Galaxy-Induced Transformation of Dark Matter Halos
Mario G. Abadi, Julio F. Navarro, Mark Fardal, Arif Babul, Matthias Steinmetz
aa r X i v : . [ a s t r o - ph . GA ] F e b Mon. Not. R. Astron. Soc. , 000–000 (2008) Printed 27 September 2018 (MN L A TEX style file v1.4)
Galaxy-Induced Transformation of Dark Matter Halos
Mario G. Abadi , Julio F. Navarro , Mark Fardal , Arif Babul and Matthias Steinmetz Observatorio Astron´omico de C´ordobai and CONICET, C´ordoba, Argentina Department of Physics and Astronomy, University of Victoria, Victoria, BC V8P 5C2, Canada Astronomy Department, University of Massachusetts at Amherst, MA 01003, US Astrophysikalisches Institut Potsdam, An der Sternwarte 16, Potsdam 14482, Germany
ABSTRACT
We use N-body/gasdynamical cosmological simulations to examine the effect of theassembly of a central galaxy on the shape and mass profile of its surrounding dark mat-ter halo. Two series of simulations are compared; one that follows only the evolutionof the dark matter component of individual halos in the proper ΛCDM cosmologicalcontext, and a second series where a baryonic component is added and followed hy-drodynamically. The gasdynamical simulations include radiative cooling but neglectthe formation of stars and their feedback. The efficient, unimpeded cooling that re-sults leads most baryons to collect at the halo center in a centrifugally-supported diskwhich, due to angular momentum losses, is too small and too massive when comparedwith typical spiral galaxies. This admittedly unrealistic model allows us, nevertheless,to gauge the maximum effect that galaxies may have in transforming their surroundingdark halos. We find, in agreement with earlier work, that the shape of the halo becomesmore axisymmetric: post galaxy assembly, halos are transformed from triaxial into es-sentially oblate systems, with well-aligned isopotential contours of roughly constantflattening ( h c/a i ∼ . how much baryonic mass has been deposited at the center of a halo matters, butalso the mode of its deposition. It might prove impossible to predict the halo responseto galaxy formation without a detailed understanding of a galaxy’s detailed assemblyhistory. Key words:
Galaxy: disk – Galaxy: formation – Galaxy: kinematics and dynamics– Galaxy: structure
Over the past couple of decades, cosmological N-body sim-ulations have established a number of important results re-garding the structure of cold dark matter (CDM) halos.In particular, broad consensus holds regarding the overallshape of the halo mass distribution and its radial mass pro-file. CDM halos are distinctly triaxial systems supported byanisotropic velocity dispersion tensors (Frenk et al., 1988;Jing & Suto, 2002; Allgood et al., 2006; Hayashi et al., 2007;Bett et al., 2007), a result that reflects the highly aspher-ical nature of gravitationally-amplified density fluctuationsas they become non linear and assemble into individual ha- los. Despite their chaotic assembly, the spherically-averagedmass profile of CDM halos is nearly “universal”, and canbe approximated fairly accurately, regardless of halo massor redshift, by scaling a simple formula (see, e.g., Navarroet al., 1996b, 1997, 2004, 2008).One major limitation is that these results have beenobtained from simulations that neglect the presence of thebaryonic, luminous component of galaxy systems. Althoughthis may be a suitable approximation in extremely darkmatter-dominated systems, it is expected to fail in regionswhere baryons contribute a substantial fraction of the mass,as is the case in the luminous regions of normal galaxieslike the Milky Way. The response of the dark halo to the c (cid:13) Abadi et al assembly of a galaxy is thus a crucial ingredient of modelsthat attempt to interpret observational constraints in termsof the prevailing CDM paradigm.There has been extensive work, both numerical and an-alytical, on the modification of the mass profile of a darkhalo induced by the assembly of a central galaxy. Early re-sults suggested that simple analytical prescriptions basedon the conservation of adiabatic invariants gave an accu-rate description of the halo response. Following the earlywork by Barnes & White (1984), Blumenthal et al. (1986)devised a simple formula to link the dark mass profilesbefore and after the assembly of a galaxy. Given the ini-tial, spherically-symmetric enclosed mass profiles of the darkmatter, M i dm ( r ), and baryons, M ib ( r ), one may derive the fi-nal dark mass profile, M f dm ( r ), once the final baryonic massprofile, M fb ( r ), is specified. The model assumes that darkmatter particles move on circular orbits before and after thecontraction, and that their initial, r i , and final, r f , radii arerelated by the condition: r f [ M fb ( r f ) + M dm ] = r i [ M dm + M ib ( r i )] , (1)where M dm = M dm ( r f ) = M dm ( r i ) is the dark mass en-closed by each