The impact of baryons on the spins and shapes of dark matter haloes

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by  S. E. Bryan

Mon.Not.R.Astron.Soc.000,1–14(2012) Printed20July2012 (MNLATEXstylefilev2.2)
The impact of baryons on the spins and shapes of dark
matter haloes
2 S. E. Bryan,1⋆ S. T. Kay,1 A. R. Duffy, 2,3 J. Schaye,4 C. Dalla Vecchia 4,5 and
1
C. M. Booth6,7,4
0
2 1Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy, The University of Manchester, Manchester M13 9PL
2School of Physics, Universityof Melbourne, Parkville, VIC 3010, Australia.
l
u 3ICRAR, Universityof WesternAustralia, WA 6009, Australia.
J 4Leiden Observatory,Leiden University,Postbus 9513, 2300 RA Leiden, The Netherlands.
9 5Max Planck Institute for Extraterrestrial Physics, Giessenbachstraße 1, 85748 Garching, Germany.
1 6Department of Astronomy & Astrophysics, The Universityof Chicago, Chicago, IL 60637, USA.
7Kavli Institute for Cosmological Physics and Enrico Fermi Institute, The Universityof Chicago, Chicago, IL, 60637, USA.

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Accepted ......Received......;inoriginalform......
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- ABSTRACT
o
We use numerical simulations to investigate how the statistical properties of dark
r
t matter(DM)haloesareaffectedbythebaryonicprocessesassociatedwithgalaxyfor-
s mation.Wefocusonhowtheseprocessesinfluencethespinandshapeofalargenumber
a
[ of DM haloes covering a wide range of mass scales, from dwarf galaxies to clusters,
at redshifts zero, one and two. The haloes are extracted from the OverWhelmingly

1 LargeSimulations, a suite of state-of-the-arthigh-resolutioncosmologicalsimulations
v
run with a range of feedback prescriptions. We find that the median spin parameter
5
in DM-only simulations is independent of mass, redshift and cosmology. Baryons in-
5
5 creasethespinoftheDMinthecentralregion(≤0.25r200)byupto50percentwhen
feedback is weak or absent. This increase can be attributed to the transfer of angular
4
. momentum from baryons to the DM. We also present fits to the mass dependence of
7 the DMhaloshape atbothlowandhighredshift.Atz =0the sphericity(triaxiality)
0
isnegatively(positively)correlatedwithhalomassandbothresultsareindependentof
2
cosmology. Interestingly, these mass-dependent trends are markedly weaker at z =2.
1
WhilethecoolingofbaryonsactstomaketheoverallDMhalomorespherical,stronger
:
v feedback prescriptions (e.g. from active galactic nuclei) tend to reduce the impact of
i baryons by reducing the central halo mass concentration. More generally, we demon-
X
strate a strongly positive (negative) correlation between halo sphericity (triaxiality)
r
a andgalaxyformationefficiency,withthelattermeasuredusingthecentralhalobaryon
fraction.Inconclusion,our results suggestthat the effects ofbaryonsonthe DM halo
spin and shape are minor when the effects of cooling are mitigated, as required by
realistic models of galaxy formation, although they remain significant for the inner
halo.
Key words: galaxies:haloes-galaxies:clusters:general-galaxies:evolution-meth-
ods: numerical - cosmology: theory
1 INTRODUCTION roundingsheetsandfilaments.Theanisotropicaccretionhis-
tory of a halo also affects its angular momentum distribu-
A natural consequence of the standard hierarchical struc-
tion,throughthepresenceofnon-zerotorques.Itistherefore
ture formation paradigm is that the shapes of dark matter
clearthatboththeshapeandspinofadarkmatterhaloare
haloes are triaxial, a property that is inherited from their
importantdiagnosticsforanaccuratedeterminationoftheir
progenitordensityperturbations(Bardeen et al.1986).This
structureand formation history.
additionally leads to aspherical growth as the halo accretes
matterfrompreferentialdirections,associatedwiththesur- While the spin and shape of a dark matter halo
are not directly observable, they have important conse-
quences for the structure and dynamics of galaxies. For
⋆ E-mail:[email protected] example, halo spin is an important parameter in galaxy
2 S. E. Bryan et al.
formation models as it affects the size of the embed- While these results are interesting, a significant
ded galactic disc (e.g. Mo et al. 1998, but see Sales et al. uncertainty is how the dark matter halo is affected by
2012). Deviations from axisymmetry in elliptical galax- the additional, non-gravitational processes acting on the
ies are likely to influence the gas kinematics of the sys- baryons. Recent work has clearly established that the
tem (deZeeuw & Franx 1989), and may be responsible for condensation of baryons to the centre of dark matter
exciting or sustaining warps and stabilising or deforming haloes tends to increase the central angular momen-
polar rings (Steiman-Cameron et al. 1992). Axisymmetry tum of the halo (see for example, Sharma & Steinmetz
may also influence the fuelling efficiency of the central 2005; Kaufmann et al. 2007; Abadi et al. 2010; Bett et al.
black hole (Franxet al. 1991). Misalignment of the angu- 2010) and to make the halo more spherical or axisym-
lar momentum of the halo and the galaxy may be re- metric (see, for example, Katz & Gunn 1991; Dubinski
sponsible for the anisotropic distribution of subhaloes and 1994; Evrard et al. 1994; Barnes & Hernquist 1996;
satellite galaxies (Holmberg 1969 and Knebeet al. 2004) Tissera & Dominguez-Tenreiro 1998; Springel et al.
and could cause galactic warps (Ostriker& Binney 1989; 2004; Kazantzidis et al. 2004; Debattista et al. 2008;
Debattista & Sellwood 1999; Bailin & Steinmetz 2004). Pedrosa et al. 2010; Tissera et al. 2010; Bryan et al. 2012;
On cluster scales, asphericity in the dark matter halo Zemp et al. 2012). This result has been used to explain
will naturally correspond to asphericity in the gas den- thediscrepancy between the strongly prolate-triaxial shape
sity and will impact the shape of X-ray isophotes and found in N-body simulations and the more spherical
the Sunyaev-Zel’dovich signal. Understanding the intrin- observed systems.
