THE IMPACT OF CLUSTER MERGERS ON ARC STATISTICS

ELENA TORRI1, MASSIMO MENEGHETTI1, MATTHIAS BARTELMANN 2,3, LAURO MOSCARDINI4,ELENA RASIA1, GIUSEPPETORMEN1

1DIPARTIMENTO DI ASTRONOMIA, UNIVERSITA DI PADOVA , VICOLO DELL’OSSERVATORIO2, I–35122 PADOVA , ITALY2MAX -PLANCK -INSTITUT FUR ASTROPHYSIK, KARL-SCHWARZSCHILD-STRASSE1, D–85748 GARCHING, GERMANY

3INSTITUT FUR THEORETISCHEASTROPHYSIK, TIERGARTENSTR. 15, D–69121 HEIDELBERG4DIPARTIMENTO DI ASTRONOMIA, UNIVERSITA DI BOLOGNA, VIA RANZANI 1, I–40127 BOLOGNA, ITALY

submitted to MNRAS

ABSTRACTWe study the impact of merger events on the strong lensing properties of galaxy clusters. Previous lensing

simulations were not able to resolve dynamical time scales of cluster lenses, which arise on time scales which areof order a Gyr. In this case study, we first describe qualitatively with an analytic model how some of the lensingproperties of clusters are expected to change during merging events. We then analyse a numerically simulated lensmodel for the variation in its efficiency for producing both tangential and radial arcs while a massive substructurefalls onto the main cluster body. We find that: (1) during the merger, the shape of the critical lines and causticschanges substantially; (2) the lensing cross sections for long and thin arcs can grow by one order of magnitudeand reach their maxima when the extent of the critical curves is largest; (3) the cross section for radial arcs alsogrows, but the cluster can efficiently produce this kind of arcs only while the merging substructure crosses the maincluster centre; (4) while the arc cross sections pass through their maxima as the merger proceeds, the cluster’sX-ray emission increases by a factor of∼ 5. Thus, we conclude that accounting for these dynamical processes isvery important for arc statistics studies. In particular, they may provide a possible explanation for the arc statisticsproblem.

1. INTRODUCTION

The abundance of strong gravitational lensing events producedby galaxy clusters is determined by several factors. Since lightdeflection depends on the distances between observer, lens andsource, gravitational lensing effects depend on the geometricalproperties of the universe. On the other hand, gravitational arcsare rare events caused by a highly nonlinear effect in clustercores. They are thus not only sensitive to the number densityof galaxy clusters, but also to their individual internal structure.Since these factors depend on cosmology and in particular onthe present value of the matter density parameterΩ0M and thecontribution from the cosmological constantΩ0Λ, arc statisticsestablishes a highly sensitive link between cosmology and ourunderstanding of cluster formation.

Using the ray-tracing technique for studying the lensing prop-erties of galaxy clusters obtained from N-body simulations,Bartelmann et al. (1998) showed that the number ofgiant arcs,commonly defined as arcs with length-to-width ratio larger than10 andB-magnitude smaller than 21.5 (Wu, 1993), which is ex-pected to be detectable on the whole sky, differs by orders-of-magnitudes between high and low-density universes, stronglydepending even on the cosmological constant. In particular, theyestimated that the number of such arcs in aΛCDM cosmologicalmodel (Ω0M = 0.3, Ω0Λ = 0.7) is of the order of some hundredson the whole sky, while roughly ten times more arcs are expectedin an OCDM cosmological model (Ω0M = 0.3, Ω0Λ = 0).

Although still based on limited samples of galaxy clusters,observations indicate that the occurence of strong lensing eventson the sky is rather high (Luppino et al., 1999; Zaritsky & Gon-zalez, 2003; Gladders et al., 2003). For example, searching forgiant arcs in a sample of X-ray luminous clusters selected fromthe Einstein ObservatoryExtended Medium Survey, Luppinoet al. (1999) found that their frequency is about 0.2− 0.4 arcsper massive cluster. Despite the obvious uncertainties in the ob-servations, the only cosmological model for which the number

of giant arcs expected from numerical simulations of gravita-tional lensing comes near the observed number is the OCDMmodel. In particular the observed incidence of strongly lensedgalaxies exceeds the predictions of aΛCDM model by about afactor of ten. On the other hand, based on the observations oftype Ia supernovae (Perlmutter et al., 1999) and the recent ac-curate measurements of the temperature fluctuations of the Cos-mic Microwave Background obtained with balloon experiments(e.g. Stompor et al. 2001; Jaffe et al. 2001; Abroe et al. 2002;Benoıt et al. 2003) or by theWMAPmission (e.g. Bennett et al.2003; Spergel et al. 2003), theΛCDM model has become thefavourite cosmogony. This is known as thearc statistics prob-lem: the mismatch between the observed number of arcs and thenumber expected in theΛCDM model preferred by virtually allother cosmological experiments hints at a lack of understandingof cluster formation.

Several extensions and improvements of the numerical simu-lations failed in finding a solution to this problem in the lensingsimulations. Meneghetti et al. (2000) studied the influence of in-dividual cluster galaxies on the ability of clusters to form largegravitational arcs, finding that their effect is statistically negligi-ble. Cooray (1999) and Kaufmann & Straumann (2000), usingspherical analytic models and the Press-Schechter formalism formodelling the lens cluster population, predicted a weaker de-pendence of arc statistics onΩ0Λ, but Meneghetti et al. (2003b),comparing numerical models of galaxy clusters and their analyt-ical fits, showed that analytic models are inadequate for quanti-tative studies of arc statistics. Finally, Meneghetti et al. (2003a)found that the presence of central cD galaxies can increase thecluster efficiency for producing giant arcs by not more than afactor of about two.

As alternative solution of the arc statistics problem, Bartel-mann et al. (2003) recently investigated arc properties in cos-mological models with more general forms of dark energythan a cosmological constant. Several studies showed thathaloes should be more concentrated in these models than in

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cosmological-constant models with the same dark energy den-sity today (Bartelmann et al. 2002; Mainini et al. 2003; Klypinet al. 2003; Dolag et al. in preparation), allowing them to bemore efficient for strong lensing. Using simple models withconstant quintessence parameter, Bartelmann et al. (2003) foundthat the relative change of the halo concentration is not sufficientto produce an increment of one order of magnitude in the ex-pected number of giant arcs. Nonetheless, other more elaboratedark-energy models need to be evaluated numerically and willbe discussed in a future paper (Meneghetti et al. in preparation).

