Mem. S.A.It. Vol. 73, 23c© SAIt 2002 Memorie della

����� �������� ������������� ���������� �!��� �"�$# %�&�'�( )* +�,�( +�-�.�

Massimo Meneghetti1, Elena Torri1, Matthias Bartelmann2,

Lauro Moscardini3, Elena Rasia1 and Giuseppe Tormen1,

1 Dipartimento di Astronomia, Universita di Padova, vicolo dell’Osservatorio 2,35122 Padova e-mail: meneghetti@pd.astro.it

2 Institut fur Theoretische Astrophysik, Tiergartenstr. 15, D–69121 Heidelberg,Germany

3 Dipartimento di Astronomia, Universita di Bologna, via Ranzani 1, I–40127Bologna, Italy

Abstract. We study the impact of merger events on the strong lensing propertiesof galaxy clusters. Previous lensing simulations were not able to resolve dynamicaltime scales of cluster lenses, which arise on time scales which are of order a Gyr.In this work, we analyse a numerically simulated lens model for the variationin its efficiency for producing both tangential and radial arcs while a massivesubstructure falls onto the main cluster body. We find that: (1) during the merger,the shape of the critical lines and caustics changes substantially; (2) the lensingcross sections for long and thin arcs can grow by one order of magnitude and reachtheir maxima when the extent of the critical curves is largest; (3) the cross sectionfor radial arcs also grows, but the cluster can efficiently produce this kind of arcsonly while the merging substructure crosses the main cluster centre; (4) while thearc 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 forthese dynamical processes is very important for arc statistics studies. In particular,they may provide a possible explanation for the arc statistics problem.

Key words. Cosmology, Dark Matter, Gravitational Lensing, Clusters of galaxies

1. Introduction

Based on the recent accurate measure-ments of the temperature fluctuationsof the CMB and the observations oftype Ia supernovae (Bennett et al. 2003;Perlmutter et al. 1999; Tonry et al. 2003),the ΛCDM model has become the favouritecosmogony. However, using the ray-tracingtechnique for studying the lensing proper-ties of galaxy clusters obtained from N-

body simulations, Bartelmann et al. (1998)showed that the number of giant arcs,which is expected to be detectable on thewhole sky in such a cosmological model,is inconsistent with the observed inci-dente of strong lensing events, as mea-sured in several gravitational arc surveys(Luppino et al. 1999; Zaritsky & Gonzalez2003; Gladders et al. 2003). This is knownas the arc statistics problem: the mismatchbetween the observed number of arcs and

24 M. Meneghetti et al.: The impact of cluster mergers on arc statistics

the number expected in the ΛCDM modelpreferred by virtually all other cosmologicalexperiments hints at a lack of understand-ing of cluster formation.

In this paper we investigate a possibleeffect which could not be properly consid-ered in the previously mentioned numeri-cal simulations of gravitational lensing bygalaxy clusters. In those works, the lensingcross sections for giant arcs of each numer-ical model were evaluated at different red-shifts, with a typical time separation be-tween two consecutive simulation outputsof approximately ∆t ∼ 1 Gyr. Thereforeall the dynamical processes arising in thelenses on time scales smaller than ∆t werenot resolved.

N-body simulations show that darkmatter haloes of different masses contin-uously fall onto rich clusters of galaxies(Tormen 1997). The typical time scale forsuch events is ∼ 1− 2 Gyr, which thereforemight be too short for having been prop-erly accounted for in the previous lensingsimulations.

2. Numerical methods

We investigate the lensing properties ofa simulated cluster undergoing a majormerger at z ≈ 0.2. At redshift z ∼ 0.25,when their viral regions merge, the maincluster clump and the infalling substruc-ture have virial masses of ∼ 7×1014h−1M�and ∼ 3× 1014h−1M�, respectively. In or-der to have a very good time resolutionfor resolving the complete merger in detail,we decided to re-simulate the cluster be-tween z = 0.25 and z = 0.15, obtaining 101equidistant outputs which we use for ourfollowing lensing analysis.

We show in Fig. 1 some projected den-sity maps of the simulated cluster at sev-eral redshifts during the merger. As is easilyseen, a large substructure crosses the clus-ter centre at z ∼ 0.2. The (comoving) sidelength of the displayed region is 3h−1 Mpc.

