Halo Mass Definition and Multiplicity Function

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by  Enric Juan

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed30January2014 (MNLATEXstylefilev2.2)
Halo Mass Definition and Multiplicity Function
⋆
Enric Juan , Eduard Salvador-Sol´e, Guillem Dom`enech and Alberto Manrique
4 Institut de Ci`encies del Cosmos. Universitatde Barcelona, UB–IEEC. Mart´ı i Franqu`es 1, E-08028 Barcelona, Spain
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30January2014

n
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J ABSTRACT

8 Comparing the excursion set and CUSP formalisms for the derivation of the halo
2 mass function, we investigate the role of the mass definition in the properties of the
multiplicity function of cold dark matter (CDM) haloes. We show that the density
]
O profile for haloes formed from triaxial peaks that undergo ellipsoidal collapse and
virialisationis suchthatthe ratiobetweenthe meaninner density and the outer local
C
density is essentially independent of mass. This causes that, for suited values of the
h. spherical overdensity ∆ and the linking length b, SO and FoF masses are essentially
p equivalent to each other and the respective multiplicity functions are essentially the
- same. The overdensity for haloes having undergone ellipsoidal collapse is the same as
o
ifthey hadformedaccordingto the sphericaltop-hatmodel,whichleadstoa valueof
tr b corresponding to the usual virial overdensity, ∆vir, equal to ∼0.2. The multiplicity
s
function resulting from such mass definitions, expressed as a function of the top-hat
a
[ height for sphericalcollapse, is very approximatelyuniversalin all CDM cosmologies.
The reason for this is that, for such mass definitions, the top-hat density contrast for

1 ellipsoidal collapse and virialisation is close to a universal value, equal to ∼0.9 times
v
the usual top-hat density contrast for spherical collapse.
4
3 Key words: methods: analytic — galaxies: haloes, formation — cosmology: theory,
3 large scale structure — dark matter: haloes
7
.
1
0
4
1 INTRODUCTION algorithm with fixed linking length b in units of the mean
1
inter-particle separation.
v: Largescalestructureharboursimportantcosmologicalinfor- The main drawback of the FoF definition is that, for
i mation. However, such a fundamental property as the halo
X large values of b, it tends to over-link haloes. Its main ad-
mass function (MF) is not well-established yet. Besides the
vantage is that it can be applied without caring about the
r lack of an accurate description of non-linear evolution of
a symmetryanddynamicalstateofhaloes.Haloesare,indeed,
density fluctuations, there is the uncertainty arising from
triaxialratherthansphericallysymmetric,harboursubstan-
the fact that the boundary of a virialised halo is a fuzzy
tial substructure and may be undergoing a merger, which
concept. Asa consequence,thehalo mass function depends
complicates the use of the SO definition. However, one can
on the particular mass definition adopted, its shape being
focusonvirialisedobjectsandconsiderthesphericallyaver-
onlyknowninafewcasesandoverlimitedmassandredshift
ageddensityprofileρ (r)andmassprofileM(r)aroundthe
ranges. h
peak-density,inwhichcasetheFoFmasscoincideswiththe
The various halo mass definitions found in the litera-
massinsidetheradiusR wherespheresofradiusbharbour
turearisefromthedifferenthalofindersusedinsimulations h
two particles in average (Lacey & Cole 1994),
(Knebeet al.2011).Forinstance,intheSphericalOverden-
sity(SO)definition(Lacey & Cole1994),themassofahalo ρ (R)= 3 b−3ρ¯(t). (2)
at thetimet isthatleadingtoatotalmean densityρ¯ (R ) h 2π
h h
equal to a fixed, constant or time-varying, overdensity ∆ Equations (1) and (2) imply the relation
times themean cosmic density ρ¯(t),
3F(c)
∆= b−3, (3)
ρ¯ (R )=∆ρ¯(t). (1) 2π
h h
between∆andbforhaloesofagivenmassM,whereF(c)
While in the Friends-of-Friends (FoF) definition ≡
ρ¯ (R )/ρ (R ) is a function of halo concentration c.
(Daviset al. 1985), the mass of a halo is the total h h h h
As c depends on M, there is no pair of ∆ and b values
mass of its particles, identified by means of a percolation
satisfying equation (3) for all M at the same time. Conse-
quently,thereisstrictlynoequivalentSOandFoFmassdef-
⋆ E-mail:[email protected] initions(More et al.2011).Yet,numericalsimulationsshow
2 Juan et al.
that, at least in the Standard Cold Dark Matter (SCDM) assume monolithic collapse or pure accretion. While in hi-
cosmology, FoF masses with b = 0.2, from now on simply erarchical cosmologies there are certainly periods in which
FoF(0.2),tightlycorrelatewithSOmasseswithoverdensity haloes evolve by accretion, major mergers are also frequent
∆ equal to the so-called virial value, ∆ 178, from now andcannot beneglected.Wewill comeback tothispointat
vir
≈
on SO(∆ ) (Cole & Lacey 1996). This correlation is often theendofthepaper.Asecondandmoreimportantissuein
vir
interpretedasduetothefactthathaloesareclosetoisother- connection with the problem mentioned above is that none
malspheres,forwhichF(c)isequalto3,soequation(3)for of these theoretical MFs makes any explicit statement on
b=0.2 implies ∆ 178. thehalo mass definition presumed,so thespecific empirical
≈
Simulations also show that, in any cold dark matter MF they are to becompared with is unknown.
(CDM)cosmology,FoF(0.2)haloeshaveamultiplicityfunc- Recently, Juan et al. (2013, hereafter JSDM) have
tion that, expressed as a function of the top-hat height for shown that,combining theCUSPformalism with theexact
sphericalcollapse,isapproximatelyuniversal(Jenkins et al. follow-up of ellipsoidal collapse and virialisation developed
2000; White 2002; Warren et al. 2006; Luki´c et al. 2007; by Salvador-Sol´e et al. (2012, hereafter SVMS), it is pos-
Tinker et al. 2008; Crocce et al. 2010) and very similar to sible to derive a MF that adapts to any desired halo mass
that found for SO(∆ ) haloes (Jenkins et al. 2000; White definition and is in excellent agreement with the results of
vir
2002). As ∆ may substantially deviate from 178 depend- simulations.
vir
ingonthecosmology,suchasimilaritycannotbeduetothe Inthepresentpaper,weusetheexcursionsetandCUSP
roughly isothermal structure of haloes as suggested by the formalisms to explain the origin of the observed properties
SCDM case. Moreover, the universality of this multiplicity of thehalo multiplicity function.
function is hard to reconcile with the dependence on cos- In Section 2, we recall the two different approaches
mology of halo densityprofile(Courtin et al. 2011).On the for the derivation of the MF. In Section 3, we investigate
other hand, haloes do not form through spherical collapse the mass definition implicitly assumed in such approaches.
but through ellipsoidal collapse. For all theses reasons, the The origin of the similarity of the multiplicity function for
originofsuchpropertiesisunknown.Havingareliabletheo- FoF(0.2) and SO(∆vir) masses and of its approximate uni-
reticalmodelofthehaloMFwouldbeveryusefulfortrying versality is addressed in Sections 4 and 5, respectively. Our
to clarify theseissues. results are discussed and summarised in Section 6.
