{
  "MAGNIFICATION BIAS IN GRAVITATIONAL ARC STATISTICS ": [
    "G. B. Caminha ",
    "Centro Brasileiro de Pesquisas F´ısicas, Rio de Janeiro, Brazil ",
    "J. Estrada ",
    "Fermi National Accelerator Laboratory, Batavia, Illinois, USA ",
    "and ",
    "M. Makler ",
    "Centro Brasileiro de Pesquisas F´ısicas, Rio de Janeiro, Brazil ",
    "Draft version September 17, 2013 "
  ],
  "ABSTRACT ": [
    "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 (σ) 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 σ with respect to the mag-niﬁcation, 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. We include the eﬀect of magniﬁcation using two methods: by considering the magniﬁcation dependent arc cross section and by assuming that all sources in the arc forming region are magniﬁed by the same amount. We show that both methods are in excellent agreement and it is thus suﬃcient to consider the eﬀect of the mean magniﬁcation in arc statistics. 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. ",
    "Subject headings: gravitational lensing: strong, galaxies: clusters: general, dark matter, cosmological parameters "
  ],
  "1. INTRODUCTION ": [
    "Gravitational arcs are highly distorted images of background galaxies (sources) due to the gravitational field of a foreground object (lens), usually a galaxy or cluster of galaxies. After the discovery of the first gravitational arcs (Lynds & Petrosian 1986; Soucail et al. 1987), it was soon realized that the statistics of arcs in galaxy clusters could provide powerful probes of both cluster structure (see, e.g., Wu & Hammer 1993; Bartelmann & Weiss 1994; Hattori, Watanabe, & Yamashita 1997; Oguri, Lee, & Suto 2003) and cosmology (see, e.g., Grossman & Saha 1994; Wu & Mao 1996; Bartelmann et al. 1998; Zieser & Bartelmann 2012). For a complete review on arc statistics, see Meneghetti et al. (2013). ",
    "With this in mind, several arc searches were undertaken, both in imaging surveys (Gladders et al. 2003; Cabanaˇc, et al. 2007; Estrada et al. 2007; Faure et al. 2008; Belokurov et al. 2009; Kubo et al. 2010; Gilbank et al. 2011; More et al. 2012; Wen et al. 2011; Bayliss 2012; Wiesner et al. 2012; Pawase et al. 2012; Erben et al. 2013, More et al., in prep, Caminha et al., in prep.), as well as in fields targeting X-ray and optical clusters, with observations from the ground (Luppino et al. 1999; Zaritsky & Gonzalez 2003; Campusano et al. 2006; Hennawi et al. 2008; Kausch et al. 2010; Furlanetto et al. 2013b) and from space (Ebeling, Edge, & Henry 2001; Smith et al. 2005; Sand et al. 2005; Horesh et al. 2010; Okabe et al. 2013). With these surveys, the homogeneous samples of arcs are reaching the order of hundreds of objects.1 In the near future, the next generation wide-field surveys such as the Dark Energy Survey 2 (DES; The Dark Energy Survey Collaboration 2005; Annis, et al. 2005; Ngeow, et al. 2006, Lahav et al., in preparation, Frieman et al., in preparation), which started its first observing season this year, are expected to lead to the discovery of thousands of gravitational arcs. ",
    "In parallel to the growing size of arc samples, the theoretical modelling of arc statistics has also experienced signiﬁcant improvements. The earliest predictions for arc abundance used idealised lenses (from axially symmetric singular isothermal models to halos from low resolution pure CDM simulations) and sources (constant surface brightness sources distributed in a single plane) and strongly underestimated the number of arcs in the now standard ΛCDM cosmology with respect to observational data (see, e.g., Wu & Hammer 1993; Wu & Mao 1996; Bartelmann et al. 1998). Several degrees of realism were added subsequently, investigating the role of each in arc statistics. For example, on the lens side, the eﬀects of cluster ellipticity, asymmetries, and substructures (e.g., Hennawi, Dalal, Bode, & Ostriker 2007; Meneghetti et al. 2007), triaxiality (Oguri, Lee, & Suto 2003; Dalal, Holder, & Hennawi 2004; Meneghetti et al. 2010), and mergers (Torri et al. 2004; Fedeli et al. 2006; Hennawi, Dalal, Bode, & Ostriker 2007; Redlich et al. 2012) were investigated. The eﬀects of baryonic processes of cooling (Puchwein et al. 2005; Rozo, Nagai, Keeton, & Kravtsov 2006; Wambsganss, Ostriker & Bode 2008), star formation rate, and AGN feedback (Mead et al. 2010; Killedar et al. 2012) on the cluster structure, as well as the contribution of cluster galaxies (Flores, Maller & Primack 2000; Meneghetti et al. 2000) to the cluster lensing potential were taken into account. The contribution of structures along the line of sight (Hilbert, White, Hartlap, & Schneider 2007; Puchwein & Hilbert 2009) was also addressed. On the sources side, the impact of their size and ellipticity was investigated (e.g., Oguri 2002), including the eﬀect of arc formation by the merging of multiple images (Rozo, Nagai, Keeton, & Kravtsov 2006). The impact of the redshift distribution of sources was also considered (Wambsganss, Bode, Ostriker 2004; Dalal, Holder, & Hennawi 2004; Li et al. 2005). ",
    "The inclusion of the eﬀects above helped to shed light on their contribution on arc statistics. Most of them increase the arc incidence up to factors of a few and their inclusion has reduced the overall discrepancy between observed and predicted arc abundances (e.g., Dalal, Holder, & Hennawi 2004; Horesh et al. 2005; Hennawi, Dalal, Bode, & Ostriker 2007). One of the key factors to reduce this discrepancy was the inclusion of realistic source distributions in the theoretical predictions (Horesh et al. 2005; Wu & Chiueh 2006; Hennawi, Dalal, Bode, & Ostriker 2007). Another key progress was the development of simulations at the image level (Meneghetti et al. 2008; Horesh et al. 2011; Boldrin et al. 2012), which led to the inclusion of new levels of realism, such as sources with surface brightness variations and noise, and allowed for a more direct comparison with the observations, including the use of the same tools for arc detection. The study of selection biases also contributed to a more accurate comparison between simulated and real data (Meneghetti et al. 2010). Overall, these developments have, if not solved, at least mitigated the longstanding arc statistics problem (Meneghetti et al. 2011). ",
    "If on one hand the simulations are generally beginning to broadly agree with the observational data, on the other hand several open issues remain, among which the variation of arc abundance with respect to the cluster redshift. For example, Gladders et al. (2003) found an over-abundance of arcs in high-redshift clusters as compared to lower redshift ones in the Red Sequence Cluster Survey (RCS). In particular, they found no arcs in clusters at z  0.6. However, theoretical investigations did not predict such an enhanced incidence of arcs at high z. For example, Hennawi, Dalal, Bode, & Ostriker (2007) found no increase in arc abundance at the redshift implied by RCS. Gonzalez et al. (2012) found arcs in a cluster at z = 1.75, which should not be present at their image depths according to their modelling. ",
