Dark Matter Equilibria in Galaxies and Galaxy Systems

1.2 MB·

by  A. Lapi

ACCEPTEDBYAPJ
PreprinttypesetusingLATEXstyleemulateapjv.03/07/07
DARKMATTEREQUILIBRIAINGALAXIESANDGALAXYSYSTEMS
A.LAPI1,2 ANDA.CAVALIERE1,3
DraftversionJanuary27,2009
ABSTRACT
Inthedarkmatter(DM)halosembeddinggalaxiesandgalaxysystemsthe‘entropy’K σ2/ρ2/3(aquantity
thatcombinestheradialvelocitydispersionσwiththedensityρ)isfoundfromintensiveN≡- bodysimulationsto
followapowerlawrunK rαthroughoutthehalos’bulk,withαaround1.25.Takingupfromphenomenology
∝
9 justthatα constapplies,wecutthroughtherichanalyticcontentsoftheJeansequationdescribingtheself-
≈
0 gravitating equilibria of the DM; we specifically focus on computing and discussing a set of novel physical
0 solutions that we name α-profiles, marked by the entropy slope α itself, and by the maximal gravitational
2 pull κ (α) required for a viable equilibrium to hold. We then use an advanced semianalytic description
crit
for the cosmological buildup of halos to constrain the values of α to within the narrow range 1.25- 1.29
n
fromgalaxiestogalaxysystems; thesecorrespondtohalos’currentmassesintherange1011- 1015M . Our
a
J rangeofα appliessince thetransitiontimethat- bothin oursemianalyticdescriptionandin state-of⊙-the-art
numerical simulations - separates two development stages: an early violent collapse that comprises a few
7
majormergersand enforcesdynamicalmixing, followedby smoothermassadditionthroughslow accretion.
2
In ourrangeof α we providea close fit for the relationκcrit(α), and discussa relatedphysicalinterpretation

] intermsofincompleterandomizationoftheinfallkineticenergythroughdynamicalmixing. Wealsogivean
h accurateanalyticrepresentationoftheα-profileswithparametersderivedfromtheJeansequation;thisprovides
p straightforwardprecisionfits to recentdetaileddata fromgravitationallensingin andaroundmassive galaxy
-
clusters,andthusreplacestheempiricalNFWformularelievingtherelatedproblemsofhighconcentrationand
o
r oldage. We finallystresshowourfindingsandpredictionsastoαandκcrit contributetounderstandhitherto
t unsolvedissuesconcerningthefundamentalstructureofDMhalos.
s
a Subjectheadings:Darkmatter–galaxies:clusters:general–galaxies:halos
[

2
v 1. INTRODUCTION lapse a substantial mass is gathered, while the major merg-
5 Galaxies, galaxy groups and clusters are widely held to ers reshuffle the gravitational potential wells and cause the
4 form under the drive of the gravitationalinstability that acts collisionless DM particles to undergo dynamical relaxation;
2 oninitialperturbationsmodulatingthecosmicdensityofthe therefromthehaloemergeswithadefinitestructureofthein-
1 nerdensityandgravitationalpotential.Duringthelaterstage,
dominantcolddarkmatter(DM)component. Theinstability
. moderate amounts of mass are slowly accreted mainly onto
0 atfirstiskeptincheckbythecosmicexpansion,butwhenthe
theoutskirts,littleaffectingtheinnerstructureandpotential,
1 localgravityprevailscollapsesetsin. Thestandardsequence
butquiescentlyrescalingupwardstheoverallsize (see Hoff-
8 runs as follows: a slightly overdense region expands more
0 slowlythanitssurroundings,progressivelydetachesfromthe manetal. 2007).
: Hubble expansion, halts and turns around; then it collapses, In the resulting simulated halos, the coarse-grainedproxy
v Q ρ/σ3tothephase-spacedensity–definedintermsofthe
i andeventuallyvirializestoforma DM‘halo’in equilibrium ≡
X under self-gravity. The amplitude of more massive pertur- DMdensityρandoftheradialvelocitydispersionσ–isem-
piricallyfoundtobestableafterthecollapse,anddistributed
r bations is smaller, so the formation is hierarchical with the
a massivestructuresformingtypicallylater(seePeebles1993). inthe formofa powerlawwith uniformslopedlogQ/dlogr
around- 1.9throughoutthehalo’sbulkwhereitisamenable
Sucha formationhistoryhasbeenconfirmedandresolved
to a considerable detail by many intensive N- body simula- to noise-free measurements (see Taylor & Navarro 2001;
Dehnen&McLaughlin2005;Ascasibar&Gottlöber2008).
tions. Earlyon(White1986)theseaddedanimportantblock
Takingupsuchanotion,thepresentpaperstarts(§2)from
of information, i.e., the growth of a halo actually includes
investigating the wide set of macroscopic equilibria for the
mergingeventswithotherclumpsofsizescomparable(‘ma-
DMunderself-gravity,asdescribedbytheJeansequation. It
jor’mergers)orsubstantiallysmaller(‘minor’mergers),down
proceeds (§ 3) to the semianalytic description of their time
tonearlysmoothaccretion(seeSpringeletal. 2006).
buildup to constrain their governingparameters, and so pre-
Morerecently,asecondroundoffeatureshasbeenincreas-
dict the halo realistic structure. It proposes (§ 4) physical
inglyrecognizedinhighlyresolvedsimulationsofindividual
interpretationsfor such parameters. The paper is concluded
halos forming in cosmological volumes (Zhao et al. 2003;
(§5)withasummaryofourfindingsandadiscussionofthe
Diemandetal.2007),totheeffectofidentifyinginthegrowth
perspectivestheyopen.
twodefinitestages:anearlyfastcollapseincludingafewvio-
Throughoutweadoptthestandard,flatcosmologicalmodel
lentmajormergers,andalatercalmerstagecomprisingmany
withnormalizedmatterdensityΩ =0.27,darkenergyden-
minor mergers and smooth accretion. During the early col- M
sityΩΛ=0.73,andHubbleconstantH0=72kms- 1Mpc- 1.
1Univ.‘TorVergata’,ViaRicercaScientifica1,00133Roma,Italy. 2. JEANSEQUILIBRIA
2AstrophysicsSector,SISSA/ISAS,ViaBeirut2-4,34014Trieste,Italy.
