Lecture 6: Dark Matter Halos

Houjun Mo

February 23, 2004

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Hierarchical clustering, the matter inthe universe condenses to form quasi-equilibrium objects (dark halos) ofincreasing masses in the passage oftime.The Goal of this lecture is tounderstand this process and theproperties of dark matter halos.

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Issues to be addressed

1. The mass function (i.e. the number density of haloes as a function of halo mass) at a giventime. The study of this issue may help us to understand the origin of the mass functions ofgalaxies and of clusters of galaxies.

2. The mass distribution of the progenitors of a given halo. The study of this issue may help usto understand the formation processes of present-day galaxies as well as galaxy populationsat high redshifts.

3. The merging rate of haloes. This issue may be related to the understanding of the populationof interacting galaxies and the substrutures in clusters of galaxies.

4. Intrinsic properties of haloes, such as density profile, shape and angular momentum. This isan important issue, because it may be related to the nature and morphology of protogalacticcollapses

5. The large-scale environment and spatial clustering of dark haloes. This may help us tounderstand the relation between galaxy population and the large scale environment.

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Halo Mass Function: the Press-Schechter Formalism

Basic idea:

• The cosmic density field is Gaussian

• The collapse of the high density regions can be described by spherical model

Consider a smoothed version of the cosmic density field δ(x):

δ(x;R)≡∫

δ(x′)W(x+x′;R)d3x′ ,

where W(x;R) is a window function of characteristic radius R.

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For a Gaussian density field, δ(x,R) has also Gaussian distribution:

P[δ(R, t)]dδ(R, t) =1√

2πσ(R, t)exp

[− δ2(R, t)

2σ2(R, t)

]dδ(R, t).

σ2(R, t) = D2(t)σ2(R); σ2(R) =≡ 〈δ2(x;R)〉=∫

∆2(k)W2(kR)dkk

,

where∆2(k)≡ 1

2π2k3P(k) ,

and W(kR) is the Fourier transform of the window function.

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The forms of the two most commonly used window functions are as follows:• Top-hat window:

Wth(x;R) =(

4π3

R3

)−1{1 (for |x| ≤ R)0 (for |x|> R)

Wth(kR) =3(sinkR−kRcoskR)

(kR)3 ;

• Gaussian window:

WG(x;R) =1

(2π)3/2R3exp

(−|x|

2

2R2

), WG(kR) = exp

[−(kR)2

2

];

The volumes of windows are related to the smoothing radius R as

V(R) ={

4πR3/3 (for top-hat)(2π)3/2R3 (for Gaussian)

.

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The fraction of points surrounded by a sphere of radius R and mean overdensityexceeds δc is:

F1/2(R, t) =∫ ∞

δc

dδ√2πD(t)σ(R)

exp

[− δ2

2D2(t)σ2(R)

]=

12

erfc[

δc/D(t)√2σ(R)

].

Press and Schechter suggested that, with δc chosen to be the thresholdovserdensity for spherical collapse (δc ≈ 1.686), this fraction be identified withthe fraction of particles which are part of collapsed lumps with masses exceedingM = 4πρa3R3/3.

A problem: as M → 0, σ(R) → ∞ (for power-law spectrum with n > −3) andF1/2 → 1/2. Only half of the particles are parts of lumps of any mass. Pressand Schechter ‘solve’ this, arbitrarily, by multiplying the mass fraction by a factorof 2.

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The number density of collapsed lumps with mass in the range M → M +dM isthen

n(M, t)dM =−2ρM

∂F1/2

∂RdRdM

dM =−√

2π

ρM

δc(t)σ2

dσdM

exp

[−δ2

c(t)2σ2

]dM ,

where δc(t) ≡ δc/D(t) is the critical overdensity linearly extrapolated to thepresent time.

Note: time enters only through D(t), mass enters through σ(R). The massfraction of the universe in objects with σ2(R) in the range from σ2 → σ2 + dσ2

is

f [σ,δc(t)]dσ2 =− 1√2π

δc(t)σ3

exp

[−δ2

c(t)2σ2

]dσ2 .

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The Press-Schechter formalism provides a useful way to understand hownonlinear structures develop in a hierarchical model. As one can see, haloeswith mass M can form in significant number only when σ(M)∼> δc(t). If we definea characteristic mass M?(t) by

σ(M∗) = δc/D(t) ,

then only haloes with M ∼< M? can form in significant number at time t. Since D(t)increases with t and σ(M) decreases with M in hierarchical models, we see thathaloes with larger and larger masses can form in the passage of time.

