A critical step in creating realistic mock catalogs that support large-scale photometric and spectroscopic sky surveys is the production of cosmological simulations that accurately model the survey observables, while taking into account the redshift-dependent galaxy formation and evolution processes. In this work, we developed an efficient framework named Halo Optimised Lightcone Constructor ( HOLCon ) for the post facto construction of dark matter halo light cones from simulations. We used these light cones to generate a mock galaxy catalogue. The framework includes a module to optimise the light cone’s orientation within the simulation box, minimising repeated structures when the survey volume exceeds a single box, which is a common challenge in modern surveys. A linear interpolation scheme tracks the evolution of halo properties across snapshots. Applied to the publicly available Uchuu simulation, we constructed a light cone of 50 deg 2 and extending up to z = 10, providing representative coverage of deep fields of Stage IV surveys. We validated the light cone for cosmological applications by comparing the dark matter halo clustering in the light cone with those from the original simulation snapshots. Subsequently, we made the galaxy-halo connection on the light cone with a redshift-extended version of the SciPIC algorithm, producing a comprehensive set of descriptive galaxy attributes. HOLCon leverages Dask , a scalable parallel computing Pythonic framework for the fast construction of dark matter halo light cones, enabling the rapid creation of multiple statistical realisations essential for robust cosmological inference. The generated galaxy mock allows us to make predictions for the clustering of H α emitters, making it a useful resource in cosmology studies.

Context. Determining cosmological parameters with high precision, as well as resolving current tensions in their values derived from low- and high-redshift probes, is one of the main objectives of the new generation of cosmological surveys. The combination of complementary probes in terms of parameter degeneracies and systematics is key to achieving these ambitious scientific goals. Aims. In this context, determining the optimal survey configuration for an analysis that combines galaxy clustering, weak lensing, and galaxy-galaxy lensing, the so-called 3 & times; 2pt analysis, remains an open problem. In this paper, we present an efficient and flexible end-to-end pipeline to optimise the sample selection for 3x2pt analyses in an automated way. Methods. Our pipeline is articulated in two main steps: we first consider a self-organising map to determine the photometric redshifts of a simulated galaxy sample. As a proof of method for stage IV surveys, we use samples from the DESC Data Challenge 2 catalogue. This allows us to classify galaxies into tomographic bins based on their colour phenotype clustering. We then explore different redshift-bin edge configurations for weak lensing only as well as 3 & times; 2pt analyses in a novel way. Our method explores multiple configurations of perturbed redshift-bin edges with respect to the fiducial case in an iterative manner. In particular, we sample tomographic configurations for the source and lens galaxies separately. Results. We show that using this method we quickly converge on an optimised configuration for different numbers of redshift bins and cosmologies. Our analysis demonstrates that for stage IV surveys an optimal tomographic sample selection can increase the figure of merit of the dark energy (DE) equation of state by a factor of similar to 2, comparable to an effective increase in survey area of similar to 4 for non-optimal photometric survey analyses.

We compute the gravity model testing parameter E-G (gravity estimator) on realistic simulated modified gravity galaxy mocks adopting the more accurate estimator described in [Z. Wang et al., J. Cosmol. Astropart. Phys. (2023) 038.]. The analysis is conducted using two twin simulations presented in [C. Arnold, M. Leo, and B. Li, Nat. Astron. 3, 917 (2019).]; one based on general relativity (GR) and the other on the f(R) Hu and Sawicki model with f 1/4 10(-5) (F5). This study aims to measure the E-G estimator in GR and f(R) models using high-fidelity simulated galaxy catalogs, with the goal of assessing how future galaxy surveys can detect deviations from standard gravity. Deriving this estimator requires precise, unbiased measurements of the growth rate of structure and the linear galaxy bias. We achieve this by implementing an end-to-end cosmological analysis pipeline in configuration space, using the multipoles of the two-point correlation function. In our analysis we estimate the scale-dependent growth rate predicted by nonstandard gravity models using COMET-VDG (cosmological observables modelled by emulated perturbation theory-velocity difference generator) fits. We split the estimation of the redshift space distortions (RSD) beta parameter over distinct scale ranges, separating large (quasilinear) and small (nonlinear) scales. We show that this estimator can be accurately measured using mock galaxies in low redshift bins (z < 1), although its discriminating power between competing theories is limited. We find that, for an all-sky galaxy survey and neglecting observational systematics, accurate and largely unbiased estimations of EG can be obtained across all redshifts. However, the error bars are too large to clearly distinguish between the theories. When measuring the scale dependence of the EG estimator, we note that state-of-the-art theory modeling limitations and intrinsic "prior volume effects" prevent high-accuracy constraints. Alternatively, we propose a null test of gravity using RSD clustering, which, if small scales are modeled accurately in future surveys, could detect departures from GR.

