Self-Interacting Dark Matter and the Excess of Small-Scale Gravitational Lenses
SSelf-Interacting Dark Matter and the Excess of Small-Scale Gravitational Lenses
Daneng Yang ID ∗ and Hai-Bo Yu ID † Department of Physics, Tsinghua University, Beijing 100084, China Department of Physics and Astronomy, University of California, Riverside, California 92521, USA (Dated: February 5, 2021)Recently, Meneghetti et al. reported an excess of small-scale gravitational lenses in galaxy clus-ters, compared to simulations of standard cold dark matter (CDM). We propose a self-interactingdark matter (SIDM) scenario, where a population of subhalos in the clusters experiences gravother-mal collapse. Using controlled N-body simulations, we show the presence of early-type galaxiesin substructures accelerates gravothermal evolution and a collapsed SIDM subhalo has a steeperdensity profile than its CDM counterpart, leading to a larger radial galaxy-galaxy strong lensingcross section and more lens images, in better agreement with the observations. Our results indicatethat strong gravitational lensing can provide a promising test of the self-interacting nature of darkmatter.
Introduction.
Strong gravitational lensing is charac-terized by the existence of giant arcs, rings, and multi-ple images caused by the deflection of lights by massiveforeground galaxies, groups, or galaxy clusters [1, 2]. Itprovides a powerful tool for testing cosmological mod-els [3, 4], determining the mass distribution of clus-ters [5–7], probing substructures [8–16] and dark mat-ter properties [17–22]. Recently, Meneghetti et al. re-ported that observed substructures in galaxy clusters aremore efficient lenses than those predicted in simulationsof standard cold dark matter (CDM) [23], indicating thatthe former are more dense and compact. Ref. [24] fur-ther confirms it. Other studies also show strong lens-ing clusters contain more substructures with high maxi-mum circular velocities than predicted in CDM simula-tions [12, 25].In this
Letter , we propose a self-interacting dark matter(SIDM) [26–28] scenario to explain the observed excess.At late stages of gravothermal evolution, an SIDM halocould experience instability and collapse, resulting in asteep density profile [29–32]. We design N-body simu-lations to model the MACS J1206.2-0847 (MACSJ1206)cluster, one of the examples studied in [23], and constructfour benchmark cases, covering a representative range ofhalo concentration. The presence of early-type galax-ies in the substructures could significantly accelerate theonset of the collapse. For all benchmarks, our simulatedSIDM subhalos experience collapse after 6 Gyr of tidalevolution and become more dense than their CDM coun-terparts, assuming a self-scattering cross section per massof σ/m = 1 cm / g, which is relatively conservative [28].We further model strong lensing observables and com-pute galaxy-galaxy strong lensing (GGSL) cross sectionsfor the benchmarks. In SIDM, the radial caustics dom-inates over the tangential one, and the predicted radialGGSL cross section is larger than that in CDM, withdetails depending on the source redshift and initial haloconcentration. We will also show mock lensing imagesand discuss their implications for future tests. Our re-sults indicate that SIDM is promising for explaining the excess of gravitational lenses in galaxy clusters. Intrigu-ingly, SIDM may also explain diverse dark matter dis-tributions in other galactic systems [27, 33–44] and theorigin of supermassive black holes [45–48]. Modeling the cluster system.
We model the hostcluster using a static spherical potential characterizedby a Navarro-Frenk-White (NFW) density profile [49]and fix its corresponding scale density and radius as ρ s = 1 . × M (cid:12) / kpc and r s = 442 kpc, respec-tively. It well reproduces the projected total mass pro-file of MACSJ1206 [50–52]. We set the apocenter ofour simulated substructures to be 400 kpc, with a tan-gential velocity of 1000 km / s. During the tidal evolu-tion, their distance to the host center oscillates in therange ∼ O (10 ) M (cid:12) . The strong lensing analysis in [23] fo-cuses on substructures within 15% of the virial radius ofthe host cluster, which is 300 kpc, and most of them havea mass in the range 10 –10 M (cid:12) . Thus our simulatedcluster system well represents those studied in [23].For the subhalos, we use an NFW profile to model theirinitial dark matter distribution. We fix the initial virialhalo mass to be M = 3 × M (cid:12) , and choose fourbenchmark values for the concentration, i.e., c = 7 . .
65, 12 . .