Assuming the spherical collapse of halo seeds, Allthequantitativeresultsgiventhroughoutthepaper
Press & Schechter (1974) derived a MF that is in fair are for the concordant ΛCDM cosmology with ΩΛ = 0.73,
agreement with the results of numerical simulations (e.g. Ωm = 0.23, Ωb = 0.045, H0 = 0.71 km s−1 Mpc−1, σ8 =
Efstathiou et al. 1988; Whiteet al. 1993; Lacey & Cole 0.81, ns = 1 and Bardeen et al. (1986, hereafter BBKS)
1994;Bond & Myers1996),although with substantialdevi- CDM spectrum with Sugiyama (1995) shape parameter.
ations at both mass ends (Lacey & Cole 1994; Gross et al.
1998; Jenkinset al. 2001; White 2002; Reed et al. 2003;
Heitmann et al. 2006).Anoutstandingcharacteristic of the 2 MASS FUNCTION
associated multiplicity function is its universal shape as a
function of the height of density fluctuations. Whetherthis All derivations of the halo MF proceed by first identifying
characteristicisconnectedwiththeapproximatelyuniversal theseedsofhaloeswithmassM atthetimetinthedensity
multiplicityfunctionofsimulatedhaloesforFoF(0.2)masses field at an arbitrary small enough cosmic time ti and then
is however hard to tell. countingthose seeds.
Bond et al. (1991) re-derived this MF making use of
the so-called excursion set formalism in order to correct
2.1 The Excursion Set Formalism
for cloud-in-cloud (nested) configurations. This formalism
wasadoptedinsubsequentrefinementscarriedoutwiththe In this approach, halo seeds are assumed to be spherical
aim to account for the more realistic ellipsoidal collapse overdense regions in the initial density field smoothed with
(Monaco 1995; Lee & Shandarin 1998; Sheth & Tormen a top-hat filterthat undergospherical collapse.
2002). The excursion set formalism has also recently been Thetimeofsphericalcollapse(neglectingshell-crossing)
modified (Paranjape et al. 2012; Paranjape & Sheth 2012) of a seed depends only on its density contrast, so there is a
to account for the fact that density maxima (peaks) in one-to-onecorrespondence between haloes with M at t and
the initial density field are the most probable halo seeds density perturbations with fixed density contrast δ at the
ci
(Hahn & Paranjape 2013). filtering radii R satisfying the relations
f
In an alternative approach, theextension to peaks was
D(t)
directly attempted from the original Press-Schechter MF δ (t)=δ (t) i (4)
ci c D(t)
(Bond 1989; Colafrancesco et al. 1989; Peacock & Heavens
1990; Appel& Jones 1990; Bond & Myers 1996; Hanami 1/3
3M
2001). The most rigorous derivation along this line was by Rf(M)= 4πρ¯ . (5)
i
Manrique and Salvador-Sol´e (1995, hereafter MSS; see also (cid:20) (cid:21)
Manrique et al. 1998), who applied the so-called ConflU- Inequations(4)and(5),ρ¯ isthemeancosmicdensityatt=
i
ent System of Peak trajectories (CUSP) formalism, based t,δ (t)isthealmostuniversaldensitycontrastforspherical
i c
ontheAnsatzsuggested bysphericalcollapse that“thereis collapse at t linearly extrapolated to that time and D(t) is
aone-to-onecorrespondencebetweenhaloesandnon-nested thecosmicgrowthfactor.IntheEinstein-deSitteruniverse,
peaks”. D(t)isequaltothecosmicscalefactora(t)andδ (t)isequal
c
A common feature of all these derivations is that they to 3(12π)2/3/20 1.686. While, in the concordant model
≈
Halo Mass Function 3
andthepresenttimet ,D(t )isafactor0.760smallerthan spectralmoment at thecurrenttimet .Thus,theresulting
0 0 0
a(t ) and δ (t ) is equal to 1.674 (e.g. Henry2000). MF (eq.[8]) is independentof thearbitrary initial time t.
0 c 0 i
≈
Equation (5) is valid to leading order in the perturba- But the assumptions made in this derivation are not
tion, theexact relation between R and M being fullysatisfactory:i)Everyoverdenseregiondoesnotcollapse
f
into a distinct halo; only those around peaks do. Unfortu-
1/3
3M nately,theextensionoftheexcursionsetformalismtopeaks
R (M,t)= . (6)
f 4πρ¯i[1+δci(t)] is not trivial (Paranjape et al. 2012; Paranjape & Sheth
(cid:26) (cid:27)
2012); ii) Real haloes (and peaks) are not spherically sym-
The interest of adopting the approx relation (5) is that the
metric but triaxial, so halo seeds do not undergo spherical
filtering radius then depends only on M. This greatly sim-
but ellipsoidal collapse. Unfortunately, the implementation
plifies themathematical treatment.
inthisapproachofellipsoidalcollapseishardtoachievedue
Following Press & Schechter (1974), every region with
to the dependence on M of the corresponding critical den-
density contrast greater than or equal to δ (t) at the scale
ci sitycontrast (Sheth & Tormen 2002;Paranjape et al.2012;
R (M)willgiveriseatttoahalowithmassgreaterthanor
f Paranjape & Sheth 2012). iii) The formation of haloes in-
equaltoM.Consequently,theMF,i.e.thecomovingnumber
volvesnot onlythecollapse of theseed,butalso theviriali-
density of haloes per infinitesimal mass around M at t, is
sation (through shell-crossing) of thesystem, which is hard
simplytheM-derivativeofthevolumefraction occupiedby
to account for. And iv) there is a slight inconsistency be-
those regions, equal in Gaussian random density fields to
tween the top-hat filter used to monitor the dynamics of
1 1 δ (t) collapse and the sharp k-space window used to correct for
V(M,t)= erfc ci , (7)
2 (cid:20)√2 σ0th(M,ti)(cid:21) nesting. The use of the top-hat filter with this latter pur-
pose is again hard to implement due to the correlation be-
divided by thevolumeM/ρ¯(t) of one single seed, tween fluctuations at different scales in top-hat smoothing
∂n (M,t) ρ¯(t) ∂V(M,t) (Musso & Paranjape 2012).
PS = . (8)
∂M M ∂M
In equation (7), σth(M,t) is the top-hat rms density fluc-
0 i
2.2 The CUSP Formalism
tuation of scale M at t.
i
Butthisderivationdoesnottakeintoaccountthatover-
In this approach, halo seeds are regions around non-nested
dense regions of a given scale may lie within larger scale
(triaxial) peaksthatundergoellipsoidal collapse and viriali-
overdenseregions, whichtranslates intoawrongnormalisa-
sation.TheuseofaGaussianfilterismandatoryinthiscase
tion1oftheMF(7)–(8).Tocorrectforthiseffect,Bond et al.
for thereasons given below.
