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Abstract:

Abstract Symbolic regression (SR) has emerged as a powerful method for uncovering interpretable mathematical relationships from data, offering a novel route to both scientific discovery and efficient empirical modelling. This article introduces the Special issue on symbolic regression for the physical sciences, motivated by the Royal Society discussion meeting held in April 2025. The contributions collected here span applications from automated equation discovery and emergent-phenomena modelling to the construction of compact emulators for computationally expensive simulations. The introductory review outlines the conceptual foundations of SR, contrasts it with conventional regression approaches and surveys its main use cases in the physical sciences, including the derivation of effective theories, empirical functional forms and surrogate models. We summarize methodological considerations such as search-space design, operator selection, complexity control, feature selection and integration with modern AI approaches. We also highlight ongoing challenges, including scalability, robustness to noise, overfitting and computational complexity. Finally, we emphasize emerging directions, particularly the incorporation of symmetry constraints, asymptotic behaviour and other theoretical information. Taken together, the papers in this Special issue illustrate the accelerating progress of SR and its growing relevance across the physical sciences. This article is part of the discussion meeting issue ‘Symbolic regression in the physical sciences’.

Abstract:

We present a measurement of the Lyα intensity mapping power spectrum from the Hobby-Eberly Telescope Dark Energy Experiment (HETDEX). We measure the cross-power spectrum of the Lyα intensity and Lyα-emitting galaxies (LAEs) in a redshift range of 1.9 ≤ z ≤ 3.5. We calculate the intensity from HETDEX spectra that do not contain any detected LAEs above a signal-to-noise ratio of 5.5. To produce a power spectrum model and its covariance matrix, we simulate the data using lognormal mocks for the LAE catalog and Lyα intensity in redshift space. The simulations include the HETDEX sensitivity, selection function, and mask. The measurements yield the product of the LAE bias, the intensity bias, the mean intensity of undetected sources, and the ratio of the actual and fiducial redshift-space distortion parameters, bgbI〈I〉F¯RSD/F¯RSDfid= (6.7 ± 3.1), (11.7 ± 1.4), and (8.3 ± 1.5) × 10−22 erg s−1 cm−2 arcsec−2 Å−1 in three redshift bins centered at z¯=2.1 , 2.6, and 3.2, respectively. The results are reasonably consistent with cosmological hydrodynamical simulations that include Lyα radiative transfer. They are, however, significantly smaller than previous results from cross-correlations of quasars with Lyα intensity. These results demonstrate the statistical power of HETDEX for Lyα intensity mapping and pave the way for a more comprehensive analysis. They will also be useful for constraining models of Lyα emission from galaxies used in modern cosmological simulations of galaxy formation and evolution.

Abstract:

We present a search for z ≃ 7 Lyman-break galaxies using the 1.72 deg 2 near-infrared (NIR) UltraVISTA survey in the COSMOS field, reaching 5 σ depths in Y of 26.2. We incorporated deep Euclid optical and Euclid + Spitzer NIR imaging for a full spectral energy distribution (SED) fitting analysis. We found 289 candidate galaxies at 6.5 ≤ z ≤ 7.5 covering −22.6 ≤ M UV ≤ −20.2, faint enough to overlap with Hubble Space Telescope studies. We conducted a separate selection by including complementary Euclid performance verification imaging (reaching 5 σ depths of 26.3), yielding 140 galaxies in 0.65 deg 2 , with 38 sources unique to this sample. We computed the rest-frame UV luminosity function (UV LF) from our samples, extending below the knee ( M ∗ = 21.14 +0.28 −0.25 ). We find that the shape of the UV LF is consistent with both a Schechter function and a double power law (DPL) at the magnitudes probed by this sample, with a DPL preferred at M UV < −22.5 when bright-end results are included. The UltraVISTA + Euclid sample provides a clean measurement of the LF due to the overlapping NIR filters identifying molecular absorption features in the SEDs of ultra-cool dwarf interlopers, and additional faint galaxies were recovered. A comparison with JWST LFs at z > 7 suggests a gentle evolution in the bright-end slope, although this is limited by a lack of robust bright-end measurements at z > 9. We forecast that in the Euclid Deep Fields, the removal of contaminant ultra-cool dwarfs as point sources will be possible at J E < 24.5. Finally, we present a high-equivalent-width Lyman- α emitter candidate identified by combining HSC, VISTA, and Euclid broadband photometry, highlighting the synergistic power these instruments will have in the Euclid Auxiliary Fields for identifying extreme sources in the epoch of reionisation.

