perturbations in the vacuole are assumed to be stationary and
axially symmetric so as to describe regions (voids in particular) in which the matter
has reached an equilibrium regime.M.M. acknowledges financial support under the projects FIS2012-30926 (MICINN) and P09-FQM-4496 (J. Andalucia-FEDER). F. M. thanks the warm hospitality from Instituto de Fisica, UERJ, Rio de Janeiro, Brasil, projects PTDC/MAT/108921/2008 and CERN/FP/123609/2011 from Fundacao para a Ciencia e a Tecnologia (FCT), as well as CMAT, Univ. Minho, for support through FEDER funds Programa Operacional Factores de Competitividade (COMPETE) and Portuguese Funds from FCT within the project PEst-C/MAT/UI0013/2011. R. V. thanks the kind hospitality from the Universidad de Salamanca, where parts of this work have been produced, and financial support from project IT592-13 of the Basque Government, and FIS2010-15492 from the MICINN
