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)

Context. The Euclid mission of the European Space Agency will deliver weak gravitational lensing and galaxy clustering surveys that can be used to constrain the standard cosmological model and extensions thereof. Aims. We present forecasts from the combination of the Euclid photometric galaxy surveys (weak lensing, galaxy clustering, and their crosscorrelations) and its spectroscopic redshift survey with respect to their sensitivity to cosmological parameters. We include the summed neutrino mass, Sigma m (v), and the e ffective number of relativistic species, N-e ff, in the standard Lambda alpha CDM scenario and in the dynamical dark energy (w (0) w(alpha)CDM) scenario. Methods. We compared the accuracy of di fferent algorithms predicting the non-linear matter power spectrum for such models. We then validated several pipelines for Fisher matrix and Markov chain Monte Carlo (MCMC) forecasts, using di fferent theory codes, algorithms for numerical derivatives, and assumptions on the non-linear cut-o ff scale. Results. The Euclid primary probes alone will reach a sensitivity of sigma(Sigma m (v) = 60 meV) = 56 meV in the Lambda CDM +Sigma m (v) model, whereas the combination with cosmic microwave background (CMB) data from Planck is expected to achieve sigma(Sigma m (v)) = 23 meV, o ffering evidence of a non-zero neutrino mass to at least the 2:6 sigma level. This could be pushed to a 4 sigma detection if future CMB data from LiteBIRD and CMB Stage-IV were included. In combination with Planck, Euclid will also deliver tight constraints on Delta N-e ff < 0:144 (95%CL) in the Lambda CDM +Sigma m (v)+N-e ff model or even Delta N-e ff < 0:063 when future CMB data are included. When floating the dark energy parameters, we find that the sensitivity to Ne ff remains stable, but for Sigma m (v), it gets degraded by up to a factor of 2, at most. Conclusions. This work illustrates the complementarity among the Euclid spectroscopic and photometric surveys and among Euclid and CMB constraints. Euclid will o ffer great potential in measuring the neutrino mass and excluding well-motivated scenarios with additional relativistic particles.

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.

We forecast the expected population of active galactic nuclei (AGN) observable in the Euclid Wide Survey (EWS) and Euclid Deep Survey (EDS). Starting from an X-ray luminosity function (XLF) we generate volume-limited samples of the AGN expected in the survey footprints. Each AGN is assigned an SED appropriate for its X-ray luminosity and redshift, with perturbations sampled from empirical distributions. The photometric detectability of each AGN is assessed via mock observation of the assigned SED. We estimate 40 million AGN will be detectable in at least one band in the EWS and 0.24 million in the EDS, corresponding to surface densities of 2.8$\times$10$^{3}$ deg$^{-2}$ and 4.7$\times$10$^{3}$ deg$^{-2}$. Employing colour selection criteria on our simulated data we select a sample of 4.8$\times$10$^{6}$ (331 deg$^{-2}$) AGN in the EWS and 1.7$\times$10$^{4}$ (346 deg$^{-2}$) in the EDS, amounting to 10% and 8% of the AGN detectable in the EWS and EDS. Including ancillary Rubin/LSST bands improves the completeness and purity of AGN selection. These data roughly double the total number of selected AGN to comprise 21% and 15% of the detectable AGN in the EWS and EDS. The total expected sample of colour-selected AGN contains 6.0$\times$10$^{6}$ (74%) unobscured AGN and 2.1$\times$10$^{6}$ (26%) obscured AGN, covering $0.02 \leq z \lesssim 5.2$ and $43 \leq \log_{10} (L_{bol} / erg s^{-1}) \leq 47$. With this simple colour selection, expected surface densities are already comparable to the yield of modern X-ray and mid-infrared surveys of similar area. The relative uncertainty on our expectation for detectable AGN is 6.7% for the EWS and 12.5% for the EDS, driven by the uncertainty of the XLF.

