Symbolically regressing dark matter halo profiles using weak lensing

TL;DR

This work tackles the bias introduced by fixed dark matter density templates in cluster mass inferences by introducing Exhaustive Symbolic Regression (ESR), which derives analytic halo profiles directly from weak-lensing data under the MDL criterion. Applying ESR to 149 HSC-XXL clusters, the study finds two-parameter profiles that statistically outperform NFW and other literature forms, though inner regions remain weakly constrained by current data. Mass estimates inferred via model averaging over ESR profiles are on average higher than NFW-based masses, highlighting a potential systematic bias when imposing a fixed profile. The results support a universal, self-similar halo profile across the sample and demonstrate ESR’s broad applicability to diverse datasets and cosmological probes, with implications for cluster-based cosmology and future surveys.

Abstract

The structure of dark matter haloes is often described by radial density profiles motivated by cosmological simulations. These are typically assumed to have a fixed functional form (e.g. NFW), with some free parameters that can be constrained with observations. However, relying on simulations has the disadvantage that the resulting profiles depend on the dark matter model and the baryonic physics implementation, which are highly uncertain. Instead, we present a method to constrain halo density profiles directly from observations. This is done using a symbolic regression algorithm called Exhaustive Symbolic Regression (ESR). ESR searches for the optimal analytic expression to fit data, combining both accuracy and simplicity. We apply ESR to a sample of 149 galaxy clusters from the HSC-XXL survey to identify which functional forms perform best across the entire sample of clusters. We identify density profiles that statistically outperform NFW under a minimum-description-length criterion. Within the radial range probed by the weak-lensing data ($R \sim 0.3 - 3$ h$^{-1}$ Mpc), the highest-ranked ESR profiles exhibit shallow inner behaviour and a maximum in the density profile. As a practical application, we show how the best-fitting ESR models can be used to obtain enclosed mass estimates. We find masses that are, on average, higher than those derived using NFW, highlighting a source of potential bias when assuming the wrong density profile. These results have important knock-on effects for analyses that utilise clusters, for example cosmological constraints on $σ_8$ and $Ω_m$ from cluster abundance and clustering. Beyond the HSC dataset, the method is readily applicable to any data constraining the dark matter distribution in galaxies and galaxy clusters, such as other weak lensing surveys, galactic rotation curves, or complementary probes.

Figures (5)

• Figure 1: Example weak-lensing ESD measurements for three clusters spanning the range of SNRs in the XXL–HSC sample. The first three panels show clusters with low (Cluster XLSSC $77$, SNR $= -0.1$), medium (Cluster XLSSC $101$, SNR $= 3.4$), and high (Cluster XLSSC $91$, SNR $= 7.4$) weak-lensing SNRs. Points denote the measured ESD ($\Delta \Sigma$) with associated shape-noise uncertainties and the blue dashed marks $\Delta\Sigma = 0$. The fourth panel shows the distribution of weak-lensing SNR for all clusters in the sample. The median SNR is $1.25$.
• Figure 2: Illustration of parameter uncertainty estimation from the conditional likelihood $\mathcal{L}(\theta_i)$. The red dashed line marks the maximum-likelihood (ML) estimate, and the shaded region corresponds to the 68% confidence interval. For our analysis, we define a symmetric uncertainty $\pm\hat{\sigma}$ (blue dashed lines), where $\hat{\sigma}$ is set to the larger of the two distances from the ML value to the boundaries of the 68% confidence interval.
• Figure 3: Best-fitting functions at ecah complexity according to the change in description length, $\Delta L (D)$, and the likelihood, $\mathcal{L}$, relative to the corresponding minima. The markers show the position in the $L(D)$ plane (red) and likelihood plane (blue) of some common dark matter halo profiles. The Burkert and Dekel-Zhao profiles are not included here for clarity, since they have very poor $L(D)$. Only the $L(D)$ values without the Katz prior are shown, since including the Katz prior produces very similar results. The two NFW points correspond to two parametrisations used in the analysis. NFW $1$ is the standard form, $\rho(r)=\theta_0 / (r(|\theta_1|+r)^2)$ and NFW $2$ is $\rho(r) = \theta_0 / (r (1/|\theta_1| + r)^2)$.
• Figure 4: Plots of the top $2$ best-fitting functions, together with the top analytically integrable functions discussed in the text. The NFW profile is also shown for comparison. The top panels correspond to cluster XLSSC $91$, which has the highest SNR in the sample, while the bottom panels show cluster XLSSC $101$, which is a medium SNR cluster. These two clusters are also shown in \ref{['fig:SNR']}. Left panels: ESD fits to the data. The horizontal dashed line marks zero. Right panels: Corresponding density profiles plotted over a wide radial range. The vertical dashed lines indicate the radial range covered by the data. Shaded regions represent the $68\%$ credible intervals derived from the posterior parameter distributions.
• Figure 5: Comparison of enclosed mass estimates, $M_{\mathrm{mESR}}$, obtained from the weighted combination of our best-fitting functions against those derived from the standard NFW profile, $M_{\mathrm{NFW}}$ for XXL clusters with a signal-to-noise ratio (SNR) higher than $2.5$. The top panel shows the results in log--log space. The dashed black line indicates the 1:1 relation, while the solid red line shows the best-fit linear relation obtained using the roxy package, with the shaded region representing the $1\sigma$ confidence interval. The bottom panel shows the stacked posterior distribution of the logarithmic mass ration $\delta_i = \log_{10}(M_{\mathrm{mESR}}/M_{\mathrm{NFW}})$. Each cluster contributes its individual posterior, built by propagating the uncorrelated uncertainties on both mass estimates under a Gaussian approximation. The black dashed line shows the $1:1$ relation and the solid red line marks median of the stacked distribution, $\langle\delta\rangle = 0.17$ dex.