(1991) introduced theexcursion set formalism. The density
The time of collapse and virialisation of triaxial seeds
contrastδatanyfixedpointtendstodecreaseasthesmooth-
depends not only on their density contrast δ at the suited
ing radius R increases, so, using a sharp k-space filter, δ
f scale R, like in spherical collapse, but also on their ellip-
traces a Brownian random walk, easy to monitor statisti-
ticity and density slope (e.g. Peebles 1980). However, all
cally. In particular, one can estimate the number of haloes
peaks with given δ at R have similar ellipticities and den-
reaching M at t by counting the excursion sets δ(R) inter-
sity slopes (JSDM). Consequently, all haloes at t can be
secting δ (t)at anyscale R (M).The important noveltyof
ci f seen to arise from non-nested triaxial peaks at t with the
i
this approach is that, whenever a halo undergoes a major
same fixed density contrast δ at any filtering radius R .
mi f
merger, δ increases instead of decreasing, so every trajec-
Then, taking advantage of the freedom in the boundary of
tory δ(R) can intersect δ (t) at more than one radius R,
ci virialised haloes, wecan adopt at t a suited mass definition
meaning that there will be haloes appearing within other
so to exactly match the one-parameter family of resulting
more massive ones. Therefore, to correct for cloud-in-cloud
halo masses M(R ,t). By doing this, we will end up with a
f
configurations,onemustsimplycounttheexcursionsetsin-
one-to-onecorrespondence between haloes with M at t and
tersecting δ (t)for thefirst timeas Rdecreases from infin-
ci non-nested (triaxial) peaks at t with δ at R , according
ity (orσth increases from zero),as if theywere absorbed at i mi f
0 to therelations
such a barrier. The MF so obtained has identical form as
D(t)
thePress-Schechterone(eqs.[8]-[7]) butwith an additional δ (t)=δ (t) i (10)
factor two, mi m D(t)
∂nes(M,t) =2∂nPS(M,t), (9) R (M,t)= 1 3M 1/3 . (11)
∂M ∂M f q(M,t) 4πρ¯
i
(cid:20) (cid:21)
yielding theright normalisation of theexcursion set MF.
Note that, as the height of a density fluctuation, de- Equations (10)–(11) are very similar to equations (4)–
fined as the density contrast normalised to the rms value (5). However, the pair of functions δm(t) and q(M,t) are
at the same scale, is constant with time, the volume nowarbitrary,fixingoneparticularhalomassdefinitioneach
V(M,t) (eq. [7]) can be written as a function of ν = (and conversely; see Sec. 3.2). For this reason the subindex
es
δ (t)/σth(M,t) = δ (t)/σth(M,t) = δ (t)/σth, where “c” for “collapse” in the density contrasts appearing in the
ci 0 i c 0 c0 0
δ (t) is δ (t)D(t )/D(t) and σth stands for the 0th order relation(4)hasbeenreplaced bythesubindex“m”indicat-
c0 c 0 0
ing that the masses of haloes at t “match” a certain defi-
nition.The only constraints thesetwofunctions mustfulfil,
1 Thenormalisationconditionreflectsthefactthatallthematter for consistency with the mass growth of haloes, are: δmi(t)
intheuniversemustbeintheformofvirialisedhaloes. must be a decreasing function of time and Rf(M,t) must
4 Juan et al.
be an increasing function of mass. That consistency condi- δ (t) is δ (t)D(t )/D(t) and σ stands for the 0th order
m0 m 0 0
tion is precisely what makes the use of the Gaussian filter spectralmomentatt .Thesubindexes“th”and“es”inthe
0
mandatory.Suchafilterisindeedtheonlyoneguaranteeing, excursion set counterparts are enough to tell between the
through the relation R 2δ =∂δ/∂R, that the density con- two sets of variables.
∇
trast δ of peaks necessarily decreases as the filtering radius Thenumberdensityofpeakswithδ perinfinitesimal
mi
Rincreases,inagreementwiththeevolutionofhalomasses. lnσ−1(M,t)att or,equivalently,withδ perinfinitesimal
0 i i m0
TheradiusR ofthespherically averagedseed (orpro- lnσ−1 at t can be readily calculated from the density of
p 0 0
tohalo) with M is now different from theGaussian filtering peaks per infinitesimal height around ν, derived by BBKS.
radiusR .Itisinsteadequaltoq R ,withq satisfying the The result is
f p f p
relation 1/3 N(σ0,δm0)= h(x2iπ(σ)02,Rδm30γ) e−ν22 , (14)
1 3M ⋆
R (M,t)= , (12)
f qp(M,t)(cid:26)4πρ¯i[1+δmthi(M,t)](cid:27) where R⋆ and γ are respectively defined as √3σ1/σ2 and
σ2/(σ σ ), being σ thej-th order (Gaussian) spectral mo-
whereδth(M,t)isthedensitycontrastofthepeakwhenthe 1 0 2 j
mi ment, and x (σ ,δ ) is the average curvature (i.e. minus
0 m0
densityfield is smoothed with a top-hat filterencompassing h i
the Laplacian scaled to the mean value σ ) of peaks with
2
the same mass M, related to the (unconvolved) spherically
δ and σ , well-fitted by theanalytic expression (BBKS)
m0 0
averaged density contrast profile of the seed δ (r) through
p
3 Rp x (ν)=γν+ 3(1−γ2)+(1.216−0.9γ4)e−γ2(γ2ν)2 . (15)
δmthi(M,t)= Rp3 Z0 drr2δp(r). (13) h i [3(1−γ2)+0.45+(γν/2)2]1/2+γν/2
But this number density is not enough for our pur-
But factors [1+δmthi(M,t)]1/3 and qp(M,t) on the right of posesbecauseweareinterestedincountingnon-nestedpeaks
equation(12)canbeabsorbedinthefunctionq(M,t).Thus, only. The homologous numberdensity of non-nested peaks,
contrarily to the relation (5), equation (11) is exact. Note Nnn(σ ,δ ), can be obtained by solving the Volterra inte-
0 m0
thatq(M,t) isthen,toleadingorderin theperturbation as gral equation
in equation (5), the radius R of the seed in units of the
p ∞
GausWsiaenemfilptehraisnigserathdaiut,soRwfi.ng to the Gaussian smoothing, Nnn(σ0,δm0)=N(σ0,δm0)−Zlnσ0−1dlnσ0′−1
theradiusofthefilterRf depends,intheCUSPformalism, M(σ′,δ )
notonlyonM butalsoontthroughq(M,t),whileδmi and, ×N(σ0,δm0|σ0′,δm0) 0ρ¯ m0 Nnn(σ0′,δm0), (16)
hence,δ arestillfunctionsoftalone. Thislatterfunction-
m
ality may seem contradictory with thefact that, as pointed where the second term on the right gives the density of
out by Sheth & Tormen (2002) and recently checked with peaks with δm0 per infinitesimal lnσ0−1 nested into peaks
