Constraining the cross-section of dark matter with giant radial arcs in galaxy clusters
J. Vega-Ferrero, J. M. Dana, J. M. Diego, G. Yepes, W. Cui, M. Meneghetti
MMNRAS , 1–12 (2020) Preprint 16 June 2020 Compiled using MNRAS L A TEX style file v3.0
Constraining the cross section of dark matter with giantradial arcs in galaxy clusters
J. Vega-Ferrero, , (cid:63) J. M. Dana, J. M. Diego, G. Yepes, , W. Cui andM. Meneghetti , IFCA, Instituto de F´ısica de Cantabria (UC-CSIC), Av. de Los Castros s/n, 39005 Santander, Spain Department of Physics and Astronomy, University of Pennsylvania, 209 S. 33rd St, Philadelphia, PA 19104, USA Almer´ıa, Spain Departamento de F´ısica Te´orica M-8, Universidad Aut´onoma de Madrid, Cantoblanco, E-28049 Madrid, Spain Centro de Investigaci´on Avanzada en F´ısica Fundamental (CIAFF), Universidad Aut´onoma de Madrid, 28049 Madrid, Spain Institute for Astronomy, University of Edinburgh, Royal Observatory, Edinburgh EH9 3HJ, United Kingdom INAF - Osservatorio di Astrofisica e Scienza dello Spazio, via Gobetti 93/3, 40129, Bologna, Italy INFN, Sezione di Bologna, viale Berti Pichat 6/2, 40127, Bologna, Italy
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We compare the statistics and morphology of giant arcs in galaxy clusters using N-body simulations within the standard cold dark matter model and simulations wheredark matter has a non-negligible probability of interaction (parametrized by its crosssection), i.e self-interacting dark matter (SIDM). We use a ray-tracing technique toproduce a statistically large number of arcs around six simulated galaxy clusters atdifferent redshifts. Since dark matter is more likely to interact in colliding clusters thanin relaxed clusters, and this probability of interaction is largest in denser regions, wefocus our analysis on radial arcs (which trace the lensing potential in the central regionbetter than tangential arcs) in galaxy clusters which underwent (or are undergoing)a major merger. We find that self-interacting dark matter produces fewer radial arcsthan standard cold dark matter but they are on average more magnified. We alsoappreciate differences in the morphology which could be used to statistically favor onemodel versus the other.
Key words: gravitational lensing; galaxies: clusters: general; cosmology: theory -dark matter
The nature of dark matter is arguably one of the biggestmysteries of modern science. Despite the wealth of observa-tional evidence for its existence (from astrophysical probes),all efforts for its direct detection have proven fruitless. Thelack of success in detecting the elusive dark matter has pro-moted the appearance of alternative explanations, includ-ing those that propose that dark matter does not exist,and all astrophysical evidences can be reinterpreted witha reformulation of the laws gravity. However, no model hasbeen as successful at reproducing cosmological observationsas Λ -Cold Dark Matter (CDM). This includes the cosmicmicrowave background power spectrum, which can be repro-duced with astonishing precision only when a precise amountof cold dark matter is included in the model. (cid:63) E-mail: [email protected]
In addition to the efforts being made for direct detec-tion (or production) of dark matter, it is important to pur-sue the detection of dark matter through indirect methodsprovided by the astrophysical probes (like decay or anni-hilation into gamma-rays, other particles or EM radiation,accumulation effects on astrophysical bodies such as starsor neutron stars, distortions in the cosmic microwave back-ground, gravitational lensing, etc. (see Profumo 2013, for areview).In this context, the CDM cosmological model has beenextremely successful in describing the large scale structurethat we observe of our Universe. However, at smaller scales,where the formation of structures becomes non-linear, theCDM scenario has found difficulties explaining some dis-crepancies (such as the core-cusp, the diversity, the miss-ing satellites and the too-big-to fail problems) that arosefrom a comparison with predictions from N-body cosmolog-ical simulations. A promising alternative to the collision-less © a r X i v : . [ a s t r o - ph . C O ] J un J. Vega-Ferrero et al.
CDM cosmological model is the self-interacting dark matter(SIDM) model, proposed by (Spergel & Steinhardt 2000) tosolve both the core-cusp and the missing satellites problems.In this framework, DM particles scatter elastically with eachother through → interactions. Assuming a cross sec-tion for the DM particles, since the scattering rate of DMparticles is proportional to the DM density and their rela-tive velocities, the SIDM model remains successful on largescales (almost identical to the CDM framework), but chang-ing the formation of structures at late times and only onsmall scales, in particular, in the inner regions of DM halos.For instance, the relatively shallow profiles in the centralregions of some galaxies and clusters (core-cusp problem),the missing satellite problem, or the too-big-to fail problemcould be all explained if the cross section of DM is around σ / m ≈ / g (see Tulin & Yu 2018, for an exhaustivereview).Among the different astrophysical probes, galaxy clus-ters have provided useful information about dark matterproperties. In the proposed SIDM cosmological model, theself-interaction rate for clusters is expected to be much largerthan for galaxy scales, since the typical scatter velocity inmassive DM halos is of the order of v rel ≈ / s . Inthis context, N-body cosmological simulations have beenan indispensable tool to study the effects of DM interac-tions in structure formation on a wide range of scales, fromdwarf galaxies to massive galaxy clusters. Recently, N-bodysimulations including SIDM with high resolution and halostatistics have reactivate the SIDM scenario as a plausiblealternative to the CDM cosmological model (Vogelsbergeret al. 2012; Rocha et al. 2013; Peter et al. 2013; Zavala et al.2013; Vogelsberger et al. 2014; Elbert et al. 2015; Fry et al.2015; Dooley et al. 2016; Wittman et al. 2018; Robertsonet al. 2019). Although more recent studies based on massiveclusters point towards smaller values of the DM cross sec-tion ( σ / m (cid:46) . / g ) on these scales (Kaplinghat et al.2014; Elbert et al. 2015), other studies based on stackedmerging clusters (Harvey et al. 2015) and stellar kinemat-ics within cluster cores (Elbert et al. 2015) suggest sometension with these values. Additionally, as shown in Ran-dall et al. (2008), the Bullet Cluster shows how dark mat-ter is consistent with the hypothesis that it is collision-less.Along with other merging clusters, the Bullet Cluster hasprovided an upper limit for the cross section for dark mat-ter at σ / m ≈ / g . Larger values of the cross sectionwould result in a shift between the peak of the dark mat-ter distribution and the centre of the distribution of galaxies(galaxies behave like truly collision-less particles even duringa major cluster merger). Such a shift has not been observed(Clowe et al. 2006) suggesting that the cross section of darkmatter must be σ / m ≈ / g at most. Contrarily to theBullet Cluster, Abell 3827 and Abell 520 exhibit large DM-stellar offsets that, if explained by the SIDM model, willrequire larger DM cross sections probably in tension withother data, such as the core sizes in galaxy clusters.Alternatively, strong gravitational lensing data has beenused to constrain the core size and density in clusters rele-vant for SIDM (Firmani et al. 2000, 2001; Meneghetti et al.2001; Wyithe et al. 2001). More recent studies have usedEinstein radii statistics in galaxy clusters to constrain theDM cross section (Robertson et al. 2019), suggesting that future wide surveys might be able to distinguish betweenCDM and SIDM cosmological models (Despali et al. 2019).In particular, Meneghetti et al. (2001) placed thestrongest constraint on the DM cross section and clustercores by examining the ability of a SIDM halo to produce”extreme” strong lensing arcs (both radial or giant tangen-tial arcs). The authors concluded that their SIDM halo sim-ulated with σ / m > / g is not dense enough (in pro-jection) to produce extreme tangential arcs with length-to-width ratios of l / w (cid:38) . . Based on the ability to produceradial arcs, the constraint found is even more severe, sincetheir SIDM halo simulated with σ / m < . / g was notable to produce radial arcs. As acknowledged in Meneghettiet al. (2001), constraints based on one single halo need tobe addressed with caution due to the variability of the den-sity profiles in SIDM halos. Moreover, this particular halois a SIDM-only simulation (no baryons were included) and,therefore, it did not account for the baryonic density due tothe central galaxy which boosts its lensing efficiency. In par-ticular, one of the major effects of the cooling and the starformation in simulations is to trigger a strong adiabatic con-traction of the baryonic component, leading to (sometimesunrealistically) denser cluster cores. Nevertheless, there havebeen found several examples of galaxy clusters that exhibitradial arcs. In particular, 12 candidate radial arcs wherefound in three of the six clusters examined by (Sand et al.2004, 2005). More recent studies by Newman et al. (2013a,b)have found radial arcs in two of the clusters analyzed bySand et al. (2004), MS2137-23 and Abell 383, with DM cores(of the order of 10 kpc) which are consistent with lensingdata once the baryonic mass is included. Another interest-ing example with several central images detected (and ∼ of its 82 multiple images classified as radial arcs) is MACSJ1206 (Caminha et al. 2017). Interestingly, as first proposedby Molikawa & Hattori (2001), it is practical to use the ra-tio of radial to tangential arcs as a statistical measure of theslope of the dark matter distribution in cluster cores (seealso Sand et al. 2005). In this study, we analyze a total of six massive galaxy clus-ters extracted from two sets of well tested, high-resolutionN-body cosmological simulations with different cosmologi-cal parameters. For each cluster we compare the results ob-tained with the simulation particles set as standard CDMparticles and an identical simulation (i.e, with the same ini-tial condition) where the DM particles are allowed to inter-act with a given probability determined by the value of σ / m (SIDM). One of the galaxy clusters used in this study is extractedfrom the MUltidark SImulations of galaxy Clusters (MU-SIC , Sembolini et al. 2013). In particular, we analyze theMUSIC-MD dataset, which consists of a set of re-simulated http://music.ft.uam.es MNRAS000