sic shape of dark matter haloes is also important for Incorporating baryonic physics in cosmological simula-
weak lensing analysis (see, for example, the discussions in tionsisanon-trivialtaskandthecomputationalcostofthis
Becker & Kravtsov 2011; Bett 2012) and it is well known process has placed limits on both the parameter space and
thatintrinsicellipticitycancontributesignificantlytoalens- the size of the sample of haloes explored to date. The de-
ing halo’s ability to form arcs (Oguriet al. 2003). tailed nature of the baryonic processes involved in galaxy
Several methods are used to constrain galaxy and formation and the precise influence of these processes on
haloshapesobservationally (see,forexample,Sackett1999; galaxies therefore remains largely uncertain. In this paper,
Merrifield 2004). Unfortunately, different observational weattempttomakeprogressonbothfronts,bystudyingthe
methods currently yield systematically different results for spin and shape distributions for a large (>1000) sample of
theshapesofdarkmatterhaloes (Olling & Merrifield 2000; dark matter haloes, spanning a range of mass (from dwarf
O’Brien et al. 2010). However, it is worth noting that the galaxies to clusters) and redshift (z = 0,1 and 2). We do
observations cover a large range of systems and vary in the this using the OverWhelmingly Large Simulations (OWLS;
extenttowhichthehaloisprobed,makingadirectcompar- Schayeet al.2010)–asuiteofcosmologicalhydrodynamical
ison somewhat difficult. Whether the discrepancies result simulations run with many different physical prescriptions
from halo-to-halo scatter or from systematic errors in the forthebaryons.Byprovidingidenticalsimulationsrunwith
observedestimatesisunclear.Giventherapidlyaccumulat- differentimplementations ofthesubgridphysics,OWLSof-
ing data set and ever increasing sophistication of the data fers the opportunityto explore theeffects of baryonsunder
analysistools,onecansoonexpecttohaveanobservational a range of physical conditions, for the same population of
datasetwhichmaybemoredirectlycomparedwiththeory. haloes. Inparticular,weusetheOWLSdatatoquantify,in
Theoretically, the predictions for the distribution of astatisticallymeaningfulway,theinfluenceoffeedbackpro-
angular momentum and halo shapes have been studied ex- cesses (from no feedback, to feedback from stars and black
tensively, primarily using cosmological N-body simulations holes) on the spin and shape distributions of dark matter
(Frenk et al.1988;Dubinski& Carlberg1991;Warren et al. haloes.
1992; Cole & Lacey 1996; Bullock 2002; Jing & Suto 2002; Thepaperisorganisedasfollows.Thesimulationsused
Bailin & Steinmetz 2005; Allgood et al. 2006; Bett et al. inthisanalysis,andthemethodsusedtoidentifyhaloesand
2007; Maccio` et al. 2008; Jeeson-Daniel et al. 2011; toestimate theirspinsandshapesareoutlinedinsection 2.
Vera-Ciro et al. 2011; Bett 2012; Zemp et al. 2012). There Our results are presented in section 3.1, including fitting
is a general consensus that cold dark matter haloes have formulae for the predicted correlations between halo shape
approximately log-normal spin distributions and are triax- and mass. The robustness of our results is demonstrated
ial, with sphericities, (c/a) 0.5 0.8 1 and elongations, via a resolution study, given in the appendix. Finally, we
≃ −
(b/a) 0.4 1. Furthermore, haloes are generally found summarise our main results in section 4.
≃ −
to be highly flattened and show a tendency toward prolate
shapes (c/b>b/a), especially in theinnerregions. There is
also general agreement that the sphericity decreases with
increasing halo mass and that the spin is independent of 2 METHODOLOGY
mass(Bullock 2002;Jing & Suto2002;Springelet al. 2004;
2.1 Simulation details
Hopkinset al. 2005; Bett et al. 2007; Maccio` et al. 2008;
Jeeson-Daniel et al. 2011). In addition, Jeeson-Daniel et al. Thehaloesusedforthisanalysiswereextractedfromasub-
(2011) have shown that sphericity is strongly correlated setoftheOWLSruns.Fordetailedinformationaboutthese
with concentration, while both triaxiality and spin are simulations the reader is referred to Schayeet al. (2010);
anti-correlated with concentration. here,themost relevantaspectsarebrieflyreviewedforcon-
venience.
For all runs, cosmological initial conditions were set
1 Wherea>b>caretheeigenvaluesofthehalo’sinertiatensor. up using a transfer function generated with cmbfast
Baryons and the spins and shapes of DM haloes 3
Table1.AlistoftheOWLSrunsusedinthisanalysis.WeusethesameidentifierforeachrunasinSchayeetal.(2010)andcomment
on the subgrid physics implemented in each case. Key global properties of the simulations are also listed, namely the number of dark
matter and baryonic particles; the comoving box length; and the dark matter particlemass. Values arepresented forall runs analysed
atz=0.The100h−1Mpcboxes arealsousedforresultsatz=1,while25h−1Mpcboxes areusedatz=2(values fortheserunsare
giveninbrackets).Themaximumforcesofteningwas0.5,2.0and8.0h−1kpcforthe25,100and400h−1Mpcboxesrespectively.
Name Description NDM Nbaryons Boxlength mDM
(h−1 Mpc) (h−1M )
⊙
DMONLY Darkmatteronlyruns
WMAP1 2163 - 50 8.6×108
WMAP3 5123 - 100(25) 4.9×108 (7.7×106)
WMAP3 5123 - 400 3.1×1010
WMAP5 5123 - 100(25) 5.3×108 (8.3×106)
WMAP5 5123 - 400 3.4×1010
NOSN NOZCOOL Nofeedback, primordialabundances forcooling 5123 5123 100(25) 4.1×108 (6.3×106)
REF Weakstellarfeedback, metalcooling 5123 5123 100(25) 4.1×108 (6.3×106)
WDENS Strongstellarfeedback, metalcooling 5123 5123 100(25) 4.1×108 (6.3×106)
AGN Weakstellar&AGNfeedback, metalcooling 5123 5123 100(25) 4.1×108 (6.3×106)
(Seljak & Zaldarriaga 1996).Initial(z=127) positions and All of the simulations include radiative
velocities were computed using the Zel’dovich (1970) ap- cooling (Wiersma et al. 2009a), star formation
proximationfromaninitialglass-likestate(White1996).All (Schaye& Dalla Vecchia 2008) and metal enrichment
simulationswererunusingamodifiedversionofGADGET-3 (Wiersma et al. 2009b), but differ in their feedback pre-
(Springel2005).Foragivenboxsizethesameinitial condi- scriptions (for more details see Dalla Vecchia & Schaye
tionswereusedineachrun,allowingustodirectlycompare 2008). All models followed the timed release of H, He
thesamehaloesevolvedusingdifferentprescriptionsforthe and 9 different heavy elements produced by massive stars,
sub-gridphysics.Asummaryofthesimulationsusedinthis AGBstars and supernovaeof typeI and II (Wiersma et al.
analysis,includingvaluesforkeynumericalparameters,can 2009b) The first run (NOSN NOZCOOL) did not include
befound in Table 1. any feedback processes2; the second (REF) includes weak
feedback from stellar winds and supernovae; the third
We begin by exploring dark matter only (DMONLY)
(WDENS) includes stronger stellar feedback with a wind-
simulations, as this allows us to validate our results by
speed that depends on the local gas density; and the final
comparing to the existing literature, as well as to set
run (AGN), feedback from both stars and active galactic
the scene for exploring the impact of baryons. We con-
nuclei (Booth & Schaye 2009). In the feedback models,
sider three sets of dark matter only simulations, each run
enhanced cooling from metal lines was also accounted for.
with different values for the cosmological parameters. Our
main results (including baryons) assume values taken from All four models were run using the 100h−1Mpc box (to
the 3rd year Wilkinson Microwave Anisotropy Probe data z = 0) and the 25h−1Mpc box (to z = 2). The same
(WMAP3; Spergel et al. 2007) with [Ωm,ΩΛ,Ωb,n,σ8] = numberof gas particles as dark matterparticles (5123) was
adopted for each box.
