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Astronomy & Astrophysics manuscript no. 12486 c ESO 2010 (cid:13) January 19, 2010 Mass functions and bias of dark matter halos P. Valageas Institut dePhysiqueThéorique, CEA Saclay, 91191 Gif-sur-Yvette,France Received / Accepted ABSTRACT Aims. Werevisit the studyof themass functions and thebias of dark matter halos. 0 Methods. Focusing on the limit of rare massive halos, we point out that exact analytical results can be obtained for 1 thelarge-mass tail of the halo mass function. This is most easily seen from a steepest-descent approach, that becomes 0 asymptotically exact for rare events. We also revisit the traditional derivation of the bias of massive halos, associated 2 with overdenseregions in theprimordial density field. n Results. We check that thetheoretical large-mass cutoff agrees with themass functions measured in numerical simula- a tions.Forhalosdefinedbyanonlinearthresholdδ =200thiscorrespondstousingalinearthresholdδL ≃1.59instead J of the traditional value ≃ 1.686. We also provide a fitting formula that matches simulations over all mass scales and 9 obeystheexactlarge-masstail.Next,payingattentiontotheLagrangian-Eulerian mapping(i.e.correctionsassociated 1 with the motions of halos), we improve the standard analytical formula for the bias of massive halos. We check that our prediction, which contains no free parameter, agrees reasonably well with numerical simulations. In particular, it ] recovers the steepening of the dependence on scale of the bias that is observed at higher redshifts, which published O fittingformulae did not capture. This behavior mostly arises from nonlinear biasing. C Key words.Cosmology: large-scale structureof Universe;gravitation; Methods: analytical . h p - 1. Introduction ciated with multiple mergings can be neglected. Moreover, o inthisregimeastrophysicalobjects,suchasgalaxiesorclus- r The distribution of nonlinear virialized objects, such as t tersofgalaxies,canbedirectlyrelatedtodarkmatterhalos, s galaxiesorclustersofgalaxies,isafundamentaltestofcos- andtheir abundance is highly sensitive to cosmologicalpa- a mologicalmodels.First,this allowsusto checkthe validity rameters thanks to the steep decline of the high-mass tail [ of the standard cosmological scenario for the formation of of the mass function (e.g., Evrard 1989). 2 large-scalestructures,wherenonlinearobjects formthanks v totheamplificationbygravitationalinstabilityofsmallpri- Thus, the computation of the halo mass function (and 7 mordial density fluctuations, built for instance by an early especially its large-mass tail) has been the focus of many 7 inflationary stage (e.g., Peebles 1993; Peacock 1998). For works,asitisoneofthemainpropertiesmeasuredingalaxy 2 cold dark matter (CDM) scenarios (Peebles 1982), where and cluster surveys that can be compared with theoretical 2 the amplitude of the initial perturbations grows at smaller predictions. Most analytical derivations follow the Press- . 5 scales, this gives rise to a hierarchical process, where in- Schechter approach(Press & Schechter 1974;Blanchardet 0 creasingly large and massive objects form as time goes on, al. 1992) or its main extension, the excursion set theory of 9 asincreasinglylargescalesturnnonlinear.Thisprocesshas Bondetal.(1991).Inthisframework,oneattemptstoesti- 0 been largely confirmed by observations, which find smaller mate the number of virialized objects of mass M from the v: galaxies at very high redshifts (e.g., Trujillo et al. 2007) probability to have a linear density contrast δL at scale M i while massive clusters of galaxies (which are the largest abovesome giventhresholdδc.Thus,one identifies current X bound objects in the Universe) appear at low redshifts nonlinear halos from positive density fluctuations in the r (e.g., Borgani et al. 2001). Second, on a more quantitative initial (linear) density field, on a one-to-one basis. This is a level, statistical properties, such as the mass function and ratherwelljustifiedforraremassiveobjects,whereonecan the two-point correlation of these objects, provide strong expect such a link to be valid since such halos should have constraints on the cosmological parameters (e.g. through remained well-defined objects until now (as they should the linear growth factor D (t) of density perturbations) have suffered only minor mergers). By contrast, small and + and on the primordial fluctuations (e.g. through the ini- typical objects have experiencedmany mergers and should tial density power spectrum P (k)). For these purposes, be sensitive to highly non-local effects (e.g. tidal forces, L themostreliableconstraintscomefromobservationsofthe mergers), so that such a direct link should no longer hold, mostmassiveobjects(rare-eventtails)atthelargestscales. as can be checked in numerical simulations (Bond et al. Indeed, in this regime the formation of large-scale struc- 1991).AsnoticedbyPress&Schechter(1974),thesimplest tures is dominated by the gravitational dynamics (bary- procedureonly yields halfthe mass ofthe Universein such onic physics, which involves intricate processes associated objects(essentiallybecauseonlyhalfoftheGaussianinitial withpressureeffects,coolingandheating,mostlyoccursat fluctuations have a positive density contrast, whatever the galactic scales and below), which further simplifies as one smoothing scale), which they corrected by an ad-hoc mul- probes quasi-linearscales or rareevents where effects asso- tiplicative factor 2. In this respect the main result of the 2 P. Valageas: Mass functions and bias of dark matter halos excursionsettheorywastoprovideananalyticalderivation wasfurtherexpandedbyBardeenetal.(1986)andBond& ofthismissingfactor2,inthesimplifiedcaseofatop-hatfil- Myers(1996),whoconsideredtheclusteringofpeaksinthe terinFourierspace(Bondetal.1991).Then,itarisesfrom Gaussianlineardensityfield.Asimplerderivation,basedon the fact that objects of mass larger than M are associated a peak-background split argument, and taking care of the with configurations such that the linear density contrast mappingfromLagrangiantoEulerianspace,waspresented goes above the threshold δ at some scale M′ M, which inMo&White (1996).Itprovidesapredictionforthe bias c ≥ includes cases missed by the Press-Schechter prescription b(M) as a function of halo mass, in the limit of large dis- wherethelineardensitycontrastdecreasesbelowthisscale tance, r , that agrees reasonably well with numerical M′ so that δ <δ atscale M (“cloud-in-cloud”problem). simulatio→ns.