dark matter particle (i.e., no shell crossing).Despite its simplicity, and the crude approximation onwhich it is based, early N-body work (e.g., Barnes & White,1984; Blumenthal et al., 1986; Jesseit et al., 2002) foundreasonable agreement between eq. 1 and the results of simu-lations, and helped to establish the “adiabatic contraction”formulation as the default procedure when considering thehalo response to the formation of a central galaxy.Semianalytic models of galaxy formation have adoptedthis formulation in order to link the (observed) dynamicalproperties of luminous galaxies to the (predicted) propertiesof the dark halos that surround them (see, e.g., Mo et al.,1998; Cole et al., 2000; Dutton et al., 2005; Croton et al.,2006; Somerville et al., 2008). This work has highlighted anumber of potential problems when attempting to reconcilethe predicted properties of galaxies in the ΛCDM cosmogonywith observed scaling relations.One particularly important challenge has been the in-ability of semianalytic galaxy formation models to matchsimultaneously the zero point of the Tully-Fisher (TF) rela-tion and the galaxy luminosity function. Successful modelsrequire that the rotation speed of disks be of the order of thevirial velocity of the halos they inhabit (see, e.g., Somerville& Primack, 1999; Croton et al., 2006). However, models thatinclude adiabatic contraction typically predict disk rotationspeeds well in excess of the virial velocity.As discussed most recently by, for example, Duttonet al. (2005); Gnedin et al. (2007) and Dutton et al. (2008),potential solutions to this problem include: (i) revising theadiabatic hypothesis so as to reduce (or even reverse!) thehalo contraction; (ii) adopting lighter stellar mass-to-lightratios in order to minimize the contribution of baryons andto allow for more dark mass enclosed within disk galaxies; or(iii) modifying the cosmological parameters so as to reducehalo concentrations. We shall concentrate our analysis hereon option (i), although we note that all three possibilitiesare probably equally important and should be treated onequal footing when addressing these issues in semianalyticmodels of galaxy formation.The possibility that the actual response of the halo to the assembly of the disk might differ substantially from thepredictions of the adiabatic contraction formalism has beennoted before. Barnes (1987) and Sellwood (1999) have re-marked that the adiabatic contraction hypothesis might leadto substantial overestimation of the halo compression. Asdiscussed by Sellwood & McGaugh (2005) and Choi et al.(2006), such deviations are likely to depend strongly on theorbital structure of a halo, performing best when most par-ticles have large tangential motions, but poorly in systemswith radially anisotropic velocity distributions.It should also be pointed out that the studies men-tioned above were carried out by perturbing simple spher-ical models with a potential term designed to imitate thegrowth of an assembling galaxy. These studies, therefore,miss the hierarchical nature of the assembly of a galaxy andof its surrounding halo. The study of Gnedin et al. (2004),on the other hand, uses the proper cosmological context,but focusses on simulations of galaxy clusters, rather thanthe galaxy-sized systems of interest for the issues discussedabove. In this respect, our study is similar to that of Gustafs-son et al. (2006), who studied four simulations of galaxy-sized halos in the ΛCDM scenario.The assembly of a central galaxy also modifies the three-dimensional shape of the surrounding dark matter halo. Thiswas already noted in the first simulations to include, in ad-dition to a dark matter halo, a dissipative baryonic compo-nent, such as the early work of Katz & Gunn (1991) andKatz & White (1993). Dubinski (1994) studied this furtherby growing adiabatically a central mass concentration insidea triaxial dark matter halo and confirmed that the steepen-ing of the potential leads to much rounder halo shapes. Thisresult was confirmed by Kazantzidis et al (2004), who alsonoticed the effect in their cluster simulations. We extend thisbody of work by focussing, as in Hayashi et al. (2007), onthe shape of the gravitational potential rather than on theaxial ratios of the inertia tensor, as well as on its dependenceon radius.The plan of this paper is as follows: § § § § § § § § § We adopt cosmological parameters consistent with the com-bined analysis of the 2dfGRS (Colless et al., 2001) and thefirst-year WMAP data (Spergel et al., 2003): a present dayvalue of the Hubble constant of H = 70 km/s/Mpc; a scale-free initial density perturbation spectrum with no tilt andnormalized by the linear rms mass fluctuations on 8 h − Mpc spheres, σ = 0 .
9. The matter-energy content of theUniverse is expressed in units of the present-day critical c (cid:13) , 000–000 ransformations of CDM Halos Figure 1.
