# On simple analytic models of microlensing amplification statistics

## TL;DR

This paper reviews four simple analytic models for microlensing amplification statistics, introduces a corrected derivation for the multiplicative model, and evaluates their accuracy against simulations, highlighting limitations at high optical depths.

## Contribution

It presents a new derivation of the multiplicative amplification distribution and compares four models' effectiveness against ray-shooting simulations.

## Key findings

- All models perform well at low optical depth.
- Models fail to capture features at high optical depth.
- Lens-lens coupling is crucial at large optical depths.

## Abstract

Gravitational microlensing is a key probe of the nature of dark matter and its distribution on the smallest scales. For many practical purposes, confronting theory to observation requires to model the probability that a light source is highly amplified by many-lens systems. This article reviews four simple analytic models of the amplification probability distribution, based on different approximations: (i) the strongest-lens model; (ii) the multiplicative model, where the total amplification is assumed to be the product of all the lenses' individual amplifications; (iii) a hybrid version of the previous two; and (iv) an empirical fitting function. In particular, a new derivation of the multiplicative amplification distribution is proposed, thereby correcting errors in the literature. Finally, the accuracy of these models is tested against ray-shooting simulations. They all produce excellent results as long as lenses are light and rare (low optical depth); however, for larger optical depths, none of them succeeds in capturing the relevant features of the amplification distribution. This conclusion emphasizes the crucial role of lens-lens coupling at large optical depths.