We present the Flagship galaxy mock, a simulated catalogue of billions of galaxies designed to support the scientific exploitation of the Euclid mission. Euclid is a medium-class mission of the European Space Agency optimised to determine the properties of dark matter and dark energy on the largest scales of the Universe. It probes structure formation over more than 10 billion years primarily from the combination of weak gravitational lensing and galaxy clustering data. The breath of Euclid's data will also foster a wide variety of scientific analyses. The Flagship simulation was developed to provide a realistic approximation to the galaxies that will be observed by Euclid and used in its scientific analyses. We ran a state-of-the-art N-body simulation with four trillion particles, producing a lightcone on the fly. From the dark matter particles, we produced a catalogue of 16 billion haloes in one octant of the sky in the lightcone up to redshift z=3. We then populated these haloes with mock galaxies using a halo occupation distribution and abundance matching approach, calibrating the free parameters of the galaxy mock against observed correlations and other basic galaxy properties. Modelled galaxy properties include luminosity and flux in several bands, redshifts, positions and velocities, spectral energy distributions, shapes and sizes, stellar masses, star formation rates, metallicities, emission line fluxes, and lensing properties. We selected a final sample of 3.4 billion galaxies with a magnitude cut of H_E<26, where we are complete. We have performed a comprehensive set of validation tests to check the similarity to observational data and theoretical models. In particular, our catalogue is able to closely reproduce the main characteristics of the weak lensing and galaxy clustering samples to be used in the mission's main cosmological analysis. (abridged)

We provide an early assessment of the imaging capabilities of the Euclid space mission to probe deeply into nearby star-forming regions and associated very young open clusters, and in particular to check to what extent it can shed light on the new-born free-floating planet population. This paper focuses on a low-reddening region observed in just one Euclid pointing where the dust and gas has been cleared out by the hot sigma Orionis star. One late-M and six known spectroscopically confirmed L-type substellar members in the sigma Orionis cluster are used as benchmarks to provide a high-purity procedure to select new candidate members with Euclid. The exquisite angular resolution and depth delivered by the Euclid instruments allow us to focus on bona-fide point sources. A cleaned sample of sigma Orionis cluster substellar members has been produced and the initial mass function (IMF) has been estimated by combining Euclid and Gaia data. Our sigma Orionis substellar IMF is consistent with a power-law distribution with no significant steepening at the planetary-mass end. No evidence of a low-mass cutoff is found down to about 4 Jupiter masses at the young age (3 Myr) of the sigma Orionis open cluster.

Cosmic shear, galaxy clustering, and the abundance of massive halos each probe the large-scale structure of the Universe in complementary ways. We present cosmological constraints from the joint analysis of the three probes, building on the latest analyses of the lensing-informed abundance of clusters identified by the South Pole Telescope (SPT) and of the auto- and cross-correlation of galaxy position and weak lensing measurements (3 x 2pt) in the Dark Energy Survey (DES). We consider the cosmological correlation between the different tracers and we account for the systematic uncertainties that are shared between the large-scale lensing correlation functions and the small-scale lensing-based cluster mass calibration. Marginalized over the remaining Lambda cold dark matter (Lambda CDM) parameters (including the sum of neutrino masses) and 52 astrophysical modeling parameters, we measure Omega(m) = 0.300 +/- 0.017 and sigma(8) = 0.797 +/- 0.026. Compared to constraints from Planck primary cosmic microwave background (CMB) anisotropies, our constraints are only 15% wider with a probability to exceed of 0.22 (1.2 sigma) for the two-parameter difference. We further obtain S-8 equivalent to sigma(8)(Omega(m)/0.3)(0.5) = 0.796 +/- 0.013 which is lower than the Planck measurement at the 1.6 sigma level. The combined SPT cluster, DES 3 x 2pt, and Planck datasets mildly prefer a nonzero positive neutrino mass, with a 95% upper limit Sigma m(nu) < 0.25 eV on the sum of neutrino masses. Assuming a wCDM model, we constrain the dark energy equation of state parameter w = -1.15(-0.17)(+0.23) and when combining with Planck primary CMB anisotropies, we recover w = -1.20(-0.09)(+0.15), a 1.7 sigma difference with a cosmological constant. The precision of our results highlights the benefits of multiwavelength multiprobe cosmology and our analysis paves the way for upcoming joint analyses of next-generation datasets.