3AccademiaNazionaledeiLincei,ViaLungara10,00165Roma,Italy ThestaticequilibriaoftheDMhalosobeytheJeansequa-
tionthat,onneglectingforthetimebeinganisotropiesandthe
2 LAPI&CAVALIERE
FIG.1.—Toppanels:theγ(α)andκ(α)relationsforthepowerlawsolutionsoftheJeansequation,see§2.1;bluedotsrefertothesolutionwithα=5/4,and
greendotsrefertothesolutionwithα=4/3(theisothermalsphere).Intherightpanelthedashedlinerepresentstherelationκ=3α/2,andthearrowindicates
howκincreasetowardthevalues(giveninthebottomrightpanel)fortheα-profiles,see§2.1-2.2fordetails.Bottompanels:theγ- γ′andκcrit(α)relationsfor
theα-profileswithαintherange1.2- 35/27,see§2.1-2.2;bluedotsandcurvesrefertothesolutionwithα=5/4,andreddotsandcurvesrefertothesolution
withα=35/27. Intheleftpanel,thesquaresmarkthecoupleofvaluesγ0,γ0′atr=r0.Notethatwewillfindin§3therelevantvaluesofαtobeconstrained
withintherange5/4- 35/27,i.e.,betweentheblueandreddots/curves.
blendintothelarge-scaleenvironmenttobediscussedin§5, 2.1. Background
reads
Webeginwithlookingfirst(afterTaylor&Navarro2001)
1d(ρσ2) =- GM(<r) . (1) atpowerlawsolutionsofEq.(2)oftheformρ r- γ,withthe
ρ dr r2 entropyrunsettoK rα;thennotonlyαis∝agivenparam-
To elicit the contents of this macroscopic-like equation, eter,butalsotheslop∝eγ=- dlogρ/dlogr isaconstanttobe
an explicit link is required between σ2(r) and ρ(r). Such a determinedfromEq.(2)intheform
link may be providedby Q(r), or by the equivalentquantity 3 3 κ r 2- α- γ/3
K(r) σ2/ρ2/3=Q- 2/3(r);thisisoftennamedDM‘entropy’ γ- α= . (3)
invie≡wofitsformalanalogywiththeadiabatofagasinther- 5 5 3- γ (cid:18)r0(cid:19)
mal equilibrium, and is increasingly used in the context of Here the ‘bare’ constant κ 4πGρ r2/σ2 = (3- γ)v2/σ2
≡ 0 0 0 0 0
DMequilibria,fromTaylor&Navarro(2001)toFaltenbacher compares the gravitational potential to the random kinetic
et al. (2007). We focuson this functionof state K(r) in the energy associated with the velocity dispersion at a refer-
configurationspace,andcommentonourchoicein§5. ence point r . At the r.h.s. κ appears as modulated by
0
In terms of the slope dlogK/dlogr α, Eq. (1) is recast r- dependencies,producedbythe integrateddynamicalmass
intotheparticularlysimpleform ≡ M(<r)toyieldv2(r) r2- γ/(3- γ)inthepowerlawcase,and
5 dlogρ dlogK v2(r) bythelocaldepencden∝ciesfromσ2(r) K(r)ρ2/3(r) rα- 2γ/3.
=- - c . (2) Forapowerlawsolutiontoactually∝holdther.h.s∝. mustbe
3 dlogr dlogr σ2(r)
constant,whichyieldsthelink
Herev2c(r)=GM(<r)/risthe(squared)circularvelocity,that γ=6- 3α, (4)
providesa naturalscaling for the depth of the DM potential
as pointed out by Taylor & Navarro (2001), Hansen (2004),
well.
andAustinetal. (2005). Giventhis,thebalanceofthethree
Takingupfromthesimulationphenomenologyonlytheno-
terms in Eq. (3) imposes on the dimensionless gravitational
tion that α be uniform throughout the halo bulk (see § 1),
pulltheconstraint
we will discuss the range of values to be expected for α as
a clue to their origin. In fact, in their pioneeringwork Tay- κ=(5γ/3- α)(3- γ)=6(α- 1)(5- 3α), (5)
lor & Navarro (2001)selected a value of the exponentclose whosenonlinearitiesreflecttheDMself-gravity. Thustheal-
to α=1.25. Other, recentstudieshaveprovidedsimilar val- lowedpowerlawsolutionsmustlieonaparabolaintheplane
uesforα: around1.27accordingto Ascasibaretal. (2004), α- κandonastraightlineintheplaneα- γ,asillustratedin
and around 1.29 according to Rasia et al. (2004). Beyond the top panels of Fig. 1; note that to ensure κ>0 the over-
the minor differences, it is the narrow empirical range that allbounds1<α<5/3musthold, implyingthat1<γ <3
constitutedanunsolvedchallenge. appliesforthepowerlawsolutions.
DMEQUILIBRIAINGALAXIESANDGALAXYSYSTEMS 3
FIG.2.—Radialrunsfortheα-profiles:densityρ,massM,circularvelocityvc,gravitationalpotentialΦ,velocitydispersionσ,andentropyK;theprofilesare
normalizedto1atthepointr0whereγ=γ0=6- 3αholds. Variouscurvesareforαintherange1.2- 35/27;theblueonereferstothesolutionwithα=5/4,
andtheredonereferstoα=35/27.
Two notable instances correspond to the values α = 4/3 otherhand,forr r itisseenthateitherγ γ 3(1+α)/2
0 b
≫ → ≡
and5/4. Theformeryieldsthestandard,singularisothermal holdsoradensitycutoffoccurs,ifthemassM(<r)istosat-
spherewithγ=2andκ=2;thelatterholdsforthevalueofα urate to a constant, implying v2(r) 1/r. In other words,
c ∝
originallyselected by Taylor & Navarro(2001), and implies powerlaws with inner γ and outer slope γ will constitute
a b
γ =9/4 and κ=15/8. Note that they are located on the in- asymptotic or effective approximationsto the full solutions.
tersectionsoftheparabolaEq.(5)withthelineκ=3α/2,see For example, for the value α=5/4 one recovers the slopes
Fig.1. γ =3/4andγ =27/8,foundbyTaylor&Navarro(2001)in
a b
The powerlaw solutions of the Jeans equation with slope theirarchetypicalfullsolution.