[0.238,0.762,0.0418,0.95,0.74]. This model was run with
dark matter only, using 5123 particles in three box sizes.
The two larger boxes (100 and 400h−1Mpc) were run to
z=0whileasmaller, high-resolution box(25h−1Mpc)was 2.2 Halo sample
run to z = 2; we use the latter to study the properties
Haloes were extracted from each simulation using the
of haloes at high redshift. The maximum physical soften-
Friends-of-Friends (FoF) algorithm (Davis et al. 1985)
ing length in the 25h−1Mpc box was 0.5 h−1kpc, while which links particles together within a fixed comoving sep-
in the 100 and 400h−1Mpc it is 4 and 16 times larger re- aration. This separation, known as the linking length, was
spectively. We also present results for the newer WMAP5
set to 0.2 times the mean inter-particle distance. The FoF
cosmology (Komatsu et al. 2009) with [Ωm,ΩΛ,Ωb,n,σ8]= groupswerethendecomposedintoself-boundsub-haloesus-
[0.258,0.742,0.0441,0.963,0.796] using the same boxes as ingthesubfindalgorithm(Springel et al.2001;Dolag et al.
before. Finally, we consider a run with the WMAP1
2008). Finally, a sphere was grown around the most bound
cosmology (Spergel et al. 2003) with [Ωm,ΩΛ,Ωb,n,σ8] = particle of the most massive sub-halo until the mean total
[0.25,0.75,0.045,1,0.9]. This simulation was runusing2163
massdensitywasequalto200timesthecriticaldensity.We
particles in a 50 h−1 Mpc box, matching the resolution of choose to define our haloes as including all particles con-
theMillenniumSimulation(Springelet al.2005),whichwas
tained within this sphere. The mass and radius of the halo
also run with theWMAP1 parameters.
are referred to as M200 and r200 respectively.
To study theeffect of varying levels of feedback on the Onlyhaloesthatcontainatleast1000particlesarecon-
spinandshapeparametersofhaloes extractedfrom ΛCDM sidered in this analysis, as this ensures that the results are
simulations, four baryon runs from the OWLS simulations
are considered, all run with theWMAP3 cosmology. These
runsmodelthegaseous componentusingsmoothed particle 2 Thissimulationalsoneglects enhancedcoolingthroughmetal-
hydrodynamics(SPH). lines.
4 S. E. Bryan et al.
fully converged (see theappendixfor adiscussion of theef- agivenradius,m isthemassofthekthparticleandr the
k k,i
fects of resolution). While estimates of the spin parameter ith component of its position vector from the halo centre.
arefoundtobewellresolved forhaloes with morethan300 Thesquarerootsoftheeigenvaluesofthemassdistribution
particles(asinBett et al.2007)alargernumberofparticles tensor,obtainedusingJacobitransformations,aredefinedas
isrequiredtoresolvethehaloshape(inparticularthetriax- a,b,c (where a>b>c) and are used to measure the shape
iality of the halo), in agreement with Maccio` et al. (2008). ofthesimulatedhaloes.Notethattheshapesobtainedusing
Thiscutimposesaminimumhalomassof4.9 1011 h−1M the inertia tensor and the mass distribution tensor M are
at z = 0 and 7.7 109 h−1M at z = 2, for×our WMAP⊙3 equivalent (Bett et al. 2007).
× ⊙
runs. Ourshaperesultsarepresentedusingthefollowing pa-
Wehaveinvestigatedwhetherourresultsareaffectedby rameters:s=c/aisusedasameasureofhalosphericity;e=
restrictingoursampletodynamicallyrelaxedhaloes.Wees- b/aasameasureofelongation; andT =(a2 b2)/(a2 c2)
− −
timatedthedynamicalstateusingthecentroidshift,defined asameasureofthetriaxialityofthehalo.Apurelyspherical
to bethedistance between theminimum potential position halowillhaves=e=1withT beingundefined.Lowvalues
(thehalocentre)andthecentreofmassofthehalo.Inaccor- of T (T 0) correspond to oblate haloes while high values
→
dancewithNetoet al.(2007),ahaloisdefinedtoberelaxed (T 1) correspond toprolate haloes.
→
ifthecentroidshiftislessthan0.07r200.Wefindthatwhile Wenotethatcomputationofthemasstensorinaspher-
relaxedhaloesaretypicallymoresphericalandlesstriaxial, ical region biases the shape towards higher sphericity; this
thisrestrictiondoesnotaffectthetrendsfoundinthiswork is corrected for, as suggested in Bailin & Steinmetz 2005,
and so we chose to present our analysis for the total halo by re-scaling the axis ratios s s√3 and e e√3. While
→ →
sample(exceptwhereexplicitly statedotherwise). Not only our adopted method described above may not be the most
does this result in larger numbers of objects, our choice is robust way of describing the physical halo shape (e.g. see
also motivated by the fact that an observational cut based the discussion in Zemp et al. 2011), we follow Bett (2012)
on the relaxation state of a halo is not a straightforward and use this simple approach as it is most directly compa-
task. rable with observations and is adequate for the comparison
The numbers of haloes in our final samples are pre- wewishtopresenthere.Wecheckedthatusingthereduced
sented in Table 2. Note that the number of haloes in mass distribution tensor, and also using an iterative tech-
DMONLY WMAP3 at z = 0 is almost twice as large as niquetodeterminethehaloshapes,donotresultinsystem-
in the baryon runs; this is due to the inclusion of haloes atically different results.