simulations (e.g., Robertson et al. 2009; Elia et al. 2012; withidenticaldensitycontrastatlargerscales,lnσ0′−1.The
Dcoenstpraalsiteftorale.ll2ip01so3i;daHlachonlla&psPeadraenpjeanpdes o2n01t3h)e, mthaessdeonfstihtye δcmon0dpiteironinafilnniutmesbimeraldelnnsσi0−ty1Nsu(bσj0e,cδtmt0o|σly0′,inδmg0i)nobfapcekagkrsouwnitdhs
perturbation.Thereishowevernocontradiction.TheCUSP with δm0 at σ0′ < σ0 can also be calculated from the con-
formalism uses a Gaussian filter instead of a top-hat filter ditional number density per infinitesimal ν in backgrounds
like in all these works and this difference is crucial because with ν′ derived by BBKS. The result is
oqfisthfiexferdeetdooomnei,ntsroocdouncseiddebryinqg(Mell,ipt)s.oIidnatlocpo-lhlaaptssemnoeoctehsisnagr-, N(σ0,δm0|σ0′,δm0)= (2πh)x2iR(σ˜30γ,√δm10) ǫ2 e−(2ν(−1−ǫνǫ′2))2 , (17)
ilytranslates intoavalueofδc dependentonM inaddition ⋆ −
tot.While,inGaussian smoothing,wecanimposethatthe where ν′ and ǫ are respectively defined as δ /σ′ and
m0 0
density contrast for collapse is independent of M and let q σ2(R )/[σ σ′], being R2 equal to the arithmetic mean of
0 m 0 0 m
depend on M and t. Note that, when the density field at the squared filtering radii corresponding to σ and σ′, and
0 0
ti is smoothed with a top-hat filter, the density contrast of where x(σ˜0,δm0) takes the same form (15) as x(σ0,δm0)
haloseeds,δth(M,t),isindeedafunctionofM andtingen- inequahtioin(14)butasafunctionofγ˜ν˜insteadohfγiν,being
mi
eral.Theexactwayδth(M,t)dependsonM willdepend,of
mi (1 r )2
course, on themass definition used. γ˜2 =γ2 1+ǫ2 − 1 (18)
WearenowreadytocalculatetheMFinthisapproach. 1 ǫ2
(cid:20) − (cid:21)
Given the one-to-one correspondence between haloes and
γ 1 r 1 ǫ2r
non-nested peaks, the counting of haloes with M at t re- ν˜(r)= − 1 ν − 1 ǫν′ , (19)
duces to count non-nested peaks with that scale at ti. In γ˜ 1−ǫ2 (cid:20) (cid:18) 1−r1 (cid:19)− (cid:21)
the original version of the CUSP formalism (MSS), such a with r equal to [σ (R )σ (R )/(σ (R )σ (R ))]2.
1 0 f 1 m 1 f 0 m
counting did not take into account the correlation between Thus, theMF of haloes at t is then
peaks at different scales. However, the more accurate ver-
∂n (M,t) ∂lnσ−1
sion later developed (Manrique et al. 1998) yielded essen- CUSP = Nnn[σ ,δ ] 0 . (20)
tially the same result, so we will follow here that simple ∂M 0 m0 ∂M
version (see the Appendix for the more accurate one). For Note that this expression of the MF is also independent of
simplicity,wewillomithereafteranysubindexintheGaus- the(arbitrary) initial time t.
i
sian rms density fluctuation σ and in the CUSP height The CUSP formalism thus solves all the problems met
0
ν δ /σ (M,t) = δ (t)/σ (M,t) = δ (t)/σ , where in the excursion set formalism: it deals with triaxial peaks
mi 0 i m 0 m0 0
≡
Halo Mass Function 5
that undergo ellipsoidal collapse and virialisation, conve- 10δ (t)a(t) 3
∆ (t) c . (24)
nientlycorrectedfornesting,andthesmoothingoftheinitial vir ≡ 3D(t)
densityfieldisalwayscarriedoutwiththesameGaussianfil- (cid:20) (cid:21)
Comparingequations(1)and(23),weseethatthehalo
ter.Theonlydrawbackofthisapproachistheneedtosolve
massdefinitionimplicitlypresumedintheexcursionsetfor-
the Volterra equation (16), which prevents from having an
malism is the SO(∆ ) one, with ∆ dependent on time
analytic expression for theresulting MF. vir vir
and cosmology. In the Einstein-de Sitter universe, where
δ (t)isequalto3(12π)2/3/20andD(t)=a(t),∆ (t)takes
c vir
the constant value 18π2 178. While, at t in the concor-
0
3 IMPLICIT HALO MASS DEFINITION ≈
dantmodel,whereδ (t )andD(t )arerespectivelyequalto
c 0 0
For any theoretical MF to be complete, the mass definition 1.674and 0.760, ∆vir(t0)takesthevalue 359(Henry
≈ ≈ ≈
itreferstomustbespecified.Inotherwords,onemuststate 2000)5.
the condition defining the total radius R or, equivalently,
h
the spherically averaged density profile for haloes with dif-
3.2 The CUSP Formalism
ferent masses at t that result from the specific halo seeds
and dynamics of collapse assumed. In ellipsoidal collapse, the total energy Ep of a sphere with
mass M is not conserved. On the other hand, for peaks
(hence,withoutwardsdecreasingdensityprofiles)shellsex-
3.1 The Excursion Set Formalism change energy as they cross each other, causing virialisa-
tion to progress from the centre of the system outwards.
Intheexcursionsetformalism,haloseedsarearbitraryover-
Thus, the spherical top-hat model does not hold. However,
denseregionswithnodefiniteinnerstructure,sotheirtypi-
asshowninSVMS,onecanstillaccuratelyderivethetypical
cal (mean) density and peculiar velocity fields are uniform.
density profile ρ (r) for haloes.
As a consequence, the density distribution in the corre- h
Thevariationintimeofthetotalenergyofasphereun-
spondingfinalvirialisedobjectsisalsouniform2.Inaddition,
dergoing ellipsoidal collapse compared to that of thespher-
thesystemissupposedtoundergosphericalcollapse.There-
ically averaged system can be accurately monitored. In ad-
fore,haloformationisaccordingtothesimplesphericaltop-
dition, during virialisation, there is no apocentre-crossing
hatmodel,inwhichcasethetypicalradiiR ofhaloeswith
h (despite there being shell-crossing), which causes virialised
different masses at t can be readily inferred (Peebles 1980).
haloes to develop from the inside out, keeping the instan-
Thevirialrelation2T+W =0holdingforthefinaluni-
taneousinnerstructureunchanged.Intheseconditions, the
form object3 togetherwithenergyconservation4 implythat
radiusrencompassinganygivenmassM inthefinaltriaxial
R is half the radius of the uniform system at turnaround.
h virialised system exactly satisfies the relation6
This leads to
3GM2 r= 3GM2 , (25)
Rh =−10Ep(M), (21) −10Ep(M)
identical, at every radius r, to the relation (21) holding for
where E (M) is the (conserved) total energy of the proto-
p
thewhole object in theexcursion set case.
