Framework for describing perturbations to the cosmic microwave
  background from a gravitational wave burst with memory
  Gravitational wave bursts with memory (BWMs) can generate measurable,
long-lived frequency shifts and permanent angular deflections in distant
sources of light. These perturbations vary across the sky with a characteristic
spatial pattern and evolve slowly over long periods of time. In this work, we
develop formalism that can be used to describe how a BWM influences the spatial
pattern of temperature fluctuations in the cosmic microwave background (CMB).
We limit our attention to planar gravitational wave fronts -- this assumption
dramatically simplifies the necessary calculations. Using toy models of the
CMB's primary temperature variation pattern, we demonstrate that a BWM can mix
power from a spherical harmonic mode of a certain degree into modes of various
other degrees with vastly different $l$ values. In other words, BWM-induced
perturbations to the CMB at any angular scale depend in detail on the
unperturbed character of the CMB on all angular scales. The tools developed
herein will greatly facilitate future analyses of BWM-induced temperature
perturbations that incorporate all of the important physics underlying the CMB.
