[Paper Review] Gravitational lensing with stochastic substructure: Effects of the clump mass function and spatial distribution

Charles R. Keeton|ArXiv.org|Aug 20, 2009

Galaxies: Formation, Evolution, Phenomena6 references3 citations

TL;DR

This paper develops a stochastic lensing framework to study how gravitational lensing observables—flux ratios, image positions, and time delays—depend on the mass function and spatial distribution of dark matter substructures in lens galaxies. It finds that magnification perturbations depend primarily on total substructure mass near images, while position and time delay perturbations are most sensitive to an effective mass scale $ m_{\text{eff}} = \langle m^2 \rangle / \langle m \rangle $, with contributions from both local and global clump populations.

ABSTRACT

Mass clumps in gravitational lens galaxies can perturb lensed images in characteristic ways. Strong lens flux ratios have been used to constrain the amount of dark matter substructure in lens galaxies, and various other observables have been considered as additional probes of substructure. We study the general theory of lensing with stochastic substructure in order to understand how lensing observables depend on the mass function and spatial distribution of clumps. We find that magnification perturbations are mainly sensitive to the total mass in substructure projected near the lensed images; when the source is small, flux ratios are not very sensitive to the shape of the clump mass function. Position perturbations are mainly sensitive to a characteristic clump mass scale, namely m_eff = /, with some mild dependence on other mass moments when the spatial distribution is not uniform. They have contributions from both "local" and "global" populations of clumps (i.e., those projected near the images, and those farther away). Time delay perturbations are sensitive to the same characteristic mass, m_eff, and mainly driven by the global population of clumps. While there is significant scatter in all lensing quantities, there are some non-trivial correlations that may contain further information about the clump population. Our results indicate that a joint analysis of multiple lens observables will offer qualitatively new constraints on the mass function and spatial distribution of dark matter substructure in distant galaxies.

Motivation & Objective

• To develop a theoretical framework for stochastic gravitational lensing that accounts for arbitrary mass functions and spatial distributions of dark matter substructures.
• To understand how different lensing observables—flux ratios, image positions, and time delays—respond to variations in substructure properties.
• To identify which lensing quantities are most sensitive to specific substructure characteristics, such as mass function shape and spatial clustering.
• To enable joint analysis of multiple observables to constrain the full population of dark matter substructures beyond mean density.

Proposed method

• Formalism based on statistical lensing theory, treating clump masses and positions as random variables with specified probability distributions.
• Derivation of expected values and covariances of lensing quantities (magnification, position, time delay) using ensemble averaging over clump populations.
• Use of the characteristic function method to compute probability distributions, accounting for non-Gaussian behavior due to divergent variances in individual terms.
• Introduction of a normalized surface mass density $ \bar{\kappa}_s(w) $ to describe the spatial distribution of clumps, enabling generalization beyond uniform distributions.
• Definition of normalized lensing response functions $ \hat{f}_i = f_i / m_i $ to separate mass dependence and isolate sensitivity to spatial and mass distribution.
• Asymptotic approximation for large $ N $ (number of clumps), showing that covariance scales with $ \langle m^2 \rangle / \langle m \rangle $, leading to the effective mass scale $ m_{\text{eff}} $.

Experimental results

Research questions

• RQ1 How do flux ratios in strong lensing systems depend on the total mass and spatial distribution of substructures near the images?
• RQ2 What is the role of the clump mass function in shaping magnification and position perturbations in stochastic lensing?
• RQ3 How do time delay perturbations depend on the effective mass scale $ m_{\text{eff}} = \langle m^2 \rangle / \langle m \rangle $ and the global population of distant clumps?
• RQ4 To what extent do correlations between different lensing observables carry information about the substructure population beyond mean density?
• RQ5 Can joint analysis of multiple observables break degeneracies in substructure parameter estimation?

Key findings

• Magnification perturbations are primarily sensitive to the total mass in substructures projected near the lensed images, with weak dependence on the shape of the mass function when the source is small.
• Position perturbations are most sensitive to the effective mass scale $ m_{\text{eff}} = \langle m^2 \rangle / \langle m \rangle $, with minor contributions from higher-order mass moments when the spatial distribution is non-uniform.
• Time delay perturbations are dominated by the global population of clumps and are also primarily sensitive to $ m_{\text{eff}} $, with minimal dependence on local substructures.
• The covariance of lensing quantities scales as $ \langle m^2 \rangle / \langle m \rangle $, confirming $ m_{\text{eff}} $ as the key characteristic mass scale for position and time delay perturbations.
• Despite significant scatter in individual observables, non-trivial correlations between them contain additional information about the substructure population’s mass function and spatial distribution.
• A joint analysis of multiple observables—flux ratios, image positions, and time delays—offers qualitatively new constraints on the physical properties of dark matter substructure in lens galaxies.