from the 400h−1Mpc box in the former case. Overall, our Ourmainresultsforspinandshapeparametersarepre-
DMONLY sample spans around three orders of magnitude sentedforthedarkmatterwithinr200,butwealsoconsider
1in01m5has−s1M(fro)matgazla=xy0t.oOculurszte=r s2caslaesm,pi.lee.cMov2e0r0s∼a s1i0m12ila−r prehsyuslitcsswpiltahyisnathmeocreentsriaglnrifiegcaionnt (r0o.l2e5. rT2h00e)cwohnevreergbeanrcyeonraic-
⊙
dynamic range but at lower mass (dwarf galaxy to group dius(see Power et al. 2003) is less than 0.25r200 for almost
scales, M200 1010 1013h−1M ). all of the haloes considered here. (The haloes that are not
∼ − ⊙
converged within 0.25r200 are excluded from the studies of
thecentral region. In all cases thisis less than oneper cent
2.3 Defining halo spin and shape
ofthesample).Resolutiontestsforthisregionarediscussed
The dimensionless spin parameter λ provides a useful mea- in the appendix.
sure of the amount of rotational support present within a
dark matter halo. We estimate this property for the dark
matter particles using the modified expression given by 3 RESULTS
Bullock et al. (2001)
Webeginbypresentingourresultsforthehalospinparam-
J
λ′(r)= , (1) eter, then go on to explore the shapes of a large numberof
√2Mvcr haloesselectedfromourcosmologicalsample.Inbothcases,
whereJ istheangularmomentumwithinasphereofradius we first discuss the results from dark matter only simula-
r, containing mass M and vc = GM(<r)/r is the halo tionsbefore studyingtheimpact of thebaryons.Quantities
circular velocity at this radius. Tphis expression reduces to are measured within the central region (0.25r200) and at
thestandardspinparameter(Peebles1969)whenmeasured r200
at the virial radius of a truncated singular isothermal halo.
For a comparison of the different definitions for the spin
3.1 Halo Spin
parameter, see Maccio` et al. (2007); hereafter we drop the
prime and refer tothe modified spin parameter as λ. 3.1.1 Dark Matter Simulations
To characterise the halo shape, we use the mass
distribution tensor M which has been used extensively Thedistribution ofdarkmatterhalospin istypicallyfound
to bewell characterised bya log-normal distribution 3
in the halo shape literature (e.g. Cole & Lacey 1996;
Bsqauilairne&maSttreiixn)maertez2005).Thecomponentsof thetensor(a P(λ)= λ√12πσ exp(cid:18)−ln22(σλ2/λ0)(cid:19), (3)
ij = mkrk,irk,j, (2)
M
Xk 3 Althoughwenotethat amoreaccurate fittingformulaforthe
wheretheindexk runsoveralldarkmatterparticleswithin distributionofhalospinsispresentedinBettetal.(2007).
Baryons and the spins and shapes of DM haloes 5
Table 2. Number of haloes (containing at least 1000 particles) and median halo mass M200 (in 1012h−1M ) for each simulation at
⊙
z=0,1and2.
z=0 z=1 z=2
Simulation Nhaloes M200 Nhaloes M200 Nhaloes M200
DMONLYWMAP1 431 1.7 - - - -
DMONLYWMAP3 7094 2.3 3765 0.77 5334 0.030
DMONLYWMAP5 8188 3.6 - - - -
AGN 3878 0.88 3375 0.83 4743 0.033
WDENS 3884 0.91 3371 0.84 4699 0.035
REF 3990 0.99 3580 0.93 4832 0.044
NOSN NOZCOOL 4634 1.0 4041 0.91 5683 0.040
0.8 0.8
0.7 z = 0 0.7 z = 2
r r
200 200
0.6 Med M = 2 ×1012 h-1 M 0.6 Med M = 3 ×1010 h-1 M
200 O• 200 O•
0.5 µ = -3.34 0.5 µ = -3.26
σ = 0.62 σ = 0.6
P 0.4 P 0.4
0.3 0.3
0.2 0.2
0.1 0.1
0 0
-7 -6 -5 -4 -3 -2 -1 -7 -6 -5 -4 -3 -2 -1
ln (λ) ln (λ)
0.8 0.8
0.7 z = 0 0.7 z = 2
0.25r 0.25r
200 200
0.6 Med M = 2 ×1012 h-1 M 0.6 Med M = 3 ×1010 h-1 M
200 O• 200 O•
0.5 µ = -3.34 0.5 µ = -3.1
σ = 0.62 σ = 0.6
P 0.4 P 0.4
0.3 0.3
0.2 0.2
0.1 0.1
0 0
-7 -6 -5 -4 -3 -2 -1 -7 -6 -5 -4 -3 -2 -1
ln (λ) ln (λ)
Figure 1.Thedistributionof loghalospinparameter fordarkmatter particles inhaloes atz =0(left) andz =2(right), taken from
the DMONLYWMAP3 runs. The top (bottom) panels show the spin distribution computed using all particles within r200 (0.25r200).
ThemeanandstandarddeviationofthedistributionarecomputedforeachsampleandarelistedinTable3.Thebest-fittingGaussian
curve,assumingtheseparameters,isoverlaid.Forcomparison,weover-plotarrowsrepresentingthemeanofthedistributionsforthetwo
baryon runs with the most extreme feedback prescriptions - the runwith no feedback (NOSN NOZCOOL; green arrow) and the AGN
feedbackrun(AGN;redarrow).ItisevidentfromthisfigurethatefficientcoolingresultsinanincreasedDMspinparameter,especially
withinthecentralregionofthehalo,whilestrong(AGN)feedbackresultsinvaluesthatareindistinguishablefromthedarkmatteronly
case.
where λ0 and σ are free parameters, determining the curve). The left panels show the results for haloes at z =0
mean and standard deviation respectively. We confirm while results for the z =2 haloes are presented in the right
this result in Fig. 1, where the distribution from the panels.Toppanelscorrespondtospinparameterscomputed
DMONLYWMAP3 haloes is shown as a histogram. Values usingalldarkmatterparticleswithinr200,whilethebottom
for the mean and standard deviation of thesample are also panels are for the inner region (0.25r200). Table 3 lists val-
given,andareusedtogetherwithEquation3,topredictthe uesforthemean,standard deviation,median and skewness
equivalent log-normal spin distribution (shown as the solid of λ for all runs,and at z =0,1 and 2.
6 S. E. Bryan et al.
Table 3. Themean,median,standarddeviationandskewnessofthedistributionofln(λ)parameters forallhaloes ineachsimulation
run at z = 0, 1 and 2. We compare the distribution of spin parameters computed using dark matter particles within r200 and that
computed usingonlythedarkmatter particleswithinthecentral 0.25r200 region.Haloesarerequiredtocontainatleast1000particles
withinr200. Hereln(λ0) denotes the mean, ln(λmed) the median and σ the standard deviation of the distributionof lnλ, see equation
(3).Errorsrepresentthe1σconfidenceintervalsandhavebeendeterminedusingbootstrapresampling(1000bootstraprealisationshave
beenused).