Dark matter (black) and gas (colored) particles for halo S02h. The color of a gas particle reflects its temperature,
T < . K(green) or
T > . K (magenta). Each panel zooms into the center of the system by consecutive factors of 3 in radius (see axis labels).The circle in the top-left panel shows the virial radius. The circle in the bottom-right panel shows the radius, r glx , used to define thecentral galaxy. Note that the colder (green) gas component inhabits the center of the main halo and of its substructures, where it formsthin, centrifugally-supported disks. The hotter component (magenta) is distributed more or less uniformly across the main halo andmakes up only 16% of all the baryons within the virial radius. density for closure, and contains a dominant cosmologicalconstant term, Ω Λ = 0 .
7, as well as contributions to thematter content, Ω M = 0 .
3, from cold dark matter (CDM),Ω
CDM = 0 . b = 0 . We have performed a suite of 13 numerical simulations of theformation of galaxy-sized halos in the ΛCDM cosmogony.Each simulation follows the evolution of a relatively smallregion of the Universe, excised from a 432 -particle simula-tion of a large 50 h − Mpc periodic box (Reed et al., 2003),and resimulated at much higher mass and spatial resolution. Each “zoomed-in” re-simulation follows the formation of asingle galaxy-sized halo of mass ∼ M ⊙ and its immedi-ate surroundings. The simulations include the tidal fields ofthe parent simulation, and follows the coupled evolution ofgas and dark matter. The hydrodynamical evolution of thegaseous component is followed using the Smooth ParticleHydrodynamics (SPH) technique. This re-simulation tech-nique follows standard practice, as described, for example,by Power et al. (2003). All our simulations were performedusing GASOLINE, a parallel N-body/SPH code describedin detail by Wadsley et al. (2004). c (cid:13) , 000–000 Abadi et al
Figure 2.
Radius containing 90% of the baryonic mass, r , ver-sus the total baryonic mass of the central galaxy, M disk , in oursimulated halos (filled circles), expressed in units of the halo virialvalues. The dashed vertical line the universal baryon fraction, f bar = Ω b / Ω M . About 70-80% of all baryons have been collectedin the central galaxy. The solid curve shows the corresponding val-ues for the Milky Way, varying the virial mass of its halo. Symbolsalong the line indicate a few different values of the virial veloc-ity, V vir = 220, 150, 110, and 91 km/s. Note that the simulatedcentral galaxies are substantially more massive and smaller thanspirals like the Milky Way. This maximizes the effect of baryonsin transforming the shape and mass profile of the dark halo. The 13 halos were selected from a list of all ∼ M ⊙ halos in the parent simulation, with a mild bias to avoidobjects that have undergone a major merger after z = 1 orthat have, at z = 0, unusually low ( λ < .
03) spin param-eter. The mass resolution of the resimulations is such thathalos are represented with ∼ ,
000 dark matter particleswithin their virial radius ⋆ at z = 0. Two sets of simu-lations are performed for each halo, one where only a darkmatter component is followed, and another where dark mat-ter and baryons are included. Dark matter-only simulations(“DMO” for short) assume that the total matter content ofthe Universe is in cold dark matter, Ω M = Ω CDM = 0 . b = 0 . CDM = 0 . ⋆ We define the virial radius, r vir , of a system as the radius ofa sphere of mean density ∆ vir ( z ) times the critical density forclosure. This definition defines implicitly the virial mass, M vir , asthat enclosed within r vir , and the virial velocity, V vir , as the circu-lar velocity measured at r vir . Quantities characterizing a systemwill be measured within r vir , unless otherwise specified. The virialdensity contrast, ∆ vir ( z ) is given by ∆ vir ( z ) = 18 π + 82 f ( z ) − f ( z ) , where f ( z ) = [Ω (1 + z ) / (Ω (1 + z ) + Ω Λ ))] − = Ω CDM + Ω b (Bryan & Norman, 1998). ∆ vir ≈
100 at z = 0. Figure 3.