0, corresponding to 0 σ , 1 σ , 2 σ and3 σ higher than the cosmological median at z = 0 [53],respectively. For each of the benchmarks, we converttheir ( M , c ) to ( ρ s , r s ) to specify the initial NFWdensity profile in our simulations. Note for a given setof ρ s and r s , the interpretation of c depends on theredshift. Consider z = 2, at which infall is expected tooccur, the concentration of the benchmarks, from low tohigh, is − . σ , − . σ , +0 . σ , and +1 . σ away fromthe median ( z = 2) [53], which are representative.We fix the initial stellar mass as M (cid:63) = 6 × M (cid:12) ,expected from the stellar-to-halo mass relation [54], andmodel its distribution with a truncated singular isother-mal profile as in [23], ρ (cid:63) ( r ) = ρ r / [ r ( r + r )], where a r X i v : . [ a s t r o - ph . GA ] F e b FIG. 1.
Left:
Evolution of V max – M sub for the simulated SIDM (solid thick) and CDM (solid thin) benchmarks with differentinitial subhalo concentrations, denoted using deviations from the cosmological median at z = 0, i.e., +3 σ (magenta), +2 σ (red), +1 σ (green) and +0 σ (blue). The arrows denote the direction of the evolution and the final snapshot is at t = 6 Gyr.CDM simulations without including the star component are also shown (dotted). For comparison, the average relation fromstrong lensing observations (black dashed) [23], the range (gray band) and best-fit model relation (gray dashed) from theirCDM simulations are displayed. Right:
SIDM radial (solid) and tangential (dashed) GGSL cross sections vs source redshift,normalized to their corresponding CDM counterparts for t = 6 Gyr. The color scheme is the same as the one in the left panel.The simulated CDM substructures without stars have a low surface density and their lensing effect is negligible. ρ is the density normalization factor and r cut is the cut-off radius. This is consistent with observations of early-type galaxies [55–58]. We take r cut = 6 .
23 kpc followingthe size-mass relation [59], and ρ = 1 . × M (cid:12) / kpc .We use live particles for the subhalo and stellar compo-nents and perform both SIDM and CDM simulations.For the former, we choose σ/m = 1 cm / g, approxi-mately the lower limit that could explain observationson galactic scales [28]. For comparison, we also performCDM simulations without including stars. As we willshow that the stellar component is important in produc-ing strong lensing observables.We use the public GADGET-2 code [60, 61], and ex-tend it with a module modeling dark matter self-interactions [32, 43], which has been validated in bothgravothermal expansion and collapse regimes with re-sults from [30, 31, 40, 46]. We use the code
SpherIC [62]to generate initial conditions for the simulated substruc-tures. The mass of the simulated particle is 10 M (cid:12) for both subhalo and stellar components, and the soften-ing length is 0 . Gravothermal collapse.
The left panel of Fig. 1 showsthe evolution of the maximum circular velocity V max vsthe total substructure mass M sub for the benchmarkswith c = 16 . . .
65 (green) and7 .
49 (blue) from our SIDM (solid thick) and CDM (solid thin) simulations, where we include both subhalo andstellar components as in [23]. The arrow on each curvedenotes the direction of the evolution. The subhalos losethe majority of their mass after tidal evolution, while thestellar mass is only reduced by an O (1) factor. Our fi-nal total stellar and subhalo masses are consistent withthose of cluster substructures from the Illustris simula-tions [66, 67]. For comparison, our CDM simulationswithout including stars are shown (dotted).The maximum velocities of the CDM substructures de-crease continuously, aside from oscillatory features due totidal interactions. For those with stars, the final V max val-ues are close to the high end of the range predicted in cos-mological hydrodynamical simulations [23] (gray band),but are all below the average value from the strong lens-ing observations (black dashed). We also see that the V max values predicted in our CDM simulations withoutstars are still within the gray band. It implies that alarge population of simulated substructures in [23] hasdiffuse baryon distributions and high dark matter frac-tions. On the other hand, our controlled simulations takean observed baryon distribution, which is compact, as aninput. We will come back to this point later.The simulated SIDM substructures follow a similartrend for most of the evolution time, but their V max val-ues spike toward the measured ones at late stages, asgravothermal collapse occurs and their central densitiesincrease. At t = 6 Gyr, all four SIDM benchmarks, eventhe one with a median concentration ( z = 0), are denserthan their CDM counterparts with stars, resulting in bet-ter agreement with the lensing observations [23]. The col-lapse occurs earlier if c is higher, leading to a higherdensity at 6 Gyr, as its timescale is extremely sensitiveto the concentration [31]. We find the presence of stellarparticles deepens potential and accelerates gravothermalevolution, as in the isolated case [48, 68, 69]. Withoutstars, the collapse would not occur within 6 Gyr unless c is 6 σ higher than the median, and a subhalo withmedian c would be nearly destroyed [43]. In the clus-ter environment, tidal stripping could also speed up theonset of gravothermal collapse [70–74]. Strong lensing observables.