∞However,asforthePSmassfunction,inorder L c The characteristic threshold δ is usually taken as the lin- toimprovetheagreementwithnumericalresultsvariousfit- c eardensitycontrastreachedwhenthesphericalcollapsedy- ting formulae have been proposed (Sheth & Tormen 1999; namics predictscollapseto a zeroradius.InanEinstein-de Hamana etal. 2001;Pillepich etal. 2009).Again,since the Sitteruniversethiscorrespondstoδ 1.686andtoanon- ellipsoidal collapse model reduces to the spherical dynam- c ≃ linear density contrast δ 177 (assuming full virialization ics for rare massive halos, it also requires free parameters ≃ in half the turn-around radius). This linear threshold only to improve its accuracy, but the latter are consistent with showsaveryweakdependenceoncosmologicalparameters. those used for the mass function (Sheth et al. 2001). Numerical simulations have shown that the Press- In this article we revisit the derivation of the mass Schechter mass function (PS) is reasonably accurate, es- function and the bias of rare massive halos, following the pecially in view of its simplicity. Thus, it correctly pre- spirit of the Press & Schechter (1974) and Kaiser (1984) dicts the typical mass scale of virialized halos at any red- approaches. That is, we use the fact that large halos can shift, as could be expected since the large-mass behavior be identifiedfromoverdensitiesinthe Gaussianinitial(lin- is rather well justified. In addition, it predicts a universal ear)density field. First,webriefly reviewin section2 some scaling that appears to be satisfied by the mass functions properties of the growth of linear fluctuations and of the measured in simulations, that is, the dependence on halo spherical dynamics in ΛCDM cosmologies. Next, we recall mass,redshiftandcosmologyisfully containedinthe ratio in section 3 that in the quasi-linear regime (i.e. at large ν = δ /σ(M,z), where σ(M,z) is the rms linear density scales),theprobabilitydistributionofthenonlineardensity c fluctuationatscaleM.However,as comparedwithnumer- contrastδ withinsphericalcellsofradiusrcanbeobtained r icalresults it overestimatesthe low-masstail and it under- fromsphericalsaddle-pointsofaspecificaction ,formod- S estimates the high-mass tail. This has led to many numer- erate values of δ where shell-crossing does not come into r ical studies which have provided various fitting formulae play.Wealsodiscussthepropertiesofthesesaddle-pointsas for the mass function of virialized halos, written in terms afunctionofmass,scaleandredshift.Then,wepointoutin of the scaling variable ν (or σ) (Sheth & Tormen 1999; section4thatthisprovidestheexactexponentialtailofthe Jenkins et al. 2001; Reed et al. 2003; Warren et al. 2006; halo mass function. This applies to any nonlinear density Tinker et al. 2008). We may note here that a theoretical contrast threshold δ that is used to define halos, provided model that attempts to improve over the PS mass func- it is below the upper bound δ where shell-crossing comes + tion is to consider the ellipsoidal collapse dynamics within into play. We compare our results with numerical simula- the excursion-setapproach,to take into accountthe devia- tionsandwegiveafittingformulathatappliesoverallmass tions from sphericalsymmetry for intermediate mass halos scalesandsatisfiestheexactlarge-masscutoff.Next,were- (Sheth et al. 2001). Note that, as described in the original call in section 5 that these results also provide the density paper (and emphasized in Robertson et al. 2009), at large profile of dark matter halos at outer radii (i.e. beyond the massthiswouldrecoverthesphericalcollapse.However,the virial radius) in the limit of large mass. Finally, we study halo mass obtained by such methods is generally underes- the bias of massive halos in section 6, paying attention to timated for non center-of-mass particles (since analytical somedetailssuchastheLagrangian-Eulerianmapping,and computations assume that particles are locatedat the cen- wecompareourresultswithnumericalsimulations.Wecon- ter of their halo, i.e. they only consider the linear densities clude in section 7. within spherical cells centered on the point of interest). To correct for this effect, in practice one treats the threshold δc asafreeparameter,closeto1.6,tobuildfittingformulae 2. Linear perturbations and spherical dynamics fromnumericalsimulations.For instance,this correctionis contained in the parameter a in the exponential cutoff of We consider in this article a flat CDM cosmology with Sheth et al. (2001). two components, (i) a non-relativistic component (dark A second property of virialized halos that can be used and baryonic matter, which we do not distinguish here) to constrain cosmological models, beyond their number that clusters through gravitational instability, and (ii) an density,is their two-pointcorrelation.Indeed, observations uniform dark energy component that does not cluster at show that galaxies and clusters do not obey a Poisson dis- the scales of interest, with an equation-of-state parameter tribution but show significant large-scalecorrelations(e.g., w =p /ρ .Forthenumericalcomputationsweshallfocus de de McCracken et al. 2008; Padilla et al. 2004). In particular, onaΛCDMcosmology,wherethedarkenergyisassociated their two-point correlation roughly follows the underlying with a cosmological constant that is exactly uniform with matter correlation, up to a multiplicative factor b2, called w = 1. However, our results directly extend to curved − the bias,thatgrowsformoremassiveandextreme objects. universes (i.e. Ω = 0) and to dark energy models with a k 6 Following the spirit of the Press-Schechter picture, Kaiser possibly time-varying w(z), as long as we can neglect the (1984)foundthatthisbehaviorarisesinanaturalfashionif dark energy fluctuations on the scales of interest, which is halosareassociatedwithlargeoverdensitiesintheGaussian validforrealisticcases.Focussingonthecaseofconstantw, initial (linear) density field, above the threshold δ . This wefirstrecallinthissectiontheequationsthatdescribethe c P. Valageas: Mass functions and bias of dark matter halos 3 dynamics of the background and of linear matter density mass M in a uniform universe with the same cosmology. perturbations, as well as the nonlinear spherical dynamics. This also implies that the density ρ writes as m The evolution of the scale factor a(t) is determined by ρ (t)=ρ (t)y(t)−3. (10) the Friedmann equation (Wang & Steinhardt 1998), m m Then, Equation (8) leads to H2(t) a˙ =Ω a−3+Ω a−3−3w, with H(t)= , (1) H02 m0 de0 a y′′+ 1 3wΩde y′+ Ωm y−3 1 y =0. (11) 2 − 2 2 − where subscripts 0 denote current values at z = 0, when (cid:20) (cid:21) (cid:0) (cid:1) a = 1, and a dot denotes the derivative with respect to Of course, we can check that in the linear regime, where cosmic time t. On the other hand, the density parameters yL = 1 δL/3, we recover Eq.