We present an analysis of Euclid observations of a 0.5 deg$^2$ field in the central region of the Fornax galaxy cluster that were acquired during the performance verification phase. With these data, we investigate the potential of Euclid for identifying GCs at 20 Mpc, and validate the search methods using artificial GCs and known GCs within the field from the literature. Our analysis of artificial GCs injected into the data shows that Euclid's data in $I_{\rm E}$ band is 80% complete at about $I_{\rm E} \sim 26.0$ mag ($M_{V\rm } \sim -5.0$ mag), and resolves GCs as small as $r_{\rm h} = 2.5$ pc. In the $I_{\rm E}$ band, we detect more than 95% of the known GCs from previous spectroscopic surveys and GC candidates of the ACS Fornax Cluster Survey, of which more than 80% are resolved. We identify more than 5000 new GC candidates within the field of view down to $I_{\rm E}$ mag, about 1.5 mag fainter than the typical GC luminosity function turn-over magnitude, and investigate their spatial distribution within the intracluster field. We then focus on the GC candidates around dwarf galaxies and investigate their numbers, stacked luminosity distribution and stacked radial distribution. While the overall GC properties are consistent with those in the literature, an interesting over-representation of relatively bright candidates is found within a small number of relatively GC-rich dwarf galaxies. Our work confirms the capabilities of Euclid data in detecting GCs and separating them from foreground and background contaminants at a distance of 20 Mpc, particularly for low-GC count systems such as dwarf galaxies.

Gravitational redshift and Doppler effects give rise to an antisymmetric component of the galaxy correlation function when cross-correlating two galaxy populations or two different tracers. In this paper, we assess the detectability of these effects in the Euclid spectroscopic galaxy survey. We model the impact of gravitational redshift on the observed redshift of galaxies in the Flagship mock catalogue using a Navarro-Frenk-White profile for the host haloes. We isolate these relativistic effects, largely subdominant in the standard analysis, by splitting the galaxy catalogue into two populations of faint and bright objects and estimating the dipole of their cross-correlation in four redshift bins. In the simulated catalogue, we detect the dipole signal on scales below 30 h(-1) Mpc, with detection significances of 4 sigma and 3 sigma in the two lowest redshift bins, respectively. At higher redshifts, the detection significance drops below 2 sigma. Overall, we estimate the total detection significance in the Euclid spectroscopic sample to be approximately 6 sigma. We find that on small scales, the major contribution to the signal comes from the nonlinear gravitational potential. Our study on the Flagship mock catalogue shows that this observable can be detected in Euclid Data Release 2 and beyond.