halo with mass M. Taking intoaccount that E is, tolead-
p
ingorderintheperturbation,equalto δ (t)GM2/R (see In equation (25), Ep(M) is the (now non-conserved)
ci f
− energydistributionofthesphericallyaveragedprotohalo.In
eqs.[30]–[29]forρ =ρ¯[1+δ (t)]),equation(21)takesthe
p i ci
theparametric form, it is given by
form
Rh = 4π∆v3iMr(t)ρ¯(t) 1/3 (22) Ep(r)=4πZ0rdr˜r˜2ρp(r˜)(cid:26)[Hir˜−2vp(r˜)]2−GMr˜(r˜)(cid:27) (26)
(cid:20) (cid:21)
r
or, equivalently, M(r)=4π dr˜r˜2ρ (r˜), (27)
p
Z0
ρ¯ (R )=∆ (t)ρ¯(t), (23)
h h vir
where ρ (r)=ρ¯[1+δ (r)] is the(unconvolved)spherically
p i p
where we have introduced the so-called virial overdensity averaged density profile of the protohalo, H is the Hubble
i
corresponding tothe spherical top-hat model, constant at t and
i
2GδM(r)
v (r)= (28)
p 3Hr2
2 AsshowninSVMS,whatcausestheoutwardsdecreasingden- i p
sity profile of virialised objects is the fact that, for seeds with is,toleadingorderintheperturbation,thepeculiarvelocity
outwardsdecreasingdensityprofiles,virialisationprogressesfrom at r induced bythe innermass excess (e.g. Peebles 1980),
the centre of the system outwards. In the case of homogeneous
r
spheresinHubbleexpansion,alltheshellscrossatthesametime δM(r)=4π dr˜r˜2ρ¯ δ (r˜). (29)
at the originof the system, sothe final object does not have an i p
Z0
outwardsdecreasingdensityprofile.
3 The effects of the cosmological constant at halo scales can be
neglected. 5 Accordingtoequation(24),∆vir(t0)=359andD(t0)=0.760
4 In the top-hat spherical model, energy cannot be evacuated imply δc(t0) = 1.621 rather than 1.674. This 3.5% error arises
outwardslikeinthevirialisationofhaloesformedbythecollapse fromtheneglectofthecosmologicalconstant inequation(24).
ofseedswithoutwardsdecreasingdensityprofiles(SVMS),sothe 6 Again, the effects of the cosmological constant at halo scales
total energyisconserved. areneglected.
6 Juan et al.
Replacing v (r) given by equation (28)–(29) into equation obtain M as a function of ρ¯ (R) and ρ¯(t) and replacing it
p h
(26),we are led to in f , we are led to a relation
1
20π r F[ρ (R),ρ¯ (R)]=ρ¯(t) (33)
E (r)= dr˜r˜ρ (r˜)GδM(r˜). (30) h h
p − 3 p
Z0 of thegeneral form of relations (1) and (2) setting thehalo
Note that the non-null (in Eulerian coordinates) peculiar massdefinitionassociatedwiththeCUSPMFwitharbitrary
velocity vp(r) introduces a factor 5/3 in the value of Ep(r) functions δm(t) and q(M,t).
with respect to the one resulting in the absence of peculiar Conversely, the functions δm(t) and q(M,t) can be in-
velocities. In the usual presentation (in Lagrangian coor- ferred from any given halo mass definition. To do this
dinates) of spherical collapse, v (r) is null, but the initial we must impose the two following consistency arguments
p
density contrast then decomposes in the growing and de- (JSDM):i)thetotalmassassociatedwiththeresultingden-
cayingmodesandthemassexcesscausingthegravitational sityprofilemustbeequaltoM andii)theresultingMFmust
pull in equation (28) has an extra factor 5/3 compared to be correctly normalised. For SO(∆vir) or FoF(0.19) masses
themassexcessδM(r)associated withthegrowingmodeδ in the concordant cosmology, JSDM found
p
contributingtothemassofthefinalhalo.Consequently,the [a(t)]1.0628
resulting value of Ep(r) is exactly the same as in equation δm(t)=δc(t) D(t) (34)
(30).
Thedensitycontrast δ(R)oftheseed oftheprogenitor and
of scale R of any accreting halo with M at t is but the σth(M,t) −2/(n+3)
(unconvolved) spherically averaged density contrast profile q(M,t) Q 0 . (35)
δp(r)oftheprotohaloconvolvedwithaGaussianwindowof ≈(cid:20) σ0(M,t) (cid:21)
radius R.We thushave Inequation(35),theratioσth(M,t)/σ (M,t)takestheform
0 0
∞
δ(R)= (2π)43π/2R3 drr2δp(r)e−12(Rr)2. (31) σ0th(M,t) =1 0.0682 D(t) 2ν, (36)
Z0 σ0(M,t) − D(t0)
(cid:20) (cid:21)
Equation (31) indicates that the trajectory δ(R) of peaks
Q is defined as
with varying scale R tracing the accretion of a halo with
mass M at t is the Laplace transform of the profile δp(r) ∞dxxn+2W2(x)
of its seed. As for haloes growing inside-out the mean den- Q2≡ ∞0 dxxn+2W2G(x), (37)
sity profile ρh(r) is determined by the mean accretion rate R0 TH
dM/dt undergone over their aggregation history, the mean whereRWth(x) and WG(x) are theFourier transforms of the
peak trajectory δ(R) tracing such an evolution is charac- top-hat and Gaussian windows of radius x/k, respectively,
terised by having the mean slope dR/dδ of peaks with δ at and nis theeffectivespectral index.Theapproximate rela-
everyR.Thus,thedesiredmeanpeaktrajectoryδ(R)isthe tion(35)followsfromthemorefundamentalone(36),taking
solution of thedifferential equation7 intoaccount therelation
dδ A ∞
dR =−hxi[R,δ(R)]σ2(R)R (32) σ02(Rf)≈ 2π2Rf−(n+3) dxW2(x)xn+2, (38)
Z0
for the boundary condition δ[R=R (M,t)]=δ (t). Once
f mi holding for the 0th order spectral moment for a filter with
thetrajectoryδ(R)hasbeenobtained,wecaninferthepro-
Fourier transform W under the power-law approximation,
fileδp(r)byinversionofequation(31)(seeSVMSfordetails) P(k) = Akn, of the CDM (linear) spectrum, and the fact
and use equations (27) and (30) to calculate E (M). Then,
p that the Gaussian and top-hat radii for a seed with M are
replacing this function into equation (25), we can infer the
respectivelyequaltoR ortoq R .Thismeansthat Qand
f p f
mass profileM(r)of thehalo, leadingtothedensityprofile
ndependonthemassrangeconsidered.Forthemassranges
ρ (r), which turns out to be in good agreement with the
h typicallycoveredbytheMFsfoundinsimulations,Qandn
results of simulations (JSDM).
takevalues around 0.5 and 1.5, respectively.
Themeansphericallyaveragedhalodensityprofilethus −
Therefore, the CUSP MF is more general than the ex-
depends, like the MF itself, on the particular mass defini-
cursion set one in the sense that it does not presume any
tion adopted through the functions δ (t) and R (M,t) or,
mi f particular mass definition; it holds for any arbitrary one,
equivalently,δ (t)andq(M,t),settingtheboundarycondi-
m adapting to it through thefunctions δ (t) and q(M,t).
m
tion for integration of equation (32).