CDM cosmological model is the self-interacting dark matter(SIDM) model, proposed by (Spergel & Steinhardt 2000) tosolve both the core-cusp and the missing satellites problems.In this framework, DM particles scatter elastically with eachother through → interactions. Assuming a cross sec-tion for the DM particles, since the scattering rate of DMparticles is proportional to the DM density and their rela-tive velocities, the SIDM model remains successful on largescales (almost identical to the CDM framework), but chang-ing the formation of structures at late times and only onsmall scales, in particular, in the inner regions of DM halos.For instance, the relatively shallow profiles in the centralregions of some galaxies and clusters (core-cusp problem),the missing satellite problem, or the too-big-to fail problemcould be all explained if the cross section of DM is around σ / m ≈ / g (see Tulin & Yu 2018, for an exhaustivereview).Among the different astrophysical probes, galaxy clus-ters have provided useful information about dark matterproperties. In the proposed SIDM cosmological model, theself-interaction rate for clusters is expected to be much largerthan for galaxy scales, since the typical scatter velocity inmassive DM halos is of the order of v rel ≈ / s . Inthis context, N-body cosmological simulations have beenan indispensable tool to study the effects of DM interac-tions in structure formation on a wide range of scales, fromdwarf galaxies to massive galaxy clusters. Recently, N-bodysimulations including SIDM with high resolution and halostatistics have reactivate the SIDM scenario as a plausiblealternative to the CDM cosmological model (Vogelsbergeret al. 2012; Rocha et al. 2013; Peter et al. 2013; Zavala et al.2013; Vogelsberger et al. 2014; Elbert et al. 2015; Fry et al.2015; Dooley et al. 2016; Wittman et al. 2018; Robertsonet al. 2019). Although more recent studies based on massiveclusters point towards smaller values of the DM cross sec-tion ( σ / m (cid:46) . / g ) on these scales (Kaplinghat et al.2014; Elbert et al. 2015), other studies based on stackedmerging clusters (Harvey et al. 2015) and stellar kinemat-ics within cluster cores (Elbert et al. 2015) suggest sometension with these values. Additionally, as shown in Ran-dall et al. (2008), the Bullet Cluster shows how dark mat-ter is consistent with the hypothesis that it is collision-less.Along with other merging clusters, the Bullet Cluster hasprovided an upper limit for the cross section for dark mat-ter at σ / m ≈ / g . Larger values of the cross sectionwould result in a shift between the peak of the dark mat-ter distribution and the centre of the distribution of galaxies(galaxies behave like truly collision-less particles even duringa major cluster merger). Such a shift has not been observed(Clowe et al. 2006) suggesting that the cross section of darkmatter must be σ / m ≈ / g at most. Contrarily to theBullet Cluster, Abell 3827 and Abell 520 exhibit large DM-stellar offsets that, if explained by the SIDM model, willrequire larger DM cross sections probably in tension withother data, such as the core sizes in galaxy clusters.Alternatively, strong gravitational lensing data has beenused to constrain the core size and density in clusters rele-vant for SIDM (Firmani et al. 2000, 2001; Meneghetti et al.2001; Wyithe et al. 2001). More recent studies have usedEinstein radii statistics in galaxy clusters to constrain theDM cross section (Robertson et al. 2019), suggesting that future wide surveys might be able to distinguish betweenCDM and SIDM cosmological models (Despali et al. 2019).In particular, Meneghetti et al. (2001) placed thestrongest constraint on the DM cross section and clustercores by examining the ability of a SIDM halo to produce”extreme” strong lensing arcs (both radial or giant tangen-tial arcs). The authors concluded that their SIDM halo sim-ulated with σ / m > / g is not dense enough (in pro-jection) to produce extreme tangential arcs with length-to-width ratios of l / w (cid:38) . . Based on the ability to produceradial arcs, the constraint found is even more severe, sincetheir SIDM halo simulated with σ / m < . / g was notable to produce radial arcs. As acknowledged in Meneghettiet al. (2001), constraints based on one single halo need tobe addressed with caution due to the variability of the den-sity profiles in SIDM halos. Moreover, this particular halois a SIDM-only simulation (no baryons were included) and,therefore, it did not account for the baryonic density due tothe central galaxy which boosts its lensing efficiency. In par-ticular, one of the major effects of the cooling and the starformation in simulations is to trigger a strong adiabatic con-traction of the baryonic component, leading to (sometimesunrealistically) denser cluster cores. Nevertheless, there havebeen found several examples of galaxy clusters that exhibitradial arcs. In particular, 12 candidate radial arcs wherefound in three of the six clusters examined by (Sand et al.2004, 2005). More recent studies by Newman et al. (2013a,b)have found radial arcs in two of the clusters analyzed bySand et al. (2004), MS2137-23 and Abell 383, with DM cores(of the order of 10 kpc) which are consistent with lensingdata once the baryonic mass is included. Another interest-ing example with several central images detected (and ∼ of its 82 multiple images classified as radial arcs) is MACSJ1206 (Caminha et al. 2017). Interestingly, as first proposedby Molikawa & Hattori (2001), it is practical to use the ra-tio of radial to tangential arcs as a statistical measure of theslope of the dark matter distribution in cluster cores (seealso Sand et al. 2005). In this study, we analyze a total of six massive galaxy clus-ters extracted from two sets of well tested, high-resolutionN-body cosmological simulations with different cosmologi-cal parameters. For each cluster we compare the results ob-tained with the simulation particles set as standard CDMparticles and an identical simulation (i.e, with the same ini-tial condition) where the DM particles are allowed to inter-act with a given probability determined by the value of σ / m (SIDM). One of the galaxy clusters used in this study is extractedfrom the MUltidark SImulations of galaxy Clusters (MU-SIC , Sembolini et al. 2013). In particular, we analyze theMUSIC-MD dataset, which consists of a set of re-simulated http://music.ft.uam.es MNRAS000 , 1–12 (2020)
M cross section from radial arcs L3 clusters extracted from the MultiDark Simulation (MDR1,Prada et al. 2012), a DM-only simulation with parti-cles in a cubic box of h − Gpc side. The MUSIC-MD simula-tion was done using the best-fit cosmological parameters toWMAP7 + BAO + SNI (Komatsu et al. 2011, Ω M = . , Ω b = . , Ω Λ = . , σ = . , n s = . , h = . ).The MUSIC-MD clusters were selected according to a masslimited selection, taking all clusters within the MDR1 sim-ulation with masses above h − M (cid:12) at z = . In total,283 different Lagrangian regions corresponding to spheres of h − Mpc radius were re-simulated with particles cen-tered on the most massive clusters found in the MDR1 sim-ulation. Therefore, the mass resolution for the re-simulatedclusters is 8 times larger than in the parent MDR1 simu-lation, that is m DM = . × h − M (cid:12) for the DM parti-cles and m SPH = . × h − M (cid:12) for the gas particles. TheMUSIC-MD clusters have been performed using the parallel gadget2 Tree-PM code (Springel 2005) with both radiativeand non-radiative (also denoted as adiabatic simulations) forthe SPH particles. By comparing simulations with differenttreatments of baryonic processes, Killedar et al. (2012) foundthat the inclusion of gas cooling, star formation and AGNfeedback together lead to lensing cross sections that are sim-ilar to those obtained from simulations including only DMand non-radiative gas (i.e., adiabatic simulations). For thisreason and in order to avoid any artificial lensing boost dueto the treatment of the baryonic processes, in this study,we only examine the non-radiative run of the MUSIC-MDsimulations.
