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MNRAS 000, 000–000 (0000) Preprint 14 June 2019 Compiled using MNRAS LATEX style file v3.0

Dark Matter Trapping by Stellar Bars: The Shadow Bar

Michael S. Petersen,1⋆ Martin D. Weinberg,1 Neal Katz11University of Massachusetts at Amherst, 710 N. Pleasant St., Amherst, MA 01003

14 June 2019

ABSTRACTWe investigate the complex interactions between the stellar disc and the dark-matter haloduring bar formation and evolution using N-body simulations with fine temporal resolutionand optimally chosen spatial resolution. We find that the forming stellar bar traps dark matterin the vicinity of the stellar bar into bar-supporting orbits. We call this feature theshadow bar.The shadow bar modifies both the location and magnitude of theangular momentum transferbetween the disc and dark matter halo and adds 10 per cent to the mass of the stellar barover 4 Gyr. The shadow bar is potentially observable by its density and velocity signature inspheroid stars and by direct dark matter detection experiments. Numerical tests demonstratethat the shadow bar can diminish the rate of angular momentumtransport from the bar to thedark matter halo by more than a factor of three over the naive dynamical friction prediction,and thus provides a possible physical explanation for the observed prevalence of fast bars innature.

Key words: galaxies: haloes—galaxies: kinematics and dynamics—galaxies: evolution—galaxies: structure

1 INTRODUCTION

Galactic bars change the internal structure of disc galaxies by re-arranging the angular momenta and energy of orbits that would beotherwise conserved. As many as two-thirds of local galaxies showbar features in the infrared (Sheth et al. 2008). Bars have been sug-gested to affect both structural properties such as bulge structure(Kormendy & Kennicutt 2004; Laurikainen et al. 2014) and stel-lar breaks (Munoz-Mateos et al. 2013; Kim et al. 2014), as well asevolutionary properties such as star formation rates (Masters et al.2012; Cheung et al. 2013) and metallicity gradients (Williams et al.2012). Weinberg & Katz(2002) also showed that bars can affectthe central profiles of dark matter haloes. Bars are the strongest ofthe persistent disk asymmetries and, thereby, provide a dynamicallaboratory for understanding the long-term evolution of galaxies.

For simplicity of analysis, many previous theoretical workshave adopted a component-by-component analysis of angularmomentum transport by adopting a fixed halo potential, a rigid bar,or both. However, the baryonic disc and dark matter halo comprisea single system that shares the same dynamics as a consequenceof their mutually generated gravitational field. Therefore, somefraction of the dark-matter orbital elements will overlap withthose in the disc and, thus, there is little reason not to suspect thatsome of the dark matter halo will respond similarly to the stellardisc. The dynamical interplay between these components is ofconsiderable interest in developing a comprehensive understandingof the disc-halo system.

⋆ mpete0@astro.umass.edu

In this paper, we present a time-dependent analysis of trappedstellar and halo orbits in a fully self-consistent simulation. We findevidence for the formation and subsequent secular evolution of atrapped population of halo orbits, ashadow bar, that arises in re-sponse to the same collective dynamics that make the classicstellarbar. The shadow bar fundamentally changes the orbital structure ofthe halo in the vicinity of the bar. These same orbits, when unper-turbed by the bar, would contribute strongly to the disc-halo torqueresponsible for slowing the bar, which can be understood in thecontext of simple perturbation theory models. We investigate anddocument the magnitude of halo trapping, finding that the trappingaffects both the dark-matter density and velocity structure.

A well-studied consequence of angular momentum trans-port in barred galaxies is the tension between the theorisedslowing of the bar pattern speed (Tremaine & Weinberg 1984;Weinberg 1985) and observations that do not show a slowingpattern speed (e.g.Aguerri et al. 2015). This is often interpretedas evidence for small amounts of dark matter at the centreof galaxies (Debattista & Sellwood 2000; Sellwood & Debattista2006; Villa-Vargas et al. 2009) or as a motivation for changing thelaw of gravity (Tiret & Combes 2007). However, the dark-matterorbits trapped by the bar are unavailable for secular evolution, sig-nificantly reducing the halo torque on the bar.

For our own Milky Way (MW)—a barred galaxy—several ex-periments may detect the influence of a shadow-bar modified darkmatter density and velocity distribution. These include ongoinganalysis of stellar surveys such as RAVE (Steinmetz et al. 2006),GAIA (Gilmore et al. 2012), and APOGEE (Allende Prieto et al.2008) since halo stars might have similar orbits to dark matterparticles. Some of these studies have already seen suggestions of

c© 0000 The Authors

2 M. Petersen et al.

rotation in a bulge-bar component (Rojas-Arriagada et al. 2014,Soto et al. 2014). Signs of a shadow bar might also figure in theinterpretation of tentative signals from direct dark matter detec-tion experiments, e.g. CoGeNT (Aalseth et al. 2013), CDMS II(Agnese et al. 2013), and DAMA/LIBRA (Bernabei et al. 2014).

This paper is organised as follows: the next section describesthe initial conditions for our fiducial simulation (section2.1.1), aseries of simulations with varying halo properties (section 2.1.2),and the details of our N-body solver (section2.2). We include twoidealised numerical experiments with modifications to the bar per-turbation to better understand the importance of the bar growth rateon orbit trapping and halo structure. Our results are presented insection3. We examine the fiducial simulation in detail, beginningwith an overall summary picture of the fiducial simulation insec-tion 3.1, followed by studies of the disc and halo orbit morpholo-gies in sections3.2.1and3.2.2, respectively. These analyses revealthat in addition to the stellar bar, a significant mass fraction of thehalo within a bar length is trapped into the bar potential. The dy-namical properties of the entire trapped orbit population are dis-cussed in section3.3and contextualised using perturbation theory.This is followed by an exploration of the overall dynamical conse-quences of the shadow bar itself in section3.4. Section4 connectswith previous work by using a suite of experiments motivatedbypublished models and further demonstrates the robust nature of theshadow bar. We conclude with a summary and discussion of thestellar disc–dark matter halo relationship and our detailed orbitalanalysis in section5.

2 METHODS

2.1 Initial conditions

This section motivates our mass models and describes the reali-sation of the initial conditions. We begin with the details of ourfiducial simulation. The phase space is realised using a method-ology that closely followsHolley-Bockelmann et al.(2005), here-after HB05. We then discuss model variants that (1) address typicalliterature treatments; (2) seek to model physical processes in theuniverse; and (3) verify the theorised dynamical implications of thefiducial experiment. Our key simulations are summarised in Table1.

In all the simulations presented here, the number of disk par-ticles and halo particles areNdisc=106 andNhalo=107, respec-tively. The disc particles have equal mass. The halo particles havea number densitynhalo∝r−α, whereα=2.5. This steep powerlaw better resolves the inner halo in the vicinity of the discbar,our subject of interest. The particle’s masses are assignedto simul-taneously reproduce the desired halo mass density and the num-ber density. In practice, the halo resolution is improved bya factorof 100 within a disc scale length (Mr<Rd

=0.005Mvir, Nr<Rd=

5×105) compared to using a fixed halo particle mass. The ef-fective halo particle number within all disc-relevant radii is thenNhalo∼109, more than satisfying the criteria ofWeinberg & Katz(2007b) for an adequate treatment of resonant dynamics.

All units are scaled to so-calledvirial units withG=1 thatdescribe the mass and radius of the halo at the point of halo col-lapse in the cold-dark-matter scenario. For comparison to the MW,we follow the estimated halo mass ofBoylan-Kolchin et al.(2013)such thatMvir=1.6×1012 M⊙, makingRvir=1.0 correspond to300 kpc, a velocity of 1.0 equivalent to 150 km s−1, and a timeof 1.0 equivalent to 2.0 Gyr. The system is constructed such thatM(Rvir)=1.0.

2.1.1 Fiducial simulation

We follow the now standard prescription for producing a strong barin an N-body simulation by choosing a fiducial model with a slowlyrising rotation curve (Fig.1). This minimises the bar-damping ef-fect of the initial inner Lindblad resonance and allows the regioninside of corotation to act as a “resonant cavity” for bar forma-tion (Sellwood & Wilkinson 1993). For a rising rotation curve, onecan show using first-order perturbation theory that the response ofa nearly circular orbit to the bar perturbation is to slow andevenreverse its apse precession rate (the so-called “donkey effect” ofLynden-Bell & Kalnajs 1972, seeBinney & Tremaine 2008). Thisresults in the orbit lingering near the position angle of thebar, caus-ing the effective bar strength, the quadrupole perturbation strengthspecifically, to increase. This growth extends the reach of the per-turbation, inducing more orbits to slow or reverse their precessionrate, resulting in an exponential growth for the bar. The fiducialsimulation is analysed in section3.

The fiducial simulation is an initially exponentially-distributedstellar disc embedded in a fully self-consistent non-rotating c=Rvir/rs≈15 NFW dark-matter halo (Navarro et al. 1997) wherers is the scale radius, andRvir is the virial radius (see the blackline and black hatched area in the left panel of Figure1 for theinitial condition rotation curve and tangential velocity dispersion,respectively):

ρ(r)=ρ0r

3s

(r+rc)(r+rs)2(1)

where ρ0 is set by the chosen mass andrc=0 is the core ra-dius for the fiducial model. The dynamical implications of a coreproduced by early evolutionary processes (Weinberg & Katz 2002,Sellwood & Debattista 2009) will be explored in section3 and 4through comparison to simulations with non-zero values ofrc. Thehalo velocities are realised from the distribution function producedby an Eddington inversion ofρ(r). Eddington inversion provides anisotropic distribution, roughly consistent with the observed distri-butions in dark-matter onlyΛCDM simulations. This method nat-urally results in a non-rotating halo; we will describe realising ro-tating haloes in section2.1.2.

