The current standard model of cosmology successfully describes a variety of measurements, but the nature of its main ingredients, dark matter and dark energy, remains unknown. Euclid is a medium-class mission in the Cosmic Vision 2015–2025 programme of the European Space Agency (ESA) that will provide high-resolution optical imaging, as well as near-infrared imaging and spectroscopy, over about 14 000 deg2 of extragalactic sky. In addition to accurate weak lensing and clustering measurements that probe structure formation over half of the age of the Universe, its primary probes for cosmology, these exquisite data will enable a wide range of science. This paper provides a high-level overview of the mission, summarising the survey characteristics, the various data-processing steps, and data products. We also highlight the main science objectives and expected performance.

Making the most of next-generation galaxy clustering surveys requires overcoming challenges in complex, non-linear modelling to access the significant amount of information at smaller cosmological scales. Field-level inference has provided a unique opportunity beyond summary statistics to use all of the information of the galaxy distribution. However, addressing current challenges often necessitates numerical modelling that incorporates non-differentiable components, hindering the use of efficient gradient-based inference methods. In this paper, we introduce Learning the Universe by Learning to Optimize (LULO), a gradient-free framework for reconstructing the 3D cosmic initial conditions. Our approach advances deep learning to train an optimization algorithm capable of fitting state-of-the-art non-differentiable simulators to data at the field level. Importantly, the neural optimizer solely acts as a search engine in an iterative scheme, always maintaining full physics simulations in the loop, ensuring scalability and reliability. We demonstrate the method by accurately reconstructing initial conditions from halos identified in a dark matter-only N-body simulation with a spherical overdensity algorithm. The derived dark matter and halo overdensity fields exhibit cross-correlation with the ground truth into the non-linear regime Mpc. Additional cosmological tests reveal accurate recovery of the power spectra, bispectra, halo mass function, and velocities. With this work, we demonstrate a promising path forward to non-linear field-level inference surpassing the requirement of a differentiable physics model.

We study the mass growth histories of the haloes of Milky Way and M31 analogues formed in constrained cosmological simulations of the Local Group. These gravity-only simulations constitute a fair and representative set of Λ cold dark matter (ΛCDM) realisations conditioned on the masses, positions, and relative velocities of the two big haloes and on the observed recession velocities at the positions of nearby isolated galaxies. Our M31 haloes have similar mass growth histories as the isolated analogues in the TNG dark-matter-only simulations, while our Milky Ways typically form earlier, with suppressed growth at late times. On average, our Milky Ways assemble half their halo mass by ⟨z₅₀⟩ = 1.4 and our M31s by ⟨z₅₀⟩ = 1.2, whereas ⟨z₅₀⟩ = 1.1 for their isolated analogues. Mass growth associated with major and minor mergers is also biased early for the Milky Way in comparison to M31. Most accretion occurs 1–4 Gyr after the Big Bang; growth at later times is relatively quiescent. Based on the mass ratio and time of infall, we find that 32% of our Milky Ways experienced a Gaia-Enceladus-Sausage-like merger, 13% host a massive Large Magellanic Cloud-like satellite at the present day, and 5% have both. In one case, a Small Magellanic Cloud analogue and a Sagittarius analogue are also present, showing that the most important mergers of the Milky Way can be reproduced in its Local Group environment in ΛCDM. We find that the material that makes up the Milky Way and M31 haloes at the present day initially collapsed onto a plane roughly aligned with the Local Sheet and super-galactic plane; after z ∼ 2, accretion occurred mostly within this plane, with the tidal effects of the heavier companion, M31, significantly impacting the late growth history of the Milky Way.

We present the first results from the Manticore Project, dubbed Manticore-Local, a suite of Bayesian constrained simulations of the nearby Universe, generated by fitting a physical structure formation model to the 2M++ galaxy catalogue using the borg algorithm. This field-level inference yields physically consistent realizations of cosmic structure, leveraging a non-linear gravitational solver, a refined galaxy bias model, and physics-informed priors. The Manticore-Local posterior realizations evolve within a parent cosmological volume statistically consistent with Lambda-cold dark matter, demonstrated through extensive posterior predictive tests of power spectra, bispectra, initial condition Gaussianity, and the halo mass function. The inferred local supervolume (⁠R<200 Mpc, or z≲0.05⁠) shows no significant deviation from cosmological expectations; notably, we find no evidence for a large local underdensity, with the mean density suppressed by only ≈5per cent relative to the cosmic mean. Our model identifies high-significance counterparts for 14 prominent galaxy clusters – including Virgo, Coma, and Perseus – each within 1 deg of its observed sky position. Across the posterior ensemble, these counterparts are consistently detected with 2σ–4σ significance, and their reconstructed masses and redshifts agree closely with observational estimates, confirming the inference’s spatial and dynamical fidelity. The peculiar velocity field recovered by Manticore-Local achieves the highest Bayesian evidence across five independent data sets, surpassing state-of-the-art non-linear models, linear theory, Wiener filtering, and machine learning approaches. Unlike methods yielding only point estimates or using simplified dynamics, Manticore-Local provides a full Bayesian posterior over cosmic structure and evolution, enabling rigorous uncertainty quantification. These results establish Manticore-Local as the most advanced constrained realization suite of the local Universe to date, offering a robust statistical foundation for future studies of galaxy formation, velocity flows, and environmental dependencies in our cosmic neighbourhood.

