  The Lemaitre-Tolman-Bondi solution has received much attention as a possible
alternative to Dark Energy, as it is able to account for the apparent
acceleration of the Universe without any exotic matter content. However, in
order to make rigorous comparisons between these models and cosmological
observations, such as the integrated Sachs-Wolfe effect, baryon acoustic
oscillations and the observed matter power spectrum, it is absolutely necessary
to have a proper understanding of the linear perturbation theory about them.
Here we present this theory in a fully general, and gauge-invariant form. It is
shown that scalar, vector and tensor perturbations interact, and that the
natural gauge invariant variables in Lemaitre-Tolman-Bondi cosmology do not
correspond straightforwardly to the usual Bardeen variables, in the limit of
spatial homogeneity. We therefore construct new variables that reduce to pure
scalar, vector and tensor modes in this limit.