For all the simulations here, we adopt an exponential radialdisc profile with an isothermal vertical distribution:

ρd(R,z)=Md

8πz0R2

d

e−R/Rdsech2(z/z0) (2)

where Md is the disc mass,Rd is the disc scale length, andz0 is the disc scale height. We setMd=0.025, Rd=0.01, andz0=0.001 throughout. We note that this simulation has a disc:halomass ratio of 1:40 inside of its virial radius1; while a compara-tively low mass ratio for studies in the literature (e.g. contrast-ing with Debattista & Sellwood 2000; Athanassoula & Misiriotis2002; Saha et al. 2012), we choose this mass ratio to mimic themass ratio for thez=0 MW, as determined through simulations(Vogelsberger et al. 2014), abundance matching (Kravtsov 2013;Moster et al. 2013), and stellar kinematics (Bovy & Rix 2013).The disc parameter values are consistent with the MW values ofBovy & Rix (2013), who foundM⋆≈4.6×1010, or 0.029 Mvir

(again using the scaling from (Boylan-Kolchin et al. 2013)); Rd=2.15 kpc andz0=370 pc, which are reasonably similar to our

1 To preserve equilibrium, the halo extends beyond a virial radius, whereit is truncated, but here we consider the mass within a virialradius for thisratio calculation.

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The Shadow Bar 3

0.00 0.015 0.0300.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Tangential Velocity

0.00 0.015 0.030Radius

0.00 0.015 0.030 0.045

Figure 1. Rotation curves (i.e. tangential velocity) of the fiducial simulation (black) and nonrotating, cored NFW simulation (red) at three times (see Table1). The shaded region encloses thep=0.33 andp=0.67 rank-ordered tangential velocity quantiles about the solid-line median (p=0.5) as a function ofcylindrical radius. Panels from left to right: the initial (T =0) rotation curves (whose means are nearly identical), rotation curves when the bar is forming(T =0.5), and at late times, when the bar is evolving slowly. Note thesteepening in the rotation curve at small radii and the flattening at larger radii as the barforms. For comparison, the velocities have been scaled suchthat the initial rotation curves have the same maximum circular velocity.

values. The purpose of the simulation was not to create an exactMW analogue, rather we sought to make a cosmologically realisticgalaxy.

Disc velocities are chosen by solving the Jeans’ equations incylindrical coordinates in the combined disc–halo potential. We setthe radial velocity dispersion from the Toomre stability equation,

σ2

r(R)=Q3.36GΣ(R)

κ(R), (3)

where G=1, Σ(R) is the stellar surface density, andκ, theepicyclic frequency (also sometimes written asΩr), is given by

κ2(R)=RdΩ(R)2

dR+4Ω(R)2 (4)

whereΩ is the azimuthal frequency (sometimes explicitly writtenas Ωφ in cylindrical coordinates). We chooseQ=0.9 to facili-tate comparisons to the literature (e.g.Athanassoula & Misiriotis2002) and to ensure the relatively rapid growth of a bar2. Thisdeparts from our previous choice in HB05 of awarm disc withQ=1.4. Also, we replace the axisymmetric velocity ellipsoid gen-erated from the epicyclic approximation in the disc plane used inHB05 by the Schwarzschild velocity ellipsoid (Binney & Tremaine2008). In practice, this latter velocity ellipsoid improves theinitialequilibrium by using the second moment of the cylindrical colli-sionless Boltzmann equation with an asymmetric drift correction:

σ2

φ=σ2

rκ(R)2

4Ω(R)2. (5)

As in HB05, the vertical velocity dispersionσz is

σ2

z(R)=1

ρd(R,z)

∫ ∞

z

ρd(R,z)∂Φtot

∂zdz. (6)

From a dynamical standpoint, our relatively low disk-to-halomass ratio accomplishes two additional goals: (1) it decreases thestrength of non-linear instabilities that allows us to focus on seculardynamics; and (2) it enables the exploration of theslow mode of bar

2 In the absence of a dark matter halo,Q=0.9 is formally unstable; in thepresence of the dark matter halo, this value results in a locally stable discthat forms a bar in several dynamical times.

growth (Polyachenko & Polyachenko 1996), seemingly not achiev-able givenmaximal discs. We examine the former in this paper andwill explore the latter in more detail in a forthcoming paper.

2.1.2 Halo model variants

We investigate two additional classes of halo models based on ourfiducial one: (1) halo models with a cored central profile instead ofa cusp3; and (2) rotating halos. These variants are inspired both bycosmological hydrodynamic simulations (Bullock et al. 2001), anda desire to connect with previous literature choices.

We construct cored halos by settingrc=0.02 in equation (1).The halo mass is matched between the cusp and core model halosat the virial radius to maintain the disc:halo mass ratio within thevirial radius. The resulting initial rotation curve for thecored sim-ulation is plotted in red in Figure1. The rotation curves have beenscaled to their maxima for comparison. The variation between thetwo rotation curves are approximately 5 per cent at any givenra-dius, yet as we shall see, the resonant dynamics are quite different,as the phase-space gradient of the distribution function isthe con-trolling parameter of the resonant density (seeSellwood 2014fora review). In a cored halo, the flattening of the central density pro-file creates a harmonic core, i.e. orbits at small radii wouldexecutesimple harmonic motion in the absence of a stellar disc. In practice,this increases the relative disc to halo density near the centre of thesimulation, leading to the well-known buckling instability. The dif-ferences between the fiducial and variant halo models are discussedin section4.1.

We construct rotating halos consistent with the cumulativemass distributionM for specific angular momentumj(M) in aspherical shell as proposed byBullock et al.(2001),

j(M)∝M1.3, (7)

normalised such that

λ=

∑

imi~ri× ~vi√

2MvirVvirRvir

=0.03. (8)

3 A cored central profile satisfiesρ(r)∝r−α whereα→0 asr→0; theunmodified NFW has a cuspy central profile withα=1.

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4 M. Petersen et al.

Simulation scale length (Rd) scale height (h) disc mass halo profile Notes

F 0.01 0.001 0.025 cuspFr 0.01 0.001 0.025 cusp,λ=0.03

C 0.01 0.001 0.025 cored,rc=0.02Cr 0.01 0.001 0.025 cored,rc=0.02, λ=0.03Ff 0.01 0.001 0.025 cusp fixed disc potentialFs 0.01 0.001 0.025 cusp shuffled halo

Table 1. List of simulations.

The valueλ=0.03 is roughly the median value in the distributioncompiled from cosmological simulations byBullock et al.(2001).We realise a distribution that satisfies equations (7) and (8) as fol-lows. First, we begin with a phase space for the initially nonrotatingspherical halo as described in section2.1.1. We randomly samplethe phase space of the initially nonrotating halo by mass as de-scribed in section2.1.1and choose particles from the distributionM(r) by rejection. Then, we change the direction of rotation for or-bits withLz<0 from the probability distribution determined fromequation (7) by changing the sign ofLz until the desired value ofλ is obtained. For ourN=107 haloes, this results in a deviationfrom the desiredλ=0.03 of less than 1 per cent. Changing thesign of any component of the angular momentum of a particularor-bit remains a valid solution to the collisionless Boltzmannequation,preserving the initial equilibrium of the system. Large values ofλmay result in an unstable halo (Kuijken & Dubinski 1994), but thisis not observed for the modestλ selected here, unsurprising giventhe results ofBarnes et al.(1986) andWeinberg(1991).

2.1.3 Two tests of the dynamical mechanism

In addition to the halo model variants, we examine two additionalmodifications to the fiducial model designed to test our physicalunderstanding of the dynamical processes. Such restrictedsimula-tions do not have analogues in the real universe, but can illuminatespecific processes.

In the first experiment, Ff in Table1, we exploit the separatebases of the stellar disc and dark matter halo in EXP (see belowfor a description of this N-body code) by allowing the dark matterhalo potential to self-consistently evolve while disallowing changesto the stellar disc potential. This allows us to investigatethe adia-batic compression of the initially spherical dark matter halo in re-sponse to the stellar disc potential. The results are compared to thatof the fiducial simulation in section3.2.2 to isolate the effect ofnon-axisymmetric disc evolution on the global halo properties.

In the second experiment, Fs in Table1, we artificially elimi-nate the trapping that leads to the shadow bar. Specifically,we shuf-fle the azimuth of halo particles in the fiducial simulation. The shuf-fling interval for an individual orbiti is chosen following a Poissondistribution with an average value of2Pφ,i wherePφ,i≡2π/Ωφ,i

is the azimuthal period for orbiti. This technique preserves the ra-dial structure of the halo and produces minimal disturbances to theequilibrium. The results of this experiment are used to quantify theeffect of trapping on classical dynamical friction torque and to cor-roborate our hypothesis of the shadow bar’s effect on the angularmomentum transport between the disc and halo in section3.4.

We verify that the shuffling process effectively eliminatestrapping by attempting to detect a trapped component using themethodology that will be described in3.2, finding that<0.1 percent of the orbits demonstrate a trapped signal, consistentwith un-certainties owing to resolution. An additional application of the

method to the fixed disc potential experiment yields a bar signalof <0.1 per cent, again reinforcing the robust nature of the methoddescribed below.

2.2 N-body simulations

We perform the system evolution using EXP (Weinberg 1999). Thiscode is designed to represent the gravitational field for multiplegalaxy components by a rapidly converging series of functions.EXP is advantageous for secular dynamics problems owing to itsabsence of high-frequency noise and its high-efficiency relative tofully adaptive Poisson solvers such as direct summation or treegravity algorithms.

