[Paper Review] A forward-modelling method to infer the dark matter particle mass from strong gravitational lenses

Qiuhan He, Andrew Robertson+8 more|arXiv (Cornell University)|Oct 25, 2020

Galaxies: Formation, Evolution, Phenomena109 references33 citations

TL;DR

This paper proposes a forward-modelling method using strong gravitational lensing residuals to constrain the mass of dark matter particles, particularly testing warm dark matter (WDM) models. By simulating lensed images with varying dark matter halo populations and comparing power spectra of residuals via Approximate Bayesian Computation, the method achieves a median 2σ constraint of 𝑚DM > 4.10 keV for a CDM-like model with a halo mass cutoff at 10⁷ M⊙, and recovers 𝑚DM ≈ 1.43–3.21 keV in a WDM universe with true 𝑚DM = 2.22 keV.

ABSTRACT

A fundamental prediction of the cold dark matter (CDM) model of structure formation is the existence of a vast population of dark matter haloes extending to subsolar masses. By contrast, other dark matter models, such as a warm thermal relic (WDM), predict a cutoff in the mass function at a mass which, for popular models, lies approximately between $10^7$ and $10^{10}~{ m M}_\odot$. We use mock observations to demonstrate the viability of a forward modelling approach to extract information about low-mass dark haloes lying along the line-of-sight to galaxy-galaxy strong lenses. This can be used to constrain the mass of a thermal relic dark matter particle, $m_\mathrm{DM}$. With 50 strong lenses at Hubble Space Telescope resolution and a maximum pixel signal-to-noise ratio of $\sim50$, the expected median 2$\sigma$ constraint for a CDM-like model (with a halo mass cutoff at $10^{7}~{ m M}_\odot$) is $m_\mathrm{DM} > 4.10 \, \mathrm{keV}$ (50% chance of constraining $m_{ m DM}$ to be better than 4.10 keV). If, however, the dark matter is a warm particle of $m_\mathrm{DM}=2.2 \, \mathrm{keV}$, our 'Approximate Bayesian Computation' method would result in a median estimate of $m_\mathrm{DM}$ between 1.43 and 3.21 keV. Our method can be extended to the large samples of strong lenses that will be observed by future telescopes, and could potentially rule out the standard CDM model of cosmogony. To aid future survey design, we quantify how these constraints will depend on data quality (spatial resolution and integration time) as well as on the lensing geometry (source and lens redshifts).

Motivation & Objective

• To develop a statistical method that extracts information about low-mass dark matter haloes from strong gravitational lensing residuals.
• To test whether the power spectrum of image residuals can constrain the half-mode mass (𝑀hf), which describes the cutoff in the halo mass function due to warm dark matter.
• To evaluate the sensitivity of the method to data quality (spatial resolution, signal-to-noise) and lensing geometry (source and lens redshifts).
• To assess the viability of using statistical lensing signatures to rule out or constrain the standard ΛCDM cold dark matter model.
• To provide a framework for future constraints using upcoming large samples of strong lenses from space telescopes like Euclid and CSST.

Proposed method

• Generate mock strong lensing images using a forward-modelling approach, simulating lensed arcs from a smooth macro model with added perturbations from low-mass dark matter haloes.
• Vary the dark matter particle mass (𝑚DM) to produce different halo mass function cutoffs, parametrized by the half-mode mass (𝑀hf).
• Apply the same lens modelling and residual analysis pipeline to both the observed data and the simulated forward models to extract image residuals.
• Compute the power spectrum of the residuals in both observed and simulated data to quantify the statistical signature of line-of-sight haloes.
• Use Approximate Bayesian Computation (ABC) to compare the power spectra of observed and simulated residuals, producing a posterior distribution for 𝑀hf.
• Combine results from 50 lensing systems to improve statistical constraints, accounting for intrinsic scatter in halo realizations.

Experimental results

Research questions

• RQ1 Can the power spectrum of image residuals in strong lensing arcs statistically constrain the half-mode mass (𝑀hf) of the dark matter halo mass function?
• RQ2 What is the expected constraint on the dark matter particle mass (𝑚DM) using 50 Hubble Space Telescope-quality strong lensing systems with a maximum signal-to-noise ratio of ~50?
• RQ3 How does the constraint on 𝑚DM depend on lensing geometry, such as source redshift and spatial extent of the Einstein ring?
• RQ4 How does data quality—specifically spatial resolution and integration time—affect the sensitivity to low-mass dark matter haloes?
• RQ5 Can this method distinguish between a cold dark matter (CDM) model (with cutoff at 10⁷ M⊙) and a warm dark matter (WDM) model (with 𝑚DM = 2.22 keV)?

Key findings

• With 50 strong lensing systems at Hubble Space Telescope resolution and a signal-to-noise ratio of ~50, the method achieves a median 2σ constraint of 𝑚DM > 4.10 keV for a CDM-like model with a halo mass cutoff at 10⁷ M⊙.
• In a WDM universe with a true particle mass of 𝑚DM = 2.22 keV (corresponding to 𝑀hf = 10⁹ M⊙), the method yields a median 2σ estimate of 𝑚DM between 1.43 keV and 3.21 keV.
• The constraint tightens significantly with higher source redshifts; a source at 𝑧 = 2.5 yields a 2σ upper limit of 5.20 keV for 𝑚DM when the true value is 8.86 keV.
• Improved data quality—such as higher resolution or longer integration times—leads to tighter constraints, with Euclid/CSST-quality data still providing meaningful constraints on 𝑀hf.
• The method remains robust even with intrinsic scatter in halo realizations, though this scatter limits precision, necessitating large sample sizes to achieve tight constraints.
• The approach is extendable to future large-scale lens surveys, offering a viable path to potentially rule out the standard ΛCDM model if low-mass haloes are found to be absent or underabundant.