    "Up to the present, most of the focus of theoretical studies in this ﬁeld — from idealized analytical treatments to fully numerical ray tracing methods in simulated mass distributions — has been on the modelling of the lens potential and on gravitational lensing aspects of arc formation in general. However, observational eﬀects such as ﬁnite PSF size and the noise present in real images play a fundamental role in arc detection. An important progress to include these eﬀects was undertaken by Horesh et al. (2005), who added noise (sky and detector backgrounds, Poisson noise, and readout noise), PSF, and light from cluster galaxies to the arc simulations (see also Meneghetti et al. 2005, 2008; ",
    "Horesh et al. 2011; Boldrin et al. 2012). However, the full image simulation process, including the generation of clusters in N-body/hydrodynamical simulations, the computation of deﬂection angles, the association of images to sources, and the addition of observational eﬀects; together with the detection of arcs in the simulated images, can be extremely expensive in terms of computational time. ",
    "An alternative to this “fully numerical” treatment is to use semi-analytical approaches, which can incorporate, to a certain degree, some of the features of the realistic modelling described above. For example, analytic density profiles can be used instead of N-body simulations, local prescriptions to define the formation of arcs instead of finding images of finite sources, etc. Although this approach is expected to be less accurate, it is useful to investigate the qualitative behavior of the arc abundance with respect to several variables. It can also provide a valuable insight to isolate the key factors that contribute to the modelling of arc abundance, being helpful for interpreting and understanding the outcome of simulations. Furthermore, if calibrated with realistic simulations, the semi-analytic modelling may provide fitting formulae for fast parameter estimation (e.g., to obtain cluster parameters from arc statistics) and for producing forecasts. ",
    "In this paper, we use a semi-analytic approach to compute the expected number of arcs per cluster as a function of cluster redshift, mass, and ellipticity. We discuss how to include some observational eﬀects and, in particular, we take into account, for the ﬁrst time in this approach, the eﬀect of the noise in the images. The signal-to-noise ratio (S/N) of the arc images is enhanced by the magniﬁcation, eﬀectively changing the limiting magnitude of the image in the region of arc formation, enhancing the eﬃciency of arc detection. This results in a kind of magniﬁcation bias and leads us to include this quantity in the arc abundance modelling. ",
    "We obtain scaling relations that allow for a fast computation of the cross section for arc formation, taking into account its variation with respect to magniﬁcation. We apply this method to an elliptical Navarro–Frenk–White matter distribution in clusters and compute the number of arcs per cluster considering a realistic redshift distribution of sources, which is eﬀectively modiﬁed due to the eﬀect of the magniﬁcation. ",
    "The paper is organized as follows. In the next section we present the semi-analytic modelling for the arc fraction in clusters. In section 3 we discuss the importance of the signal-to-noise ratio for arc detection and how to incorporate the eﬀect of S/N enhancement through the magniﬁcation in gravitational arc statistics. In section 4 we review the semi-analytic computation of the arc cross section and the lens model that will be used. In section 5 we obtain the scaling of the cross section with respect to the magniﬁcation, which is needed to take the noise into account in the modelling. Finally, in section 6 we apply these results to compute the expected number of arcs per cluster — showing the importance of magniﬁcation — and discuss its evolution with respect to the cluster redshift. We summarize our results and present some concluding remarks in section 7. In the Appendix we derive useful expressions for the lensing quantities of elliptical lens models for any parameterisation of the ellipticity. "
  ],
  "2. SEMI-ANALYTICAL MODELLING FOR ARC ABUNDANCE: INCLUDING OBSERVATIONAL EFFECTS ": [
    "Our aim is to compute the average number of arcs per cluster farcs, for clusters with given properties, such as mass and ellipticity (see, e.g., Cypriano et al. 2001; Hattori, Watanabe, & Yamashita 1997; Oguri, Lee, & Suto 2003; Horesh et al. 2005, for other works using this same statistical measure). For a sample of clusters selected with uniform criteria, for example a given richness and redshift interval, this fraction is easily calculated as farcs = Narc/Ncluster, where Ncluster is the number of clusters in the sample and Narcs is the total number of arcs found in this sample. While many studies on arc statistics have focused on predictions for the total number of arcs, this quantity is not very suitable to probe properties associated with gravitational lensing (such as lens mass proﬁles or cosmological distances), since it is extremely sensitive to the cosmological parameters in a way not related to lensing at all. More speciﬁcally, the total number of arcs depends on cluster abundance, which is in turn very sensitive to σ8 and the growth factor. Some inconsistencies among arc abundance predictions were due to the diﬀerent choices of the cosmological parameters, rather than the modelling of the lensing aspects of the problem (see, e.g., Li et al. 2005). By focusing on farcs we may decouple the generation of arcs from cluster abundance, minimizing the explicit dependence on cosmology. Moreover, this statistic is also well suited for targeted surveys, were a sample of clusters is selected to be observed with better imaging to search for arcs. Finally, the fraction of arcs is less dependent on the cluster selection function than the total arc number (of course, as long as the selection is not based on the presence of arcs). A careful selection of the clusters for this statistic could also mitigate the selection biases (Meneghetti et al. 2010). ",
    "A direct method to evaluate the expected number of arcs per cluster is to perform simulations where a given mass distribution is used to lens objects from a real astronomical image. The resulting images of these sources are identiﬁed as arcs, according to their properties. As mentioned in section 1, the source redshift distribution must be taken into account in computing the arc abundance. For this sake, redshifts can be associated to each source in the image, as in Horesh et al. (2005). A similar approach is to lens sources distributed in multiple planes (Hennawi, Dalal, Bode, & Ostriker 2007). Another possibility, which we refer to as “semi-analytic modelling for arc abundance”, consists in computing the cross section for arc formation σ for a given cluster (i.e., the eﬀective area in the source plane where a source gives rise to an arc in the lens plane) as a function of source redshift zS and convolving it with the source surface number density ΣS as a function of zS . ",