In this paper we investigate one other possible effect whichcould not be properly considered in the previously mentionednumerical simulations of gravitational lensing by galaxy clus-ters. In those works, the lensing cross sections for giant arcs ofeach numerical model were evaluated at different redshifts, witha typical time separation between two consecutive simulationoutputs of approximately∆t ∼ 1 Gyr. Therefore all the dynami-cal processes arising in the lenses on time scales smaller than∆twere not resolved.

N-body simulations show that dark matter haloes of differentmasses continuously fall onto rich clusters of galaxies (Tormen,1997). The typical time scale for such events is∼ 1− 2 Gyr,which therefore might be too short for having been properly ac-counted for in the previous lensing simulations.

However, the effects of mergers on the lensing properties ofgalaxy clusters may potentially be very important. As discussedby Bartelmann et al. (1995) and Meneghetti et al. (2003b), sub-structures play a very important role for determining the clusterefficiency for lensing. Indeed, analytic models, where substruc-tures and asymmetries in the lensing mass distributions are notproperly taken into account, systematically underestimate thelensing cross sections of the numerical models. The main rea-son is that mass concentrations around and within clusters en-hance the shear field, increasing the length of the critical curves,and consequently the probability of forming long arcs becomeshigher.

Given the strong impact of substructures on the lensing prop-erties of galaxy clusters, it is reasonable to expect that duringthe passage of a massive mass concentration through or near thecluster centre, the lensing efficiency might sensitively fluctuate.While the substructure is approaching the main cluster clump,the intensity of the shear field and, consequently, the shape ofthe critical curves may substantially change. Moreover, whilethe infalling dark matter halo gets closer to the cluster centre,the projected surface density increases, making the cluster muchmore efficient for strong lensing.

This paper describes a case study on how much the lensingcross sections change during the infall of a massive dark matterhalo on the main cluster progenitor. For doing so, we investigatethe lensing properties of a numerically simulated galaxy clusterduring a redshift interval when a major merging event occurs.The general aim is to understand if mergers can enhance thecluster lensing efficiency sufficiently to provide a solution to thearc statistics problem.

The plan of the paper is as follows. In Section 2 we use a sim-ple analytic model based on the NFW density profile for com-puting the strong lensing properties produced by cluster merg-ers. In Section 3 we describe our lensing simulations utilising anumerical model obtained from a high-resolution N-body simu-lation; a method for evaluating the dynamical state of the clusteris also introduced. In Section 4 we present our results for criti-cal lines and caustics, and we estimate the expected number oftangential and radial arcs. Section 5 is devoted to a discussion ofthe observational implications of our results, and we summarisethe results and present our conclusions in Sect. 6.

2. ANALYTIC MODELS

We begin by investigating with the help of analytic models howthe lensing properties of a cluster lens are expected to changewhile a massive substructure passes through its centre. We willmodel both the main cluster clump and the infalling substructureas spherical bodies with the density profile found in numericalsimulations by Navarro et al. (1996, hereafter NFW).

2.1. NFW model

The NFW density profile is given by

ρ(r) =ρs

(r/rs)(1+ r/rs)2 , (1)

whereρs and rs are the characteristic density and the distancescale, respectively (see Navarro et al. 1997). The logarithmicslope of this density profile changes from−1 at the centre to−3at large radii.

We briefly summarise here the main features of NFW haloes.First, ρs and rs are not independent. Second, the ratio be-tween the radiusr200, within which the mean halo density is 200times the critical density, andrs defines the halo concentration,c ≡ r200/rs. Numerical simulations show that this parametersystematically changes with the halo virial mass, which is thusthe only free parameter. Several algorithms exist for computingthe concentration parameter from the halo virial mass (Navarroet al., 1997; Eke et al., 2001; Bullock et al., 2001). In the fol-lowing analysis we adopt the method proposed by Navarro et al.(1997). Third, the concentration also depends on the cosmolog-ical model, implying that the lensing properties of haloes withidentical mass are different in different cosmological models ifthey are modelled as NFW spheres. The lensing properties ofhaloes with NFW profiles have been discussed in several papers(Bartelmann, 1996; Wright & Brainerd, 2000; Li & Ostriker,2002; Wyithe et al., 2001; Perrotta et al., 2002; Bartelmann et al.,2002; Meneghetti et al., 2003b,a).

Due to axial symmetry, lensing by an NFW sphere reduces toa one-dimensional problem. We define the optical axis as thestraight line passing through the observer and the centre of thelens, and introduce the physical distances perpendicular to theoptical axis on the lens and source planes,ξ andη, respectively.We introduce dimension-less coordinatesx= ξ/ξ0 andy= η/η0by means of the coordinate scalesξ0 = rs andη0 = (Ds/Dl)η0,whereDs andDl are the angular diameter distances to the sourceand lens planes, respectively.

Using this notation, the density profile (1) implies the surfacemass density (Bartelmann, 1996)

Σ(x) =2ρsrs

x2−1f (x) , (2)

with

f (x) =

1− 2√

x2−1arctan

√x−1x+1 (x > 1)

1− 2√1−x2

arctanh√

1−x1+x (x < 1)

0 (x = 1)

. (3)

The lensing potential is given by

Ψ(x) = 4κsg(x) , (4)

where

g(x) =12

ln2 x2

+

2arctan2

√x−1x+1 (x > 1)

−2arctanh2√

1−x1+x (x < 1)

0 (x = 1)

, (5)

2

d=1.0 d=0.8 d=0.6

d=0.4 d=0.2 d=0

FIG. 1.—Illustration of the analytic model described in Sect. 2.2.: Maps of the tangential eigenvalueλt for all configurations of the main clusterbody relative to the infalling substructure. The critical lines are over-plotted in each map. The (comoving) size of each panel is∼ 1.2h−1Mpc.Different panels refer to different distancesd (in units of the scale radius of the larger halo) between the two haloes, as indicated by the top label.