Angular momentum and mass of the central galaxy inour simulations, expressed in units of the virial values. The curvelabeled j disk = j vir corresponds to central galaxies with the same specific angular momentum as the surrounding halo (a commonassumption of semianalytic models). Those on the curve labeled j disk /j vir = m d /f bar have specific angular momenta (in units ofthe halo’s) that scale in proportion to the fraction of baryonswithin the virial radius that have collected in the central galaxy.Note that the simulated central galaxies have even lower angularmomenta than in the latter assumption, highlighting the largeangular momentum losses that accompany the assembly of thecentral galaxy, and explaining the small sizes of the disks shownin Fig. 2. ing SPH including, besides the self-gravity of gas and darkmatter, pressure gradients and shocks. Baryons are allowedto cool radiatively according to the cooling function of a gasof primordial composition down to a temperature of 10 K,below which cooling is disabled. No star formation or feed-back is included. As we discuss below, this choice maximizescooling and favors the collection of most baryons at the cen-ter of each dark halo. Although unrealistic as a model forgalaxy formation, this choice has the virtue of simplicity andalso allows us to examine the halo response when the effectof the baryonic component is maximal.Pairwise gravitational interactions are softened adopt-ing a spline softening length kept fixed in comoving coordi-nates. We test for numerical resolution effects by simulatingeach halo with 8 times fewer particles. In the interest ofbrevity we do not present results from this low-resolutionseries here, but we have checked explicitly that none of theconclusions we present here are modified by this change innumerical resolution. We have also increased the number ofparticles in one case. Halo S02h is equivalent to S02, butwith ∼ . z = 0 this halo has ∼ ,
000 particles per component within the virial radius.Table 1 lists the main properties of the simulated halos. c (cid:13) , 000–000 ransformations of CDM Halos Figure 4.
Dark matter particles corresponding, at z = 0, to the two resimulations of halo S02h, “dark matter only” (left panels)and “dark matter plus baryons” (right panels). Particles are colored according to the value of the gravitational potential (binned inlogarithmic units, top panels) or the local density (bottom panels). Note that the halo responds to the assembly of the central galaxy(shown in black in the right-hand panels) by becoming noticeably more spherical. Projections are chosen so that the rotation axis of thecentral disk coincides with the z-axis. The central disk is shown with black dots, to illustrate the orientation of the disk relative to thehalo shape. In general, the central disk is well aligned with the minor axis of the halo. Because our simulations neglect the formation of stars andtheir feedback, the evolution of the baryonic component ischaracterized by the rapid cooling and collapse of baryons at the center of the early collapsing progenitors of the fi-nal halo. As a result, the main mode of galaxy assembly ismergers: up to 70% of all baryons in the central galaxy wereaccreted in the form of dense, cold, gaseous clumps that sinkto the center through dynamical friction.As shown in earlier work, (see, e.g., Navarro & Benz, c (cid:13) , 000–000 Abadi et al
Figure 5. ”Radial” ( r = √ a + b + c ) dependence of axialratios b/a (open symbols) and c/a (filled symbols) for our high-resolution run S02h. Halo shapes are measured by fitting 3D el-lipsoids to the position of dark matter particles in narrow loga-rithmic bins of the gravitational potential. The potential is com-puted using only dark matter particles, so it actually correspondsto the contribution of the dark matter component to the overallpotential. Red (lower) symbols correspond to the DMO run; blue(upper) symbols to the DM+B run. Note that the central galaxyturns a rather triaxial, nearly prolate halo into an axisymmetric,nearly oblate one. Curves are fits using eq. 2. Fit parameters forthis halo and the mean value computed over the sample are listedin Table 2. T < . K) is shown in green, whereashot gas (
T > . K) is shown in magenta. Note that mostof the cold gas inhabits the center of the main halo and itssubstructures, where it forms easily identifiable thin, mas-sive disks. All cold baryons associated with the central diskare contained within a sphere of radius r glx = 10 kpc, whichwe shall use hereafter to define the central galaxy in all runs. Figure 6. ”Radial” ( r = √ a + b + c ) dependence of axialratios b/a (upper panel) and c/a (lower panel) for all simulatedhaloes. The curves for the “dark matter only” runs are computedby fitting eq. 2 to the radial dependence of the axial ratios of par-ticles along isopotential contours. The contours in all cases referto the potential contributed solely by the dark matter compo-nent. Note that, as a result of the assembly of the central galaxy,(i) halos become nearly oblate and (ii) axial ratios become ap-proximately independent of radius. The effect on the shape of thehalo extends almost to the virial radius, far beyond the actualsize of the central galaxy. Thick lines correspond to mean valuescomputed over the sample. The central galaxy contains most of the baryons within thevirial radius of the system (74% in the case of S02h, seeTable 1). This is shown in Fig. 2, where we plot, for all oursimulations, the baryonic mass of the central galaxy, M disk ,versus the radius, r , that contains 90% of its mass, scaledto the virial mass and radius of the system, respectively. Thevertical dashed line indicates the universal baryon fractionin the simulations, f bar = Ω bar / Ω M = 0 . r vir are found in fairly smallcentral disks that are fully contained within a radius of order ∼