γ 6- 3αwillturnouttobetangenttothefullsolutionsat Thus all full solutions may be conveniently visualized as
0
≡
r =r so as to provide a useful reference slope in the mid- bentdownfromthecorrespondingtangentpowerlawsoasto
0
dle of the halo. However, they do not qualify as relevant meettheaboveasymptoticformsatbothsmallandlargeradii.
solutions since at both extremes they suffer of unphysical These solutions will be analytically studied and constrained
features: since 1<γ <3 holds as shown above, the force next. In fact, the solution space of the Jeans equationis an-
GM(< r)/r2 r1- γ would diverge at the center, while the alyticallyrich,so we referthe interestedreadertothe works
massM(<r)∝r3- γ woulddivergetowardlarger. by Austin et al. (2005), Dehnen& McLaughlin(2005), and
To avoid th∝ese drawbacks, in all physically relevant solu- Barnes et al. (2006, 2007) for extensive coverage. Here we
tionstherunningslopeγ(r)istobenddownfromthemiddle summarizethemainresultsandproceedtofocusonthe‘criti-
valueingoingtowardlargeandsmallr.Infact,forr r one cal’solutionsintheterminologyofTaylor&Navarro(2001).
0
caninferfromEq.(3)thatasymptoticallyγ γ 3≪α/5isto To highlight the properties of these solutions with run-
a
holdifthegravitationalforceistobefinite,w→hich≡impliesthat ningγ(r),itisconvenient(seeDehnen&McLaughlin2005;
both sides of Eq. (3) vanish (see Austin et al. 2005); on the Austinetal. 2005;alsoBarnesetal. 2006,2007)toreplace
4 LAPI&CAVALIERE
FIG. 3.—Energydistributions fortheα-profiles: specificpotential energyEpot,randomkineticenergyEkin,virialratio2Ekin/|Epot|,andratioofcircular
velocitytothevelocitydispersionv2/σ2appearinginEq.(2).LinesasinFig.2.
c
theoriginalJeansequationγ- 3α/5=3v2/5σ2 withitsfirst and(7);theformeralsoprovidesthefollowingexhaustivede-
c
andsecondderivative,andusethedifferentialmassdefinition scriptionofthesolutions.
dM/dr=4πρr2,toobtain In the inner region only three situations possibly arise
(Williams et al. 2004, Dehnen & McLaughlin 2005): (i) γ
2 3 r 2- α ρ 1/3 attains the value γ asymptotically for r r , which is un-
γ′- 3(γ- γa)(γ- γb)= 5κ r ρ , physicalsincesuch0asolutionwouldendt≪here0withadiverg-
(cid:18) 0(cid:19) (cid:18) 0(cid:19) inggravitationalforce;(ii)thesolutionattainsthevalueγ at
(6) a
afiniteradius,andthencontinuestoflattendowntothepoint
γ - γ γ- 2γa+2γb- γ0 = 2(γ- γ )(γ- γ )(γ- γ ). ofdevelopingacentralhole,whichisobviouslytobeavoided;
′′ ′ a 0 b
3 9 (iii)γapproachesthevalueγ asymptoticallyforr r ,that
(cid:18) (cid:19) a ≪ 0
is the only viable possibility. Thus physical solutions must
Hereγ andγ denotethefirstandsecondlogarithmicderiva-
′ ′′ approachtheinnerslopeγ ,andmuststeepenmonotonically
a
tive of γ with respectto r; in addition, r is the pointwhere
0 outwards.
the density slope matches γ 6- 3α (see Eq. [4]), while
0 However,notallofthemaretenableinviewoftheirouter
γ 3α/5andγ 3(1+α)/≡2areshorthandsfor theinner
a b behavior(Austin et al. 2005, Dehnen& McLaughlin2005).
≡ ≡
andtheoutereffectiveslopesanticipatedabove.
Here,againthreesituationsformallyarisethatdependonthe
In the second equation the normalization constant κ
value of α: (i) γ approachesthe value γ asymptoticallyfor
4πGρ r2/σ2 does not appear explicitly; however, from th≡e 0
0 0 0 r r0, whichoccursforα>35/27andis unphysicalsince
firstEq.(6)evaluatedatr itiseasilyrecognizedthat ≫
0 with such a solution the mass would diverge (in practice, it
κ=6(α- 1)(5- 3α)+5γ0′/3 (7) wfinoiuteldrabdeiuilsl-adnedfintheedn,sgeoee§st5o);an(iio)uγtearttcauitnosfft,htehsaltoopcecγubrsaftoar
isrelatedtoγ0′,andinafullsolutionspecificallymeasuresthe α<35/27andisphysicallyviablesincethecutoffsactually
upwarddeviationofκfromthecorrespondingpowerlaw,that set in at large radii exterior to the halo’s bulk, as discussed
is,abovetheparabolaexpressedbyEq.(5),seealsoFig.1. below;(iii)γ approachestheslopeγ asymptoticallyforr
b
Infact,asthevalueofκincreasesatgivenα,thesolutions asitoccursforα=35/27,whichalsoprovidesaphysical→ly
will be progressivelycurveddownboth inward and outward ∞viablesolution.
of r , to constitute a bundletangent to the related powerlaw
0
with slope given by Eq. (4). But, as stressed by Taylor &
2.2. Viablefullsolutions
Navarro (2001), only a definite maximal value κ can be
crit
withstoodbythehaloatequilibrium,lestthe solutiondevel- Summingup, the criticalsolutionssatisfy bothsets of cri-
opspathologieslike a centralhole; this valuemarksa ‘criti- teria, but exist only for α 35/27=1.296 (which excludes
≤
cal’solution. Correspondingly,thevaluesforκ (α)maybe the case corresponding to the isothermal powerlaw); here-
crit
foundbya painstakingtrial-and-errorprocedure,asdoneby afterthesewill benamed‘α-profiles’. We computethemby
Taylor&Navarro(2001)fortheparticularcaseα=5/4.More integrating the second of Eqs. (6) with boundary conditions
conveniently,theymaybecomputeddirectlythroughEqs.(6) γ γ andγ 0forr 0. Theresultsintheplaneγ- γ
a ′ ′
→ → →
DMEQUILIBRIAINGALAXIESANDGALAXYSYSTEMS 5
FIG. 4.—Characteristicradii(inunitsofr0)fortheα-profilesasafunctionofα: radiusrpwherethecircularvelocitypeaks,radiusrmwherethevelocity
dispersionpeaks,radiusr3wheretheslopeofthedensityprofileequals3,andradiusrbwheretheslopeofthedensityprofilereadsγb=3(1+α)/2. Theblue
dotsrefertothesolutionwithα=5/4,andthereddotstoα=35/27.Inthebottomrightpaneltheredarrowindicatesthatthesolutionwithα=35/27attainsthe
slopeγbonlyasymptoticallyinapowerlawtail.Notethatasαincreases,thepeakofthevelocitydispersiondisplacesrelativelymorethanthecircularvelocity’s.