andκs≡ ρsrsΣ−1cr . hereΣcr = [c2/(4πG)] [Ds/(DlDls)] is the crit-

ical surface mass density for strong lensing andDls is the angu-lar diameter distance between lens and source. This implies thedeflection angle

α(x) =dψdx

=4κs

xh(x) , (6)

with

h(x) = lnx2

+

2√

x2−1arctan

√x−1x+1 (x > 1)

2√1−x2

arctanh√

1−x1+x (x < 1)

1 (x = 1)

. (7)

2.2. Expectations

In this analytic computation, the main cluster body has a mass ofMmain = 1015 h−1M. It merges with a massive substructure ofmassMsub= Mmain/4= 2.5×1014h−1M, which is a fairly typ-ical ratio for merging halo masses. In this simple test, we keepthe lens plane at the fixed redshiftzl = 0.3, while the sourceplane is placed at redshiftzs = 1. We assume a backgroundΛCDM cosmological model, withΩ0M = 0.3, Ω0Λ = 0.7, andHubble constant (in units of 100 km/s/Mpc)h = 0.7.

We simulate the infall of the substructure onto the mainclump, starting from the initial configuration when the smallerhalo is placed at distancers from the larger one, wherers isthe scale radius of the most massive clump. For a halo of mass1015h−1M in aΛCDM model,rs corresponds to∼ 310h−1kpc.Then, we reduce the distance between the two haloes by movingthe smaller towards the larger one by 0.2rs per time step, untilthe two haloes are exactly aligned.

For any new configuration, we trace a bundle of 512× 512light rays through a regular grid on the lens plane which cov-ers a region whose side length is 4rs. This is large enough toencompass the critical lines of both the clumps.

Using (6), we compute the contributions~αmain and~αsub ofboth the main clump and the substructure to the deflection angleof each ray. Being linear in mass, the total deflection angle ofa ray passing through the mass distribution is the sum of thecontributions from each mass element of the deflector. Thus,the deflection angle of each ray parametrised by its grid indices(i, j) on the lens plane is given by

~αi j =~αmaini j +~αsub

i j . (8)

The convergence and the shear maps of the lens system arereconstructed from the deflection angles. The true position ofthe source on the source plane,~y, and its observed position onthe lens plane,x, are related by the lens equation

~y =~x−~α(~x) . (9)

The local properties of the lens mapping are described by theJacobian matrix of (9),

Ahk≡∂yh

∂xk= δhk−

∂αh

∂xk, (10)

whereδhk is the Kronecker symbol. The convergenceκ and theshear componentsγ1 andγ2 are found fromAhk through the stan-dard relations

κ(~x) =12[A11(~x)+A22(~x)] , (11)

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FIG. 2.—Profiles of convergenceκ (left panel) and shearγ (right panel) along the axis connecting the centres of the main cluster body and theinfalling substructure illustrated in Fig. 1. Different curves refer to different distancesd (in units of the scale radius of the larger halo) between thetwo haloes. Notice the increased shear level at fixedx during the merger.

FIG. 3.—Caustics for all configurations of the main cluster body relative to the infalling substructure shown in Fig. 1. The size of each panel is∼ 200h−1kpc on a side. Different panels refer to different distancesd (in units of the scale radius of the larger halo) between the two haloes.

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FIG. 4.—Left panel: caustic area and length (solid and dashed lines, respectively) as functions of the distance between the merging halos (in unitsof the scale radius of the more massive halo). The lens redhift iszl = 0.3. The thick and thin lines refer to source redshiftszs = 1 andzs = 2,respectively. Curves are normalized to the maximum value of area and length for sources atzs = 2. Right panel: as in the left panel, but for lensesat redshiftzl = 0.8

γ1(~x) = −12[A11(~x)−A22(~x)] , (12)

γ2(~x) = −12[A12(~x)+A21(~x)] . (13)

The Jacobian matrix is symmetric and can be diagonalised. Itstwo eigenvalues are

λt = 1−κ− γ and λr = 1−κ+ γ , (14)

whereγ =√

γ21 + γ2

2. Radial and tangential critical lines are lo-cated where the conditions

λt = 0 and λr = 0 (15)

are satisfied, respectively. The corresponding caustics in thesource plane, close to which sources are imaged as large arcs,are obtained by applying the lens equation to the critical curves.

We are particularly interested in understanding by how muchthe cluster efficiency for producing tangential arcs changes dur-ing the merger phase. The size of the lensing cross sections forthe formation of tangential arcs is strictly connected to the ex-tent of the tangential critical curves and caustics: the longer thecritical curves and the corresponding caustics are, the larger arethe lensing cross sections. Therefore, we first investigate theshape of the tangential critical lines and caustics. In Fig. 1, weshow the maps of the tangential eigenvalueλt for all configura-tions of the main clump relative to the infalling substructure. Ineach map, we over-plot the tangential critical lines. As the twohaloes approach each other, their critical lines widen and stretchalong the approaching direction. When the distance is∼ 0.6rs,the critical lines touch and merge to form a single critical line.While the substructure overlaps with the main cluster body, itshrinks in the direction along which the merger proceeds, whileit widens perpendicular to it.

We can explain the evolution of the critical lines as follows.As the two haloes approach each other, the shear in the regionbetween them grows. This stretches the critical lines in the di-rection along which the two haloes are approaching. When thecritical lines merge, the shear continues growing in the regioninside the unique critical line. At this point the critical curvestops growing in a privileged direction. It shrinks only in the di-rection along which the haloes merge because of the decreasing

distance between their centres. On the other hand, it expandsalmost isotropically by the effect of the increasing convergence.We show in Fig. 2 how the convergence and the shear evolvealong the axis connecting the halo centres, where shrinking andstretching effects are most evident. The main clump is kept atx= 0. Different curves show the results for different distancesdbetween the two haloes, in units of the scale radius of the largestclump.

The caustics are shown in Fig. 3. Their evolution reflects thatof the critical lines. As the shear grows between the two clumps,they are stretched and develop cusps. Then, after they merge, thesingle caustic shrinks and reduces to a point when the two haloesare exactly aligned.

