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UNIVERSITY OF PADUA

Department of Astronomy

ASTROMUNDUS MASTERS COURSE INASTRONOMY AND ASTROPHYSICS

I CICLE

Galaxy Dynamics

Authors: Luigi Secco, Daniele Bindoni

Contents

Acknowledgements v

1 The Metric 11.1 Transformation by Covariance

and Contravariance . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.1 General Postulate of Covariance . . . . . . . . . . . . . 4

1.3 Exemplifications . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 The metric of Robertson-Walker . . . . . . . . . . . . . . . . . 51.5 On the geometric curvature k . . . . . . . . . . . . . . . . . . 7

2 On Einstein’s Equations 92.1 General Relativity as Geometric Theory . . . . . . . . . . . . 92.2 On Einstein’s equations by analogy . . . . . . . . . . . . . . . 112.3 On the meaning of Einstein’s Equations . . . . . . . . . . . . 12

3 Recombination 133.1 Recombination Epoch . . . . . . . . . . . . . . . . . . . . . . 13

3.1.1 When does it begin? . . . . . . . . . . . . . . . . . . . 133.2 Typical photon-electron interaction time . . . . . . . . . . . . 143.3 Ionization degree . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3.1 Typical time for ionization branch . . . . . . . . . . . . 153.3.2 The other branch . . . . . . . . . . . . . . . . . . . . . 16

3.4 Expansion Time . . . . . . . . . . . . . . . . . . . . . . . . . . 163.5 The last scattering surface . . . . . . . . . . . . . . . . . . . . 18

4 The thermodynamical point of view 214.1 The first Principle . . . . . . . . . . . . . . . . . . . . . . . . 21

4.1.1 Integrability condition . . . . . . . . . . . . . . . . . . 21

i

ii CONTENTS

4.1.2 Entropy integration in thermodynamic coordinates . . 23

4.2 Cosmological application . . . . . . . . . . . . . . . . . . . . . 24

4.2.1 Physical meaning . . . . . . . . . . . . . . . . . . . . . 25

5 Jeans’ Gravitational Instability 27

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.2 Jeans’ Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.2.1 Dump time . . . . . . . . . . . . . . . . . . . . . . . . 28

5.3 Collapse time . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.4 Jeans’ Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.5 Gravitational Jeans’ Instabilityin an expanding Universe . . . . . . . . . . . . . . . . . . . . . 34

5.6 Increasing rate in radiation/matter dominated Universe . . . . 35

5.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.7.1 Lagrangian and eulerian point of view . . . . . . . . . 36

6 Density Perturbations 39

6.1 Density perturbations . . . . . . . . . . . . . . . . . . . . . . . 39

6.2 To help the comprehension . . . . . . . . . . . . . . . . . . . . 41

6.3 Mass variance - Io filter . . . . . . . . . . . . . . . . . . . . . 42

6.4 The scale-free power spectrum . . . . . . . . . . . . . . . . . . 43

6.5 Transfer Function - IIo filter . . . . . . . . . . . . . . . . . . . 44

6.6 Initial power spectrum . . . . . . . . . . . . . . . . . . . . . . 44

6.7 Effective index . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6.7.1 CDM scenario . . . . . . . . . . . . . . . . . . . . . . . 45

6.7.2 ΛCDM scenario . . . . . . . . . . . . . . . . . . . . . . 46

7 Mini-universe 49

7.1 Condition to form a structure . . . . . . . . . . . . . . . . . . 50

7.2 The closed mini-universe . . . . . . . . . . . . . . . . . . . . . 52

7.3 Perturbation density at maximum expansion vs. Universe den-sity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

7.4 Perturbation density at virialization vs. Universe density . . . 55

8 Oscillations of Relaxation 57

8.1 Oscillations of Relaxation . . . . . . . . . . . . . . . . . . . . 57

8.2 Relaxation period and Virial dimension . . . . . . . . . . . . . 59

8.3 Relaxation times: Trelax, τrelax, τff . . . . . . . . . . . . . . . . 60

8.3.1 Item a) . . . . . . . . . . . . . . . . . . . . . . . . . . 60

8.3.2 Item b) . . . . . . . . . . . . . . . . . . . . . . . . . . 62

CONTENTS iii

8.4 Statistical Mechanics:Boltzmann’s Equation for a collisionless particle system . . . . 62

8.5 Analogy stars-pendulums in phase space . . . . . . . . . . . . 638.6 Movement of phase points in µ-space . . . . . . . . . . . . . . 66

9 Violent relaxation in phase-space 699.1 The Violent Relaxation problem . . . . . . . . . . . . . . . . . 699.2 Distribution Function . . . . . . . . . . . . . . . . . . . . . . . 719.3 Stellar Dynamical Equilibria - Boltzmann Equation . . . . . . 72

9.3.1 Jeans Theorem . . . . . . . . . . . . . . . . . . . . . . 729.4 Information Theory and Statistical Mechanics . . . . . . . . . 739.5 Lynden-Bell . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

9.5.1 Incomplete Relaxation . . . . . . . . . . . . . . . . . . 819.6 Shu’s Criticism . . . . . . . . . . . . . . . . . . . . . . . . . . 81

9.6.1 Shu’s approach . . . . . . . . . . . . . . . . . . . . . . 829.6.2 Incomplete Relaxation . . . . . . . . . . . . . . . . . . 87

9.7 Mass Segregation . . . . . . . . . . . . . . . . . . . . . . . . . 879.8 Kull, Treumann & Bohringer’s Criticism . . . . . . . . . . . . 89

9.8.1 Kull, Treumann & Bohringer’s approach . . . . . . . . 899.9 Nakamura’s Criticism . . . . . . . . . . . . . . . . . . . . . . . 93

9.9.1 Nakamura’s approach . . . . . . . . . . . . . . . . . . . 959.9.2 To get Lynden-Bell statistics . . . . . . . . . . . . . . . 100

9.10 Inconsistency in theories of violent relaxation . . . . . . . . . . 1019.11 The phase-space structure of Dark Matter halos . . . . . . . . 1039.12 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

iv CONTENTS

Acknowledgements

Thanks to ....

v

vi ACKNOWLEDGEMENTS

Chapter 1

The Metric

1.1 Transformation by Covariance

and Contravariance

1.1.1 Vectors

We consider two systems of coordinates (Sokolnikoff, 1964):

X : xi = (x1, x2, .....xn)Y : yi = (y1, y2, .....yn)

(1.1)

and the transformation between them:

T : xi = xi(y1, y2, .....yn) (1.2)

We form the set of partial derivatives:

∂f

∂x1,

∂f

∂x2, ......

∂f

∂xn(1.3)

of a continuously differentiable function f(x1, x2, .....xn) that is the compo-nents of a gradient of potential functions.

The same vector has in the system Y the components:

∂f

∂y1,

∂f

∂y2, ......

∂f

∂yn(1.4)

linked to the previous ones by the rule for differentiation of composite func-tions, namely:

∂f

∂yi=

∂f

∂xα

∂xα

∂yi(1.5)

1

2 CHAPTER 1. THE METRIC

If, in general, the components of a vector in X: A1(x), ....An(x), transformin the system Y as:

Bi(y) =∂xα

∂yiAα(x) (1.6)

we name that a law of covariant transformation and use by convenction thesubscripts for sets that transform in this manner. Another law of transfor-mation of vectors which is quite different from the previous one refers to theinfinitesimal displacement vector: P1P2, where

P1 ≡ P1(x1, x2, .....xn), P2 ≡ P1(x

1 + dx1, x2 + dx2, .....xn + dxn) (1.7)

Due to the differentation law we have:

dyi =∂yi

∂xαdxα; (i, α = 1, 2, ....n) (1.8)

which yields:

Bi(y) =∂yi

∂xαAα(x) (1.9)

we name that a law of contravariant transformation and use by convenctionthe superscripts for sets that transform in this manner.

1.1.2 Tensors

A covariant tensor of rank one is a set of quantities: A(1; x), A(2; x),...,A(n; x) which transforms from the X-coordinate system into the Y-one, ac-cording to:

B(i; y) =∂xα

∂yiA(α, x) (1.10)

which means (by convention);1

Bi =∂xα

∂yiAα (covariant law) (1.11)

Moreover a covariant tensor of rank two is a set of quantities: A(i, j; x) whichtransforms from the X-coordinate system into the Y-one, according to:

B(i, j; y) =∂xα

∂yi

∂xβ

∂yjA(α, β; x) (1.12)

1The only exception to this convention is the use of superscripts to identify the variablesxi, yi, etc. These quantities do not transform according to a covariant or contravariantlaw (see, Sokolnikoff, pg. 60).

1.2. INVARIANTS 3

which by convention will be denoted:

Bij =∂xα

∂yi

∂xβ

∂yjAαβ (1.13)

On the contrary a set of quantities A(i; x) which transforms from theX-coordinate system X into the Y-one, according to:

B(i; y) =∂yi

∂xαA(α; x) (1.14)

which by convention means:

Bi =∂yi

∂xαAα (contravariant law) (1.15)

define a contravariant tensor of rank one.Moreover for contravariant tensor of rank two we have:

Bij =∂yi

∂xα

∂yj

∂xβAαβ (1.16)

In the case of a tensor which transform according to:

Bji (y) =

∂xα

∂yi

∂yj

∂xβAβ

α(x) (1.17)

we will speak of a mixed tensor, covariant of rank one and contravariant ofrank one.

The extension to higher ranks is manifest.

1.2 Invariants

The aim of General Relativity is to write down relationships between physicalquantities which are invariants that is which are not altered by a transfor-mation of coordinates.

To this purpose the tensors are perfectly suitable tools. Indeed once aphysical property is written by an equation between two tensors of the sameranks (covariants and/or contravariants) on both sides, e.g., of this kind inthe system X:

A(α; x) = C(α; x) (1.18)

it transforms inside the coordinate system Y into:

B(i; y) = D(i; y) (1.19)


The problem is: the last equation means the same property told by Eq.(1.18)or something else? Owing to the way in which: A(α; x) and C(α; x) trans-form:

B(i; y) =∂xα

∂yiA(α; x) (1.20)

D(i; y) =∂xα

∂yiC(α; x) (1.21)

it follows that:(A = C) ⇒ (B = D) (1.22)

1.2.1 General Postulate of Covariance

According to Pauli (....,pg. 223) the following General Postulate has to hold:”The general laws of nature have to be written in the same way in any gaus-sian2 coordinate system so that they have to be ”covariants” in respect to anarbitrary transformation of coodinates”. 3

1.3 Exemplifications

We take into account the invariance of ds2 under Lorentz’ transformationsin special relativity. We pass from the :

ds2 = c2 dt2 − dx2 − dy2 − dz2 (1.23)

in the X frame of reference to the new one Y ≡ (ct′, x′, y′, z′) by Lorentz. Itmeans :

yα = Λαβxβ (1.24)

where: Λαβ =

γ γv1 0 0γv1 γ 0 00 0 γ 00 0 0 γ

.

and: γ = 1

(1− v2

c2)1/2

.

Then it follows:

ds2 = ηαβ dxα dxβ = η′µν dyµ dyν (1.25)

2Gaussian Geometry includes the geometries which violate the V o Euclide’s Postulate.A more general formulation of them is due to Riemann.

3It is to be noted that in this context the meaning of ”covariance” is the same of:invariant in form.

1.4. THE METRIC OF ROBERTSON-WALKER 5

For each fixed α we define:

Aα(x) = ηαβ dxβ (1.26)

so that dividing by dxα, the Eq.(1.25) becomes:

Aα(x) =dyµ

dxαBµ(y) (1.27)

defining: Bµ(y) = η′µν dyν. In conclusion a Lorentz’ invariance produces a

covariance transformation. Moreover we underline that, generally speaking,the ordinary derivative (usualy written as a ”,” subscript) of a tensor doesn’tproduce a tensor. On the contrary the Covariant Derivative (written as a”;” subscript) always yields a tensor (see, e.g., the energy-momentum tensorin Coles & Lucchin, 1995, pg.4).

