Multi-Probe Mass Modelling

• Multi-Probe Mass Modelling is a technique that combines varied observables such as strong/weak lensing, X-ray, and SZ data to constrain mass distributions.
• It employs both parametric and non-parametric Bayesian methods to reduce degeneracies and distinguish dark matter from baryonic matter.
• Applications range from galaxy-scale lenses to cluster-scale reconstructions, enhancing robustness and enabling direct comparisons with cosmological simulations.

Multi-probe mass modelling is the joint inference of mass distributions from multiple observables that constrain different physical components, different projections of the same potential, or both. In the recent literature, the term encompasses galaxy-scale strong-lens analyses with multi-wavelength imaging and flexible lens potentials, cluster-scale reconstructions that combine strong or weak lensing with X-ray, Sunyaev–Zel’dovich, and stellar-kinematic data, and tomographic large-scale analyses that calibrate mass-related systematics such as the hydrostatic mass bias. Across these settings, the common objective is to reduce degeneracies, improve robustness against model inadequacy, and, in some cases, disentangle dark matter from baryonic mass at cluster and galaxy scales (Lange et al., 2024, Beauchesne et al., 9 Sep 2025, Ibitoye et al., 23 Feb 2026, Umetsu, 2013).

1. Observational content and probe complementarity

Multi-probe mass modelling relies on the fact that different datasets respond to different components of the mass distribution. In galaxy-scale strong lensing with JWST/NIRCam, multi-wavelength imaging is used to separate lens and source galaxy light more precisely and to model the lens mass distribution with increased accuracy and fewer degeneracies; the analysis of SPT0418-47 and SPT2147-50 uses several filters and is carried out with PyAutoLens (Lange et al., 2024). In cluster applications, strong lensing constrains the total projected mass distribution through positions of multiple images, X-ray surface brightness constrains the intra-cluster gas, and stellar kinematics constrain the stellar and total enclosed mass of cluster members, the brightest cluster galaxy (BCG), and the intra-cluster light (ICL) (Beauchesne et al., 9 Sep 2025, Beauchesne et al., 9 Sep 2025).

At larger scales, multi-probe tomographic analyses combine the thermal Sunyaev–Zel’dovich effect, weak lensing, and galaxy clustering. In that setting, the observables are the auto- and cross-angular power spectra

$$\left\{C_\ell^{yy},\ C_\ell^{gg},\ C_\ell^{\gamma\gamma},\ C_\ell^{y\gamma},\ C_\ell^{yg},\ C_\ell^{g\gamma}\right\},$$

computed for tomographic redshift-bin combinations. The tSZ signal is sensitive to hot ICM pressure, weak lensing measures gravitational potential via coherent distortions in background galaxy shapes, and galaxy clustering traces the underlying dark matter distribution with redshift-selected galaxy samples (Ibitoye et al., 23 Feb 2026).

A related lensing-specific formulation combines weak-lensing distortion, magnification bias, and strong-lensing projected mass estimates. In the model-free reconstruction of cluster mass profiles, magnification bias is measured with multiple populations of background sources, including red galaxies with strong depletion and blue galaxies with enhancement or mild decrement toward the cluster center, while strong lensing provides central aperture-mass constraints. This combination is designed to break the mass-sheet degeneracy and to improve the statistical precision of cluster mass measurements (Umetsu, 2013).

The complementarity extends to cluster thermodynamics. In CL J1226.9+3332, a joint analysis of the tSZ effect with the NIKA2 camera and X-ray data from XMM-Newton reconstructs thermodynamical properties and mass under hydrostatic equilibrium, while CLASH convergence maps provide an independent lensing mass estimate for comparison (Muñoz-Echeverría et al., 2021). A plausible implication is that “multi-probe” is less a single algorithm than a design principle: each additional observable is chosen because it carries a distinct dependence on the mass field or on one of its baryonic components.

2. Core modelling architectures

A first class of models augments otherwise smooth gravitational-lens potentials. In the JWST strong-lens analysis of SPT0418-47 and SPT2147-50, the baseline is a smooth elliptical profile perturbed by multipole terms of orders $m=1,3,4$. The perturbation is written as

$$\psi(\theta) = ... + A_m R^m \cos(m(\phi - \phi_m)),$$

with $A_m$ the amplitude, $m$ the multipole order, and $\phi_m$ the phase. The adopted interpretation is explicit: $m=1$ represents lopsidedness or offset, $m=3$ represents threefold asymmetries, and $m=4$ represents boxy/disky structures. Candidate subhaloes are introduced as NFW halos in the lens plane, parameterized by mass, position, and concentration (Lange et al., 2024).

