Gravitational lensing shear by an exotic lens object with negative convergence or negative mass

Gravitational lens models with negative convergence (surface mass density projected onto the lens plane) inspired by modified gravity theories, exotic matter, and energy have been recently discussed in such a way that a static and spherically symmetric modified spacetime metric depends on the inverse distance to the power of positive $n$ ($n=1$ for Schwarzschild metric, $n=2$ for Ellis wormhole) in the weak-field approximation [T. Kitamura, K. Nakajima, and H. Asada, Phys. Rev. D 87, 027501 (2013)], and it has been shown that demagnification of images could occur for $n>1$ lens models associated with exotic matter (and energy), though they cause the gravitational pull on light rays. The present paper considers gravitational lensing shear by the demagnifying lens models and other models such as negative-mass compact objects causing the gravitational repulsion on light rays like a concave lens. It is shown that images by the lens models for the gravitational pull are tangentially elongated, whereas those by the repulsive ones are radially distorted. This feature of lensed image shapes may be used for searching (or constraining) localized exotic matter or energy with gravitational lensing surveys. It is suggested also that an underdense region such as a cosmic void might produce radially elongated images of background galaxies rather than tangential ones.

Figure 1

$\kappa$, ${\lambda }_{+}$, and ${\lambda }_{-}$ for $\epsilon >0$. They are denoted by solid (blue), dotted (purple), and dashed (red) curves, respectively. The horizontal axis denotes the image position $\theta$ in the units of the Einstein radius. Top left: $n=0.5$. Top right: $n=1$. Bottom left: $n=2$. Bottom right: $n=3$.

Figure 2

Numerical figures of lensed images for attractive ($\epsilon >0$) and repulsive ($\epsilon <0$) cases. They are denoted by dashed curves. We take $n=2$. The source for each case is denoted by solid circles, which are located on the horizontal axis and the vertical one for $\epsilon <0$ and $\epsilon >0$, respectively.

Figure 3

Repulsive lens model ($\epsilon <0$). Solid curves denote $1/{\widehat{\theta }}^n$ and straight lines mean $\widehat{\theta }-\widehat{\beta }$. Their intersections correspond to image positions that are roots for the lens equation. There are three cases: No image for a small $\widehat{\beta }$ (dot-dashed line), a single image for a particular $\widehat{\beta }$ (dotted line), and two images for a large $\widehat{\beta }$ (dashed line). The two images are on the same side of the lens object.

Figure 4

$\kappa$, ${\lambda }_{+}$, and ${\lambda }_{-}$ for $\epsilon <0$. They are denoted by solid (blue), dotted (purple) and dashed (red) curves, respectively. The horizontal axis denotes the image position $\theta$ in the units of the Einstein radius. Top left: $n=0.5$. Top right: $n=1$. Bottom left: $n=2$. Bottom right: $n=3$.