Withinr200 Within0.25r200
Simulation ln(λ0) ln(λmed) σ skew ln(λ0) ln(λmed) σ skew
z=0
DMONLYWMAP1 -3.41 ±0.03 -3.38 ±0.03 0.60 +0.02 -0.47 +0.15 -3.50 ±0.03 -3.51 +0.05 0.63 ±0.02 -0.19 +0.16
0.03 0.12 0.02 0.14
DMONLYWMAP3 -3.34 ±0.01 -3.29 ±0.01 0.62 ±−0.01 -0.49 ±−0.04 -3.34 ±0.01 -3.31 ±−0.01 0.62 +0.00 -0.36 −+0.05
0.01 0.04
DMONLYWMAP5 -3.33 ±0.01 -3.30 ±0.01 0.62 ±0.01 -0.46 ±0.03 -3.37 ±0.01 -3.33 ±0.01 0.63 ±−0.01 -0.40 +−0.03
0.04
AGN -3.30 ±0.01 -3.26 ±0.01 0.61 ±0.01 -0.50 ±0.06 -3.32 ±0.01 -3.28 ±0.01 0.63 ±0.01 -0.49 +−0.06
0.07
WDENS -3.29 ±0.01 -3.24 ±0.01 0.60 ±0.01 -0.58 +00..0065 -3.28 ±0.01 -3.24 ±0.01 0.62 ±0.01 -0.46 ±−0.04
REF -3.27 ±0.01 -3.23 ±0.01 0.61 ±0.01 -0.63 ±−0.07 -3.20 ±0.01 -3.15 ±0.01 0.61 ±0.01 -0.53 ±0.04
NOSN NOZCOOL -3.20 ±0.01 -3.14 ±0.01 0.61 ±0.01 -0.64 ±0.06 -2.96 ±0.01 -2.88 ±0.01 0.61 ±0.01 -0.66 ±0.04
z=1
DMONLYWMAP3 -3.26 ±0.01 -3.22 ±0.01 0.62 ±0.01 -0.62 +0.06 -3.18 ±0.01 -3.14 ±0.01 0.60 ±0.01 -0.34 ±0.05
0.07
AGN -3.26 ±0.01 -3.21 ±0.01 0.60 ±0.01 -0.51 ±−0.06 -3.16 ±0.01 -3.11 ±0.01 0.60 ±0.01 -0.50 ±0.06
WDENS -3.26 ±0.01 -3.21 ±0.01 0.60 ±0.01 -0.58 +0.06 -3.16 ±0.01 -3.13 ±0.01 0.61 ±0.01 -0.53 +0.08
0.07 0.07
REF -3.26 ±0.01 -3.21 ±0.01 0.60 ±0.01 -0.64 +−0.08 -3.10 ±0.01 -3.06 ±0.01 0.57 ±0.01 -0.43 ±−0.07
0.07
NOSN NOZCOOL -3.21 ±0.01 -3.15 ±0.01 0.59 ±0.01 -0.56 ±−0.05 -2.95 ±0.01 -2.91 ±0.01 0.57 ±0.01 -0.57 ±0.05
z=2
DMONLYWMAP3 -3.26 ±0.01 -3.22 ±0.01 0.60 ±0.01 -0.54 ±0.05 -3.10 ±0.01 -3.07 ±0.01 0.60 ±0.01 -0.40 ±0.05
AGN -3.28 ±0.01 -3.24 ±0.01 0.58 ±0.01 -0.47 ±0.04 -3.12 ±0.01 -3.07 ±0.01 0.60 ±0.01 -0.54 ±0.05
WDENS -3.27 ±0.01 -3.23 ±0.01 0.59 ±0.01 -0.53 +0.06 -3.12 ±0.01 -3.08 ±0.01 0.58 ±0.01 -0.48 +0.06
0.05 0.05
REF -3.27 ±0.01 -3.23 ±0.01 0.59 ±0.01 -0.53 ±−0.05 -3.12 ±0.01 -3.07 ±0.01 0.59 ±0.01 -0.55 +−0.05
0.04
NOSN NOZCOOL -3.25 ±0.01 -3.21 ±0.01 0.59 ±0.01 -0.62 ±0.11 -2.96 ±0.01 -2.90 ±0.01 0.59 ±0.01 -0.71 ±−0.06
Atz=0wefindln(λ0)= 3.34 0.03(orλ0 =0.036) Ourresultsarebroadlyinagreement: atz=0wefindare-
− ±
and σ =0.62 0.01, for our DMONLYWMAP3 run, when duction in λ0 of around 12 per cent while at z = 2, this
±
all dark matter particles within r200 are considered. These increases to 17 per cent.
values are in good agreement with those found in previous At z = 0, we find that the inner region (r < 0.25r200,
analyses such as Bullock et al. (2001), who obtained best- showninthebottomleftpanelofFig.1)ischaracterisedby
fit values of λ0 = 0.035 and σ = 0.5; Bailin & Steinmetz nearly the same distribution as the overall halo. However,
(2005), who measured λ0 = 0.035 and σ = 0.58; and this does not hold at z = 2 (bottom right panel) where we
Maccio` et al.(2008),whofoundameanvalueofλ0=0.034 find that the mean spin parameter is significantly higher
and σ = 0.59. Our median spin parameter (λmed = 0.037) (ln(λ0)= 3.10 0.01) thanthatof thewholehalo(ln(λ0)
− ±
is also in good agreement with the analysis of Bett et al. = 3.26 0.01).Interestingly,thisdifferenceisevenstronger
− ±
(2007), who found λmed = 0.0367 0.0429 (depending on fortherelaxedsampleatz=2,wheretheinnerregionofthe
−
thedefinitionofthehalo).Thespinparametersofthehigher relaxed dark matter haloes exhibits a mean spin, ln(λ0) =
redshift(z=1and2)samplesarefoundtobeslightlyhigher 3.21 0.01 while themean spin computed overthewhole
− ±
than those at z =0. For example, at z =2, the mean spin halo is found to beln(λ0)= 3.42 0.01 .
parameter of the DMONLYWMAP3 haloes is λ0 = 0.038, Wehavealsoconsidered−theeff±ectsofcosmologyonthe
around 8 per cent higher. At least part of this (statistically
spin parameter distribution of dark matter haloes, compar-
significant) shift can be explained by a very weak depen-
ingthespindistributionsfromdarkmatteronlysimulations
dence of spin with halo mass (see below), given that the runwiththeWMAP1,WMAP3andWMAP5cosmological
mean mass of ourhaloes at z =2 is lower than at z=0.