These series of functions are the eigenfunctions of the Lapla-cian. For conic coordinate systems, the Laplacian is a special caseof the Sturm-Liouville (SL) equation,

d

dx

[

p(x)dΦ(x)

dx

]

−q(x)Φ(x)=λω(x)Φ(x) (9)

whereλ is a constant andω(x) is a known function, called eitherthe density or weighting function. These eigenfunctions have manyuseful properties. For systems with a finite domain, the eigenval-ues are real, discrete, and bounded from below; their correspondingeigenfunctions are mutually orthogonal, and oscillate more rapidlywith increasing eigenvalue. We will assume that the functions areindexed in order of increasing eigenvalue. For example, theeigen-functions of the Laplacian over a periodic finite interval are sinesand cosines; in cylindrical coordinates, they are Bessel functions.For a spherical geometry, the eigenfunctions satisfy Poisson’s equa-tion such that

1

4πG

∫

drr2d∗k(r)uj(r)=δjk (10)

where dk is the density anduj is the potential that togetherform a biorthogonal pair. A variety of analytic biorthgonalpairsare available (Clutton-Brock 1973, Clutton-Brock et al. 1977,Hernquist & Ostriker 1992, Earn 1996). Here, we followWeinberg(1999) and numerically solve equation (9) to obtain a basis whoselowest-order pair (n=0, l=m=0) matches the equilibrium profileof the initial conditions. Higher-order terms then represent devia-tions about this profile with successive higher spatial frequency,which can account for the evolution of the system.

The overall power owing to discreteness noise scales as1/N , and we may limit small-scale fluctuations by truncatingthe series, effectively providing a low-pass spatial filterthat re-moves high-frequency noise from the discrete particle distribution(Weinberg 1998). Although it is possible to derive an analogousthree-dimensional basis from the cylindrical Laplacian, the bound-ary conditions are complex and difficult to match to the approx-imately spherical domain of the halo. After much trial and error,we found that a empirical orthogonal function decomposition of ahigh-dimensional spherical basis produces better results. Weinberg

MNRAS 000, 000–000 (0000)

The Shadow Bar 5

−0.04

−0.02

0.00

0.02

0.04

−0.04

−0.02

0.00

0.02

0.04

Y virial

−0.04−0.02 0.00 0.02 0.04−0.04

−0.02

0.00

0.02

0.04

1 2 3 4

−0.04−0.02 0.00 0.02 0.04

Xvirial

1 2 3

Figure 2. The surface density of the fiducial simulation for the stellar disc (left panels) and the dark matter halo (right panels).The upper panels show thein-plane density (|z|<0.003), the middle panels show the in-plane density of the trappedcomponent (the stellar bar on the left and the shadow bar on theright), and the lower panels show the in-plane density of theuntrapped component, atT =2.0 (≈ 4 Gyr for the MW scalings).

(1999) and HB05 describe this method in detail. The field approachis also advantageous as it enables a direct comparison with pertur-bation theory. However, we are aware of two significant disadvan-tages of the field approach as well: 1) the technique cannot befullyadaptive and limit high frequencies simultaneously; and 2)it can besusceptible tom=1 (dipole) modes induced by unphysical ‘slosh-ing’ of the expansion centre against the centre-of-mass. While oursimulations show nonzerom=1 power, we verify that the expan-sion centre deviates from the centre-of-mass by less than0.1z0 atall times. We also note thatm=1modes are theoretically predicted(Weinberg 1994) and appear in real galaxies (e.g.Zaritsky et al.

2013) and hence do not necessarily belie a numerical fault. We willinvestigate these phenomena further in a future paper.

For this class of Poisson solver, the computational effort scaleslinearly with particle number, making low-noise simulations withhighN possible. A number of basis terms are kept to allow for as-tronomically realistic perturbations to the equilibrium profile whileminimising the linear least-squares fit to a Monte Carlo realisationof initial particle positions. We retain terms in the halo basis up tol=m=6, and a maximum radial index ofn=20 correspondingto the largest eigenvalues for the spherical harmonics of the halo.For the disc, we selectm=6, n=12 for the empirical orthogonal

MNRAS 000, 000–000 (0000)

6 M. Petersen et al.

functions. This basis selection allows for variations of0.1Rd (300pc for a MW-sized galaxy) in the vicinity of the bar, decreasing to5×10−5 (15 pc for a MW-sized galaxy) near the centre. In thisway, the solver naturally suppresses diffusion (i.e. two-body relax-ation) that can result in unphysically rapid evolution at small radii4.While possible, we do not recondition the basis on the evolvingequilibrium during simulations, relying instead on the initial Pois-son variance to allow for sufficient degrees of freedom.

Particles are advanced using a leapfrog integrator with anadaptive time step algorithm. Briefly, we begin by partitioningphase spaces ways such that each partition containsnj particlesthat require a time stepδt=2−jδT , whereδT is the largest timestep andj=0, . . . , s−1. The time step forj=s−1 corresponds toa single time-step simulation. Since the total cost of a timestep isproportional to the number of force evaluations, the speed up fac-tor is given by:S=

∑s−1

j=0nj/

∑s−1

j=0nj2

−j . For ac=15 NFW

dark-matter profile withN=107 multimass particles as describedin section2.1.1, we find thats=8 andS≈30, an enormous speedup! Forces in our algorithm depend on the basis coefficients andthe leap frog algorithm requires linear interpolation of these coef-ficients to maintain second-order error accuracy per step. The ex-pansion coefficients are partially accumulated and interpolated ateach levelj to preserve continuity as required by the numericalintegration scheme for finer time steps. This interpolationand thebookkeeping required for the successive bisection of the time inter-val is straightforward. The symmetric structure of the leap-frog al-gorithm allows the coefficient values to be interpolated rather thanextrapolated. This algorithm will be discussed in more detail in aforthcoming paper.

To motivate our choice of timesteps, we first define a charac-teristic force scale asdi≡Φi/|∂Φi/∂xi| whereΦi is the currentscalar potential of the particle. The time step level,j, for each or-bit is then assigned by the minimum of three criteria: (1)vi/ai,the characteristic force timescale, (2)di/vi, the characteristic worktimescale, and (3)

√

di/ai, the characteristic escape timescale. Thequantityvi is the current scalar velocity of the particle andai is thecurrent scalar acceleration of the particle. The criteria have pref-actorsǫf , ǫw, ǫe, respectively, which can be tuned to achieve thedesired time resolution. Using the initial distribution ofparticles inthe fiducial model, we tune the ratios of the prefactors basedonthe mean of the ratio of the calculated timescales for all thediscparticles. We find thatǫf/ǫw≈10, andǫw/ǫe≈2.5.

Weinberg & Katz(2007a) use numerical perturbation theoryto determine a scaling for particle timestep criteria, finding that1/100 of the period of oscillation appropriately recovers the res-onant dynamics.Weinberg & Katz(2007a) defined particle num-ber and time-step criteria to help ensure that the dynamics in thevicinity of the homoclinic trajectory are accurate for a very slowlyevolving system. The formal requirement of a slowly evolving sys-tem is implicit in the time ordering that enables the perturbativeanalytic estimates that underpin secular evolution. In this stan-dard formulation of secular evolution (e.g. dynamical friction),changes in otherwise conserved actions occur during their homo-clinic passage. For strong (typically low-order) resonances, theseregions are large and the criteria are easily satisfied. For weaker(typically higher-order) resonances, large particle numbers are re-quired. Using the fiducial model, we find that at a scale height,

4 Note that this method does not remove all the relaxation fromfluctuationsin the basis; this method will only remove relaxation on resolution scalessmaller than those probed by the basis.

r=0.001, the appropriate timestep ish61.8×10−4, similar to thetimestep chosen in HB05,h=2×10−4. To comply with the find-ings of Weinberg & Katz(2007a), we set the smallest timestep tobeh=1.25×10−4. We then tune the prefactors such that nearly allthe disc particles (>99.5 per cent) as well as halo particles in thevicinity of the disk reside in this level, while allowing other haloparticles to occupy larger timestep levels to take advantage of thecomputational speedup. We allow for four multistep levels (s=5)so that the maximum timestep is 16 times larger than the minimumtime step (hmax=2×10−3). We note that while many halo parti-cles at large radii do not even requireh=2×10−3 for 1/100th ofan oscillation, we truncate the multistep levels to match the outputfrequency,δT =0.002. While this decreases the speed up factor,we find that withs=5, S≈5, still a considerable speed up.

3 RESULTS FROM THE FIDUCIAL SIMULATION

This section characterises the fiducial simulation. We begin with adescription of the structure and long-term evolution of thebar pro-file and pattern speed in section3.1. We then motivate a new time-dependent tool for characterising orbits trapped into resonance andapply this tool in section3.2.1and3.2.2to the disc and halo pop-ulations, respectively. We show that the evolution of a stellar barin the presence of a dark-matter halo traps a large fraction of darkmatter orbits in the vicinity of the bar, resulting in a shadow bar.The implications and dynamical consequences of this trapping onthe evolution of the galaxy are presented in section3.3. This isfollowed by a discussion of the observational consequencesof theshadow bar in section3.4.

3.1 Formation and evolution of the bar

Figure 1 shows the rotation curves determined by rank orderingthe tangential velocities of particles in a narrow annulus at eachradiusR and selecting quantile values atp=0.33,0.5,0.67. Wedisplay the resulting curves for three characteristic times: the ini-tial state (T =0), immediately after initial bar formation (T =0.5)while the bar is still rapidly growing, and well-after bar formation(T =2.0). The fiducial curve is shown in black; for later compari-son, the cored, nonrotating halo model is shown in red (see section4.1). The heavy solid curve is the median (p=0.5) and the hatchedband describes the envelope betweenp=0.33 andp=0.67 to indi-cate the dispersion. At late times and for radiiR<Rbar(≈0.01),the curves withp=0.33 andp=0.67 approximately coincide withthe rotation curve along the bar major and minor axes, respectively.The bulk of our analyses focus on the evolution after a discerniblebar feature has formed, i.e.T >0.5. The rotation curve continuesto evolve after bar formation, as evidenced by the comparison be-tween the second and third panels of Figure1.