(4) for the linear growth − vary with time as of δL. Then, to obtain the nonlinear density contrast, δ(z) = ρ /ρ 1, associated with the linear density con- Ω (a)= Ωm0 , Ω (a)= Ωde0 .(2) trast, δLm(z),mat−a given redshift z, we solve Eq.(11) with m Ωm0+Ωde0a−3w de Ωm0a3w+Ωde0 the initial condition y(zi) = 1−δLi/3 and y′(zi)= −δLi/3 at some highredshift z, with δ /δ =D (z)/D (z). For i Li L + i + Next, introducing the matter density contrast, δ(x,t) = any redshiftz this defines a mapping δ δ = (δ ) that L L (ρ (x,t) ρ )/ρ ,whereρ (t)isthemeanmatterdensity, 7→ F m − m m m fullydescribesthesphericaldynamicsbeforeshell-crossing. linear density fluctuations grow as a˙ δ¨ +2 δ˙ 4π ρ δ =0, (3) L a L− G m L where the subscript L denotes linear quantities. For nu- mericalpurposesitisconvenienttousethelogarithmofthe scalefactorasthetimevariable.Then,usingtheFriedmann equation (1) the linear growth factor D (t) evolves from + Eq.(3) as 1 3 3 D′′ + wΩ D′ Ω D =0, (4) + 2 − 2 de +− 2 m + (cid:20) (cid:21) where we note with a prime the derivative with respect to lna, as D′ = dD /dlna. Then, the normalized linear + + growth factor, g(t), defined as D (t) + g(t)= and g(t=0)=1, (5) Fig.1.Thefunction (δ )thatdescribesthesphericaldy- a(t) L F namics when there is no shell-crossing, at z =0. The solid obeys line corresponds to the ΛCDM cosmology (with Ω = m0 0.27) and the dashed line to the Einstein-de Sitter case 5 3 3 g′′+ wΩ g′+ (1 w)Ω g =0, (6) Ωm0 =1.Thesecondpeakcorrespondstoasecondcollapse de de 2 − 2 2 − to the center,but for our purposes we only need (δ ) be- (cid:20) (cid:21) F L fore first collapse. with the initial conditions g 1 and g′ 0 at a 0. → → → In the following we shall also need the dynamics of sphericaldensityfluctuations.Forsuchsphericallysymmet- We compare in Fig. 1 the function (δL) obtained at F ricinitialconditions,the physicalradiusr(t),thatcontains z =0withintheΛCDMcosmologythatweconsiderinthis the constant mass M until shell-crossing,evolves as article (solid line) with the Einstein-de Sitter case (dashed line), where it has a well-known parametric form (Peebles 4π 3M 1980).As is wellknown,we cancheckthatthe dependence r¨= G r[ρ +(1+3w)ρ ] with ρ = . (7) − 3 m de m 4πr3 on Ω is very weak until full collapse to a point, which m0 occurs at slightly lower values of δ for low Ω . We show Notethatρ isthemeandensitywithinthesphereofradius L m0 m in Fig. 2 the linear density contrasts, δ = −1(δ), asso- r (and not the local density at radius r). Using again the L F ciated with three nonlinear density contrasts, δ =100,200 Friedmann equation (1) this reads as and300,asafunctionofthecosmologicalparameterΩ (z). m 3 ρ +(1+3w)ρ In agreement with Fig. 1, they show a slight decrease for r′′− 2(1+wΩde)r′+ m2(ρ +ρ ) de r =0. (8) smaller Ωm. For δ = 200 a simple fit (dashed line) is pro- m de vided by As for the linear growth factor D+(t), it is convenient to δ =200: δ 1.567+0.032(Ω +0.0005)0.24, (12) introduce the normalized radius y(t) defined as L ≃ m which agrees with the exact curve to better than 5 10−4 r(t) × for Ω >0.2, which covers the range of practical interest. y(t)= and q(t) a(t), y(t=0)=1. (9) m q(t) ∝ In this article we consider Gaussianinitial fluctuations, whicharefullydefinedbythelineardensitypowerspectrum Thus, q(t) is the Lagrangian coordinate of the shell r(t), that is, the physical radius that would enclose the same δ˜ (k )δ˜ (k ) =δ (k +k )P (k ), (13) L 1 L 2 D 1 2 L 1 h i 4 P. Valageas: Mass functions and bias of dark matter halos scale r can be derived from a steepest-descent method (Valageas 2002a). In agreement with the alternative approach of Bernardeau (1994a), this shows that rare-event tailsaredominatedbysphericalsaddle-points,whichweuse in sections 4-6 to obtain the properties of massive halos. Since the system is statistically homogeneous we can considerthesphereofradiusrcenteredontheoriginx=0. Then, we first introduce the cumulant generating function ϕ(y), ∞ e−ϕ(y)/σr2 = e−yδr/σr2 = dδre−yδr/σr2 (δr), (19) h i P Z−1 which determines the distribution (δ ) through the in- r P verse Laplace transform P(δr)= +i∞ 2πdiyσ2 e[yδr−ϕ(y)]/σr2. (20) Fig.2.ThelineardensitycontrastδL = −1(δ)associated Z−i∞ r F with the nonlinear density contrast δ, through the spheri- InEq.(19)werescaledthecumulantgeneratingfunctionby cal dynamics, as a function of the cosmological parameter a factor σ2 so that it has a finite limit in the quasi-linear r Ωm(z) at the redshift of interest. We show the three cases limit σr 0 for the Gaussian initial density fluctuations δ =100,200and 300. The dashed line is the fit (12) to the that we s→tudy in this article (Bernardeau et al. 2002). In case δ =200. particular, its expansion at y =0 reads as ∞ ( y)p δp where δ is the Dirac distribution and we normalized the ϕ(y)= − h ric . (21) FourierDtransform as −p=2 p! σr2(p−1) X δ(x)= dkeik.xδ˜(k), (14) Then, the average (19) can be written as the path integral where xZ and k are the comoving spatial coordinate and e−ϕ(y)/σr2 =(detCL−1)1/2 DδL(q)e−S[δL]/σr2, (22) wavenumber. This gives for the linear density two-point Z correlation whereC−1istheinversematrixofthetwo-pointcorrelation L (15) and the action reads as C (x ,x ) = δ (x )δ (x ) S L 1 2 L 1 L 2 h i σ2 = 4π dkk2PL(k)sink(kx|x2−xx1|). (15) S[δL]=yδr[δL]+ 2r δL.CL−1.δL (23) Z | 2− 1| Here δ [δ ] is the nonlinear functional that affects r L Wealsointroducethesmootheddensitycontrast,δr(x), to the initial condition defined by the linear density withinthesphereofradiusrandvolumeV aroundposition field δ (q) the nonlinear density contrast δ within L r x, the sphere of radius r, and we note δ .C−1.