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Euclid is the second M-class mission of the ESA Cosmic Vision programme, withthe principal science goal of studying dark energy. Euclid is also expected toundertake additional Legacy Science programmes. One proposal is the ExoplanetEuclid Legacy Survey (ExELS) which will be the first survey able to measure theabundance of exoplanets down to Earth mass for host separations from ~1AU outto the free-floating (unbound) regime. The cold and free-floating exoplanetregimes represent a crucial discovery space for testing planet formationtheories. ExELS will use the gravitational microlensing technique and willdetect 1000 microlensing events per month over 1.6 deg^2 of the Galactic bulge.We assess how many of these events will have detectable planetary signaturesusing a detailed multi-wavelength microlensing simulator (MABuLS) whichincorporates the Besancon Galactic model with 3D extinction. MABuLS is thefirst theoretical simulation of microlensing to treat the effects of pointspread function (PSF) blending self-consistently with the underlying Galacticmodel. We use MABuLS, together with current numerical models for the EuclidPSFs, to explore a number of designs and de-scope options for ExELS, includingthe exoplanet yield as a function of filter choice and slewing time, and theeffect of systematic photometry errors. Using conservative extrapolations ofcurrent empirical exoplanet mass functions determined from ground-basedmicrolensing and radial velocity surveys, ExELS can expect to detect a fewhundred cold exoplanets around mainly G, K and M-type stellar hosts, including~45 Earth-mass planets and ~6 Mars-mass planets for an observing programmetotalling 10 months. ExELS will be capable of measuring the cold exoplanet massfunction down to Earth mass or below, with orbital separations from ~1AU to thefree-floating regime. (Abridged)

ExELS: an exoplanet legacy science proposal for the ESA Euclid mission I. Cold exoplanets