parametersatz=0.Themean,median,standarddeviation
We also checked whether excluding unrelaxed haloes andskewness forthedistributionsof thelog ofthespin pa-
from our sample made any difference to our results, as rameters are also listed in Table 3. It is evident from this
Maccio` et al. (2006) found that this reduced themean spin
table(asinMaccio` et al. 2008)thatthespindistribution is
parameterby 15percentandJeeson-Daniel et al.(2011) notsensitivetotheexactchoiceofcosmologicalparameters.
found that spi∼n correlates with therelaxednessof thehalo4 Wenotethatthelargest changeinthemodelswehavecon-
(they demonstrated that this is due to the strong anti- sideredisthevalueofσ8,whichvariesfrom0.7inWMAP3
correlationofbothspinandrelaxednesswithconcentration). to 0.9 in WMAP1. Haloes in a lower σ8 cosmology are ex-
pected to form later and be less concentrated than those in
a higher σ8 cosmology (Navarro et al. 1997). The spin dis-
4 Alowvalueofrelaxedness correspondstoarelaxedhalo. tribution itself remains almost unchanged for the range of
Baryons and the spins and shapes of DM haloes 7
0.09 0.09
WMAP1
WMAP3
0.08 WMAP5 0.08
0.07 0.07
0.06 0.06
0.05 0.05
λ λ
0.04 0.04
0.03 0.03
0.02 0.02
0.01 slope = 0 0.01 slope = 0
z = 0 z = 2
intercept = 0.037 intercept = 0.04
0 0
11.5 12 12.5 13 13.5 14 14.5 15 9.5 10 10.5 11 11.5 12 12.5 13 13.5
log (M (h-1 M )) log (M (h-1 M ))
10 200 O• 10 200 O•
Figure2.Theleft(right)panelshowsthespinparameterversusM200 inthedarkmatteronlysimulationsatz=0(2).Inthesepanels
we show the median value within each mass binand errors represent the 1σ scatter inthe mass bin. Haloes are requiredto contain at
least 1000 particles and mass bins at least 20 haloes. We plot the results from the three cosmologies we have considered: WMAP1 is
depictedusingredsquares,WMAP3usinggreencirclesandWMAP5usingbluetriangles.Forclarityweshowthemedianofthewhole
WMAP3sampleasasolidlineandpresentthisvalueineachpanel.Wenotethatthespinparameterremainsunchangedaswevarythe
cosmologicalparametersandthatthereisalsonocleardependence onmass.
0.1 0.1
DMONLY
0.09 AGN 0.09
WDENS
0.08 REF 0.08
NOSN_NOZCOOL
)00 0.07 )00 0.07
2 2
5r 5r
0.2 0.06 0.2 0.06
< <
λ (r 0.05 λ (r 0.05
0.04 0.04
0.03 0.03
z = 0 z = 2
0.02 0.02
0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Baryon fraction (r < 0.05r ) Baryon fraction (r < 0.05r )
200 200
Figure 3.Thisfigureillustrates how theDMhalo spindepends onthe central baryon fractionofthe halo(usedas aproxyforgalaxy
formation efficiency). The z =0 (2) haloes are shown in the left (right) panel. We plot the median values and error bars show the 1σ
halo-to-halo scatter. The DM spin parameters are computed within 0.25r200 while the baryon fraction is calculated within 0.05r200.
The two quantities are positively correlated at z = 0 but the correlation is absent at z = 2, where only the run without feedback
(NOSN NOZCOOL)showsasignificantincreaseinthemedianλ.
parametersexploredhere,althoughwenoteaslightdecrease spin parameter is insensitive to halo mass; a weak trend is
in the spin parameter for the WMAP1 haloes, consistent seen(whichmayexplaintheslightredshiftdependenceseen
with theanti-correlation of spin with concentration (as dis- earlier, due to our two samples covering a different mass
cussedinMaccio` et al.2008andJeeson-Daniel et al.2011). range), but is not statistically significant (we find the best-
Haloes formed in this cosmology are expected to have the fittingslope to beconsistent with zero at the1σ levelin all
highest concentrations and hence may be expected to have cases - where the 1σ confidence is determined using boot-
thelowest spin values. strap resampling). Once again, we can clearly see that the
effect of cosmology on thespin parameter is negligible.
We explore the relationship between spin and mass in
Fig. 2. In this figure we show the median spin value and
the 1σ halo-to-halo scatter for several mass bins, for each
3.1.2 Effect of baryons on DM halo spin
of the three cosmologies that we have considered at z = 0
(left panel) and for the WMAP3 cosmology only at z = 2 Following on from the dark matter only case, we now con-
(right panel). We use least-squares fitting of the median in siderthedistributionofDMhalospinparameterscomputed
each mass bin to determine the slope and intercept of the from the baryon runs. We only include dark matter par-
λa −pivlootg1m0(aMss20o0f)Mrepliavtoiton=i1n01e2achh−1siMmu.laIttioins csleeta,ratshsuamt tinhge fteiccltessoifnboauryrondsetoenrmthineadtiaornkomfaλttetrocdoimrepcotlnyenats.sTeshsetmheeaenf-,
⊙
8 S. E. Bryan et al.
median, standard deviation and skewness for the distribu-
tion of spin parameters in each run is presented in Table 3. 2.2 AGN
For ease of comparison with the dark matter only case, we WDENS
show in Fig. 1 the mean value for two of the most extreme )0 2 REF
feedback schemes we have considered: the no feedback case < r 1.8 NOSN_NOZCOOL
( o
(NOSN NOZCOOL;green arrow) and thestellar and AGN m 1.6
d
feedback run (AGN;red arrow). ) / j0 1.4
0
2
5r 1.2
2
< 0. 1
(g
m 0.8
d
j
0.6
From Fig. 1 and Table 3 we see that, at z = 0, the
0.4
baryons induce a small increase in the mean (and median) 0.8 0.9 1 1.1 1.2 1.3
value of λ within r200. For NOSN NOZCOOL (where feed- Mdmg ( < 0.25r200) / Mdmo ( < 0.25r200)
back is absent), the increase is around 15 per cent, but for Figure 4. The ratio of specific angular momentum of the dark
the more realistic5 AGN run (where feedback is strongest), matter within 0.25r200 from the baryon runs jdmg (< 0.25r200)
it is only 5 per cent. At higher redshift (z = 1 and 2), the to that of the corresponding dark matter only run within a ra-
effect of baryons on the mean spin parameter is weak or dius containing an equivalent mass (jdmo(< r0)). This ratio is
absent in all cases except for NOSN NOZCOOL. It is also shown against the DM mass ratio within 0.25r200 for the 500
most massive haloes that have been matched between runs at
interestingtonotethat,withinthecentralregion(0.25r200),
z = 0. Symbols depict the median values while the error bars
thebaryonrunshavedistributionsthatareskewedtolower
representthe1σ scatter.