We construct an antihalo void catalogue of 150 voids with radii 𝑅>10ℎ⁻¹Mpc in the Local Super-Volume (⁠<135ℎ⁻¹Mpc from the Milky Way), using posterior resimulation of initial conditions inferred by field-level inference with Bayesian Origin Reconstruction from Galaxies (BORG). We describe and make use of a new algorithm for creating a single, unified void catalogue by combining different samples from the posterior. The catalogue is complete out to 135ℎ⁻¹Mpc⁠, with void abundances matching theoretical predictions. Finally, we compute stacked density profiles of those voids which are reliably identified across posterior samples, and show that these are compatible with Λ cold dark matter expectations once environmental selection (e.g. the estimated ∼4 per cent underdensity of the Local Super-Volume) is accounted for.

Analysing next-generation cosmological data requires balancing accurate modelling of non-linear gravitational structure formation and computational demands. We propose a solution by introducing a machine learning-based field-level emulator, within the Hamiltonian Monte Carlo-based Bayesian Origin Reconstruction from Galaxies (BORG) inference algorithm. Built on a V-net neural network architecture, the emulator enhances the predictions by first-order Lagrangian perturbation theory to be accurately aligned with full N-body simulations while significantly reducing evaluation time. We test its incorporation in BORG for sampling cosmic initial conditions using mock data based on non-linear large-scale structures from N-body simulations and Gaussian noise. The method efficiently and accurately explores the high-dimensional parameter space of initial conditions, fully extracting the cross-correlation information of the data field binned at a resolution of Mpc. Percent-level agreement with the ground truth in the power spectrum and bispectrum is achieved up to the Nyquist frequency. Posterior resimulations-using the inferred initial conditions for N-body simulations-show that the recovery of information in the initial conditions is sufficient to accurately reproduce halo properties. In particular, we show highly accurate halo mass function and stacked density profiles of haloes in different mass bins. As all available cross-correlation information is extracted, we acknowledge that limitations in recovering the initial conditions stem from the noise level and data grid resolution. This is promising as it underscores the significance of accurate non-linear modelling, indicating the potential for extracting additional information at smaller scales.

We present a novel approach based on Bayesian field-level inference that provides representative ΛCDM initial conditions for simulation of the Local Group (LG) of galaxies and its neighbourhood, constrained by present-day observations. We extended the Bayesian Origin Reconstruction from Galaxies (BORG) algorithm with a multi-resolution approach, allowing us to reach the smaller scales needed to apply the constraints. Our data model simultaneously accounts for observations of mass tracers within the dark haloes of the Milky Way (MW) and M31, for their observed separation and relative velocity, and for the quiet surrounding Hubble flow, represented by the positions and velocities of 31 galaxies at distances between one and four megaparsec. Our approach delivers representative posterior samples of ΛCDM realisations that are statistically and simultaneously consistent with all of these observations, leading to significantly tighter mass constraints than found if the individual datasets are considered separately. In particular, we estimate the virial masses of the MW and M31 to be log₁₀(M_200c/M_⊙) = 12.07 ± 0.08 and 12.33 ± 0.10, respectively, their sum to be log₁₀(ΣM_200c/M_⊙) = 12.52 ± 0.07, and the enclosed mass within spheres of radius R to be log₁₀(M(R)/M_⊙) = 12.71 ± 0.06 and 12.96 ± 0.08 for R = 1 Mpc and 3 Mpc, respectively. The M31-MW orbit is nearly radial for most of our ΛCDM realisations, and most of them feature a dark matter sheet aligning approximately with the supergalactic plane, despite the surrounding density field not being used explicitly as a constraint. High-resolution, high-fidelity resimulations from initial conditions identified using the approximate simulations of our inference scheme continue to satisfy the observational constraints, demonstrating a route to future high-resolution, full-physics ΛCDM simulations of ensembles of LG look-alikes, all of which closely mirror the observed properties of the real system and its immediate environment.