To obtain the halo mass definition that corresponds to
anygivenpairofδ (t)andq(M,t)functions,wemustcalcu-
m
latethedensityprofileforhaloesofdifferentmassesandfind 4 SIMILARITY OF SO AND FOF MASSES
the relations f [ρ (R),M] = ρ¯(t) and f [ρ¯ (R),M] = ρ¯(t)
1 h 2 h The fact that the CUSP formalism distinguishes between
betweenρ (R)andρ¯ (R)andthemeancosmicdensityρ¯(t)
h h different mass definitions can be used to try to understand
forhaloeswithdifferentM.Then,inverting,say,f soasto
2 theoriginofthesimilarity betweenSOandFoFmassesand
theirrespective mass and multiplicity functions.
Equations (27) and (30) imply
7 ThedistributionofpeakcurvaturesxgivenintheAppendixA
isaquitepeakedsymmetricfunction,sotheinverseofthemean dEp= 5GδM(Rp)= 5 4πρ¯i 1/3GM2/3δth(M,t),(39)
inversecurvature,hx−1i,isclosetothemeancurvature, hxi. dM − 3Rp −3 3 mi
h i
Halo Mass Function 7
wherewehavetakenintoaccountthattheradiusR ofthe F(c)
p δth(t)=δ (t)6 1 . (45)
protohalo is equal to qpRf with qp satisfying equation (12). mi ci − 6
(cid:20) (cid:21)
ComparingwiththeM-derivativeofequation(25)andtak-
ing intoaccount theidentityM =4πρ¯ (R )R3/3, equation ThetypicalvalueofF(c)forSO(∆vir)haloescanbeinferred
h h h from equation (45) for δ (t) given by equation (4) and δth
(39) leads to the relation ci mi
givenbyequation(13)forseedsofanyarbitrarymass.How-
5 ρ¯i 1/3δth(M,t)= ρ¯h(Rh) 1/3 1 ρ¯h(Rh) , (40) ever,thedensityprofileδp(r)ofprotohaloes isnotaccurate
9 ρ¯(t) mi ρ¯(t) − 6ρh(Rh) enough (owing to the inverse Laplace transform of eq. [31])
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) forδth tobeinferredwiththerequiredprecision.Therefore,
mi
which,makinguseofthedefinitionofF(c),canberewritten
as the CUSP formalism recovers, to leading order in the
in thetwo following forms
perturbation, the typical spherically averaged density pro-
5δth(M,t)a(t) 3 F(c) −3 fileforsimulatedhaloes,wecanestimateF(c)directlyfrom
ρ¯h(R)=ρ¯(t)(cid:20) mi9a(ti) (cid:21) (cid:20)1− 6 (cid:21) (41) sthuechNeFmWpirfoicraml p(Nroafivlaersr.oAestwale.ll1-9k9n7o)wanndth,ehseenpcero,fisaletsisfayrethoef
and relation
ρ (R)=ρ¯(t) 5δmthi(M,t)a(t) 3 1 F(c) −3 1 . (42) F(c) ρ¯h(Rh) =3(1+c)2 ln(1+c) c . (46)
h 9a(ti)] − 6 F(c) ≡ ρh(Rh) c2 − 1+c
(cid:20) (cid:21) (cid:20) (cid:21) h i
For SO and FoF masses, these expressions therefore imply For c spanning from 5 to 15 as found in simulations
∼ ∼
oftheconcordant cosmology forSO(∆ ) haloes at t (and
∆= 5δmthi(M,t)a(t) 3 1 F(c) −3 (43) approximately at any other time andvcirosmology), w0e find
9a(ti) − 6 F(c) 5.1 0.1. And, bringing this value of F(c) and
(cid:20) (cid:21) (cid:20) (cid:21) ∼ ±
∆ = ∆ 359 into equation (3), we arrive at b 0.19,
and vir ≈ ∼
in full agreement with the results of numerical simulations.
b= 2π −1/3 5δmthi(M,t)a(t) −1 1 F(c) , (44) Of course, the exact typical value of F(c) may vary with
3F(c) 9a(ti) − 6 timeandcosmology.But,accordingtotheresultsofnumer-
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
ical simulations, we donotexpect anysubstantial variation
respectively.
in this sense, so we have that FoF(0.2) masses are approxi-
Equation(43)seemstoindicatethat,intheSOcase,the
mass dependence of δth must cancel with that coming from mately equivalentto SO(∆vir) ones, in general.
Fin(tco).eqBuuattieoqnua(2ti5o)na(t1mr)i=imRplhie,sleRadhs∝toME1p/3(M, w)hichM, r5e/p3laacnedd, 5.1 ±As0.a1bimypprloiedsuctthewerehlaatvieonthδamtth(etq)ua∼tio0n.9(δ4c5()t)f.orInF(oct)he≈r
hence,todE /dM M2/3,implying (seeeq.[∝39]) that δth words, in thecase of SO(∆vir) orFoF(0.2) masses, thetop-
p ∝ mi hatdensitycontrast forellipsoidal collapse andvirialisation
isafunctionoftalone.Thesolutiontothisparadoxisthat,
would take an almost universal value independent of M,
toleadingorderintheperturbationasusedinthederivation
of thedensity profile(see eq.[28]),δth and F(c) are, in the justalittlesmallerthanthealmostuniversalvalueδc(t)for
mi spherical collapse. This result thus suggests that it should
SO case, independent of M. (Likewise, eq. [44] multiplied
be possible to modify the excursion set formalism in order
by thecubic root of F(c) leads in theFoF case to a similar
toaccountforellipsoidalcollapseandvirialisationbysimply
paradox, with identical solution.) Consequently, to such an
decreasing the usual density contrast for spherical collapse
order of approximation, the SO and FoF mass definitions
byafactor 0.9. Wewill comeback tothisinteresting pre-
with ∆ and b satisfying equation (3) are equivalent to each ∼
diction below.
other.
Wethusseethattheoriginofthisapproxequivalenceis
the inside-out growth of accreting haloes, crucial to obtain
equation(25)settingthetypicalsphericallyaverageddensity 5 MULTIPLICITY FUNCTION
profileforhaloesarisingfrompeaksthatundergoellipsoidal
The multiplicity function associated with any given MF,
collapse and virialisation. But this is not all. We can go a
∂n(M,t)/∂M, is defined as
step further and infer the value of b leading to FoF masses
equivalent to SO(∆ ) ones. M ∂n[M(σth),t]
vir f(σth,t)= 0 . (47)
The relation between the two functions (of t) δth and 0 ρ¯ ∂ln[(σth)−1]
mi 0
1 F(c)/6 can be readily derived for the particular case
− Intheexcursionsetcase,thisleadstoafunctionofthe
ofSO(∆ )haloes.Comparingequations(21)and(25),the
vir simple form
latteratr=R ,wehavethathaloesarisingfromellipsoidal
h
cwoiltlhapaseGoafupsesaiaknswfilittehrδomfirinadtihuesdRefn,sictoyufildeldhaavtetifosrmmoeodthaecd- fes(σ0th,δc0)= π2 1/2 νese−ν2e2s , (48)
cordingto thespherical top-hatmodel from thesame seeds (cid:16) (cid:17)
while, in theCUSP case, it leads to (see eqs. [20] and [47])
with δth when the density field is smoothed with a top-hat
mi
filterof radius qRf.8 Equations (24) and (43), thelatter for f (σ ,δ )= M(σ0,δm0)Nnn(σ ,δ ). (49)
∆=∆ , then imply CUSP 0 m0 ρ¯ 0 m0
vir
8 Theoutwardsdecreasingdensityprofileofseeds forpurely ac- peaks.TheuseofaGaussianwindowisonlymandatory,asmen-
creting haloes ensures the possibility to use any spherical win- tioned, if haloes can also undergo major mergers (see MSS and
dowtodefinetheone-to-onecorrespondencebetweenhaloesand SVMS).