Using each snapshot, we per-form ray-tracing simulations. Ourmethod is described in detail elsewhere

(Meneghetti et al. 2000, 2001, 2003a,b).We produce surface density fields Σ byprojecting the three-dimensional densityρ along three coordinate axes. We choseour reference frame such that one axis isperpendicular to the direction of merging.By projecting along this coordinate axis,we minimize the impact parameter ofthe infalling substructure with respect tothe main cluster clump. Hereafter thisprojection will be called “optimal”. Forcomparison, we also investigate the projec-tion along a second axis, where the minimaldistance between the merging clumps isnever smaller than ∼ 250 h−1kpc. Thetwo-dimensional density fields are thenused as lens planes.

The lensing simulations are performedby tracing a bundle of 2048 × 2048 lightrays through the central quarter of the lensplane and computing for each of them thedeflection angle. Then, a large number ofsources is distributed on the source plane.We place this plane at redshift zs = 1.After calculating the deflection angles foreach ray, we reconstruct the distorted im-ages of all the background sources and mea-sure their properties.

3. Results

3.1. Critical lines and caustics

As expected, the critical lines and the caus-tics of the numerical lens evolve stronglyduring the merger.

First, we discuss the results for the“optimal” projection. We show the criticallines at some relevant redshifts in Fig. 2(left panel). At redshift z = 0.230, the maincluster clump and the infalling substruc-ture develop separate critical lines. Thelargest mass concentration also produces asmall radial critical line (enclosed by themore extended tangential critical line). Asthe merger proceeds, the tangential criti-cal lines are stretched towards each other.This is due to the increasing shear in the re-gion between the mass concentrations. Thecritical lines merge approximately at red-

M. Meneghetti et al.: The impact of cluster mergers on arc statistics 25

z=0.230 z=0.214 z=0.211 z=0.203 z=0.199 z=0.190 z=0.180

Fig. 1. Surface density maps of the numerically simulated cluster at several redshiftsbetween z = 0.230 and z = 0.180, during the merger. The scale of each panel is 3 h−1Mpc.

shift z = 0.214. After that, there existsa single critical line, which, after a shortphase of shrinking, expands isotropicallywhile the two clumps overlap. This happensat z ∼ 0.203.

Note that the radial critical line growsduring the merging event, reaching themaximum extension when the clumps areexactly aligned and the surface density ishighest. Indeed, in order to develop a ra-dial critical line, the lens must reach a suf-ficiently high central surface density.

When the substructure moves to the op-posite side of the main cluster body, thetangential critical line stretches again andreaches its maximum elongation at z ∼0.190. Then, separate critical lines appeararound each clump. Their size decreases fordecreasing z because both the shear andthe convergence between the two mass con-centration decrease as their distance grows.Similarly, the radial critical lines shrink.

In the second projection we have con-sidered, the substructure does not passthrough the cluster centre. In this case itsdistance from the main cluster clump is al-ways >∼ 250h−1kpc. We show in the rightpanel of Fig. 2 the critical lines for thisprojection. As in the case of the “optimal”projection, while the merger proceeds, thecritical lines of the two merging clumpsare stretched towards each other by theeffect of the increasing shear in the re-gion between the two mass concentrations.The largest elongation is reached at red-shift ∼ 0.198. However, the substructuredoes not cross the region enclosed by thecritical lines of the main cluster clump andtherefore the critical lines do not shrink inthe direction of merging and never expand

isotropically. After reaching the maximalelongation, it shrinks while the distance be-tween the two clumps grows.

3.2. Tangential and radial arcs

Since the extent of the high-magnificationregions and critical lines changes during theinfall of the substructure onto the maincluster clump as shown in the previous sec-tion, we expect that the lensing efficiencyfor producing both tangential and radialarcs changes accordingly. We quantify theinfluence of merging on the cluster’s stronglensing efficiency by measuring its cross sec-tions for different arc properties.