The Three Hundred project
As a second set of simulated galaxy clusters, we also an-alyzed 324 spherical regions centered on each of the mostmassive clusters ( M > h − M (cid:12) ) identified at z = within the DM-only MultiDark simulation (MDPL2, Klypinet al. 2016). The MDPL2 simulation was performed usingthe cosmological parameters presented by Planck Collabo-ration et al. (2016) ( Ω M = . , Ω b = . , Ω Λ = . , σ = . , n s = . , h = . ). The MDPL2 is a peri-odic cube with a comoving length of h − Gpc containing DM particles. DM particles within the highest reso-lution Lagrangian regions are split into DM and gas par-ticles, according to the assumed cosmological baryon frac-tion. The re-simulated clusters have a mass resolution of m DM = . × h − M (cid:12) for the DM particles and m SPH = . × h − M (cid:12) for the gas particles. The radius of thespherical regions where the re-simulated clusters are cen-tered is h − Mpc and, therefore, much larger than theirvirial radius. The 324 galaxy clusters within
The Three Hun-dred project were re-simulated using two different codessmooth-particle hydrodynamics (SPH) to follow the evolu-tion of the gas component: the gadget-music code (Sem-bolini et al. 2013) and ’modern’ SPH code gadget-x (Mu-rante et al. 2010; Rasia et al. 2015). Both codes are basedon the gravity solver of the gadget3 Tree-PM code (anupdated version of the gadget2 code; Springel 2005), but http://the300-project.org they apply different SPH techniques as well as rather dis-tinct models for the sub-resolution physics (see Cui et al.2018, for more details). Basically, the gadget-music run isperformed using the classic entropy-conserving SPH formu-lation and, therefore, it can be considered as an adiabatic(non-radiative) simulation, while the gadget-x includes animproved SPH scheme with AGN feedback and black holeseeding and growth. In this study, we only examine the non-radiative run ( gadget-music ) of the clusters within the TheThree Hundred project.
The aim of this study is to examine the effects caused by DMself-interactions on cluster-size DM halos extracted from cos-mological numerical simulations. More specifically, we aimat a fiducial comparison of several morphological, dynamicand gravitational lensing features of cluster-scale DM halos.Using a ray-shooting pipeline (see Meneghetti et al.2010, and references therein) we derive the gravitationallensing properties (such as deflection angles, convergence,shear and magnification maps) of all the DM halos withM vir > × h − M (cid:12) within the MUSIC-MD dataset andof the most massive DM halos in the 324 re-simulated re-gions within the gadget-music run of The Three Hun-dred project. For our lensing analysis of the MUSIC-MDdataset, we analyze the simulation snapshots at redshift z = ( . , . , . , . ) for 500 random projectionsalong the line of sight (see Meneghetti et al. 2014, for adetailed description). We select the cluster 11 (hereafterclus11) for being the cluster with the largest effective Ein-stein radius ( θ E ) within the MUSIC-MD dataset, which is agood estimate of the lensing efficiency of a cluster lens. Thisselection is motivated not only by the fact that clus11 hasa strong lens at z = . , but also by the fact that galaxyclusters at redshifts . (cid:46) z (cid:46) . are the most efficientgravitational lenses for sources at redshifts z s (cid:38) . Clus11 inthe MUSIC-MD simulations shows an Einstein radius with θ E (cid:39) arcsec for a cluster redshift of z l = . and asource redshift of z s = . .For the lensing analysis performed over the gadget-music dataset, we use the same ray-shooting pipeline as forthe MUSIC-MD dataset to analyze four different projectionsalong the line of sight (three of them are arbitrary, x -axis, y -axis and z -axis, and one corresponds to the cluster’s ma-jor axis projected along the line of sight) for each of the 324clusters at redshifts z = ( . , . , . , . ) . Generally,we expect a larger strong lensing signal when the clustermass distribution is projected along its major axis. Follow-ing the previous procedure, we select the clusters with thelargest Einstein radii at each given redshift. Some of themare selected not only at one redshift and, therefore, we endup with a total of five galaxy clusters extracted form the gadget-music dataset, which are labelled as clus2, clus7,clus9, clus30 and clus82 .Then, we re-simulate clus11 and clus2, clus7, clus9,clus30 and clus82 for both CDM and SIDM cosmologicalmodels using the N-body/SPH framework GIZMO (Hop-kins 2015) with the same initial conditions and cosmologi-cal parameters as for the MUSIC-MD and The Three Hun-dred simulations, respectively. It is important to note thatwe do not expect significant differences in the gravitational
MNRAS , 1–12 (2020) J. Vega-Ferrero et al. lensing properties of the cluster-size halos here presenteddue to the differences in the cosmological parameters be-tween the clus11 extracted from the MUSIC-MD datasetand the other five halos extracted from the
The Three Hun-dred simulations. We also checked that the results obtainedwith GIZMO for the six mentioned cluster-size halos withinthe CDM model are consistent with the original simulationsperformed with the gadget code. In the SIDM model, DMparticles scatter elastically with each other with a veloc-ity independent cross section of σ / m = cm / g. Clus11 isa cluster-size DM halo that is undergoing a major mergerbetween z = . and z = . , with two DM clumps sepa-rated less than 500 h − kpc at those redshifts. This situationis of particular interest for a detailed study of the DM selfinteractions, given the high rate of interactions that are ex-pected to take place in the collision or merger of massive DMhalos. Therefore, we simulate clus11 at three different red-shifts, z = ( . , . , . ) , for both cosmological models.Moreover, clus2 is a strong lens at the four redshifts ana-lyzed, z = ( . , . , . , . ) ; clus7 is one of the mostefficient lenses at z = . ; clus9 is one of most efficientlenses at z = ( . , . , . ) ; clus30 is one of the mostefficient lenses at z = . ; and finally, clus82 is a stronglens at z = . . To characterize the halo in terms of its mass and shape wecompute its triaxial shape following a similar procedure asthe one described in Despali et al. (2013). First, we assume aspherical overdensity (SO) criterion to find the virial radiusand mass enclosing an average overdensity ∆ c ρ c ( z ) , with ∆ c defined as a certain overdensity value and ρ c ( z ) being thecritical density of the Universe at a given redshift ( z ). Then,for particles found within the SO virial radius, we derive themass tensor M αβ as follows: M αβ = M vir N vir (cid:213) i = m i r i ,α r i ,β , (1)where M vir is the SO virial mass, N vir is the number of par-ticles within the SO virial radius, m i is the particle mass, r i is the position vector of the i th particle and α and β arethe tensor indices ( x , y and z components of the three coor-dinate axes). The mass tensor defined in this way allows todetermine the halo shape for different types of particles (i.e.,with different masses). Therefore, it is possible to computethe halo shape for the DM and GAS distribution separately,or the