In the case of merging halos, where very complicated struc-tures develop in the caustics, it is difficult to say whether thecross sections for long and thin arcs scale as the area enclosedby the caustics or as the caustic length. In Fig. 4 we show theevolution of both the area (solid lines) and the length (dashedlines) of the caustics while the merger proceeds. Thick lines inthe left panel refer to the configuration of lenses and sources de-scribed earlier. For completeness, in the same panel we showalso the corresponding curves for sources at redshiftzs = 2 (thinlines). In the right panel, the corresponding curves for lensesat zl = 0.8 are shown. As expected, the size of the caustics islarger for sources more distant from the lens because of its in-creased convergence. All curves in each panel are normalizedto the maximum value of length and enclosed area for sourcesat redshiftzs = 2. Of course the area enclosed by the causticsgrows more steeply compared to the length of the caustics. It isinteresting that caustics reach their maximum length before theyreach their maximum area. This is due to the stretching of thecaustics of the two individual merging halos towards each other:the cusps become very peaked and pronunced until they touch,with large change of length. The maximum value of the area en-closed is reached only shortly thereafter, when the caustics havealready started to enlarge isotropically.

5

z=0.230 z=0.214 z=0.211 z=0.203

z=0.199 z=0.190 z=0.180

FIG. 5.—Surface density maps of the numerically simulated cluster at several redshifts betweenz= 0.230 andz= 0.180, during the merger. Thescale of each panel is 3h−1Mpc.

3. NUMERICAL METHODS

3.1. Numerical model

We now turn to a numerical cluster model in order to quantifyhow the strong lensing efficiency changes during a merger of arealistic cluster mass distributions embedded into the tidal fieldof the surrounding large-scale structure.

The numerical model we investigate here is part of a set of 17objects obtained using the technique of re-simulating at higherresolution a patch of a pre-existing cosmological simulation.The re-simulation method is described in (Tormen, 1997). A de-tailed discussion of the dynamical properties of the whole sam-ple of these simulated clusters is presented elsewhere (Tormenet al., 2003).

The parent N-body simulation, with 5123 particles in a boxof side 479h−1 Mpc, has been produced by N. Yoshida for theVirgo Consortium (see also Jenkins et al. 1998). It assumesa flat universe withΩ0M = 0.3, Ω0Λ = 0.7 andh = 0.7. Theinitial conditions correspond to a cold dark matter power spec-trum normalised to haveσ8 = 0.9 today. The particle massis 6.86× 1010h−1M, which allows to resolve a cluster-sizedhalo by several thousand particles; the gravitational softening is15h−1 kpc. From the final output of this simulation we randomlyextracted some spherical regions containing a cluster-sized darkmatter halo. Each of these regions was re-sampled to build newinitial conditions for a higher number of particles - on average106 dark matter particles and the same number of gas particles.The mass of the dark matter particles is∼ 5×109 h−1M, whilethe mass of the gas particles is∼ 10 times smaller. The soften-ing length for dark matter and gas is 3.6 h−1kpc and 7.1 h−1kpc,respectively. These initial conditions were evolved using thepublicly available code GADGET (Springel et al., 2001) froma starting redshiftzin = 35−50 to redshift zero.

In our sample of 17 objects we selected a simulated cluster

undergoing a major merger atz≈ 0.2. At redshift z∼ 0.25,when their viral regions merge, the main cluster clump and theinfalling substructure have virial masses of∼ 7× 1014h−1Mand∼ 3× 1014h−1M, respectively. In order to have a verygood time resolution for resolving the complete merger in detail,we decided to re-simulate the cluster betweenz= 0.25 andz=0.15, obtaining 101 equidistant outputs which we use for ourfollowing lensing analysis.

3.2. Dynamical state of the cluster

We show in Fig. 5 some projected density maps of the simulatedcluster at several redshifts during the merger. As is easily seen, alarge substructure crosses the cluster centre atz∼ 0.2. The (co-moving) side length of the displayed region is 3h−1Mpc. Wequantify the dynamical state of the cluster using the multipoleexpansion technique of the surface density field discussed byMeneghetti et al. (2003b). Briefly, we place the origin of ourreference frame on the centre of the main cluster clump. Start-ing from the particle positions in the numerical simulations, wecompute the surface densityΣ at discrete radiirn and positionanglesφk taken from[0,1.5]h−1Mpc and[0,2π], respectively.For anyrn, each discrete sample of dataΣ(rn,φk) is expandedinto a Fourier series in the position angle,

Σ(rn,φk) =∞

∑l=0

Sl (rn)e−ilφk , (16)

where the coefficientsSl (rn) are given by

Sl (rn) =∞

∑k=0

Σ(rn,φk)eilφk , (17)

and can be computed using fast-Fourier techniques.

6

FIG. 6.—Total integrated powerPint as function of redshift for five different distancesr (in units ofh−1Mpc) from the centre of the main clusterclump, as indicated in each panel.

7

FIG. 7.—Critical lines of the numerically simulated galaxy cluster (“optimal” projection) at the same redshifts shown in Fig. 5. The scale of eachpanel is 375′′.

We define the power spectrumPn(l) of the multipole expan-sion l asPn(l) = |Sl (rn)|2. As discussed by Meneghetti et al.(2003b), the amount of substructure and the degree of asymme-try in the mass distributions of the numerically simulated clusterat any distancern from the main clump can be quantified bydefining an integrated powerPint(rn) as the sum of the powerspectral densities over all multipoles, from which we subtractthe monopole and the quadrupole in order to remove the axiallysymmetric and elliptical contributions,

Pint(rn) =∞

∑l=0

Pn(l)−Pn(0)−Pn(2) . (18)

This quantity measures the deviation from an elliptical distribu-tion of the surface mass density at a given distancern from thecluster centre.

The results are shown in Fig. 6: the total integrated power isshown as a function of redshift for five different distancesr fromthe centre of the main cluster body. The peaks in each plot arisewhen the infalling substructure enters or exits a circle of radiusr.Therefore, through the location of the peaks, we can determinewith precision the distance between the two merging clumps ateach redshift. The merger occurs betweenzin = 0.250 andzfin =0.150. In order to follow in detail this event, we pickNsnap=100 simulation snapshots equidistant in redshift betweenzin andzfin. The redshift interval between two consecutive snapshotstherefore is∆z= 0.001. In aΛCDM model, this corresponds toa time interval of approximately∆t ∼ 10 Myr.