    "The cross section is generally defined in terms of a given set of parameters of a lensed object. For example, a standard criterium, which we shall apply in this work, is to use the length-to-width ratio (R := L/W ) of the images. Whenever this ratio is above a given threshold (Rth) the image is said to form an arc. Other parameters that are relevant for detection include the magnitude and, possibly, the curvature. Naturally, σ is a function of the lens properties, such as mass M, ellipticity e, slope of the density proﬁle γ, and characteristic radius rs, and also depends on the lens and ",
    "source redshifts, zL and zS, through the cosmological distances. It may also depend on source properties, such as ellipticity eS and angular size θS. Therefore, we may write the cross section as σ = σ (ΠL, zL; ΠS, zS; QP ), where ΠL is a set of cluster properties (M, rs, γ, e, etc.), ΠS is a set of source properties (θS , eS ), and QP is a set of arc properties (L/W , S/N, µ, etc.).3 ",
    "Besides its dependence on zS, the source distribution ΣS is a function of θS , eS , and magnitude m. In principle, ΣS (zS, ΠS) can be estimated from deep space-based surveys, such as the COSMOS ﬁeld4 (see, e.g., Scoville 2007). If we do not take into account the source sizes and ellipticies, ΣS can be estimated from ground based observations. ",
    "A third key ingredient is related to the observational eﬀects, which transform the “proper” arc parameters QP , that would be measured in noise free images without PSF eﬀects, into their observed ones QO. For example, the seeing will change the width of the arcs in a way that can be modelled approximately by WO = WP2 + σs2 , where σs is the seeing FWHM. These eﬀects are encoded in the “transfer” function T (QP |QO). ",
    "The ﬁnal important ingredient in the arc fraction calculation is the detection eﬃciency, which depends only on the observed arc properties QO (i.e., of the objects identiﬁed in real images, which are PSF convolved and include all sources of noise) in a given survey. Usually it is assumed that all arcs with observed (L/W )O and peak surface brightness above some threshold will be detected in a given image. However, in realistic arc ﬁnding methods the eﬃciency is a more complicated function of these quantities. The eﬃciency P (QO) can be determined, for each survey, using image simulations containing fake arcs, as in Estrada et al. (2007) (see also Horesh et al. 2005; Furlanetto et al. 2013a). ",
    "Taking into account the above aspects, we may express the arc fraction for clusters with properties ΠL as ",
    ". ",
    "(1) ",
    "In practical applications this expression can be simpliﬁed. For example, we shall see in section 4 that a scaling of σ with zL, and zS can be taken into account analytically. Besides, in section 5 we obtain a good approximation for the scaling of the cross section with an arc parameter QP (in this case, the magniﬁcation µ). Finally, in our modelling of section 6, σ will be independent of ΠS . Therefore, the number of degrees of freedom in equation 1 can be substantially reduced, and some of the remaining ones can be included in an analytic form. Furthermore, in many cases we may explicitly write QP in terms of QO (as the relation among WP and WQ discussed above), instead of using T (QP |QO). ",
    "The formalism described above allows us to take into account some observational eﬀects together with the detection eﬃciency, which are usually considered only in full image simulations. A useful aspect of this approach is that it naturally separates the problem into various components, enabling investigations on the inﬂuence of each part in the arc statistics predictions. ",
    "In this work we shall discuss the eﬀects of the signal-to-noise ratio in the images, which leads us to consider the role of the magniﬁcation in the computation of the arc fraction in clusters. "
  ],
  "3. THE ROLE OF MAGNIFICATION ": [
    "Since gravitational lensing conserves surface brightness, it is commonly assumed that the magnification is not important for the detection of extended objects such as arcs. This would indeed be true if no noise was present in the images. However, in a real astronomical situation the total flux may have an important role in the detection, due to the signal-to-noise ratio of the arcs. In fact, arcs will not be identified by their peak surface brightness, but rather by the coherence of the signal produced by that object compared to the noise in the image. Therefore, the signal-to-noise ratio (S/N) must play an important role in arc detection. ",
    "Here we assume that the dominant sources of noise are Poisson noise from the background and from the object counts. The signal is simply given by the total flux of the object (counts) and the noise is the integrated noise on the object from the Poisson noise of the background and the object. For an object with surface brightness F and occupying the area A in an image whose background has a surface brightness b, this ratio is ",
    "(2) ",
    "The gravitational lensing eﬀect changes only the image area, with the new area being Alensed = µ × A, where µ is the magniﬁcation. Taking this into account, the lensed object has ",
    "(3) ",
    "The enhancement of the signal-to-noise of the lensed objects will lead to an increase of the eﬀective depth of the image in the magniﬁed regions, which will be translated into an eﬀective change of the distribution of sources that can be mapped into arcs. Let us assume that the objects in an image are identiﬁed and selected in terms of their S/N (e.g.,√imposing a cut of 5σ above noise for detection). In this case, the eﬀective ﬂux limit will decrease by a factor of 1/ µ (c.f. Eq. 3) in comparison to a non-lensed region, due to the eﬀect of magniﬁcation. In other words, the eﬀective limiting magnitude will be given by ",
    "(4) ",
    "To include this eﬀect in gravitational arc statistics, we use the formalism of section 2. The arc detection eﬃciency P (QO), which will be a function of the arc S/N, is connected to the source distribution in an intricate way through Eq. (1). The cross section will depend on arc and source properties and the transfer function will map the intrinsic to observed arc properties. Here we will make a few simplifying assumptions as a ﬁrst approach to the problem. First, we assume that the total signal-to-noise and the length-to-width ratios are the dominant factors to determine the detectability of the arcs. We will further assume that the sensitivity to these quantities is uncorrelated and that all arcs with L/W above a given threshold Rth and with S/N above a threshold (S/N)th can be detected. In other words, the selection function will be given by ",