1.4 The metric of Robertson-Walker

The most general metric element in a four-dimensional Riemann’s space4 is:

ds2 = gαβ dxα dxβ; (α, β = 0, 1, 2, 3) (1.28)

where the tensor: gαβ = gβα, is symmetric and: gαβ = gαβ(P ).Let us (see, Schutz) re-write (1.28) as follows:

ds2 = goo dxo dxo + 2goi dxo dxi − hij dxi dxj ; (i, j = 1, 2, 3) (1.29)

where (i, j) refer to spatial components.By choosing the universal time t, i.e., the proper time any observer, at

rest with his local matter distribution, synchronizes on the value of localdensity ρ at the time t (the synchronous gauge), the metric element (1.29)becomes:

ds2 = c2 dt2 + 2goi dxo dxi − hij dxi dxj ; (i, j = 1, 2, 3) (1.30)

We wonder: goi = 0?The answer is: YES, because if not, we would have:

2go1 dxo dx1 + 2go2 dxo dx2 + 2go3 dxo dx3 (1.31)

4Named riemaniann variety, i.e., a continuum and derivable space-time where all theeuclidean entities are present but extended to curved geometries.


which means, in turn, that it exists on the point P an infinitesimal spatialvector:

dP ≡ ( da dx1, db dx2, dc dx3) (1.32)

with:

da = 2go1 dxo; db = 2go2 dxo; dc = 2go3 dx3

which identifies at each point P a privileged direction, against the Cosmo-logical Principle.

The same result may be reached using the Equivalence Principle:”Experiments in a sufficiently small freely falling laboratory, over a sufficienlyshort time, give results that are indistinguishable from those of the sameexperiments in an inertial frame in empty space” (Hartle, pg. 119).

In each point P we may choose to describe what happens, a local inertialframe, i.e., locally indistinguishable from that of a flat space-time related tothe Special Relativity. In other words: G.R. → S.R. as ds → 0. So we havethat:

g′αβ(x′

P ) = ηαβ = diag.(+1,−1,−1,−1) (1.33)

which means:

g′oi(P ) = ~eo · ~ei = 0 (1.34)

We wonder moreover: gij =?At the instant t = to, any three-dimension hyper-surface, embedded into afour-dimension space, will have the line element:

dl2(to, Pk) = hij(to, Pk) dxi dxj , (i, j = 1, 2, 3) (1.35)

Thanks to Cosmological Principle, again Pk doesn’t appear, so that (1.35)becomes:

dl2(to) = hij(to) dxi dxj (1.36)

Moreover, if expansion has to occur conserving isotropy, at time t1 > to wewill have:

dl2(t1) = hij(t1) dxi dxj (1.37)

where:

hij(t1) = hij(to)f(t1, to) (1.38)

with a function f without indices, i.e., all the hij have to increase of the samefactor. By generalization, Eq.(1.38) becomes indeed:

hij(t) = hij(to)f(t, to) = hij(to)a(t) (1.39)

1.5. ON THE GEOMETRIC CURVATURE K 7

and Eq.(1.37) reads:dl2(t) = a2(t) dl′2(to) (1.40)

where into the square of the expansion factor a(t), the dimensions of dxi

and dxj have been transfered. The coordinates xi, xj become comovingdimensionless coordinates.

In spherical coordinates (1.40) transforms into:

dl2 = grr dr2 + r2 dΩ2 = a2(t)[grr(r′) dr′2 + r′2 dΩ2]; dΩ2 = dθ2 + sin2θ dφ2

(1.41)where r has its dimension and r′ is, on the contrary, without it.

In its most general expression, grr(r′) = e2Λ(r′); by constraining it with

the condition that the geometric curvature k does be the same in each point,we obtain:

grr(r′) =

1

1 − kr′2(1.42)

In conclusion (1.29) turns to be the following Robertson-Walker metricelement:

ds2 = c2 dt2 − a2(t)

[

1

1 − kr′2+ r′2 dΩ2

]

(1.43)

1.5 On the geometric curvature k

We may recognize the following association between the normalized valueof the geometric curvature k with the different types of geometries which,except the Euclidean, violate the V o Euclide’s Postulate:

Hyperbolic-Space (Lobacewskij):

k = −1; Ωo < 1; Open Universe

Parabolic-Space (Euclide):

k = 0; Ωo = 1; Critical Einstein − de Sitter Universe

Elliptic-Space (Riemann):

k = +1; Ωo > 1; Closed Universe

As soon as the Einstein’s Eqs. (see next Chap.) are put inside Robertson-Walker’s Metric, they yield Friedmann’s Eqs.:

a = −4πG3

(ρ + 3pc2

)a (Io)

a2 + kc2 = 8πG3

ρa2 (IIo)


Figure 1.1: The R.W. metric includes with the Euclidean geometric space(k = 0) also that of Lobacewskij (k = −1) and the Riemann’s one (k = +1)which violate the V o Euclide’s Postulate. The density parameter Ωo in thecorresponding Friedmann’s models are also indicated.

From the second one we obtain that the critical density in order to obtaink = 0 is:

ρcrit(t) = H2 3

8πG(1.44)

where the Hubble constant H = aa.

Moreover, dividing by a2 the same equation and defining the densityparameter Ω = ρ

ρcritwe may easily link the different Friedmann’s models,

characterized by Ω, with k and Hubble constant (see, Fig.1.1):

k = (Ω − 1)H2a2

c2(1.45)

Chapter 2

On Einstein’s Equations

2.1 General Relativity as Geometric Theory

Let us remember the Hamilton Principle for a free particle in special Rela-tivity:

I = −α

∫ b

a

ds (2.1)

where the line element ds is a relativistic invariant:

ds = c · dt′ = c ·√

1 − v2

c2dt (2.2)

and t′, proper time, t in lab. The Hamilton Principle (2.1) transforms into:

I ≃∫ tb

ta

(

−moc2 +

1

2mov

2

)

dt (2.3)

as soon as vc≪ 1 and α = moc.

The meaning of (2.3) is that the Lagrangian of a free particle at a nonrelativistic velocity is:

L =1

2mov

2 − moc2 (2.4)

We wish now to prove as Newtonian gravity can be expressed completely ingeometric terms in the curved space-time. Indeed if there is a gravitationalpotential φ constraining the free particle, the Lagrangian becomes:

L = T − V =1

2mov

2 − moc2 − moφ (2.5)

9

10 CHAPTER 2. ON EINSTEIN’S EQUATIONS

T,V being respectively kinetic and potential energy.By inserting φ in the metric element ds as follows:

ds2 = c2

(

1 +2φ

c2

)

dt2 −(

1 − 2φ

c2

)

dr2 (2.6)

= c2

(

1 +2φ

c2

)

dt2 −(

1 − 2φ

c2

)

(

dx2 + dy2 + dz2)

(2.7)

and puting it into (2.1), we obtain:

I = −α

∫ b

a

ds (2.8)

= −α

∫ b

a

c dt

[(

1 +2φ

c2

)

− 1

c2

(

1 − 2φ

c2

)(

dx2

dt2+

dy2

dt2+

dz2

dt2

)]1/2

(2.9)

It means in turn:

[(

1 +2φ

c2

)

− 1

c2

(

1 − 2φ

c2

)

v2

]1/2

≃ 1 +1

2

(

2φ

c2− v2

c2

)

(2.10)

which yields at the end:

I = −moc

∫ tb

ta

c

[

1 +φ

c2− 1

2

v2

c2

]

dt =

∫ tb

ta

(

−moc2 +

1

2mov

2 − moφ

)

dt

(2.11)

Figure 2.1: Newton’s view of gravity: one body attracts another by meansof a force.

2.2. ON EINSTEIN’S EQUATIONS BY ANALOGY 11

Figure 2.2: Einstein’s view of gravity: a massive body (the Sun) significantlydistorts the space-time in its vicinity. This distorsion influences the motionof the Earth, giving the impression that a force is acting.

2.2 On Einstein’s equations by analogy

Without pretending to derive the Einstein’s Equations let us discuss on themby analogy with the electromagnetic field (E-H). By applying the Euler-Lagrange equation to the (E-H) lagrangian density L and using the subsidiaryLorentz’ condition, we obtain the following equations in the (E-H) potentials

Aµ ≡ ( ~A, iφ), inside the Minkowski space xµ ≡ (~x, ict); (µ = 1, 2, 3, 4):

Aµ = −4π

cjµ (2.12)

where the current surface density tetra-vector is (~j, icρ), with ρ the electricspatial density.The meaning of Eqs.(2.12), which are the Maxwell’s Equations of Io and IIo

Groups, is to link (E-H) potential field with its sources: static charges, by ρ,and moving charges, by current density ~j.We are now looking for some equations which are able to play the samerole of the previous ones in the gravitational field, i.e., to link the space-time derivatives of an object containing the gravitational potential φ withits sources. These lasts are all the forms of energy the matter has in orderto produce the gravitational potential: matter density, by ρ, inner energydensity, by p and movement energy by the matter velocity. In other wordswe are looking for Eqs. of this kind (see, M. Roos, 1994, pg. 41):

φµν = kTµν (2.13)

12 CHAPTER 2. ON EINSTEIN’S EQUATIONS

where Tµν is the energy-momentum tensor, given by:

Tµν =(

ρc2 + p)

uµuν − pgµν (2.14)

By puting ourselfs in a rest-frame by a Lorentz’ transformation and assumingthe following approximations:

• weak field: φc2

≪ 1;

• stationary field: φµν → ∇2φµν

the time component (xµ ≡ (ct, ~x); µ = 0, 1, 2, 3) of Eq.(2.13) becomes:

∇2φoo = kToo (2.15)

where:Too = (ρc2 + p) c2 − pgoo.In the limit of flat-Minkowski space, at the origin we obtain: goo → ηoo =

c2; ∇2goo = ∇22φ according to Eq.(2.6), so that Eq.(2.15) transforms into:

∇2φ = 4πGρ (2.16)

which is the Poisson’s Equation, as soon as: k = 8πGc4

.To come back to the (E-H) Eqs.(2.12), they are characterized by:

• the term on the left is linear in the second derivatives of (E-H) potentialcomponents;

• the member on the right has a tetra-divergency: ∂µjµ = 0.

A simply translation of charge conservation.

Referring now to Eq.(2.13), we may pass from φµν → Gµν which is the Ein-stein tensor, linear in the second derivatives of gµν and, less than a constant,equal to Tµν characterized by a: Tµν;ν = 0. It may be proved that the tensorwhich satisfy this two conditions is a unique one.

In conclusion, it may be recognized that the Einstein tensor is the mostgeneral, the simplest one, the unique one (Longair, pg. 451; Straneo, pg.104,50 anni di Relativita’).

2.3 On the meaning of Einstein’s Equations

We turn again on the Einstein’s Equations:

Gµν(gµν) = kTµν (2.17)

Looking from the right side, Eq.(2.17) may read in this meaningful way:matter with all its form of energy tells to the space-time as it has to bend.

By reading it from the left side, Eq.(2.17) reads: it is the space-time whichtells to the matter how it has to move (Straneo) (see, Fig.2.2).

Chapter 3

Recombination

3.1 Recombination Epoch

At this epoch the breaking of thermodynamical symmetry occurs due tothe decoupling between matter and radiation. So the small initial densityperturbations may begin to increase.

3.1.1 When does it begin?

The bound energy for neutral H is:

BH ≃ 13.6eV → TH ≃ 1.6 · 105 K (3.1)

When T > TH the following reaction goes in both directions:

H+ + e− H + γ (3.2)

With the complicity of Universe’s expansion, as T < TH we lose this sym-metry. But when recombination starts the release of the bound energy causesthe heating of the plasma’s electrons, then T has to decrease in a meaningfulway in order that the symmetry effectively breaks. So the initial Ti of thephenomenon is:

Ti(≃ 0.3 eV ) →≈ 4000 K → z ≃ 1100 ÷ 1000 (3.3)

The degree of ionization is:

x =ne

ntot=

ne

np + nH(3.4)

where ni= number of particles per unit volume. It begins to decrease from1 (when nH = 0) downwards.

13

14 CHAPTER 3. RECOMBINATION

3.2 Typical photon-electron interaction time

How much is the number of electrons with which the incoming photon doesinteract in the time dt?During this infinitesimal time it spans the infinitesimal volume equal to:

σT · ds (3.5)

where the σT is the cross-section of the photon-electron interaction. Theinfinitesimal volume (3.5) contains a total number of interacting electronsequal to: σT · ds · ne. Then, the mean time for one reaction is given by:

τrem =dt

σT · ds · ne=

1

σT · c · ne(3.6)

At the energy levels we are considering the cross-section of the Compton’sscattering photon-electron, is that of Thomson:

σT =8 π

3

(

e2

me c2

)

≃ 0.665 · 10−28 m2 (3.7)

To be noted that in the plasma, protons and electrons are linked by theelectro-magnetic coupling constant, α, and in turn photons and electronsare coupled by the Thomson scattering so that matter and radiation get thesame temperature which is that of the radiation owing to the heat capacityof the radiation very much greater than that of the matter.