We study the constraint on f(R) gravity that can be obtained by photometric primary probes of the Euclid mission. Our focus is the dependence of the constraint on the theoretical modelling of the nonlinear matter power spectrum. In the Hu-Sawicki f(R) gravity model, we consider four different predictions for the ratio between the power spectrum in f(R) and that in Lambda cold dark matter (Lambda CDM): a fitting formula, the halo model reaction approach, ReACT, and two emulators based on dark matter only N-body simulations, FORGE and e-Mantis. These predictions are added to the MontePython implementation to predict the angular power spectra for weak lensing (WL), photometric galaxy clustering, and their cross-correlation. By running Markov chain Monte Carlo, we compare constraints on parameters and investigate the bias of the recovered f(R) parameter if the data are created by a different model. For the pessimistic setting of WL, one-dimensional bias for the f(R) parameter, log10|fR0|, is found to be 0.5 sigma when FORGE is used to create the synthetic data with log10|fR0| = -5.301 and fitted by e-Mantis. The impact of baryonic physics on WL is studied by using a baryonification emulator, BCemu. For the optimistic setting, the f(R) parameter and two main baryonic parameters are well constrained despite the degeneracies among these parameters. However, the difference in the nonlinear dark matter prediction can be compensated for the adjustment of baryonic parameters, and the one-dimensional marginalised constraint on log10|fR0| is biased. This bias can be avoided in the pessimistic setting at the expense of weaker constraints. For the pessimistic setting, using the Lambda CDM synthetic data for WL, we obtain the prior-independent upper limit of log10|fR0| < -5.6. Finally, we implement a method to include theoretical errors to avoid the bias due to inaccuracies in the nonlinear matter power spectrum prediction.

Stage-IV surveys will enable unprecedented tests of gravity on cosmological scales. However, assuming General Relativity in the analysis of large-scale structure could introduce systematic biases if gravity deviates from GR at these scales. Modified gravity theories, such as the Hu-Sawicki formulation of f(R) gravity, offer an alternative explanation for cosmic acceleration without invoking a cosmological constant, while remaining consistent with Solar System tests through screening mechanisms. In this work, we quantify the cosmological parameter biases that arise when using a combination of galaxy clustering and weak-lensing data-vectors, the so-called 3x2pt analysis, from an f(R) galaxy mock under the incorrect assumption of GR, using for the first time high-fidelity full-sky galaxy mock catalogues. We employ a pair of twin simulations: one with GR and one with Hu–Sawicki f(R) gravity with |f_R0| = 10^-5. The mocks are built using an HOD method to populate the dark matter haloes with galaxies, calibrated against SDSS observations at low redshift. Using conservative scale cuts to minimise modelling uncertainties, we perform 3x2pt analyses and infer cosmological parameters through nested sampling, validating our pipeline with the GR mock. Our results show that when analysing the f(R) galaxy mock assuming GR, the recovered cosmological parameters are very significantly biased, even when considering conservative scale cuts: the Figure of Bias reaches ∼12σ for both {Ω_ m, σ_8} and S_8. These biases persist even when marginalising over the galaxy bias and baryonic feedback, demonstrating that nuisance parameters cannot absorb the effects of modified gravity. We conclude that incorrectly assuming GR in a universe governed by f(R) gravity leads to severe and detectable biases in cosmological inference for Stage-IV surveys.

Euclid is poised to make significant advances in the study of nearby galaxies in the local Universe. Here we present a first look at 6 galaxies observed for the Nearby Galaxy Showcase as part of the Euclid Early Release Observations acquired between August and November, 2023. These targets, 3 dwarf galaxies (HolmbergII, IC10, NGC6822) and 3 spirals (IC342, NGC2403, NGC6744), range in distance from about 0.5 Mpc to 8.8 Mpc. Our assessment of the surface brightness depths in the stacked Euclid images confirms previous estimates in 100 arcsec^2 regions of 1sigma=30.5 mag/arcsec^2 for VIS, but slightly deeper than previous estimates for NISP with 1sigma=29.2-29.4 mag/arcsec^2. By combining Euclid HE, YE, and IE into RGB images, we illustrate the large field-of-view covered by a single Reference Observing Sequence, together with exquisite detail on parsec scales in these nearby galaxies. Radial surface brightness and color profiles demonstrate galaxy colors in agreement with stellar population synthesis models. Standard stellar photometry selection techniques find approximately 1.3 million stars across the 6 galaxy fields. Euclid's resolved stellar photometry allows us to constrain the star-formation histories of these galaxies, by disentangling the distributions of young stars, as well as asymptotic giant branch and red giant branch stellar populations. We finally examine 2 galaxies individually for surrounding satellite systems. Our analysis of the ensemble of dwarf satellites around NGC6744 reveals a new galaxy, EDwC1, a nucleated dwarf spheroidal at the end of a spiral arm. Our new census of the globular clusters around NGC2403 yields 9 new star-cluster candidates, 8 of which with colors indicative of evolved stellar populations. In summary, our investigation of the 6 Showcase galaxies demonstrates that Euclid is a powerful probe of the anatomy of nearby galaxies [abridged].