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Excursion Set Derivation of the Press-Schechter Formula

How to explain the fudge factor of 2 in the Press-Schecther formalism? Thederivation by Bond et al. (1991).

Instead with the spherical top-hat filter, smooth δ(x) a sharp-k filter:

WSk(k;kc) ={

1 (for |k| ≤ kc)0 (for |k|> kc)

.

The smoothed field is

δs(x, t;kc) =∫

d3kδk(t)WSk(k;kc)e−ik·x =∫

k<kc

d3kδk(t)e−ik·x .

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Consider an increase in kc by ∆k:

δs(x, t;kc +∆k) = δs(x, t;kc)+4πδk(t)e−ik·xk2∆k.

As kc is increased the value of δs at a given point x executes a random walk.

The advantage of using sharp-k filter:

〈∆δsδs(x;kc)〉= 0

where ∆δs = δs(x, t;kc +∆k)−δs(x, t;kc)and so ∆δs is independent of δs(x, t;kc). The variance

〈(∆δs)2〉= σ2(kc +∆k, t)−σ2(kc, t) = ∆σ2.

So as we change the window size, the mean density with it execute a randomwalk governed by a simple Gaussian distribution independent of early steps.

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At given time t we can identify all thepoints, {x}, where δs(x, t;Kc) = δc ; andδs(x, t;kc) < δc for all kc < Kc(x) . Kcis the value of kc where δs(x, t;kc) firstcrosses the barrier at δs = δc

Different points on the random walkcan be divided into three catagories:

1. Points with δs > δc for kc = Kc;

2. Points with δs < δc for kc = Kc but δs >δc for some kc < Kc;

3. Points with δs < δc for all kc ≤ Kc.

We want the fraction of mass elementsin class (iii) since this is the fraction ofmass elements with first upcrossing atkc > Kc.

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Note that for every random walk leading to an element with δs = δ0 > δc in class(i) there is an equally probable walk leading to an element in class (ii) with δs =δc−(δ0−δc) = 2δc−δ0. Hence the distribution of δs for points with first upcrossingat kc > Kc is

PFU(δs)dδs =1√2π

dδs

D(t)σ(Mc)

{exp

[− δ2

s

2D2(t)σ2(Mc)

]−exp

[− (2δc−δs)2

2D2(t)σ2(Mc)

]}and the fraction of mass elements with first upcrossing at kc > Kc is

F(> Kc) =∫ δc

−∞PFUdδs =

∫ νc

−∞

dx√2π

e−x2/2−∫ ∞

νc

dx√2π

e−x2/2,

where νc ≡ δc(t)/σ(Mc).

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Thus,

f [σ(Mc),δc(t)]dS=∂F∂S

dS=1√2π

δc(t)[S(Mc)]3/2

exp

[− δ2

c(t)2S(Mc)

]dS.

The form of f is exactly the same as given by the Press-Schechter formalism,except that the variance S are calculated in two different types of windows. Theorigin of the famous factor of 2 in the Press-Schechter formalism is now clear.The original treatment included only the first of the two terms in PFU; an equalcontribution should come from the second term.

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Interpretation of the Excursion Set

According to spherical model, a region will collapse at a time when itsoverdensity reaches the critical overdensity δc. One might therefore think thatall the mass elements which have first upcrossing at mass scale in the rangecorresponding to (S,S+dS) are parts of collapsed haloes with this mass at timet, and so f in the above equation could be interpreted as the fraction of masselements which are part of such collapsed objects.

The first part of the inference is not correct while the second part is.

The simplest way to understand this is to consider a uniform spherical region ofmass M and radius R, with critical overdensity at some time t, which is embbededin a larger region with lower overdensity. According to spherical model, all masselements in the spherical region will be part of a collpased halo of mass M at timet. Now consider an interior point x which is at a distance r from the surface of thespherical region, and calculate the mean overdensities within spheres centred onthis point. Clearly, the overdensity reaches the critical density only for spheres

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with radii smaller than r. Thus, the mass element at x has its first upcrossingat a mass near M(r/R)3 which is smaller than M if x is not at the centre of M.From this simple example, we see that the mass at the first upcrossing of a masselement only gives the lower limit of the collapsed object which the mass elementis part of. It then follows that the integration of f over mass,

F(M, t) =∫ ∞

Mf [S(M′),δc(t)]

dS(M′)dM′ dM′ ,

is the fraction of mass elements which are part of collapsed objects of massexceeding M, and f itself is just the fraction of mass elements which are part ofcollapsed objects of the mass in the range corresponding to (S,S+dS). Thus, theexcursion-set approach predicts how much mass ends up in collapsed objects ofcertain mass at a given time in a statistical sense, but it does not predict directlythe host object of a particular mass element.