Extragalactic globular clusters (EGCs) are an abundant and powerful tracer of galaxy dynamics and formation, and their own formation and evolution is also a matter of extensive debate. The compact nature of globular clusters means that they are hard to spatially resolve and thus study outside the Local Group. In this work we have examined how well EGCs will be detectable in images from the Euclid telescope, using both simulated pre-launch images and the first early-release observations of the Fornax galaxy cluster. The Euclid Wide Survey will provide high-spatial resolution VIS imaging in the broad IE band as well as near-infrared photometry (YE, JE, and HE). We estimate that the galaxies within 100 Mpc in the footprint of the Euclid survey host around 830 000 EGCs of which about 350 000 are within the survey's detection limits. For about half of these EGCs, three infrared colours will be available as well. For any galaxy within 50Mpc the brighter half of its GC luminosity function will be detectable by the Euclid Wide Survey. The detectability of EGCs is mainly driven by the residual surface brightness of their host galaxy. We find that an automated machine-learning EGC-classification method based on real Euclid data of the Fornax galaxy cluster provides an efficient method to generate high purity and high completeness GC candidate catalogues. We confirm that EGCs are spatially resolved compared to pure point sources in VIS images of Fornax. Our analysis of both simulated and first on-sky data show that Euclid will increase the number of GCs accessible with high-resolution imaging substantially compared to previous surveys, and will permit the study of GCs in the outskirts of their hosts. Euclid is unique in enabling systematic studies of EGCs in a spatially unbiased and homogeneous manner and is primed to improve our understanding of many understudied aspects of GC astrophysics.

The Euclid mission is generating a vast amount of imaging data in four broadband filters at a high angular resolution. This data will allow for the detailed study of mass, metallicity, and stellar populations across galaxies that will constrain their formation and evolutionary pathways. Transforming the Euclid imaging for large samples of galaxies into maps of physical parameters in an efficient and reliable manner is an outstanding challenge. Here, we investigate the power and reliability of machine learning techniques to extract the distribution of physical parameters within well-resolved galaxies. We focus on estimating stellar mass surface density, mass-averaged stellar metallicity, and age. We generated noise-free synthetic high-resolution (100 pcx100 pc) imaging data in the Euclid photometric bands for a set of 1154 galaxies from the TNG50 cosmological simulation. The images were generated with the SKIRT radiative transfer code, taking into account the complex 3D distribution of stellar populations and interstellar dust attenuation. We used a machine learning framework to map the idealised mock observational data to the physical parameters on a pixel-by-pixel basis. We find that stellar mass surface density can be accurately recovered with a <= 0.130 dex scatter. Conversely, stellar metallicity and age estimates are, as expected, less robust, but they still contain significant information that originates from underlying correlations at a sub-kiloparsec scales between stellar mass surface density and stellar population properties. As a corollary, we show that TNG50 follows a spatially resolved mass-metallicity relation that is consistent with observations. Due to its relatively low computational and time requirements, which has a time-frame of minutes without dedicated high performance computing infrastructure once it has been trained, our method allows for fast and robust estimates of the stellar mass surface density distributions of nearby galaxies from four-filter Euclid imaging data. Equivalent estimates of stellar population properties (stellar metallicity and age) are less robust but still hold value as first-order approximations across large samples.

Context. Cluster cosmology can benefit from combining multi-wavelength studies. In turn, these studies benefit from a characterisation of the correlation coefficients among different mass-observable relations. Aims. In this work, we aim to provide information on the scatter, skewness, and covariance of various mass-observable relations in galaxy clusters in cosmological hydrodynamic simulations. This information will help future analyses improve the general approach to accretion histories and projection effects, as well as to model mass-observable relations for cosmology studies. Methods. We identified galaxy clusters in Magneticum Box2b simulations with masses of M-200c > 10(14) M-circle dot at redshifts of z = 0.24 and z = 0.90. Our analysis included Euclid-derived properties such as richness, stellar mass, lensing mass, and concentration. Additionally, we investigated complementary multi-wavelength data, including X-ray luminosity, integrated Compton-y parameter, gas mass, and temperature. We then examined the impact of projection effects on mass-observable residuals and correlations. Results. We find that at intermediate redshift (z = 0.24), projection effects have the greatest impact of lensing concentration, richness, and gas mass in terms of the scatter and skewness of the log-residuals of scaling relations. The contribution of projection effects can be significant enough to boost a spurious hot- versus cold-baryon correlations and consequently hide underlying correlations due to halo accretion histories. At high redshift (z = 0.9), the richness has a much lower scatter (of log-residuals), while the quantity that is most impacted by projection effects is the lensing mass. The lensing concentration reconstruction, in particular, is affected by deviations of the reduced-shear profile shape from that derived using a Navarro-Frenk-White (NFW) profile; the amount of interlopers in the line of sight, on the other hand, is not as important.