δ = L L L δr(x)= dx′ δ(x+x′)= dkeik.xδ˜(k)W˜(kr), (16) dodeqs1ndoqt2δdLe(pqe1n)dCoL−n1(tqh1e,qn2o)rδmLa(qli2z)a.tiNonotoeftthhaet ltihneeaarctpioonweSr V ZV Z sRpectrum since both σ2 and C are proportional to P . r L L with a top-hat window that reads in Fourier space as In the quasi-linear limit, σ 0, as shown in Valageas r → (2002a) the path integral (22) is dominated by the mini- W˜(kr)= dVxeik.x =3sin(kr)(−krk)r3cos(kr). (17) mum1 of the action S[δL], ZV σ 0: ϕ(y) min [δ ]. (24) r L Then, in the linear regime, the cross-correlation of the → →δL(q)S smoothed linear density contrast at scales r1 and r2 and Usingthesphericalsymmetryofthetop-hatwindowW,in positions x1 and x2 =x1+x reads as agreement with the approach of Bernardeau (1994a), one obtains a spherical saddle-point with the radial profile σ2 (x) = δ (x )δ (x +x) r1,r2 h Lr1 1 Lr2 1 i σ2 = 4π dkk2PL(k)W˜(kr1)W˜(kr2)sink(xkx). (18) δLq′ =δLq σq,q2q′, (25) Z In particular, σr = σr,r(0) is the usual rms linear density whereσq,q′ =σq,q′(0,0)andq istheLagrangiancoordinate contrast at scale r. associated with the Eulerian radius r through the spherical dynamics recalled in section 2, and q′ is a dummy 3. Distribution of the density contrast 1 As described in details in Valageas (2002a), depending on the slope of the linear power spectrum the saddle-point (25) We recall here that in the quasi-linear limit, σr 0, the may only be a local minimum or maximum, but it still governs → distribution (δr) of the nonlinear density contrast δr at therare-event limit, in agreement with physical expectations. P P. Valageas: Mass functions and bias of dark matter halos 5 mass scale M of the saddle point for a curved CDM linear powerspectrum,butnotonredshift.WeshowinFig.3the profile(25)obtainedforseveralmassesM.Forapower-law linear power spectrum, of slope n, Eq.(25) leads to δLq′ ∝ q′−(n+3) at large radii, q′ . Then, since for a CDM → ∞ cosmology n increases at larger scales, the profile shows a steeper falloff at large radii for larger mass, in agreement withFig.3(inthissectionweconsideraΛCDMcosmology with (Ω ,Ω ,σ ,n ,h)=(0.27,0.73,0.79,0.95,0.7)). m0 Λ0 8 s Fig.3.Theradialprofile(25)ofthelineardensitycontrast δLq′ ofthesaddlepointoftheaction [δL(q)].Weshowthe S profiles obtained with a ΛCDM cosmology for the masses M = 1011,1012,1013,1014 and 1015h−1M . A larger mass ⊙ correspondstoalowerratioδLq′/δLq atlargeradiiq′/q>1. Lagrangian coordinate along the radial profile (hereafter we denote by the letter q Lagrangian radii, which are associated with the linear density field, whereas r denotes Eulerian radii associated with the nonlinear density contrast δ ). Thus, the radii q′ and r′ are related by Fig.4. The Lagrangian map q′ r′ given by the spheri- r 7→ caldynamics(neglectingshell-crossing)forthesaddle-point q′3 =(1+δr′)r′3, with δr′ = (δLq′), (26) (25) with a nonlinear density contrast δ = 200 at the F Eulerian radius r. We plot the curves obtained at z = 0 where the function δr′ =F(δLq′), that describesthe spher- for several masses, as in Fig. 3. Inner shells have already ical dynamics, was obtained below Eq.(11) and shown in gone once through the center (hence their dynamics is no Fig. 1. longer exactly given by Eq.(11)) but they have not crossed We can note that the profile (25) is also the mean con- the radius r yet. ditionalprofileof the linear density contastδLq′,under the constraint that is it equal to a given value δ at a given Lq radiusq (e.g.,Bernardeau1994a).Thereasonwhythenon- We show in Fig. 4 the Lagrangian map, q′ r′, given 7→ linear dynamics gives back this result is that the nonlinear by the spherical dynamics (i.e. the function (δ ) where L F density contrastδ only depends onthe linear density con- we neglect shell-crossing) for the saddle-point (25), with a r trastδ attheassociatedLagrangianradiusq,throughthe nonlinear density contrast δ = 200 at the Eulerian radius Lq mappingδ = (δ ).Indeed,aslongasshell-crossingdoes r, at redshift z = 0. Inner shells have already gone once r Lq F notmodifythemassenclosedwithintheshellofLagrangian through the center but they have not reachedradius r yet. coordinate q, its dynamics is independent of the motion of Even though their dynamics is no longer exactly given by innerandoutershells(thanks toGausstheorem).Then,in Eq.(11), an exact computation would give the same prop- orderto obtainthe minimum ofthe action we couldpro- erty as the increasing mass seen by these particles, as they S ceed in two steps. First, for arbitrary Lagrangian radius q passoutershells,shouldslowthemdownascomparedwith and linear contrastδ , we minimize with respect to the the constant-mass dynamics. In agreement with Fig. 3, for Lq profile δLq′ over q′ = q. From the preSvious argument, only larger masses, which have a larger central linear density 6 the secondGaussiantermin(23)variessothatthispartial contrast,shell-crossinghas moved to larger radii (the local minimizationleadstotheprofile(25)(indeed,forGaussian maximum of r′/r, to the left of r′ =0, is higher). integrals the saddle-point method is exact). Second, we FromtheLagrangianmap,q′ r′,wedefinethenonlin- 7→ minimizeovertheLagrangianradiusq(orequivalentlyover eardensitythreshold, δ ,asthe nonlineardensity contrast + δ orδ ),whichleadstoEq.(29)below.Herewealsousethe δ reached within radius r at the time when inner shells L r r fact that a spherical saddle-point with respect to spherical first cross this radius r. Then, up to δ , the mass within + fluctuationsisautomaticallyasaddle-pointwithrespectto the Lagrangian shell q has remained constant, so that the arbitrary non-spherical perturbations and it can be seen saddle-point(25)-(26)isexact(thisslightlyunderestimates that for small y one obtains a minimum (then we assume δ as the expansion of inner shells should be somewhat + that at finite y strong deviations from spherical symmetry sloweddownbythemassoftheoutershellstheyhaveover- donotgiverisetodeeperminima,whichseemsnaturalfrom taken).WeshowinFig.5thedependenceonredshiftofthis physicalexpectations).We referto Valageas(2002a,2009b) thresholdδ ,forseveralmasses.InagreementwithFigs.3, + for more detailed derivations. 4, this threshold is smaller for larger masses. It shows a Note that the shape of the linear profile (25) depends slight decrease at higher redshift as Ω (z) grows to unity. m on the shape of the linear power spectrum, whence on the We can see that for massive clusters at z = 0, which have 6 P. Valageas: Mass functions and bias of dark matter halos power spectrum throughthe ratio oflinear power σ /σ at r q Eulerian and Lagrangian scales r and q. Finally, from the inverse Laplace transform (20), a second steepest-descent integration over the variable y gives (Valageas 2002a) σr 0: (δr) e−τ2/(2σr2) =e−δL2q/(2σq2). (30) → P ∼ Thus,inthequasi-linearlimit,thedistribution (δ )isgov- r P ernedatleading orderby the Gaussianweightofthe linear fluctuationδ thatisassociatedwiththenonlineardensity Lq contrast δ through the spherical dynamics. We can note r thatEq.(30)couldalsobeobtainedfromaLagrangemulti- plier method, without introducing the generating function ϕ(y). Indeed, in the rare-event limit, where (δ) is gov- P erned by a single (or a few) initial configuration, we may write Fig.5. The nonlinear density contrast δ+, beyond which rare events: (δ) max e−12δL.CL−1.