forvariousαbetween1.2and35/27areplottedasdifferent alistic beingnowherenegative(see alsoZhao1996;Widrow
linesin Fig. 1 (leftbottompanel); therewe can readoutthe 2000).
valuesof γ atr=r (squaresin the figure),and findthe re- In Fig. 3 we show the ratio v2(r)/σ2(r) entering Eq. (2);
0′ 0 c
latedκ afterEq.(7).Therelationκ (α)isplottedinFig.1 we also show the energy distributions associated to the
crit crit
(rightbottompanel);theresultingcorrelationisfittedbythe α-profiles: the specific potential energy E = 4π rdr
expressionκcrit(α) 8.228- 4.443αtobetterthan1%. r2ρ(r)v2(r )/M(< r), the random kinetic peontergy E0 =′
Oursolutionsinc≃ludetheprototypicalonecorrespondingto 4′π rd′r cr2′ρ(r)3σ2(r)/2M(<r), and the local virialRkriantio
α=5/4(highlightedinblue),firstfoundbyTaylor&Navarro 0 ′ ′ ′ ′
2E /E . Notethatthelattertendsto1atlarger,asisex-
(2001)andshowntobewellfitted,ifonlyinaregionaround Rkin | pot|
pectedforstableself-gravitatinghalos(seeŁokas&Mamon
r , by the NFW formula (Navarro et al. 1997); for such a
0 2001);infact,itcloselyapproachesunityalreadyforr r .
solution the precise value of κcrit reads 2.674, see Dehnen In Fig. 4 we plot as a function of α various charact≈eris0tic
& McLaughlin (2005). Our solutions also include the one
radii (in units of r ) of the α-profiles; specifically, we show
corresponding to the limiting value α= 35/27 (highlighted 0
theradiusr wherethecircularvelocityv (r)peaks,theradius
in red) studied by Dehnen & McLaughlin (2005); for this p c
κ =200/81isfound. rm where the velocity dispersion σ2(r) peaks, the radius r3
crit
In Fig. 2 we illustrate the radial dependencies of vari- where the slope of the density profile reads γ = 3, and the
ous interesting quantities for the α-profiles: density ρ and radiusrbwheretheslopeoftheprofiletakesonthevalueγb=
its logarithmic slope γ, circular velocity v2, velocity disper- 3(1+α)/2.Notethatrbisalwayslarge,andformallyrecedes
c to infinity (as represented in the plot with an arrow) for the
sionσ2, entropyK, massM(<r), andgravitationalpotential
Φ=- r∞dr′GM(<r′)/r′2; the profiles are normalizedto 1 limFiotirntghecassaekαe =of3c5l/a2ri7tyt,hianththaessaepFoiwguerrelaswwfealhloafvfe. confined
atthepointr withinthehalo’smiddle. Outwardsofthis,so-
lutionsRcorres0pondingtolargervaluesofαhavelargeroverall thevaluesofαtobetween1.2and35/27;althoughtheupper
boundisalreadyrequiredforhavingnocentralholeandafi-
masses, flatter density profiles, higher velocity dispersions,
nite mass asdiscussed above, limited informationexisted so
anddeeperpotentialwells.
far concerning lower bounds from the isotropic Jeans equa-
As mentionedabove, the α-profileswith α<35/27show
tion.
anoutercutoff,to whichthefollowingremarksapply. First,
thecutoffsgenerallyoccurbeyondthevirialradius(see§5). 3. THEENTROPYSLOPESFROM
Second, the velocity dispersion σ2 ρ2/3K rα- 2γ/3 falls COSMOGONICBUILDUP
∝ ∝
rapidly outside of r and vanishesat the density cutoff, thus
0 Here we show that the cosmogonichistories of DM halos
preventinganyfinite outflowrelatedto residualrandommo-
enforceadditional,stringentlimitationstotherangeofα;on
tions of DM particles. Third, the microscopic distribution
the high side the resulting bound turns out to be pleasingly
function as obtained through the Eddington formula (e.g.,
close to the value 35/27 derived above from the analysis of
Binney& Tremaine1987)frompairsof ρ(r) and Φ(r) is re-
the static Jeans equation, but on the low side it constitutesa
6 LAPI&CAVALIERE
newstringentresult. oftheinitialperturbations,thatalsoyieldsasequenceofcol-
Considera DM halo with currentboundingradiusR, den- lapse times. With a realistic bell-shaped perturbation ǫ 1
sity ρ, mass M ρR3, velocity scales σ2 v2 M/R, and still applies to its middle; in the outer regionsǫ&1 imp≈lies
∝ ∝ c ∝
entropyK σ2/ρ2/3. Duringthehierarchicalbuilduptherun- theaccretiontobecomeslower,andcorrespondinglytheden-
ninggrowt∝hratescalesasM˙ ρvR2 withtheinflowspeedv sity slope γ to steepen there relative to the middle one. The
setbythedepthoftheDMwe∝llandhenceproportionaltov2; centralregionsaredescribedbyǫ 1,correspondingtofast
c collapse;thescalinglawthensugg≪eststhedensityslopetobe
thustheabovequantitiesmaybeexpressedintermsofMand
˙ flatterthere.
Mtoprovidethescalinglaws:
However, this simple approachcannotbe carried very far.