The 2-point correlation function of the galaxy spatial distribution is a major cosmological observable that enables constraints on the dynamics and geometry of the Universe. The Euclid mission aims at performing an extensive spectroscopic survey of approximately 20–30 million Hα-emitting galaxies up to about redshift two. This ambitious project seeks to elucidate the nature of dark energy by mapping the 3-dimensional clustering of galaxies over a significant portion of the sky. This paper presents the methodology and software developed for estimating the 3-dimensional 2-point correlation function within the Euclid Science Ground Segment. The software is designed to overcome the significant challenges posed by the large and complex Euclid data set, which involves millions of galaxies. Key challenges include efficient pair counting, managing computational resources, and ensuring the accuracy of the correlation function estimation. The software leverages advanced algorithms, including kd-tree, octree, and linked-list data partitioning strategies, to optimise the pair-counting process. The implementation also includes parallel processing capabilities using shared-memory open multi-processing to further enhance performance and reduce computation times. Extensive validation and performance testing of the software are presented. The results indicate that the software is robust and can reliably estimate the 2-point correlation function, which is essential for deriving cosmological parameters with high precision. Furthermore, the paper discusses the expected performance of the software during different stages of the Euclid Wide Survey observations and forecasts how the precision of the correlation function measurements will improve over the mission's timeline, highlighting the software's capability to handle large data sets efficiently.

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.

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 near-infrared calibration unit (NI-CU) on board NISP is the first astronomical calibration lamp based on LED to be operated in space. is a mission in ESA's Cosmic Vision 2015--2025 framework to explore the dark universe and provide a next-level characterisation of the nature of gravitation, dark matter, and dark energy. Calibrating photometric and spectrometric measurements of galaxies to better than 1.5 accuracy in a survey homogeneously mapping sim \,14\,000\,deg$^2$ of extragalactic sky requires a very detailed characterisation of NIR detector properties as well as constant monitoring of them in flight. To cover two of the main contributions -- relative pixel-to-pixel sensitivity and non-linearity characteristics -- and to support other calibration activities, NI-CU was designed to provide spatially approximately homogeneous ($<$\,12 variations) and temporally stable illumination (0.1--0.2 over 1200\,s) over the NISP detector plane with minimal power consumption and energy dissipation. NI-CU covers the spectral range sim \,nm -- at cryo-operating temperature -- at five fixed independent wavelengths to capture wavelength-dependent behaviour of the detectors, with fluence over a dynamic range of gtrsim \,100 from sim $. For this functionality, NI-CU is based on LED . We describe the rationale behind the decision and design process, the challenges in sourcing the right LED and the qualification process and lessons learned. We also provide a description of the completed NI-CU, its capabilities, and performance as well as its limits. NI-CU has been integrated into NISP and the satellite, and since launch in July 2023, it has started supporting survey operations.

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.

Euclid will provide deep near-infrared (NIR) imaging to similar to 26.5 AB magnitude over similar to 59 deg(2) in its deep and auxiliary fields. The Cosmic DAWN survey combines dedicated and archival UV-NIR observations to provide matched depth multiwavelength imaging of the Euclid deep and auxiliary fields. The DAWN survey will provide consistently measured Euclid NIR-selected photometric catalogues, accurate photometric redshifts, and measurements of galaxy properties to a redshift of z similar to 10. The DAWN catalogues include Spitzer IRAC data that are critical for stellar mass measurements at z greater than or similar to 2.5 and high-z science. These catalogues complement the standard Euclid catalogues, which will not include Spitzer IRAC data. In this paper, we present an overview of the survey, including the footprints of the survey fields, the existing and planned observations, and the primary science goals for the combined data set.

To date, galaxy image simulations for weak lensing surveys usually approximate the light profiles of all galaxies as a single or double Sérsic profile, neglecting the influence of galaxy substructures and morphologies deviating from such a simplified parametric characterisation. While this approximation may be sufficient for previous data sets, the stringent cosmic shear calibration requirements and the high quality of the data in the upcoming survey demand a consideration of the effects that realistic galaxy substructures and irregular shapes have on shear measurement biases. Here we present a novel deep learning-based method to create such simulated galaxies directly from Hubble Space Telescope (HST) data. We first build and validate a convolutional neural network based on the wavelet scattering transform to learn noise-free representations independent of the point-spread function (PSF) of HST galaxy images. These can be injected into simulations of images from optical instrument VIS without introducing noise correlations during PSF convolution or shearing. Then, we demonstrate the generation of new galaxy images by sampling from the model randomly as well as conditionally. In the latter case, we fine-tune the interpolation between latent space vectors of sample galaxies to directly obtain new realistic objects following a specific Sérsic index and half-light radius distribution. Furthermore, we show that the distribution of galaxy structural and morphological parameters of our generative model matches the distribution of the input HST training data, proving the capability of the model to produce realistic shapes. Next, we quantify the cosmic shear bias from complex galaxy shapes in simulations by comparing the shear measurement biases between a sample of model objects and their best-fit double-Sérsic counterparts, thereby creating two separate branches that only differ in the complexity of their shapes. Using the Kaiser, Squires, and Broadhurst shape measurement algorithm, we find a multiplicative bias difference between these branches with realistic morphologies and parametric profiles on the order of $(6.9± 0.6) $ for a realistic magnitude-Sérsic index distribution. Moreover, we find clear detection bias differences between full image scenes simulated with parametric and realistic galaxies, leading to a bias difference of $(4.0± 0.9) $ independent of the shape measurement method. This makes complex morphology relevant for stage IV weak lensing surveys, exceeding the full error budget of the Euclid Wide Survey (Δμ_