δL. (31) P ∼ shell-crossing must be taken into account, as a function of {δL[q]|δr[δL]=δ} redshift for several mass scales. Thatis, (δ)is governedbythe maximumoftheGaussian a mass of order 1015h−1M⊙, the density contrast δ 200 weight e−P(δL.CL−1.δL)/2 subject to the constraint δr[δL] = δ shouldseparateoutershellswithradialaccretionfrom≃inner (assuming there are no degenerate maxima). Then, we can shellswithasignificanttransversevelocitydispersion,built obtain this maximum by minimizing the action S[δL]/σr2 bytheradial-orbitinstabilitythatdominatesthedynamics of Eq.(23), where y plays the role of a Lagrange multi- after shell-crossing, see Valageas 2002b (in particular the plier. This gives the saddle-point (25), and the amplitude appendix A). This agrees with numerical simulations, see δLq and the radius q are directly obtained from the con- forinstancethelowerpanelofFig.3inCuestaetal.(2008), straint δ = (δLq), as in Eq.(26). Then, we do not need F which show that rare massive clusters exhibit a strong ra- the explicit expressionofthe Lagrangemultiplier y,asthis dial infall pattern, with a low velocity dispersion, beyond is sufficient to obtain the last expression of the asymptotic the virial radius (where δ 200), while inner radii show tail(30).Nevertheless,itisusefultointroducethegenerat- r a largevelocity dispersion(e∼venthoughwe candistinguish ing function ϕ(y), which makes it clear that the Lagrange close to the virial radius the outward velocity associated multiplier y is also the Laplace conjugate of the nonlinear with the shells that have gone once through the center). density contrast δ as in Eq.(19), since it is also of inter- Small-mass halos do not show such a clear infall pattern est by itself, as it yields the density cumulants throughthe and the velocity dispersion is significant at all radii (e.g., expansion (21). upper panel of Fig.3 in Cuesta et al. 2008). This expresses The asymptotic (30) holds in the rare-eventlimit. This the factthatsuchhalos,associatedwithtypicaleventsand corresponds to both the quasi-linear limit, σr 0 at fixed → moderate density fluctuations, are no longer governed by density contrast δr, and to the low-density limit, δr 0 → the spherical saddle-point (25). Indeed, at low mass and at fixed σr, as long as there is no shell-crossing. This lat- small Lagrangian radius q, σq is no longer very small so terrequirementgivesalowerboundaryδ−,forlinearpower that the path integral (22) is no longer dominated by its spectrawithaslopen> 1,andanupperboundaryδ+,for − minimum and one must integrate over all typical initial any linear spectrum (Valageas 2002b). This upper bound- conditions, including large deviations from spherical sym- arywasshowninFig.5foraΛCDMcosmology,forseveral metry. masses. Next, the amplitude δ and the minimum ϕ are given Indeed,atlargepositive density contrast,shell-crossing Lq asfunctionsofy bytheimplicitequations(Legendretrans- always occurs, as seen in Figs. 4-5. This invalidates the form) mapping δL δ = (δL) obtained from Eq.(11), as mass 7→ F is no longer conserved within the Lagrangian shell q, so dτ τ2 that the asymptotic behavior (30) is no longer exact. In y = τ and ϕ=y + , (27) − d G 2 fact,as showninValageas(2002b),aftershell-crossingit is G nolongersufficienttofollowthesphericaldynamics,evenif where the function τ( ) is defined from the spherical dy- G wetakeintoaccountshell-crossing.Indeed,astrongradial- namics through the parametric system (Valageas 2002a; orbit instability develops so that the sensitivity to initial Bernardeau 1994b) perturbations actually diverges when particles cross the σr center of the halo. Then, the functional δ [δ (q)] is sin- =δ = (δ ) and τ = δ . (28) r L r Lq Lq G F − σq gular at such spherical states (i.e. it is discontinuous as infinitesimal deviations from spherical symmetry lead to a The system (27) also reads as finite change of δ ) and the path integral (22) is no longer r τ2( ) governed by spherical states that have a zero measure. As ϕ=min y + G . (29) noticed above, this also means that, in the limit of rare G G 2 (cid:20) (cid:21) massivehalos,thenonlineardensitythresholdδ separates + Note that the function τ( ), whence the cumulant gener- outershellswithasmoothradialflowfrominnershellswith G ating function ϕ(y), depends on the shape of the linear asignificanttransversevelocitydispersion.Thus,δ marks + P. Valageas: Mass functions and bias of dark matter halos 7 the virialization radius where isotropization of the veloc- prefactorsthatwouldariseasonegoesfromthedistribution itytensorbecomesimportant,inagreementwithnumerical (δ ) to the mass function n(M). However,this expansion r P simulations (e.g., Cuesta et al. 2008). would not allow one to derive the shape of the mass func- It is interesting to note that a similar approach can be tionatlowmasses,asthisregimeisfarfromthequasi-linear developedforthe“adhesionmodel”(Gurbatovetal.1989), limitandinvolvesmultiplemergersfarfromsphericalsym- where particles move according to the Zeldovich dynamics metry.Untilanewmethodisdevisedtohandlethisregime, (Zeldovich 1970) but do not cross because of an infinitesi- one must treat the prefactors as free parameters to be fit- mal viscosity (i.e. they follow the Burgers dynamics in the ted to numericalsimulations,in orderto describe the mass inviscidlimit).Moreover,intheone-dimensionalcase,with function from small to large masses. a linear power-spectrum slope n = 2 or n = 0 (i.e. the Here we can note that in the Press-Schechter approach − linear velocity is a Brownian motion or a white noise), it (and in most models) the mass function (34) is obtained is possible to derive the exact distribution (δ ) by other from the linear density reached within the sphere centered r P techniques and to check that it agreeswith the asymptotic on each mass element. That is, a particle is assumed to be tail (30), as seen in Valageas (2009a,b). part of a halo of mass larger than M if the sphere of mass M (or a sphere of mass larger than M in the excursionset approachofBondetal.1991),centeredonthisparticle,has 4. Mass function of collapsed halos a linear density contrast above a threshold δ . As pointed L Thequasi-linearlimit(30)ofthe distribution (δ )clearly out in Sheth, Mo & Tormen (2001), and discussed in their r governs the large-masstail of the mass functioPn n(M)dM, section 3, since all particles are not located at the cen- wherewedefinehalosassphericalobjectswithafixedden- ter of their parent halo, the correct criteria should rather sity contrast δ = δ. Indeed, since the Lagrangian radius be that the particle belongs to a sphere, not necessarily r q is related to the halo mass by M = ρ 4πq3/3, massive centered on this point, of mass greater than M, that has m halos correspond to large Eulerian and Lagrangian radii, collapsed by the redshift of interest. Within the excursion andthe limitM correspondstothe quasi-linearlimit set approach of Bond et al. (1991), this means that one σ 0. Then, g→oin∞g from the distribution (δ ) to the should consider the first-crossing distributions associated r r mas→s function n(M) can introduce geometricPal power-law with center-of-mass particles, rather than with randomly prefactors since halos are not exactly centered on the cells chosenparticles,asarguedinSheth, Mo& Tormen(2001). of a fixed grid, as discussed in Betancort-Rijo & Montero- Then, the latter suggestthat this couldmodify the numer- Dorta (2006), but the exponential cutoff remains the same icalfactorintheexponentialtailofthemassfunction,that as in Eq.