Abstract
In gravitational lensing, magnification cross sections characterize the probability that a light source will have magnification greater than some fixed value, which is useful in a variety of applications. The (area) cross section is known to scale as μ − 2 for fold caustics and μ − 2.5 for cusp caustics. We aim to extend the results to higher-order caustic singularities, focusing on the elliptic umbilic, which can be manifested in lensing systems with two or three galaxies. The elliptic umbilic has a caustic surface, and we show that the volume cross section scales as μ − 2.5 in the two-image region and μ − 2 in the four-image region, where μ is the total unsigned magnification. In both cases our results are supported both numerically and analytically.
Highlights
IntroductionKnowing the magnification cross section allows one to determine the probability that a light source will have magnification greater than some fixed value
Magnification cross sections are an important tool in gravitational lensing
It is well known that for the so-called “fold” and “cusp” caustic singularities, the area cross sections scale asymptotically as μ−2 and μ−2.5, respectively, where μ is the total unsigned magnification of a lensed source
Summary
Knowing the magnification cross section allows one to determine the probability that a light source will have magnification greater than some fixed value This in turn gives information about the accuracy of cosmological models, since these predict different probabilities regarding source magnifications For multiple-plane lensing, the fold and cusp scalings were determined by Petters et al ([4] Chapter 13). The asymptotic scaling of the elliptic umbilic volume cross section will depend on whether the source gives rise to two or four lensed images locally. In our derivation of μ−2 in the four-image region, we make use of a certain magnification relation that holds for higher-order caustic singularities, and the elliptic umbilic in particular, that was shown to hold in Aazami and Petters 2009 [19]
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