8 Juan et al.
Figure 1. Multiplicity function at t0 derived from the CUSP Figure 2. Same as Figure 1 but in a much wider mass range,
(redlines)andexcursionset(greenlong-dashedline)formalisms, correspondingtocurrenthaloeswithmassesfrom1M⊙to3×1016
compared to Warrenetal. (2006) analytic fit to the multiplicity M⊙.
functionofsimulatedhaloes(bluedottedline)overthemaximum
massrange(2×1010 M⊙,2×1015 M⊙)coveredbysimulations.
FortheCUSPcase,weplotboththetheapproximatesolutionnot that follows from equations (34) and (36). This is a mere
accounting for the correlation between peaks of different scales change of variable; it does not presume any modification
(dashed line) and the more accurate solution given in Appendix in the assumptions entering the derivation of the different
(solidline).Ratiosinthebottompanel arewithrespecttofW. multiplicity functions.
As can be seen, while f shows significant deviations
es
To obtain equation (49) we have taken the partial deriva- fromfW atbothmassends,fCUSP isinexcellentagreement
tive of nCUSP with respect to σ0 instead of σoth as pre- withfWalloverthemassrangecoveredbysimulations.This
scribed in equation (47). But this is irrelevant for SO(∆ ) istrueregardlessofwhetherweconsidertheapproximateor
vir
or FoF(0.19) masses in the concordant cosmology as here- moreaccurateversionsoffCUSP.Thedeviation(ofopposite
after assumed, given the relation (36) between the two 0th signinbothcases) islessthan6.5%.Westressthatthereis
order spectral moments. nofreeparameterintheCUSPformalism,sothisagreement
is really remarkable.
It might be argued that f cannot be trusted at
CUSP
5.1 Comparison with Simulations smallν’sbecausepeakswiththoseheightshavebigchances
to be destroyed by the gravitational tides of neighbouring
In Figure 1 we compare these two multiplicity functions at
massive peaks. Although this possibility exists, peaks suf-
t to the Warren et al. (2006) analytic expression, of the
0
fering strong tides are expected to be nested within such
Sheth & Tormen (2002) form,
neighbours and, hence, they should not be counted in the
fW(νes)=0.3303 νe1s.625+0.5558 e−0.4565νe2s, (50) MF corrected for nesting. The correction for nesting be-
comes increasingly important, indeed, towards the small ν
fitting the multip(cid:0)licity function(cid:1)of simulated haloes with end.Ontheotherhand,f iswell-normalised9 andstill
FoF(0.2)massesatt inallCDMcosmologies.f isusually CUSP
0 W predictstherightabundanceofmassivehaloes,whichwould
expressed as a function of σth instead of ν ; theexpression
0 es hardlybethecaseiff overestimatedtheabundanceof
(50) has been obtained from that usual expression assum- CUSP
low-mass objects. Therefore, we do not actually expect any
ing δ (t ) = 1.674 (taking the value 1.686 would make no
c 0 major effect of that kind.
significant difference).InFigure1,all themultiplicityfunc-
Itisthusworthseeinghowf comparestof out-
tions are expressed as functions of the Gaussian height for CUSP W
side the mass range covered by simulations. In Figure 2
ellipsoidal collapse and virialisation, ν, instead of the top-
we represent the same multiplicity functions as in Figure
hatheightforsphericalcollapse,ν .Thechangeofvariable
es
from ν toν has been carried out using therelation
es
2 −1 9 The CUSP MF is well-normalised by construction as this is
D(t) D(t)
νes = [a(t)]1.0628 ν(1−0.0682(cid:20)D(t0)(cid:21) ν) (51) qo(nMeo,tf)t.he conditions imposedtoobtain the functions δm(t) and
Halo Mass Function 9
Figure 3. CUSP multiplicity functions at z = 0, 5, 10 and 20, Figure4.SameasFigure3butforfCUSPexpressedasafunction
from left to right, in red (solid line), orange (long-dashed line), ofvariableνesinsteadofν.Thegreenlong-dashedlinerepresents
gold (dashed line) and brown (dotted line), respectively. Ratios themultiplicityfunctionthatwouldbeobtainedfromtheexcur-
inthebottom panelarewithrespecttothemultiplicityfunction sionsetformalismtakingadensitycontrastforsphericalcollapse
atz=20. equalto0.89timestheusualvalue.
1 over a much wider range. Surprisingly, the agreement be- gies (see eqs. [14], [17] and [15])10. Certainly, these number
tween fCUSP and fW is still very good. At very small ν’s, densities also depend on γ, γ′ and R that involve spec-
⋆
f shows a slight trend tounderestimatetheabundanceof
W tral moments of different orders and, hence, depend on the
haloes predicted by f , but the difference is small. It
CUSP cosmology through the exact shape of the (linear) power
increasesmonotonouslyuntilreaching,inthecaseoftheac-
spectrum. However, in all CDM cosmologies, the effective
curateversionoff ,aratioof 0.70( 30%deviation)
CUSP ∼ ∼ spectralindexntakesessentiallythesamefixedvalue,with
at M ∼5×104 M⊙. less than 20% error over the whole mass range (2 1010
M⊙, 2 1015 M⊙) of interest, implying that γ ×γ′ and
R /R [3×(1 γ2)]1/2 takesalmost “universal” value≈srespec-
5.2 Approx Universality ⋆ f −
tively equal to 0.6 0.1 and 1.4 0.1. Thus, those number
± ±
Theexcursionset multiplicity functionexpressed asafunc- densities are indeed very approximately universal functions
tion of νes, fes(νes), is cosmology-independent (it takes the of ν and ν′ but for a factor Rf−3. Moreover, if we multi-
same form [48] in all cosmologies) and time-invariant (the ply the Volterra equation (16) by M/ρ¯ = 4πρ¯(qR )3/(3ρ¯)
i f
height is constant). Hence, it is universal in a strict sense. sothat its solution is directly f (seeeq.[49]),then the
CUSP
Suchauniversality is in fact what has motivated theuse of factor R −3 in the two number densities cancels with the
f
themultiplicity functiondefinedinequation (47) instead of factor R 3 coming from the mass. Therefore, the solution
f
the (non-universal) MF. Unfortunately, fes does not prop- fCUSP of such a Volterra equation will have very approxi-
erly recoverthe multiplicity function of simulated haloes. mately the same expression of ν in all CDM cosmologies,
But fCUSP does, so the question rises: is fCUSP also provided only thefunction q(M,t) does.
universal?Certainly,sincetheCUSPMF(aswellasthereal But, according to equations (35)–(36) holding for
MFofsimulatedhaloes)dependsontheparticularhalomass SO(∆ ) and FoF(0.19) haloes, q(M,t) involves the ratio
vir
definitionwhileσ0th doesnot,fCUSP willnecessarily depend σ0th/σ0 which is not a function of ν alone, but also depends
(like the multiplicity function of simulated haloes; see e.g. on t through the cosmology-dependent ratio D(t)/D(t ).