R M/M˙2/3 v2 σ2 M˙2/3 ρ M˙2/M2 K M4/3/M˙2/3. Atsmallradiialsoangularmomentumeffectscomeintoplay
∝ c∝ ∝ ∝ ∝ (8) tolowerthedensityprofile;theyintroduceanotherradialscal-
ing related to centrifugal barriers, as discussed in detail by
Lu et al. (2006); these authors show how a density slope
3.1. Aheuristicapproach
around- 1 is enforced if efficient isotropization of particles’
Primarilywelookforalinkoftheentropyslopeαwiththe orbits takes place during the fast collapse, while still flat-
massgrowthrateinsimpleterms;thisanalysiswillhaveonly ter slopes occur with more even angular momentum distri-
heuristic scope, introductory to the full study performed in butions. These,however,mustbeconsistentwiththeoverall
§3.2.Towardourscope,wemakecontactwiththeclassicline equilibrium requirements; these orbital features are actually
ofdevelopments(Fillmore&Goldreich1984;Luetal. 2006) subsumed into the macroscopic Jeans Eq. (2) with K rα
thatanalyticallyrelatethecollapsehistoriestotheshapeand andprovidetherunningγ(r)=3α/5+3v2(r)/5σ2(r)(th∝elast
c
growthrateoftheprimordialDMperturbations. Theformer termisshowninFig.3)thatistoreplacethestiffscalingvalue
maybeexpressedintermsofδM/M M- ǫ;butsinceahalo γ=6- 3αabove,butstillrecoversitatthesingleradiusr (cf.
∝ 0
virializeswhenδM/Mattainsacriticalthreshold(e.g.,1.686 Eqs.[4]and[6]).
in the critical universe), the shape parameter ǫ =- dlog(1+ So we will depart from the above heuristic approach, and
z)/dlogM effectivelygovernsalsothe(inverse)massgrowth focusoncomputingthevaluesofαtobeinsertedintoJeans,
ratedefinedintermsofthescalingM (1+z)- 1/ǫ. alsoimplementingprecisemassandtimedependenciesofthe
∝
Thenwe use the standardrelationsbetween redshiftz and growthrateǫ.Tofocusthesesensitiveissuesarefinedanalysis
cosmic time t, that may be expressed in the form (1+z) andacarefuldiscussionarerequiredandperformednext.
t- q. Inthecriticalcosmologyq=2/3holds;inthecanonic∝al
Concordance Cosmology the value q 2/3 still applies to 3.2. Advanced,semianalyticanalysis
≈
z&1,butaszdecreasesqprogressivelyrisestovaluesq 0.8
Wederivearobusttimeevolutionandmassdependenceof
≈
forz.0.5. ThenthemassgrowthhistorywritessimplyM ˙
˙ ∝ MfromanupdatedversionoftheExtendedPress&Schechter
tq/ǫ and the growth rate reads M/M =q/ǫt; from Eqs. (8),
formalism(EPS,seeLacey&Cole1993).
simplealgebrayieldsthescalinglawsR t(2+q/ǫ)/3, ρ t- 2, We are mainly interested in the average evolutionary be-
andK t2(1+q/ǫ)/3. Oneliminatingt one∝obtainsK R∝αand haviorofhalos,sowecomputethemeangrowthrateM˙ ofthe
ρ R-∝γ withslopes ∝ currentmassMas
∝
α=1+1+21ǫ/q and γ= 2ǫ6+ǫq , (9) M˙ = MdM′ (M- M′)d2dPMM′→dtM , (10)
Z0 ′
the latter being related to the former by the simple scaling in terms of the differential merger rate d2PM′ M/dM′dt
link γ =6- 3α, as expected at r r0 from Eq. (4). From giveninAppendixA.Thisincorporatesthefullco→smological
≈
Eq.(9)αisseentobeslightlylargerthanunitybyaquantity framework,andthedetailedcoldDMperturbationspectrum;
slowlydependingonǫ/q;thisisconsistentwithK RM1/3 astotheformerweimplementtheConcordanceCosmology,
R2- γ/3, thestraightforwardmessagefromEqs.(8)∝withM∝ andasforthelatterweadopttheshapegivenbyBardeenetal.
R3- γ. ∝ (1986)correctedfor baryons(Sugiyama1995), and normal-
These relationshipsfor α and γ embodyin a simple form ized as to yield a mass variance σ8 =0.8 on a scale of 8h- 1
somerelevantcosmogonicandcosmologicalinformation.For Mpc. We improve over previous studies of this kind (e.g.,
z & 1 where q 2/3 applies, one explicitly has α (2+ Milleretal. 2006;Lietal. 2007;Salvador-Soléetal. 2007)
3ǫ)/(1+3ǫ) and≈relatedly γ 9ǫ/(1+3ǫ), the latter≈being byusingthemergerrateappropriateforellipsoidalcollapses
the well-knownexpressionfo≈rthe similarity solutionsto the (Sheth& Tormen2002)asprovidedbyZhangetal. (2008);
sphericalinfall modelin a critical Universe. For z.0.5 in- this is shown by the latter authors to yield formation times
stead when q 4/5 applies, one has α = (4+5ǫ)/(2+5ǫ) and growth rates in much closer agreement with numerical
and γ 15ǫ/(≈2+5ǫ). As said, the remaining parameter ǫ simulationsthanthestandardEPStheory. Furtherdetailsare
govern≈s both the shape of the initial perturbations and the presentedinAppendixA.
˙
times of their collapse. For instance, if the standard top-hat Eq.(10)providestherelationM(M,t)illustratedin Fig.5.
shape appliedwith ǫ 1 correspondingto a monolithiccol- This may be viewed as a first order differentialequation for
≈
lapse,fromtheaboverelationsonewouldobtainα 1.25and M(t), that is numerically solved once the mass M(t ) at the
0
≈
γ 2.25forhalosthatvirializeatz&1,aspointedoutearly present time t has been assigned in the way of a boundary
0
≈
onbyBertschinger(1985);formoremassivehalosthatvirial- condition. In terms of M(t) so computed, the evolution of
izeatz.0.5,fromthescalinglawsoneanticipatesaslightly thehalo propertiesare obtainedthroughEqs. (8); the results
steeperα 1.29andaflatterγ 2.15. are illustrated in Fig. 6, where we plot the time and redshift
≈ ≈
Qualitatively,differentvaluesoftheslopeγmayberelated evolutionsfortheinversegrowthrateǫ, theoverallradiusR,
toothervaluesofǫ,suchasoccurringinanarticulatedshape the mass M, the characteristic velocities v σ, the entropy
∝
DMEQUILIBRIAINGALAXIESANDGALAXYSYSTEMS 7
˙
FIG.5.—TherelationM(M,z)fromtheaveragecosmogonicevolution,see§3.2fordetails.
