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The next generation of large sky photometric surveys will finally be able touse arc statistics as a cosmological probe. Here we present the first of aseries of papers on this topic. In particular, we study how arc counts aresensitive to the variation of two cosmological parameters: the (total) matterdensity parameter, $\Omega_m$, and the normalisation of the primordial powerspectrum, expressed in terms of $\sigma_8$. Both these parameters influence theabundances of collapsed structures and their internal structure. We compute theexpected number of gravitational arcs with various length-to-width ratios inmock light cones, by varying these cosmological parameters in the ranges$0.1\leq\Omega_m\leq0.5$ and $0.6\leq\sigma_8\leq 1$. We find that the arccounts dependence on $\Omega_m$ and $\sigma_8$ is similar, but not identical,to that of the halo counts. We investigate how the precision of the constraintson the cosmological parameters based on arc counts depends on the survey area.We find that the constraining power of arc statistics degrades critically onlyfor surveys covering an area smaller than $10\%$ of the whole sky. Finally, weconsider the case in which the search for arcs is done only in frames wheregalaxy clusters have been previously identified. Adopting the selectionfunction for galaxy clusters expected to be detected from photometric data infuture wide surveys, we find that less than $10\%$ of the arcs will be missed,with only a small degradation of the corresponding cosmological constraints.

Cosmology through arc statistics I: sensitivity to $\Omega_m$ and $\sigma_8$