    "(5) ",
    "where θ(x) is the step function. In reality the situation is a bit more complicated since other factors, such as curvature and surface brightness, may impact the detection eﬃciency and also the sensitivity to S/N and L/W may be correlated. Furthermore, in realistic situations the selection function P (QO) will be dependent on the arc ﬁnding method. Here we assume that the object pixels are selected as part of the object identiﬁcation or arc ﬁnding methods. This can be achieved, for example, by requiring a low signiﬁcance for each object pixel with respect to the background. The S/N is then computed taking into account all object pixels and a higher signiﬁcance is required for the object to be selected as an arc (provided that the L/W restriction is met). As for the hard cut in L/W in Eq. (5), this is implicitly assumed in most arc statistics calculations. In any case, the novel aspect that we shall address in this work is the dependency on S/N, which will be shown to have a large impact on the arc abundance predictions. ",
    "We shall also neglect for now the explicit dependence of the arc L/W and S/N on the image PSF, such that the transfer function is given simply by ",
    "(6) ",
    "where δ is the Dirac delta function, and the quantities with O refer to the observed ones and those with P correspond to what would be measured without the eﬀects of the PSF (but same instrument and other observing conditions). ",
    "A third ingredient in the arc abundance calculation is the source distribution. In this paper we will neglect any dependence of the cross section on the properties of ﬁnite sources, such as size and ellipticity (see Sect. 4). Therefore, we only need to consider its dependence on redshift and source S/N, i.e. ΣS (zS, (S/N)source). The condition (6) implies that we need to consider all arcs with S/N above a given threshold. This will imply that all sources above a certain S/N (as would be seen in the same observing conditions, but no lensing) may give rise to arcs and the connection  √ among these thresholds is given by equation (3). In other words, we need to consider ΣS zS, > (S/N)th/ µ . Here we are of course assuming the same observing conditions for the determination of the source density and the arcs, such that we can use Eq. (3). In practice we only need to know how ΣS(zS) scales in terms of an S/N threshold. Usually the source distribution determined from imaging surveys is expressed in terms of zS and a limiting magnitude mlim (as long as mlim is well below the detection limit of the images), not a S/N threshold. However, this limiting magnitude is dependent on the threshold. We mimic this eﬀect following Eq. (4). Therefore, after integrating on the arc properties QO in Eq. (1) and applying the selection function (6), we only need to consider the surface density of sources per unit redshift, for a given limiting magnitude, An explicit form for this function will be used in Sect. 6. ",
    "Finally, we need to consider the cross section and its dependence on source and arc properties. As mentioned above, for simplicity we will not consider the eﬀect of ﬁnite sources. Besides, the cross section will not depend on S/N. The only dependence on the arc parameters will be on L/W and µ. Also, as in the case of the integral over (S/N)O, the integral over (L/W )O will lead to a threshold in (L/W )P , Rth. Thus we only need to compute the cross section for (L/W ) > Rth, which we will refer to as σ (ΠL, zL; zS; µ, Rth). ",
    "Inserting all these ingredients in Eq. (1) and integrating over the arc parameters µ, (L/W )O, and (S/N)O we ﬁnally obtain the expression for the arc fraction including the eﬀect of magniﬁcation ",
    "(7) ",
    "In the next section we discuss the computation of the cross section, and in particular the inclusion of its explicit dependence on the magnification, employing a semi-analytic approach. "
  ],
  "4. SEMI-ANALYTICAL CROSS SECTION ": [
    "To compute the cross section for the production of arcs, a simple approach is to resort to the local mapping given by the magnification tensor, which determines the distortion of infinitesimal sources. For this sake, we evaluate the two eigenvalues of the magnification tensor to determine the region in the image plane in which their ratio exceeds a certain threshold Rth. It is in this region that arcs with L/W exceeding Rth are expected to be formed. ",
    "The magnification eigenvalues are given by ",
    "(8) ",
    "where the convergence κ and the shear strength, γ = γ12 + γ22 , can be obtained from the derivatives of the lens potential Ψ: ",
    "(9) ",
    "where the subindices 1 and 2 refer to derivatives with respect to θ1 and θ2 respectively and Ψ = 2ψ DLS/(DOSDOL), where ψ is the projected gravitational potential of the lens, DOS denotes the angular diameter distance from the observer to the source, DOL that from the observer to the lens, and DLS that from the lens to the source 5 (see, e.g., Schneider, Ehlers, & Falco 1992; Mollerach & Roulet 2002). ",
    "Usually, µt corresponds to a deformation approximately in the tangential direction, while µr corresponds to the deformation along the radial direction. Therefore tangential arcs are expected to be formed in regions where R := |µt/µr| ≫ 1. For inﬁnitesimal circular sources, the length to width ratio (L/W ) of tangentially distorted images will be approximately given by the local deformation R (Hattori, Watanabe, & Yamashita 1997; Hamana & Futamase 1997). Arcs are usually deﬁned as objects with axial ratio above a given threshold L/W > Rth (a typical value considered is Rth = 10) and, according to the approximation discussed above, they will be produced in a region ΩL around the lens for which the local eigenvalue ratio satisﬁes the condition R > Rth. ",
    "This region in the image plane is mapped through the lens equation into a region in the source plane. The cross section for arc production σ (which we will refer to as deformation cross section) will be given by the corresponding solid angle in the source plane. Since the solid angles in the image and source planes are related by dΩS = dΩL/|µ(θ)| (where |µ| = |µt × µr| is the magniﬁcation) we hence have (see, e.g., Blandford & Narayan 1986; Fedeli et al. 2006; D´umet-Montoya et al. 2012) ",
    "(10) ",
    "Notice that, through this procedure, two points in the lens plane with R > Rth that are mapped into the same point in the source plane are accounted for twice in the computation of σ. This is the appropriate deﬁnition for computing farcs since this region in the source plane will give rise to two arcs. Hence σ is an eﬀective area in the source plane, already accounting for the multiplicity of the images. ",
    "Throughout this work we model the lenses by an elliptical Navarro, Frenk, & White (1996, 1997, hereafter NFW) mass distribution, whose radial (i.e. angle averaged) profile is given by ",
