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**Session 27 - Gravitational Lensing & Dark Matter.**

*Oral session, Monday, January 13*

*Frontenac Ballroom, *

## [27.03] Statistics of Gravitational Microlensing Magnification

*M. H. Lee (Queen's), L. Kofman (IfA, Hawaii), N. Kaiser (CITA), A. Babul (NYU)*
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**