Probing the integrated Sachs-Wolfe effect using embedded lens models
  The photometry profile of the integrated Sachs-Wolfe (ISW) effect recently
obtained by the Planck consortium by stacking patches of Cosmic Microwave
Background (CMB) sky maps around a large number of cosmic voids, contains a
cold ring at about half the void's effective radius surrounded by a hot ring
near the void's boundary. The source of the temperature structure is assumed to
be the ISW effect but the exact cause of the ringed structure is not currently
well understood, particularly the outer hot ring. Numerical simulations have
suggested that hot/cold ring structures can be produced by motions associated
with nonlinear growths of cosmic structures whose gravitational potentials
produce the ISW effect. We have recently developed the embedded lens theory and
the Fermat potential formalism which can be used to model the ISW effect caused
by intervening individual lens inhomogeneities evolving arbitrarily. This
theory only requires knowledge of the void's projected mass profile as a
function of the passing CMB photons' impact radius and the rate of change of
that mass distribution at passage. We present two simple embedded void lens
models with evolving mass densities and investigate the ISW effect caused by
these lenses. Both models posses expanding mass shells which produce hot rings
around central cold regions, consistent with the recent observations. By adding
a small over-density at the void's center we can produce the slight positive
temperature excess hinted at in Planck's photometric results. We conclude that
the embedded lens theory and the Fermat potential formalism is well suited for
modeling the ISW effect.
