Magnification Bias in Gravitational Arc Statistics
  The statistics of gravitational arcs in galaxy clusters is a powerful probe
of cluster structure and may provide complementary cosmological constraints.
Despite recent progresses, discrepancies still remain among modelling and
observations of arc abundance, specially regarding the redshift distribution of
strong lensing clusters. Besides, fast "semi-analytic" methods still have to
incorporate the success obtained with simulations. In this paper we discuss the
contribution of the magnification in gravitational arc statistics. Although
lensing conserves surface brightness, the magnification increases the
signal-to-noise ratio of the arcs, enhancing their detectability. We present an
approach to include this and other observational effects in semi-analytic
calculations for arc statistics. The cross section for arc formation ({\sigma})
is computed through a semi-analytic method based on the ratio of the
eigenvalues of the magnification tensor. Using this approach we obtained the
scaling of {\sigma} with respect to the magnification, and other parameters,
allowing for a fast computation of the cross section. We apply this method to
evaluate the expected number of arcs per cluster using an elliptical
Navarro--Frenk--White matter distribution. Our results show that the
magnification has a strong effect on the arc abundance, enhancing the fraction
of arcs, moving the peak of the arc fraction to higher redshifts, and softening
its decrease at high redshifts. We argue that the effect of magnification
should be included in arc statistics modelling and that it could help to
reconcile arcs statistics predictions with the observational data.