Cosmic shear is a powerful probe of cosmological models and the transition from current Stage-III surveys such as the Kilo-Degree Survey (KiDS) to the increased area and redshift range of Stage IV surveys such as Euclid will significantly increase the precision of weak lensing analyses. However, with increasing precision, the accuracy of model assumptions needs to be evaluated. In this study, we quantify the impact of the correlated clustering of weak lensing source galaxies with the surrounding large-scale structure, known as source-lens clustering (SLC), which is commonly neglected. We include the impact of realistic scatter in photometric redshift estimates, which impacts the assignment of galaxies to tomographic bins and increases the SLC. For this, we use simulated cosmological datasets with realistically distributed galaxies and measure shear correlation functions for both clustered and uniformly distributed source galaxies. Cosmological analyses are performed for both scenarios to quantify the impact of SLC on parameter inference for a KiDS-like and a Euclid-like setting. We find for Stage III surveys such as KiDS, SLC has a minor impact when accounting for nuisance parameters for intrinsic alignments and shifts of tomographic bins, as these nuisance parameters absorb the e ffect of SLC, thus changing their original meaning. For KiDS (Euclid), the inferred intrinsic alignment amplitude A(IA) changes from 0.11(-0.46)(+0.44)(0.009(-0.080)(+0.079)) for data without SLC to 0.28(-0.44)(+0.42)(0.022(-0.082)(+0.081)) with SLC. However, fixed nuisance parameters lead to shifts in S-8 and Omega(m), emphasizing the need for including SLC in the modelling. For Euclid we find that sigma(8), Omega(m), and w(0) are shifted by 0.19, 0.12, and 0.12 sigma, respectively, when including free nuisance parameters, and by 0.20, 0.16, and 0.32 sigma when fixing the nuisance parameters. Consequently, SLC on its own has only a small impact on the inferred parameter inference when using uninformative priors for nuisance parameters. However, SLC might conspire with the breakdown of other modelling assumptions, such as magnification bias or source obscuration, which could collectively exert a more pronounced e ffect on inferred parameters.

The Euclid mission will measure cosmological parameters with unprecedented precision. To distinguish between cosmological models, it is essential to generate realistic mock observables from cosmological simulations that were run in both the standard Lambda-cold-dark-matter (Lambda CDM) paradigm and in many non-standard models beyond Lambda CDM. We present the scientific results from a suite of cosmological N-body simulations using non-standard models including dynamical dark energy, k-essence, interacting dark energy, modified gravity, massive neutrinos, and primordial non-Gaussianities. We investigate how these models affect the large-scale-structure formation and evolution in addition to providing synthetic observables that can be used to test and constrain these models with Euclid data. We developed a custom pipeline based on the Rockstar halo finder and the nbodykit large-scale structure toolkit to analyse the particle output of non-standard simulations and generate mock observables such as halo and void catalogues, mass density fields, and power spectra in a consistent way. We compare these observables with those from the standard Lambda CDM model and quantify the deviations. We find that non-standard cosmological models can leave large imprints on the synthetic observables that we have generated. Our results demonstrate that non-standard cosmological N-body simulations provide valuable insights into the physics of dark energy and dark matter, which is essential to maximising the scientific return of Euclid.

We perform cosmological parameters estimation on Planck Cosmic Microwave Background (CMB) maps masking the recently discovered foreground related to nearby spiral galaxies. In addition, we also analyse the association between these foreground regions and recent claims of cosmological causal horizons in localized CMB parameter estimates. Our analysis shows consistent cosmological parameter values regardless of the masking approach, though reduced sky areas introduce larger uncertainties. By modelling the new extragalactic foreground, we identify a resemblance with local parameter variation maps with a statistical significance at the 3 sigma level, suggesting that a simplified foreground model partially accounts, (40-50)% correlation with 15% uncertainty, for the observed causal horizons. These findings add new evidence to the existence of the new foreground associated with large spiral galaxies and show that estimates of cosmological parameters on smaller patches on the sky can be largely affected by these foregrounds, but that the parameters taken over the full sky are unaltered.