As soon as the ionized matter becomes neutral, the Thomson cross-sectionbecomes very low:

σT → σ′T =

(

me

mH

)2

σT (3.8)

me

mH

= β =1

1836≃ 0.5 · 10−3 → β2 ≃ 0.25 · 10−6 (3.9)

σ′T ≃ 3 · 10−7 σT (3.10)

Then we are dealing with two typical reaction times:τrem (σT , ne) τ ′

rem (σ′T , nH) corresponding to two parallel reaction channels

yielding a total reaction time:

1

(τrem)tot

=1

τrem

+1

τ ′rem

(3.11)

3.3. IONIZATION DEGREE 15

3.3 Ionization degree

The number of free electrons per unit volume ne is a function of the ionizationdegree x. How do we calculate x?The first idea is to use the Saha’s law, i.e.:

x2

1 − x=

1

ntot

(

me kB T

2 π ~2

)3/2

exp (− BH

KB T) (3.12)

But that is not a good idea, is a wrong idea!Saha is indeed derived from the Boltzmann’s statistics at the presence ofThermodynamical Equilibrium (T.E.). On the contrary we are looking forthe quantity x when the T.E. doesn’t hold. Under this condition Zeldovich& Sunyaev (1970) found:

x(z) ≃ 5.9 · 106 (Ωoh2)−1/2(1 + z)−1exp (−14590

z) ; 900 ≤ z ≤ 2000 (3.13)

The difference in respect to the results given by Saha’s equation are relevant:(3.13) produces a value of x greater also of a factor 102.

3.3.1 Typical time for ionization branch

We take now into account only the ionization branch of Eq. (3.11). For thatthe following equations hold:

ne = x(z)ntot(z); (3.14)

ntot =ρom

mp(1 + z)3; (3.15)

τrem ∼ 1

ne=

1

x(z)ntot(z)∼ 1

(Ωoh2)−1/2

1

(1 + z)−1exp(− constz

) · (1 + z)3(3.16)

∼ 1

z2exp(− constz

); 1 + z ≃ z; Ωoh

2 ≈ 1 (3.17)

• degree of ionization effect: x(z) ≃ 5.9 · 106 · 1zexp(− const

z);

• x ≃ 6 · 10−4 ÷ 1 in the range: 900 ≤ z ≤ 2000;

• τrem ր, about of 2 · 104 considering also that the Universe expands.


3.3.2 The other branch

As soon as z ց, x → 0, τrem → ∞ and then the R in the parallel goes to∞. τ ′

rem is decreasing, at fixed expansion, because nH increases so that theother branch of parallel begins to work. At the end is τ ′

rem which dominateswith the following value:

τ ′rem(900) ≃ 3 · 106τrem(2000) (3.18)

The interaction time is indeed longer for a factor of about 3 ·106 (see, Fig.3.1in the case Ωoh

2 = 1).

Figure 3.1: Interaction times, τrem, τ ′rem for Thomson scattering and expan-

sion time τexp of Universe vs. redshift z in the case: Ωoh2 = 1.

3.4 Expansion Time

We need to compare the interaction time with the expansion time in order tounderstand if matter and radiation have time enough to mantain the same

3.4. EXPANSION TIME 17

temperature. By definition we have that:

τexp =1

H=

a(t)

a(t)(3.19)

It means: texp at time t, is the time the Universe needs to expand itself withthe rate a(t), from 0 to the value a(t). We know (Coles & Lucchin, pg.43,1995) that the trend of Hubble constant is given by:

H2(z) = H2o (1 + z)2[Ωoω(1 + z)1+3ω + (1 − Ωo)] (3.20)

ω = 0 for matter; ω = 1/3 for radiation; (3.21)

Ωoω being the contribution to Ωo from ingredient ω (e.g., Ωor = 10−5). Bydisregarding, at first approximation, the term (1 − Ωo), we obtain:

•ω = 0; H2(z) ≃ H2

o · z2Ωom · z ≃ H2o · z3Ωom

Then:

τexp =1

H≃ 1

HoΩ1/2omz3/2

so that τexp ր as z ց.

τexp(z = 900)

τexp(z = 2000)≈ 3.3

•ω = 1/3; H2(z) ≃ H2

o · z2Ωor · z2 ≃ H2o · z4Ωor

which yields:τexp(z = 900)

τexp(z = 2000)≈ 5.0

In conclusion τexp increases less than an order of magnitude in respect tothe interaction time matter-radiation, τ ′

rem, which grows of more than sixorder of magnitude. But when expansion goes faster than the time matterand radiation have to speak together, their temperature can not remain thesame. The thermodynamical symmetry then no longer holds. The decouplingof matter from radiation occurs so that gravity may begin to act on matterwithout the ”‘viscosity”’ due to radiation. The small density perturbationsmay grow producing the macro structures in the Universe.


3.5 The last scattering surface

As soon as the recombination process is complete the photons do not prac-tically scatter against the neutral hydrogen atoms. Their mean free pathincreases more than the radius of Hubble sphere at this epoch. The Uni-verse becomes transparent to radiation. It occurs the contrary in respect tothe situation before recombination (see, Fig.3.2) when the numerous scattersconstrain the photons to have a small free path increasing the radiation opac-ity. It would have been impossible to do astronomy if this situation wouldhave persisted until our days.The last photon scattering occurs during the cooling of Universe due toexpansion, at about T = 5000 ÷ 3000K. We will refer at Trec ≃ 4000K(photon’s energy ≃ 0.26 eV , corresponding to trec ≃ 180000(Ωoh

2)−1/2 yearsand z ≃ 1100 even if the last scattering surface has a thickness of about∆z ≃ 0.07z.

3.5. THE LAST SCATTERING SURFACE 19

Figure 3.2: Mean free path of photons before and after recombination.

Chapter 4

The thermodynamical point ofview

4.1 The first Principle

The most general formulation of the Io Thermodynamical Principle in thedifferential form reads as:

dU = /dQ − /dL (4.1)

where:

• U=internal energy of system looking at it as a budget of energy;

• Q= amount of heat with sign + if accounted and − when extracted;

• L=work performed with sign + when done from the system and −when done on the system;

• /dQ, /dL non exact differentials.

4.1.1 Integrability condition

To handle with an exact differential, dσ(x, y), means that:

∫ 2

1

dσ(x, y) = σ(x2, y2) − σ(x1, y1)

i.e., does exist a primitive function σ(x, y) which solves the integral withno-dependence on the path between 1 and 2. If the contrary occurs this

21

22 CHAPTER 4. THE THERMODYNAMICAL POINT OF VIEW

kind of function doesn’t exist. The integrability condition for dσ(x, y) =(∂σ

∂x) dx + (∂σ

∂y) dy, is given by:

∂X

∂y=

∂Y

∂x→ ∂2σ

∂y∂x=

∂2σ

∂x∂y(4.2)

beeing:

X = (∂σ

∂x); Y = (

∂σ

∂y) (4.3)

In our case, e.g., for the work, the no-exactness of the differential, /dL, which,for an idrostatic system may be evaluated as p dV , it means:

∫ 2

1/dL doesn’t

solved by a function L(p, V ) which doesn’t exist. The value of integral maybe evaluated by many ways depending on the paths considered.

To clarify it, we consider the Clapeyron (1799-1864, French engineer)representation in the coordinates (p,V) (see, Fig.4.1). There are different

P0

2P0

V0 2V0

P

V

isocora

fb

isobaraai

Figure 4.1: The work done from the system when it moves from i to f dependson the different paths along which the integral L =

∫ Vf

Vip dV is performed.

paths to arrive to f starting from i:

• path: ia (isobar, P=const.)+ af (isocor, V=const.) → L =∫ Vf

Vip dV =

2PoVo;

• ” : ib (isocor)+ bf (isobar) → L =∫ Vf

Vip dV = PoVo;

4.1. THE FIRST PRINCIPLE 23

• ” : if → L =∫ Vf

Vip dV = (2Po + Po) · Vo

2, trapezium area;

In conclusion doesn’t exist a function of the thermodynamical coordinates:p, V, T of which /dL is the differential.

4.1.2 Entropy integration in thermodynamic coordi-

nates

Owing to the existence of an equation of state:

f(p, V, T ) = 0

only two of the thermodynamical coordinates p, V, T , are independent. If wedivide by T Eq.(4.1) the two differentials: /dL

T,

/dQT

become exact differentials;the second one that of entropy dS. Then Eq. (4.1) reads:

dS =1

TdU +

1

T/dL (4.4)

where S = S(V, T ), U = U(V, T ) so that the followings relations hold:

dS =1

T

[(

∂U

∂V

)

T

dV +

(

∂U

∂T

)

V

dT

]

+p

TdV (4.5)

=

[

1

T

(

∂U

∂V

)

T

+p

T

]

dV +1

T

(

∂U

∂T

)

V

dT (4.6)

dS = [1] dV + [2] d dT (4.7)

=

(

∂S

∂V

)

T

dV +

(

∂S

∂T

)

V

dT (4.8)

Constraining the function S by its integrability condition:

∂2S

∂T∂V=

∂2S

∂V ∂T,

we obtain:

∂

∂T

[

1

T

(

∂U

∂V

)

T

+p

T

]

=∂

∂V

[

1

T

(

∂U

∂T

)

V

]

(4.9)

− 1

T 2

[(

∂U

∂V

)

T

+ p

]

+1

T

[(

∂2U

∂T∂V

)

+

(

∂p

∂T

)

V

]

=1

T

∂2U

∂V ∂T(4.10)

Owing to the exactness of dU we have:

∂2U

∂T∂V=

∂2U

∂V ∂T(4.11)


and then the Eq.(4.10) becomes:(

∂U

∂V

)

T

= T

(

∂p

∂T

)

V

− p (4.12)

If u(T ) is the internal energy per unit of comoving volume, then U(T ) =u(T ) · V so that: ∂U

∂V= u(T ). Eq.(4.12) transforms into:

u(T ) + p = T

(

∂p

∂T

)

V

(4.13)

The last step to do is the entropy integration using Eq.(4.13) and startingfrom:

dS =1

TdU +

1

Tp dV (4.14)

According to the equation of state, p = p(V, T ), it follows:

dp =

(

∂p

∂V

)

T

dV +

(

∂p

∂T

)

V

dT (4.15)

from which we recognize that: ( dp)V =(

∂p∂T

)

VdT . Eq.(4.13) then becomes:

( dp)V =1

T(u(T ) + p) dT (4.16)

In Eq.(4.14) the following substitution may be performed: p dV = d(V ·p)−V dp, so that the Eq.(4.14) by Eq.(4.16) at the end becomes:

dS =1

Td(u · V ) +

1

Td(V · p) − 1

T 2(u(T ) + p) dT (4.17)

=1

Td[(u + p)V ] − V

T 2(u(T ) + p) dT (4.18)

Then the main result is:

dS = d[(u + p)V

T] (4.19)

4.2 Cosmological application

Entropy is an (extensive thermodynamical quantity so that when we handlewith two ingredients: matter and radiation, the total entropy, S, and its cor-responding entropy density per unit of comoving volume, s, is, respectively:

S = Sr + Sm (4.20)

s = sr + sm (4.21)

4.2. COSMOLOGICAL APPLICATION 25

taking into account Eq.(4.19), we have for the radiation:

sr =σT 4

r + 13σT 4

r

Tr

=4

3σT 3

r (4.22)

and for the matter:

sm =

pm

γ−1+ pm

Tm=

γ

γ − 1

pm

Tm=

5

2

ρm

mpkB (4.23)

owing to the equation of state: pm = kB

mpρmT with: γ = 5/3, for a monoatomic

gas. Eqs.(4.22,4.23) yields the ratio:

sr

sm=

8

3σT 3

r

mp

ρmkB· 1

5= 2 · σr

1

5(4.24)

The main result being that:

sr

sm

=(ρcV )r

(ρcV )m

· 1

5(4.25)

which tells us why is so great the radiation thermal capacity per unit volumein respect to that of matter. It is because the corresponding entropy densityratio is very high (∼ 108 ÷ 109)!

4.2.1 Physical meaning

We begin with the dimensional analysis of the terms involved in the ratio(4.25). sm ∝ ρm

mp· kB it means nprot. per unit volume time kB ≃, i.e., the

entropy of a proton (kB = 1.38 · 10−16erg/K). Then the ratio (4.25) giveshow many times, for each proton in each unit of volume the radiation entropyovercomes that of a single proton.