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Halo bias: halo number density is modulated in space

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N[M1|δ0(R)]dM1 =M0

M1f (1|0)

dS1

dM1dM1, f (1|0) =

δ1−δ0√2π(S1−S0)3/2

exp

[− (δ1−δ0)2

2(S1−S0)

]where S1 = σ2(M1), S0 = σ2(M0).

δh(1|0) = bh(M1,δ1)δm, δh = ∆nh/nh; δm = ∆ρm/ρm

Halo bias (Mo & White 1996):

bh = 1+D(t1)D(t0)

(ν2

1−1δc

), ν1 = δc(t1)/σ(M1).

Linear bias of the halo-mass, halo-halo correlations:

ξhm(r;M) = bhξm(r) ξhh(r;M) = b2hξm(r) .

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Tests by numerial simulations

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An improvement of the PS formalism

• Basic idea (Sheth, Mo, Tormen 2001):

– Collapse of dark halos dependsnot only on overdensity, but alsoon the shape of the shear field:so ellipsoidal collapse instead ofspherical collapse

– In a Gaussian density fieldthe ellipticity of the shear fielddepends on the mass of theregion in consideration

– The threshold over-density forcollapse depends on the massscale in consideration

δec(M)δsc

= 1+β[

σ2(M)δ2

sc

]γ

where β ≈ 0.5, γ = 0.6.

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• Mass function

n(M,z) = A

(1+

1ν′2q

)√2π

ρM

dν′

dMexp

(−ν′2

2

)where ν′ =

√aν, ν = δc/σ, a = 0.707, q = 0.3, A = 0.322.

• Bias factor

b = 1+1δc

[ν′2+bν′2(1−c)− ν′2c/

√a

ν′2c+b(1−c)(1−c/2)

], (for halos)

b0 = 1+D(z)

δc

[ν′2+bν′2(1−c)− ν′2c/

√a

ν′2c+b(1−c)(1−c/2)

], (for descendants)

where b = 0.5 and c = 0.6, D(z) is the linear growth factor.

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Mass functions and bias factors:

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Some applications

The correlation of clusters at z= 0 (Mo,Jing & White 96)

d [Mpc/h]

The correlation of Lyman-breakgalaxies at z∼ 3 (Mo, Mao & White 99)

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Some interesting predictions (Mo &White 2002)

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Halo Progenitor Distribution

An advantage of the excursion set approach is that it provides a neat way tocalculate the properties of the progenitors which give rise to any given class ofcollapsed objects. For example one can calculate the mass function at z= 5 ofthose clumps (progenitors) which are today part of rich clusters of mass 1015M�.This is important, because it may give a model to describe how a collapsedobject was build up by the accretion and merger of small objects.

It is easy to show the fraction, among all of the mass elements of all (M2,δ2)patches, which was in collapsed objects of mass M1 at the earlier time t1:

f (S1,D1|S2,D2)dS1 =1√2π

δ1−δ2

(S1−S2)3/2

exp

[− (δ1−δ2)

2

2(S1−S2)

]dS1 ,

where S1 ≡ σ2(M1), S2 ≡ σ2(M2), δ1 ≡ δc/D(t1), δ2 ≡ δc/D(t2).

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The mass distribution of the progenitors of objects of mass M2 is therefore

n(M1, t1|M2, t2)dM1 =M2

M1f (S1,D1|S2,D2)

dS1

dM1dM1 .

Note that D, t and δ = δc/D(t) are equivalent variables, so are S and M. Fromthis we can calculate, for example, the fraction of the material in present-dayrich clusters, which at z= 3 was in haloes with mass exceeding 1012M�. Sincea mass of this order must be assembled to make a bright galaxies, this clearlylimits when the bright galaxies currently observed in clusters have formed.

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Other derived quantities

Straightforward manipulations using the calculus of probabilities allow us toconstruct models for the distributions of many interesting properties of darkhaloes.

Halo Formation Time

A convenient operational definition for the formation time of a halo of mass M isthe time when the largest of its progenitors first reaches mass M/2.

P (tf < t1;M2, t2) =∫ M2

M2/2

M2

M1f (S1,D1|S2,D2)

dS1

dM1dM1 .

The differential probability distribution for tf is then given by

dPdtf

(tf;M2, t2) =∫ M2

M2/2

M2

M1

∣∣∣∣ ∂∂tf

[ f (S1,Df|S2,D2)]∣∣∣∣ dS1

dM1dM1 .