K, and finally the corresponding slope α dlogK/dlogR. ingthisstagetheN- bodyexperiments(seeZhaoetal. 2003;
≡
Thecurvesarefordifferentcurrentvaluesofthehalomasses Peirani et al. 2006; Hoffman et al. 2007) highlight turmoil
M(t )rangingfrom1011to1015M . associatedwithfastcollapseandmajormergers,effectivedy-
0
In fact, in these detailed mass a⊙ccretion histories one can namical relaxation and mixing of DM particles. Taking up
recognizeatransitionepochwhenthecircularvelocityv2 of from § 3.1, it is during these events that the orbits’ angular
c
the halo attains its highest value along an accretion history momentummustsettoanearisotropicdistributionsuchasto
(see Li et al. 2007). The transitionseparatestwo stages: an lowertheinnerdensities.
early stage of fast collapse, when ǫ.0.4 corresponds to a Butastheturmoilassociatedwiththefastcollapsesubsides,
growth rate still high on the running Hubble timescale, and the orbital structure is bypassed and replaced by the macro-
nearlyconstant;andalaterphaseofsloweraccretionwhenǫ scopicentropydistributionK rα withuniformvaluesofα,
∝
takes off from the value 0.4, i.e., the growth rate is low and asphenomenologicallyfoundtoholdthroughoutthenowsta-
decreasing. Small halos with M .1012M have high tran- blehalos’bulk. Thenthe valuesof αatthetransitionepoch
sitionredshifts,hencearenowwelladvanc⊙edintotheirslow and the corresponding edge R=r can be derived from the
0
accretion stage; on the other hand, very massive ones with scalinglawsandappliedtothewholeinnerregion,toobtain
M&1015M havelowtransitionredshifts,sotodaytheyhave γ(r)=3α/5+3v2(r)/5σ2(r),takingonthevalue6- 3αonly
c
just comple⊙tedtheir fast accretion stage. A similar behavior atthepointr accordingtotheresultsof§2.1.
0
has been identified in a number of recent numericalsimula- Thevalueofαisexpectedtopersistinthebulkduringthe
tions(Zhaoetal. 2003;Diemandetal. 2007),andhasbeen subsequent development in the slow accretion stage, when
recognizedalsoinothersemianalyticcomputationsbasedon mass is added smoothly and mainly onto the outskirts, with
thesimpler,lesspreciseEPStheory(seeLietal. 2007;Neis- littlemixing. Suchadevelopmentisborneoutbythewidely
teinetal. 2006). sharedscenarioofhalobuildupfromtheinsideout,champi-
oned by Bertschinger (1985), Taylor & Navarro (2001), and
3.3. Results
manyothers up to Salvador-Soléet al. (2007). Specifically,
From Fig. 6 we also see that the values of α at the tran- theanalysisbytheformerauthorsconcerningtherandomiza-
sition are narrowlyconstrained to within 1.27 0.02,i.e., a tionandthestratificationofparticles’orbitsduringsecondary
±
1.5% range. We note that this range explainswhy narrowly infallindicatesthattheirinternaldispositionislittle affected
constrained phenomenologicalvalues of α are found within by subsequent, smoother mass additions on top of a closely
the halo bulk from many and differentsimulations since the staticpotentialwell.Thusduringthelate,slowaccretionstage
year 2000. In particular, our values turn out to agree with thestructureforr.r willbeunaffected.
0
thebestfittingexponentofQ(r) [orK(r)]withits scatterre- Meanwhile, the outer scale R M/M˙2/3 beyond r is in-
0
centlygivenindetailbyAscasibar&Gottlöber(2008)within creasingly stretched out to subs∝tantially larger values (see
the halo bulk during calm infall; these authors warn against Fig. 6, top right panel); correspondingly, we find the outer
extrapolatingtheirresultstoearlytimesandouterradii. slope α(R) dlogK/dlogR at R > r to decline in time
0
Correspondingly,inourfindingsthevaluesofαbeforethe (see Fig. 6,≡bottom right panel). This translates into a ra-
transitionshould notbe overinterpretednor insertedinto the dial decrease α(r) on maintaining entropy stratification also
Jeansequation. Infact,attheseearlyepochsthehalosexpe- intotheseregions(seeBertschinger1985,Taylor& Navarro
riencerapidchangesandareclearlyoutofequilibrium;dur-
8 LAPI&CAVALIERE
FIG. 6.—Evolutionofthevarious quantities alongtheaverage accretion historyofDMhalos: massM,radius R,velocity v, (inverse) massgrowthrate
ǫ=- dlog(1+z)/dlogM,entropyK,andslopeα≡dlogK/dlogR. Thecurvesarefordifferentcurrenthalomassesintherange1011- 1015M⊙;fromtopto
bottom,bluelinesrefersto1011M⊙,cyanto1012M⊙,greento1013M⊙,orangeto1014M⊙,andredto1015M⊙.Linesaredashedinthefastaccretionregime,
anddotted intheslowaccretion phase; thesolid linemarksvalues atthetransition epoch, see§3.3fordetails. Inthebottomrightpanel, theshaded area
highlightsthenarrowrangeexpectedforthevaluesofαatthetransition.
2001). Suchadecreaseistobeexpectedalsofromconsider- transitionfrom1.25to1.29inmovingfromgalaxy-tocluster-
ingthatK(r) r2- γ/3 rapplies,onthebasisoftherelated sizedcurrentmasses. Afitaccuratetobetterthan0.1%tothis
asymptotic be∝haviors→of ρ(r) r- γ and σ2(r) r2- γ where averagerelationisprovidedbyα 2.71- 0.23log(M/M )+
γ 3. We find thevaluesof∝α to decreaseby∝less than2% 0.009[log(M/M )]2;inparticular≃forgalaxyclusters,this⊙im-
ou→ttor 2r ,andlessthan10%outtor 5r . pliesvaluesofα⊙ 1.27forAbellrichnessclassesr 0and
0 0
Conce≈rningtheJeansequation,thistoo≈(withitsboundary α 1.29forrichn≈essclassr&3. Thecorrelationα(≈M)may
≈
conditionssetatr=0, see §2.2)technicallyworksfromthe help to understand the differences in the values of α found
insideout,andmaybeextendedtotheranger>r withde- in numerical simulations; in fact, the highest value 1.29 to
0
creasingslopeα(r). Thecorrespondingdensityrunissome- date has been reported by Rasia et al. (2004) in looking at
whatsteepenedforr>r ,asifshiftingfromtheuppertothe cluster-sizedhalos,whilethevalue1.25firstfoundbyTaylor
0
lowercurvesinFig.2;wehavecheckedthattheouterdensity &Navarro(2001)referstogalaxy-sizedhalos. Observational
profilelowersby lessthan 20%forr.5r . State-of-the-art evidencesupportingtheabovepictureispresentedin§5.
