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Session 27 - Gravitational Lensing & Dark Matter.
Oral session, Monday, January 13
For a low optical depth (\tau \ll 1) planar distribution of point mass lenses, we derive the macroimage magnification distribution P(A) at high magnification (A-1 \gg \tau^2) by modeling the illumination pattern as a superposition of the patterns due to individual ``point mass plus weak shear'' lenses. By convolving the magnification cross-section of the point mass plus weak shear lens with the shear distribution, we obtain a caustic-induced feature in P(A) which exhibits a simple scaling property and results in a 20% enhancement at A \approx 2/\tau. We also derive P(A) for low magnification (A-1 \ll 1), taking into account the correlations in the magnification of the microimages. The low-A distribution has a peak of amplitude \sim 1/\tau^2 at A-1 \sim \tau^2. We combine the low- and high-A results and obtain a practical semi-analytic expression for P(A).
For a low optical depth three-dimensional lens distributions. we show that the multiplane lens equation near a point mass can be reduced to the single plane equation of a point mass perturbed by weak shear. This allows us to calculate the caustic-induced feature in P(A) as a weighted sum of the feature derived in the planar case. The resulting semi-analytic feature is similar to the feature in the planar case, but it does not have any simple scaling properties, and it is shifted to higher magnification.
The semi-analytic distributions are compared to previous numerical results for \tau \sim 0.1. They are in better agreement in the three-dimensional case. We explain this by re-examining the criterion for low optical depth. For \tau \approx 0.1, a simple argument shows that the fraction of caustics of individual lenses that merge with those of their neighbors is \approx 20% for the three-dimensional case, much smaller than the \approx 1-\exp(-8 \tau) \approx 55% for the planar case.
Program listing for Monday