valuesthantheDMONLYrun(also trueforthewhole halo
atz =0).Itispossiblethatthisisaresultofthetransferof
angularmomentumfrominfallingsatellites.Asinthemodel
3.1.3 Origin of excess spin in the DM
ofMaller et al.(2002),ifgasisabletocoolandformadense
clump,itwillloseangularmomentumviadynamicalfriction Fromequation1,itisclearthattheincreaseinλmustbedue
and will play an important role in shaping the lower end of toanincreaseofthespecificangularmomentumofthedark
theangular momentum distribution. matter, or a decrease in the enclosed mass. As discussed in
Bett et al.(2010),anincreaseinspecificangularmomentum
can result from two factors. The first is the transfer of an-
gular momentum from gas to dark matter via tidal torques
anddynamicalfriction.Thesecondisthecontractionofthe
dark matter in response to the deepened potential well of
Theeffectsofbaryonsarestrongerinrunswithweakor thesystem (assuming angular momentum is conserved).
nofeedback,when weconsiderdarkmatterparticleswithin To explore this further, we apply the approach of
thecentralregion(consistentwiththefindingsofBett et al. Bett et al. (2010) to our haloes at z = 0. We begin by
2010, who studied the effects of baryons using a more re- matching haloes between the DMONLY and baryon runs;
strictedsetofsimulations).There,themeanspinparameter a match is identified by selecting the halo from the baryon
increases by 50 per cent in NOSN NOZCOOL (but again, run that contains the most of the 1000 most-bound parti-
is almost unchanged in AGN). This effect is shown more clesfrom thedarkmattersimulation.Wethencomputethe
clearly inFig.3,whereweplot6 themedianspinparameter DM mass ratio (within 0.25r200) for each matched pair of
for the central region (0.25r200) against the baryon frac- haloes.IftheDMhalohascontractedduetothepresenceof
tion within theverycentre(0.05r200).Thelatter isused as baryons, themass should increase with respect to themass
a proxy for galaxy formation efficiency, as weaker feedback in the DMONLY halo. We then compare the specific an-
leadstostrongergascoolingandalargercentralbaryonfrac- gular momentum of the dark matter within 0.25r200 from
tion,associated withthegalaxyatthecentreofthehalo.It thebaryonrunjdmg(<0.25r200),tothecorrespondinghalo
is clear that there is a positive correlation between the two in DMONLY, within a radius r0 containing the equivalent
quantities at z = 0 but this correlation has all but disap- mass, jdmo(<r0). If the halo contracts due to the presence
peared at z=2.
5 The AGN run reproduces the z = 0 observed relations be-
tween black holes and the mass and velocity dispersion of their
hostgalaxies(Booth&Schaye2009),aswellastheobservedop-
tical and X-ray properties of the groups in which they reside
(McCarthyetal.2010).
Wefindthat baryonsdo not influencethemass depen-
6 Note: To ensure that mass dependent trends do not influence
dence of the spin parameter (in almost all cases, the slope
ourcomparisonwedonotincludethe400h−1Mpcboxdarkmat-
is consistent with zero at 1σ and in all cases it is consistent
tersimulationwhendirectlycomparingrunsatz=0.Thisleaves
with zero at 2σ). The trendsfound in this analysis are also us with a sample of 4329 haloes in the DMONLY simulation at
preservedwhenonlythesub-setofrelaxed haloesisconsid- z = 0 with a median mass of 8.3×1011h−1M comparable to
ered. themedianmassinthebaryonruns. ⊙
Baryons and the spins and shapes of DM haloes 9
ofbaryonsandconservesitsangularmomentum(i.e.ifthere haloes of mass > 1014 h−1M . Since e = b/a decreases
⊙
isnotransferfromthebaryons)thenthisratiowillbeunity. fasterwithmassthans=c/adoes,thetriaxialityparameter
The results for each baryon run are shown in Fig. 4, increaseswithmass;overthesamerangeinmass,thetriaxi-
whereweshowtheratioof specificangularmomentumver- alityparameterincreasesfrom 0.6to 0.8.Inshort,high-
∼ ∼
susthemassratiowithin0.25r200 forthe500most massive mass haloes are less spherical and more prolate than their
haloes that we are able to match7 between runs at z = 0. lower-masscounterparts.Thisislikelyaconsequenceoftheir
From this figure, it can clearly be seen that the AGN feed- more recent formation time (as discussed in Springel et al.
back acts to reduce the mass in the central regions, but as 2004; Jeeson-Daniel et al. 2011). Our results are in agree-
galaxy formation efficiency increases, we see the effect of ment with previous studies (Bett et al. 2007; Maccio` et al.
halo contraction. This effect was studied in detail for the 2008; Jeeson-Daniel et al. 2011).
same set of runsby Duffyet al. (2010). At z = 2, we find that the correlations between the
Ifwenowconsidertheratioofspecificangularmomen- shape parameters and mass are much weaker, but still
tum,wefindthattheAGNresultisconsistentwithnotrans- present.Forexample,s∝M2−000.055atz=0buts∝M2−000.035
fer, but as galaxy formation (and thus halo contraction) is at z = 2. A similar trend is seen in the evolution of the
more efficient, a net transfer from the baryons to the dark concentration-mass relation (Zhao et al. 2003; Duffy et al.
matterdoesoccur(thisisalsoseeninKaufmann et al.2007; 2008)andagain,islikelytoreflectthathaloeshavehadless
Abadiet al. 2010; Bett et al. 2010). In summary, the small time to collapse, so exhibit a narrower range in formation
increaseinspinparameter(within0.25r200)intheAGNrun times (and therefore internal properties). This explanation
appears to be driven by a slight decrease in halo mass, but isconsistentwiththefindingsofJeeson-Daniel et al.(2011),
in all other runs with more efficient galaxy formation, the whousedaprincipalcomponentsanalysistoshowthatcon-
increaseisduetoanettransferofangularmomentumfrom centration, which they found to be essentially equivalent to
thebaryons. formation time, is the most fundamental property of dark
matter haloes in that scatter in this quantity accounts for
muchof thescatterin theotherdimensionless propertiesof
3.2 Halo Shapes
darkmatterhaloes,includingtheshapeparametersinvesti-
3.2.1 Dark matter only simulations gated here.
Beforeturningourattentiontotheeffectofbaryonsonhalo
shape, we will analyse the DMONLY runs. Of particular 3.2.2 Effect of baryons on DM halo shape
interest is the relationship between the shape parameters,
Baryonsareknowntohaveasignificanteffectontheshapes
e = b/a, s = c/a and T = (a2 b2)/(a2 c2), and mass,
− − of haloes. As gas cools and condenses within the centre of
M200, of the dark matter halo. Previous work has shown a halo, it reduces the fraction of box orbits present in the
thateandsarenegativelycorrelatedwithmass,andT pos-
dark matter and stars, resulting in a more spherical struc-
itively correlated (e.g. Bett et al. 2007; Maccio` et al. 2008;
ture. Feedback prevents cooling and acts to expel gas from
Jeeson-Daniel et al.2011).Ourdarkmatteronlyresultsex-
thecentralregion,resulting inalowercentralmassconcen-
tendthisworkbystudyingthedependenceofthesecorrela-
tration, more box orbits and a more triaxial shape (see, for
tionsonbothcosmologyandredshift,aswellasestablishing
example, Debattista et al. 2008; Bryan et al. 2012). Here,
thebaseline for the runswith baryons.