overall halo shape including both particles types.By diagonalizing the mass tensor M αβ we obtain an ini-tial guess for the shape and orientation of the DM halo.The principal axes of the best-fitting ellipsoid are definedas the square roots of the mass tensor eigenvalues, whiletheir orientations are given by the corresponding eigenvec-tors. Secondly, we identify the particles located within theellipsoid defined by the first set of eigenvalues and eigen-vectors, and which encloses an ellipsoidal overdensity (EO)equal to ∆ c ρ c ( z ) . Then we re-compute the mass tensor forthe new distribution of particles to obtain a new set of eigen-values and eigenvectors that improve the characterization ofthe halo shape. Finally, we repeat the procedure iteratively until a convergence of a . in the axis ratios. We denotethe minor-to-major axis ratio as a / c and the intermediate-to-major axis ratio as b / c , with a < b < c .Along with the redshift at which each cluster has beensimulated, in table 1, we show the spherical and ellipticalvirial masses and the axis ratios at the typical overdensityof ρ ( < r ) = ρ c ( z ) . As found by Despali et al. (2013), differ-ences between M SO and M EO are on average about (with M EO being systematically larger than M SO ). We found upto a 15 % difference for clus2 at z = . for CDM frame-work, evidencing the presence of substructures at large radiior an interaction/merger with another DM halo at the givenredshift.Interactions between DM particles are expected to pro-duce more spherical DM halo configurations than collision-less CDM halos, more importantly towards the cluster cen-ter where the scattering rate of DM particles is larger (Pe-ter et al. 2013). On average, both the minor-to-major andthe intermediate-to-major axis ratios at an overdensity of ρ c ( z ) are systematically larger in the SIDM than in theCDM cosmological model. The median minor-to-major axisratio of SIDM halos is 7% higher than in CDM halos. Whenlooking at higher overdensities, i.e., ρ c ( z ) , this ratioincreases up to a median value of 1.40, confirming thatSIDM simulated cluster-size halos are more rounder thantheir CDM counterparts. As expected, the axis ratios de-rived at a radius corresponding to an overdensity of ρ c ( z ) ,where the effects of interactions between DM particles areless important than at the inner regions of the cluster (i.e., ρ c ( z ) ). There are some exceptions to this, for instancethe SIMD simulation of clus2 at z = . for and overden-sity of ρ c ( z ) is less round, with ( a / c ) = . and ( b / c ) = . , than its CDM counterpart, with ( a / c ) = . and ( b / c ) = . .In figure 1, we show the spherical overdensity profilesof the six massive clusters in our sample up to r = r atfour different redshifts. DM particles self-interactions clearlytransform more cuspy cores, like those formed in CDM sim-ulations, into flat cores. The differences are more obvious atlower redshift, when DM particles in massive halos have ex-perienced a larger number of self-interactions, but also DMhalos have evolved to undergo a significant number of merg-ers. The total mass within a radius equal to r remainsalmost identical in both CDM and SIDM scenarios. Interactions between DM particles affect the structure ofgalaxy clusters, making them more spherical and reducingthe number of substructures around them. The same inter-actions can also transform cuspy cores into flat cores. Sincethe gravitational lensing is proportional to the projectedmass distribution along the line of sight, we expect thesedifferences to arise between the CDM and SIDM halos whencomparing the distribution and formation of strong gravita-tional arcs. Radial arcs can allow us to characterize the sizeand compactness of the cluster lens cores (Narayan & Bartel-mann 1996). Moreover, given the more prominent differencesbetween CDM and the SIDM mass profiles at the center ofthe clusters (as shown in figure 1), radial arcs statistics are
MNRAS , 1–12 (2020)
M cross section from radial arcs L5 Table 1.
Summary of halo properties and redshifts. First column corresponds to the cluster label. Second column denotes the cosmologicalmodel used for the simulation, either CDM or SIDM. Third column shows the redshift at which the different clusters have been analyzed.Fourth and fifth columns show the masses (in h − M (cid:12) units) enclosed within a sphere (SO) and an ellipsoid (EO) of an overdensity of ρ c ( z ) , respectively. Sixth and seventh columns correspond to the intermediate-to-major, ( b / c ) , and minor-to-major, ( a / c ) , axisratios of the best-fitting ellipsoid of an overdensity of ρ c ( z ) , while the eighth and ninth columns correspond to the intermediate-to-major, ( b / c ) , and minor-to-major, ( a / c ) , axis ratios of the best-fitting ellipsoid of an overdensity of ρ c ( z ) . For the masses andaxis ratios we take into account all particle types (i.e., DM+GAS particles).cluster model z M SO M EO ( b / c ) ( a / c ) ( b / c ) ( a / c ) clus2 CDM 0.250 .
88 2 . .
91 2 . .
04 2 . .
05 2 . .
10 2 . .
07 2 . .
82 1 . .
80 1 . .
38 1 . .
36 1 . .
48 1 . .
49 1 . .
36 1 . .
37 1 . .
31 1 . .
32 1 . .
41 1 . .
40 1 . .
31 1 . .
31 1 . .
25 1 . .
24 1 . .
03 1 . .
03 1 . .
03 1 . .
04 1 . in principle more sensitive to possible interactions betweenDM particles.In order to compare the strong lensing properties forboth the CDM and SIDM models, we examine each cluster-size DM halo shown in table 1 for 1,000 random orientationsalong the line of sight to figure out if the halo is super-critical(i.e., if it is able to form critical lines) at each simulatedredshift. To do so we use the consolidated ray-tracing codedescribed in Meneghetti et al. (2010) and follow the lens-ing simulation pipeline assumed in Meneghetti et al. (2017).All particles belonging to each individual halo are projectedalong the line-of-sight on the lens plane, while a bundle oflight-rays is traced through a regular grid of × cov-ering a region of (cid:48)(cid:48) × (cid:48)(cid:48) around the halo center. Thedeflection angle is computed at each light-ray position af-ter accounting for the contributions from all particles on thelens plane within a box volume of . h − Mpc centered at thecluster’s center. Then, the deflection field is used to derivelensing quantities, such as the convergence ( κ ), the shear ( γ ) and the magnification ( µ ). As described in Schneider et al.(1992), the lens critical lines are defined as the curves alongwhich the determinant det A = µ − = µ − t µ − r = ( − κ − γ )( − κ + γ ) = , (2)where A is the Jacobian of the lensing potential, µ is the to-tal magnification, and µ t and µ r are the tangential and ra-dial magnification, respectively. In particular, the tangentialcritical line is defined by the condition µ − t = ( − κ − γ ) = ,while the radial critical line occurs when µ − r = ( − κ + γ ) = .Hereafter, we will use the term effective Einstein radius torefer to the size of the tangential critical line which is definedas follows: θ E ≡ D ( z l ) (cid:114) S π , (3)where S is the area enclosed by the tangential critical lineand D ( z l ) is the angular-diameter distance to the lens plane MNRAS , 1–12 (2020) J. Vega-Ferrero et al.
Figure 1.