In Figure2, we plot the surface density of the disc and halo at alate time (T =2.0), well after the bar has formed. An in-plane den-sity slice of the halo reveals an approximately elliptical distributionaligned with the stellar bar, up to a maximum ellipticity ofe=0.3in the fiducial model. Because the disc bar dominates the gravita-tional potential at these radii, by itself, this is not particularly sur-prising (e.g.Colın et al. 2006; Athanassoula 2007; Debattista et al.2008). However, as we shall see in section3.2.2, a large fraction ofthis dark matter is dynamically trapped in the bar potential; that is,it now part of the bar! The dynamical nature of this halo feature andits evolutionary implications has not been identified previously. Ad-ditionally, a three-dimensional analysis of the disc showsthat the

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Figure 3. Upper panel: the pattern speed of the stellar bar as a function oftime for the fiducial simulation. Middle panel: the rate of change in the pat-tern speed as a function of time. Bottom panel: the characteristic radius ofthe bar that encloses 90 per cent of the trapped stellar orbits (R90) deter-mined byk-means analysis. The bar slows nearly linearly at all times whilethe bar lengthens.

initially spherical halo has, at late times, been flattened by the pres-ence of the disc potential particularly atR<0.01 (see section3.2.2for additional discussion).

We first describe the stellar bar by fitting ellipses to the pro-jected disk surface density at each time. The bar changes appre-ciably with time, both elongating and slowing over the course ofthe simulation. In the upper panel of Figure3, we plot the patternspeed of the bar,Ωp as a function of time. The pattern speed mono-tonically decreases with time, nearly linearly asΩp changes onlyslightly (middle panel).

This global, phenomenological view of bar evolution does notreveal the underlying dynamical details and their implications forthe long-term evolution of galaxies. Advancing beyond the tradi-tional ellipse and Fourier determinations of the bar is possible inN-body simulations, albeit difficult. The centrepiece of our analysisbelow is the ability to accurately examine the dynamical character-istics of individual orbits to construct a more nuanced understand-ing of bar-disc-halo dynamics.

3.2 Orbit taxonomy

Orbits mediate the transfer of angular momentum between large-scale components (e.g. the disc and halo). For systems dominatedby regular orbits, for which the derivatives of the potential canbe computed exactly (i.e. separable and regular systems), Fourier

transformations of particle coordinates can determine theorthogo-nal frequencies of orbits in cylindrical coordinates (Ωr, Ωφ, Ωz)and these frequencies allow for rapid and unambiguous determina-tion of orbits trapped into potential features. The most obvious ofthese features are resonances, commensurabilities of frequencieswith the pattern speed of the bar that satisfy the relation

lrΩr+ lφΩφ=mΩp (11)

Previous studies have analysed N-body simulations by fixingor‘freezing’ the gravitational potential after a period of self-consistentevolution and characterising the time-independent orbital struc-ture (e.g.Binney & Spergel 1982, Martinez-Valpuesta et al. 2006,Saha et al. 2012). If the rate of bar evolution is sufficiently smallcompared to the time required to cross the resonance zone (Ωp=O(ǫ) andIbar=O(ǫ) asǫ→0, whereIbar is the time derivative ofthe moment of inertia of the bar), the dynamics will be approx-imately described by secular theory (e.g.Tremaine & Weinberg1984, Weinberg 1985, Hernquist & Weinberg 1992). However, re-alistic potentials are continuously evolving, so one must attempt afinite-time analysis of secular processes to investigate the nonlin-ear effects. In other words, one must devise techniques thatcharac-terise orbits while the bar pattern speed and geometry change. Tothis end, we have developed ak-means classifier for orbits, quicklyand accurately decomposing particles into orbital components clas-sified by their commensurabilities. Briefly, thek-means technique(Lloyd 1982) iteratively sorts a collection of points intok clustersgiven a distance metric. For our application, we use a time seriesin ther andθ (azimuthal angle) positions of apsides (radial turningpoints,rmax) for a given orbit to determine its membership in thebar. As a first application of the method in this work, we limitouranalysis tok=2.

In a frame corotating with the bar, the apsides of orbits trappedby the bar do not precess through2π. That is, the azimuthal posi-tion of outer turning points for trapped orbits oscillate about somefixed position angle in the bar frame of reference. We exploittheseslowly changing apse positions for trapped orbits to identify coher-ent bar-supporting orbits over a finite time period. Specifically, wedefineδθ‖ to be the absolute value of the difference in azimuthalangle between the outer turning point of a particular orbit and thebar position angle. Let〈δθ‖〉20 be the average value ofδθ‖ in 20azimuthal periods for a given orbit. We use thek-means techniqueto identify orbits with〈δθ‖〉20<π/8. Orbits that are periodic andaligned with the bar in its corotating frame have〈δθ‖〉20=0 whilethose with no correlation have〈δθ‖〉20=π/4. Additionally, rele-vant resonances trap orbital apsides perpendicular to the bar. In thiscase, we define an analogousδθ⊥. In practice, this technique allowsfor rapid determinations of bar membership over time windows thatare too brief for frequency determinations using Fourier techniques.In many cases, the bar evolution is so rapid that an analysis in afrozen potential would not be relevant.

After testing a range of period windows, the choice of 20 az-imuthal periods was chosen as a balance between minimizing sam-pling noise while retaining temporal resolution of the evolution-ary time scale. Our method is applicable to all but the most ex-tremely centrally confined particles whose azimuthal periods aretoo small to satisfy an approximate Nyquist criterion. For the simu-lations here, this critical azimuthal period isTθ=0.004 and affectsapproximately 0.1 per cent of the orbits.

For this paper, we will abbreviate our discussion by assumingthat all orbits trapped by the bar potential are 2:1 orbits–those thatexhibit two radial periods in every azimuthal period, identified byk=2. In reality, the primary resonance bifurcates into 4:1 and 2:1

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Figure 4. Trapped populations by fraction of mass interior toR=0.01 (the initial scale length), as determined byk-means power, for the fiducial simulation.Left panel: disc trapping as a function of time. Middle panel: halo trapping as a function of time. The trapped halo fraction (the shadow bar) monotonicallyincreases at all times. Right panel: The ratio of the shadow bar to stellar bar (by total mass) as a function of time. The halo is trapping at a faster rate than thedisc, owing to the consistent increase in the reservoir of untrapped matter as the halo contracts in response to the bar growth.

orbits at high bar strength near the ends of the bar (Athanassoula1992). An in-depth discussion of thek-means identifier and its ap-plication to orbital family determination will be presented in a laterpaper.

3.2.1 Disc orbits

Since our understanding of secular evolution is informed byper-turbation theory, we describe the orbital behaviour in the disc byconserved quantities; here we choose energy and the in-plane an-gular momentum,E andLz . At radii larger than the corotation ra-dius (hereafter, CR)5, orbits have low ellipticity (e<0.1). Inside ofCR, the ellipticity increases owing to the bar. The bar-parenting or-bit family (x1 in the notation ofContopoulos & Papayannopoulos1980) is an orbit family of precessing ellipses aligned with the bar.These arise from the inner Lindblad resonance (ILR)6 in pertur-bation theory. Such orbits are easily identified with thek-meansidentifier.

As a first application of thek-means method, we identify theradius that encloses 90 per cent of the bar-trapped disc orbits asthe characteristic bar radius,R90. The bottom panel of Figure3shows that the radius of trapped orbits increases monotonicallywith time, corroborating previous works indicating the lengthen-ing of bars over time. The trapped fraction of disc orbits satisfyingR<0.01 is an explicit measure of the bar mass and, therefore, thebar potential (see the left-hand panel of Figure4). The fiducial sim-ulation exhibits a monotonic increase in the trapped fraction withtime. Further, the growth of the bar can be roughly divided intothree phases: formation, fast growth, and slow growth. The first,formation (T <0.5 in Figure 4), is not studied in this work. Weinstead begin our study of orbits at times after which the barcanbe detected through ellipse fitting, the fast growth phase7. In thisphase (0.5<T <1.2), the bar rapidly gains mass and significant

5 The CR resonance hasm= lφ=2, lr=0 in the notation of equation11.The CR radius is≈0.02 in the fiducial simulation atT =2.0.6 The ILR resonance hasm= lφ=2, lr=−1 in the notation of equation117 The trend at0.5<T <1.0 cannot be extrapolated to T=0.0, suggest-ing that the growth rate at0.0<T <0.5 is more rapid than in the range

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Figure 5. The halo wake ((m>0)+(m=0,n 6=0), filled colour) with thebar position and length indicated as the thick black line. The wake lags be-hind the bar (rotating clockwise) atR>0.5Rd, but is generally alignedwith the bar within0.5Rd, where we observe significant trapping (seeFig. 4). Note the resemblance to the untrapped component of the disc inFig. 2.

m=2,4,6 patterns are observed beyondRd. In the third phase(T >1.2), the bar continues to add mass (albeit at a diminishedrate), but the disc is relatively devoid of other (m 6=2) features.

3.2.2 Halo orbits

The structure of the dark matter distribution in the stellardisc hasrecently been debated, with particular discussion regarding the cre-ation of a co-rotating ‘dark disc’ (Read et al. 2008; Bruch et al.2009; Read et al. 2009; Purcell et al. 2009; Pillepich et al. 2014).This dark disc is formed by the accretion of satellites that settle intoa rotating structure through dynamical friction, affecting the localdark matter density by contributing up to 60 per cent in some simu-lations (Ling 2010), but very little in others (Pillepich et al. 2014).