0
Tinker et al.2008)onthemassdefinitionadopted.Thus,we Nevertheless, the term with the ratio σth/σ responsible of
0 0
will focus on the SO(∆vir) or FoF(0.2) mass definitions, as theundesiredfunctionalityofq(M,t)issmallingeneral(ex-
suggested bytheresults of simulations (see theform [50] of cept for large ν’s), particularly at high-z where D(t)/D(t )
0
fW(νes)). becomes increasingly small. There, q(M,t) becomes con-
By construction, the unconditioned and conditional
peak number densities, N(σ ,δ ) and N(σ ,δ σ′,δ ) en-
0 m 0 m| 0 m
tering the Volterra equation (16) take thesame form of σ0, 10 ǫ takes the form 2(n+3)/2(ν/ν′)[1+(ν/ν′)4/(n+3)]−(n+3)/2,
σ0′ and δm0, through the heights ν and ν′, in all cosmolo- wherenistheeffectivespectralindexintherelevantmassrange.
10 Juan et al.
stant (equal to Q−2/(n+3)) and f (ν) becomes essen-
CUSP
tiallyuniversal.However,atlow-z thisisonlytrueforsmall
enough ν’s.
InFigure3weshowf (ν)intheconcordantmodel
CUSP
for variousredshifts (see JSDMfor thecorresponding MFs,
in full agreement with the results of simulations). The
deviations from universality or, more exactly, from time-
invarianceathigh-zaresmallasexpected,butatlow-zthey
are very marked.Thus, f (ν) is far from universal (!).
CUSP
But this result was not unexpected.Given the relation
(51)betweenν andν ,wecannotpretendthatf (ν)is
es CUSP
universalasafunctionofbothargumentsatthesametime.
Inspired by the universality of f (ν ), most efforts in the
es es
literature have been done in trying to find one mass defini-
tion rendering the multiplicity function of simulated haloes
approximately universal as a function of the top-height for
spherical collapse,notasafunctionofthe(unknown)Gaus-
sian height for ellipsoidal collapse and virialisation. There-
fore,whatweshouldactuallycheckiswhetherf isuni-
CUSP
versal asa function of ν andnot of ν.Asshown in Figure
es
4,whenthechangeofvariablefromν toν ismade,f
es CUSP
becomes indeed almost fully time-invariant. Strictly, it still
shows slight deviations from universality at large ν , but
es
these deviations are in full agreement with those found in
simulations (see Fig. 14 in Luki´cet al. 2007).
Figure 5.Modified excursionset multiplicityfunction resulting
Thanks to the CUSP formalism, we can determine the
fromadensitycontrastforcollapseequalto0.889timestheusual
Gaussianheightforellipsoidalcollapseandvirialisationcor-
respondingtoanydesiredmassdefinition.Thus,wecanseek value(solidgreenline),comparedtofWforFoF(0.2)haloes(dot-
ted blue line). Both multiplicity functions are strictly universal,
thehalomassdefinitionforwhichf expressedasafunc-
CUSP sothetwocurvesholdforanyarbitraryredshift.
tionofν takesauniversalform.Accordingtothereasoning
above,forthistobepossibletheratioσth/σ shouldbeequal
0 0
to 1+cν, with c equal to an arbitrary universal constant. cidence” is that the top-hat density contrast for ellipsoidal
ThiswouldensureboththatthepartialderivativeofnCUSP collapse and virialisation for SO(∆vir) masses, δmth, takes a
withrespecttoσ0 coincideswiththepartialderivativewith universal value, independent of M, approximately equal to
respecttoσthandthatthefunctionq(M,t)isafunctionofν 0.9timesthetop-hatdensitycontrastforsphericalcollapse,
0
alone: q(ν) [Q (1+cν)]−2/(n+3). Consequently, following δ (t). Given this relation, changing the latter density con-
≈ c
theproceduregiveninSection3.2,wecaninfer,fromsucha trast by the former in the excursion set formalism, f (ν )
es es
function q(ν) and any arbitrary function δm(t), the desired shouldkeeponbeinguniversaland,inaddition,recoverthe
halomassdefinition.Unfortunately,despitethefreedomleft realmultiplicityfunctionofhaloesformedbyellipsoidalcol-
in those two functions, the mass definition so obtained will lapse and virialisation. As shown in Figure 5, this is fully
hardly coincide with any of thepractical SO and FoF ones. confirmed. One must just renormalise the resulting modi-
Thus, it is actually preferable to keep on requiring the uni- fied excursion set multiplicity function in therelevant mass
versalityofthemultiplicityfunctionintermsofνesasusual. range by multiplying it by 0.714. But this is simply due to
But this does not explain why the FoF mass defini- thefactthatthecorrectionfornestingachievedintheexcur-
tion with linking length 0.2 is successful in giving rise to sion set formalism is inconsistent with top-hat smoothing,
∼
a universal multiplicity function expressed as a function of which yields an increasing deviation of the predicted func-
νes.Clearly,whatmakesthismassdefinitionspecialisthat, tion at low-masses, the most affected by such a correction.
for thereasons explained in Section 4, it coincides with the (Therightnormalisation shouldnaturallyresultifwecould
SO(∆vir) definition. In fact, as mentioned there, the exact implementtheexcursionsetcorrectionfornestingwithtop-
value of the linking length may somewhat vary with time hat smoothing.)
andcosmology,sothecanonicalmassdefinitionwouldbethe Therefore, the ultimate reason for the success of the
SO(∆vir) definition rather than the FoF(0.2) one. But why SO(∆vir) mass definition, and by extension of the FoF(0.2)
shouldamass definitionthatinvolvesthevirialoverdensity one, is that, as a consequence of the inside-out-growth of
∆virarisingfromtheformalsphericaltop-hatmodelsuccess- accretinghaloes,thecorrespondingtop-hatdensitycontrast
fully lead to a universal multiplicity function expressed as forellipsoidalcollapseandvirialisationisessentiallypropor-
a function of νes if haloes actually form from peaks that tional to the formal top-hat density contrast for spherical
undergo ellipsoidal collapse and virialisation? The reason collapse.
for this is that, as a consequence of the inside-out growth To end up we want to mention that the previous re-
of haloes formed from ellipsoidal collapse and virialisation, sultsuggestswhatisactuallythemostnaturalargumentfor
they satisfy the relation (25), identical to the relation (21) the halo multiplicity function to take a universal form: the
satisfied by objects formed in thespherical top-hat model. top-hat height for ellipsoidal collapse and virialisation, νth,
Asmentioned,aninterestingconsequenceofthis“coin- defined as δth(t )/σth 0.889δ (t )/σth. This expression
m0 0 0 ≈ c0 0 0