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.

We propose a universal description of dark energy and modified gravity that includes all single-field models. By extending a formalism previously applied to inflation, we consider the metric universally coupled to matter fields and we write in terms of it the most general unitary gauge action consistent with the residual unbroken symmetries of spatial diffeomorphisms. Our action is particularly suited for cosmological perturbation theory: the background evolution depends on only three operators. All other operators start at least at quadratic order in the perturbations and their effects can be studied independently and systematically. In particular, we focus on the properties of a few operators which appear in non-minimally coupled scalar-tensor gravity and galileon theories. In this context, we study the mixing between gravity and the scalar degree of freedom. We assess the quantum and classical stability, derive the speed of sound of fluctuations and the renormalization of the Newton constant. The scalar can always be de-mixed from gravity at quadratic order in the perturbations, but not necessarily through a conformal rescaling of the metric. We show how to express covariant field-operators in our formalism and give several explicit examples of dark energy and modified gravity models in our language. Finally, we discuss the relation with the covariant EFT methods recently appeared in the literature.

The observation of GW170817 and its electromagnetic counterpart implies that gravitational waves travel at the speed of light, with deviations smaller than a few×10^{-15}. We discuss the consequences of this experimental result for models of dark energy and modified gravity characterized by a single scalar degree of freedom. To avoid tuning, the speed of gravitational waves must be unaffected not only for our particular cosmological solution but also for nearby solutions obtained by slightly changing the matter abundance. For this to happen, the coefficients of various operators must satisfy precise relations that we discuss both in the language of the effective field theory of dark energy and in the covariant one, for Horndeski, beyond Horndeski, and degenerate higher-order theories. The simplification is dramatic: of the three functions describing quartic and quintic beyond Horndeski theories, only one remains and reduces to a standard conformal coupling to the Ricci scalar for Horndeski theories. We show that the deduced relations among operators do not introduce further tuning of the models, since they are stable under quantum corrections.