A second class of models performs component-level decomposition in galaxy clusters. In Abell S1063, the intra-cluster gas is modelled as a sum of three free-form dual pseudo-isothermal elliptical (dPIE) potentials, with only large cut radii $r_{\rm cut}=1.25$ Mpc fixed. Each cluster member is modelled with two coincident dPIE profiles: a stellar component whose geometric parameters are fixed by the light profile, and a dark-matter component scaled relative to the baryonic one through

$m=1,3,4$0

The stellar-to-subhalo mass relation is parameterized as a double power law,

$m=1,3,4$1

For the BCG+ICL, the stellar component is described by a Multi-Gaussian Expansion fitted to HST photometry, the dark-matter component is a dPIE, and JAM is used to model stellar kinematics. Cluster-scale dark matter is represented by two dPIE halos, a main component and a North-East substructure (Beauchesne et al., 9 Sep 2025).

The preparatory measurements for the same system use dPIE light profiles for 289 cluster members in the HST F160W filter and an MGE model for the BCG+ICL. The dPIE light profile is written as

$m=1,3,4$2

while the generic MGE surface brightness is

$m=1,3,4$3

Those measurements provide the stellar mass and kinematic constraints required for the full multi-probe model (Beauchesne et al., 9 Sep 2025).

A third class of models addresses cluster geometry explicitly. In the CHEX-MATE CLUMP-3D analysis, the three-dimensional shape is described by

$m=1,3,4$4

with Euler angles specifying orientation. The gas density is modelled with a modified $m=1,3,4$5-model,

$m=1,3,4$6

and the pressure with a gNFW profile,

$m=1,3,4$7

Rather than assuming an ellipsoidal NFW density, the gravitational potential is taken to have isosurfaces that follow the ICM shape (Gavidia et al., 14 Jul 2025).

A fourth class is explicitly non-parametric in the recovered mass profile. In model-free multi-probe lensing, the unknown signal vector is a set of binned convergence values and the mean convergence inside the innermost aperture. The combined likelihood is

$m=1,3,4$8

with separate terms for shear, magnification bias, and strong lensing (Umetsu, 2013).

Finally, analytical halo-model formalisms extend multi-probe mass modelling to three-dimensional statistics of matter and baryonic fields. In that setting,

$m=1,3,4$9

with

$$\psi(\theta) = ... + A_m R^m \cos(m(\phi - \phi_m)),$$0

$$\psi(\theta) = ... + A_m R^m \cos(m(\phi - \phi_m)),$$1

This framework is used to jointly model weak lensing, tSZ, kSZ, X-rays, and FRB dispersion measures, with dark matter, gas, and stars maintained as physically connected components (S. et al., 17 Jul 2025).

3. Inference, evidence, and model selection

The statistical structure of multi-probe mass modelling is predominantly Bayesian. In gravitational-lens model selection, the core object is the model evidence

$$\psi(\theta) = ... + A_m R^m \cos(m(\phi - \phi_m)),$$2

and competing models are compared by the Bayes factor

$$\psi(\theta) = ... + A_m R^m \cos(m(\phi - \phi_m)),$$3

The explicit motivation is to account not only for goodness of fit but also for the size of the prior parameter space, thereby enforcing Occam’s razor quantitatively. The 2011 Bayesian lens-selection analysis shows that more complicated lens models may not be justified given the level of accuracy of the available data (Balmès, 2011).

The same logic is central in recent strong-lens substructure searches. In the JWST analysis, the evidence for a subhalo is quantified with a Bayes factor comparing models with and without substructure,

$$\psi(\theta) = ... + A_m R^m \cos(m(\phi - \phi_m)),$$4

and the calculation is performed with UltraNest (Lange et al., 2024).

In cluster multi-probe inference, likelihood construction is probe-specific. In Abell S1063, the strong-lensing term uses Gaussian positional errors $$\psi(\theta) = ... + A_m R^m \cos(m(\phi - \phi_m)),$$5 with $$\psi(\theta) = ... + A_m R^m \cos(m(\phi - \phi_m)),$$6; the X-ray surface-brightness term uses a negative binomial distribution with intrinsic scatter proportional to the predicted flux; cluster-member velocity dispersions and BCG/ICL kinematics enter through dynamical likelihoods, with the latter added by importance sampling after an initial tractable run. Model selection by WAIC favors the model that frees the BCG and ICL mass-to-light ratios separately, denoted “BCG-ML2” (Beauchesne et al., 9 Sep 2025).