What is the further meaning of the ratio: σT 3

nprot.kB?

Its dimensions are: [ σT 3

nprot.kB] = erg

cm3·T 4 · T 3 · 11

cm3ergT

=1

cm31

cm3=

nfotons

nprot.;

σ =8π5k4

B

15·c3h3 = 7.55 · 10−15 ergcm3T 4 , being the Stefan-Boltzmann’s constant.

Then we have a so great unbalance between radiation/matter thermal ca-pacity and entropy per unit volume because we are in the presence of about108 ÷ 109 fotons for baryon in each unit volume.To be noted: the calculations of (ρcV )r, sr have been performed as we wouldhandle with a black body radiation. But as soon as we are at z < zrec matterand radiation are no longer in thermodynamical equilibrium so that it wouldbe no more correct to speak of black body radiation. But on the contrarythat is not true. Owing to the uge amount of the ratio of radiation thermalcapacity and entropy in respect to that of matter, radiation mantains itselfautonomous respect to matter which it contains like it would be a thermostat.

Chapter 5

Jeans’ Gravitational Instability

5.1 Introduction

The recombination epoch marks not only the breaking of thermodynamicalsymmetry but also the transition from a homogeneous cosmological fluidto a disomogeneous one on the macroscopic scale. The process starts fromthe very small fluctuations we may observe on the cosmological backgroundradiation map by COBE or WMAP (or Planck) space probes.

What is a possible mechanism to amplify these so tiny density fluctua-tions?At the beginning of twentieth century Jeans proposed his theory to explainthe stars and planets formation. Even though the theory is not completelycorrect when good re-interpreted (from Bonnor, 1957) it leads to exact re-sults.

5.2 Jeans’ Idea

By a quick compression of gas inside a piston a density perturbation doesarise. Two opposite mechanisms will be present: one to dump it, connectedwith the sound velocity, the other one to increase it. Indeed by getting nearthe molecules of gas during compression they begin to feel the mutual gravi-tational attraction. It is indeed very weack (among the four forces of nature,the gravitational one has the lowest coupling constant ≃ 10−39) but the at-tractive weackness may be compensated by a very long time available to itseffect. That may cause the beginning of an increasing center of coagulation.To understand the fate of the initial perturbation we need to compare thetypical times of the two mechanisms.

27

28 CHAPTER 5. JEANS’ GRAVITATIONAL INSTABILITY

5.2.1 Dump time

If λ is the typical spatial lenght involved by perturbation we may assumeit is locally dumped as soon as it has removed of one amount equal to theperturbed dimension by the sound wave itself generates.To do that the typical dumping time needed by sound velocity vso is:

τso =λdsdt

=λ

vso

(5.1)

where:

vso =

(

∂p

∂ρ

)1/2

S=const.

(5.2)

and the adiabaticity condition of compression is reflected by entropy S =constant.As well known in an adiabatic process, according to Io Thermod. Prin., itholds:

pV γ = cost. (5.3)

If v = 1ρ

is the specific volume (i.e., Vm

, the volume inside which there is the

mass unit), we have:

p = const.ργ −→(

∂p

∂ρ

)1/2

S=const.

=(

γργ−1const.)1/2

S=const.=

(

γp

ρ

)1/2

(5.4)

At fixed ρ :⇒ vso = const. ⇒ τso ր with λ ր.To be noted that τso is also the pressure response time:

τso =λ

(

γ pρ

)1/2∼ λ

(γT )1/2(5.5)

5.3 Collapse time

We learn from the cosmology what is the typical time a gas mass needs tocollapse under its auto-gravity. For a closed model of Universe we know thatthe time a homogeneous sphere without pressure, of density ρ at dimensionam takes to reduce itself to a point, i.e. free-fall time, is given by:

τff = tm =

√

3π

32Gρ(5.6)

5.3. COLLAPSE TIME 29

We may consider that as the shortest reaction time of gravity when it’s freeto act.1

We consider now the dynamical time as the time a unit mass point P layingon the surface of a spherical homogeneous mass distribution with density ρ,takes to reach the center free falling down according to the equation:

d2r

dt2= −GM(r)

r2= −4

3πGρr (5.7)

and pulsation given by:

ω =

√

4

3πGρ (5.8)

with oscillation period:

T =2π

ω=

2π√

43πGρ

=

√

4π2

43πGρ

=

√

3π

Gρ(5.9)

The result is:

tdyn =T

4=

1

4

√

3π

Gρ=

√

3π

16Gρ(5.10)

From Eq.(5.10) we recognize:

τff =1√2

T

4=

√

3π

32Gρ=

τdyn√2

(5.11)

It means the dynamical time is a multiple of τff which in turn is a submultiple

of√

πGρ

.

Whithout losing generality, we may assume here as typical collapse time:

τcoll =

√

π

Gρ(5.12)

We draw (Fig.5.1) as function of λ the two typical times for the two oppositemechanisms: τso, τcoll. A special value of spatial dimension arises: the Jeans’λ:

λj = vso

√

π

Gρ(5.13)

which corresponds to the condition τcoll = τso.Two possibilities are then available:

1Though we will meet a time shorter than τff during the violent relaxation phe-nomenon.


λλ j

τcoll

τdu

Figure 5.1: Typical dumping and collapse times as function of λ (τdu = τso).

• λ < λj , the pertubation may not increase, the dumping mechanism isfaster than collapse time;

• λ > λj , the pertubation has enough time to increase: collapse time isfaster than dumping time.

The existence of this last possibility is the great merit of Jeans.

5.4 Jeans’ Mechanism

To treat rigorously the problem from a idrodynamical point of view we needto consider the following equations:

∂ρ∂t

+ ~∇· (ρ~u) = 0∂~u∂t

+(

~u· ~∇)

~u + 1ρ~∇p + ~∇φ = 0

∇2φ = −4πGρ

(5.14)

which, in newtonian approximation for a perfect gas are: Eq. of continuity,Eq. of Eulero, Eq. of Poisson, respectively. To these we add the Eq. ofentropy conservation:

∂s

∂t+(

~u· ~∇)

s = 0 (5.15)

5.4. JEANS’ MECHANISM 31

where:

ρ = fluid density~u = velocity (eulerian)s = specific entropyφ = gravitational potential

(5.16)

and:~∇ =

∂

∂x~i +

∂

∂y~j +

∂

∂z~k

The set of Eqs.(5.14) has the static solution:

p = po = const.; ρ = ρo = const.; ~u = 0; ~∇φ = ~g = 0; (5.17)

To be noted that the last term, which corresponds to φ = const., shows amanifest contradiction with Poisson’s Eq. spoiling of correctness this solu-tion.Nevertheless we take into account the following perturbed solutions of thestatic ones:

p′ = po + δp; ρ′ = ρo + δρ; ~u′ = ~u1 6= 0; ~g′ = ~g1 6= 0; (5.18)

under the following conditions:

δp

po≪ 1;

δρ

ρo≪ 1; u1, g1 = infinitesimals (5.19)

and we wonder under which conditions the perturbed quantities (5.18) aresolutions of the set (5.14).The condition turns to be that, δ = δρ

ρhas to be solution of the following

equation:∂2δ

∂t2− v2

so∇2δ − 4πGρoδ = 0 (5.20)

What is its physical meaning?We look at it by isolating the first and third terms:

∂2δ

∂t2− 4πGρoδ = 0 (5.21)

Its general solution is:δ = C1e

q1t + C2eq2t (5.22)

where (q1, q2) are solutions of the characterist algebric equation:

q2 − 4πGρo = 0 −→ q1,2 = ±√

4πGρo (5.23)


Then the general integral reads:

δ = C1e√

4πGρot + C2e−√

4πGρot (5.24)

where the two terms represent the increasing mode (the first one) and thedecreasing mode (the second one) of perturbation. The last one practicallydisappears after a given time, the first one allows the coagulation process.Then we may regard at Eq.(5.20) as an equation of collapse disturbed bythe presence of a laplacian term which typically enters the wave propagationequation.By regarding only to the first and second term of Eq.(5.20), we are indeedin front of exactly the d’Alembert equation:

∂2δ

∂t2− v2

so∇2δ = 0 (5.25)

Then we may look at the Eq.(5.20) also in a complementary way , i.e., asto an equation of sound wave propagation perturbed by a typical collapseterm 4πGρoδ. Owing to the possibility to develop any wavy perturbation inFourier series of planar waves we look for what is the condition a function ofthis kind:

δ = δo exp[

i(

~k · ~r)

− iωt]

(5.26)

may be a solution of Eq.(5.20).The wave vector is:

|~k| =2π

λ;

the pulsation: ω = 2πT

. To be noted that: ω 6= ωso = 2πτso

that is the purepulsation of a sound wave solution of Eq.(5.20) without the collapse term.It turns that the planar wave (5.26) is solution of Eq. (5.20) if and only if:

ω2 = v2sok

2 − 4πGρo (5.27)

The Eq. (5.27) is named dispersion relation or secular equation.What is its physical meaning?

We re-write (5.27) as follows:

(2π)2

T 2= v2

so

(2π)2

λ2− 4πGρo (5.28)

and multiply the previous Eq. (5.28) by the factor: 1(2π)2

, obtaining:

1

T 2=

1λ2

v2so

− Gρo

π(5.29)

5.4. JEANS’ MECHANISM 33

Finally, the dispersion Eq. (5.27) becomes:

1

T 2=

1

τ 2so

− 1

τ 2coll

(5.30)

that is a simple relationship among typical times.Writing down Eq. (5.30) as follows:

τ 2so

T 2= 1 − τ 2

so

τ 2coll

→ T 2

τ 2so

=1

1 − τ2so

τ2coll

(5.31)

which tells us:τso ≪ τcoll → T ≃ τso (5.32)

τso → τcoll (i.e., λ → λj) T → ∞ (5.33)

It means the period of sound wave is only a little bit disturbed when λ isvery short in comparison with λj, on the contrary the dumping time increasesenormously near λj and the pulsation: ω = 2π

T→ 0.

To put in evidence the pulsation’s trend as function of λ, we start again fromEq. (5.30) manipulating it as follows:

(2π)2

T 2=

(2π)2

τ 2so

− (2π)2

τ 2coll

· λ2

λ2(5.34)

Remembering that: λ = vsoτso and taucoll =λj

vso, Eq. (5.30), defining ωso =

2πτso

, reads:

ω2 = ω2so −

(2π)2

λ2j

v2so

· λ2

v2soτ

2so

(5.35)

which in turn yields:

ω2 = ω2so −

(2π)2

τ 2so

· λ2

λ2j

=⇒ ω2 = ω2so

[

1 −(

λ

λj

)2]

(5.36)

The pulsation ω is real only when λ < λj. In this range the term(

λλj

)2

plays a role to deviate ω from ωso, affecting it more and more as λ increasestoward λj , where ω becomes 0 and T goes to ∞.A meaningful trend for ω may be also deduced starting from Eq. (5.27)which easily becomes:

ω2 = −4πGρo

(

1 − v2sok

2

4πGρo

)

(5.37)


After a bit of manipulation it follows:

ω2 = −4πGρo

(

1 − v2so4π

2

λ24πGρo

)

= −4πGρo

[

1 −(

λj

λ

)2]

(5.38)

where it appears as the collapse term 4πGρo has been affected less and lessat increasing λ and as an imaginary value of ω is reached as soon as λ > λj:

ω = ±i√

4πGρo

[

1 −(

λj

λ

)2]1/2

(5.39)

In this case, coming back to the Eq.(5.26) we obtain:

δ = δo exp

i(

~k · ~r)

− i2

±√

4πGρo

[

1 −(

λj

λ

)2]1/2

t

(5.40)

which in a more meaningful way transforms into:

δ = δo exp[

i(

~k · ~r)

± |ω|t]

(5.41)

showing how a planar wave transforms into a stationary wave with an expo-nentially increasing amplitude (increasing mode; or exponentially decreasingone, decreasing mode). The net result is the collapse of the initial densityperturbation.