3% of the virial radius.We may compare this with a typical spiral galaxy likethe Milky Way (MW), where the mass and size of the bary-onic component can be estimated accurately. Assuming thatthe disk is exponential with total mass 4 . × M ⊙ andradial scalelength R MW d = 2 . M bulge = 4 . × M ⊙ , we estimate that 90% of theMilky Way baryons are confined within r MW90 = 9 . ∼
1% of the virialmass, or roughly 7% of all baryons within r vir . The simulated c (cid:13) , 000–000 ransformations of CDM Halos disks are thus much smaller than a typical spiral like theMW. They are also comparatively more massive.Only if the virial velocity is as low as 110 km/s do theMW bulge+disk make up a fraction of available baryonsas high as in our simulations ( ∼ r MW90 ∼ . r vir , a factor of ∼ J d ≡ J disk /J vir = M disk j disk /M vir j vir , versus themass fraction, m d ≡ M disk /M vir . These are the parameterscommonly adopted in semianalytic models of disk formation,such as those of Mo et al. (1998).Since baryons acquire during the expansion phase asmuch angular momentum as the dark matter, and it is un-likely that M disk /M vir will exceed the universal baryon frac-tion, then we do not expect the specific angular momentumof the disk, j disk , to exceed that of the system as a whole, j vir . Therefore, central galaxies are unlikely to populate theshaded areas of Fig. 3.Most semianalytic work assumes, for simplicity, that j disk = j vir or, equivalently, that J d = m d , regardless of thevalue of m d . On the other hand, Navarro & Steinmetz (2000)argue that it is unlikely that a galaxy where m d << f bar mayhave the same specific angular momentum as the whole sys-tem, and propose that j disk , as a fraction of j vir , should becomparable to the fraction of baryons that make up the cen-tral galaxy; i.e., j disk /j vir = M disk / ( f bar M vir ) = m d /f bar .This is illustrated in Fig. 3 by the lower diagonal line.Our simulated galaxies are well below both lines. Thisindicates that baryons have specific angular momenta muchlower than their surrounding halos and explains their smallsizes compared to typical spirals. These are therefore unreal-istic models of spiral galaxy formation, but the combinationof large mass and small size allows us to probe the responseof the dark halo in the case where the deepening of the po-tential well due to the central galaxy is maximal. As anticipated in §
1, the halo responds to the presence ofthe central galaxy by becoming significantly more spherical.This is illustrated in Fig. 4, where we compare the shapeof isodensity and isopotential contours for the “dark matteronly” and “dark matter plus baryons” runs of halo S02h.Panels on the left correspond to the DMO run, those onthe right to the DM+B run. Particles are colored accordingto their local values of their gravitational potential or localdensity (binned in a logarithmic scale), computed using only the dark matter particles. Note that using the gravitationalpotential leads to much more stable estimates of the shapeof the halo, since it is much less affected by the presence ofsubstructures and other transient fluctuations in the massdistribution.The isopotential contours are well approximated by el-lipsoidal surfaces, and we use the axial ratios of such el-lipsoids to measure, as a function or radius, the change inshape of halo S02h. We show the result in Fig. 5. As dis- cussed by Hayashi et al. (2007), the radial dependence ofthe axial ratios may be approximated by the formula,log( ba or ca ) = α h tanh (cid:16) γ log rr α (cid:17) − i , (2)Here α parameterizes the central value of the axial ratio,( b/a ) or ( c/a ) , by 10 − α ; r α indicates the characteristicradius at which the axial ratio increases significantly from itscentral value; and γ regulates the sharpness of the transition.The presence of the central galaxy turns the halo from atriaxial, nearly prolate system into an axisymmetric, nearlyoblate system where the axial ratio is nearly constant withradius. As a result, eq. 2 is not adequate for the nearly fea-tureless radial dependence of the axial ratios in the DM+Bruns. We fit the latter with a simple power-law,log( ba or ca ) = α log r + β, (3)The best-fit parameter to the mass and isopotential con-tour profile shapes are given in Table 2.The profile fits for all simulations are compiled in Fig. 6,and illustrate a few important points. As a result of the as-sembly of the central galaxy: (i) halos become nearly oblate;(ii) axial ratios are roughly independent of radius; and (iii)halo shapes are affected well beyond the size of the centralgalaxy, and nearly as far out as the virial radius. Fig. 7 shows the enclosed mass profile of the DMO andDM+B runs corresponding to halo S02h. DMO dark masseshave been scaled by (1 − f bar ) so that the total dark massin both the DMO and DM+B runs are the same. The DMOmass profile is shown by the thick red solid curve and, as ex-pected, it is well approximated by the NFW (Navarro et al.,1996b, 1997) formula (thin solid line). The central disk thatassembles in the DM+B run leads to a contraction of thedark halo: there is more dark mass in the DM+B run at allradii compared with the DMO run. This is a result com-mon to all our simulations; in no case do we see the halo“expand” as a results of the assembly of the central galaxy.The contraction, however, is not as pronounced as whatwould be expected from the “adiabatic contraction” modeldiscussed in §
1. The adiabatic contraction prediction (eq. 1)is shown by the thick dotted line in Fig. 7. The discrepancybetween model and numerical results is not small. At ∼ ∼ . times more dark massthan found in our simulations . Even at a radius of 10 kpc,eq. 1 overestimates the halo contraction by more than ∼ r i ) and the DM+B run ( r f ) versus theratio between M i , the total mass within r i in the DMO run,and M f , the total mass within r f in the DM+B run. Withthis choice, the adiabatic-contraction prediction is simply r f /r i = M i /M f , which is traced by the 1:1 line in Fig. 8. c (cid:13) , 000–000 Abadi et al
Figure 7.