At redshift z∼ 0.3, the virial mass of our numerical halo is∼ 7× 1014h−1M. In order to increase the lensing efficiencyand thus to reduce uncertainties in the numerically determined

arc cross sections, we artificially rescale the cluster mass by mul-tiplying the particle masses by a factorf = 2.5. Recalling thatthe virial radiusRvir of a halo scales asRvir ∝ M1/3, whereM isthe virial mass, to obtain a halo of massf ×M which is dynam-ically stable, we also need to rescale the distances by a factorf 1/3. This means that while the three-dimensional densityρ re-mains fixed, the halo surface density is enhanced by a factorf 1/3. For this reason we expect that, increasing its mass, the nu-merical cluster will become much more able to produce stronglensing effects.

3.3. Lensing simulations

Using each of theNsnapsnapshots, we perform ray-tracing simu-lations. Our method is described in detail elsewhere (Meneghettiet al., 2000, 2001, 2003b,a), but some parameters differ from thesimulations described earlier.

For each snapshot, we select again from the simulation boxa cube of 3h−1Mpc side length, containing the high-density re-gion of the cluster. The centre of the cube is defined such that theselected region contains both merging clumps. A third smallersubstructure enters the field atz∼ 0.23. In the lensing simula-tions performed using the snapshots at higher redshift the effectof this small clump of matter is not taken into account, becauseit is out of the considered region. We checked its influence onthe cluster strong lensing properties by selecting a larger region.We found that the effect of this substructure is negligible whenits distance from the main clump is larger than∼ 1.5h−1Mpc.

We then determine the three-dimensional density fieldρ ofthe cluster from the particle positions, by interpolating the mass

8

z=0.230 z=0.214 z=0.211 z=0.203

z=0.199 z=0.190 z=0.180

200 400 600 800

FIG. 8.—Maps of the magnification on the source plane produced by the numerically simulated galaxy cluster at the same redshifts shown inFig. 5. The scale of each panel is 1.5 h−1Mpc.

density within a regular grid of 2563 cells, using theTriangu-lar Shaped Cloudmethod Hockney & Eastwood (1988). Wethen produce surface density fieldsΣ by projectingρ along threecoordinate axes. We chose our reference frame such that oneaxis is perpendicular to the direction of merging. By project-ing along this coordinate axis, we minimise the impact parame-ter of the infalling substructure with respect to the main clusterclump. Hereafter this projection will be called “optimal”. Forcomparison, we also investigate the projection along a secondaxis, where the minimal distance between the merging clumpsis never smaller than∼ 250h−1kpc. The third projection, whichis not interesting for the purpose of this paper, since the sub-structure moves almost exactly along the line of sight, will beneglected in the following analysis.

The lensing simulations are performed by tracing a bundleof 2048× 2048 light rays through the central quarter of thelens plane and computing for each of them the deflection an-gle. Then, a large number of sources is distributed on the sourceplane. We place this plane at redshiftzs = 1. Keeping all sourcesat the same redshift is an approximation justified for the pur-poses of the present case study, but the recent detections of arcsin high-redshift clusters (Zaritsky & Gonzalez, 2003; Gladderset al., 2003) indicate that more detailed simulations will have toaccount for a wide source redshift distribution.

The sources are elliptical with axis ratios randomly drawnfrom [0.5,1]. Their equivalent diameter (the diameter of the cir-cle enclosing the same area of the source) isre = 1′′. Unlikeprevious studies, we do not distribute the elliptical sources onthe central quarter of the source plane, since we need to inves-tigate a region large enough to contain the caustics of both themerging clumps. Nevertheless, we keep the same spatial reso-

lution of the source grids in the previous simulations, becausesources are initially distributed on a regular grid of 64×64 in-stead of 32×32 cells. Then, we adaptively increase the sourcenumber density in the high magnification regions of the sourceplane by adding sources on sub-grids whose resolution is in-creased towards the lens caustics. This increases the probabilityof producing long arcs and thus the numerical efficiency of themethod. In order to compensate for this artificial source-densityincrease, we assign for the following statistical analysis to eachimage a statistical weight proportional to the area of the grid cellon which the source was placed. We refine the source grid fourtimes, and the total number of sources is typically∼ 30,000.

Using the ray-tracing technique, we reconstruct the images ofbackground galaxies and measure their length and width. Ourtechnique for image detection and classification was describedin detail by Bartelmann & Weiss (1994) and adopted by Bartel-mann et al. (1998) and Meneghetti et al. (2000, 2001, 2003b,a).It results in a catalogue of simulated images which is subse-quently analysed statistically.

4. RESULTS

In the following sections we show how the lensing properties ofour numerical model change during the merger. We focus onthe “optimal” projection of the numerical cluster, i.e. the projec-tion along the axis perpendicular to the direction of the merger,where the effects of the merger are expected to be strongest. Werefer to the second projection only in passing.

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FIG. 9.—Caustics of the numerically simulated galaxy cluster at the same redshifts as shown in Fig. 5. The scale of each panel is∼ 375′′. Notethe appearance of thin, long caustic structures which are not seen in the analytic model shown in Fig. 3 due to the lack of internal asymmetries andexternal shear.

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4.1. Critical lines and caustics

As expected, the critical lines and the caustics of the numericallens evolve strongly during the merger.

First, we discuss the results for the “optimal” projection. Weshow the critical lines at some relevant redshifts in Fig. 7. Atredshiftz= 0.230, the main cluster clump and the infalling sub-structure develop separate critical lines. The largest mass con-centration also produces a small radial critical line (enclosedby the more extended tangential critical line). As the mergerproceeds, the tangential critical lines are stretched towards eachother. As discussed in Sect. 2.2., this is due to the increasingshear in the region between the mass concentrations. The criti-cal lines merge approximately at redshiftz= 0.214. After that,there exists a single critical line, which, after a short phase ofshrinking, expands isotropically while the two clumps overlap.This happens atz∼ 0.203.