Spherical overdensity profiles of the six massive clus-ters in our sample. Each panel corresponds to a different red-shift, and each color indicates a different cluster. Solid and dashedlines correspond to CDM and SIDM cosmological models, respec-tively. The total density within a given radius is normalized bythe critical density of the Universe at the given redshift, ρ c ( z ) .Overdensities are computed up to a radius equal to r , i.e., ρ ( < r ) = ρ c ( z ) . (see Meneghetti et al. 2013 and also Redlich et al. 2012 formore details on the definition of the effective Einstein ra-dius).From these low-resolution maps ( × pix ) for eachof the 1,000 random projections, we select those showingat least one pixel with a magnification µ r > , to en-sure that radial critical curves are formed. Given that forreal lenses κ and γ are positive quantities, the conditionfor forming tangential critical lines is less restrictive thanthe condition for forming radial critical lines (equation 2).Therefore, the formation of radial critical curves implies theformation of tangential critical curves, while the contrary isnot true. Consequently, those cases with µ r > , are la-belled as super-critical (i.e., they show both tangential andradial critical lines).It is important to note that, for some clusters, none ofthe projections is super-critical for the SIDM cosmologicalmodel, while some of them are super-critical for the CDMcosmology. As we describe below, this effect is more evidentfor SIDM simulated cluster-size halos as redshift decreases.DM particles interactions throughout cosmic time will grad-ually dilute the cluster cores. In some massive galaxy clus-ters, which are formed hierarchically from mergers of DMhalos where the number of DM particles interactions is ex-pected to be high, we expect to find cores that are not denseenough to surpass the critical surface mass density for lens-ing and, therefore, not able to produce either radial or tan-gential critical lines. In particular, clus2 at z = . and z = . , clus9 at z = . and clus11 at z = . are not super-critical for any of the 1,000 projections along theline of sight for the SIDM cosmological model, while thereare some projections for the CDM cosmological model inwhich they are super-critical. For those cases, the compar-ison between both cosmological frameworks is not feasibleand, therefore, we will not include them in the subsequentanalysis. For the remaining clusters and redshifts (i.e., thosewith super-critical projections in both cosmological models),we re-computed the deflection field and the rest of lensingquantities at higher resolution, × pix (which trans-lates into an angular resolution of ≈ . arcsec in the lensplane), but keeping the rest of parameters constant.In figure 2, we show the magnification maps in thelens plane for six randomly selected projections of clus2 at z = . : the projections in the top panels correspond tothe CDM simulations, while the ones in the bottom pan-els correspond to the SIDM simulations). There is not a di-rect correspondence between the top and the bottom panels,however differences between both cosmological models areevident. In particular, for the three projections of the CDMmodel both tangential and radial critical curves are visible.In the top-left panel, more than one radial critical curve isvisible within only one tangential critical curve. This config-uration may be explain as two (even three) massive clumpsvery close in projection, but not enough for the two radialcritical lines to merge together (as it is clearly visible inthe top-middle panel). On the other hand, when the sametwo clumps are separate away in projection, it is possibleto find two distinct tangential critical lines (with two radialcritical lines enclosed by them) due to the two clumps con-forming clus2 at z = . . For the projections of the SIDMmodel, two of them show both tangential and radial criticallines (i.e., they are super-critical), while the projection inthe bottom-right panel is not dense enough to produce tan-gential or radial critical lines (i.e., it is not super-critical).In particular, for the SIDM simulation of clus2 at z = . ,only 77 projections from the 1,000 random projections pro-duced are super-critical, while the CDM simulation of thesame cluster produces 361 super-critical projections out ofthe 1,000 random projections produced (see table 2). Addi-tionally, it is clearly visible how the magnification in areasvery close to the radial critical line is higher in SIDM thanin CDM projections (wider redder regions). This is due tothe expected shallower projected mass profile in the innedregions of SIDM simulations compared to CDM simulations.This flattening in the projected mass profiles also leads toless de-magnified areas within the radial critical line. As before mentioned, the size of the Einstein radius is pro-portional to the total area within the tangential critical line.The tangential critical line is defined by the positions in thelens plane that satisfy µ − t = − κ − | γ | = − ¯ κ = , where ¯ κ is defined as the mean convergence ¯ κ = Σ ( < R ) Σ c ( z l , z s ) , (4)with Σ ( < R ) is the total projected mass within a given radius(R) normalized by the critical surface density for lensing, Σ c , which depends on the lens and source redshifts ( z l and MNRAS000
Spherical overdensity profiles of the six massive clus-ters in our sample. Each panel corresponds to a different red-shift, and each color indicates a different cluster. Solid and dashedlines correspond to CDM and SIDM cosmological models, respec-tively. The total density within a given radius is normalized bythe critical density of the Universe at the given redshift, ρ c ( z ) .Overdensities are computed up to a radius equal to r , i.e., ρ ( < r ) = ρ c ( z ) . (see Meneghetti et al. 2013 and also Redlich et al. 2012 formore details on the definition of the effective Einstein ra-dius).From these low-resolution maps ( × pix ) for eachof the 1,000 random projections, we select those showingat least one pixel with a magnification µ r > , to en-sure that radial critical curves are formed. Given that forreal lenses κ and γ are positive quantities, the conditionfor forming tangential critical lines is less restrictive thanthe condition for forming radial critical lines (equation 2).Therefore, the formation of radial critical curves implies theformation of tangential critical curves, while the contrary isnot true. Consequently, those cases with µ r > , are la-belled as super-critical (i.e., they show both tangential andradial critical lines).It is important to note that, for some clusters, none ofthe projections is super-critical for the SIDM cosmologicalmodel, while some of them are super-critical for the CDMcosmology. As we describe below, this effect is more evidentfor SIDM simulated cluster-size halos as redshift decreases.DM particles interactions throughout cosmic time will grad-ually dilute the cluster cores. In some massive galaxy clus-ters, which are formed hierarchically from mergers of DMhalos where the number of DM particles interactions is ex-pected to be high, we expect to find cores that are not denseenough to surpass the critical surface mass density for lens-ing and, therefore, not able to produce either radial or tan-gential critical lines. In particular, clus2 at z = . and z = . , clus9 at z = . and clus11 at z = . are not super-critical for any of the 1,000 projections along theline of sight for the SIDM cosmological model, while thereare some projections for the CDM cosmological model inwhich they are super-critical. For those cases, the compar-ison between both cosmological frameworks is not feasibleand, therefore, we will not include them in the subsequentanalysis. For the remaining clusters and redshifts (i.e., thosewith super-critical projections in both cosmological models),we re-computed the deflection field and the rest of lensingquantities at higher resolution, × pix (which trans-lates into an angular resolution of ≈ . arcsec in the lensplane), but keeping the rest of parameters constant.In figure 2, we show the magnification maps in thelens plane for six randomly selected projections of clus2 at z = . : the projections in the top panels correspond tothe CDM simulations, while the ones in the bottom pan-els correspond to the SIDM simulations). There is not a di-rect correspondence between the top and the bottom panels,however differences between both cosmological models areevident. In particular, for the three projections of the CDMmodel both tangential and radial critical curves are visible.In the top-left panel, more than one radial critical curve isvisible within only one tangential critical curve. This config-uration may be explain as two (even three) massive clumpsvery close in projection, but not enough for the two radialcritical lines to merge together (as it is clearly visible inthe top-middle panel). On the other hand, when the sametwo clumps are separate away in projection, it is possibleto find two distinct tangential critical lines (with two radialcritical lines enclosed by them) due to the two clumps con-forming clus2 at z = . . For the projections of the SIDMmodel, two of them show both tangential and radial criticallines (i.e., they are super-critical), while the projection inthe bottom-right panel is not dense enough to produce tan-gential or radial critical lines (i.e., it is not super-critical).In particular, for the SIDM simulation of clus2 at z = . ,only 77 projections from the 1,000 random projections pro-duced are super-critical, while the CDM simulation of thesame cluster produces 361 super-critical projections out ofthe 1,000 random projections produced (see table 2). Addi-tionally, it is clearly visible how the magnification in areasvery close to the radial critical line is higher in SIDM thanin CDM projections (wider redder regions). This is due tothe expected shallower projected mass profile in the innedregions of SIDM simulations compared to CDM simulations.This flattening in the projected mass profiles also leads toless de-magnified areas within the radial critical line. As before mentioned, the size of the Einstein radius is pro-portional to the total area within the tangential critical line.The tangential critical line is defined by the positions in thelens plane that satisfy µ − t = − κ − | γ | = − ¯ κ = , where ¯ κ is defined as the mean convergence ¯ κ = Σ ( < R ) Σ c ( z l , z s ) , (4)with Σ ( < R ) is the total projected mass within a given radius(R) normalized by the critical surface density for lensing, Σ c , which depends on the lens and source redshifts ( z l and MNRAS000 , 1–12 (2020)
M cross section from radial arcs L7 Figure 2.
Magnification maps in the lens plane for six randomly selected projections of clus2 at z = . (CDM simulations are shownin the top panels, while SIDM simulations are shown in the bottom panels). Color coding denotes the logarithmic of the absolute valueof the magnification in the lens plane from − . < log | µ | < . (and saturated for values of log | µ | > ). Both the tangential (outer)and the radial (inner) critical curves are clearly visible for five projections (which are denoted as super-critical), with the exception ofthe last panel (bottom-right) which shows no critical curves (i.e., it is not super-critical). Moreover, there are some projections showingmore than one radial critical curve (e.g., there are three distinguishable radial critical lines in the top-left panel). z s , respectively). Therefore, the mean convergence inside atangential critical curve must equal unity. Consequently, forcircularly symmetric lenses, the size of the Einstein radius isproportional to the square-root of the total projected masswithin the tangential critical curve.We compute the two-dimensional maps of ¯ κ and derivethe size of the Einstein radius using equation 3 for each of the1,000 random projections along the line of sight produced foreach cluster. Hereafter, we fix the source redshift at z s = . .A direct comparison is only feasible for those clusters andredshifts with at least one super-critical projection in bothcosmologies. A summary of the Einstein radii statistics isshown in table 2. For the CDM simulations, the fraction ofprojections with tangential and radial critical lines for eachcluster is above 73% (clus2 at z = . ). This fraction dropsdown to a 6% (clus9 at z = . ) for the SIDM simulation.The largest Einstein radius is found to be θ E ≈ (cid:48)(cid:48) in theCDM simulation of clus11 at z = . , while its SIDM coun-terpart shows a slightly smaller value of θ E ≈ (cid:48)(cid:48) . The formation and location of radial arcs by galaxy clustersdepends both on the slope of the projected mass profile andon the central density of the lens (Narayan & Bartelmann1996; Meneghetti et al. 2013). Overall, the conclusion holdsthat the presence of radial arcs indicate that clusters havedense cores with fairly flat density profiles (see e.g. Kor-mann et al. 1994). Moreover, the steeper the density profile,the closer to the centre the radial arcs tend to be located(Williams et al. 1999). As shown in figure 1, DM particlesself-interactions favor the formation of flat cores in massiveclusters and, therefore, the formation of large radial arcs ifthey are dense enough (i.e., µ − r = − κ + | γ | ≈ ). Backgroundobjects, such as high- z galaxies, located close to the caustics(i.e., the analogs to the critical lines in the source plane) willappear strongly distorted (and/or multiply imaged) in thelens plane. The caustics can be computed by mapping thecritical lines onto the source plane using the lens equation.In this subsection, we present an analysis on the dis-tribution of large radial arcs produced by the clusters andredshifts with at least one super-critical projection in bothcosmological models. We follow a procedure similar to the MNRAS , 1–12 (2020) J. Vega-Ferrero et al.