    "(11) ",
    "where the parameters rs and ρs can be expressed in terms of the mass M200 and the concentration parameter c (see Sect. 6). ",
    "The lensing properties may be obtained from the projected mass distribution, which in the case of the NFW profile is determined by two parameters (see, e.g., Bartelmann 1996): rs and ",
    "(12) ",
    "where Σcr = DOS/(4πGDOLDLS). ",
    "An elliptical surface density distribution can be obtained from the projected z2)dz (Bartelmann 1996) by replacing the radial coordinate ξ by ",
    "(13) ",
    "where the parameter e denotes the ellipticity. With this choice, the projected mass contained inside contours of constant ξ is independent of e. The relevant quantities for the lensing mapping can be expressed in terms of integrals, which are given in the Appendix. ",
    "Since the lens equation can be written in terms of an arbitrary length scale (see, e.g., Schneider, Ehlers, & Falco 1992), we may work with all distances in units of rs and eliminate this quantity from the analysis, such that the (rescaled) cross section σ˜ becomes a function of κs and e only (for a given L/W threshold). The cross section in steradians is then given by6 (D´umet-Montoya et al. 2012) ",
    "(14) ",
    "This already simplifies the calculation of the cross section by eliminating one degree of freedom from the problem (see, e.g., Oguri 2002; Oguri, Lee, & Suto 2003; Fedeli et al. 2006, for similar transformations). We will refer to σ˜ as the dimensionless cross section. To recover the cosmological and mass dependence of the parameters, one has to write κs and rs in terms of the NFW model parameters M200 and c, and the cosmological distances (see Sect. 6). ",
    "A similar procedure as the one described above for computing the arc cross section is often used for computing the magnification cross section for point-like sources (although, of course, the integration domain is defined by the constraint µ > µth, rather than R > Rth). For finite-size objects such as arcs, the validity of this method has to be checked. Indeed, due to the extended nature of the images, variations of the local eigenvalue ratio throughout the sources may be important. Source ellipticity will break the direct relation between L/W and R. Moreover, it is well ",
    "Fig. 1.— Dependence of the cross section with κs. A steep reduction in σ˜(κs) is seen for low values of κs. ",
    "known that arcs can be formed as the result of the merging of multiple images. The criterium based on the local eigenvalue ratio can only take into account arcs generated through the high distortion of single images, produced when sources are close to any caustic, but not those formed by image merging (Rozo, Nagai, Keeton, & Kravtsov 2006). Some of these ﬁnite source eﬀects can be accounted for in the semi-analytic computation of the cross section. For example Fedeli et al. (2006) introduce a method to include the gradients of R across the source. Keeton (2001c) proposed an approximation to obtain L/W from the magniﬁcation eigenvalues considering source ellipticity and orientation. ",
    "Any detailed prediction of arc abundance should take into account these eﬀects. Nevertheless, it has been show that including ﬁnite size and source ellipticity does not change the cross section by large factors (Hattori, Watanabe, & Yamashita 1997; Bartelmann et al. 1998; Oguri 2002; Fedeli et al. 2006) and their impact is considerably smaller than lens ellipticity, for example (Oguri 2002). Li et al. (2005) made a comparison of the cross section of lensing clusters with randomly oriented elliptical sources of ﬁxed area. They found that the cross section derived from |µt/µr| agrees within ∼ 30% with the ray-tracing simulations for Rth = 10 for an average of many simulated clusters and for a generalized NFW proﬁle. As will be shown, the impact of the magniﬁcation on arc statistics is much stronger than the above mentioned eﬀects. Therefore, for simplicity, we will ignore the eﬀects of ﬁnite size throughout this work and use Eq. (10) to compute the arc cross section. ",
    "In figure 1 we show the dimensionless cross section σ˜ as a function of κs for three ellipticities. From this ﬁgure we see that the cross section has a sharp drop for low values of κs and is, in general, not very sensitive to the ellipticity, which is in agreement with Rozo, Nagai, Keeton, & Kravtsov (2006) for distortion arcs. However, the sensitivity to e increases for lower κs. Therefore, the ellipticity will play a more important role for clusters of lower masses and those at lower redshifts (see the results of Sect. 6). "
  ],
  "5. SCALING OF THE CROSS SECTION WITH MAGNIFICATION ": [
    "As discussed in Sect. 3, the magniﬁcation may have an important role in determining the observed abundance of arcs by enhancing their signal-to-noise ratio in the images. To include the eﬀect of the magniﬁcation in farcs following the method of section 3 we need to determine the cross section as a function of µ. For this sake, we obtain the cross section as a function of a magniﬁcation threshold µth, σ˜(κs, e; µth), which is computed using the method described in section 4 above (Eq. 10) by adding the constraint that µ > µth in the integration region (besides requiring that R > Rth). In ﬁgure 2 we show σ˜(κs, e; µth), for e = 0.2, and κs = 0.5, 0.75, 1 (full lines). We see that for low values of µth the cross section is insensitive to this parameter. This is easy to understand, since all sources which lead to images ",
    "with R ≥ Rth will have a magniﬁcation above some minimum value µmin. For µth > µmin the cross section begins to decrease (since not all regions that lead to R ≥ Rth will have µ ≥ µth), exhibiting an approximately quadratic decay with µth for large magniﬁcations. This is exactly what would be expected if the cross section is dominated by arcs whose sources are close to fold caustics. Indeed, it is well known that the magniﬁcation cross section (i.e., with no constraint on R) scales as σ ∝ µ−2 for high µ (see, e.g., Chang & Refsdal 1979; Blandford & Narayan 1986; Blandford & Kochaneck 1987). In fact, for a source approaching a tangential (radial) fold, the tangential (radial) magniﬁcation µt (µr) is inversely proportional to the square root of the distance between the sources and the caustic (β − βc) ",
    "(15) ",
    "(see, e.g., Mollerach & Roulet 2002), which leads to the quadratic scaling for the cross section.7 A simple approximation for the dependence of the cross cross section on µth is given by ",
    "(16) ",
    "where σ˜(κs, e) is the cross section computed with no restriction on µ and µcr(κs, e) is a critical value above which the scaling is quadratic. Notice that, by construction, the expression above is exact for µ < µmin. ",
    "(17) ",
    "where d˜σ(κs, e; µ) is the cross section for images with magniﬁcation between µ and µ + dµ. Using approximation (16) we have ",