Note that the radial critical line grows during the mergingevent, reaching the maximum extension when the clumps are ex-actly aligned and the surface density is highest. Indeed, in orderto develop a radial critical line, the lens must reach a sufficientlyhigh central surface density.

When the substructure moves to the opposite side of themain cluster body, the tangential critical line stretches again andreaches its maximum elongation atz∼ 0.190. Then, separatecritical lines appear around each clump. Their size decreasesfor decreasingz because both the shear and the convergencebetween the two mass concentration decrease as their distancegrows. Similarly, the radial critical lines shrink.

During the merger phase, the magnification pattern on thesource plane changes as shown in Fig. 8. The highest magni-fications are reached during the phase of maximum overlap, butthe extent of the highest magnification regions is largest at red-shiftsz= 0.214 andz= 0.190. Therefore, at these redshifts eventhe caustics are most elongated. In Fig. 9 we show the causticsat the same redshifts as in Fig. 8. Their evolution reflects that ofthe critical lines. Before redshiftz= 0.214, two separated caus-tics exist; as the distance between the two clumps shrinks, theirelongation in the direction of merging grows. Then the causticsmerge into a single line, shrink along their long axis and widenperpendicular to it. When the two mass concentrations overlap,it has a diamond shape with four pronounced cusps. Finally,when the distance between the substructure and the main clusterclump grows again, the caustic shrinks, elongates in the direc-tion of relative motion of the substructure, and finally splits intotwo small separate caustics.

In the second projection we have considered, the substructuredoes not pass through the cluster centre. In this case its dis-tance from the main cluster clump is always& 250h−1kpc. Weshow in Fig. 10 the critical lines for this projection. As in thecase of the “optimal” projection, while the merger proceeds, thecritical lines of the two merging clumps are stretched towardseach other by the effect of the increasing shear in the region be-tween the two mass concentrations. The largest elongation isreached at redshift∼ 0.198. However, the substructure does notcross the region enclosed by the critical lines of the main clusterclump and therefore the critical lines do not shrink in the direc-tion of merging and never expand isotropically. After reachingthe maximal elongation, it shrinks while the distance betweenthe two clumps grows. The caustics (not shown) reflect the sameevolution.

4.2. Tangential and radial arcs

Since the extent of the high magnification regions and causticschanges during the infall of the substructure onto the main clus-ter clump as shown in the previous section, we expect that the

lensing efficiency for producing both tangential and radial arcschanges accordingly. We quantify the influence of merging onthe cluster’s strong lensing efficiency by measuring its cross sec-tions for different arc properties.

First, we consider tangential arcs, which can be identifiedamong the lensed images due to their large length-to-width ra-tio L/W. By definition, the lensing cross section is the area onthe source plane where a source must be located in order to beimaged as an arc with the specified property. As explained inSect. 3.3., each source is taken to represent a fraction of thesource plane. We assign a statistical weight of unity to thesources which are placed on the sub-grid with the highest res-olution. These cells have areaA. The lensing cross section isthen measured by counting the statistical weights of the sourceswhose images satisfy a specified property. If a source has multi-ple images with the required characteristics, its statistical weightis multiplied by the number of such images. Thus, the formulafor computing cross sections for arcs with a propertyp, σp, is

σp = A ∑i

wi ni , (19)

whereni is the number of images of thei-th source satisfying therequired conditions, andwi is the statistical weight of the source.Using this method, we compute the lensing cross sections forarcs withL/W ≥ 5,7.5 and 10, respectively.

The results are shown in the first three panels of Fig. 11. Thesolid lines refer to the “optimal” projection. All curves exhibitthe same redshift evolution. Their main properties can be sum-marised as follows:

• The cross sections grow by a factor of two betweenz∼0.240 andz∼ 0.220, as shown in Fig. 6. Then, the crosssection further increases by a factor of five betweenz∼0.220 andz∼ 0.200, i.e. within∼ 0.2 Gyr.

• The curves have three peaks, located at redshiftsz1 =0.214, z2 = 0.203 andz3 = 0.190. The peaks atz1 andz3 correspond to the maximum extent of caustics and criti-cal curves along the merging directionbeforeandafter themoment when the merging clumps overlap; the peak atz2occurs when the distance of the infalling substructure fromthe merging clump is minimal.

• Two local minima arise between the three maxima at red-shiftsz4 = 0.211 andz5 = 0.199, where the cross sectionsare a factor of two smaller than at the peaks. At these red-shifts, the critical lines have shrunk along the merging di-rection and the caustics are less cuspy than atz2.

• The cross sections reduce by more than one order of mag-nitude afterz = 0.190. At redshiftz∼ 0.180, when thedistance between the merging clumps is& 1.5 h−1Mpc,the cross section sizes are comparable to those beforez∼0.240.

Therefore, during the merger, our simulated cluster becomesextremely more efficient in producing tangential arcs. Theinfalling substructure starts affecting the cross sections forlong and thin arcs when its distance from the main clusterclump is approximately equivalent to the cluster virial radius(∼ 1.5 h−1Mpc), and the largest effects are seen at three dif-ferent times:

• When the critical lines (and the corresponding caustics)merge, i.e. when the shear between the mass concentra-tions is sufficient to produce the largest elongation of thecritical lines along the direction of merging. This happens∼ 100 Myr before and after the substructure crosses thecluster centre;

11

FIG. 10.—Critical lines of the numerically simulated galaxy cluster (second projection) at the same redshifts shown in Fig. 5. The scale of eachpanel is 375′′.

• when the two clumps overlap, i.e. when the projectedsurface density or convergence is maximal, producing thelargestisotropic expansion of the critical lines and of thecaustics.

A different behaviour is found in the second projection. Theresults are shown again in the first three panels of Fig. 11 asdashed lines. In this case the critical lines and caustics reach amaximal elongation atz6 = 0.198 and shrink at lower redshifts.Again the lensing cross sections reflect the evolution of the crit-ical curves and caustics: they reach a maximum extent atz6 andthen their size decreases. A very important result is that, evenif the merging clumps never get closer than∼ 250h−1kpc, thecross sections still grow by roughly a factor of five within ap-proximately 50 Myr.