Table 2.
Summary of gravitational lensing properties. First, second and third columns indicate the cluster, the cosmological modeland the clusters’ redshift. Fourth column shows the median Einstein radius ( θ E , in arcsec) along with the first and third quantiles.Fifth column corresponds to the largest Einstein radius for each cluster, cosmological model and given redshift. Sixth column indicatesthe number of projections with tangential critical lines (n t ) from a total of 1,000 projections per cluster and redshift. Seventh columnindicates the number of projections with radial critical lines (n r ) from a total of 1,000 projections per cluster and redshift. Eighth columnshows the logarithmic of the median length of the radial arcs (log l , in arcsec) along with the first and third quantiles. Ninth columncorresponds to logarithmic of the length for the largest radial arc (in arcsec). Tenth column shows the logarithmic of the median widthof the radial arcs (log w , in arcsec) along with the first and third quantiles. Eleventh column corresponds to logarithmic of the widthfor the widest radial arc (in arcsec). Last column indicates the number of radial arcs identified for each cluster and redshift in the n r projections with radial critical curves. All the lensing properties have been derived for a source redshift of z s = . .cluster model z θ E ( (cid:48)(cid:48) ) max ( θ E ) n t n r log l ( (cid:48)(cid:48) ) max(log l ) log w ( (cid:48)(cid:48) ) max(log w ) n arcs clus11 CDM 0.300 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . one presented by Meneghetti et al. (2001) to produce lensedimages of background sources. The sources are initially dis-tributed on a regular grid in the source plane. Subsequently,the spatial density of the sources iteratively increases nearthe caustics in order to obtain a larger number of themplaced on regions where the probability of being stronglydistorted is higher. Sources are assumed to be elliptical, withaxis ratios uniformly distributed in the interval ( . , ) , andwith an area equal to a circle of 1 arcsec diameter. The reso-lution of . arcsec of the gravitational lensing maps is largeenough to properly resolve the lensed images of the back-ground sources. Then, we classify as strong radial arcs thoselensed images of background sources containing at least onepixel for which µ r > and µ r / µ t > . These choices weremade to avoid miss-classifications of radial arcs, such as arcsthat can be considered both radial and tangential at thesame time as the radial and critical curves overlap. Finally,we characterize the lensed images classified as radial arcs interms of their lengths and widths following the procedureintroduced by Bartelmann & Weiss (1994) and described in Meneghetti et al. (2013). The lengths are defined as themaximum length of the circular segment passing throughthe lensed image, while the widths are found by fitting theimage with several geometric forms (such as ellipses, circles,rectangles and rings).Given that SIDM cluster-size halos generally have shal-lower profiles than CDM ones towards their inner parts, weexpect them to produce more elongated radial arcs. How-ever, as we showed in section 2.3.1, CDM cluster-size halosare more elliptical at their inner parts, compensating forthe shallower mass profiles of SIDM halos and explainingwhy the lengths of radial arcs are comparable in both cos-mological models. In figure 3, we show the distributions inlengths ( l ), widths ( w ) and length-to-width ratios ( l / w ) forthe radial arcs in each cluster and redshift. Each panel in-cludes all the radial arcs identified in each of the projectionsalong the line of sight having radial critical lines. The totalnumber of radial arcs for each cluster, redshift and cosmo-logical model along with a summary with their statistics oflengths and widths are shown in table 2. The number of MNRAS000
Summary of gravitational lensing properties. First, second and third columns indicate the cluster, the cosmological modeland the clusters’ redshift. Fourth column shows the median Einstein radius ( θ E , in arcsec) along with the first and third quantiles.Fifth column corresponds to the largest Einstein radius for each cluster, cosmological model and given redshift. Sixth column indicatesthe number of projections with tangential critical lines (n t ) from a total of 1,000 projections per cluster and redshift. Seventh columnindicates the number of projections with radial critical lines (n r ) from a total of 1,000 projections per cluster and redshift. Eighth columnshows the logarithmic of the median length of the radial arcs (log l , in arcsec) along with the first and third quantiles. Ninth columncorresponds to logarithmic of the length for the largest radial arc (in arcsec). Tenth column shows the logarithmic of the median widthof the radial arcs (log w , in arcsec) along with the first and third quantiles. Eleventh column corresponds to logarithmic of the widthfor the widest radial arc (in arcsec). Last column indicates the number of radial arcs identified for each cluster and redshift in the n r projections with radial critical curves. All the lensing properties have been derived for a source redshift of z s = . .cluster model z θ E ( (cid:48)(cid:48) ) max ( θ E ) n t n r log l ( (cid:48)(cid:48) ) max(log l ) log w ( (cid:48)(cid:48) ) max(log w ) n arcs clus11 CDM 0.300 . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . . + . − . one presented by Meneghetti et al. (2001) to produce lensedimages of background sources. The sources are initially dis-tributed on a regular grid in the source plane. Subsequently,the spatial density of the sources iteratively increases nearthe caustics in order to obtain a larger number of themplaced on regions where the probability of being stronglydistorted is higher. Sources are assumed to be elliptical, withaxis ratios uniformly distributed in the interval ( . , ) , andwith an area equal to a circle of 1 arcsec diameter. The reso-lution of . arcsec of the gravitational lensing maps is largeenough to properly resolve the lensed images of the back-ground sources. Then, we classify as strong radial arcs thoselensed images of background sources containing at least onepixel for which µ r > and µ r / µ t > . These choices weremade to avoid miss-classifications of radial arcs, such as arcsthat can be considered both radial and tangential at thesame time as the radial and critical curves overlap. Finally,we characterize the lensed images classified as radial arcs interms of their lengths and widths following the procedureintroduced by Bartelmann & Weiss (1994) and described in Meneghetti et al. (2013). The lengths are defined as themaximum length of the circular segment passing throughthe lensed image, while the widths are found by fitting theimage with several geometric forms (such as ellipses, circles,rectangles and rings).Given that SIDM cluster-size halos generally have shal-lower profiles than CDM ones towards their inner parts, weexpect them to produce more elongated radial arcs. How-ever, as we showed in section 2.3.1, CDM cluster-size halosare more elliptical at their inner parts, compensating forthe shallower mass profiles of SIDM halos and explainingwhy the lengths of radial arcs are comparable in both cos-mological models. In figure 3, we show the distributions inlengths ( l ), widths ( w ) and length-to-width ratios ( l / w ) forthe radial arcs in each cluster and redshift. Each panel in-cludes all the radial arcs identified in each of the projectionsalong the line of sight having radial critical lines. The totalnumber of radial arcs for each cluster, redshift and cosmo-logical model along with a summary with their statistics oflengths and widths are shown in table 2. The number of MNRAS000 , 1–12 (2020)