Enclosed mass profile of various components of haloS02h. The “dark matter only” profile is shown with a thick curve,after scaling masses by (1 − f bar ), so that, within r vir , the totaldark mass of the DMO and DM+B runs will be comparable. Thethin line shows a Navarro-Frenk-White halo fit to the DMO pro-file. Other colors and line types correspond to the various compo-nents of the DM+B run, as specified in the figure labels. The thickdashed blue curve shows the dark mass profile for the DM+B run.The assembly of most baryons into a central galaxy (dotted ma-genta curve) has clearly led to a contraction of the dark massprofile. The profile predicted by the “adiabatic contraction” for-mula (eq. 1; dot-dashed green curve) overpredicts the responseof the halo. The modified adiabatic contraction model of Gnedinet al. (2004), shown by the dotted cyan curve, also overestimatesthe halo response. The numerical simulations are shown by the uppercurves in this figure. Near the center, the baryons dominateover the DMO mass profile, and therefore M i /M f ≪
1. Fur-ther out, the contribution of baryons to the total enclosedmass decreases, approaching the universal baryon fraction atthe virial radius, where M i /M f tends to unity. The inner-most radius plotted in each case corresponds to that enclos-ing 1000 dark matter particles. This is in all cases comfort-ably larger than the gravitational softening, minimizing thepossibility that our results are unduly influenced by numer-ical artifact. The open circles with error bars in Fig. 8 tracethe median and rms scatter of all our simulation results.The ratio r f /r i tends to a constant for M i /M f ≪ r f /r i = 1 + a [( M i /M f ) n − , (4)with a = 0 . n = 2. As an application, we shall use thisexpression below to explore what constraints this implies forthe mass and concentration of the halo of the Milky Way. Figure 8.
Dark halo response to the assembly of a central galaxy.The ordinate shows the ratio, r f /r i , between the radius contain-ing a given amount of dark mass in the DMO and DM+B runs,respectively, after scaling DMO masses by 1 − f bar . The smaller r f /r i the stronger the halo contraction. The x-axis shows the ratiobetween the total mass, M i , contained within r i (in the DMO run)and M f , that enclosed within r f (in the DM+B run). The adia-batic contraction formula (eq. 1) predicts that r f /r i = M i /M f ;this is shown by the 1:1 line in the figure. Numerical results areshown for individual halos; from the radius that contains 1000dark particles outwards in order to minimize numerical uncer-tainties. The adiabatic-contraction formula overestimates the haloresponse. Gnedin et al. (2004)’s modified adiabatic contraction(“Contra”) does better but still overpredicts the halo responseat most radii, and especially near the center. Symbols with errorbars trace the median and quartiles of the numerical results. Thethick upper curve is a fit using eq. 4. The Milky Way offers a case study for the results describedabove. Because the mass and radial distribution of the bary-onic component, as well as the rotation speed of the localstandard of rest (LSR), are relatively well known, the totaldark mass contained within the solar circle is firmly con-strained. Adopting the same quantities for the bulge anddisk of the Milky Way adopted in § ∼
171 km/s to the circular ve-locity of the LSR, which we assume to be 220 km/s. Thisimplies that the dark mass of the Milky Way within the solarcircle ( R ⊙ = 8 kpc) is ∼ . × M ⊙ . In order to allow forthe possibility that some of this dark mass may be baryonic,we shall treat this a formal upper limit in the analysis thatfollows.The second constraint comes from the total baryonicmass of the Milky Way, under the plausible assumption thatthe mass of the central galaxy cannot exceed the total massof baryons within the virial radius, ≈ f bar M vir . Thus, the minimum virial mass allowed for the MW halo by this con- c (cid:13) , 000–000 ransformations of CDM Halos straint is 3 . × M ⊙ , which corresponds to a virial ve-locity of ∼
92 km/s.Are these constraints compatible with ΛCDM halos?The top left panel of Fig. 9 shows the dark mass ex-pected within 8 kpc vs halo virial velocity. Each dot in thispanel correspond to an NFW halo with concentration drawnat random from the mass-concentration relation (includingscatter) derived by Neto et al. (2007) from the MillenniumSimulation (Springel et al., 2005). As expected, the masswithin 8 kpc increases with the virial velocity (mass) of thehalo, modulated by the fairly large scatter in concentrationat given halo mass.The black “wedge” in this panel indicates the region al-lowed by the MW constraints discussed above. As discussedby Eke et al. (2001), most ΛCDM halos satisfy the con-straints, and would be consistent with the MW if they weresomehow able to avoid contraction. Note as well that thelarger the MW halo virial velocity the lower the allowed con-centration. For V vir = 220 km/s (the value required by semi-analytic models to match the Tully-Fisher relation and theluminosity function, see §
1) only halos with c vir < . V vir = 220 km/s halos is h c vir i ∼ . σ log c = 2 .
