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Abstract:

Dark matter haloes are typically characterized by radial density profiles with fixed forms motivated by simulations (e.g. Navarro-Frenk-White [NFW]). However, simulation predictions depend on uncertain dark matter physics and baryonic modelling. Here, we present a method to constrain halo density profiles directly from observations using Exhaustive Symbolic Regression (ESR), a technique that searches the space of analytic expressions for the function that best balances accuracy and simplicity for a given dataset. We test the approach on mock weak lensing excess surface density (ESD) data of synthetic clusters with NFW profiles. Motivated by real data, we assign each ESD data point a constant fractional uncertainty and vary this uncertainty and the number of clusters to probe how data precision and sample size affect model selection. For fractional errors around 5%, ESR recovers the NFW profile even from samples as small as approximately 20 clusters. At higher uncertainties representative of current surveys, simpler functions are favoured over NFW, though it remains competitive. This preference arises because weak lensing errors are smallest in the outskirts, causing the fits to be dominated by the outer profile. ESR therefore provides a robust, simulation-independent framework both for testing mass models and determining which features of a halo's density profile are genuinely constrained by the data. This article is part of the discussion meeting issue 'Symbolic regression in the physical sciences'.

Abstract:

Physics seeks to uncover the laws of Nature and express them through mathematical equations . Despite the vast diversity of natural phenomena, physical equations exhibit structural regularities that set them apart from arbitrary mathematical expressions. While principles such as dimensional analysis have long guided the formulation of physical models, the exploration of more subtle statistical patterns within the equations of physics remains an open question. Here, by analysing four corpora of physics equations and applying advanced implicit-likelihood techniques, we find that the frequency of mathematical operators follows an exponential decay law, in contrast to Zipf's power law for word frequencies in natural languages. This reveals a statistical meta-law of physics, possibly reflecting a combination of communication efficiency and constraints imposed by Nature itself. The meta-law offers practical benefits for symbolic regression by drastically narrowing down the space of physically plausible expressions. More broadly, it may inform the development of language models that can generate coherent mathematical representations, advancing the automation of physical law discovery. This article is part of the discussion meeting issue 'Symbolic regression in the physical sciences'.

Abstract:

Symbolic regression (SR) is the machine learning (ML) method for learning functions from data. After a brief overview of the SR landscape, I will describe the two main challenges that traditional algorithms face: they have an unknown (and probably significant) probability of failing to find any given good function, and they suffer from ambiguity and poorly justified assumptions in their function-selection procedure. To address these, I propose an exhaustive search and model selection by the minimum description length (MDL) principle, which allows accuracy and complexity to be directly traded off by measuring each in units of information. I showcase the resulting publicly available Exhaustive Symbolic Regression (ESR) algorithm on three open problems in astrophysics: the expansion history of the universe, the effective behaviour of gravity in galaxies and the potential of the inflaton field. In each case, the algorithm identifies many functions superior to the literature standards. This general-purpose methodology should find widespread utility in science and beyond. This article is part of the discussion meeting issue 'Symbolic regression in the physical sciences'.

Abstract:

Measuring galaxy rotation curves is critical for inferring the properties of dark-matter haloes in the Lambda Cold Dark Matter (ΛCDM) paradigm. We present H i rotation curves and mass models for 20 galaxies from the MIGHTEE survey. Using extended H i kinematics, we construct resolved mass models that include stellar, gaseous, and dark-matter components. Stellar masses are derived using 3.6 μm imaging under fixed mass-to-light ratio (ϒ* = M/L) assumptions and are complemented, for the first time for a H I-selected sample, by spatially resolved M/L, obtained from multi-wavelength SED fitting. We examine the ratio of baryonic to observed rotation velocity (Vbar/Vobs) at the characteristic radius R2.2. Adopting a fixed ϒ⋆ = 0.5 M⊙/L⊙ yields a clear dependence of V2.2/Vobs on galaxy luminosity, while adopting ϒ⋆ = 0.2 M⊙/L⊙ substantially weakens this trend. In contrast, the resolved M/L analysis preserves the luminosity dependence while modifying the stellar contribution on a galaxy-by-galaxy basis, providing a more accurate representation of the underlying relation. We model the dark-matter haloes using Navarro–Frenk–White profiles and find that the different assumptions for a fixed a M/L systematically shift galaxies relative to the theoretical stellar-to-halo mass and baryonic-to-halo mass relations, while the spatially varying M/L yields the closest agreement with theoretical benchmarks within ΛCDM. We therefore demonstrate that future investigations of the dark matter properties of galaxies using rotation curves need to account for varying M/L across individual galaxy profiles and between galaxies in order to obtain accurate measurements of the dark matter, and therefore test ΛCDM.

Abstract:

If dark matter annihilates into standard model particles with a cross section which is velocity dependent, then Local Group dwarf galaxies will not be the best place to search for the resulting gamma ray emission. A greater flux would be produced by more distant and massive halos, with larger velocity dispersions. We construct full-sky predictions for the gamma ray emission from galaxy- and cluster-mass halos within $∼200\mathrm{Mpc}$ using a suite of constrained $N$ -body simulations () based on the Bayesian Origin Reconstruction from Galaxies algorithm. Comparing to observations from the Large Area Telescope and marginalizing over reconstruction uncertainties and other astrophysical contributions to the flux, we obtain constraints on the cross section which are 2 (7) orders of magnitude tighter than those obtained from dwarf spheroidals for $p$ -wave ( $d$ -wave) annihilation. We find no evidence for either type of annihilation from dark matter particles with masses in the range $m_{\mathrm{χ}}=2\mathrm{–}500\mathrm{GeV}/c^2$ , for any channel. As an example, for annihilations producing bottom quarks with $m_{\mathrm{χ}}=10\mathrm{GeV}/c^2$ , we find $a_1<2.4×{10}^{−21}{\mathrm{cm}}^3{\mathrm{s}}^{−1}$ and $a_2<3.0×{10}^{−18}{\mathrm{cm}}^3{\mathrm{s}}^{−1}$ at 95% confidence, where the product of the cross section, $\mathrm{σ}$ , and relative particle velocity, $v$ , is given by $\mathrm{σ}v=a_{ℓ}(v/c{)}^{2ℓ}$ and $ℓ=1$ , 2 for $p$ - and $d$ -wave annihilation, respectively. Our bounds, although failing to exclude the thermal relic cross section for velocity-dependent annihilation channels, are among the tightest to date.