Formation, structure and clustering of CDM halos

Houjun Mo (UMass)

June, 2008

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Structure formation in CDM Scenario

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Hierarchical formation, simple consideration

Consider a perturbation spectrum (or a perturbation profile):

δM(r)M(r)

∝ M−ε , typically ε =neff +3

6(neff : effective power index) .

Spherical collapse model: δM(r)M(r) ≈ 1.68 to collapse

For a single perturbation, the mass accretion history is related to the lineargrowth factor D(z) by

M(z) ∝ D1/ε(z) .

Hierarchical formation requires ε > 0, or neff >−3.

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According to spherical model:

ρv(Rv) = ∆vΩm(z)ρcrit(z) , ∆v∼ constant .

Defining halo virial (circular) velocity:

V2c =

GMv

Rv,

we have

Vc ∝[H(z)Ω1/2

m (z)Mv(z)]1/3

.

At high-z:Vc(z)∼ (1+z)(1/2−1/3ε) .

If ε < 2/3 (i.e. neff < 1, Vc increases with time (potential well builds up with time)

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CDM structure formation

For CDM, −3< neff < 1, structure grows hierarchically to form CDM halos, and Vc

of each halo must grow from zero. There must be a fast accretion regime, whereMv(z) grows faster than 1/H(z) in order to establish halo potential well.

For individual halos, the growth of Vc is determined by the mass accretion history.

Fast and slow accretions:

• If Mv(z) increases faster than 1/H(z), then Vc increases with time;

• If Mv(z) increases as fast as 1/H(z) then Vc ramains constant;

• If Mv(z) increases slower than 1/H(z), then Vc decreases with time.

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Halo accretion histories

The fast/slow accretions are notcompletely random. In general, theformation of a halo can roughtlydivided into two regimes: fast andslow.[Zhao et al. 2003]

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Halo Properties in the fast/slow regimes

Circular velocities at virial radius Rv

and the scale radius rs

ρ(r) ∝ 1/[r(rs+ r)2] (NFW profile) Changes of Rv and rs with time

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Energy ranking preserved, especiallyin the slow accretion regime;completely mixing is not achieved.

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What determine the fast/slow regimes?

Several possibilities:

• Shapes of density peaks of different heights;

• Truncations by large-scale tidal fields;

• Change of local background (due to halo bias).

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Halo structure and formation history

[Zhao et al. 2003]

Explanation in terms of halo formationtime:NFW model: ρ(rs) ∝ ρ(zf ); zf : at which half ofthe halo mass is in progenitors more massivethan 1% of the final mass;Bullock et al. model: c= 9(M/M?)−0.13(1+z)−1;Eke et al. model: c3 = ∆(zc)Ω(z)

∆(z)Ω(zc)

(1+zc1+z

)3;

σ(zc,M) = 1/28;Wechsler et al. model: c = 4.1a/ac; M(a) =M0exp[−2ac(a0/a−1)];Zhao et al. model: inner region established infast accretion with c= 4, and c increases as Rv

increases.

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NFW profile and halo accretion history

Simple spherical model (Lu et al. 2006)For a given mass accreion history, M =M(z), the initial perturbation profile is

δi(r i) = 1.686D(zi)D(z)

,

where r i is related to M by

r i(M) =[

3M4πρ(zi)

]1/3

.

Assuming radial orbits

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Effect of velocity isotropization

Assuming particles accreted in the fastaccretion regime have random orbits:

σ2t

σ2r= 2

[1+

(Rt

ra

)β]−1

,

where ra is the characteristic scaledemarcating fast and slow accretion.Effective acceleration includingcentrifugal force in spherical model:

a = g+J2

r3

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Model predictions

Halo concentration: c = Rv/rs

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Interpretations

To produce the inner r−1-profile, twoconditions:(1) Fast collapse: 0 < ε < 1/6, so thatVc increases faster than H−1;(2) Orbit isotropisation in the fastaccretion regime

Fast accretion allows particles with awide range of energies to contributeto the central profile, if permited byangular momentum;

Velocity isotropisation allows mixing ofthese particles, resulting in ρ(r) ∝ r−γ

with γ≈ 1.

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Outer profile

In the slow accretion regime, the growth is inside-out:

ρ(r) =1

4πr2

dMdt

(drdt

)−1

∝Mr3

µ2+µ

,

where µ≡ d lnM/d ln t.

Write M = Me+∆M, Me: the mass of the halo at t−∆t; ∆M is the mass accretedbetween t−∆t and t.

If ∆M increases as a power law of t, and if ∆M Me, then ρ ∝ r−3.

If M(a) continues to grow as M(a) = M0exp[−2ac(a0/a−1)], ρ ∝ r−4 for a a0.

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The ‘universal’ CDM halo profile seems to be determined by initial conditions ofcollapse provided by the CDM model.

• Halo profile depends on formation history.

• How is velocity field isotropized?

• How is velocity structure correlated with halo profile?

• What do the results tell us about the mass profile of elliptical galaxies?

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Halo bias

ξhh = b2ξm

[Gao et al. 05]

Spherical collapse: does notdependent of large-scale environment;Ellipsoidal collapse: depends on localtidal field (Sheth, Mo, Tormen, 01)

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Age-dependence of halo bias

[Jing, Suto, Mo, 07]

Ellipsoidal collapse: halo formationdepends on large-scale tidal field. Isthis the only reason?

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Tidal heating in high density regions

[Wang, Mo, Jing 07]

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• It is clear that halo clustering depends on halo formation.

• Tidal heating and truncation may be the origin of such dependence.

• It is interesting to examine the implications for galaxy assembly in denseenvironments.

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Connection to Galaxies

Dark matter halos: hierarchical formation; Galaxies: not necessarily! [Li, Mo,Gao 08]

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Constrain from Galaxy-Galaxy Lensing

[R. Li et al. 2008]

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Environmental Effects

[Y. Wang wt al. 2008]

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Summary

CDM halos show regularities in their assembly histories: fast accretion followedby slow accretion.

Halo structure depends on formation history: not only concentration, but also theprofile itself may be determined by accretion histoy.

Halo formation depends on environments, likely due to large-scale tidal fields.

Subhalos: number and radial distribution quite well understood in pure DMsimulations, but the connection to reality is not clear.

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