wequantifytheimpactoftheseeffectsforourrangeoffeed-
Themassdependenceoftheshapeparametersisshown
back prescriptions, over a wide range in halo mass, and at
inFig.5.Ineachpanelweshowthemedianvaluefore,sand
low and high redshift.
T, within several mass bins, while the error bars represent
The mass dependence of the halo shape parameters in
the 1σ halo-to-halo scatter. Again, haloes are required to
the baryon runs is summarised in Table 4. Again, this is
containatleast1000 particlesandthemassbinsatleast 20
for the shape determined within r200 using only the dark
haloes. We use least-squares fitting to determine the slope
matter particles. From this we see that the trends seen in
and intercept of the mass relation for each parameter, us-
theDMONLYrunsarestillpresentintherunswithbaryons.
ing a pivot mass of Mpivot = 1012h−1M⊙. In the left pan- However,itisinterestingtonotethatthebaryonsareableto
els we show the median values for all three cosmological
affecttheshapeofthehaloouttor200,albeitinaminorway.
models at z = 0, but for clarity, only plot the fit to the
The axis ratio intercepts (both b/a and c/a) systematically
DMONLY WMAP3 haloes. The corresponding results for
increase with increasing galaxy formation efficiency, while
theDMONLYWMAP3runatz=2areshownintheright
thetriaxiality parameter decreases accordingly.
panels. The best-fit parameters for the mass dependence of
The baryons have a larger impact on the shape of the
all simulation sets are presented in Table 4.
centralregion of thedark matterhalo. Toquantify this, we
As with the spin parameter, it is clear from the left
compare the median values for e,s and T within 0.25r200
panels of Fig. 5, that there is no significant difference be-
to the central baryon fraction (within 0.05r200) for each of
tween the different cosmological models studied here. At
the simulation runs. Note: To ensure that mass dependent
z = 0 there is a strong trend for more massive haloes to
trends do not influence our comparison we do not include
fhraovmes∼ma0l.l6erfaoxrishraalotieoss.oQf umaanstsita1t0i1v2elhy,−t1hMes⊙p,hteoric∼ity0v.5arfioesr dthireec4t0l0yhc−o1mMpaprcinbgoxrudnasr.kTmheasteterressuimltsulaarteiopnreastenzt=ed0inwFhiegn.
6.Fromthisfigurewecanseethatthereisacleartrendfore
7 Amatchisfoundin96%ofthecases(andatleast85%ofthese ands toincrease with increasing baryon fraction, while the
matches have m200 ratios within 20% and minimum potential triaxiality decreases with increasing baryon fraction. This
positionswithin20%ofr200 ofeachother). trend is apparent both at z = 0 and z = 2. At z = 0,
10 S. E. Bryan et al.
1 1
z = 0 z = 2
0.8 0.8
0.6 0.6
a a
b/ b/
= =
e 0.4 e 0.4
0.2 0.2
slope = -0.066 WMAP1 slope = -0.028
WMAP3
intercept = 0.81 WMAP5 intercept = 0.69
0 0
11.5 12 12.5 13 13.5 14 14.5 15 9.5 10 10.5 11 11.5 12 12.5 13 13.5
log (M (h-1 M )) log (M (h-1 M ))
10 200 O• 10 200 O•
1 1
s = 1 ⇒ SPHERICAL z = 0 s = 1 ⇒ SPHERICAL z = 2
0.8 0.8
0.6 0.6
a a
c/ c/
= =
s 0.4 s 0.4
0.2 0.2
slope = -0.055 slope = -0.035
intercept = 0.63 intercept = 0.5
0 0
11.5 12 12.5 13 13.5 14 14.5 15 9.5 10 10.5 11 11.5 12 12.5 13 13.5
log (M (h-1 M )) log (M (h-1 M ))
10 200 O• 10 200 O•
1 1
T → 1 ⇒ PROLATE z = 0 T → 1 ⇒ PROLATE z = 2
0.8 0.8
0.6 0.6
T T
0.4 0.4
0.2 0.2
slope = 0.063 slope = 0.019
T → 0 ⇒ OBLATE intercept = 0.62 T → 0 ⇒ OBLATE intercept = 0.73
0 0
11.5 12 12.5 13 13.5 14 14.5 15 9.5 10 10.5 11 11.5 12 12.5 13 13.5
log (M (h-1 M )) log (M (h-1 M ))
10 200 O• 10 200 O•
Figure 5.Thepanelsshow,fromtoptobottom,howtheaxisratios(e=b/a,s=c/a)andtriaxiality(T)ofdarkmatterhaloesinthe
DMONLYruns scalewith halomass (M200)at z =0(left) andat z =2(right). We show the median valuewithineach mass binand
the error bars represent the 1σ intrinsic scatter. The results from the WMAP1 haloes are depicted using red squares, WMAP3 using
green circles and WMAP5using bluetriangles. Haloes are requiredto contain at least 1000 particles and mass bins at least20 haloes.
Least-squares linesof best-fit (assuming apivot mass of1012h−1M )arecomputed foreach simulationandarepresented inTable4.
Thebest-fitlinefortheWMAP3haloesisshownineachpanel. ⊙
the median sphericity is increased by 25 per cent when ical haloes. It also drives down the value of the triaxiality
∼
comparingtheextreme(DMONLYandNOSN NOZCOOL) parameter,tendingtomakethehaloesconsiderablylesspro-
cases, while the triaxiality is reduced by approximately the late than seen in dark matter only runs. Feedback reduces
same factor. Thechangeisstill significant in themorereal- the amount of gas that is able to cool and condense to the
istic AGN model, where the sphericity increases by around centre of the halo (resulting in a lower central baryon frac-
10-15percentandthetriaxialitydecreasesbyasimilarfac- tion)andhencereducestheimpactontheshapeofthehalo.
tor. As expected, these results are consistent with the idea
thatgascoolingtothecentreofhaloesresultsinmorespher-