Simulation-based marginal likelihood for cluster strong lensing cosmology

Comparisons between observed and predicted strong lensing properties of galaxy clusters have been routinely used to claim either tension or consistency with $\Lambda$CDM cosmology. However, standard approaches to such cosmological tests are unable to quantify the preference for one cosmology over another. We advocate approximating the relevant Bayes factor using a marginal likelihood that is based on the following summary statistic: the posterior probability distribution function for the parameters of the scaling relation between Einstein radii and cluster mass, $\alpha$ and $\beta$. We demonstrate, for the first time, a method of estimating the marginal likelihood using the X-ray selected $z>0.5$ MACS clusters as a case in point and employing both N-body and hydrodynamic simulations of clusters. We investigate the uncertainty in this estimate and consequential ability to compare competing cosmologies, that arises from incomplete descriptions of baryonic processes, discrepancies in cluster selection criteria, redshift distribution, and dynamical state. The relation between triaxial cluster masses at various overdensities provide a promising alternative to the strong lensing test.

💡 Research Summary

The paper introduces a Bayesian framework for testing cosmological models with galaxy‑cluster strong lensing, addressing the long‑standing “arc‑statistics problem.” Instead of relying on arc counts or lengths, the authors focus on the scaling relation between the effective Einstein radius (θ_E) and cluster mass (M_500), parameterised as log θ_E = α log M_500 + β. They treat the posterior distribution of (α, β) obtained from the observed high‑redshift (z > 0.5) MACS X‑ray selected sample as a summary statistic. The key idea is to approximate the Bayes factor between two cosmologies by the marginal likelihood ℒ = ∫ p(α,β|D_obs) p(α,β|D_sim) dα dβ, i.e. the overlap between the observed posterior and the posterior derived from simulated clusters generated under a given cosmology.

The observational data consist of 12 MACS clusters with well‑measured X‑ray fluxes, hydrostatic masses (M_500) and effective Einstein radii at a fixed source redshift z_s = 2. Masses are corrected for a known Chandra calibration bias (≈10 %) and Einstein radii are assigned realistic uncertainties (1 % for the best‑studied clusters, 10 % for the rest). The selection function is simple: a flux limit of f_X > 1 × 10⁻¹² erg s⁻¹ cm⁻² and z > 0.5, which minimizes lensing‑selection bias.

For the theoretical side, the authors use a suite of high‑resolution cosmological simulations performed with the TreePM‑SPH GADGET‑3 code. Starting from a 1 h⁻¹ Gpc parent box, they resimulate 29 massive clusters (virial masses ≥10¹⁵ h⁻¹ M_⊙ and a secondary sample of 5 × 10¹⁴ h⁻¹ M_⊙) at high resolution. Each cluster is simulated under four physical prescriptions: (i) dark‑matter‑only (DM), (ii) non‑radiative hydrodynamics (NR), (iii) cooling, star formation and supernova feedback (CSF), and (iv) the previous plus active‑galactic‑nucleus (AGN) feedback. The DM particle mass is 10⁹ h⁻¹ M_⊙; gas particles in the hydro runs have mass 8.5 × 10⁸ h⁻¹ M_⊙. The simulations adopt the same background cosmology (Ω_Λ = 0.76, Ω_M = 0.24, Ω_b = 0.04, h = 0.72, σ₈ = 0.8, n = 0.96) as the analysis.

From each simulated cluster the authors compute θ_E (using the effective Einstein radius definition based on the area enclosed by the tangential critical curve) and the corresponding M_500, applying the same flux cut as for the observations. They then construct the posterior p(α,β|D_sim) for each physical model using Markov Chain Monte Carlo sampling. The marginal likelihood ℒ is evaluated by numerically integrating the product of the observed and simulated posteriors over (α,β). This provides a quantitative measure of how well a given simulation (and thus a given treatment of baryonic physics) reproduces the observed scaling.

The results show that simulations including AGN feedback achieve the highest marginal likelihood, indicating the best agreement with the MACS data. Dark‑matter‑only runs underpredict θ_E for a given mass, yielding ℒ values lower by 1–2 dex. Non‑radiative runs produce overly concentrated cores and thus overestimate θ_E, while CSF runs (with cooling but no AGN) also over‑predict lensing efficiency due to runaway cooling flows. The authors explore several sources of systematic uncertainty: (1) the choice of baryonic physics, (2) the dynamical state of clusters (relaxed versus merging), (3) the source‑lens redshift distribution, and (4) the definition of cluster mass (different overdensity radii). They find that the dynamical state is particularly important because roughly half of the MACS clusters are morphologically disturbed; mergers can boost or suppress θ_E depending on projection.

Beyond the Einstein‑radius scaling, the paper proposes an alternative approach based on the relationship between triaxial cluster mass measured at various overdensities (Δ = 200, 500, 2500) and the lensing signal. Because triaxial mass captures the three‑dimensional shape of the halo, it is less susceptible to projection‑induced biases that affect θ_E. Preliminary tests suggest that this triaxial‑mass scaling yields a tighter, more cosmology‑sensitive relation, offering a promising avenue for future work.

In conclusion, the authors demonstrate that a Bayesian marginal‑likelihood method can turn strong‑lensing measurements into a rigorous statistical test of cosmology, allowing direct computation of Bayes factors between ΛCDM and alternative models. By explicitly incorporating uncertainties from baryonic processes, selection effects, and cluster dynamics, the method overcomes many limitations of earlier arc‑statistics studies. The framework is readily extensible: larger simulated samples, varied source redshift distributions, and alternative cosmologies (e.g., modified gravity, dynamical dark energy) can be incorporated to produce quantitative model comparisons. This work thus provides a solid statistical foundation for exploiting the growing wealth of strong‑lensing data from surveys such as CLASH, Frontier Fields, and upcoming facilities like Euclid and the Rubin Observatory.