Tomographic cluster-cosmology forecasts employ a Fisher matrix,

$$\psi(\theta) = ... + A_m R^m \cos(m(\phi - \phi_m)),$$7

with $$\psi(\theta) = ... + A_m R^m \cos(m(\phi - \phi_m)),$$8, and marginalization over cosmological, astrophysical, and nuisance parameters, including per-bin galaxy bias perturbations, photometric-redshift shifts, intrinsic alignments, and baryonic feedback modelled with HMCode2020 (Ibitoye et al., 23 Feb 2026).

Other implementations use MCMC-based posterior sampling. The model-free lensing reconstruction is sampled with Metropolis–Hastings MCMC (Umetsu, 2013), the CHEX-MATE triaxial fit uses affine-invariant MCMC with emcee (Gavidia et al., 14 Jul 2025), and the CL J1226.9+3332 analysis combines posteriors from different transfer-function and pressure-profile choices to propagate systematic uncertainty (Muñoz-Echeverría et al., 2021).

4. Representative systems and quantitative results

The galaxy-scale strong-lens analysis of SPT2147-50 and SPT0418-47 provides a clear demonstration of how multi-probe mass modelling addresses degeneracy between angular mass complexity and dark-matter substructure. In SPT2147-50 there is a strong preference for including multipoles, with amplitudes of $$\psi(\theta) = ... + A_m R^m \cos(m(\phi - \phi_m)),$$9 for $A_m$0 and $A_m$1 for $A_m$2. Without multipoles, the Bayes factor for a subhalo is $A_m$3; with multipoles, it decreases to $A_m$4, which still corresponds to a $A_m$5 detection of a subhalo with an NFW mass

$A_m$6

SPT0418-47 shows no statistically significant evidence for angular mass complexity and no significant evidence for dark matter substructure (Lange et al., 2024).

In Abell S1063, the best-fitting multi-probe mass model achieves an RMS of $A_m$7 on the multiple image positions. The kinematic profiles of the BCG+ICL, as well as the X-ray surface brightness of the intra-cluster gas, are accurately reproduced within observational uncertainties, although a $A_m$8 scatter is required for the cluster member line-of-sight dispersions. The inferred stellar-to-subhalo mass relation is in $A_m$9 agreement with the prediction from IllustrisTNG, and the ICL stellar mass derived from the model is consistent with estimates from stellar population modelling (Beauchesne et al., 9 Sep 2025).

The same cluster also illustrates the measurement burden required before a full component-level decomposition can be attempted. For Abell S1063, the first paper in the series measures the light distribution, stellar mass, and kinematics of the cluster members, BCG, and ICL; obtains light profiles for $m$0 cluster members; derives stellar masses using three different SED models; and measures line-of-sight velocity dispersions at half-light radii for cluster members, with elliptical annular apertures for the BCG+ICL (Beauchesne et al., 9 Sep 2025).

At cluster scale, the NIKA2/XMM-Newton/CLASH analysis of CL J1226.9+3332 yields

$m$1

and

$m$2

implying a hydrostatic-to-lensing bias consistent with $m$3 within $m$4 (Muñoz-Echeverría et al., 2021).

The CHEX-MATE triaxial analysis of Abell 1689 shows how geometry changes the inferred mass. The cluster is elongated along the line of sight relative to the plane of sky by

$m$5

The triaxial fit gives

$m$6

whereas the spherically symmetric fit gives

$m$7

The concentration from the triaxial fit is

$m$8

consistent with the spherical value of

$m$9

The non-thermal pressure fraction is measured between $\phi_m$0 and $\phi_m$1 Mpc, with a minimum of approximately $\phi_m$2 per cent at intermediate radii and near $\phi_m$3 per cent at both the smallest and largest radii, with a typical measurement precision of $\phi_m$4 per cent (Gavidia et al., 14 Jul 2025).

In model-free cluster lensing, the combination of weak-lensing tangential shear, magnification-bias profiles from red and blue source populations, and strong-lensing aperture masses was applied to CLASH data for MACS J1206.2-0847. The reconstruction uses $\phi_m$5 radial bins, two strong-lensing constraints, and two color-selected magnification samples, for a total of

$\phi_m$6

constraints (Umetsu, 2013).