Abstract:

Context. This paper presents new results from the ESO Large Programme Looking into the faintEst WIth MUSE (LEWIS). The LEWIS sample consists of low-surface-brightness galaxies (LSBs) and ultra-diffuse galaxies (UDGs) located inside 0.4 R vir of the Hydra I cluster. Integral-field spectroscopy is acquired for 24 galaxies with the MUSE spectrograph mounted on the Very Large Telescope (VLT). Aims. Our main objectives are to analyse possible correlations between the environment and the integrated stellar population properties of our targets, based on which we infer clues about their formation. Methods. For each galaxy in the sample, we extracted the 1D stacked spectrum in an aperture of one effective radius R e and adopted previously published stellar kinematics to derive the age, metallicity, and [Mg/Fe] through a full spectral fitting technique. Results. We find that the analysed LEWIS sample has a mean metallicity of ⟨[M/H]⟩= − 0.9 ± 0.2 dex and a mean age of 10 ± 2 Gyr, comparable to previous results of UDGs in other clusters. According to their position in the projected phase space, galaxies can be classified into two groups: very early infaller galaxies, which on average have slightly higher metallicities (⟨[M/H]⟩ early = −0.8 ± 0.1 dex), and late infaller galaxies, with slightly lower values (⟨[M/H]⟩ late = −1.0 ± 0.1 dex). According to their properties, late-infallers tend to be rotation-91̽»¨ed systems. Conversely, two types of galaxies are found in the early-infall region. Roughly half have metallicities consistent with the dwarf galaxy mass–metallicity relation. The other half show higher metallicities (with ⟨[M/H]⟩≥ − 1.0 dex) and are located outside the 1 σ scatter of the mass-metallicity relation. The two subgroups of early-infallers also display different timescales for stellar mass assembly. Metal-rich galaxies reached 50% of their stellar mass in less than 1 Gyr and show a prolonged and almost constant star formation over more than 12 Gyr. The other galaxies exhibit a star formation history similar to that found for galaxies in the late-infall region. Both early and late infallers show solar-like α abundances. Conclusions. From the analysis of stellar population properties presented in this work and of stellar kinematics previously obtained from LEWIS, we identified different classes of UDGs within the Hydra I cluster – as shown by metallicities, quenching timescales, and kinematics – which suggest different formation mechanisms. Almost all of the UDGs and LSBs in this cluster are consistent with the puffed-up dwarf formation scenario, having dwarf-like metallicities and being consistent with the mass-metallicity relation for dwarfs. In the innermost regions of the cluster, where more metal-rich UDGs are found, tidal effects or the environment might have influenced their formation and evolution.

Abstract:

Context. Nearby galaxies exhibit a variety of structures, including so-called central or (circum-)nuclear rings that are similar to the Milky Way (MW) Central Molecular Zone (CMZ). These rings are common in barred galaxies and can be gas-rich and highly star-forming. Aims. We aim to study the molecular gas content and star formation rate of central rings within nearby galaxies and link them to global galaxy properties, especially the bar morphology. Methods. We utilized 1″(≲100 pc) resolution CO(2–1) observations from the PHANGS-ALMA survey, visually identifying 20 central rings and determine their properties. For 14 of these rings, MUSE observations tracing star formation rate (SFR) surface density were available. We derived the rings’ geometry, integrated molecular gas masses, SFRs, depletion times, and compared them to host galaxy and bar properties from the literature. Results. Molecular gas is an effective tracer for central rings. Previous studies have used ionized gas and dust tracers to identify central rings in galaxies of similar morphological types as the PHANGS galaxies (numerical Hubble type T ∼ −3 to T ∼ 9). In comparison, molecular gas yields similar fractions of galaxies hosting central rings and similar radii distributions. The gaseous central rings have typical radii of ∼ 400 +250 −150 pc, molecular gas masses of log( M / M ⊙ ) ∼ 8.1 +0.17 −0.23 , and SFRs of ∼ 0.21 +0.15 −0.16 M ⊙ /yr. As a result, they contribute 5.6 +4.5 −2.1 % and 13 +10 −5 % to their host galaxies’ molecular gas mass and SFR, respectively. While the MW CMZ sits at the lower end of the radius, molecular gas mass, and SFR distribution, it matches well in terms of ring molecular gas mass and SFR fraction, and depletion time. Longer bars contain more massive molecular central rings, but there is no correlation between the classical bar strength parameters ( Q b , ε bar , A 2 max ) and the ring’s molecular gas content. Conclusions. Although absolute central ring properties (ring radius, molecular gas mass, SFR) likely depend on host galaxy properties, the similarities between the MW CMZ and PHANGS central rings in relative parameters (molecular gas and SFR fraction, depletion time) suggest that the processes of gas inflow and star formation are similar for central rings across nearby galaxies.