    "(18) ",
    "if µ > µcr, and zero otherwise. ",
    "Expression (16) provides an excellent approximation for σ˜(κs, e; µ) if µcr is determined as follows. First we note that the mean magniﬁcation for images with R > Rth is given by ",
    "(19) ",
    "Then, using equation (18) we obtain ",
    "(20) ",
    "The mean magnification µ¯ (κs, e) is computed using the deﬁnition (19). In ﬁgure 3 we show µ¯ as a function of κs for several values of the ellipticity. The mean magniﬁcation has a steep increase for lower values of κs and is very sensitive to the ellipticity. As for the dimensionless cross section, the dependence with ellipticity is inverted close at κs ≃ 0.75. ",
    "In figure 2 we plot the scaling of σ˜ with µ, computed numerically for each magniﬁcation threshold µth, together with the approximation deﬁned by equations (16) and (20). We see that they provide a good approximation in the whole µ range displayed.8 Therefore, to obtain the cross section for a given Rth and for any speciﬁed magniﬁcation threshold we only need two functions: σ˜(κs, e) and µ¯(κs, e). In the remaining of this work we shall use this approximation to account for the behavior of the cross section as a function of the magniﬁcation threshold. "
  ],
  "6. THE EFFECT OF MAGNIFICATION ON ARC STATISTICS ": [
    "Now we are ready to compute the expected number of arcs per cluster including the eﬀect of magniﬁcation following Eq. (7). The ﬁrst ingredient is the dimension-full magniﬁcation-dependent cross section, which is obtained by combining Eqs. (14), (18) and (20): ",
    "(21) ",
    "The explicit dependence of κs and rs in terms of the cluster mass and concentration parameter, lens and source redshifts, and cosmological model is obtained as follows. First rs and ρs are written in terms of the concentration parameter c and mass9 M200 as ",
    "(22) ",
    "(23) ",
    "Fig. 2.— Scaling of the cross section σ˜(κs, e; µth), with respect to the magniﬁcation threshold µth, for the elliptic NFW model with e = 0.2. The full lines show the numerical computation using the conditions R > Rth and µ > µth in equation (10). The dashed lines show the results of the approximation deﬁned by equations (16) and (20). The vertical lines indicate the values of µmin and µcrit for κs = 0.5. ",
    "where g(c) = c2/ (ln (1 + c) − c/ (1 + c)). ",
    "Obviously the relation (22) among the mass and radius holds for a spherical cluster. Our elliptical clusters are obtained by “deforming” the mass distribution in the direction perpendicular to the line of sight according to equation (13). In other words, a region of constant density would deﬁne a shell ξ2 + z2 = const. Since, by construction, the area of a ξ = const. curve is independent of e, the mass is independent of the ellipticity and equation (22) will hold even in the elliptical case. It is worth stressing that, by this procedure, the elliptical clusters are oblate spheroids with the smaller axis perpendicular to the line of sight. We are thus neglecting the triaxiality of these objects. However, this situation is somewhat intermediate to having a triaxial cluster aligned along the major or minor axes. For a full treatment of triaxiality, see e.g., Oguri, Lee, & Suto (2003). For the purposes of this paper it is suﬃcient to proceed with the approximation described above. ",
    "Then, using Eqs. (12), (22), and (23), and inserting the relevant units, the projected NFW profile parameters are ",
    "Fig. 3.— Average magnification within the region with |µt/µr | > Rth = 10 as a function of κs for several values of the ellipticity. ",
    "finally given by ",
    "(24) ",
    "(25) ",
    "where ILS is the transverse commoving distance from lens to source in units of the Hubble radius (and equivalently for IOL and with and sinh χ, for ",
    "0, respectively. Here we adopt a ﬂat ΛCDM model, such that E(z) = Ωm(1 + z)3 + ΩΛ and Ωk = 1 − Ωm − ΩΛ = 0, with parameters Ωm = 0.3, ΩΛ = 0.7. Notice that if the mass is expressed in units of h−1 , the results will be independent on the Hubble parameter. ",
    "For a given background cosmological model (and fixed zL and zS), κs and rs are functions of M200 and c only. Although c is in principle independent of M200, these two quantities are strongly correlated, as seen in simulations (see, e.g. Klypin, et al. 1999) and in observational data (Comerford & Natarajan 2007). Here we use the expression for c(M200) from Neto et al. (2007), with the redshift dependence given in Macci`o et al. (2008). Although the distribution of halo concentrations for a given mass and redshift follows a log-normal distribution, we shall use only the mean ",
    "relation in this work. ",
    "The second ingredient to compute the arc fraction is the source distribution as a function of redshift and limiting magnitude. The redshift dependence is modelled as in Baugh & Efstathiou (1993) and we take the total surface density of sources and the scaling with mlim from the COMBO-17 survey (Taylor et al. 2007), such that ",
    "(26) ",
    "where z∗ = zm/1.412 and zm is the median redshift, which depends on the limiting magnitude mlim in the relevant band. For an R-band survey zm ≃ 0.23(mlim − 20.6) (Brown et al. 2003). ",
    "Although we do not explicitly take into account the eﬀects of the PSF in this work, we point out that the distribution above was obtained from ground-based observations. Therefore, our arc fraction computation will include, to some extent, the eﬀect of the seeing. Ideally one would determine the source distribution from deep space based observations and include the eﬀects of the PSF through the formalism of Sect. 2. This is left for future work. ",
    "With the relations above, we may finally compute the expected number of arcs per cluster, which is given by a convolution of the cross section with the source surface density, integrated over all sources behind the cluster and all the magnifications (Eq. 7): ",
    "(27) ",
    "We also test an approximation by which the whole area defined by the condition R > Rth has the mean magnification µ¯ (Eq. 19), such that σ˜(κs, e; µth) = σ˜(κs, e)θ (µ − µ¯). In this case, Eq. (27) simpliﬁes to ",
    "(28) ",
    "where σ is the dimension-full cross section without imposing any cut in magniﬁcation. In other words, all sources are assumed to be magniﬁed by the mean magniﬁcation µ¯. As above, the dependence on zL, zS, and M enters on both σ and µ¯(κs, e; Rth) through κs. ",
    "For the sake of comparison, we also compute the arc fraction neglecting the eﬀect of magniﬁcation, i.e., using mlim eﬀ = mlim in the equation above, which is the more standard computation. In ﬁgure 4 we show the results of these 3 computations for the arc fraction farcs as a function of lens redshift for a cluster with M = 5 × 1014h−1M⊙ and e = 0.5 and a survey with limiting magnitude Rlim = 24. We see that the magniﬁcation has a dramatic impact on arc abundance, enhancing the fraction of arcs by more than an order of magnitude. Furthermore, the inclusion of the eﬀect of magniﬁcation shifts the peak of the arc incidence to higher redshifts and modiﬁes the slope of the decay at large zL, showing a softer decrease with lens redshift. ",