We now consider the effects of merging on the cross sectionfor radial arcs. These are identified from the complete sampleof distorted images using the technique described in Meneghettiet al. (2001). It consists in selecting those arcs for which themeasured radial magnification at their position exceeds a giventhreshold.

The cross section for this type of arcs as a function of redshiftis shown in the fourth panel of Fig. 11. In both projections, atredshiftsz& 0.22, the cross section for radial arcs keeps constantand fluctuates around 10−3.5 h−2Mpc2. Then, in the “optimal”projection, it grows by a factor of five, reaching the highest valueat z= 0.203. This is the same redshift where the cross sectionsfor tangential arcs peak. The enhancement of the convergencedue the overlapping of the merging clumps thus makes the clus-ter substantially more efficient for producing radial arcs.

Then, the cross section rapidly drops to zero for smaller red-shifts. Note that the redshift interval where our cluster is very

efficient for producing radial arcs is quite small (∆zrad . 0.04).Radial arcs have so far been reported in only five galaxy clusters(MS 2137, Fort et al. 1992; A 370, Smail et al. 1996; MS 0440,Gioia et al. 1998; AC 114, Natarajan et al. 1998; A 383, Smithet al. 2001). Our results contribute to explaining why radialarcs are so rare: if they form preferentially during the cross-ing of large substructures through the cluster centre, when thecentral lens surface density is higher, the visibility window ofsuch events is very narrow (. 100 Myr per merger).

Note that, in the second projection (dashed line), the crosssection for radial arcs has some strong fluctuations during themerger but never reaches very high values. As for the “optimal”projection, it peaks at the same redshift where the cross sectionfor tangential arcs is largest, but it changes by less than a factorof two with respect to redshiftsz & 0.21. This confirms thatonly large enhancements of the central surface density produce asubstantial increase of the cluster’s efficiency for forming radialarcs.

As mentioned before, assuming a single source redshift isjustified for the purpose of this work. Indeed, the critical sur-face density changes by∼ 10% when moving the sources fromredshift one to redshift two, while keeping the lens redshift atzl ∼ 0.2. Thus the extent of the critical lines and caustics (andconsequently of the lensing cross sections) is not expected tochange very much for sources at redshifts above unity. Never-theless, we show in Fig. 11 the cross sections obtained by plac-ing the source plane atzs = 2 for some characteristic snapshotsof our simulations. Filled and open circles refer to the “optimal”and the second projection, respectively. As expected the rela-tive change in the amplitude of the cross sections is modest (lessthan a factor of 2) for both tangential and radial arcs. Moreover,

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FIG. 11.—Lensing cross sections for tangential and radial arcs as functions of redshift. Top left panel: arcs with length-to-with ratioL/W > 5,σ5; top right panel: arcs with length-to-with ratioL/W > 7.5, σ7.5; bottom left panel: arcs with length-to-with ratioL/W > 10,σ10; bottom rightpanel: radial arcs,σrad. The solid lines refer to the “optimal” projection, i.e. where the substructure crosses the centre of the main cluster clump,while the dashed lines refer to the second projection of the cluster. Note the three peaks in the top and bottom left panels; their origin is discussedin the text.

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“Optimal” projection

Second projection

FIG. 12.—Simulated observations with theChandrasatellite of the X-ray emission by our numerically modelled galaxy cluster. The plots showthe distributions of detection events on the CCD ACIS-S3. The exposure time is 30,000 sec, while the scale of each figure is∼ 2.1 h−1Mpc. Theupper and lower panels refer to the “optimal” and to the second projections, respectively.

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the amplitude of the fluctuations induced by the merger event onthe lensing cross sections seems to depend very weakly on thesource redshift.

5. OBSERVATIONAL IMPLICATIONS

Our results show that cluster mergers could play an importantrole for arc statistics. In particular, since the lensing efficiencygrows by one order of magnitude during mergers, they may offera solution for thearc statistics problem.

It is quite important to notice that mergers might have someother important observational implications to account for. In factthe largest sample of clusters used for arc statistics studies (Lup-pino et al., 1999) was selected in the X-ray band, where the lu-minosity is due to bremsstrahlung emission. This is very sensi-tive to the dynamical processes going on in the cluster, since itis proportional to the square gas density. Therefore, we expectthat the cluster X-ray luminosity has large variations during amerging phase.

In Fig. 12 we show simulated X-ray observations of our clus-ter. These were obtained by using a code for simulating datataken with theChandrasatellite (Gardini, 2002). The clusteremissivity is calculated integrating over the gas density and tem-perature distribution within the cluster simulation, adopting theMekal plasma model (Kaastra & Mewe, 1993) with a metallicityof 0.3Z. The upper panels present the results in the “optimal”projection at six different redshifts. The upper left panel showsthe cluster atz = 0.333, before the merger starts. The secondpanel shows the cluster atz = 0.250, when the virial regionsmerge. The third, fourth and fifth panels show the cluster justbefore, at and justafter the maximum overlap of the mergingclumps, respectively. Finally, in the sixth panel, the cluster isobserved after the end of the merger. Since the colour scale isthe same for all images, it can be easily seen that the X-ray lumi-nosity is highest when the merging clumps are closest. Simularresults for the second projection are shown in the bottom panels.

In Fig. 13 we show the observed X-ray luminosity of the nu-merical cluster as function of redshift. The curve has a nar-row and almost symmetric peak located atz∼ 0.200. The X-ray luminosity grows by more than a factor of four betweenz∼ 0.300 andz∼ 0.200, by roughly a factor of∼ 2.5 betweenz∼ 0.230 andz∼ 0.200 and by roughly a factor of∼ 1.55 be-tweenz∼ 0.210 andz∼ 0.200. The width at half maximum ofthe peak is approximately∆z. 0.05. Note that the X-ray lumi-nosity peak is wider than the maxima in the arc cross sections,thus the X-ray emission increases earlier and decreases later thanthe arc cross sections during the merger.

If a cluster sample is built by collecting all the objects with X-ray luminosityLX larger than a given threshold, we expect thatmany merging clusters are present among them, since they arestronger X-ray emitters. Since these are all very efficient clus-ters for producing gravitational arcs, this could introduce a biasin the observationally determined frequency ofgiant arcs, whichcould become larger than predicted by previous numerical lens-ing simulations inΛCDM model. However, it is quite difficultto make more robust conclusions here since only one cluster hasbeen analysed. Further investigations are needed here. In anycase, our results show that much caution must be applied whenselecting clusters for arc statistics studies through their X-rayemission (cf. also Bartelmann & Steinmetz 1996).