To further see implica-tions for strong lensing observations, we compute GGSLcross sections for the simulated substructures. We adoptthin-lens approximation and project the mass distribu-tion of the host cluster and substructure, assumed to bespherical, onto the lens plane, which is perpendicular tothe line of sight. The distance between the substructureand the host center is fixed to be 300 kpc. We denotethe angular positions as θ and β on the lens and sourceplanes, respectively. It is convenient to introduce effec-tive lensing potential as [1, 2]:Ψ( θ ) ≡ π (cid:90) d θ ln | θ − θ (cid:48) | κ ( θ (cid:48) ) , where κ ( θ ) = Σ( D L θ ) / Σ cr is the scaled projected densityand Σ cr = c D S / πGD L D LS is the critical density. D L and D S are lens and source angular diameter distances,respectively, and D LS the distance between the two. Wecalculate these quantities in a flat universe with matterenergy density Ω m = 0 . h = 0 .
671 [75]. The lensequation is β = θ − D LS ˆ α /D S , where ˆ α is the deflectionangle.For each simulated substructure plus the host cluster,we determine the lensing potential by solving the Poissonequation ∇ θ Ψ( θ ) = 2 κ ( θ ), where we use the fast Fouriertransformation method. After obtaining Ψ, we calcu-late the shear matrix as A ≡ ∂β i /∂θ j = ( δ ij − Ψ ij ) andthe pseudo-vector shear γ = (cid:112) [(Ψ − Ψ ) / + Ψ ,where Ψ ij ≡ ∂ Ψ /∂θ i ∂θ j and i, j are indices of thetwo spatial coordinates. The tangential and radial crit-ical lines are contours of λ t = 1 − κ − γ = 0 and λ r = 1 − κ + γ = 0, respectively. We obtain their corre-sponding caustic lines by mapping them onto the sourceplane using the lens equation. We compute radial andtangential GGSL cross sections defined as the area en-closed by the secondary caustic [76].The right panel of Fig. 1 shows ratios of SIDM to CDMradial (solid) and tangential (dashed) GGSL cross sec-tions as a function of the source redshift z s . Apparently,the SIDM substructures have larger radial cross sectionsthan their CDM counterparts, by a factor of ∼ z s (cid:38)
1, and the significance increases with the concen-tration. The tangential cross sections are comparable forboth cases. For the CDM substructures without stars,
FIG. 2. Total (solid) and dark matter (dashed) density pro-files for the SIDM (red) and CDM (gray) benchmarks with c = 12 . σ ) at t = 6 Gyr. The red and gray arrowsdenote their Einstein radii, respectively, assuming z s = 3.The dark matter density profile from CDM simulations with-out stars is also shown (dotted); its lensing effect is negligible. Insert:
Evolution of V max – r max for the benchmarks with stars. their surface density is low and the lensing effect is neg-ligible.Ref. [23] reports the measured GGSL cross section thatsums over contributions from individual substructures ismore than one order magnitude larger than the one pre-dicted in CDM simulations. We note that the number ofthe observed secondary caustics is a factor of 3 largerthan that from their simulations. Thus the requiredboost factor is (cid:38) Density profiles.
We take the benchmark with c =12 . σ ) and perform a detailed case study. In Fig. 2,we show its dark matter (dashed) and total (solid) den-sity profiles with (red) and without (gray) dark matterself-interactions. After 6 Gyr of evolution, the collapse FIG. 3.
Left:
Radial (solid) and tangential (dashed) caustics for the simulated SIDM and CDM substructures with c = 12 . σ ), together with four mock sources (circles). The corresponding critical lines and lens images are shown in the Middle and
Right panels, respectively, assuming z s = 3. For the lens redshift z l = 0 .