5.5 Gravitational Jeans’ Instability

in an expanding Universe

The theory of Jeans’ instability has been reformulated by Bonnor (1957) whoembedded it in an expanding Universe so taking off the inconsistency presentinside the original formulation.The zero-order approximation is no longer a static solution with ~∇φ = 0 butthe following one characterized by the expansion parameter a(t):

ρ = ρo

(

ao

a

)3

~v = aa~r

φ = 23πGρr2

(5.42)

where r is the physical spatial coordinate linked to the comoving one ao, by:

~r = ~roa(t)

ao(5.43)

5.6. INCREASING RATE IN RADIATION/MATTER DOMINATED UNIVERSE35

The set (5.42) describe an expanding homogeneous and isotropic fluid.Let us perturb this zero-solution of infinitesimal variations: δρ, δ~v, δφ, andlook again for the condition under which the perturbed quantities are solu-tions of equation set (5.14) once translated for an expanding Universe. Thecondition is that: δ = δρ

ρhas to obey instead of Eq.(5.20), to the following

one:

δ − v2so∇2δ + 2

a

aδ − 4πGρoδ = 0 (5.44)

where it is manifest how the sign due to expansion term is opposite of thatwhich is in front of the collapse one. Expansion appears to be antagonistin respect to the coagulation process. So we have to expect that also themodality to obtain the collapse of a density perturbation, will be mathe-matically expressed by an amplitude of the stationary wave, which increasessofter than the previous exponential one.The solution of Eq.(5.44) in the case of λ greater than the new Jeans’ λ equal

to λ′j =

√245

λj , turns indeed to be:

δ = δo

[

exp(

i~k · ~r)]

tn (5.45)

where the exp |ωt| is substituted by the power term tn.

5.6 Increasing rate in radiation/matter dom-

inated Universe

The way according which the density perturbation increases in an expandingUniverse depends on the density parameter Ω, characterizing the Friedmann’smodel, and by the ingredient (radiation or matter) which dominates.It turns indeed that:

n = n(λ, Ω, w) (5.46)

where w enters the state equation: p = wρc2, with w = 0 for matter andw = 1

3for radiation.

So in the case:

n = n(λ > λ′j, 1, 0) (5.47)

we have:

n = −1

6± 5

6

[

1 −(

λ′j

λ

)2]1/2


It means that in a EdS model (Ω = 1) characterize by:

a = ao

(

tto

)2/3

ρ = (6πGt2)−1

aa

= 23t−1

(5.48)

the density perturbation (with λ >> λ′j) in the matter (w = 0) does increase:

δ ∼ t2/3 (increasing mode) exactly as a = a(t)

δ ∼ t−1 (decreasing mode)

When λ < λ′j , the sound waves become:

δ ∝ 1

t1/6· exp

(

i~k · ~r)

± i5

6

[

(

λ′j

λ

)2

− 1

]1/2

· ln t

(5.49)

where the amplitude decreases instead to remain constant as in a static Uni-verse.In the radiation dominated Universe (w = 1/3), the density perturbations(with λ >> λ′

j) increase as:

δ ∼ t+1 (increasing mode) (5.50)

δ ∼ t−1 (decreasing mode) (5.51)

When λ < λ′j , δ(t) remains about constant.

5.7 Appendix

5.7.1 Lagrangian and eulerian point of view

When we are handling with a velocity and/or acceleration field, we mayobserve it from two possible points of view.If we follow the fluid element as a moving point P and then consider itsvelocity as:

~vL = ~vL (P (t)) ≡ ~vL (x(t), y(t), z(t)) ≡ dP

dt≡ (x, y, z) (5.52)

we are looking at it from the lagrangian point of view.On the contrary, if we put ourself on a given fixed point P and associate atit the velocity of the fluid element which at time t passes across it, then:

~uE = ~uE (P (x, y, z), t) (5.53)

5.7. APPENDIX 37

and we are looking at the velocity field from an eulerian point of view.But how to derive the acceleration lagrangian field?

From Eq.(5.52), immediately it follows:

~aL =d~vL

dt=

d2P

dt2≡ (x, y, z) (5.54)

Starting from Eq.(5.53), we obtain:

~aL =

(

∂~uE

∂t

)

P=const.

+ ∆ (5.55)

where ∆ has to give the contribution to the acceleration for the variation of~uE when the fluid element moves from P → P + dP , during the time dt.Then we have to perform the total derivative of ~uE respect to time t. Ityields:

~aL =∂~uE

∂t+

∂~uE

∂x

∂x

∂t+

∂~uE

∂y

∂y

∂t+

∂~uE

∂z

∂z

∂t(5.56)

The previous equation may be also put in the form:

~aL =∂~uE

∂t+

∂~uE

∂x

dPx

dt+

∂~uE

∂y

dPy

dt+

∂~uE

∂z

dPz

dt(5.57)

By remembering that:

dP = ~uEdt → dP ≡ [(~uE)xdt, (~uE)ydt, (~uE)zdt]

the Eq.(5.57) reads as:

~aL =∂~uE

∂t+(

~uE · ~∇)

~uE (5.58)

with:

~uE = ux~i + uy

~j + uz~k

~∇ = ∂∂x

~i + ∂∂y

~j + ∂∂z

~k(5.59)

In other words, e.g., the x-lagrangian acceleration component is:

ax =∂ux

∂t+

(

ux∂

∂x+ uy

∂

∂y+ uz

∂

∂z

)

ux (5.60)

or equivalently:

ax =∂ux

∂t+ ux

∂ux

∂x+ uy

∂ux

∂y+ uz

∂ux

∂z(5.61)

Chapter 6

Density Perturbations

6.1 Density perturbations

The Jeans’ mechanism is able, under some constraints, to amplify the initialdensity perturbations.

But we wonder: what is the spectrum of primordial density fluctuations?In other words: how would be they distributed as function of λ from thebeginning?It is resonable (see, notes of: Cosmology, Modulus A, of G.Tormen.) toassume at Planck time, tp, an initial perturbation spectum as:

P (k, tp) = Apknp ; k =

2π

λ; Ap, np = const. (6.1)

where np is the primordial spectrum index. What is the physical meaning ofP (k)?We noted that: P (k)d3k → δρ

ρ, so that, by dimensional point of view:

[P (k)] = k−3, it means a density in Fourier space, or a power of fluctua-tions per unit volume in the same space.We choose now a point: ~x. Let us calculate in it the density contrast as:

δ(~x) =ρ(~x) − ρbg

ρbg(6.2)

that is the relative difference between the value of density ρ(~x) due to allfluctuations in ~x, and the umperturbed density value of the background: ρbg.To evaluate the mean quadratic square of δ(~x) we have to know the varianceσ2 of the random processes acting in ~x, i.e., we would need to measure δ(~x)for different realizations of Universe. To be noted that we are looking forσ2(~x) which, owing to the Cosmological Principle, is independent of ~x.

39

40 CHAPTER 6. DENSITY PERTURBATIONS

But available is an unique Universe, i.e., an unique realization of it, then weneed of ergodic hypothesis:regions of Universe which are enough separated from each other are statisti-cally independent.Then referring to them is equivalent to consider different realizations of thesame stocastic process. We divide the Universe in a great number of volumes(equivalent to dealing with independent realizations) under the condition:ΣVi = V∞, and perform the evaluation of σ2 as follows. In the generic vol-ume Vi we consider:

δi(~x) =ρi(~x) − 〈ρ〉i

〈ρ〉i(6.3)

where 〈ρ〉i is the density mean value inside the volume Vi.Although in the same volume δi(~x) 6= δi(~x′), the following linear mean in itturns to be:

∫

δi(~x) d3~x∫

d3~x= 0 (6.4)

To obtain a mean entity of the contrast in Vi we have to pass to the quadraticmean density contrast as:

δ2im =

∫

δ2i (~x) d3~x∫

d3~x6= 0 (6.5)

where ~x drops. After that we perform the weighted mean of δ2im taking as

weights the respective volume Vi as:

∑

i Vi

∫

δ2i (~x) d3~x∫

d3~x∑

i Vi = V∞=

∑

i

∫

δ2i (~x) d3~x

∑

i Vi = V∞(6.6)

Passing in Eq.(6.6) from the summation over i:∑

i to the integral:∫

d3~x,i drops and the result becomes:

σ2 = 〈 1

V∞

∫

δ2(~x) d3~x〉 (6.7)

The meaning of 〈〉 is wheighted mean over∑

or∫

. So the meaning ofσ2 is the mean weighted value of

∫

δ2i (~x) d3~x, over all the realizations (or

independent volumes) of Universe. σ2 is named punctual variance and V∞the volume of Universe.Each contrast value relative to the volume Vj , may be developed in Fourierseries as:

δj(~x) =∑

~k

δj~k(~x) ei~k·~x (6.8)

6.2. TO HELP THE COMPREHENSION 41

with the following periodicity conditions on the considered volume L3 :

kx = nx2πL

ky = ny2πL

kz = nz2πL

(6.9)

It is to be noted that:

• the∑

~k means: over all the possible directions and once fixed one ofthem, over all the λ values.

• The δj~k

is a complex number characterized by modulus and phase in

this way: δj~k

=∣

∣

∣δj~k

∣

∣

∣eiφ~k . Moreover:

δj~k

=1

Vj

∫

Vj

δj(~x)e−i~k·~x d~x (6.10)

they are the Fourier transform of δj(~x).

If we are dealing with a gaussian field, phases and modules are randomdistributed (stocastic field) and according to the Cosmological Principle: ~k →k. Taking into account the Fourier transform: δ(~k) of δ(~x) and its conjugate:

δ∗(~k) = δ(−~k), due to the reality of the field δ(~x), at the end the Eq. (6.7)of punctual variance transforms into:

σ2 =1

(2π)3

∫

P (k) d3k =1

2π2

∫ ∞

0

P (k)k2 dk (6.11)

remembering that in Fourier space: d3k = 4πk2dk.

6.2 To help the comprehension

Without ambition to prove the result (see, e.g., Coles & Lucchin, 1995), onlyintuitively we may understand what there is inside P (k).To build up Eq. (6.6) we need of products of this kind : δj(~x)2 ⇒ δj

k ·(δjk)

∗ byusing the Fourier transform of δj(~x). Similarly the Fourier transform enters

(6.7) to transform 〈δ2(~x)〉 ⇒ 〈δ(~k)δ∗(~k)〉 so that, in Fourier space we obtain:

σ2 =1

(2π)3V∞

∫

d3k〈δ(~k)δ∗(~k)〉 =1

(2π)3

∫

d3kP (k) (6.12)

That means:

P (k) =〈δ(~k)δ∗(~k)〉∑

i Vi = V∞(6.13)


6.3 Mass variance - Io filter

It is to stress again that the variance given by (6.11): σ2 = σ2(~x, t) becomesσ2(t) thanks to the Cosmological Principle. It is a measurement of how muchis perturbed the Universe on any point due to the contribution of the densityperturbations on all λ scales.We need now of an other statistical quantity which allow us to understandwhat is the mean square entity of density fluctuations on a fixed sphericalvolume of radius R. In other words, the fluctuations on the scale R cause themass value M(R) to change. To build up the mass variance at the lenght Rwe will follow the same line of reasoning used for the punctual mass variance.Again taking into account the ergodic hypothesis we perform the followingquantities in the spherical volume V1:

〈M〉1 =4

3π〈ρ〉1R3 (6.14)

M1 =

∫

V1

ρ(~x) dV (6.15)

(

δM

M

)

1

= δM1 =

M1 − 〈M〉1〈M〉1

(6.16)

Then in the generic volume Vj we have:

〈M〉j =4

3π〈ρ〉jR3 (6.17)

Mj =

∫

Vj

ρ(~x) dV (6.18)

(

δM

M

)

j

= δMj =

Mj − 〈M〉j〈M〉j

(6.19)

The next step is: consider all the mean square values (δMj )2 and make their

mean square weighting each of them by the volume Vj to obtain:

∑(

δMj

)2 · Vj∑

Vj = V∞=

∑(

δMM

)2

j· Vj

V∞= σ2

M (6.20)

σ2M is the mass variance on the scale R (or M)1. It is a measurement of the

contribution given by the perturbations on the assigned scale. To extract σ2M

1Sometimes the same name is used improperly for σM .

6.4. THE SCALE-FREE POWER SPECTRUM 43

from P (k), one needs of a filter by which to select, among the perturbationson the all lambda-scales, those relevants on the dimension R. It is indeed:

σ2M =

1

(2π)3

∫

P (k)W 2(kR) d3k (6.21)

The filter W 2(kR) is named the Window Function with the following math-ematical expression:

W (kR) =3(sin kR − kR cos kR)

(kR)3(6.22)

It is to be noted that:

• W 2 ց, when, kR ր, i.e., the λ ≪ R get destructive interference;

• σ2M → σ2(~x) as soon as R → 0.