M cross section from radial arcs L9 projections with radial critical lines and the total numberof radial arcs are systematically larger for the CDM thanfor the SIDM cosmological models. Nevertheless, the overalltrend is that the distribution of lengths for both cosmologi-cal models is equivalent within errors for the six cluster-sizehalos and each corresponding redshift here presented. Themedian values for the lengths of radial arcs are slightly largerfor the SIDM than for the CDM model. The largest radialarc, l ≈ arcsec, is produced by one of the projections ofclus11 at z = . simulated with CDM cosmology. Thisextremely elongated radial arcs is the result of the merger offive multiple images into a single one due to the superposi-tion of several radial caustics along the source. In particular,for the projection of clus11 producing this particular radialarc it is possible to identify three different radial critical lineswhich are similar (in the lens plane) to the top-left panel infigure 2. Although radial arcs are more difficult to be char-acterized than tangential arcs since they appear close to thecluster centers and, therefore, their light is usually screenedby the BCG light, there have already been identified largeradial arcs in observed clusters, such as the radial arc foundby Caminha et al. (2017) in MACS J1206 (system 4b withan approximately length of 10 arcsec).Contrarily to the findings of Meneghetti et al. (2001)for only one simulated cluster-sized halo, which constrained σ / m < . cm / g for it to produce extreme strong lens-ing arcs, each of the clusters here presented is able to pro-duce giant radial arcs (also large Einstein radii) assuming σ / m = cm / g within the SIDM cosmological model. Oneof the reasons of the simulated halo in Meneghetti et al.(2001) for not being dense enough to produce strong lens-ing features may be the redshift at which the halo has beenanalyzed, z = . . As mentioned above, we find that someof the clusters at lower redshifts (i.e., z = . for clus2,clus9 and clus11, and z = . for clus2) are not massiveand dense enough to produce tangential and radial criti-cal curves. Depending on the mass accretion and mergerhistories of each cluster-size halo, as they evolve with red-shift, DM self-interactions lead to a dilution of the DM coresand, consequently, prevents them from being strong gravita-tional lenses (i.e., super-critical). However, for all the halosanalyzed at higher redshifts ( z = . and z = . ), wefind that both the CDM and the SIDM cosmological modelslead to the formation of extremely large Einstein radii andgiant radial arcs. When comparing at the distributions ofthe width of radial arcs in the two cosmological models, thesituation is slightly different. Radial arcs produced by thesame cluster-size halo are on average wider for the SIDM,with median values in the range log w = (− . , . ) , thanfor the CDM cosmological model, with median values in therange log w = (− . , . ) . In order to highlight the differences between the radial arcsformed by the different clusters in the CDM and SIDM cos-mological models, we also derived the convergence ( κ ), theshear ( γ ) and, as a combination of both, the radial magnifi-cation ( µ − r = − κ + | γ | ) for each radial arc. The values of κ and γ are computed in the central pixel of the lensed imagesin the lens plane for each of the detected radial arcs.In figure 4, we show the distributions in κ , γ and log µ r for all the radial arcs identified in the six clusters and thedifferent redshifts (the total number of radial arcs is equal tothe sum of the column n arcs in table 2 for each cosmologicalmodel). The radial arcs formed by clusters simulated withSIDM cosmological model show values of κ systematicallysmaller than those produced by the same clusters simulatedwithin the CDM framework. The median values of the con-vergence (along with the 1st and 3rd quartiles) for the radialarcs are ¯ κ Λ CDM (cid:39) . + . − . and ¯ κ SIDM (cid:39) . + . − . for CDMand SIDM cosmological models, respectively. The maximumvalues of the convergence for CDM and SIDM cosmologicalmodels are approximately 1.96 and 1.59, respectively. Whenlooking at the values of the shear, radial arcs identified inclusters simulated with SIDM cosmological model tend tobe formed in regions with smaller values of γ . More pre-cisely, the median values of the shear (along with the 1st and3rd quartiles) for the radial arcs are ¯ γ Λ CDM (cid:39) . + . − . and ¯ γ SIDM (cid:39) . + . − . for CDM and SIDM cosmological mod-els, respectively. Finally, the pdfs of the radial magnification( µ r ) are consistent in both cosmological models, with a pref-erence for slightly higher values ( log µ r > . ) in the SIDMframework. We compare the statistics and morphology of extremelylarge radial arcs produced by a set of six simulated cluster-size DM halos. The simulated galaxy clusters of study are se-lected for being the most efficient gravitational lenses foundin two datasets of re-simulated galaxy clusters with slightlydifferent cosmological parameters but the same simulatedcubic box volume of h − Gpc side. The six selected galaxyclusters are then simulated using the N-body/SPH frame-work GIZMO assuming a CDM and a SIDM cosmologicalmodel with a with a velocity independent cross section forthe DM particles of σ / m = cm / g. Finally, we study thegravitational lensing properties for 1,000 random orienta-tions along the line sight of each simulated cluster-size halousing a ray-tracing pipeline by selecting those producingboth radial and tangential critical lines for a given sourceredshift ( z s = . ). To produce the lensing images of back-ground sources by these simulated halos we populate thesource plane behind them with elliptical sources. We then se-lect those lensed images that are classify as strong radial arcsaccording to their radial and tangential magnifications, andderive the probability distributions of their lengths, widthsand radial magnifications.By looking at the overall properties of the cluster-sizehalos (see table 1), we found that the axis ratios measuredat an overdensity of ρ c ( z ) are systematically (but onlyslightly) larger in SIDM than in CDM simulations of thesame cluster-size halos. If we refer to the same axis ratiosbut measured at an inner radius (e.g., at an overdensityof ρ c ( z ) ) the differences are significantly larger, witha value of 1.4 for the median of the ratio of the minor-to-major axis ratios of SIDM and CDM, confirming thatSIDM simulated cluster-size halos are on average rounderthan their CDM counterparts at an overdensity of ρ c ( z ) .As expected, DM particles self-interactions clearly transformcuspy cores (like those formed in CDM simulations) into flat MNRAS , 1–12 (2020) J. Vega-Ferrero et al.
Figure 3.
Probability distribution functions of lengths ( l , left-hand panels), widths ( w , middle panels) and length-to-width ratios ( l / w ,right-hand panels) in log-scale for the selected radial arcs. Each panel corresponds to one cluster and redshift (indicated by the label inthe top-left corner of each panel). Grey and red histograms show the results for CDM and SIDM cosmological models, respectively, whilethe black and red arrows correspond to the maximum value found in each case. MNRAS000
Probability distribution functions of lengths ( l , left-hand panels), widths ( w , middle panels) and length-to-width ratios ( l / w ,right-hand panels) in log-scale for the selected radial arcs. Each panel corresponds to one cluster and redshift (indicated by the label inthe top-left corner of each panel). Grey and red histograms show the results for CDM and SIDM cosmological models, respectively, whilethe black and red arrows correspond to the maximum value found in each case. MNRAS000 , 1–12 (2020) M cross section from radial arcs
L11
Figure 4.