(30), whence is,leadto a smallerfactor δL inEq.(32)than the one asso- ciatedwithspherical(orellipsoidal)collapsedynamics.We δ2 point out here that this effect should only give subleading M : ln[n(M)] L , with δ = −1(δ), (32) →∞ ∼−2σ2 L F correctionstothe tail(32),sothatthe factorδL inEq.(32) remains exactly given by the spherical collapse dynamics. where σ = σ with M = ρ 4πq3/3. A simple approxi- q m This can be seen from the fact that the same effect mation for the mass function that satisfies this large-mass would apply to the density probability distribution (δ ), r falloff can be obtained from the Press-Schechter method P as randomly placed cells of radius r are typically not cen- (PS), see Press & Schechter (1974), but using the actual teredonhaloprofiles.Nevertheless,theresults(27)-(31)are linear threshold δ = −1(δ) associated with the nonlin- L exact in the quasi-linear limit (as can also be checked by F ear threshold δ that defines the halo radius r, rather than the comparison with standard perturbation theory for the the linear threshold δ 1.686 associated with the spheri- c cumulant generating function ϕ(y)), and off-center effects ≃ cal collapse down to a point. This yields for the number of would be included in the subleading terms, computed as halos in the range [M,M +dM] per unit volume, usualbyexpandingthepathintegral(22)arounditssaddle- point,whichwouldtypicallygivethepower-lawprefactorto ρ dσ n(M)dM = m f(σ) , (33) the tail (30). In terms of the halo mass function itself, this M σ (cid:12) (cid:12) point was also studied in Betancort-Rijo & Montero-Dorta (cid:12) (cid:12) with (cid:12) (cid:12) (2006), who found as expected that at large mass such a (cid:12) (cid:12) geometrical factor only modifies that power-law prefactor. f(σ)= 2 δL e−δL2/(2σ2). (34) AsfortheprobabilitydistributionP(δr),itisinteresting π σ to note that a similar high-mass tail canbe derivedfor the r “adhesion model”, where halos are defined as zero-size ob- The mass function (34) includes the usual prefactor 2 that jects (shocks).Again,inthe one-dimensionalcase,forboth gives the normalization n = 2 and n = 0, where the exact mass function can be − ∞ dσ obtainedby othermeans,onecancheckthatitagreeswith f(σ)=1, (35) theanalogoftheasymptotictail(32)(Valageas2009a,b,c). σ Z0 Thus, we can check that in these two non-trivial examples which ensures that all the mass is contained in such halos. the leading-order terms for the large-mass decays of (δ ) r P However,wemustnotethatthepower-lawprefactorin(34) andn(m)areexactlysetbysaddle-pointpropertiesandare has no strong theoretical justification since only the expo- not modified by the off-center effects discussed above. nentialcutoff(32)wasexactlyderivedfrom(30).Inprinci- We cannote that the explanationofthe large-masstail ple,itcouldbepossibletoderivesubleadingterms(whence (32) by the exact asymptotic result of the steepest-descent power-lawprefactors)inthequasi-linearlimitfor (δ ),by method describedinsection3 agreesnicely withnumerical r P expanding the path integral (22) around the saddle-point simulations. This is most clearly seen in Figs. 3 and 6 of (25). Then, one would also need to take more care of the Robertson et al. (2009), who trace back the linear density 8 P. Valageas: Mass functions and bias of dark matter halos contrast δ of the Lagrangian regions that form halos at L z = 0. Their results show that the distribution of linear contrastsδ , measuredas a function ofmass (or ofσ(M)), L hasaroughlyconstantlowerboundδ−,withδ− 1.6,and L L ∼ an upper bound δ+that grows with σ(M). We must note L howeverthatthedifferencebetween1.59and1.6754(which wouldbethestandardthresholdintheirΛCDMcosmology with Ω = 0.27) is too small to discriminate both values m0 fromtheresultsshownintheirfigures,sothatthesenumer- ical simulations alone do not give the asymptotic value δ− L tobetterthanabout0.2.Theyobtainthesameresultswhen they define halos by nonlinear density contrastsδ =100or δ =600,witha lowerbound δ− that growssomewhatwith L δ. In terms of the approach described above, this behavior expresses the fact that the most probable way to build a massivehaloofnonlineardensitycontrastδ istostartfrom aLagrangianregionoflineardensitycontrastδ = −1(δ), L F which obeys the spherical profile (25) and corresponds to thesaddle-pointoftheaction(23).Asrecalledinsection3, thepathintegral(22)isincreasinglysharplypeakedaround Fig.6. The mass function at redshift z = 0 of halos de- this initial state at large mass scales, which explains why finedbythenonlineardensitycontrastδ =200.The points thedispersionoflineardensitycontrastsδ measuredinthe L arethe fits to numericalsimulations fromSheth & Tormen simulations decreases with σ(M). At smaller mass, one is (1999),Jenkins et al. (2001),Reed et al. (2003),Warrenet sensitivetoanincreasinglybroadregionaroundthesaddle- al.(2006)andTinkeretal.(2008).Thedashedline(“δ ”)is point, which mostly includes non-spherical initial fluctua- c theusualPress-Schechtermassfunction,whilethesolidline tions.Since these initialconditions areless efficientto con- that goes close to the Press-Schechter prediction at small centrate matter in a small region one needs a larger linear mass is the mass function (34) with the exact cutoff (32). density contrast δ to reach the same nonlinear threshold L Here we have δ 1.59. The second solid line that agrees δ within the Eulerian radius r, which is why the distribu- L ≃ with simulations over the whole range is the fit (36), that tionisnotsymmetricandmostlybroadensbyincreasingits obeys the same large-mass exponential cutoff. typical upper bound δ+. In principles, it may be possible L to estimate the width of this distribution, at large masses, by expanding the action [δL] around its saddle-point. (34) closely follows the usual Press-Schechter prediction S and shows the same level of disagreement with numerical We compare in Fig. 6 the prediction (34) (solid line simulations.Thisisexpectedsinceonlytheexponentialcut- labeled as δL = −1(δ)) with results from numerical sim- off (32) has been exactly derived from section 3. Since at F ulations, for the nonlinear threshold δ = 200 at redshift largemassesthe power-lawprefactor1/σ in Eq.