6.4 The scale-free power spectrum

Starting from a scale- free power spectrum like that of Eq. (6.1), where theprimordial exponent np is indpendent of the λ scale, we obtain for the massvariance on scale R at the time tp:

σ2M(tp) =

1

2π2

∫ ∞

0

P (k, tp)W2(kR)k2 dk (6.23)

which by manipulations transforms into:

σM (tp) = Kp

(

M

MH(tp)

)− (3+np)

6

∝ M−αp (6.24)

Kp ≃ 10−5 = proposed by Zeldovic;MH(tp) = mass entering the horizon at time tp;αp = (3 + np)/6 = mass index.

(6.25)

Generally speaking, when Jeans’ instability works without effects due tomicrophysical processes, it turns to be:

σM(t) ∝ D(t)M− (np+3)

6 (6.26)

where D(t) is the linear growing factor, so that:

δ(~x, t) = δ(~x)D(t) (6.27)

It means, in the linear regime (i.e., δ < 1), as soon as λ ≫ λj, the perturba-tions grow in a self-similar way on all λ scales.


6.5 Transfer Function - IIo filter

The initial power spectrum (6.1) has tuned by the microphysical processeswhich occurs at different rates for the different λ scales during the cosmicevolution. To get information about the modified P (k, t) we introduce theTransfer Function T 2(k, t), which works as follows:

P (k, t) =

[

D(t)

D(tp)

]2

T 2(k, t)P (k, tp) (6.28)

In the case of EdS’ Universe (Ω = 1), when it is matter dominated (ω = 0),the linear growing factor: D(t) ∼ t2/3 ∼ a(t). So, after ricombination, wehave again:

P (k, t) = cost. · kne (6.29)

with an effective index ne, no more the same for the all wave numbers k, asin (6.1), but with different values in a given range ∆k (→ ∆M) (see, nextsection).

6.6 Initial power spectrum

Different values for the primordial np may be chosen:

• np = 0 → white noise spectrum. The amplification of perturbations onthe galaxy scale turns to be in excess.

• np = 1 Harrison-Zeldovic spectrum.

The mass index α becomes 2/3, so that the mass variance turns out to be:

σM( δMM

) ∼ M− 23 . Following the line of reasoning to reach Eq. (6.20) by

using instead of δMM

the corresponding variation of gravitational potentialδφ = φ δM

M, owing to the following links:

φ =GM

R−→ δφ

φ=

δM

M(6.30)

we obtain the variance of gravitational potential:

σ(δφ) = φσ

(

δM

M

)

∼ M

RM− 2

3

Once fixed the mean density value 〈ρ〉 of Universe at the time t, it occursthat

〈ρ〉 ∼ M

R3⇒ R ∼ M

13 (6.31)

6.7. EFFECTIVE INDEX 45

σ(δφ) ∼ M

M13

M− 23 ∼ M

23− 2

3 = cost. (6.32)

It means that the gravitational potential variance turns to be the same on allthe mass scales. Due to this property the Harrison-Zeldovic spectrum turnsthen to be not only scale -free but also scale-invariant.

6.7 Effective index

6.7.1 CDM scenario

To examplificate how to determine the effective spectrum index ne at recom-bination epoch, we will take into consideration the mass variance σM(trec, M)for a CDM scenario (Gunn, 1987) (Fig.6.1). The mass index α(trec) turns tobe linked to ne through the local logarithmic slope of σM , as follows:

α(trec) = − d ln σM(trec)

d ln M=

(ne + 3)

6(6.33)

Given an interval ∆M the value ne ≃ const., so that again it holds:

σM ∼ M−α (6.34)

Figure 6.1: The trend of mass variance σMD, at recombination epoch, as

function of MD, for a CDM scenario. The normalization is to the presentepoch. The ticks along the line mark values of the effective spectral index(Gunn, 1987).


6.7.2 ΛCDM scenario

We will refer to the cosmological standard model derived from WMAP preci-sion data only (Spergel et al., 2003; Spergel et al., 2007; Binney & Tremaine,Chapt.9, 2008) for a flat Λ-dominated universe: i.e., ΛCDM model definedby the following parameters: σ8 = 0.9± 0.1; h = 0.72± 0.05; matter density,Ωmh2 = 0.14±0.02, baryon density, Ωbh

2 = 0.024±0.001 and primordial spec-tal index ns = 0.99 ± 0.04 which is consistent with the Harrison-Zel’dovichscale-invariant value (ns = 1). The trend of mass-variance spectrum σMD

atequivalence between matter and radiation (after that epoch the microphysicsmay never affect the DM fluctuations) is shown in Fig.6.2 as function of MD.

1

10

Figure 6.2: The trend of mass variance σMD, at equivalence epoch, as function

of MD, for a flat Λ−dominated universe derived from WMAP precision data(ΛCDM model defined by the following parameters: σ8 = 0.9 ± 0.1; h =0.72± 0.05; Ωmh2 = 0.14± 0.02; Ωbh

2 = 0.024± 0.001; ns = 0.99± 0.04 (see,text).

6.7. EFFECTIVE INDEX 47

Again in this self-similar model, an effective final spectrum index, ne, maybe obtained by local slope of σMD

at a given MD value, as follows:

αeq = − d log σMD(teq)

d log MD= (ne + 3)/6 (6.35)

Moreover a meaningful quantity is also:

1/γ′ =(5 + ne)

6=

3αeq + 1

3(6.36)

0.1

0.12

0.14

0.16

0.18

0.2

1.8

1.9

2

2.1

2.2

2.3

Figure 6.3: The trend of the local slope, αeq, Eq. (6.35) (top panel) and ofγ′, Eq. (6.36) (bottom panel) as function of DM halo mass in the mass rangehere considered.

Chapter 7

Mini-universe

The matter1 is able to form structures only when z < zeq, i.e., if radiationdoesn’t dominate, owing to Mezsaros effect. Moreover the density contrastδ = δm may increase only after the epoch of decoupling between matter andradiation (trec), owing to Jeans’ mechanism is able to work without radiationdrag, in the linear regime (δ < 1).

The problem is how to follow the density perturbation when it enters thenon-linear regime?

The method is that of autosolution based on a paper of Barenblatt &Zeldovic (1958) (see: Coles & Lucchin, 1995, pg. 212). Dealing with aperturbation on a scale λ >> λj, the dumping mechanism is negligible (p ≃0) respect to the self-gravity, so that it becomes possible to simulate theperturbation evolution by a mini-universe.

It means: the density perturbation, increased by Jeans’ instability, maybe assimilated to a spherical, homogeneous and isotropic perturbation fol-lowing the top hat approximation (Fig.7.1) which, in turn, evolves as a Fried-mann’s model characterized by parameters different from those of the unper-turbed Universe containing it.

To this aim we have to remember the existence of Birkhoff’s theorem (see:Coles & Lucchin, 1995, pg. 212), the relativistic analogue of the Newton’sfirst theorem, which preserves the mini-universe from the influence of externalmatter and energy distribution.

The density parameter of this mini-universe, corresponding to the top-hatperturbation, is given by:

Ωp(ti) =ρu(ti) + δiρu(ti)

ρc(ti)= Ωu(ti)(1 + δi) (7.1)

1To be intended the dark matter (hereafter DM) in a DM dominated scenario.

49

50 CHAPTER 7. MINI-UNIVERSE

Figure 7.1: Top-hat approximation. The evolution of the typical overdensityon the scale ap is assimilated to a closed mini-universe homogeneous andisotropic of density ρp embedded in the Universe of density ρu.

where subscripts u, p mean: Universe and perturbation, respectively. Thedensity contrast δi at time ti has been produced by Jeans’ mechanism startingfrom a given initial perturbation value δ(tio), as follows:

δi = δ+(tio)

(

titio

)23

+ δ−(tio)

(

titio

)−1

; δi < 1 (7.2)

where δ+ is the increasing mode and δ− the decreasing one until tio, so thatafter a short time:

δi ≃ δ+(tio)

(

titio

) 23

(7.3)

In this context the main aim is to supplement the linear power spectrumregime by approximate analytic arguments to follow structure formation intothe non-linear regime. So we may think at δi simply as to a typical overden-sity on a given scale ai without any other details (see, Binney & Tremaine,2008, pg.733).

7.1 Condition to form a structure

Looking for a model of a proto-structure collapse we ask that, at given time,the region involved by perturbation has to stop its expansion with the Uni-verse, i.e., to follow the Hubble flow, for beginning an inverse collapse process

7.1. CONDITION TO FORM A STRUCTURE 51

under its self-gravity. In other words we ask that the mini-universe is closed,which means: Ωp > 1.

From Eq.(7.1), the condition reads:

Ωu(ti)(1 + δi) > 1; Ωu(ti) 6= Ωp(ti) (7.4)

which becomes:

δi >1 − Ωu(ti)

Ωu(ti)(7.5)

The (7.5) doesn’t be a constraint in both cases: Ωu > 1; Ωu = 1. Only in thecase of an open Universe it becomes actually a constraint. Indeed, by simplyalgebric manipulation of IIo Friedmann equation, we know that:

1 − Ωu(ti)

Ωu(ti)=

1 − Ωo

Ωo(1 + zi)1+3ω(7.6)

where Ωo is the actual value of density parameter and ω refers, as usual, tothe ingredient we are dealing with at the time ti corresponding to zi. Becausewe are in the post-recombination phases: ω ≃ 0. Then (7.5) transforms into:

δiΩo(1 + zi) > 1 − Ωo (7.7)

which becomes:

Ωo >1

δi(1 + zi) + 1(7.8)

or:

Ωo >1

δizi + 1(7.9)

due to the great value of zi. Its physical meaning is that the structureformation constrains how much open may be the Universe. Indeed let usmade this exercise: zi ≃ 1500 ≈ zrec. If :

δi ≃ 10−2 ⇒ Ωo >1

16≃ 0.06 (7.10)

δi ≃ 10−3 ⇒ Ωo >1

1 + 1.5≃ 0.4 (7.11)

δi ≃ 10−4 ⇒ Ωo > 0.87 (7.12)

The message is: as soon as lower is the value of δi less open has to be theUniverse in order to produce structures.


7.2 The closed mini-universe

To assimilate a top-hat perturbation with a closed mini-universe character-ized at the time ti by a density parameter (7.1), we have to translate thesolution of Friedmann equation for a closed Universe, given in parametricform, as follows:

a(θ) = aiΩp(ti)

2(Ωp(ti)−1)(1 − cos θ); 0 ≤ θ ≤ 2π

t(θ) = 12Hi

Ωp(ti)

(Ωp(ti)−1)32(θ − sin θ)

(7.13)

which is the parametric equation of a cycloid, i.e., the curve describes bya point on a circumference rolling without slithering along the t axis. Themaximum of the expansion parameter a(t) is reached at tm corresponding toθ = θm = π (see, Fig.7.2). By puting Ωp(ti) = Ωpi, the maximum expansionphase is given by:

am = a(θm) = aiΩpi

2(Ωpi−1)· 2

tm = t(θm) = 12Hi

Ωpi

(Ωpi−1)32· π

(7.14)

The aim is now to link tm at the maximum expansion density value of per-turbation ρm and then to look for the ratio between this density and that ofthe Universe at the same time tm.

Figure 7.2: Evolution of the density parameter of the closed mini-universecharacterized by the value ai(ti). The maximum expansion phase occurs atam(tm).

7.2. THE CLOSED MINI-UNIVERSE 53

Again from the IIo Friedmann equation, we obtain:

(

a

ai

)2

= H2i

(

Ωpiai

a+ 1 − Ωpi

)

= 0 (7.15)

Remembering that: am=0, the (7.15) yields:

Ωpiai

am+ 1 − Ωpi = 0 (7.16)

In the matter dominated mini-universe:

ω ≃ 0 → p ≃ 0 → ρ

ρi=(ai

a

)3

(7.17)

which, using (7.16), transfroms into:

ai

am=

(

ρm

ρi

)13

=Ωpi − 1

Ωpi(7.18)

Then by definition of the density parameter: Ωpi = ρu(ti)+δiρu(ti)ρc(ti)

= ρi

ρc(ti), we

obtain that:

ρm = ρc(ti)Ωpi

(

Ωpi − 1

Ωpi

)3

(7.19)

To link it to tm given by the second equation of system (7.14), we buildup the quantity:

ρ12m · tm = ρc(ti)

12 Ω

12pi

(

Ωpi − 1

Ωpi

)32 π

2Hi

Ωpi

(Ωpi − 1)32

(7.20)

= ρc(ti)12

π

2Hi(7.21)

By definition:

ρc(ti) = H2i

3

8πG−→ ρ

12c

Hi=

(

3

8πG

) 12

(7.22)

Then in conclusion:

ρ12m · tm =

(

3

8πG

)12 π

2(7.23)

which yields the definition of the free fall time, τff :

tm =1

ρ12m

(

3π2

32πG

)12

=

(

3π

32Gρm

)12

= τff (7.24)


Then we recover the relationships with dynamical time, τdyn and collapsetime, τcoll:

τff ≃ 1√2τdyn ≃ 0.3τcoll (7.25)

already met in Chapter 5.

