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.

Type Ia supernova (SN Ia), galaxy clustering, and cosmic microwave background anisotropy (CMB) data provide complementary constraints on the nature of the dark energy in the universe. We find that the three-year Wilkinson Microwave Anisotropy Probe (WMAP) observations give a CMB shift parameter of R ≡ Ω m H 2 0 1/2 zCMB 0 dz ′ /H(z ′ ) = 1.70 ± 0.03. Using this new measured value of the CMB shift parameter, together with the baryon acoustic oscillation (BAO) measurement from the Sloan Digital Sky Survey (SDSS), and SN Ia data from the HST/GOODS program and the first year Supernova Legacy Survey, we derive model-independent constraints on the dark energy density ρ X (z) and the cosmic expansion rate H(z). We also derive constraints on the dark energy equation of state w X (z) = w 0 + w ′ z (with cutoff at z = 2) and w X (a) = w 0 + (1 − a)w a .We find that current data provide slightly tighter constraints on ρ X (z) and H(z) as free functions in redshift, and roughly a factor of two improvement in constraining w X (z). A cosmological constant remains consistent with data, however, uncertainties remain large for model-independent constraints of dark energy. Significant increase in the number of observed SNe Ia between redshifts of 1 and 2, complemented by improved BAO and weak lensing cosmography measurements (as expected from the JEDI mission concept for the Joint Dark Energy Mission), will be required to dramatically tighten model-independent dark energy constraints.

Continued photometric monitoring of the gravitational lens system 0957+561A,B in the g and r bands with the Apache Point Observatory (APO) 3.5 m telescope during 1996 shows a sharp g band event in the trailing (B) image light curve at the precise time predicted in an earlier paper. The prediction was using gravitational lenses and some other possible implications and uses of the 0957+561A,B light curves.

Using supernova, cosmic microwave background, and galaxy clustering data, we make the most accurate measurements to date of the dark energy density rho(X) as a function of cosmic time, constraining it in a rather model-independent way, assuming a flat universe. We find that Einstein's simplest scenario, where rho(X)(z) is constant, remains consistent with these new tight constraints and that a big crunch or big rip is more than 50 Gyr away for a broader class of models allowing such cataclysmic events. We discuss popular pitfalls and hidden priors.

Current observational bounds on dark energy depend on our assumptions about the curvature of the universe. We present a simple and efficient method for incorporating constraints from Cosmic Microwave Background (CMB) anisotropy data, and use it to derive constraints on cosmic curvature and dark energy density as a free function of cosmic time using current CMB, Type Ia supernova (SN Ia), and baryon acoustic oscillation (BAO) data.We show that there are two CMB shift parameters, R ≡ ΩmH 2 0 r(zCMB) (the scaled distance to recombination) and la ≡ πr(zCMB)/rs(zCMB) (the angular scale of the sound horizon at recombination), with measured values that are nearly uncorrelated with each other. Allowing nonzero cosmic curvature, the three-year WMAP data give R = 1.71 ± 0.03, la = 302.5 ± 1.2, and Ω b h 2 = 0.02173 ± 0.00082, independent of the dark energy model. The corresponding bounds for a flat universe are R = 1.70 ± 0.03, la = 302.2 ± 1.2, and Ω b h 2 = 0.022 ± 0.00082. We give the covariance matrix of (R, la, Ω b h 2 ) from the three-year WMAP data. We find that (R, la, Ω b h 2 ) provide an efficient and intuitive summary of CMB data as far as dark energy constraints are concerned.Assuming the HST prior of H0 = 72 ± 8 (km/s)Mpc −1 , using 182 SNe Ia (from the HST/GOODS program, the first year Supernova Legacy Survey, and nearby SN Ia surveys), (R, la, Ω b h 2 ) from WMAP three year data, and SDSS measurement of the baryon acoustic oscillation scale, we find that dark energy density is consistent with a constant in cosmic time, with marginal deviations from a cosmological constant that may reflect current systematic uncertainties or true evolution in dark energy. A flat universe is allowed by current data: Ω k = −0.006 +0.013 −0.012 +0.025 −0.025 for assuming that the dark energy equation of state wX (z) is constant, and Ω k = −0.002 +0.018 −0.018 +0.041 −0.032 for wX (z) = w0 + wa(1 − a) (68% and 95% confidence levels). The bounds on cosmic curvature are less stringent if dark energy density is allowed to be a free function of cosmic time, and are also dependent on the assumption about the early time property of dark energy. We demonstrate this by studying two examples. Significant improvement in dark energy and cosmic curvature constraints is expected as a result of future dark energy and CMB experiments.