First, we consider tangential arcs, whichcan be identified among the lensed im-ages due to their large length-to-width ra-tio L/W . In the left panel of Fig. 3, weshow the lensing cross section for arcs withL/W > 7.5 as a function of the clusterredshift. The solid lines refer to the “op-timal” projection, the dashed ones to thesecond projection. As can be easily seen,our simulated cluster becomes extremelymore efficient in producing tangential arcs.In the “optimal” projection, the infallingsubstructure starts affecting the cross sec-tions for long and thin arcs when its dis-tance from the main cluster clump is ap-proximately equivalent to the cluster virialradius (∼ 1.5 h−1Mpc), and the largesteffects are seen at three different times:i) when the critical lines (and the cor-responding caustics) merge, i.e. when theshear between the mass concentrations issufficient to produce the largest elongationof the critical lines along the direction ofmerging. This happens ∼ 100 Myr before

26 M. Meneghetti et al.: The impact of cluster mergers on arc statistics

Fig. 2. Critical lines of the numerically simulated galaxy cluster at the same redshiftsshown in Fig. 1. The scale of each panel is 375′′. The left and the right panels refer tothe “optimal” and to the second projection, respectively.

and after the substructure crosses the clus-ter centre; ii) when the two clumps over-lap, i.e. when the projected surface densityor convergence is maximal, producing thelargest isotropic expansion of the criticallines. During the merger, the cross sectiongrows by approximately one order of mag-nitude.

A different behaviour is found in thesecond projection. The results are shownagain in Fig. 3 as dashed lines. In this casethe critical lines reach a maximal elonga-tion at z6 = 0.198 and shrink at lowerredshifts. Again the lensing cross sectionsreflect the evolution of the critical curves:they reach a maximum extent at z6 andthen their size decreases. A very importantresult is that, even if the merging clumpsnever get closer than ∼ 250 h−1kpc, thecross sections still grow by roughly a factorof five within approximately 50 Myr.

We now consider the effects of merg-ing on the cross section for radial arcs.The cross section for this type of arcs asa function of redshift is shown in the rightpanel of Fig. 3. In both projections, at red-shifts z >∼ 0.22, the cross section for radialarcs keeps constant and fluctuates around10−3.5 h−2Mpc2. Then, in the “optimal”projection, it grows by a factor of five,reaching the highest value at z = 0.203.This is the same redshift where the crosssections for tangential arcs peak. The en-hancement of the convergence due the over-

lapping of the merging clumps thus makesthe cluster substantially more efficient forproducing radial arcs. Then, the cross sec-tion rapidly drops to zero for smaller red-shifts. Note that the redshift interval whereour cluster is very efficient for producing ra-dial arcs is quite small (∆zrad <∼ 0.04). Ourresults contribute to explaining why radialarcs are so rare: if they form preferentiallyduring the crossing of large substructuresthrough the cluster centre, when the cen-tral lens surface density is higher, the vis-ibility window of such events is very nar-row (<∼ 100 Myr per merger). Note that,in the second projection (dashed line), thecross section for radial arcs has some strongfluctuations during the merger but neverreaches very high values. As for the “opti-mal” projection, it peaks at the same red-shift where the cross section for tangentialarcs is largest, but it changes by less thana factor of two with respect to redshiftsz >∼ 0.21. This confirms that only large en-hancements of the central surface densityproduce a substantial increase of the clus-ter’s efficiency for forming radial arcs.

4. Observational implications

Our results show that cluster mergers couldplay an important role for arc statistics. Inparticular, since the lensing efficiency growsby one order of magnitude during mergers,

M. Meneghetti et al.: The impact of cluster mergers on arc statistics 27

Fig. 3. Lensing cross sections for tangential and radial arcs as functions of redshift. Leftpanel: arcs with length-to-with ratio L/W > 7.5; right panel: radial arcs, σrad. The solidlines refer to the “optimal” projection, i.e. where the substructure crosses the centreof the main cluster clump, while the dashed lines refer to the second projection of thecluster; in both cases the sources are placed at redshift zs = 1. Note the three peaksin the top and bottom left panels; their origin is discussed in the text. Cross sectionscomputed considering sources placed at zs = 2 are shown for the “optimal” and secondprojections by filled and open circles, respectively. As expected the relative change in theamplitude of the cross sections is modest (less than a factor of 2) for both tangential andradial arcs. Moreover, the amplitude of the fluctuations induced by the merger event onthe lensing cross sections seems to depend very weakly on the source redshift.

they may offer a solution for the arc statis-tics problem.