0
simulations still yield little information on the outskirts for Tosum up, fromthe cosmologicalbuildupwe find forthe
r&r owingto lowdensitiesandnoisydatatherediscussed entropyslopethenarrowallowedrange
0
by Ascasibar & Gottlöber(2008). We presentelsewhere the α 1.27 0.02; (11)
complete solutions outside r as a testbed for future simula- ≈ ±
0 this we use in our Jeans-like description of DM equilibria
tions aimed at resolving the outskirts, and at fitting gravita-
within the halos’ bulk. The boundson α are relevantin two
tionallensingobservations(seediscussionin§5).
respects: theupperonestrengthenstheformallimit35/27=
In a similar vein, we predict a minor increase of α at the
1.296requiredforthesolutionsofthestaticJeansequationto
DMEQUILIBRIAINGALAXIESANDGALAXYSYSTEMS 9
haveviableprofilesinthefaroutskirts(see§2.2anddiscus-
sionin§5);theloweronecomplementarilyrestrictstherange
oftheviablevalues.
4. APHYSICALMEANINGFORκcrit
Here we propose to discuss the physical meaning of the
quantity κ (defined in § 2.1 following Taylor & Navarro
crit
2001)in terms of a direct energeticargument. We first note
thatgiventheasymptoticsoftheα-profiles,thevelocitydis-
persionv2(r) mustpeakat a pointr &r (see Fig. 4) to the
c p 0
valuev2 =4πGρ(r )r2;atthiseasilyidentifiedpointthecon-
p p p
stantκ alsowrites4πGρ(r )r2/σ2 =v2/σ2, andthusis seen
p p p p p
tocomparetheestimateσ2fortherandomkineticenergywith
p
theestimatev2 forthegravitationalpotential. Wethenrecall
p
from§2.1thatahalointheviableequilibriumconditionfor
agivenαismarkedbyamaximalvalueκ (α),thatis,bya
crit
minimallevelofrandomizedenergyσ2=v2/κ .
p p crit
Wenowshowthattoaverygoodapproximationthevalue FIG. 7.—Theκcrit(α)relation,asinFig.1(bottomrightpanel). Crosses
for κ may be obtained from considering incomplete con- illustratethegoodnessofourapproximationtoκcrit(α)basedonadirecten-
crit ergeticargument,see§4fordetails.
versionoftheinfallkineticenergyv2 /2ofDMparticlesinto
inf
their dynamically randomized energy 3σ2/2 during forma-
p computevaluesforκ asafunctionofα;notethatthesede-
tion; a process introduced by Bertschinger (1985), different crit
creaseforincreasingαwhichimplieslarger∆φ ,seeFig.2.
from the complete violent relaxation proposed by Lynden- pt
Fig.7 showshowwellEq.(13)recoversthe κ (α) relation
Bell (1967)and discussed widely, includingthe recentArad crit
obtainedthroughnumericalintegrationoftheJeansequation.
& Lynden-Bell(2005), Trenti & Bertin (2005)and Trentiet
This excellent fit to κ means that the expression for η
al. (2005). crit
inEq.(12),morethanasuccessfulguess,isdescribingwhat
We propose to derive the critical value of κ=v2/σ2 from
p p actuallyoccursinthemacroscopicequilibriaexpressedbythe
therelation
Jeansequationinkeepingwiththenumericalsimulations. At
3σ2 ηv2 , with η= 3 v2p/v2inf . (12) thereferee’ssuggestionweaddthatwhatultimatelythe first
p≈ inf 2 1+v2/v2 Eq. (12) describes with the isotropic form of its l.h.s. is a
p inf
partial effective conversion at r r from radial infall into
p
This expression for the effective conversionefficiency η has ≈
tangentialmotions.
beenconstructedonconsideringthat: (i)ithasto dependon
theratioofthe(1- D)infallvelocitysquaredv2 tothedepth
inf 5. DISCUSSIONANDCONCLUSIONS
ofthepotentialwellasmeasuredbythe(2- D)circularveloc-
itysquaredv2;(ii)theformallimitv2 =2σ2 correspondingto The general picture of DM halo formation as it emerges
p p p from recent N- body simulations envisages a progressive
κ=2(theisothermalpowerlaw,see§2.1)mustbeapproached
for a fast inflow with v2 v2, which quantifiesthe incom- buildup through a two-stage development. At early epochs
inf ≫ p majormergersassociatedwithfastreshapingofthebulkgrav-
pletenessoftheenergyconversionduringthetransitofabullet
itational potential cause particles’ orbits to dynamically re-
withcrossingtimeshortcomparedwiththehalo’sdynamical
lax and mix into a definite equilibrium disposition. At later
time.
epochsthehalooutskirtsprogressivelybuildupthroughrare
As a simple example guided by the canonical theory of
if persisting minor mergers and by smooth mass accretion,
gravitational interactions (see Saslaw 1985; Cavaliere et al.
1992)wemayconsiderthatv2 /v2 3istoholdforhaving settlingtoaquasi-staticequilibrium.
inf p≈ The stable equilibria soon after the early stage are effec-
considerableenergytransfer;usingthissimpleapproximation
tivelydescribedintermsofa phenomenologicalentropyrun
inEq.(12)yieldsκ 2.6,avaluepleasinglycloseto2.64
crit≈ K rαwithuniformslopeαsomewhatlargerthan1.Spurred
as obtained numerically in § 2.2 for the particular α-profile ∝
bythispowerlawbehavior,onemightlookforpowerlawso-
withα=1.25.Ontheotherhand,thissimpleargumentcannot
lutionsoftheJeansequationfortheequilibriumdensitypro-
pinpointvalues of κ to the precision requiredto discrimi-
crit files, only to find these just tangent to the actual solutions.
natedifferentvaluesofα.
