We have rigorously developed the embedded gravitational lensing theory for point mass lenses in a series of recent papers (Chen et al. 2010, 2011, 2015; Kantowski et al. 2010, 2012, 2013) including the embedded lens equation, time delays, lensing magnifications, shears, etc. We successfully extended the lowest-order embedded point mass lens theory to arbitrary spherically symmetric distributed lenses in Kantowski et al. (2013). The gravitational correctness of the theory follows from its origin in Einstein’s gravity. The embedded lens theory is based on the Swiss cheese cosmologies (Einstein & Straus 1945; Schücking 1954; Kantowski 1969). The idea of embedding (or Swiss cheese) is to remove a co-moving sphere of homogeneous dust from the background Friedmann–Lemaître–Robertson–Walker (FLRW) cosmology and replace it with the gravity field of a spherical inhomogeneity, maintaining the Einstein equations. In a Swiss cheese cosmology the total mass of the inhomogeneity (up to a small curvature factor) is the same as that of the removed homogeneous dust sphere. For a galaxy cluster, embedding requires the overdense cluster to be surrounded by large underdense regions often modeled as vacuum. For a cosmic void, embedding requires the underdense interior to be “compensated” by an overdense bounding ridge, i.e., a compensated void (Sato & Maeda 1983; Bertschinger 1985; Thompson & Vishniac 1987; Martínez-González et al. 1990; Amendola et al. 1999; Lavaux & Wandelt 2012). A low-density region without a compensating overdense boundary, or with an overdense boundary not containing enough mass to compensate the interior mass deficit, has a negative net mass (with respect to the homogeneous background) and is known as an “uncompensated” or “undercompensated” void (Fillmore & Goldreich 1984; Bertschinger 1985; Sheth & van de Weygaert 2004; Das & Spergel 2009).3

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This dichotomy between compensated and uncompensated voids is slightly different from one based on the classification of the small initial perturbations from which voids are thought to be formed. The initial perturbation can be compensated or uncompensated, which leads to different void growth scenarios (Bertschinger 1985), but if the evolved void formed from either perturbation is surrounded by an overdense shell that “largely” compensates the underdense region (i.e., the majority of the void mass is swept into the boundary shell in the snowplowing fashion when the void is growing), we still call it compensated because the small mass deficit originating in the initial perturbation is unimportant for gravitational lensing.
 Similarly, an overcompensated void has positive net mass with respect to the homogeneous FLRW background. Numerical or theoretical models of over-or undercompensated voids do commonly exist (e.g., Sheth & van de Weygaert 2004; Cai et al. 2010, 2014; Ceccarelli et al. 2013; Hamaus et al. 2014). We focus on compensated void models in this paper, given that uncompensated void models do not satisfy Einstein’s equations. The critical difference between an embedded lens and a traditional lens lies in the fact that embedding effectively reduces the gravitational potential’s range, i.e., partially shields the lensing potential because the lens mass is made a contributor to the mean mass density of the universe and not simply superimposed upon it. At lowest order, this implies that the repulsive bending caused by the removed homogeneous dust sphere must be accounted for when computing the bending angle caused by the lens mass inhomogeneity and legitimizes the prior practice of treating negative density perturbations as repulsive and positive perturbations as attractive. In this paper we investigate the gravitational lensing of cosmic voids using the lowest-order embedded lens theory (Kantowski et al. 2013). We introduce the embedded lens theory in Section 2, build the simplest possible lens model for a void in Section 3, and study the lensing of the CMB by individual cosmic voids in Section 4. Steps we outline can be followed for many void models of current interest.
Citation: Bertschinger 1985
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