    "We also notice that the results including the dependence of the cross section with µ as in Eqs. (18) and (27) (full line in Fig. 4) or considering only the mean magniﬁcation, as in Eq. (28) (dashed line), are very similar. Indeed, the relative diﬀerence is less than 10% for a broad range of parameters. Therefore, from now on we will use the constant magniﬁcation approximation given by (28), since it makes calculations much faster, by avoiding the integration in µ. ",
    "In Fig. 5 we show the impact of the depth of the survey on farcs(z) for the same lens parameters as in the previous figure. The bottom line corresponds to Rlim = 22, typical of the Sloan Digital Sky Survey,10 while the upper line corresponds to Rlim = 28, close to what would be achieved from the final Large Synoptic Survey Telescope11 (LSST) coadded data (Ivezi´c et al. 2008). As expected, the change in limiting magnitude has a large impact on arc incidence, which could explain the diﬀerences in arc abundance found in the various surveys. Also, by comparing Figs. 4 and 5, it is clear that the eﬀect of magniﬁcation is qualitatively akin to changing the limiting magnitude of the survey. ",
    "In Fig. 6 we show the result of the integration of Eq. (28) as a function of zL for several values of M and e. The qualitative behaviour is similar in all cases, with the expected number of arcs increasing as a function of cluster redshift at low redshifts and decreasing at higher redshifts. These results can be simply understood in terms of the scaling of the cross section with the characteristic convergence κs (Fig. 1), the scaling of κs with mass and redshift (Eq. 24), and the source redshift distribution (Eq. 26). First, for a given mass and ellipticity, the maximum value of κs increases with zL. In other words, the lensing eﬃciency has the potential to grow with zL, as long as there are far enough sources to be lensed. Therefore, for low lens redshifts, farcs will always increase with zL. On the other hand, for high zL the exponential decay of sources will end up causing a fall down in farcs. The arc fraction is more sensitive to zL for lower masses, ellipticities, and lens redshifts. This occurs because the cross section is much more sensitive to κs for smaller values of this parameter and smaller ellipticites. ",
    "From these figures we see that the arc fraction is more sensitive to the ellipticity than to the mass, with roughly a linear dependence on mass at the peak and exponential with ellipticity. The transition redshift (i.e. the redshift at which farcs starts to decrease) is weakly dependent on M and e. ",
    "The dependence of arc abundance on the source distribution and its dependence on the limiting magnitude highlight the importance of the magniﬁcation eﬀects for farcs. For example, for κs ≃ 1, we have µmin ≃ 20 (see Fig. 3), which implies that the eﬀective limiting magnitude is increased by at least 1.25 (see Eq. 4), changing both the normalization and shape of farcs(zL). ",
    "Fig. 4.— The expected fraction of arcs as a function of lens redshift computed in three diﬀerent ways: ignoring the magniﬁcation (dotted line), taking into account the mean magniﬁcation (dashed line), and considering the quadratic scaling of the cross section with magniﬁcation (full line). The results are shown for a NFW cluster with M = 5 × 1014h−1M⊙ and e = 0.5, assuming Rlim = 24. "
  ],
  "7. CONCLUDING REMARKS ": [
    "In this paper we have introduced a method to include the eﬀect of noise and other observational eﬀects in semi-analytic computations of gravitational arc statistics. We argue that the expected fraction of arcs per cluster farcs is a well suited indicator to study arc statistics. On the theoretical side this quantity is less dependent on cosmological parameters not directly related to lensing (such as σ8) than the total arc number. On the observational side farcs is less aﬀected by selection eﬀects. The procedure to compute farcs involves the arc cross section σ, the source distribution, and a combination of observational eﬀects with the selection function (see Eq. 1). In particular, to include the eﬀect of the increased S/N of the arcs due to the magniﬁcation on their detectability, we are lead to consider the magniﬁcation dependent cross section. This cross section is obtained by adding an additional constraint on the image properties. Besides requiring that L/W > Rth, we also impose that the sources must be magniﬁed by a minimum factor µth. ",
    "The cross section was computed through a semi-analytic method based on the local eigenvalue ratio of the magniﬁ-cation tensor. This method has been used in previous studies (see, e.g., Hattori, Watanabe, & Yamashita 1997; Fedeli et al. 2006), and was validated using simulations of extended sources. For distortion arcs, the local eigenvalue ratio gives a good approximation for the cross section and is much faster than ray tracing ﬁnite sources. Using this method we investigated the scaling of σ with respect to the magniﬁcation threshold µth. We found that σ scales approximately as µ−th2 for high magniﬁcations — which is expected from the properties of the lens mapping close to folds — and obtained a good approximation for all values of µth using the mean magniﬁcation µ¯. This approximation is expected to be valid for generic lens models (as long as the scaling is close to the one expected by fold arcs). ",
    "Given a cluster mass model, the cosmological dependence of the cross section is encoded on the projected characteristic density and characteristic scale. In the case of the elliptical NFW proﬁle, the problem of computing σ(M, e, zL; zS) is simpliﬁed to the one of obtaining σ˜(κs, e). Further, if one wishes to approximate the cross section for arbitrary µth it suﬃces to obtain σ˜(κs, e) and µ¯(κs, e). Obviously, the scaling of σ with projected mass and (arbitrary) length scale is a property of the lensing geometry. This feature could be used to scale the cross sections of simulated clusters for diﬀerent lens and source redshifts if the lens parameters are scaled accordingly. ",
    "To obtain the arc fraction we assume that the objects are identiﬁed and selected in terms of their S/N in the images. The arc selection function is given by a constant threshold in L/W and S/N. The enhanced detectability due to the magniﬁcation acts eﬀectively as an increased depth of the image in the magniﬁed region, which translates into a ",
    "Fig. 5.— The expected fraction of arcs as a function of lens redshift for diﬀerent limiting magnitudes for a NFW cluster with M = 5 × 1014 h−1M⊙ and e = 0.5. The bottom line shows the results for Rlim = 22, increasing in steps of unity, to reach Rlim = 28 for the upper curve. ",
