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.

Abstract. We present accurate and efficient computations of large scale structure observables, obtained with a modified version of the class code which is made publicly available. This code includes all relativistic corrections and computes both the power spectrum C (z 1 , z 2 ) and the corresponding correlation function ξ(θ, z 1 , z 2 ) of the matter density and the galaxy number fluctuations in linear perturbation theory. For Gaussian initial perturbations, these quantities contain the full information encoded in the large scale matter distribution at the level of linear perturbation theory. We illustrate the usefulness of our code for cosmological parameter estimation through a few simple examples.

Cosmological models with Galileon gravity are an alternative to the standard ΛCDM paradigm with testable predictions at the level of its self-accelerating solutions for the expansion history, as well as large-scale structure formation. Here, we place constraints on the full parameter space of these models using data from the cosmic microwave background (CMB) (including lensing), baryonic acoustic oscillations (BAO) and the Integrated Sachs-Wolfe (ISW) effect. We pay special attention to the ISW effect for which we use the crossspectra, C Tg , of CMB temperature maps and foreground galaxies from the WISE survey. The sign of C Tg is set by the time evolution of the lensing potential in the redshift range of the galaxy sample: it is positive if the potential decays (like in ΛCDM), negative if it deepens. We constrain three subsets of Galileon gravity separately known as the Cubic, Quartic and Quintic Galileons. The cubic Galileon model predicts a negative C Tg and exhibits a 7.8σ tension with the data, which effectively rules it out. For the quartic and quintic models the ISW data also rule out a significant portion of the parameter space but permit regions where the goodness-of-fit is comparable to ΛCDM. The data prefers a non zero sum of the neutrino masses ( m ν ≈ 0.5eV) with ∼ 5σ significance in these models. The best-fitting models have values of H 0 consistent with local determinations, thereby avoiding the tension that exists in ΛCDM. We also identify and discuss a ∼ 2σ tension that Galileon gravity exhibits with recent BAO measurements. Our analysis shows overall that Galileon cosmologies cannot be ruled out by current data but future lensing, BAO and ISW data hold strong potential to do so.

Abstract. We determine the number counts to second order in cosmological perturbation theory in the Poisson gauge and allowing for anisotropic stress. The calculation is performed using an innovative approach based on the recently proposed "geodesic light-cone" gauge. This allows us to determine the number counts in a purely geometric way, without using Einstein's equation. The result is valid for general dark energy models and (most) modified gravity models. We then evaluate numerically some relevant contributions to the number counts bispectrum. In particular we consider the terms involving the density, redshift space distortion and lensing.