0.5<T <1.0. However, this cannot be determined through thek-meanstechnique due to time resolution limits.

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The Shadow Bar 9

Additionally, the halo is shown to be prolate at small radii in thepresence of a stellar bar, referred to as the ‘halo bar’ or ‘dark mat-ter bar’ (Colın et al. 2006; Athanassoula 2007; Athanassoula et al.2013; Saha & Naab 2013), thoughDebattista et al.(2008) demon-strated that if the disc were evaporated, the prolate inner region ofthe halo would recover its initially spherical shape.

In our fiducial simulation, the dark-matter halo becomesoblate near the disc at late times. We also refer to this as the‘dark disc’, while noting the difference in formation mechanismfrom previous authors. This dark disc is independent of previousstatements about the dark disc: the feature we observe in oursim-ulations is a necessary consequence of the gravitational potentialof the disk instead of dependent on accretion history. We there-fore expect this dark matter feature to be a generic consequenceof stellar discs. AtT =2.0, long after the bar has formed, and atR=0.03, approximately the radius where the disc and halo con-tributions to the rotation curves are equal, a determination of theellipsoidal shape of the halo following the method ofAllgood et al.(2006) findsS≡c/a=0.45 (as well asQ≡b/a=0.8), wherea, b,andc are the lengths of the major, intermediate, and minor axes, re-spectively. While some of this flattening is attributable toadiabaticcompression of the halo in response to the presence of the stellardisc, a test simulation that allows the halo to achieve equilibriumwhile holding the initial disc potential fixed demonstratesthat adi-abatic compression alone, which givesS=0.6 andQ>0.99, can-not account for their deviation from the initially spherical distribu-tion. In the plane of the disc, the effect is even more pronounced;at R=0.03 for all T >1.0, the azimuthal average of the densityin the disc plane is>30 per cent larger than the initially sphericaldistribution8.

Examining the evolution of the halo flattening in the fiducialsimulation reveals that the initial response of the halo to the pres-ence of the disc is complete byT =0.3, in the midst of bar forma-tion; atT =0.3 andR=0.03 in the fully self-consistent simulationwe find thatS=0.6 andQ=0.95. The similarity of these values tothat of the settled halo in the fixed disc potential test at late timessuggests that the evolution of the disc is responsible for the contin-ued compression and the move to triaxiality. Further, the continuedevolution of the halo shape suggests that simply including adiabaticcompression in halo models is insufficient to recover the dynamicsof the stellar disc and halo.

The distribution ofcosβ≡Lz/L also illuminates the flatten-ing of the halo. A perfectly spherical halo will exhibit uniformcoverage incosβ owing to the random orientations of the orbitalplanes. However, at high binding energy (small radii), where thedisc dominates the potential, the halo shows enhanced populationsat cosβ≈1. Taken together, the continued flattening of the haloand the distribution ofcosβ suggest that the continued depositionof angular momentum into the halo owing to secular torques fromthe bar is the cause of the dark disc in our simulations.

Clearly, the perturbative effect of the stellar bar on the halocannot be ignored. To investigate the secular response of the haloorbits to the bar, we apply thek-means analysis to these orbits aswell and find a class of 2:1 orbits that respond similarly to the baras their disc analogues. In other words, the presence of the stel-lar disc traps initially spherical halo orbits into not onlyplanar or-

8 The density along the polar axis at all times is nearly equal to the ini-tial distribution atR>0.01. At R<0.01, the potential of the disk causesspherical adiabatic contraction such that the halo densityalong the polaraxis is also enhanced relative to the initial distribution.

bits, but into bar-supporting orbits that we call theshadow bar. Weemphasise that this result is qualitatively different fromthose ofDebattista et al.(2008), whose orbits are necessarily untrapped ifthey retain their box-like nature relative to the triaxial halo uponevaporation of the disc. We cannot comment on the likelihoodof asimilar box orbit-retaining outcome for other studies thatpoint outa bar-like feature in the halo (Colın et al. 2006; Athanassoula 2007;Athanassoula et al. 2013).

The growth rate of trapped halo orbits also follows that of thestellar bar, albeit with a smaller fraction of the halo mass avail-able for trapping atR<0.01 (the middle panel of Figure4). Thecontinued growth of the trapped fraction during the simulation cor-roborates that halo trapping is secular and not merely a result ofthe initial settling of the halo in response to the stellar disc. In therightmost panel of Figure4, we plot the ratio of dark matter trappedmass to stellar trapped mass (the total for the simulation rather thanthat scaled to the mass interior toRd). This also rises monotoni-cally throughout the simulation, owing to both the presenceof alarger reservoir of planar material as the dark disc grows with timeand the increasing strength of the bar potential with time. The resultis that the shadow bar grows at a faster rate than the stellar bar.

Overall, the trapped halo orbits exhibit the same behaviourastheir trapped disc counterparts: bar-supporting orbits are found atlarger radii through time. The trapped orbits in the halo are5-10per cent of the trapped mass of disc orbits, and 3-5 per cent ofthetotal trapped angular momentum. That is, the shadow bar is not alarge contributor to the mass of the visible stellar bar. However, theexistence of shadow bar modifies the rate and location of angularmomentum exchange between the bar and halo. Simple perturba-tion theory calculations following the model presented inWeinberg(1985) demonstrate that the halo orbits that acceptLz from the diskare preferentially located nearcosβ=1 and these orbits are theones that are efficiently trapped into the shadow bar. This suggeststhat the shadow bar will decrease the angular momentum transportfrom the bar to the halo relative to that predicted by dynamicalfriction theory. We will estimate the magnitude of these effects insection3.4. In the case of a dark disc formed through satellite ac-cretion, whose distribution would be concentrated nearcosβ=1rather than evenly distributed incosβ, we would expect to observeenhanced trapping relative to what is presented here.

Moreover, the overall contribution of the dark matter wakehas a very long reach. To see additional effects of the shadowbar on the halo at large scales, we plot the in-plane wake (as inWeinberg & Katz 2007a) at T =2.0 in Figure5. We find the halowake by removing all the trapped orbits as found by thek-meansanalysis and comparing the resulting distribution to the lowest orderaxisymmetric density profile. To accomplish this, we constructed anew basis using only the untrapped orbits (as described in section2.2) and then removed the lowest order (m=0, n=0) componentto find the wake9. Despite removing the trapped orbits, we observea strong quadrupole feature that follows the stellar bar (along thex-axis) and extends to the end of the stellar bar (the extent ofthestellar bar is shown as the black line). Beyond the extent of the bar,anm=2 pattern is observed, a direct result of the secular evolutiondetailed inWeinberg(1985). If the disc scale length of the simula-tion is scaled to that of the MW, this wake extends out to the Solarcircle at approximately the 10 per cent level. This scaling of thesimulation may not be the most appropriate due to the apparent dis-

9 Note that in addition to the non-axisymmetricm 6=0 components, weincludem=0, n 6=0 terms in the wake calculation.

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10 M. Petersen et al.

agreement between the bar length-scale length relationship in oursimulation and the MW. If the simulation is instead scaled tothebar length ofWegg et al.(2015), 4.6 kpc, the density enhancementfrom the wake may be as large as 25 per cent!

3.3 Dynamical properties of orbits

To understand the dynamics of the trapped halo orbits, we inves-tigate the distribution of conserved quantities in the various orbitpopulations. We characterise the orbits by energyE, an isolating in-tegral for all regular orbits, andLz/Lmax(R), where we normalisethe angular momentum perpendicular to the disc,Lz, by the angu-lar momentum of a circular orbit with the same energy,Lmax

10.Figure6 shows the distribution of〈δθ‖〉20 for stellar orbits fromthe fiducial simulation atT =2.0. Orbits with 〈δθ‖〉20<π/8 re-liably indicate bar-trapped orbits. As expected, the strongest sig-nal comes from mildly elliptical orbits withLz/Lmax(R)=0.5near the end of the bar withE=−9.5 (R=0.01). Many of the or-bits within a characteristic radius of the bar are trapped (exhibiting〈δθ‖〉20<π/8), though Figure4 indicates that not all orbits withinR=0.01 are trapped.

A similar exercise performed for the halo (Fig.7) reveals thattrapped halo orbits occupy the same region inE−Lz parameterspace as the trapped disc orbits. Because the fiducial halo isinitiallyisotropic the quantityLz/Lz,max(R) can have values from -1 to1. The shadow bar reveals itself by the prominent〈δθ‖〉20<π/8feature in the upper left corner of Figure7 at E<−9 (R<0.01).This reinforces our finding that both dark-matter and stellar orbitsare trapped by their mutual gravitational potential, and that the discand the halo cannot be treated as distinct dynamical components.

In addition to the shadow bar, Figure7 reveals two pla-nar retrograde families exhibiting〈δθ‖〉20>π/3 (or equivalently〈δθ⊥〉20<π/6), which correlate with overdensities in Figure5 andare associated with the ILR and CR resonances. Similarly, weob-serve the outer Lindblad resonance (OLR,m= lφ=2, lr=1 in thenotation of equation11) as a prograde planar family atE=−6.5(R=0.04), observable as an overdense region with〈δθ‖〉20>π/3in Figure5. The approximate position of the resonances were lo-cated by using the monopole potential field of the disc and halofrom the basis so that equation11could be solved numerically as afunction ofE andκ when combined with the instantaneous patternspeed of the bar (see Figure3)11. We further confirmed the resonantnature of orbits in theseE−κ regions by examining individual or-bits.