7.3 Perturbation density at maximum expan-

sion vs. Universe density

In the case of an EdS Universe, the trend of the density is given by:

ρu(t) =1

6πGt2; z < zrec (7.26)

According to the (7.26), at time tm,u evaluated from the singularity, Universedensity is:

(ρu)m ≃ 0.0531

Gt2m,u

(7.27)

On the other side at the maximum expansion phase the mini-universe reaches:

(ρp)m ≃ 0.291

Gt2m,p

(7.28)

according to (7.24), where the tm,p has its origin at trec from which theJeans’ mechanism may amplify the perturbations without radiation drag.Disregarding the difference between the origins of the two times, tm,u andtm,p, because the typical time for the expansion phase is of the order ofabout 108 years in comparison with the trec of the order of about 100.000years, the density ratio between (7.28) and (7.27) becomes:

(ρp)m

(ρu)m

≃ 5.55 (7.29)

for all the proto-structures. To be noted that tm,p changes by changing thescale ai at which the typical overdensity δi refers, because (ρp)m changes.But, e.g., if the mean perturbation density increases, tm,p and tm,u have bothto decrease and this last, in turn, produces a greater value of (ρu)m.

Following the perturbation according to the linear approximation, theresult would be:

(ρp)m

(ρu)m

≃ 2.06 (7.30)

7.4. PERTURBATION DENSITY AT VIRIALIZATION VS. UNIVERSE DENSITY55

corresponding to a density contrast at the maximum expansion, equal to:

(ρp)m − 〈ρu〉m〈ρu〉m

≃ 1.06 (7.31)

7.4 Perturbation density at virialization vs.

Universe density

In Chapter 8 we will see as virialization of the proto-structure occurs at aradius equal about one half of its maximum expansion radius am = Rm. Tofind the contrast between perturbation and Universe density, we need to knowat which time the mini-universe reaches the value of expansion parameterequal to 1

2am = 1

2Rm.

We start again from (7.13). For sake of simplicity we put:

A = aiΩp(ti)

2(Ωp(ti)−1)

B = 12Hi

Ωp(ti)

(Ωp(ti)−1)32

(7.32)

so that:

Rm = 2A

tm = πB(7.33)

We are looking for the t (see, Fig.7.3), corresponding to θ, at which a = 12Rm

or, in other words, for the solution of the system:

Rm

2= A(1 − cos θ)

t = B(θ − sin θ)(7.34)

Dividing the first and second equation of (7.33) by the first and secondone of (7.34), respectively, we obtain:

2 = 2(1−cos θ)

→ θ = 3π2

t/tm = 1.5 + π−1 → t ≃ 1.82 tm

(7.35)

Then the perturbation density at virialization phase (t = tvir) vs. that ofUniverse may be written as:

ρp(t)

ρu(t)=

ρp(t)

ρp(tm)· ρp(tm)

ρu(tm)· ρu(tm)

ρu(t)= 23 · 5.55 ·

(

t

tm

)2

≃ 150


Figure 7.3: The closed mini-universe reaches the value am/2 of the expansionparameter at t after tm = τff (see, text).

if the mass profile of density perturbation remains the same during the tran-sition phase tm → tvir.

Even if the result corresponds to a very idealized representation, a similarresult, rounded to 200, comes out from N-body simulations, so that the virialradius of a structure is usually put in the form:

Rvir ≃ R200(zvir) =

(

3

4π

)1/3(M

200ρu(zvir)

)1/3

Moreover, going on by linear approximation from tm to tvir, we wouldobtain from (7.31) for density contrast at virialization:

δvir =

(

δρ

ρ

)

vir

=

(

δρ

ρ

)

m

·(

tvir

tm

)2/3

≃ 1.686

That explain why in the linear field of δ, the condition for considering acollapsed perturbation is assumed to be: δ ≥ 1.686.

Chapter 8

Oscillations of Relaxation

8.1 Oscillations of Relaxation

Following Lynden-Bell (1967) we assume a politropic density distribution1,of index np, for the proto-galactic structure of mass M and initial radius R.Its inertial moment is:

I = λ2MR2 (8.1)

where λR, known as gyration radius, is depending on the mass distribution(λ = 3

5in the case of an homogeneous sphere). Its potential gravitational

energy is:

Ω = −νΩGM2

R= − 3

5 − np

GM2

R(8.2)

as soon as in νΩ we highlight np. The general condition of non-stationaryvirial equilibrium does hold when it moves from the maximum expansionphase, at time tm, towards virialization, at time tvir.

1

2λ2M

(

R2)

= 2T + Ω (8.3)

where T is its kinetic energy. We look at this proto-structure as a mini-universe inside which the stars have been formed very early. At the maximumexpansion phase it stops to follow the Hubble flow so thah we may assume,at first approximation, its kinetic energy T (tm) ≃ 0. 2

Due to the collisionless condition for a star-gas, during the transitionphase, the following equation of total energy conservation holds:

tm < t < tvir −→ T + Ω = Tv + Ωv = Ev; (v = virialization) (8.4)

1The distribution is politropic but the equilibrium is not hydrostatic.2Ordered motions will arise during Relaxation.

57

58 CHAPTER 8. OSCILLATIONS OF RELAXATION

Then at the generic time t of the interval, the kinetic energy T is given by:

T = Ev − Ω = Ev + νΩGM2

R(8.5)

So Eq.(8.3) becomes:

1

2λ2M

(

R2)

= 2

(

Ev + νΩGM2

R

)

− νΩGM2

R(8.6)

Dividing by M , Eq.(8.6) reads:

1

2λ2(

R2)

=2Ev

M+ νΩ

GM

R= F (R) (8.7)

In the case of oscillations not too big in comparison with the virial radiusRv, we may develop F (R) from Rv to obtain, at the first order:

F (R) ≃ F (Rv) +

(

dF

dR

)

R=Rv

· (R − Rv) (8.8)

which means:

F (R) ≃ 2Ev

M+ νΩ

GM

Rv− νΩ

(

GM

R2

)

Rv

· (R − Rv) (8.9)

By considering that:

2Ev − Ωv = 2Tv + 2Ωv − Ωv = 0 (8.10)

and multiplying again by M it follows:

F (R) · M ≃ 0 − νΩGM2

R2v

δR; δR = R − Rv (8.11)

Then Eq. (8.6) transforms into:

1

2λ2M

(

R2)

≃ −νΩGM2

R2v

δR (8.12)

Developing again R2 in Taylor series around Rv,

R2 ≃ R2v +

(

dR2

dR

)

Rv

δR = R2v + 2RvδR (8.13)

and puting the result (8.13), at the first order, into Eq.(8.12) as follows:

1

2λ2M

d2

dt2R2 =

1

2λ2M2Rv(δR) ≃ −νΩ

GM2

R2v

δR (8.14)

we obtain the differential harmonic equation:

( ¨δR) ≃ −νΩ

λ2

GM

R3v

δR (8.15)

which describes the oscillations of relaxation process.

8.2. RELAXATION PERIOD AND VIRIAL DIMENSION 59

8.2 Relaxation period and Virial dimension

If the mass distribution remains homogeneous, as it was according to thetop-hat approximation, then:

np = 0

λ2 = 0.40νΩ = 0.60

−→ νΩ

λ2≃ 1.5 (8.16)

The square of harmonic pulsation may be approximated by:

ω2 =νΩ

λ2

GM

R3v

≃ GM

R3v

(8.17)

so that the relaxation period turns to be:

Trelax ≃ 2π

ω= 2π

(

R3v

GM

)12

(8.18)

By using the homogeneous density ρv at virial configuration:

ρv =M

43πR3

v

−→ M

R3v

=4

3πρv (8.19)

Eq. (8.18) transforms into:

Trelax = 2π

(

1

G43πρv

)12

= 2

(

3π2

4πGρv

)12

= 2

(

3

4

π

Gρv

)12

(8.20)

Moreover, by remembering that the transition from maximum expansionphase to virialization has occurred by conserving total energy: Em = Evir

with Tm ≃ 0, that means:

Em ≃ Ωm = − 3

5 − np

GM2

Rm

(8.21)

and:

Ev = Ωv + Tv = −2Tv + Tv = −Tv =Ωv

2(8.22)

By conserving np = 0, we obtain:

Em = −3

5

GM2

Rm= Ev = −3

5

GM2

2Rv(8.23)

which implies in turn the following link between maximum expansion radiusand the virialization one:

Rm = 2Rv → Rv =1

2Rm (8.24)


The consequence, at this level of approximation, is: ρv = 8ρm, which oncesubstituted into Eq. (8.20), yields:

Trelax = 2

(

3π

4G8ρm

)12

= 2 tff (8.25)

8.3 Relaxation times: Trelax, τrelax, τff

If we look at the simulations we may observe that:

• during the relaxation process the limit R → 0 ⇒ ρ → ∞ has neverreached;

• there is something which damps the oscillations.

How may both items to be accounted?

8.3.1 Item a)

Although the total energy is conserved during transition towards virializa-tion, the relaxing system undergoes a re-distribution of it. That means thesingle star may change its energy. We may prove that when the gravitationalpotential φ changes, it occurs:

e∗ = φ (8.26)

e∗ being the energy per unit star mass. Then we have to introduce a furthertypical time which has to characterize not the time scale of the global process,as already done with Trelax, but the time a single star needs to change itsenergy of the same amount it initially owns. According to Eq. (8.26), thatoccurs when the gravitational potential changes significantly as:

∆φ ≃ φ → ∆φ

∆t≃ φ

τrelax(8.27)

During the lenght of the process, by φ = φ(t) we may build up the followingmean square value of τrelax:

〈τ 2relax〉

12 = 〈φ

2

φ2〉 1

2 (8.28)

More simply we may evaluate of what fraction of Trelax one needs to varythe potential in a significant way. Let us develop φ in Taylor series from

8.3. RELAXATION TIMES: TRELAX , τRELAX , τFF 61

the maximum expansion value φm. At the first order of approximation weobtain:

φ ≃ φm +

(

dφ

dR

)

m

∆R (8.29)

where the gravitational potential used corresponds to that of a homogeneoussphere of radius R on its surface3. We get the value at the virialization phaseφv as follows:

φv − φm = ∆φ =

[

d

dR

(

−GM

R

)]

Rm

(Rv − Rm) (8.30)

=GM

R2m

(Rv − 2Rv) (8.31)

= −GM

4R2v

Rv (8.32)

= −1

2

GM

2Rv(8.33)

= −1

2

GM

Rm=

1

2φm (8.34)

That means: as soon as R = Rv, the gravitational potential has changed itsinitial value of an amount equal to one half of its previous value. The timeelapsed is only:

τrelax =1

4Trelax =

1

2τff (8.35)

owing to Eq. (8.25). It is thanks to the shortness of the time during whichthe unit mass of a star may change its energy via the engine (8.26), that thephenomenon is called violent relaxation.

In Fig.8.1 the two typical relax times are shown with their comparisonwith the τff . During this time also the energy per unit star mass e∗ haschanged in a meaningful way its value, by a transfer of the initial potentialenergy into disordered kinetic energy, which plays a role of macroscopic pres-sure (see, simulations). It is thanks to this presence that the collapse to amass point is inhibited. The structure will stop to collapse by reaching aminimum value of R not too far from Rv. It should to be noted that thevariations δR, considered in the Lynden-Bell’s theory, become in this waymore suitable to compare in Taylor series developments. Otherwise varia-tions from Rm = 2Rv to R = 0 would have spoiled the theory of correctness(some analogy with Jeans’ theory).

3The star we are dealing with is then located on this surface.


Figure 8.1: Comparison between the two typical relax times: Trelax, τrelax =14Trelax = 0.5τff and τff as function of R during the relaxation process (see,

text). The cycloidal collapse (dashed line) crosses the virial dimension Rv att = 0.82τff . The harmonic oscillation of relaxation process (continue line),on the contrary, at 0.5τff .

