Embedded Lensing Time Delays, the Fermat Potential, and the Integrated
  Sachs-Wolfe Effect
  We derive the Fermat potential for a spherically symmetric lens embedded in
an FLRW cosmology and use it to investigate the late-time integrated
Sachs-Wolfe (ISW) effect, i.e., secondary temperature fluctuations in the
cosmic microwave background (CMB) caused by individual large scale clusters and
voids. We present a simple analytical expression for the temperature
fluctuation in the CMB across such a lens as a derivative of the lens' Fermat
potential. This formalism is applicable to both linear and nonlinear density
evolution scenarios, to arbitrarily large density contrasts, and to all open
and closed background cosmologies. It is much simpler to use and makes the same
predictions as conventional approaches. In this approach the total temperature
fluctuation can be split into a time-delay part and an evolutionary part. Both
parts must be included for cosmic structures that evolve and both can be
equally important. We present very simple ISW models for cosmic voids and
galaxy clusters to illustrate the ease of use of our formalism. We use the
Fermat potentials of simple cosmic void models to compare predicted ISW effects
with those recently extracted from WMAP and \emph{Planck} data by stacking
large cosmic voids using the aperture photometry method. If voids in the local
universe with large density contrasts are no longer evolving we find that the
time delay contribution alone predicts values consistent with the measurements.
However, we find that for voids still evolving linearly, the evolutionary
contribution cancels a significant part of the time delay contribution and
results in predicted signals that are much smaller than recently observed.