To summarise the previous sections, the initially nearlyisotropic dark matter halo evolves significantly in the presence ofan evolving disk. The effects are three-fold: (1) the trapping of darkmatter orbits into the shadow bar; these orbits subsequently behavejust like stellar bar orbits; (2) the dark disc, a response tothe pres-ence of the stellar disc as well as secular torques from the stellarand shadow bar; these orbits do not resemble the stellar discper se,but more importantly do not resemble initial halo orbits; and (3) thedark matter wake, created by the influx ofLz from the stellar disc.

10 The quantity, Lz/Lmax, is equivalent to (cosβ)κ, wherecosβ≡Lz/L and κ≡L/Lmax, as defined in the literature (e.g.Tremaine & Weinberg 1984)11 Owing to the choice of a spherically symmetric system for theresonancedetermination, the tilt of the rotation plane, and thus our approximation forthe location of resonances, does not depend oncosβ.

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Figure 7. As in Fig. 6 for the dark matter halo. Retrograde orbits are nowpresent and theLz/Lmax(R) axis now runs from -1 to 1. The dark blueregion in the upper left, with an x-hatched region overlaid,indicates orbitstrapped into the shadow bar, which occupies the same region of E−Lz

space as the stellar bar (compare to Fig.6). Additional features from thek-means analysis are seen, including two retrograde populations atE=−9.5andE=−8.2 that are transiently moving through the ILR and the CR,respectively. A prograde population is observed at OLR,E=−6.5. Thesepopulations correspond to the wake overdensities in Fig.5.

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Figure 8. Left panel: the in-plane density ratio for axisymmetric densityprofiles at three pairs of times. The black line comparesT =2.0 withT =0.0 (the total change in the profile during the simulation), the red linecomparesT =0.3 to T =0.0 (the change in the halo distribution inducedby bar formation), and the grey line comparesT =2.0 to T =0.3 (changescaused by secular evolution). The halo has dramatically redistributed itsdensity in response to the presence of a stellar disc byT =0.3, but continuesto evolve secularly owing to the presence of the bar. Outsideof R=0.05,the fractional density difference rapidly falls to zero. Right panel: the in-plane fractional density ratio parallel to the bar and perpendicular to thebar, compared to the axisymmetrised profile, for two times. The on-bar ra-dial cut shows an enhancement at a>5 per cent level withinR=0.02 fol-lowing bar formation (T =0.3), and a>10 per cent enhancement withinR=0.02 after further evolution (T =2.0).

3.4 Observational consequences of the shadow bar

Although the trapped dark-matter mass fraction is small, even asmall change inLz for halo 2:1 orbits affects both the structureof the dark matter halo and the evolution of the stellar disc in oursimulations, and these dynamics are reflected in observableproper-ties of each. In this subsection, we discuss the global implicationsof several facets of the dark-matter halo response, focusing first onchanges to the density and velocity structure, effects thatmight betraced by halo stars (e.g. observable by Gaia), and then turnto im-plications for the fast bar–slow bar formation scenarios.

3.4.1 Density and velocity structure

The bar changes the density and velocity structure of the darkmatter halo. The dark disc in the fiducial simulation appearsasan in-plane overdensity that grows with time. However, the non-axisymmetric structure of the shadow bar also contributes to den-sity variations in the plane. Figure8 demonstrates these two ef-fects at play. In the left panel, we plot the axisymmetric density(azimuthally averaged) ratios for three pairs of times to directlyexplore the temporal evolution of the planar density without theshadow bar. Over the course of the entire simulation, fromT =0to T =2 (δT =4 Gyr scaled to the MW; black curve), the in-planedensity out toR=0.05 (15 kpc for the MW) increases everywhereby >8 per cent . Inside ofR=0.01, the density increases by up to20 per cent. For comparison, the evolution fromT =0 toT =0.3 isshown in red, demonstrating that the majority of the azimuthally av-eraged evolution atR>0.015 (4.5 kpc for the MW) happens as thebar is forming. The evolution fromT =0.3 to T =2 (δT =3.4 Gyrfor the MW), shown in grey, corroborates this, and shows thatthein-plane density atR<0.015 increases dramatically as the bar con-tinues to grow.

The right-hand panel of Figure8 examines the density onand off the bar axis, relative to the axisymmetric density, at two

different times. The dark matter distribution is significantly non-axisymmetric atT =0.3, in the midst of bar formation, includingan enhancement of up to 17 per cent atR=0.005 (1.5 kpc for theMW). At T =2.0, the asymmetry is more pronounced enhancingthe overdensity in the direction of the bar to 10 per cent out toR=0.02 (6 kpc for the MW) and to 5 per cent out toR=0.035(10.5 kpc for the MW), as well as shifting the peak overdensity toa larger radius as compared toT =0.3. Although the azimuthallyaveraged profile remains roughly unchanged atR>0.015, the barproduces significant non-axisymmetric halo evolution at all rele-vant disc radii.

Further, the isotropic nature of the dark matter halo velocityin the plane of the disc has been significantly altered through aninfusion of Lz from the disc. In the left panel of Figure9, weplot the tangential velocity distributions on- and off-barfor par-ticles (black and red curves, respectively) near the end of the bar(0.008<R<0.01) atT =2. For comparison, the dashed black lineindicates a normal distribution with the same tangential velocitydispersion, centred at zero velocity. Along the bar axis, the peakof the tangential velocity (Vθ) distribution is shifted byδVθ=0.2(30 km s−1 for the MW). Even off the bar axis, the peak ofthe tangential velocity distribution has been shifted byδVθ=0.1(15km s−1 for the MW). Additionally, the tails of both the on- andoff-bar distributions are enhanced relative to a normal distributionowing to secular processes reshaping the halo.

In the right panel of Figure9 we again plot the tangential ve-locity distribution for on- and off-bar populations, but now compareit to the disc. We plot the on-bar disc distribution in the same an-nulus near the end of the bar as the solid grey line and the axisym-metric velocity distribution of velocities in the disc as the dashedgrey line12. The on-bar disc particles are shifted to higher tangen-tial velocities (Vθ,bar=0.8=120 km s−1 for the MW, including ashift between peaks ofδVθ=0.5=75 km s−1 for the MW), whichthe halo reflects, albeit at a smallerδV (as discussed above). Theon-bar disc distribution also shows an enhancement relative to theaxisymmetric distribution atVθ=1.4 that is mildly reflected in thehalo. The particles trapped into the bar potential librate around thepattern speed as the flattening and trapping of dark matter particlescorrelate the velocities to create a non-zero mean velocitydistribu-tion that is particularly enhanced along the bar axis (in addition tothe generic rotation seen in the off-bar population). In other words,the theLz distribution has been modified, with the enhancementmost clearly seen along the bar major axis. We make predictionsfor the net effect of the above considerations on dark matterdirectdetection experiments inPetersen et al.(2016).

Although our fiducial model is motivated byΛCDM simu-lations (halo) and direct observations (disc), it represents a sin-gle realisation from a particular model family. We emphasise thatthese observational conclusions are likely to vary even within amodel family. However, we expect that the dynamical nature ofthe density and velocity differences driven by the shadow bar willbe generic. To reduce the uncertainty of these simulation-derivedquantities for application to Earth-based dark matter detection ex-periments, at least three additional pieces of informationare criti-cal. First, both the density and velocity signatures will beaffectedby the assumed length of the MW bar and its evolution as a func-tion of time. Relative to the disc scale length, our bar is shorter

12 5 per cent of the disc orbits at the end of the bar are retrograde, reflectingprecession toward the bar that causes the tangential velocity in the inertialframe to be negative.

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12 M. Petersen et al.

−2 −1 0 1 2Tangential Velocity

0.00

0.02

0.04

0.06

0.08

0.10Frequency

−2 −1 0 1 2

Figure 9. Left panel: Tangential velocity distributions atT =2.0 on- (solidblack) and off-bar (red) for the dark matter halo, atR=0.008 (near theend of the bar). The dashed black line indicates a normal distribution. Thetangential velocity distribution on the bar axis (solid black) shows both anenhancement centred nearVc=0.8(120 km s−1 for the MW ), approxi-mately the bar velocity at this radius, as well as a shift in the overall velocitypeak. Both curves show enhanced tails relative to the normaldistribution, atvelocitiesVc<−1.5. Right panel: The same dark halo tangential velocitydistributions as the left panel are plotted, along with the disc on-bar distri-bution and axisymmetric distribution shown as the solid grey and dashedgrey lines, respectively (scaled by 0.6).

than the observed MW bar (Wegg et al. 2015); also, the MW barmay have been longer in the past depending on its formation mech-anism. Therefore, dark matter at the solar radius maybe havebeeninfluenced more readily. Secondly, we have demonstrated that thevelocity structure depends on the position angle of the bar,so weneed an accurate determination of the MW bar position angle.Fi-nally, we need the formation time of the MW bar. The shadow barcontinues to grow in time relative to the mass of the stellar bar.Therefore, an older MW bar would have more trapped dark mat-ter material relative to the stellar bar. An older MW bar may alsohave a different velocity structure due to trapped dark matter orbitscontributing to the torque on the untrapped halo component and theefficiency ofLz transfer from the enhanced mass of the shadow bar.