(34)is also z = 0. In our case, this corresponds to a linear density unlikely to be correct,as discussed above,we give a simple contrast δL 1.59, that is obtained from Fig. 2 or the fit that matches the numerical simulations from small to ≃ fit (12). As in section 3, we consider a ΛCDM cosmol- large masses (solid line that runs through the simulation ogy with (Ωm0,ΩΛ0,σ8,ns,h) = (0.27,0.73,0.79,0.95,0.7). points) while keeping the exact exponential cutoff: This corresponds to the cosmological parameters of the largest-box numerical simulations of Tinker et al. (2008), f(σ)=0.5 (0.6ν)2.5+(0.62ν)0.5 e−ν2/2, (36) which allow the best comparison with the theoretical predictions, as they also define halos by the density thresh- with (cid:2) (cid:3) old δ = 200 with a spherical-overdensity algorithm. The δ numerical results are the fits to the mass function given ν = L, δL = −1(δ) 1.59 for δ =200, Ωm =0.27(37) σ F ≃ in Sheth & Tormen (1999) (“ST”), Jenkins et al. (2001) (“J”),Reedetal.(2003)(“R”),Warrenetal.(2006)(“W”) At higher redshift or for other Ω one simply needs to use m and Tinker et al. (2008)(“T”). Note that these mass func- the relevant mapping −1(δ), shown in Fig. 2 or given by F tions are defined in slightly different fashions, using ei- the simple fit (12) for δ = 200. This mass function also theraspherical-overdensityorfriends-of-friendsalgorithm, satisfies the normalization (35), whatever the value of the and density contrast thresholds that vary somewhat about threshold δ . L δ = 200. However, they agree rather well, as the depen- Thus,wesuggestthatmassfunctions ofvirializedhalos dence on δ is rather weak. This can be understood from should be defined with a fixed nonlinear density threshold, Fig. 1, which shows that around δ = 200 the linear den- such as δ = 200, and fits to numerical simulations should sity contrast δ = −1(δ) has a very weak dependence use the exact exponential cutoff of Eq.(32), with the ap- L on δ. We also plot Fthe usual Press-Schechter prediction propriate linear density contrast δ = −1(δ), rather than L F (dashed line labeled δ ), that amounts to replace δ by treatingthisasafreeparameter.Thiswouldautomatically c L δ = 1.6754 in Eq.(34) (since Ω = 0.27 at z = 0). We ensurethatthelargemasstailhastherightform(uptosub- c m can see that using the exact value δ = −1(δ) signifi- leadingtermssuchaspower-lawprefactors),asemphasized L F cantly improves the agreement with numerical simulations bythereasonableagreementwithsimulationsofEq.(34)at at large masses. Note that there are no free parameters large masses. Moreover,it is best no to introduce unneces- in Eq.(34). Of course, at small masses the mass function sary free parameters that become partly degenerate. Here P. Valageas: Mass functions and bias of dark matter halos 9 (32) should also improve the robustness of the mass func- tionwithrespecttochangesofcosmologicalparametersand redshifts. Next,wecompareinFig.7themassfunctionsobtained at z = 0 for the density thresholds δ = 100 and δ = 300 with the numerical simulations from Tinker et al. (2008), who also consideredthese density thresholds. We show the usual Press-Schechter mass function (dashed line) and the results obtained from Eqs.(34) and (36) (solid lines), with now δ = −1(100) 1.55 and δ = −1(300) 1.61. L L F ≃ F ≃ Wecanseethatthelargemasstailremainsconsistentwith the simulations, but for the case δ =100 it seems that the shape of the mass function at low andintermediate masses ismodifiedandcannotbeabsorbedthroughtherescalingof δ .ItappearstofollowEq.(34)ratherthanthefit(36),but L this is likelytobe a merecoincidence.We shouldnotethat Fig.5showsthatδ <300formassivehalos,sothatshell- + crossing should be taken into account and the tail (32) is nolongerexactforδ =300,althoughitshouldstillprovide a reasonable approximation, as checked in Fig. 7. Thus, eventhoughTinkeretal.(2008)alsostudiedhigherdensity thresholds, we do not consider such cases here as the tail (32) no longer applies. 5. Halo density profile As for the mass function n(M), the analysis of section 3 showsthatthe densityprofileofraremassivehalosisgiven by the sphericalsaddle-point(25), see also Barkana (2004) and Prada et al. (2006). This holds for halos selected by some nonlinear density threshold δ in the limit of rare events,providedshell-crossinghasnotoccurredbeyondthe Fig.7. The mass functions at z = 0 of halos defined by associated radius r. In particular, this only applies to the outer part of the halo since in the inner part, at r′ r, the nonlinear density contrasts δ = 100 (upper panel) and ≪ shell-crossing must be taken into account. Then, as dis- δ =300 (lower panel). The points show the numerical sim- cussed in section 3, a strong radial-orbit instability comes ulationsofTinkeretal.(2008).Thedashedlineistheusual into play and modifies the profile in this inner region, as Press-Schechter mass function, while the solid lines correspond to Eqs.(34) and (36), with now δ = −1(100) deviations from spherical symmetry govern the dynamics L 1.55 (upper panel) and δ = −1(300) F1.61 (lowe≃r and the virialization process (Valageas 2002b). L F ≃ We compare in Fig. 8 the nonlinear density profile ob- panel). tained from Eq.(25) with fits to numerical simulations. We plot the overdensity within radius r′, 1+δ = ρ (< r m r′)/ρ , as a function of radius r. This is again obtained m wemustnotethatBarkana(2004)hadalreadynoticedthat, from Eq.