439 of MACSJ1206, one arcsec corresponds to 5 .
76 kpc. leads to an overdense region within 1 . r (cid:38) . V max and r max for SIDM and CDM sub-structures with stars, their r max values decrease overalldue to tidal mass loss. The SIDM one becomes furthersmaller at late stages, as the collapse occurs and the cen-tral density increases; see also [74].The left panel of Fig. 3 shows the corresponding sec-ondary radial (solid) and tangential (dashed) caustics,assuming a source at z s = 3 for the SIDM (red) andCDM (gray) benchmarks ( c = 12 . r − , and hence its associated radialcaustic dominates over the tangential one. Mock lensed images.
The left panel of Fig. 3 displaysfour mock sources at four representative locations; themiddle (SIDM) and right (CDM) panels show their cor-responding lens images, together with the critical linesmapped from the caustics using the lens equation. Theinnermost blue source is inside radial and tangential caus-tics predicted in SIDM and CDM, and it has four imagesin both cases [79, 80]. The source in orange sits on thesecond fold caustic in SIDM while it only crosses the tan-gential caustic in CDM, thus it has one more image inthe former case. Similarly, the sources in green and ma-genta have one more image in SIDM than in CDM. Ourexample explicitly illustrates that the collapsed SIDMsubstructure has higher capability of producing multiple images, which could be further tested statistically.We compute the Einstein radii as r E = θ E D L ≈ . . r E is defined as the radius of the circle with thesame area as that enclosed by the critical line [76]. It’snot surprising that they are comparable. In our setup,the stellar component dominates the inner region andboth cases have similar stellar distributions after tidalevolution. In addition, during the collapse process, theSIDM central density increases, but the total mass doesnot change. As indicated in Fig. 2, r max ∼ r E for CDM,while r max < r E for SIDM. Thus the mass distributioninduced by gravothermal collapse does not produce anappreciable change in the enclosed mass within r E . Thisis a unique prediction of SIDM. Discussion and Conclusions.
Our simulations assumea compact stellar distribution motivated by observationsof early-type galaxies. Hydrodynamical simulations showthat for SIDM halos with masses (cid:38) M (cid:12) stars coulddominate the inner region [81–84]. Thus our assumptionis well justified. In addition, the SIDM halo structureis more resilient to feedback than its CDM counterpart,because of rapid energy redistributions induced by theself-interactions; see [85, 86]. We expect our overall pre-dictions are robust, and it would be interesting to furthertest them with cosmological simulations. In summary, wehave shown that a collapsed SIDM substructure couldhave a steeper density profile, resulting in a larger radialGGSL cross section and more lens images, compared toCDM, a trend favored in explaining the observed excessof strong gravitational lenses in galaxy clusters. The fea-tures of strong lensing observables predicted in the SIDMscenario could be further tested using existing data [87],and upcoming observations [88]. ACKNOWLEDGMENTS
We thank Haipeng An and Seong Chan Park foruseful discussion. DY is supported by NSFC un-der Grant No. 11975134 and the National Key Re-search and Development Program of China under GrantNo.2017YFA0402204. HBY was supported by the U.S.Department of Energy under Grant No. de-sc0008541and the John Templeton Foundation under Grant ID ∗ [email protected] † [email protected][1] P. Schneider, J. Ehlers, and E. E. Falco, GravitationalLenses (Springer-Verlag Berlin Heidelberg New York,1992).[2] J.-P. Kneib and P. Natarajan, Astron. Astrophys. Rev. , 47 (2011), 1202.0185.[3] S. Cao, Y. Pan, M. Biesiada, W. Godlowski, and Z.-H.Zhu, JCAP , 016 (2012), 1105.6226.[4] J. Merten et al., Astrophys. J. , 4 (2015), 1404.1376.[5] A. B. Newman, T. Treu, R. S. Ellis, D. J. Sand, C. Nipoti,J. Richard, and E. Jullo, Astrophys. J. , 24 (2013),1209.1391.[6] M. Annunziatella et al., Astrophys. J. , 81 (2017),1711.02109.[7] M. Bonamigo, C. Grillo, S. Ettori, G. B. Caminha,P. Rosati, A. Mercurio, E. Munari, M. 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