In this paper we estimate the number of gravitational arcs detectable in a
wide-field survey such as that which will be operated by the Euclid space
mission, assuming a {\Lambda}CDM cosmology. We use the publicly available code
MOKA to obtain realistic deflection angle maps of mock gravitational lenses.
The maps are processed by a ray-tracing code to estimate the strong lensing
cross sections of each lens. Our procedure involves 1) the generation of a
light-cone which is populated with lenses drawn from a theoretical
mass-function; 2) the modeling of each single lens using a triaxial halo with a
NFW (Navarro-Frenk-White) density profile and theoretical concentration-mass
relation, including substructures, 3) the determination of the lensing cross
section as a function of redshift for each lens in the light-cone, 4) the
simulation of mock observations to characterize the redshift distribution of
sources that will be detectable in the Euclid images. We focus on the so-called
giant arcs, i.e. gravitational arcs characterized by large length-to-width
ratios (l/w > 5, 7.5 and 10). We quantify the arc detectability at different
significances above the level of the background. Performing 128 different
realizations of a 15,000 sq. degree survey, we find that the number of arcs
detectable at 1{\sigma} above the local background will be 8912,2914, and 1275
for l/w>5, 7.5 and 10, respectively. The expected arc numbers decrease to 2409,
790, and 346 for a detection limit at 3{\sigma} above the background level.
From our analysis, we find that most of the lenses which contribute to the
lensing optical depth are located at redshifts 0.4<zl<0.7 and that the 50% of
the arcs are images of sources at zs > 3. This is the first step towards the
full characterization of the population of strong lenses that will be observed
by Euclid. [abridged]Comment: replaced to match the accepted version by MNRAS, 12 pag, 10 fig -
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