Fromhereouranalysisgoesbeyondpowerlawdensities,and
So, we may test our proposed η as given in Eq. (12) for
providestworelatedblocksofnovelresults.
differentvaluesofαbyusinginsteadenergyconservationto
express the infall velocity v2 2∆φ in terms of the adi- First,wehavecomputedandillustratedawidesetofnovel
mensional potential drop ∆iφnf ≈(normaptlized to v2) from the solutions to the isotropic Jeans equation, that we name α-
pt p profiles, includingas two particular instances the prototypes
turnaround radius to r (see Bertschinger 1985; Lapi et al.
p investigated by Taylor & Navarro (2001) and Dehnen &
2005).InsertingthisintoEq.(12)yieldstheexpression
McLaughlin(2005). Wehavestressedthatatgivenαtheso-
v2 1 lutionsformatangentbundlewithρ(r)progressivelycurved
κcrit≡ σp2 =2+∆φ ; (13) down as the gravitational pull measured by κ = v2p/σ2p in-
p pt creasesabovethe value correspondingto the pure powerlaw
this highlights the deviation of κ from the value 2 that profileγ=6- 3α;fromsuchbundlestheα-profilesstandout
crit
wouldapplyfortheisothermalpowerlaw(see§3).Usingthe forhavingashapemaximallycurvedtothepointofcomply-
runsofφappropriatefortheα-profileasgivenin Fig.2,we ing with sensible outer and inner behaviors, i.e., finite total
10 LAPI&CAVALIERE
FIG.8.—Analyticfitstothedensityruns(toppanels)andresiduals(bottompanels)fortheα-profileswithα=1.25(leftpanels)andα=35/27(rightpanels).
SolidlinesillustratestheexactsolutionsfromnumericalintegrationoftheJeansequation,dashedlinesshowouranalyticfittingformulaEq.(B1),anddotted
line(onlyinleftpanels)isthefittingformulabyTaylor&Navarro(2001,theirEq.[10]);notethatEq.(B1)forα=35/27coincideswiththeexactsolutiongiven
byDehnen&McLaughlin(2005).
mass and no central hole. This occurs for a maximal value 2000;Barnesetal. 2006),thatreads
κ (α)ofthegravitationalpullv2relativetotherandomcom-
crit p 1
preosnpeonntdσs2pt,oi.teh.e,fmoarxaimmianlismloapleσγ2pc=om3pαa/r5ed+tκov2pa;tththiseaplosionctorr-. ρ(r)∝ r/r0 γa 1+w r/r0 u q, (14)
p crit p
Suchsolutionsstillconstitutealargeset,beingparameterized withγa governingthe(cid:0)inne(cid:1)rpr(cid:2)ofile,q(cid:0)the(cid:1)ou(cid:3)terone,wandu
intermsofawideoverallrange1<α 1.296forthevalues the position and sharpness of the radial transition. Here for
≤
ofα. the first time such a profile is substantiated with parameters
Second, we have shown that toward further constraining expressed in terms of the derivatives of the Jeans equation
thesevaluesthekeystepisprovidedbythehalos’cosmolog- (see § 2.2) and of the values for α as detailed in Appendix
icalbuildup;infact,thescalinglawsof§3narrowlyrestrict B. We stress that the Jeans equlibrium (see Fig. 2) requires
theviablerangetoα 1.25- 1.29 1.27 0.02,correspond- that the central slopes are always flatter, and the outer ones
ing to current masse≈s M 1011- ≈1015M±. We have found alwayssteeper thanintheempiricalNFWfit. Thegoodness
theseboundstoapplyatth∼eepochsoftran⊙sitionfromfastcol- ofthe abovefittingformulaisdiscussedinAppendixB, and
lapsetoslowaccretion,whenthepotentialwellssettletotheir illustratedinFig.8.
maximaldepthsalonganaccretionhistory(see§3). Infact, Weprobethephysicalrelevanceoftheα- profilesbycom-
oursemianalyticanalysisnotonlyrecoversthetwo-stagede- paring in Fig. 9 the related projected density runs with the
velopmentobservedinmanyrecentN- bodysimulations,but results from recent, exquisite observations of strong and
also explains why only such narrowly constrained values of weakgravitationallensinginandaroundmassivegalaxyclus-
αareactuallyfoundinmanydifferentnumericalworkssince ters, which just require flat inner, and steep outer behaviors
theyear2000.Thevaluesκcritiscorrespondinglyconstrained (Broadhurstetal. 2008,Umetsu&Broadhurst2008,andref-
to the range 2.7&κcrit &2.5; we trace back these valuesto erencestherein).Itisseenhowstraightforwardareourfitsin
limitedrandomizationfortheinfallkineticenergyofDMpar- termsofhigh-massα- profilesthatnaturallyexhibitjustsuch
ticlesduringaccretion,whichleadstominimalvaluesofσ2. features, compared to those in terms of the NFW empirical
Theα- profilesfromtheJeansequilibriumwithαandκcrit formula that not only has problems of ill-defined mass, but
soconstrainedimplythatgalaxieshaveagenerallymorecom- also would require particularly high concentrations and old
pactcentralstructurethangroupsandclusters,withapprecia- ages. Ourfindingssubstantiatetheα-profiles,andstresstheir
bledeviationfromself-similarity;ontheotherhand,thefor- value as to specific predictions for future simulations aimed
mer have currently developed extensive outskirts, while the atresolving,and forweak gravitationallensingobservations
veryrichclustersdonotyet. aimedatprobingtheouterclusterdensities. Afulldiscussion
Toembodyinahandyexpressionthebasicfeaturesofthe oftheseissuesdeservesadedicatedpaper.
density runs for the α- profiles, we provide here an analytic Two commentsare in order concerningthe validity of the
fittingformulaoftheformoftenadvocatedonphenomenolog- Jeans equation in the simple form of Eq. (2). First, reason-
ical grounds(see Zhao1996; Kravtsovet al. 1998; Widrow able anisotropies in the DM are described by the standard