5. Mass decomposition, degeneracy control, and physically meaningful outputs

A central use of multi-probe mass modelling is the separation of dark matter from baryonic components. In Abell S1063, the method is explicitly designed to disentangle the dark matter distribution from the baryonic mass at both cluster and galaxy scales by combining constraints on total mass and on individual baryonic components. The recovered two-dimensional mass distributions of dark matter, ICL+BCG stars, and gas exhibit spatial offsets and shapes in line with physical expectations. In radial terms, baryons dominate within $\phi_m$7 kpc, gas is dominant at $\phi_m$8 kpc, and baryons contribute up to $\phi_m$9 in the center, decreasing to $m=1$0 at $m=1$1 kpc. Relative to previous models, total-mass errors decrease by $m=1$2, and the dark-matter-only profile has $m=1$3 error across the mapped region (Beauchesne et al., 9 Sep 2025).

A distinct but related degeneracy is the one between subhaloes and inadequacies in the smooth mass model. The JWST strong-lens analysis is explicit that both angular multipoles and dark matter subhaloes can produce lensing anomalies such as flux-ratio or image-position perturbations. The adopted procedure is therefore to fit models both including and excluding multipole terms; if the evidence for substructure remains strong after angular terms are included, this suggests a real subhalo, whereas a major loss of evidence suggests that the anomaly may primarily arise from complex angular mass structure (Lange et al., 2024).

Projection effects introduce another class of degeneracy. The CHEX-MATE result for Abell 1689 shows that weak-lensing masses obtained under a spherical assumption can be significantly biased for a cluster elongated along the line of sight, while a triaxial fit mitigates that bias by jointly fitting X-ray, SZ, and weak-lensing data (Gavidia et al., 14 Jul 2025). A related mechanism operates in model-free lensing: magnification transforms differently from shear under the mass-sheet transformation, so adding magnification bias and strong lensing to distortion data yields an absolute, non-parametric mass determination (Umetsu, 2013).

Multi-probe outputs are also designed for direct comparison to simulations. The Abell S1063 analysis states that its results, including the stellar-to-subhalo mass relation and the distribution of each mass component, can be directly compared to hydrodynamical cosmological simulations such as IllustrisTNG (Beauchesne et al., 9 Sep 2025). This suggests that the distinctive value of multi-probe models lies not only in lower residuals or tighter error bars, but in the fact that the inferred quantities map onto physically interpretable components.

6. Cosmological calibration, performance forecasts, and methodological limits

In cluster cosmology, multi-probe mass modelling is used to calibrate the hydrostatic mass bias,

$m=1$4

which is identified as a leading systematic uncertainty and a principal source of degeneracy with $m=1$5 and $m=1$6. The forecast framework based on tSZ, galaxy clustering, and weak lensing finds marginalized constraints on $m=1$7 of $m=1$8 for SO+LSST, $m=1$9 for CMB-S4+LSST, and $m=3$0 for CMB-S4+CSST. Tomography improves $m=3$1 precision by factors of approximately three relative to non-tomographic analyses, and optical-only probes provide no direct constraint on $m=3$2, whereas inclusion of tSZ-containing spectra enables percent-level calibration under realistic systematic assumptions. The same analysis reports that $m=3$3 contributes over $m=3$4 of the Fisher information for $m=3$5 when joint with tomography (Ibitoye et al., 23 Feb 2026).

Analytical halo-model studies extend this calibration logic to baryonic feedback and to the interpretation of multi-wavelength datasets. Tested against the Magneticum hydrodynamical simulation, the assessed models reproduce the relevant three-dimensional power spectra at sub-percent to few-percent accuracy, depending on tracer combination and number of free parameters. Their ability to recover underlying halo properties varies considerably. The best-case models predict gas and stellar fractions within $m=3$6 RMS error when power spectra and mass fractions are both fit, and thermodynamic profiles of group- and cluster-scale halos within $m=3$7 RMS error, while performance degrades for other models or with insufficient constraints (S. et al., 17 Jul 2025).

The literature is correspondingly careful about methodological limits. The Bayesian lens-selection study emphasizes that the assumption of more complicated lens models may not be justified given the level of accuracy of the available data (Balmès, 2011). The JWST strong-lens work states that even with incredibly high-quality JWST data, distinguishing subhaloes from angular mass features is nontrivial, and further analysis is needed to confirm that the SPT2147-50 signal is not due to systematics associated with the lens mass model (Lange et al., 2024). The halo-model validation study likewise notes that high fit accuracy does not guarantee physical realism and concludes that the models require further refinement and testing for reliable interpretation of multi-wavelength datasets (S. et al., 17 Jul 2025).

Taken together, these results define multi-probe mass modelling less by a single preferred parameterization than by a methodological standard: combine observables with different physical dependencies, propagate probe-specific and modelling uncertainties through a unified inference scheme, and treat model complexity as something that must be earned by the data.