Euclid is a European Space Agency medium-class mission selected for launch in 2020 within the cosmic vision 2015–2025 program. The main goal of Euclid is to understand the origin of the accelerated expansion of the universe. Euclid will explore the expansion history of the universe and the evolution of cosmic structures by measuring shapes and red-shifts of galaxies as well as the distribution of clusters of galaxies over a large fraction of the sky. Although the main driver for Euclid is the nature of dark energy, Euclid science covers a vast range of topics, from cosmology to galaxy evolution to planetary research. In this review we focus on cosmology and fundamental physics, with a strong emphasis on science beyond the current standard models. We discuss five broad topics: dark energy and modified gravity, dark matter, initial conditions, basic assumptions and questions of methodology in the data analysis. This review has been planned and carried out within Euclid’s Theory Working Group and is meant to provide a guide to the scientific themes that will underlie the activity of the group during the preparation of the Euclid mission.

Euclid is a European Space Agency medium-class mission selected for launch in 2019 within the Cosmic Vision 2015–2025 program. The main goal of Euclid is to understand the origin of the accelerated expansion of the universe. Euclid will explore the expansion history of the universe and the evolution of cosmic structures by measuring shapes and red-shifts of galaxies as well as the distribution of clusters of galaxies over a large fraction of the sky.Although the main driver for Euclid is the nature of dark energy, Euclid science covers a vast range of topics, from cosmology to galaxy evolution to planetary research. In this review we focus on cosmology and fundamental physics, with a strong emphasis on science beyond the current standard models. We discuss five broad topics: dark energy and modified gravity, dark matter, initial conditions, basic assumptions and questions of methodology in the data analysis.This review has been planned and carried out within Euclid’s Theory Working Group and is meant to provide a guide to the scientific themes that will underlie the activity of the group during the preparation of the Euclid mission.

The description of the abundance and clustering of haloes for non‐Gaussian initial conditions has recently received renewed interest, motivated by the forthcoming large galaxy and cluster surveys, which can potentially yield constraints of the order of unity on the non‐Gaussianity parameter fNL. We present tests on N‐body simulations of analytical formulae describing the halo abundance and clustering for non‐Gaussian initial conditions. We calibrate the analytic non‐Gaussian mass function of Matarrese, Verde & Jimenez and LoVerde et al. and the analytic description of clustering of haloes for non‐Gaussian initial conditions on N‐body simulations. We find an excellent agreement between the simulations and the analytic predictions if we make the corrections and , where q≃ 0.75, in the density threshold for gravitational collapse and in the non‐Gaussian fractional correction to the halo bias, respectively. We discuss the implications of this correction on present and forecasted primordial non‐Gaussianity constraints. We confirm that the non‐Gaussian halo bias offers a robust and highly competitive test of primordial non‐Gaussianity.

We analyse the clustering features of Large Scale Structures (LSS) in the presence of massive neutrinos, employing a set of large-volume, high-resolution cosmological N-body simulations, where neutrinos are treated as a separate collisionless fluid. The volume of 8 h −3 Gpc 3 , combined with a resolution of about 8 × 10 10 h −1 M for the cold dark matter (CDM) component, represents a significant improvement over previous N-body simulations in massive neutrino cosmologies. In this work we focus, in the first place, on the analysis of nonlinear effects in CDM and neutrinos perturbations contributing to the total matter power spectrum. We show that most of the nonlinear evolution is generated exclusively by the CDM component. We therefore compare mildly nonlinear predictions from Eulerian Perturbation Theory (PT), and fully nonlinear prescriptions (halofit) with the measurements obtained from the simulations. We find that accounting only for the nonlinear evolution of the CDM power spectrum allows to recover the total matter power spectrum with the same accuracy as the massless case. Indeed, we show that, the most recent version of the halofit formula calibrated on ΛCDM simulations can be applied directly to the linear CDM power spectrum without requiring additional fitting parameters in the massive case. As a second step, we study the abundance and clustering properties of CDM halos, confirming that, in massive neutrino cosmologies, the proper definition of the halo bias should be made with respect to the cold rather than the total matter distribution, as recently shown in the literature. Here we extend these results to the redshift space, finding that, when accounting for massive neutrinos, an improper definition of the linear bias can lead to a systematic error of about 1-2% in the determination of the linear growth rate from anisotropic clustering. This result is quite important if we consider that future spectroscopic galaxy surveys, as e.g. Euclid, are expected to measure the linear growth-rate with statistical errors less than about 3% at z 1.