Galaxies of different types are not equally distributed in the Local Universe. In particular, the supergalactic plane is prominent among the brightest ellipticals, but inconspicuous among the brightest disk galaxies. This striking difference provides a unique test for our understanding of galaxy and structure formation. Here we use the SIBELIUS DARK constrained simulation to confront the predictions of the standard Lambda Cold Dark Matter (ΛCDM) model and standard galaxy formation theory with these observations. We find that SIBELIUS DARK reproduces the spatial distributions of disks and ellipticals and, in particular, the observed excess of massive ellipticals near the supergalactic equator. We show that this follows directly from the local large-scale structure and from the standard galaxy formation paradigm, wherein disk galaxies evolve mostly in isolation, while giant ellipticals congregate in the massive clusters that define the supergalactic plane. Rather than being anomalous as earlier works have suggested, the distributions of giant ellipticals and disks in the Local Universe and in relation to the supergalactic plane are key predictions of the ΛCDM model.

Primordial features, in particular oscillatory signals, imprinted in the primordial power spectrum of density perturbations represent a clear window of opportunity for detecting new physics at high-energy scales. Future spectroscopic and photometric measurements from the Euclid space mission will provide unique constraints on the primordial power spectrum, thanks to the redshift coverage and high-accuracy measurement of nonlinear scales, thus allowing us to investigate deviations from the standard power-law primordial power spectrum. We consider two models with primordial undamped oscillations superimposed on the matter power spectrum described by 1 + 𝒜_X sin (ω_XΞ_X + 2 πϕ_X), one linearly spaced in k space with Ξ_lin ≡ k/k_* where k_* = 0.05 Mpc⁻¹ and the other logarithmically spaced in k space with Ξ_log ≡ ln(k/k_*). We note that 𝒜_X is the amplitude of the primordial feature, ω_X is the dimensionless frequency, and ϕ_X is the normalised phase, where X = {lin, log}. We provide forecasts from spectroscopic and photometric primary Euclid probes on the standard cosmological parameters Ω_m, 0, Ω_b, 0, h, n_s, and σ₈, and the primordial feature parameters 𝒜_X, ω_X, and ϕ_X. We focus on the uncertainties of the primordial feature amplitude 𝒜_X and on the capability of Euclid to detect primordial features at a given frequency. We also study a nonlinear density reconstruction method in order to retrieve the oscillatory signals in the primordial power spectrum, which are damped on small scales in the late-time Universe due to cosmic structure formation. Finally, we also include the expected measurements from Euclid’s galaxy-clustering bispectrum and from observations of the cosmic microwave background (CMB). We forecast uncertainties in estimated values of the cosmological parameters with a Fisher matrix method applied to spectroscopic galaxy clustering (GC_sp), weak lensing (WL), photometric galaxy clustering (GC_ph), the cross correlation (XC) between GC_ph and WL, the spectroscopic galaxy clustering bispectrum, the CMB temperature and E-mode polarisation, the temperature-polarisation cross correlation, and CMB weak lensing. We consider two sets of specifications for the Euclid probes (pessimistic and optimistic) and three different CMB experiment configurations, that is, Planck, Simons Observatory (SO), and CMB Stage-4 (CMB-S4). We find the following percentage relative errors in the feature amplitude with Euclid primary probes: for the linear (logarithmic) feature model, with a fiducial value of 𝒜_X = 0.01, ω_X = 10, and ϕ_X = 0: 21% (22%) in the pessimistic settings and 18% (18%) in the optimistic settings at a 68.3% confidence level (CL) using GC_sp+WL+GC_ph+XC. While the uncertainties on the feature amplitude are strongly dependent on the frequency value when single Euclid probes are considered, we find robust constraints on 𝒜_X from the combination of spectroscopic and photometric measurements over the frequency range of (1,  10^2.1). Due to the inclusion of numerical reconstruction, the GC_sp bispectrum, SO-like CMB reduces the uncertainty on the primordial feature amplitude by 32%–48%, 50%–65%, and 15%–50%, respectively. Combining all the sources of information explored expected from Euclid in combination with the future SO-like CMB experiment, we forecast 𝒜_lin ≃ 0.010 ± 0.001 at a 68.3% CL and 𝒜_log ≃ 0.010 ± 0.001 for GC_sp(PS rec + BS)+WL+GC_ph+XC+SO-like for both the optimistic and pessimistic settings over the frequency range (1,  10^2.1).

Normalizing Flows (NF) are Generative models which transform a simple prior distribution into the desired target. They however require the design of an invertible mapping whose Jacobian determinant has to be computable. Recently introduced, Neural Hamiltonian Flows (NHF) are Hamiltonian dynamics-based flows, which are continuous, volume-preserving and invertible and thus make for natural candidates for robust NF architectures. In particular, their similarity to classical Mechanics could lead to easier interpretability of the learned mapping. In this paper, we show that the current NHF architecture may still pose a challenge to interpretability. Inspired by Physics, we introduce a fixed-kinetic energy version of the model. This approach improves interpretability and robustness while requiring fewer parameters than the original model. We illustrate that on a 2D Gaussian mixture and on the MNIST and Fashion-MNIST datasets. Finally, we show how to adapt NHF to the context of Bayesian inference and illustrate the method on an example from cosmology.