6. SUMMARY AND CONCLUSIONS

In this paper we have investigated how the lensing properties of agalaxy cluster change during merging events. Similar dynamicalprocesses were not resolved in the previous lensing simulations

FIG. 13.—X-ray luminosityLx of the numerically simulated cluster asfunction of redshift. The curve is accurately sampled in the redshiftrangez= 0.15÷0.25. The X-ray luminosity has been measured evenatz= 0.1 andz= 0.333, where is indicated by the filled circles.

but they might play a relevant role for determining the stronglensing efficiency of cluster lenses.

We address this problem first by using analytic models. Whensimulating a collision between spherical haloes with NFW den-sity profiles, we find that both the critical lines and the causticsof the lens system strongly evolve during the merger. This be-haviour is explained by the change of the shear and of the con-vergence induced by the infalling clump. Indeed, while the dis-tance between the merging mass concentrations decreases, theshear intensity grows in the region between the halo centres. Theindividual critical lines and caustics of the main cluster clumpand the infalling substructure are stretched towards each otheruntil they merge.

To obtain a quantitative estimate of the impact of mergerson the lensing cross sections, we have studied the strong lens-ing properties of a numerically simulated galaxy cluster. Withinthis numerical model, a massive substructure falls onto the maincluster clump betweenzin = 0.250 andzfin = 0.150. We havestudied the merger in detail, picking a large number of simula-tion snapshots within this redshift range. The time separationbetween consecutive snapshots is approximately∼ 0.01 Gyr.Two different projections of the cluster where analysed: in the“optimal” projection, the substructure passes through the centreof the main cluster clump, while in the second projection thedistance between the two mass concentrations is always largerthan∼ 250h−1kpc.

The main results of this study can be summarised as follows:

• As expected from the results of the analytic tests, the shapesof critical lines and caustics substantially change during themerger. At the beginning, the two clumps develop individ-ual critical lines and caustics. These are stretched towardseach other while the distance between the mass concen-trations decreases, because the intensity of the shear fieldgrows in the region between the approaching clumps. In the“optimal” projection, when they merge, the resulting singlecritical line and caustic shrink along the merging directionand then expand isotropically, because of the increasingconvergence. The same behaviour is observed when thesubstructure moves far away from the main cluster clumpafter crossing its centre. In the other projection the max-imum extent of the critical lines is reached when the dis-tance between the mass concentrations is such that the ef-

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fect of the shear is largest. After that, the size of the criticallines drops.

• In the “optimal” projection, the lensing cross sections fortangential arcs change by one order of magnitude duringthe merger. The effect of the infalling substructure starts tobe relevant when its distance from the main cluster clump is∼ 1.5 h−1Mpc. The cross sections have three peaks locatedat the redshifts where the critical lines have the largest ex-tent along the merging direction, or when the shear effectinduced by the infalling substructure is largest, and at theredshift where the two clumps overlap and consequentlythe maximum convergence is reached. Although the effectsof the merger on the lensing cross sections are importantwithin a time interval of∼ 1 Gyr, the strongest impact isthus observed during the central part of the merging phase,on a time scale of∼ 200 Myr. In the second projection,the lensing cross section for long and thin arcs change by afactor of five within a time interval of∼ 100 Myr.

• The numerical cluster is highly efficient in producing radialarcs only during the merger. The cross section for this typeof arcs has only one peak, located at the redshift were theinfalling substructure crosses the centre of the main clusterclump.

Thus, our results show that mergers have a strong impact onthe strong lensing efficiency of galaxy clusters. Since the lens-ing cross sections for long and thin arcs change by one order ofmagnitude during the mergers, these dynamical processes couldbe a possible solution to the arc statistics problem.

This picture is in principle supported by the fact that sam-ples of clusters used in arc statistics studies are selected throughtheir X-ray luminosity, which is very sensitive to the dynam-ical processes arising in the cluster. In particular, we expectthat many merging clusters are present in these samples, sincethey are strong X-ray emitters. For example, Randall et al.(2002) estimate that the number of clusters with luminositiesLX > 5×1044h−2 erg/sec can be increased by a factor of 8.9 dueto merger boosts.

In addition, by surveying clusters in the LCDCS and in theRCS, Zaritsky & Gonzalez (2003) and Gladders et al. (2003)have recently found a high incidence of giant arcs in clusters athigh redhsift. Their results are particularly interesting since alarge number of clusters merging at high redshift are predictedby the commonly accepted theory of structure formation. In par-ticular, Gladders et al. (2003) speculate that a subset of clusterswith low-mass and large arc cross sections may be responsiblefor large numbers of arcs in distant clusters. One possibility isthat such “super-lenses” are clusters in the process of merging.

Detailed conclusions are, however, pending on further studiesquantifying the frequency of mergers and the dependence on arccross sections on the detailed merger parameters, such as theimpact parameter of the collision, the mass ratio of the merginghaloes and so forth. Such studies are now under way.

In recent studies, Wambsganss et al. (2003) and Dalal et al.(2003) suggest that arc statistics inΛCDM models can be rec-onciled with the observed abundance of gravitational arcs byadopting a broader distribution of source redshifts. Certainly,cross sections can change substantially for a cluster of a givenmass depending on its dynamical state, which makes earlier andcurrent statements about the theoretical expectation highly in-secure. Moreover, numerous observational effects need to betaken into account in addition for a reliable comparison betweennumerical simulations and observations.

ACKNOWLEDGEMENTS

This work has been partially supported by Italian MIUR (Grant2001, prot. 2001028932, “Clusters and groups of galaxies: theinterplay of dark and baryonic matter”), CNR and ASI. MMthanks EARA for financial support and the Max-Planck-Institutfur Astrophysik for the hospitality during the visits when part ofthis work was done. We are grateful to A. Gardini for the possi-bility of using the code simulating Chandra observations, and toS. Matarrese, N. Dalal and J. Kempner for useful discussions.

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