(25) with the mapping q′3 = (1 + δr′)r′3 and tloaskibnyg tahenosnplhinereiacraltchorlelsahposledaδt f(ahcee vchalouseeaδnd=d1e8fiπn2in),gohnae- δr′ = F(δLq′). Since this only applies to the limit of rare events, we choose for each mass M a redshift z such that should use the linear threshold δ = −1(δ) for the Press- L σ(M,z) = 0.5. Thus, smaller masses are associated with F Schechter mass function. Unfortunately, noticing that the higherredshifts.The resultsdonotsignificantlydependon value of δL given by fits to numerical simulations was even theprecisevalueofthecriteriumusedtodefinerareevents, lower,he concludedthatthiswasnotsufficienttoreconcile here σ(M,z)=0.5.The points in Fig.8 are the results ob- theoreticalpredictionswithnumericalresults.Asshownby tained from a Navarro, Frenk & White profile (Navarro et Fig. 6, this is not the case, as the parameter δL used in al. 1997, NFW), fitting formulae is partly degenerate with the exponents of ρ the power-law prefactors, so that it is possible to match ρ(r′)= s , (38) numerical simulations while satisfying the large-mass tail (r′/r )(1+r′/r )2 s s (32). Of course, the actual justification of the asymptote or an Einasto profile (Einasto 1965), (32) is provided by the analysis of section 3, which shows thatbelow the upper boundδ the sphericalcollapseisin- + 2 deed relevant and asymptotically correct at large masses, ln[ρ(r′)/ρ ]= [(r′/r )α 1]. (39) −2 −2 −α − as it corresponds to the saddle-point of the action (23). Note that the result (32) also implies that the mass For the NFW profile, the characteristic radius r is ob- s function f(σ) is not exactly universal, since the mapping tained from the concentration parameter, c(M,z) = r/r , s δ δ = −1(δ) shows a (very) weak dependence on where as in the previous sections r is the Eulerian ra- L 7→ F cosmological parameters, see Fig. 2. Using the exact tail dius where δ = 200 (i.e. r = r ). Then, we use in r 200 10 P. Valageas: Mass functions and bias of dark matter halos imumshell-crossingradius,foreachEulerianradiusr′ there are two Lagrangian radii q′, a large one that corresponds toshellsthatarestillfalling in,andasmalleronethatcor- respondsto shells that havealreadygoneonce throughthe center (note that in agreement with Fig. 4, shell-crossing appears at slightly smaller relative radii for smaller mass). Then, the theoretical prediction only holds to the right of this second branch, where there is only one branch and no shell-crossing. Note that the theoretical prediction (25) explicitly shows that the halo density profiles are not universal. Within the phenomenological fits (38)-(39) this is parameterized through the dependence on mass and redshift of the concentration parameter and of the exponent α.However,wecanseethatoverthe regimewhereEq.(25) applies the local slope of the halo density is ρ(r′) r′−2, ∼ which explains the validity of the fits (38)-(39) in this do- main.Unfortunately,ourapproachcannotshedlightonthe innerdensityprofile,whereδr′ 200,whichistheregionof ≫ interestformostpracticalapplicationsofthefits(38)-(39). Thesameapproach,basedonthesphericalcollapse,was studied in greater details in Betancort-Rijo et al. (2006) and Prada et al. (2006). They consider the “typical” profile (25) that we study here, as well as “mean” and “most probable” profiles. In agreement with the steepest-descent approachofsection3,theyfindthatthemostprobablepro- Fig.8. The density profile of rare massive halos, for sev- filecloselyfollowsthetypicalprofile(25)andtheyobtaina good match with numerical simulations for massive halos, eral masses M. For each mass the redshift is such that payingparticularattentiontoouterradii.Therefore,wedo σ(M,z) = 0.5, as the theoretical prediction from Eq.(25) not further discuss halo profiles here. (solid line) only applies to rare events. The second branch thatappearsatsmallradiiisduetotheshell-crossing,hence the theoretical prediction only holds to the right of this 6. Halo bias branch.The points are fits to numericalsimulations,based onanNFW profile(Dolaget al.2004;Duffy etal.2008)or In additionto their multiplicity and their density profile,a an Einasto profile (Duffy et al. 2008). Note that we show key property of virialized halos is their two-point correla- the mean density within radius r′, ρm(< r′)/ρm, rather tion function. At large scales it is usually proportional to than the local density at radius r′, so that the NFW and the matter density correlation, up to a multiplicative fac- Einasto profiles are integrated once. tor b2, called the bias of the specific halo population. We revisit in this section the derivation of the bias of massive halos, following Kaiser (1984), and we point out that pay- Fig. 8 the two fits obtained in Dolag et al. (2004) and ing attention to some details it is possible to reconcile the Duffy et al.(2008)fromnumericalsimulations,ofthe form theoreticalpredictionswithnumericalsimulations,without c(M,z)=a(M/M )b(1+z)c.FortheEinastoprofile,weuse 0 introducing any free parameter. thefitsobtainedinDuffyetal.(2008)fortheconcentration As seeninthe previoussections,raremassivehaloscan parameter, c(M,z)=r/r , and for the exponent α (writ- −2 be identified with rare spherical fluctuations in the initial ten as a quadratic polynomial over ν = δ /σ(M,z) as in c (linear)densityfield.Moreprecisely,asinsection3wemay Gao et al. (2008)). Then, we integrate over r′ the profiles consider the bivariate density distribution, (δ ,δ ), of (38)-(39), to obtain the overdensity profiles ρm(< r′)/ρm the density contrasts δ ,δ , in the spheres Pof rar1diirr2 ,r , showninFig.8(asρ (<r′)isthe meandensitywithinra- r1 r2 1 2 m centeredat points x ,x . Thus, we introduce as in Eq.(19) diusr′ andnotthelocaldensityatradiusr′ asinEqs.(38)- 1 2 the double Laplace transform ϕ(y ,y ), (39)). 1 2 WecancheckinFig.8thatourresultsagreereasonably e−ϕ(y1,y2)/σ2 = e−(y1δr1+y2δr2)/σ2 wellwiththese fits to numericalsimulationsoverthe range h i ∞ whahserneobfroetehpparreadmicettioenr,ssainreceviatliids.gNivoetnebthyatthoeusradpdreled-ipctoiionnt = dδr1dδr2e−(y1δr1+y2δr2)/σ2P(δr1,δr2), (40) Z−1 profile (25). At large radii we recover the mean density of the Universe, while the numerical profiles (38)-(39) go to where σ2 =σ2 (0) is the linearvariance(18)at somescale r,r zero, but this is an artefact of the forms (38)-(39), that r. If r = r we may take r = r = r , otherwise it can 1 2 1 2 were designed to give a sharp boundary for the halos and be any intermediate scale, as the results do not depend on aremostlyusedforthe high-densityregions.Forastudyof this factor. The only requirement is that σ should scale numerical simulations at outer radii, see Figs.8 and 10 of as σ and σ , which is obtained by choosing for instance r1 r2 Prada et al. (2006) which recover the mean density of the a fixed ratio r/√r1r2. Then, the probability distribution Universe at large scales. At small radii, the second branch (δ ,δ ) can be obtained from the cumulant generating P r1 r2 thatmakesaturnsomewhatbelowrisduetoshell-crossing function ϕ(y ,y ) through a double inverse Laplace trans- 1 2 thatmakesthefunctionρ (<r′)bivaluate.Belowthemax- form,asinEq.(20).Next, ϕ(y ,y )canbewritteninterms m 1 2