Probability distribution functions of convergence ( κ ,top panel), shear ( γ , middle panel) and the logarithmic of theradial magnification ( log µ r , bottom panel) for the selected radialarcs in all clusters and redshifts. Grey and red histograms showthe results for CDM and SIDM cosmological models, respectively. cores (as found in SIDM simulations), as is clearly seen infigure 1.The gravitational lensing properties are derived by ex-amining 1,000 random projections for each cluster, redshiftand cosmological model, and assuming a source redshift z s = . . We define as super-critical the projections forwhich both tangential and radial critical lines are present.We found none super-critical projections for any of theSIDM simulations at z = . , indicating that some mas-sive galaxy clusters for which the number of DM particlesinteractions is expected to be high are not able to produceeither radial or tangential critical lines. Contrarily, even forthe lower redshift ( z = . ), at least one of the projectionsof the CDM simulations is super-critical. To illustrate thiseffect we show the magnification maps for six random pro-jections of clus2 at z = . in figure 2, where (depending onthe projection) one, two and even three radial critical curvescould be detected. Additionally, the values of the magnifica-tion along and within the radial critical lines for the SIDMsimulation are larger than in the CDM simulations.The clusters and redshifts for which super-critical pro-jections in both cosmological models are found are then sim- ulated with higher resolution ( ≈ . arcsec) to study indetail the Einstein radii and the radial arc statistics. Theoverall statistics of the gravitational lensing properties areshown in table 2. The main conclusions are summarized asfollows: • For the CDM simulations, the fraction of projectionswith tangential and radial critical lines for each cluster isabove 73%. This fraction drops down to a 6% for the SIDMsimulation. • The largest Einstein radius is found to be θ E ≈ (cid:48)(cid:48) inthe CDM simulation of clus11 at z = . , while its SIDMcounterpart shows a slightly smaller value of θ E ≈ (cid:48)(cid:48) . • The number of projections with radial critical lines andthe total number of radial arcs are systematically larger forthe CDM than for the SIDM cosmological models. • The distribution of lengths for both cosmological mod-els is equivalent within errors for the six cluster-size halosand each corresponding redshift here presented, with me-dian values for the lengths of radial arcs slightly larger forthe SIDM than for the CDM model. • The largest radial arc, l ≈ arcsec, is produced byone of the projections of clus11 at z = . simulated withCDM cosmology. • Each of the clusters here presented is able to producegiant radial arcs (also large Einstein radii) assuming σ / m = cm / g within the SIDM cosmological model. • We find that some of the clusters at lower redshifts (i.e., z = . for clus2, clus9 and clus11, and z = . for clus2)are not massive and dense enough to produce tangential andradial critical curves. However, for all the halos analyzed athigher redshifts ( z = . and z = . ), we find that boththe CDM and the SIDM cosmological models lead to theformation of extremely large Einstein radii and giant radialarcs. • Median lengths of radial arcs are comparable in bothcosmological models. • Radial arcs produced by the same cluster-size halo areon average wider for the SIDM, with median values in therange log w = (− . , . ) , than for the CDM cosmologicalmodel, with median values in the range log w = (− . , . ) . • Radial arcs formed by clusters simulated with SIDMcosmological model show values of κ systematically smallerthan those produced by the same clusters simulated withinthe CDM framework. The median values of the convergence(along with the 1st and 3rd quartiles) for the radial arcs are ¯ κ Λ CDM (cid:39) . + . − . and ¯ κ SIDM (cid:39) . + . − . for CDM and SIDMcosmological models, respectively. The maximum values ofthe convergence for CDM and SIDM cosmological modelsare approximately 1.96 and 1.59, respectively. • Radial arcs identified in clusters simulated with SIDMcosmological model tend to be formed in regions with smallervalues of γ . More precisely, the median values of the shear(along with the 1st and 3rd quartiles) for the radial arcsare ¯ γ Λ CDM (cid:39) . + . − . and ¯ γ SIDM (cid:39) . + . − . for CDM andSIDM cosmological models, respectively. • Finally, the pdfs of the radial magnification ( µ r ) areconsistent in both cosmological models, with a preference forslightly higher values ( log µ r > . ) in the SIDM cosmologicalmodel. MNRAS , 1–12 (2020) J. Vega-Ferrero et al.
ACKNOWLEDGMENTS
The authors would like to thank the “Red Espa˜nola deSupercomputaci´on” for granting us computing time at theMareNostrum Supercomputer of the BSC-CNS where someof the cluster simulations presented in this work havebeen performed. GY acknowledges financial support by theMINECO/FEDER under project grant AYA2015-63810-Pand MICIU/FEDER under project grant PGC2018-094975-C21. WC acknowledges the supported by the European Re-search Council under grant number 670193. MM acknowl-edges support from PRIN-MIUR “Cosmology and Funda-mental Physics: illuminating the dark universe with Eu-clid”, and from ASI through contract Euclid Phase D1.05.04.37.01.
REFERENCES
Bartelmann M., Weiss A., 1994, A&A, 287, 1Caminha G. B., et al., 2017, A&A, 607, A93Clowe D., Bradaˇc M., Gonzalez A. H., Markevitch M., RandallS. W., Jones C., Zaritsky D., 2006, ApJ, 648, L109Cui W., et al., 2018, MNRAS, 480, 2898Despali G., Tormen G., Sheth R. K., 2013, MNRAS, 431, 1143Despali G., Sparre M., Vegetti S., Vogelsberger M., Zavala J.,Marinacci F., 2019, MNRAS, 484, 4563Dooley G. A., Peter A. H. G., Vogelsberger M., Zavala J., FrebelA., 2016, MNRAS, 461, 710Elbert O. D., Bullock J. S., Garrison-Kimmel S., Rocha M.,O˜norbe J., Peter A. H. G., 2015, MNRAS, 453, 29Firmani C., D’Onghia E., Avila-Reese V., Chincarini G., Hern´an-dez X., 2000, MNRAS, 315, L29Firmani C., D’Onghia E., Chincarini G., Hern´andez X., Avila-Reese V., 2001, MNRAS, 321, 713Fry A. B., et al., 2015, MNRAS, 452, 1468Harvey D., Massey R., Kitching T., Taylor A., Tittley E., 2015,Science, 347, 1462Hopkins P. F., 2015, MNRAS, 450, 53Kaplinghat M., Keeley R. E., Linden T., Yu H.-B., 2014, PhysicalReview Letters, 113, 021302Killedar M., Borgani S., Meneghetti M., Dolag K., Fabjan D.,Tornatore L., 2012, MNRAS, 427, 533Klypin A., Yepes G., Gottl¨ober S., Prada F., Heß S., 2016, MN-RAS, 457, 4340Komatsu E., et al., 2011, ApJS, 192, 18Kormann R., Schneider P., Bartelmann M., 1994, A&A, 284, 285Meneghetti M., Yoshida N., Bartelmann M., Moscardini L.,Springel V., Tormen G., White S. D. M., 2001, MNRAS, 325,435Meneghetti M., Fedeli C., Pace F., Gottl¨ober S., Yepes G., 2010,A&A, 519, A90Meneghetti M., Bartelmann M., Dahle H., Limousin M., 2013,Space Sci. Rev., 177, 31Meneghetti M., et al., 2014, ApJ, 797, 34Meneghetti M., et al., 2017, MNRAS, 472, 3177Molikawa K., Hattori M., 2001, ApJ, 559, 544Murante G., Monaco P., Giovalli M., Borgani S., Diaferio A., 2010,MNRAS, 405, 1491Narayan R., Bartelmann M., 1996, arXiv Astrophysics e-prints,Newman A. B., Treu T., Ellis R. S., Sand D. J., Nipoti C., RichardJ., Jullo E., 2013a, ApJ, 765, 24Newman A. B., Treu T., Ellis R. S., Sand D. J., 2013b, ApJ, 765,25Peter A. H. G., Rocha M., Bullock J. S., Kaplinghat M., 2013,MNRAS, 430, 105Planck Collaboration et al., 2016, A&A, 594, A13 Prada F., Klypin A. A., Cuesta A. J., Betancort-Rijo J. E., Pri-mack J., 2012, MNRAS, 423, 3018Profumo S., 2013, arXiv e-prints,Randall S. W., Markevitch M., Clowe D., Gonzalez A. H., BradaˇcM., 2008, ApJ, 679, 1173Rasia E., et al., 2015, ApJ, 813, L17Redlich M., Bartelmann M., Waizmann J.-C., Fedeli C., 2012,A&A, 547, A66Robertson A., Harvey D., Massey R., Eke V., McCarthy I. G.,Jauzac M., Li B., Schaye J., 2019, MNRAS, 488, 3646Rocha M., Peter A. H. G., Bullock J. S., Kaplinghat M., Garrison-Kimmel S., O˜norbe J., Moustakas L. A., 2013, MNRAS, 430,81Sand D. J., Treu T., Smith G. P., Ellis R. S., 2004, ApJ, 604, 88Sand D. J., Treu T., Ellis R. S., Smith G. P., 2005, ApJ, 627, 32Schneider P., Ehlers J., Falco E. E., 1992, Gravitational LensesSembolini F., Yepes G., De Petris M., Gottl¨ober S., Lamagna L.,Comis B., 2013, MNRAS, 429, 323Spergel D. N., Steinhardt P. J., 2000, Physical Review Letters,84, 3760Springel V., 2005, MNRAS, 364, 1105Tulin S., Yu H.-B., 2018, Phys. Rep., 730, 1Vogelsberger M., Zavala J., Loeb A., 2012, MNRAS, 423, 3740Vogelsberger M., et al., 2014, MNRAS, 444, 1518Williams L. L. R., Navarro J. F., Bartelmann M., 1999, ApJ, 527,535Wittman D., Golovich N., Dawson W. A., 2018, ApJ, 869, 104Wyithe J. S. B., Turner E. L., Spergel D. N., 2001, ApJ, 555, 504Zavala J., Vogelsberger M., Walker M. G., 2013, MNRAS, 431,L20This paper has been typeset from a TEX/L A TEX file prepared bythe author. MNRAS000