9, so this con-dition effectively excludes only 57% of such halos from theallowed pool.The situation is rather different if halos are adiabati-cally contracted (panel labeled “AdiabCont” in Fig. 9). Thecontraction increases substantially the dark mass containedwithin 8 kpc, so that very few halos satisfy the MW con-straints. For example, essentially no halo with V vir ≈ c vir < .
2, many sigma awayfrom the average concentration of halos of that mass. Thehalo of the Milky Way would need to be a very special halo ofunusually low concentration if adiabatic contraction holds.Adopting the halo contraction of our numerical simula-tions improves matters. This may be seen in the bottom-leftpanel of Fig. 9, where we have contracted each halo usingeq. 4. The range of allowed halo masses and concentrationsis broader; even some halos with V vir ∼
220 km/s couldbe consistent with the Milky Way, provided that c vir < . V vir ∼
130 km/shalos satisfy the MW constraints. The possibility that thevirial velocity of the Milky Way is significantly lower than220 km/s has indeed been advocated by a number of recentobservational studies (Smith et al., 2007; Sales et al., 2007;Xue et al., 2008).Lowering the average concentration at given mass alsohelps. This is shown in the bottom-right panel of Fig. 9,where we have repeated the exercise, but lowering all con-centrations by 20%. According to Duffy et al. (2008), thisis approximately the change in average concentration whenmodifying the cosmological parameters from the values weadopt here (which are consistent with the first-year analy-sis of the WMAP satellite data) to those favoured by thelatest 5-year WMAP data analysis. With this revision, 18%of V vir ∼
220 km/s halos and roughly half of all V vir ∼ Figure 9.
Dark mass within 8 kpc vs virial velocity for halosdrawn at random from the ΛCDM mass-concentration relation(including scatter) of Neto et al. (2007). The DMO panel (topleft) shows the results neglecting the effects of baryons and as-suming an NFW profile for the dark halos. The black “wedge”highlights the region of parameter space compatible with obser-vations of the Milky Way. The panel labeled “AdiabCont” showsthe result of contracting each halo adopting the adiabatic contrac-tion formula (eq. 1). Very few halos would be consistent with theMilky Way if this formula holds. The bottom-left panels shows theresult of applying our simulation results for the halo contraction,as captured by eq. 4. The bottom-right panel repeats the sameexercise, but after reducing all concentrations by 20% in order tomimic the expected concentration change resulting from adopt-ing the latest cosmological parameters from the 5-year analysisof WMAP satellite data (Duffy et al., 2008).