The Euclid mission of the European Space Agency will provide weak gravitational lensing and galaxy clustering surveys that can be used to constrain the standard cosmological model and its extensions, with an opportunity to test the properties of dark matter beyond the minimal cold dark matter paradigm. We present forecasts from the combination of the Euclid weak lensing and photometric galaxy clustering data on the parameters describing four interesting and representative non-minimal dark matter models: a mixture of cold and warm dark matter relics; unstable dark matter decaying either into massless or massive relics; and dark matter undergoing feeble interactions with relativistic relics. We modelled these scenarios at the level of the non-linear matter power spectrum using emulators trained on dedicated N-body simulations. We used a mock Euclid likelihood and Monte Carlo Markov chains to fit mock data and infer error bars on dark matter parameters marginalised over other parameters. We find that the Euclid photometric probe (alone or in combination with cosmic microwave background data from the Planck satellite) will be sensitive to the effect of each of the four dark matter models considered here. The improvement will be particularly spectacular for decaying and interacting dark matter models. With Euclid, the bounds on some dark matter parameters can improve by up to two orders of magnitude compared to current limits. We discuss the dependence of predicted uncertainties on different assumptions: the inclusion of photometric galaxy clustering data, the minimum angular scale taken into account, and modelling of baryonic feedback effects. We conclude that the Euclid mission will be able to measure quantities related to the dark sector of particle physics with unprecedented sensitivity. This will provide important information for model building in high-energy physics. Any hint of a deviation from the minimal cold dark matter paradigm would have profound implications for cosmology and particle physics.

The Near-Infrared Spectrometer and Photometer (NISP) on board the satellite provides multiband photometry and $R slitless grism spectroscopy in the 950--2020\,nm wavelength range. In this reference article, we illuminate the background of NISP's functional and calibration requirements, describe the instrument's integral components, and provide all its key properties. We also sketch the processes needed to understand how NISP operates and is calibrated as well as its technical potentials and limitations. Links to articles providing more details and the technical background are included. The NISP's 16 H2RG detectors with a plate scale of $ deliver a field of view of 0.57\,deg$^2$. In photometric mode, NISP reaches a limiting magnitude of sim \,24.5\,AB\,mag in three photometric exposures of about 100\,s in exposure time for point sources and with a S/N of five. For spectroscopy, NISP's point-source sensitivity is a SNR = 3.5 detection of an emission line with flux sim \,$2 $ integrated over two resolution elements of 13.4\ in 3times 560\,s grism exposures at 1.6\ (redshifted Halpha ). Our calibration includes on-ground and in-flight characterisation and monitoring of the pixel-based detector baseline, dark current, non-linearity, and sensitivity to guarantee a relative photometric accuracy better than 1.5 and a relative spectrophotometry better than 0.7. The wavelength calibration must be accurate to 5\ or better. The NISP is the state-of-the-art instrument in the near-infrared for all science beyond small areas available from HST and JWST -- and it represents an enormous advance from any existing instrumentation due to its combination of field size and high throughput of telescope and instrument. During six-year survey covering 14\,000\,deg$^2$ of extragalactic sky, NISP will be the backbone in determining distances of more than a billion galaxies. Its near-infrared data will become a rich reference imaging and spectroscopy data set for the coming decades.

Data from the Euclid space telescope will enable cosmic shear measurements with very small statistical errors, requiring corresponding systematic error control level. A common approach to correct for shear biases involves calibrating shape measurement methods using image simulations with known input shear. Given their high resolution, Hubble Space Telescope (HST) galaxies can, in principle, be utilised to emulate Euclid observations. In this work, we employ a GalSim-based testing environment to investigate whether uncertainties in the HST point spread function (PSF) model or in data processing techniques introduce significant biases in weak-lensing (WL) shear calibration. We used single Sérsic galaxy models to simulate both HST and Euclid observations. We then `Euclidised' our HST simulations and compared the results with the directly simulated Euclid-like images. For this comparison, we utilised a moment-based shape measurement algorithm and galaxy model fits. Through the Euclidisation procedure, we effectively reduced the residual multiplicative biases in shear measurements to sub-percent levels. This achievement was made possible by employing either the native pixel scales of the instruments, utilising the Lanczos15 interpolation kernel, correcting for noise correlations, and ensuring consistent galaxy signal-to-noise ratios between simulation branches. However, the Euclidisation procedure requires further analysis on the impact of the correlated noise, to estimate calibration bias. Additionally, we conducted an in-depth analysis of the accuracy of TinyTim HST PSF models using star fields observed in the F606W and F814W filters. We observe that F606W images exhibit a broader scatter in the recovered best-fit focus, compared to those in the F814W filter.