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.

In this paper we present the framework for measuring angular power spectra in the Euclid mission. The observables in galaxy surveys, such as galaxy clustering and cosmic shear, are not continuous fields, but discrete sets of data, obtained only at the positions of galaxies. We show how to compute the angular power spectra of such discrete data sets, without treating observations as maps of an underlying continuous field that is overlaid with a noise component. This formalism allows us to compute the exact theoretical expectations for our measured spectra, under a number of assumptions that we track explicitly. In particular, we obtain exact expressions for the additive biases ('shot noise') in angular galaxy clustering and cosmic shear. For efficient practical computations, we introduce a spin-weighted spherical convolution with a well-defined convolution theorem, which allows us to apply exact theoretical predictions to finite-resolution maps, including HEALPix. When validating our methodology, we find that our measurements are biased by less than 1% of their statistical uncertainty in simulations of Euclid's first data release.

This paper presents the specification, design, and development of the Visible Camera (VIS) on the ESA Euclid mission. VIS is a large optical-band imager with a field of view of 0.54 deg^2 sampled at 0.1" with an array of 609 Megapixels and spatial resolution of 0.18". It will be used to survey approximately 14,000 deg^2 of extragalactic sky to measure the distortion of galaxies in the redshift range z=0.1-1.5 resulting from weak gravitational lensing, one of the two principal cosmology probes of Euclid. With photometric redshifts, the distribution of dark matter can be mapped in three dimensions, and, from how this has changed with look-back time, the nature of dark energy and theories of gravity can be constrained. The entire VIS focal plane will be transmitted to provide the largest images of the Universe from space to date, reaching m_AB>24.5 with S/N >10 in a single broad I_E~(r+i+z) band over a six year survey. The particularly challenging aspects of the instrument are the control and calibration of observational biases, which lead to stringent performance requirements and calibration regimes. With its combination of spatial resolution, calibration knowledge, depth, and area covering most of the extra-Galactic sky, VIS will also provide a legacy data set for many other fields. This paper discusses the rationale behind the VIS concept and describes the instrument design and development before reporting the pre-launch performance derived from ground calibrations and brief results from the in-orbit commissioning. VIS should reach fainter than m_AB=25 with S/N>10 for galaxies of full-width half-maximum of 0.3" in a 1.3" diameter aperture over the Wide Survey, and m_AB>26.4 for a Deep Survey that will cover more than 50 deg^2. The paper also describes how VIS works with the other Euclid components of survey, telescope, and science data processing to extract the cosmological information.

Galaxy proto-clusters are receiving an increased interest since most of the processes shaping the structure of clusters of galaxies and their galaxy population are happening at early stages of their formation. The Euclid Survey will provide a unique opportunity to discover a large number of proto-clusters over a large fraction of the sky (14 500 square degrees). In this paper, we explore the expected observational properties of proto-clusters in the Euclid Wide Survey by means of theoretical models and simulations. We provide an overview of the predicted proto-cluster extent, galaxy density profiles, mass-richness relations, abundance, and sky-filling as a function of redshift. Useful analytical approximations for the functions of these properties are provided. The focus is on the redshift range z= 1.5 to 4. We discuss in particular the density contrast with which proto-clusters can be observed against the background in the galaxy distribution if photometric galaxy redshifts are used as supplied by the ESA Euclid mission together with the ground-based photometric surveys. We show that the obtainable detection significance is sufficient to find large numbers of interesting proto-cluster candidates. For quantitative studies, additional spectroscopic follow-up is required to confirm the proto-clusters and establish their richness.