The failure of the adiabatic contraction formalism to matchthe results of our simulations suggest that the response ofthe halo is more complex than what can be captured with asimple model for the contraction of spherical shells. This isconfirmed by comparing our results with earlier work.Our results disagree not only with the traditional adi-abatic contraction formula, but also with the modified for-malism proposed by Gnedin et al. (2004) (shown with thindash-dotted green lines in Fig. 8). These authors calibratedtheir results with numerical simulations not dissimilar toours, except for the mass scale—they used mainly galaxycluster halos. The mass scale, on the other hand, does notseem to be the reason for the discrepancy. Our halo contrac-tion is also less pronounced than found in two galaxy-sizedhalo simulations, as shown by the lines labeled A03&M03in Fig. 8. These correspond to gasdynamical simulations ofgalaxy formation presented by Abadi et al. (2003a,b) andMeza et al. (2003), and differ from ours mainly in their in-clusion of star formation and feedback effects. Those two c (cid:13) , 000–000 Abadi et al simulations seem to agree better with the “Contra” predic-tions than with our simulation results.Barring numerical artifact, these results seem to sug-gest that the halo response does not depend solely on theinitial and final distribution of baryons . One possible expla-nation is that not only how much baryonic mass has beendeposited at the center of a halo matters, but also the mode of its deposition. It is certainly plausible that central galax-ies assembled through merging of dense subclumps may leadto different halo response than galaxies assembled througha smooth flow of baryons to the center. Mergers of bary-onic subsystems may in principle pump energy into the darkhalo (through dynamical friction), altering its central struc-ture and softening its contraction. This possibility has beenargued before (see, e.g., El-Zant et al., 2001; Ma & Boylan-Kolchin, 2004; Mo & Mao, 2004), and seems to find favorin our simulations, where mergers between massive baryonicclumps are frequent and plentiful and halo contraction is lessstrong than reported in earlier work.Recent work has also speculated that mergers may leadto halo expansion and that this would help to reconcile theproperties of disk galaxies with ΛCDM halos (Dutton et al.,2007). Our halos always contract, and it seems safe to con-clude that our results reflect the maximum effect of mergerson the halo response. We conclude therefore that mergersalone are unlikely to result in halo expansion. If such expan-sion is truly needed to reconcile disk properties with ΛCDMhalos, it should come as a consequence of other processes notconsidered here, such as feedback-driven winds that may re-move substantial fraction of baryons from the central galaxy(see, e.g., Navarro et al., 1996a; Babul & Ferguson, 1996).
We have used a suite of cosmological N-body/gasdynamicalsimulations to examine the modifications to the dark halostructure that result from the assembly of a central galaxy.The formation of 13 ΛCDM halos is simulated twice, withand without a baryonic (gaseous) component. For simplic-ity, the gasdynamic simulations include radiative cooling butneglect star formation and feedback. This favors the forma-tion of massive central baryonic disks at the center of theearly collapsing progenitors of the final halo. As these sys-tems merge, the gaseous clumps merge and re-form a diskat the center of the remnant.The merger process leads to the transfer of a large frac-tion of the angular momentum from the baryons to the halo.At z = 0, the simulated central galaxies are too massive andtoo small to be consistent with observed spiral galaxies. Al-though unrealistic as a disk galaxy formation model, thesesimulations allow us to probe the dark halo response in theinteresting case where the deepening of the potential wellresulting from the formation of the central galaxy is maxi-mized. Our main conclusions may be summarized as follows. • Dark halos become significantly more spherical as aresult of the assembly of the central galaxy. The triaxial,nearly prolate systems that form in the absence of a baryoniccomponent are transformed into essentially oblate systemswith a roughly constant axial isopotential ratio h c/a i ≈ . • Halos always contract in response to the formation ofthe central galaxy. The “adiabatic contraction” formalism overestimates the halo contraction in our simulations. Thediscrepancy increases toward the centre, where the effect ofbaryons is larger. A simple empirical formula (eq. 4) de-scribes our numerical results. • The halo contraction in our simulations is also less pro-nounced than found in earlier numerical work (see, e.g.,Gnedin et al., 2004), and suggest that the response of a halodoes not depend on the final mass and radial distribution ofbaryons in the central galaxy, but also on the mode of theirassembly. • We apply these results to the Milky Way, where accu-rate estimates of the mass of baryons and dark matter insidethe solar circle exist. These allow us to probe the range ofhalo virial mass and concentration consistent with the con-straints. Only halos of unusually low mass and concentra-tion would match such constraints if “adiabatic contraction”holds. • This restriction is less severe if one uses the halo con-traction reported here (eq. 4): although few ΛCDM halos ofvirial velocity ∼
220 km/s would be eligible hosts for theMilky Way, the situation improves for lower virial veloci-ties. Halos of average concentration with virial velocity aslarge as V vir = 135 km/s, would be consistent with the MWconstraints.The dark halo response to the formation of a galaxyseems inextricably linked to the full coupled evolution ofbaryons and dark matter and may vary from system to sys-tem. Progress in our understanding of the distribution ofdark matter in a baryon-dominated system will thus likelydevelop in step with our understanding of the particular as-sembly history of each individual system. ACKNOWLEDGMENTS
We thank James Wadsley, Joachim Stadel and Tom Quinnfor allowing us to use their excellent GASOLINE code forthe numerical simulations reported here.
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