Euclid: performance on main cosmological parameter science

Euclid will observe 15 000 deg$^2$ of the darkest sky, in regions free of contamination by light from our Galaxy and our Solar System. Three "Euclid Deep Fields" surveys covering around 40 deg$^2$ in total will extend the scientific scope of the mission to the high-redshift Universe. The complete survey will be constituted by hundreds of thousands of images and several tens of petabytes of data. About 10 billion sources will be observed. With these images Euclid will probe the expansion history of the Universe and the evolution of cosmic structures. This will be achieved by measuring the effect on galaxy shapes due to dark matter gravitational lensing, and by reconstructing the three-dimensional distribution of cosmic structures from the measured spectroscopic redshifts of galaxies and clusters of galaxies. These proceedings present the implications for cosmology and cosmological constraints of this unprecedented data set. Of particular interest are the expected constraints on the nature of dark energy.