An embedded lens is a gravitational lens such that the mass of the lens is a part of the mean mass density of the background universe and not simply superimposed upon it as is done in the standard gravitational lensing theory.
Contents
For a homogeneous background Universe, a spherical sphere is removed and a lens of mass equal to the removed dust sphere is placed at the center of the void. The mass condensation can be either a point mass or distributed mass, but should be spherically symmetric with respect to the center of the void. If the background universe also contains a non-vanishing cosmological constant Λ, then Λ is required to be the same inside and outside of the void. The metric describing the geometry within the void can be Schwarzschild or Kottler depending on whether there is a non-zero cosmological constant.
Embedding a lens effectively reduces the gravitational potential's range, i.e., partially shields the lensing potential produced by the lens mass condensation. For example, a light ray grazing the boundary of a Kottler/Schwarzschild void will not be bended by the lens mass condensation (i.e., does not feel the gravitational potential of the embedded lens) and travels along a straight line path in a flat background universe.
Properties
In order to be an analytical solution of the Einstein's field equation, the embedded lens has to satisfy the following conditions:
- The mass of the embedded lens (point mass or distributed), should be the same as that from the removed sphere.
- The mass distribution within the void should be spherically symmetric.
- The cosmological constant should be the same inside and outside of the embedded lens.
History
A universe with inhomogeneities (galaxies, clusters of galaxies, large voids, etc.) represented by spherical voids containing mass condensations described as above is called a Swiss Cheese Universe. The concept of Swiss Cheese Universe was first invented by Einstein and Straus in 1945. Swiss Cheese model has been used extensively to model inhomogeneities in the Universe. For an example, effects of large scale inhomogeneities (such as superclusters) on the observed anisotropy of the temperatures of cosmic microwave background radiation (CMB) was investigated by Rees and Sciama in 1968 using Swiss cheese model (the so-called Rees-Sciama effect). Distance redshift relation in Swiss cheese universe has been investigated by Ronald Kantowski in 1969, and Dyer & Roeder in the 1970s. The gravitational lensing theory for a single embedded point mass lens in flat pressure-less Friedman-Lemaître-Robertson-Walker (FLRW) background universe with non-zero cosmological constant has been built by Ronald Kantowski, Bin Chen, and Xinyu Dai in a series papers.
Embedded Lens vs. Classical Gravitational Lens
The key difference between an embedded lens and a traditional lens is that the mass of an embedded lens contributes to the mean of the cosmological density, whereas that of a standard lens does not. Consequently, the gravitational potential of an embedded lens has a finite range, i.e., there is no lensing effect outside of the void. This is different from a standard lens where the gravitational potential of the lens has an infinite range.
As a consequence of embedding, the bending angle, lens equation, image amplification, image shear, and time delay between multiple images of an embedded lens are all different from those of a standard linearized lens. For example, the potential part of the time delay between image pairs, and the weak lensing shear of embedded lens can differ from the standard gravitational lensing theory by more than a few percents.
For an embedded point mass lens, the lens equation to the lowest order can be written
where