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.