    "Fig. 6.— Expected fraction of arcs as a function of lens redshift for diﬀerent masses and ellipticities for a NFW cluster in a survey with Rlim = 24. Left: ﬁxed ellipticity (e = 0.5) and varying the mass in steps of 1014 h−1M⊙, from M = 1 × 1014h−1M⊙ (lower curve) up to M = 1015 M⊙ (upper curve). Right: ﬁxed mass (M = 5 × 1014 h−1M⊙) and varying ellipticity, from e = 0 (bottom line), increasing in steps of 0.2 up to e = 0.8 for the upper curve. ",
    "(magniﬁcation dependent) modiﬁcation of the source distribution. We account for this eﬀect by eﬀectively changing the limiting magnitude used to determine the source distribution as in equation (4). We take the source surface density as a function of redshift and its scaling with mlim from the COMBO-17 survey. ",
    "The resulting arc fraction is shown as a function of zL in figure 4. We see that the inclusion of the magnification has a very strong impact on the arc incidence, both enhancing the arc fraction by over an order of magnitude, as well as changing the shape of its redshift distribution. In particular, the peak in the arc fraction shifts towards higher zL and the tail of farcs has a less steep decrease with the cluster redshift. All these features are qualitatively in the right direction for improving the agreement between the predictions and observations of arc abundance, though the modelling used in this work is still oversimplified to allow for a direct comparison. Also, the detailed form and magnitude of the ",
    "arc fraction is sensitive to the redshift variation of the mass-concentration relation, the mass deﬁnition, etc. Besides, it must be pointed out that the impact of the magniﬁcation may be a bit overemphasised in these results, given our assumption on the direct dependence of detection on the arc S/N . Other quantities may also play an important role, such as the surface brightness, which is not directly aﬀected by the magniﬁcation. In any case, we emphasise the importance of including the eﬀect of magniﬁcation, which is often ignored in semi-analytic predictions of arc abundance and is not accounted for in simulations that use constant surface brightness sources. ",
    "The magniﬁcation bias considered in this work is diﬀerent from the standard magniﬁcation bias in the statistics of multiple imaging of point sources (see, e.g., Turner 1980; Turner et al. 1984), in that the eﬀect scales with the square root of magniﬁcation (i.e. S/N enhancement) and not the magniﬁcation itself. The magniﬁcation has been considered in previous arc statistics studies (see, e.g., Oguri, Lee, & Suto 2003), but through the enhancement of the total ﬂux, which should not be the dominant factor for arc detection. ",
    "As far as we know, this is the ﬁrst work to take into account the eﬀect of the magniﬁcation through the S/N on arc statistics and to consider the dependence of the cross section on the magniﬁcation for computing the arc abundance. We compute the arc fraction using two methods: by using the magniﬁcation dependent cross section (Eq. 27) and by using the cross section with no cut in µth but associating to all arcs the mean magniﬁcation µ¯ (Eq. 28). Both methods turn out to be in very good agreement. Therefore, it is suﬃcient to consider the magniﬁcation independent cross section and carry out the integration only on the source redshift, as in the more standard approach, as long as we consider the eﬀect of the mean magniﬁcation (which depends on the lens parameters and source redshift) on the source distribution. The inclusion of magniﬁcation on arc statistics through this procedure requires only the additional computation of µ¯ in addition to the cross section. ",
    "Several aspects of this work could be improved to yield more realistic predictions for the arc cross section and the expected number of arcs per cluster. Regarding the lens model, two key features that should be considered are the triaxial mass distribution (see, e.g., Oguri, Lee, & Suto 2003; Hennawi, Dalal, Bode, & Ostriker 2007) and the mass-concentration relation c(M ), including its scatter. More sophisticated mass distributions could also be dealt with, including diﬀerent radial proﬁles, external perturbations, asymmetries, and substructures. Regarding the sources, the eﬀects of ﬁnite sizes and ellipticities could be included with the methods discussed in section 4. Finally, given the observed morphological properties of the sources (e.g. their size distribution), the eﬀects of the seeing should be taken into account, as outlined in section 2. This is left for future work. In any case, we argue that the enhancement of the S/N of the arcs due to the magniﬁcation must be included in more sophisticated semi-analytic modelling of arc abundance. Furthermore, the inclusion of this eﬀect could help to bring a better agreement between observations and theoretical modelling of arc abundance. ",
    "With the upcoming wide-field surveys such as DES, which has just started its first observing season, and LSST, and future space based projects such as EUCLID12 (Laureijs et al. 2011), the observed number of gravitational arcs is expected to reach ∼ 103 −104 . This will become a promising new era for doing precision statistical studies of the cluster inner matter distribution and its evolution through strong lensing. For such a program to be successful, end-to-end simulations to generate fake images are starting to be extensively used in combination with arcﬁnding codes, allowing for a direct comparison of theoretical predictions with observational data, taking into account the physical processes involved and the observational eﬀects. As in many areas of cosmology, the interplay between detailed simulations and simple semi-analytic models is a useful and necessary step. The semi-analytic approach allows for a qualitative understanding of the problem and provides a framework to interpret the outcome of simulations. Furthermore, the semi-analytic modelling provides a way to make fast computations allowing a large parameter space to be probed for performing parameter estimations and forecasts. "
  ],
  "ACKNOWLEDGMENTS ": [
    "The authors are grateful to Silvia Mollerach and Esteban Roulet for providing fundamental contributions to this work in its earlier stages. We thank Eduardo Rozo for numerous remarks that lead to a significant improvement of this work. The authors acknowledge Chuck Keeton for making his gravlens code publicly available, which was used to test our numerical codes. MM wishes to thank Eduardo Cypriano, Laerte Sodr´e, and Joan-Marc Miralles for useful comments and suggestions. JE and MM acknowledge the hospitality of Instituto Balseiro, where this work was initiated. MM acknowledges the hospitality of the Particle Astrophysics Center at Fermilab were part of this work was done. GBC is partially funded by CNPq and CAPES. MM is partially supported by CNPq (grant 309804/2012-4) and FAPERJ (grant E-26/110.516/2012). "
  ]
}