It is quite important to notice thatmergers might have some other importantobservational implications to account for.In fact the largest sample of clusters usedfor arc statistics studies (Luppino et al.1999) was selected in the X-ray band, wherethe luminosity is due to bremsstrahlungemission. This is very sensitive to the dy-namical processes going on in the cluster,since it is proportional to the square gasdensity. Therefore, we expect that the clus-ter X-ray luminosity has large variationsduring a merging phase.

In Fig. 4 we show the observed X-ray lu-minosity of the numerical cluster as func-tion of redshift. The curve has a narrowand almost symmetric peak located at z ∼

0.200. The X-ray luminosity grows by morethan a factor of four between z ∼ 0.300and z ∼ 0.200, by roughly a factor of∼ 2.5 between z ∼ 0.230 and z ∼ 0.200and by roughly a factor of ∼ 1.55 betweenz ∼ 0.210 and z ∼ 0.200. The width athalf maximum of the peak is approximately∆z <∼ 0.05. Note that the X-ray luminositypeak is wider than the maxima in the arccross sections, thus the X-ray emission in-creases earlier and decreases later than thearc cross sections during the merger.

If a cluster sample is built by collectingall the objects with X-ray luminosity LXlarger than a given threshold, we expectthat many merging clusters are presentamong them, since they are stronger X-rayemitters. Since these are all very efficientclusters for producing gravitational arcs,

28 M. Meneghetti et al.: The impact of cluster mergers on arc statistics

Fig. 4. X-ray luminosity Lx of the numer-ically simulated cluster as function of red-shift. The curve is accurately sampled inthe redshift range z = 0.15÷ 0.25. The X-ray luminosity has been measured even atz = 0.1 and z = 0.333, where is indicatedby the filled circles.

this could introduce a bias in the observa-tionally determined frequency of giant arcs,which could become larger than predictedby previous numerical lensing simulationsin ΛCDM model. However, it is quite diffi-cult to make more robust conclusions heresince only one cluster has been analysed.Further investigations are needed here. Inany case, our results show that much cau-tion must be applied when selecting clus-ters for arc statistics studies through theirX-ray emission.

5. Conclusions

In this paper we have investigated howthe lensing properties of a galaxy clusterchange during merging events.

Our results show that mergers have astrong impact on the strong lensing effi-ciency of galaxy clusters. In fact, the lens-ing properties of clusters change substan-tially during the merging phase. For ex-ample, the size of the critical curves, along

which highly magnified images form growsby a considerable amount while a substrc-ture falls into the main cluster clump. Asa consequence, both the lensing cross sec-tions for tangential and radial arcs changeby up to one order of magnitude duringmergers.

Therefore, these dynamical processescould be a possible solution to the arcstatistics problem.

Acknowledgements. Part of this work was sup-ported by the German Deutsche Forschungsge-meinschaft, DFG project number Ts 17/2–1.

References

Bartelmann, M., Huss, A., Colberg, J.,Jenkins, A., & Pearce, F. 1998, A&A,330, 1

Bennett, C., Bay, M., Halpern, M.,Hinshaw, G., et al. 2003, ApJ, 583, 1

Gladders, M., Hoekstra, H., Yee, H., Hall,P., & Barrientos, L. 2003, ApJ, 593, 48

Luppino, G., Gioia, I., Hammer, F., LeFevre, O., & Annis, J. 1999, A&AS, 136,117

Meneghetti, M., Bartelmann, M., &Moscardini, L. 2003a, MNRAS in press;preprint astro-ph/0302603

Meneghetti, M., Bartelmann, M., &Moscardini, L. 2003b, MNRAS, 340, 105

Meneghetti, M., Bolzonella, M.,Bartelmann, M., Moscardini, L., &Tormen, G. 2000, MNRAS, 314, 338

Meneghetti, M., Yoshida, N., Bartelmann,M., et al. 2001, MNRAS,, 325, 435

Perlmutter, S., Aldering, G., Goldhaber,G., Knop, R., et al. 1999, ApJ, 517, 565

Tonry, J., Schmidt, B., Barris, B., Candia,P., et al. 2003, ApJ, 594, 1

Tormen, G. 1997, MNRAS, 290, 411Zaritsky, D. & Gonzalez, A. 2003, ApJ, 584,

691