3.4.2 Angular momentum transport

The shadow bar contains a large fraction of the dark-matter or-bits at low inclination:>50 per cent of orbits withR<0.01 and0.9<cosβ61 are trapped. These are the same orbits responsi-ble for accepting angular momentum and slowing the bar as de-termined by examining the model ofWeinberg(1985). We expectthat this will diminish the torque relative to the canonicalpertur-bation theory results (e.g.Tremaine & Weinberg 1984). We testthis speculation by azimuthally shuffling the halo orbits asde-scribed in section2.1.3. The purpose of this simulation is to pre-vent the shadow bar from forming and mimic the standard dark-matter halo dynamical friction scenario. Figure10 compares theresults of the fiducial simulation to the azimuthally shuffled ex-periment. The azimuthally-shuffled halo is able to transferangu-lar momentum continuously, evident through the rapidly decreas-ing pattern speed during the evolution. Given the assumption that∆L≈∆Ωp for a stellar bar with the same geometry, we findthat∆Ωp,shuffled/∆Ωp,self−consistent≈3, an appreciable increasein torque. Clearly, inhibiting the formation of the shadow bar re-sults in a strong torque on the bar by the halo. We only presentthe results of the simulation forT <1.2 because after this time theresultant stellar bars no longer qualitatively resemble each other

0

20

40

60

80

100

Ωp (ra

d/t)

0.0 0.4 0.8 1.2Time

−100

−80

−60

−40

−20

0

Ωp

Figure 10. Upper panel: the pattern speed for the fiducial simulation (black)and an azimuthally-shuffled halo to suppress shadow bar formation (Fs; thedashed blue line). The simulation time is limited toT <1.2 so that thestellar bars that form are still roughly equivalent. Lower panel: Change inpattern speed, following the same colour scheme. The shuffled simulationslows much more rapidly during the simulation, indicative of an increasedLz acceptance by the halo when there is no shadow bar.

(Ibar,shuffled 6=Ibar,self−consistent whereI is the moment of inertiaof the bar), making the comparison of the simulations unfair.

The shadow bar does not eliminate the torque altogether, ofcourse. A massive bar, enhanced by the dark matter, will couplemore strongly to higher order resonances, even though theseres-onances are much weaker. The relative strength can be roughlyapproximated by considering the inverse of the largest windingnumber in the resonant equation (the maximum of[m, lφ, lr]in equation 11) for a particular resonance. For example, the4:2:1 resonance will be approximately half as strong as the2:2:1 resonance, and so on, for all realistic distribution functions(Binney & Tremaine 2008). However, the primary (low-order) res-onances are weakened by the reduction in available dark matter toaccept angular momentum as more halo material becomes trapped.An enhanced dark disc, fed through satellite accretion, could actas a larger reservoir for material to trap, with presently unexploredimplications for the torque.

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The Shadow Bar 13

0.00.20.40.60.81.0

Disc

0.000.050.100.150.20

Halo

0.000.050.100.150.20

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Fr

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Trapped Fraction, R<0.01

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C

0.0 0.5 1.0 1.5 2.00.00.20.40.60.81.0

0.0 0.5 1.0 1.5 2.0Time

0.000.050.100.150.20

0.0 0.5 1.0 1.5 2.00.000.050.100.150.20

Cr

Figure 11. From left to right: stellar disc trapping, dark matter trapping, and the ratio of dark-matter halo to stellar disc trapping by the bar insideR=0.01.Compare these to Fig.4. From top to bottom the simulations are: the cusped rotatinghalo (Fr in Table1), the cored nonrotating (C), and the cored rotating(Cr) models. Black lines indicate the total mass trapped scaled by the mass interior toR=0.01 and grey lines indicate the corresponding curve for the fiducialmodel.

4 COMPARISON BETWEEN OUR FIDUCIAL ANDOTHER MODELS

4.1 Halo initial conditions

Our fiducial model is a simple representation of a disc galaxyinthe ΛCDM formation scenario: a cuspy dark-matter halo profilewith an exponential stellar disc. However, there are a variety ofalternatives that are commonly explored in the literature.These in-clude models whose density approaches a constant at some char-acteristic core radiusrc and rotating halo models (models with anon-zero value ofLz). The former, which have some observational(e.g. rotation curves,Breddels & Helmi 2013) and theoretical mo-tivation (e.g. satellite accretion producing cores, blackholes, andfeedback from star formation or supernova;Tremaine & Ostriker1999, Mashchenko et al. 2006, Governato et al. 2012) have oftenbeen used in N-body bar simulations (Athanassoula & Misiriotis2002, Martinez-Valpuesta et al. 2006, Saha et al. 2010). The lat-ter are motivated by both evolutionary processes such as merg-ing (Dekel et al. 2003) and the hierarchical formation process it-self (Navarro et al. 1997, Bullock et al. 2001). In this section, webriefly summarise the variation in the properties of the discand theshadow bar for a representative member of each of these two modelclasses.

Table1 describes the halo models we tested in addition to thefiducial cuspy non-rotating model. The resultant disc and shadowbar evolutions are shown in Figure11, the analogue of Figure4.The qualitative evolution is similar between all four simulations,suggesting that the results presented in section3 are generally rel-evant. In particular, the shadow bar continues to grow relative tothe stellar bar in all the simulations (the right panel of Figure11).The fiducial model exhibits the strongest stellar and shadowbars atlate times as a fraction of the available mass, but the trend in eachmodel suggests that an equilibrium has not been reached atT =2.0,the end of the simulations. (Recall that one system time unitis ap-proximately 2 Gyr scaled to the MW, so thatT =2.0 is roughly 4

Gyr or 40 per cent of the age of the Galaxy.) The cored halo mod-els exhibit a buckling instability atT =0.6 in the cored nonrotatingmodel, and atT =1.0 in the cored rotating model. This instabilityreleases disc orbits trapped during the bar formation phase, albeita relatively small fraction, and does not appear to reduce the dark-matter fraction relative to the trapped disc fraction.

Overall, the differences in evolution between the models canbe described as follows: (1) the cored halo increases the initialstrength of the stellar bar through a stronger instability (rapidgrowth atT <0.6). However, the relative strength of the shadowbar is smaller and stagnant at late times owing to the reducedcen-tral density, and consequently, a smaller population of halo orbitsthat are available to trap; and (2) the increased rotation ofa haloaffects models by reducing the strength of the bar at early times(diminished the growth betweenT =0.5 andT =1.0); the trappingof the bar and shadow bar takes longer to dynamically mature and,consequently, has not evolved to be as massive over the same spanof time. This effect is smaller in the cored rotating halo compared tothe cusped rotating halo, suggesting that the cuspy or corednatureof the halo dominates the evolution when compared to rotation inthese models. We will discuss these trends further in a future paper.

4.2 Disc Maximality

Our fiducial model is a submaximal disc, chosen to correspondtocosmological simulations. Simulations with maximal discsare sub-ject to a violent disc instability that results in a qualitatively differ-ent bar formation scenario than that of our fiducial model. The vi-olent disc instability may create a noisy bar formation process thatinhibits the formation of the dark disc required for the stellar bar tothen trap into the shadow bar.

It is also likely that the relative contributions to the potentialfrom the disc and halo will change the formation efficiency oftheshadow bar, an effect we have already seen qualitatively in the con-trast between the cusp and core simulations. In the simulations with

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14 M. Petersen et al.

cores, the halo material interior toRd is approximately half that inthe cusp simulation, so that even when the same fraction of ma-terial is trapped (e.g. the middle panel of the cored simulation inFigure11), the contribution to the overall bar potential is low (e.g.the right panel of the cored simulation in Figure11). Further, amore violent formation of the bar may change the structure ofthedark matter wake. In this bar formation scenario, the formation ofthe dark disc and subsequent shadow bar trapping, both of whichare slow processes, may be disrupted. A study of the implicationsof stellar disc maximality for the shadow bar and dark matterwakewill be investigated in future work.

5 CONCLUSION

Our main findings are as follows:

(1) The stellar bar traps dark matter into ashadow bar. Thisshadow bar has a mass that is>6 per cent of the stellar bar massafter the bar forms, and this ratio increases with time throughout thesimulations to values>9 per cent by mass. The halo is deformedby the presence of the disc as well as the continued torque fromthe stellar and shadow bar, creating a population of disc-like orbits(with a preferred angular momentum axis). We suggest that thisreservoir increases the trapping rate of dark matter orbits.

(2) Trapping in the dark matter halo and the stellar disc takesplace both at bar formation and throughout the simulation.

(3) The existence and strength of the shadow bar does notchange appreciably in the presence of a core or rotation in the halo.However, the trapping rate of both the dark matter halo and thestellar component strongly depends on halo profile and the initialangular momentum distribution in the halo.

(4) The dark matter halo exhibits a density and velocity signa-ture indicative of a reaction to the presence of the stellar disc atradii much larger than that of the bar radius. The density andve-locity structure change with the angle to the bar, even at several barradii. This could have important implications for direct dark matterdetection experiments.

(5) Approximately 12 per cent of the halo inside of the bar ra-dius13 is trapped after 4 Gyr. These trapped orbits are preciselythe ones that dominate the secular angular momentum transfer tothe halo in the absence of trapping. The trapping changes thesecu-lar angular momentum transfer rates that one would estimatewith-out trapping. We demonstrate this point with a simulation that sup-presses trapped orbits by artificially decorrelating the halo orbit ap-sides with the bar position. This increases the halo torque on the barafter formation by a factor of three! Stronger bars are likely pro-portionally larger fractions of their halos. This suggeststhat halotorques may be much smaller than theoretically predicted, espe-cially for strong bars, helping to resolve the tension between theΛCDM scenario and the observational evidence for rapidly rota-tion bars.

In concept, the idea of trapped dark matter is striking in itssimplicity: when disc orbits and halo orbits occupy similarplacesin phase space, they will respond similarly to a perturbation. Wehave only begun to explore the parameter space necessary to char-acterise dark matter trapping, as well as the implications for theevolution of isolated galaxies. Therefore, we expect that other sim-ulations have shadow bars as well, but previous studies havenot

13 Using the spherical distribution of dark matter; the percent of dark mat-ter satisfyingcosβ>0.8 that is trapped is much greater.

investigated or identified the halo trapping and have not undertakena detailed study of trapped populations in the disc component. Ourk-means orbit finding algorithm can be used as a technique to fur-ther understand the dynamics of strongly time-dependent evolution,such as the bar-disc-halo system. In future papers, we will presenta detailed breakdown of the orbit populations in the disc andhalocomponents using thek-means technique.

