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.

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.

Context. Accurate mock halo catalogues are indispensable data products for developing and validating cosmological inference pipelines. A major challenge in generating mock catalogues is modelling the halo or galaxy bias, which is the mapping from matter density to dark matter halos or observable galaxies. To this end, N-body codes produce state-of-the-art catalogues. However, generating large numbers of these N-body simulations for big volumes, especially if magnetohydrodynamics are included, requires significant computational time.

Aims. We introduce and benchmark a differentiable and physics-informed neural network that can generate mock halo catalogues of comparable quality to those obtained from full N-body codes. The model design is computationally efficient for the training procedure and the production of large mock catalogue suites.

Methods. We present a neural network, relying only on 18 to 34 trainable parameters, that produces halo catalogues from dark matter overdensity fields. The reduction in network weights was realised through incorporating symmetries motivated by first principles into our model architecture. We trained our model using dark-matter-only N-body simulations across different resolutions, redshifts, and mass bins. We validated the final mock catalogues by comparing them to N-body halo catalogues using different N-point correlation functions.

Results. Our model produces mock halo catalogues consistent with the reference simulations, showing that this novel network is a promising way to generate mock data for upcoming wide-field surveys due to its computational efficiency. Moreover, we find that the network can be trained on approximate overdensity fields to reduce the computational cost further. We also present how the trained network parameters can be interpreted to give insights into the physics of structure formation. Finally, we discuss the current limitations of our model as well as more general requirements and pitfalls of approximate halo mock generation that became evident from this study.

Detecting and measuring a non-Gaussian signature of primordial origin in the density field is a major science goal of next-generation galaxy surveys. The signal will permit us to determine primordial-physics processes and constrain models of cosmic inflation. While traditional approaches use a limited set of statistical summaries of the galaxy distribution to constrain primordial non-Gaussianity, we present a field-level approach by Bayesian forward modelling the entire three-dimensional galaxy survey. Since our method includes the entire cosmic field in the analysis, it can naturally and fully self-consistently exploit all available information in the large-scale structure, to extract information on the local non-Gaussianity parameter, fnl. Examples include higher order statistics through correlation functions, peculiar velocity fields through redshift-space distortions, and scale-dependent galaxy bias. To illustrate the feasibility of field-level primordial non-Gaussianity inference, we present our approach using a first-order Lagrangian perturbation theory model, approximating structure growth at sufficiently large scales. We demonstrate the performance of our approach through various tests with self-consistent mock galaxy data emulating relevant features of the SDSS-III/BOSS-like survey, and additional tests with a Stage IV mock data set. These tests reveal that the method infers unbiased values of fnl by accurately handling survey geometries, noise, and unknown galaxy biases. We demonstrate that our method can achieve constraints of σ_fnl≈8.78 for SDSS-III/BOSS-like data, indicating potential improvements of a factor ∼2.5 over current published constraints. We perform resolution studies on scales larger than ∼16h⁻¹ Mpc showing the promise of significant constraints with next-generation surveys. Furthermore, the results demonstrate that our method can consistently marginalize all nuisance parameters of the data model. The method further provides an inference of the three-dimensional primordial density field, providing opportunities to explore additional signatures of primordial physics. This first demonstration of a field-level inference pipeline demonstrates a promising complementary path forward for analysing next-generation surveys.

For a detailed understanding of many processes in nature involving, for example, energy or electron transfer, the theory of open quantumsystems is of key importance. For larger systems, an accurate description of the underlying quantum dynamics is still a formidable task, and,hence, approaches employing machine learning techniques have been developed to reduce the computational effort of accurate dissipativequantum dynamics. A downside of many previous machine learning methods is that they require expensive numerical training datasets forsystems of the same size as the ones they will be employed on, making them unfeasible to use for larger systems where those calculationsare still too expensive. In this work, we will introduce a new method that is implemented as a machine-learned correction term to the socalled Numerical Integration of Schrödinger Equation (NISE) approach. It is shown that this term can be trained on data from small systemswhere accurate quantum methods are still numerically feasible. Subsequently, the NISE scheme, together with the new machine-learnedcorrection, can be used to determine the dissipative quantum dynamics for larger systems. Furthermore, we show that the newly proposedmachine-learned correction outperforms a previously handcrafted one, which, however, improves the results already considerably.