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Merging Trees

The growth of a halo as the resultof a series of mergers. Timeincreases from top to bottom, and thewidth of the tree branches representthe masses of the individual parenthaloes. A slice throught the treehorizontally gives the distribution ofmasses in the parent haloes at agiven time. The present time t0 andthe formation time tf are marked byhorizontal lines, where the formationtime is defined as the time at which aparent halo containing in excess of halfof the mass of the final halo was firstcreated. [after Lacey & Cole (1993)]

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How to construct a merger tree?

To construct a merger tree of a halo, start from the trunk and work upwards.

Consider a halo with mass M2 at a time t2.

At a slightly earlier time t1 = t2−∆t, the progenitor distribution is given by theconditional mass function If ∆t is chosen small enough, the halo will either remainto be one halo or split into two haloes. The probability of there being a merger att1, with the smaller progenitor having mass M1 (the larger one has M′

1≡M2−M1)can be taken to be

dP = n(M1, t1|M2, t2)dM1 ,

where n(M1, t1|M2, t2) is the conditional mass function.

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Internal Structures of Dark Matter HaloesHalo Density Profiles

From high-resolution simulations:

ρ(r) = ρcritδch

(r/rs)(1+ r/rs)2,

where rs is a scale radius.[NFW profile (Navarro et al. 1997)].Concentration c≡ rvir/rs.

Jing et al.

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Asymmetry and Substructure

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Angular Momentum

The angular momentum of a dark halo is represented by its spin parameter:

λ =|E|1/2JGM5/2

,

where J, E and M are the total angular momentum, energy and mass. Foran isolated system, these quantities are all conserved during dissipationlessgravitational evolution, and so is λ itself.

λ = rotational energy/total energy

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Numerical simulations show that λ has approximately a lognormal distribution

p(λ)dλ =1√

2πσλexp

[−ln2(λ/λ)

2σ2λ

]dλλ

,

with λ ≈ 0.05 and σλ = 0.5.

Detail simulations show that the λ declines weakly with increasing halo mass andthat the distribution is similar for haloes in different large-scale environments andin different cosmogonic models.

Other issues:

(1) The distribution of angular momentum in individual halos

(2) Physical reasons: torque by large-scale structure

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Observational Tests of Massive halos

Individual galaxies: Groups and clusters:

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Halo Model of Mass Clustering in the Universe

Ingredient:

• halo mass function

• halo correlation functions

• halo density profiles

We can obtain the correlationfunctions of mass (original ideadue to Neyman and Scott (1952)

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An example: the power spectrum(Ma & Fry 2000)

P(k) = P1h(k)+P2h(k)

P1h(k) =1

ρ2 ∑M

ρ2(M,k),

P2h(k)=1

ρ2 ∑M1,M2

ρ(M1,k)ρ(M2,k)Phh(k;M1,M2).

Note that ρ can either be the massdensity profile of dark halos, or thegalaxy number density profile in halos

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For simplicity, we writeρ(x) = mu(x|m) , (1)

x is the position relative to the halo center, and∫

u(x|m)d3x = 1.

Dividing the space into many small cells, ∆Vi (i = 1,2, · · ·), so small that none ofthem contains more than one halo center. The occupation number, Ni, is either1 or 0, so Ni = N 2

i = N 3i = · · ·

ρ(x) = ∑i

Nimiu(x−xi|mi) , (2)

mi is the mass of the halo in ∆Vi. Note that

〈Nimiu(x−xi|mi)〉=∫

dmn(m)m∆Viu(x−xi|m)

where n(m) is the halo mass function.

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It is more convenient to work in Fourier space:

ρ(k) = ∑i

Nimiu(k|mi)eik·xi , (3)

u(k|mi) is the Fourier transform of the density profile.

Using the properties of Ni,

P(k)≡Vu〈|δk|2〉= P1h(k)+P2h(k) , (4)

P1h(k) =1

ρ2

∫dmm2n(m) |u(k|m)|2 , (5)

P2h(k) =1

ρ2

∫dm1m1n(m1)u(k|m1)

∫dm2m2n(m2)u(k|m2)Phh(k|m1,m2) , (6)

with Phh(k|m1,m2) the power spectrum of halos of mass m1 and m2.

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Models can be constructed for all low-order statistics of clustering:

• Two-point correlation function and power spectrum

• Higher-order correlation functions

• Redshift-space correlation functions

• Redshift-distortions and velocity statistics

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