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 deg^2 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 study the intracluster light (ICL) and intracluster globular clusters (ICGCs) in the nearby Perseus cluster of galaxies using Euclid's Early Release Observations. By modelling the isophotal and iso-density contours, we mapped the distributions and properties of the ICL and ICGCs out to radii of 200-600 kpc (up to similar to 1/3 of the virial radius, depending on the parameter) from the brightest cluster galaxy (BCG). We find that the central 500 kpc of the Perseus cluster hosts 70 000 +/- 2800 globular clusters, and 1.7 x 10(12) L-circle dot of diffuse light from the BCG+ICL in the near-infrared H-E. This accounts for 38 +/- 6% of the cluster's total stellar luminosity within this radius. The ICL and ICGCs share a coherent spatial distribution which suggests that they have a common origin or that a common potential governs their distribution. Their contours on the largest scales (>200 kpc) are not centred on the BCG's core, but are instead offset westwards by 60 kpc towards several luminous cluster galaxies. This offset is opposite to the displacement observed in the gaseous intracluster medium. The radial surface brightness profile of the BCG+ICL is best described by a double S & eacute;rsic model, with 68 +/- 4% of the H-E light contained in the extended, outer component. The transition between these components occurs at approximate to 60 kpc, beyond which the isophotes become increasingly elliptical and off-centred. Furthermore, the radial ICGC number density profile closely follows the profile of the BCG+ICL only beyond this 60 kpc radius, where we find an average of 60-80 globular clusters per 10(9) M-circle dot of diffuse stellar mass. The BCG+ICL colour becomes increasingly blue with radius, consistent with the stellar populations in the ICL having subsolar metallicities [Fe/H] similar to -0.6 to -1.0. The colour of the ICL, and the specific frequency and luminosity function of the ICGCs suggest that the ICL+ICGCs were tidally stripped from the outskirts of massive satellites with masses of a few x10(10) M-circle dot, with an increasing contribution from dwarf galaxies at large radii.

Measurements of galaxy clustering are affected by redshift-space distortions (RSDs). Peculiar velocities, gravitational lensing, and other light-cone projection effects modify the observed redshifts, fluxes, and sky positions of distant light sources. We determined which of these effects leave a detectable imprint on several two-point clustering statistics to be extracted from the Euclid wide spectroscopic survey (EWSS) on large scales. We generated 140 mock galaxy catalogues with the survey geometry and selection function of the EWSS and made use of the LIGER (LIght cones with GEneral Relativity) method to account for a variable number of relativistic RSDs to linear order in the cosmological perturbations. We estimated different two-point clustering statistics from the mocks and used the likelihood-ratio test to calculate the statistical significance with which the EWSS could reject the null hypothesis that certain relativistic projection effects can be neglected in the theoretical models. We find that the combined effects of lensing magnification and convergence imprint characteristic signatures on several clustering observables. Their signal-to-noise ratio (S/N) ranges between 2.5 and 6 (depending on the adopted summary statistic) for the highest-redshift galaxies in the EWSS. The corresponding feature due to the peculiar velocity of the Sun is measured with a S/N of order one or two. The multipoles of the power spectrum from the catalogues that include all relativistic effects reject the null hypothesis that RSDs are only generated by the variation in the peculiar velocity along the line of sight with a significance of 2.9 standard deviations. As a by-product of our study, we demonstrate that the mixing-matrix formalism to model finite-volume effects in the multipole moments of the power spectrum can be robustly applied to surveys made of several disconnected patches. Our results indicate that relativistic RSDs, in particular the contribution from weak gravitational lensing, cannot be disregarded when modelling two-point clustering statistics extracted from the EWSS.