Many simulations have shown that galactic bars effi-ciently transport angular momentum from the disc to the halo,causing the bar to slow, lengthen, and increase in strength(Debattista & Sellwood 2000; Athanassoula & Misiriotis 2002;Debattista et al. 2006; Weinberg & Katz 2007b). Many of thesesame authors have argued that bars are long-lived, as no mech-anism for their destruction has been observed in simulations ofisolated systems (e.g.Athanassoula et al. 2013). The results ofHernquist & Weinberg(1992) then suggest that the bar either can-not be strongly coupled to the halo, or requires fine-tuning to elim-inate lower-order resonances to reduce the strong torque. The sug-gestion that bars cannot be strongly coupled to the halo seems tobe in conflict with the angular momentum transfer rates observedin simulations depicting a rapidly slowing bar. The production of ashadow bar, and the trapping of halo orbits by the bar more gener-ally, offers a possible way out of this dynamical conundrum.With-out a sink of halo particles that are distributed asymmetrically inphase space relative to the bar, there can be no torque. Our discus-sion in section4.1suggests that the existence of the shadow bar isrobust, but that its magnitude will depend on the galaxy profile andthe details of the subsequent bar formation. If a significantdark discexists as the remnant of shredded satellites accreted by thegalaxy,the magnitude of the shadow bar could be significantly enhanced.

Although the shadow bar is kinematically similar to the stel-lar bar, the findings presented here provide new insight intothekinematics of the galactic centre. Observationally, stellar surveysmay detect rotation in the stellar halo toward the inner Galaxy thatwould be indicative of trapping in a spherical component. For starsand stellar systems that may have formed prior to the formationof the MW bar (such as metal-poor halo stars, thick-disc stars, orglobular clusters), the findings presented here indicate that thesecomponents could also be trapped into the bar potential. Sugges-tions in the literature dating as early asBlitz & Spergel(1991) ofa triaxial rotating component at the centre of the galaxy could bethe result of an originally spheroidal component trapped bythebar potential. Recent observations byBabusiaux et al.(2014) andRossi et al.(2015) have pointed out kinematic signatures (e.g. ananomalous tangential and radial velocity dispersion) thatcould beconsistent with this interpretation. New stellar survey data availablein the coming years will provide further insights.

Finally, evidence for a shadow bar in our own Milky Waycould be directly detected through future direct dark matter de-tection experiments, including directional direct detection experi-ments, which would provide significant dynamical insights into theunseen dark matter component.

REFERENCES

Aalseth C. E., et al., 2013,PhysRevD, 88, 012002Agnese R., et al., 2013,PhysRevD, 88, 031104Aguerri J. A. L., et al., 2015, preprint, (arXiv:1501.05498)Allende Prieto C., et al., 2008,Astronomische Nachrichten, 329, 1018Allgood B., Flores R. A., Primack J. R., Kravtsov A. V., Wechsler R. H.,

Faltenbacher A., Bullock J. S., 2006,MNRAS, 367, 1781Athanassoula E., 1992, MNRAS,259, 328

MNRAS 000, 000–000 (0000)

The Shadow Bar 15

Athanassoula E., 2007,MNRAS, 377, 1569Athanassoula E., Misiriotis A., 2002, MNRAS, 330, 35Athanassoula E., Machado R. E. G., Rodionov S. A., 2013,MNRAS,

429, 1949Babusiaux C., et al., 2014,A&A , 563, A15Barnes J., Hut P., Goodman J., 1986,ApJ, 300, 112Bernabei R., et al., 2014,Nuclear Instruments and Methods in Physics Research A,

742, 177Binney J., Spergel D., 1982, ApJ, 252, 308Binney J., Tremaine S., 2008, Galactic Dynamics: Second Edition. Prince-

ton University PressBlitz L., Spergel D. N., 1991,ApJ, 379, 631Bovy J., Rix H.-W., 2013,ApJ, 779, 115Boylan-Kolchin M., Bullock J. S., Sohn S. T., Besla G., van der Marel R. P.,

2013,ApJ, 768, 140Breddels M. A., Helmi A., 2013,A&A , 558, A35Bruch T., Read J., Baudis L., Lake G., 2009,ApJ, 696, 920Bullock J., Dekel A., Kolatt T., Kravtsov A., Klypin A., Porcianai C., Pri-

mack J., 2001, ApJ, 555, 240Cheung E., et al., 2013,ApJ, 779, 162Clutton-Brock M., 1973,Ap&SS, 23, 55Clutton-Brock M., Innanen K. A., Papp K. A., 1977,Ap&SS, 47, 299Colın P., Valenzuela O., Klypin A., 2006,ApJ, 644, 687Contopoulos G., Papayannopoulos T., 1980, A&A, 92, 33Debattista V. P., Sellwood J. A., 2000,ApJ, 543, 704Debattista V., Mayer L., Carollo C., Moore B., Wadsley J., Quinn T., 2006,

ApJ, 645, 209Debattista V. P., Moore B., Quinn T., Kazantzidis S., Maas R., Mayer L.,

Read J., Stadel J., 2008,ApJ, 681, 1076Dekel A., Devor J., Hetzroni G., 2003,MNRAS, 341, 326Earn D. J. D., 1996,ApJ, 465, 91Gilmore G., et al., 2012, The Messenger,147, 25Governato F., et al., 2012,MNRAS, 422, 1231Hernquist L., Ostriker J. P., 1992,ApJ, 386, 375Hernquist L., Weinberg M. D., 1992,ApJ, 400, 80Holley-Bockelmann K., Weinberg M., Katz N., 2005, MNRAS, 363, 991Kim T., et al., 2014,ApJ, 782, 64Kormendy J., Kennicutt Jr. R. C., 2004,ARA&A , 42, 603Kravtsov A. V., 2013,ApJL, 764, L31Kuijken K., Dubinski J., 1994, MNRAS,269, 13Laurikainen E., Salo H., Athanassoula E., Bosma A., Herrera-Endoqui M.,

2014,MNRAS, 444, L80Ling F.-S., 2010,PhysRevD, 82, 023534Lloyd S., 1982, IEEE Transactions on Information Theory, 28, 2Lynden-Bell D., Kalnajs A. J., 1972, MNRAS,157, 1Martinez-Valpuesta I., Shlosman I., Heller C., 2006,ApJ, 637, 214Mashchenko S., Couchman H. M. P., Wadsley J., 2006,Nature, 442, 539Masters K. L., et al., 2012,MNRAS, 424, 2180Moster B. P., Naab T., White S. D. M., 2013,MNRAS, 428, 3121Munoz-Mateos J. C., et al., 2013,ApJ, 771, 59Navarro J. F., Frenk C. S., White S. D. M., 1997, ApJ,490, 493Petersen M., Weinberg M., Katz N., 2016, PhysRevDPillepich A., Kuhlen M., Guedes J., Madau P., 2014,ApJ, 784, 161Polyachenko V., Polyachenko E., 1996, AstL, 22, 302Purcell C. W., Bullock J. S., Kaplinghat M., 2009,ApJ, 703, 2275Read J. I., Lake G., Agertz O., Debattista V. P., 2008,MNRAS, 389, 1041Read J. I., Mayer L., Brooks A. M., Governato F., Lake G., 2009, MNRAS,

397, 44Rojas-Arriagada A., et al., 2014,A&A , 569, A103Rossi L. J., Ortolani S., Barbuy B., Bica E., Bonfanti A., 2015, MNRAS,

450, 3270Saha K., Naab T., 2013,MNRAS, 434, 1287Saha K., Tseng Y.-H., Taam R. E., 2010,ApJ, 721, 1878Saha K., Martinez-Valpuesta I., Gerhard O., 2012,MNRAS, 421, 333Sellwood J., 2014, RevModPhys, 86, 1Sellwood J. A., Debattista V. P., 2006,ApJ, 639, 868Sellwood J. A., Debattista V. P., 2009,MNRAS, 398, 1279

Sellwood J. A., Wilkinson A., 1993,Reports on Progress in Physics,56, 173

Sheth K., et al., 2008,ApJ, 675, 1141Soto M., Zeballos H., Kuijken K., Rich R. M., Kunder A., Astraatmadja T.,

2014,A&A , 562, A41Steinmetz M., et al., 2006,AJ, 132, 1645Tiret O., Combes F., 2007,A&A , 464, 517Tremaine S., Ostriker J. P., 1999,MNRAS, 306, 662Tremaine S., Weinberg M., 1984, MNRAS, 209, 729Villa-Vargas J., Shlosman I., Heller C., 2009,ApJ, 707, 218Vogelsberger M., et al., 2014,MNRAS, 444, 1518Wegg C., Gerhard O., Portail M., 2015,MNRAS, 450, 4050Weinberg M. D., 1985, MNRAS,213, 451Weinberg M. D., 1991,ApJ, 368, 66Weinberg M. D., 1994,ApJ, 421, 481Weinberg M. D., 1998,MNRAS, 297, 101Weinberg M., 1999, AJ, 117, 629Weinberg M., Katz N., 2002, ApJ, 580, 627Weinberg M. D., Katz N., 2007a,MNRAS, 375, 425Weinberg M. D., Katz N., 2007b,MNRAS, 375, 460Williams M. J., Bureau M., Kuntschner H., 2012,MNRAS, 427, L99Zaritsky D., et al., 2013,ApJ, 772, 135

MNRAS 000, 000–000 (0000)