8.3.2 Item b)

To give answer to this question it is necessary to take into account whathappens to the stars in the phase space. It is indeed surprising that ina star-gas, typically collisionless then without dissipation, a damping mayoccur. Essentially the reason is the following: a collapse is the phenomenonwhich asks the stars to be tightly coordinated. Even though this coordinationdoes exist at the beginning, due to the initial small phase displacementsin few dynamical times the system as a whole loses it. In this way theoscillations become damped by the Landau damping otherwise called phasemixing, analogous to the phenomenon of free-streaming.

8.4 Statistical Mechanics:

Boltzmann’s Equation for a collisionless

particle system

To describe a set of collisionless particles (e.g. stars inside a galaxy), movingunder the effect of a gravitational potential φ(~r, t), one needs to give at eachtime t the distribution function (DF), f (~r, ~v, t) in the µ−phase space4. Its

4It is the 6-dimensions space of coordinates: (x, y, z, vx, vy, vz). In µ−space to N -objects→ N -points. To distinguish from the other phase space: Γ−space which has 6N-dimensions: (q1, q2, .....q3N , p1, p2, .....p3N ) where (q, p) are canonical variables, coordinatesand coniugate momenta (or generalised momenta; see, Landau....), respectively. In the

8.5. ANALOGY STARS-PENDULUMS IN PHASE SPACE 63

meaning is to be a density in µ space, i.e., it gives the number of particlesper unit volume of phase space, so that:

f (~r, ~v, t) d3~r d3~v (8.36)

gives the number of particles which have positions inside the infinitesimalvolume d3~r with center ~r and velocities in the range d3~v with center ~v.Moreover the DF has to be a definite positive function, so that:

f (~r, ~v, t) ≥ 0 (8.37)

The following fundamental equation, due to Boltzmann5(derived in 1872),holds to describe how flow the collisionless particles in µ space:

df

dt=

∂f

∂t+

3∑

i=1

(

∂f

∂xi

xi +∂f

∂vi

vi

)

= 0 (8.38)

To understand the physical meaning we need to consider initially the term: ∂f∂t

which means to look, at a fixed µ phase point, the evolution of f only in time(Eulerian point of view). As time goes on at the same place the DF in generalchanges. But if we move with the µ phase element changing ~r → ~r ± d~r and~v → ~v ± d~v and take into account the corresponding variations (Lagrangian

point of view): term∑3

i=1

(

∂f∂xi

xi

)

and term∑3

i=1

(

∂f∂vi

vi

)

, respectively, the

three variations altogether give 0. It means that the phase-space densityaround a phase point of a given star, always remains the same. In other wordsthe profile of f (~ro, ~vo, to) shifts rigidly by changing (~ro, ~vo, to) into (~r, ~v, t).The flow of stellar phase points through phase spase is incompressible.

8.5 Analogy stars-pendulums in phase space

We associate at each star a simple pendulum (Fig.8.2) of unit mass andlenght. The lagrangian coordinate is q = θ and the coniugate momentumis p = θ. Their initial phases and initial random velocities are uniformlydistributed (see, Fig.8.5 (a)) as follows:

θo ± ∆θo

∆θo ≪ θo

θo = 0 ± ∆θo

(8.39)

Γ−space to N -objects→ 1-point which describes the state of the whole system.5Alternative name is: Vlasov’s equation. When collisions are present Eq. (8.38) be-

comes: df

dt= Γ(f)coll, Fokker-Planck’s Equation.


mg

O

θL

Figure 8.2: A simple pendulum of unit mass and length is associated to eachstar.

The dynamical of oscillations is in general (m 6= 1, L 6= 1) given by:

θ = − gL

sin θ ≃ − gLθ

θ = θo cos ωtω =

√

gL

(8.40)

where g is the gravity acceleration and the initial conditions are: t = 0, θ =θo, θo = 0. Let us define:

a = θo

q = θ

q = θ = pm

(8.41)

Then the canonical variables q and p become:

q = a cos ωtp = mq = −maω sin ωt

(8.42)

and the sum of their squares produces the line drawn by the simple har-monic oscillator in the phase space (Fig.8.3):

q2

a2 = cos2 ωtp2

(maω)2= sin2 ωt

−→ p2

(maω)2+

q2

a2= 1 (8.43)

The ellipse of Eq.(8.43) has the following area A:

A =

∮

p dq (8.44)

= πac = πamaω (8.45)

= 2π2ma2ν (8.46)

8.5. ANALOGY STARS-PENDULUMS IN PHASE SPACE 65

p

qa

−maω

c=maω

Figure 8.3: The line drawn by a simple harmonic ascillator in the phasespace.

being: ω = 2πν.To get the amount of the total conserved energy (kinetic+ potential) :

Etot = T + Ω, we consider it at the following step:

q = 0 → q = −aω → θ = max → Ω = 0 → Etot = T =1

2mq2 (8.47)

=1

2ma24π2ν2(8.48)

= 2π2ma2ν2(8.49)

By comparison with Eq.(8.46) the result is that:

A =Etot

ν(8.50)

We wish remember shortly that the strong Planck’s assumption for a quanticoscillator was its energy had quantized, i.e.:

Etot = nhν (n = 1, 2, .....) (8.51)

h being the Plank’s constant. In other words it means that only eclipsesclosing an area (Fig.8.4):

A =

∮

p dq = nh (8.52)


p

q

2

3

1h

2h3h

Figure 8.4: Planck’s assumption for a simple quantized oscillator. Theeclipses in phase space have to close an area which is a multiple of h (see,text).

are allowed. An absorbtion process is corresponding to:

n → n + 1 ⇒ ∆A = h; ∆Etot = hν (8.53)

on the contrary, an emission one produces:

n → n − 1 ⇒ ∆A = −h; ∆Etot = −hν (8.54)

8.6 Movement of phase points in µ-space

Once coming back to the assumption (m = 1; L = 1) the frequency turnsto be the same for all pendulums and their energies become, according toEqs.(8.46), (8.50):

Etot = A · ν = 2π2θ2oν

2 (8.55)

depending on the θo distribution given by (8.39). Generally speaking thetrajectory will be closed but not a circle (also due to θo 6= 0). For sake ofsemplicity let us assume it as a circle. Then at θo ր Etot ր so that the move-ments of all phase points are bordered by the two limit circles correspondingto: (Etot)max ÷ (Etot)min corresponding to θomax ÷ θomin.

It is well known from the classical Mechanics that the pendulum periodPo for small oscillations is totally independent from the initial amplitude θo

8.6. MOVEMENT OF PHASE POINTS IN µ-SPACE 67

Figure 8.5: a) At the beginning the phase points cover the sector surfacearound θ-axis. Then they flow between the inner circle of lowest energy andthe outer one of maximum energy depending on the initial value of θo. Thereis a delay in their movement which has the effect to stretch the point set. b)After some dynamical times the initil area becomes a thin band of decreasingwidth (see, text).

, and that it does increase as soon as θo increases, following the law:

P = Po + Po

∞∑

1

[

1 · 2 · ... · (2n − 1)

2 · 4 · ... · 2n

]2

k2n (8.56)

being: k = sinθo

2. The consequence is shown in the Fig.8.5 (a). The phase

points starting with larger θo flow in the µ− space slower than the ones withsmaller θo. The delay increases as time goes on. According to BoltzmannEq.(8.38) the total area covered on the phase space has to be conserved asa liquid surface. That means the initial box full of phase points (the sectorin Fig.8.5 (a)) has to stretch becoming increasingly thinner. After somedynamical times (t ≃ nPo) the point phase distribution is approaching aband of ever decreasing width wrapped many times as a spiral. This annulusstretches from the inner circle towards the outer one.

Now the question is: how many pendulums are in phase at a given timet?That means we have to count how many of them have θ,

¯θ in the ranges:

θ ± ∆θ¯θ ± ∆

¯θ

(8.57)

We refer now to a finite volume in phase space, then to a mean value f of finside it, named coarse-grained phase space density. At the beginning it was:f = f in the black box of Fig.8.6 (A), at the end, in the same box: f ≪ f ,


θ

−θ −θ

θ

(B)(A)

Figure 8.6: Coarse- grained phase space density at the beginning (A) and atthe end (B) of the dynamical process of phase mixing.

Fig.8.6 (B). The observational box (B) contains indeed either segments ofspirals, full of phase points, or empty bands.The conclusion is that: the number of stars (pendulums) which are going todo the same things, i.e., which are coordinates, are decreasing in time. Themechanism get the name of : Landau damping or phase mixing.

Chapter 9

Violent Relaxation inphase-space

In this chapter the problem of violent relaxation mechanism in collisionlesssystems from the point of view of the distribution function (DF) in µ-spaceis reviewed. The literature run starts from the seminal paper of Lynden-Bell(1967) and is closed by that of the same author (Arad & Lynden-Bell, 2005).After some introductive sections on the stellar dynamical equilibria and onthe Shannon’s information theory, the different approaches follow each ac-companied with its criticism on the previous works. Different coarse-grainedDFs proposed by different authors have been taken into account. It appearsthat for a collisionless gas of a unique mass specie there is not significantdiscrepancies among the different approaches which converge to the sameDF at the end of relaxation process. The main problem is to avoid the nonobserved mass segregation in the case of multi-species composition, e.g., in astar-dominated galaxy component. On this topic the results are very differ-ent and are depending on the shape and size one chooses for µ-space tiles. Agreat effort has been spent into the visualization of the different partitions inphase-space in order to understand clearly from what the differences arise.

9.1 The Violent Relaxation problem

It has long been realized that galaxies, and self-gravitating systems in general,follow a kind of organization despite the diversity of their initial conditionsand their environement. At galaxy scale, this organization is illustrated bymorphological classification schemes such as the Hubble sequence and by sim-ple rules which govern their structure as individual self-gravitating systems.For example, elliptical galaxies display a quasi-universal luminosity profile

69

70 CHAPTER 9. VIOLENT RELAXATION IN PHASE-SPACE

described by de Vaucouleurs R1/4 law and on a different mass scale, a stellarsystem as a globular cluster is generally well fitted by the Michie-King model(Binney & Tremaine, 1987). The question that naturally emerges is, whatdetermines the particular configuration to which a self-gravitating systemsettles? It is possible that their present configuration crucially depends onthe conditions that prevail at their birth and on the details of their evolution.However, in view of their apparent regularity, it is tempting to investigatewhether their organization can be favoured by some fundamental physicalprinciples like those of thermodynamics and statistical physics. We won-der therefore if the actual states of self-gravitating systems are not simplymore probable than any other possible configuration, i.e., if they cannot beconsidered as maximum entropy states.

For most stellar systems, including the important class of elliptical galax-ies, the relaxation time due to close two-body encounters is larger than theHubble time by several orders of magnitude (Binney & Tremaine, 1987).Therefore, close encounters are negligible and the fundamental dynamics isthat of a collisionless system in which the constituent particles (stars) moveunder the influence of the mean potential generated by all the other par-ticles. Mathematically, the dynamics of stellar systems is described by theself-consistent Vlasov-Poisson system (see, Section 9.2 and Section 9.3). Theevolution of the Vlasov-Poisson system is extremely complicated. Althoughthe dynamics is collisionless, the fluctuations of the gravitational potentialare able to redistribute energy between stars and provide an effective relax-ation mechanism on a very short timescale (less then the free-fall time). Thisprocess is referred to it as violent relaxation.

Lynden-Bell was the first who introduced the violent relaxation theory in1967. Despite his formulation at the first order, the main features of themechanism have been outlined in his pioneering paper (Lynden-Bell, 1967).

Due to the complexity of the problem, no analitycal satisfactory treatmentof the dynamics of violent relaxation has been successful. Instead, ideas fromclassical statistical mechanics have been used to calculate the most proba-ble final state for a system of particles conserving quantities such as totalmass M, energy E, and angular momentum J. The theoretical foundations ofthe statistical mechanics of violent relaxation were set again by Lynden-Bell(1967) in the same paper, using a continuum approach for the distributionfunction, and re-derived by Shu (1978) with a particulate approach to thesame distribution. These analytical studies are now considered classical, de-spite the fact that the so-derived equilibrium distribution functions are far tobe able in accounting for the properties of observed systems produced insteadby more realistic N-Body simulations.

Before analysing the contributions to the discussion about Violent Relax-

9.2. DISTRIBUTION FUNCTION 71

ation we wish to underline that in order to compare the approaches and tounderstand clearly the criticisms of the different authors a great help comesfrom the capability to visualize how they tile the phase-space.The theories we will discussing assume time independent particle masses un-like the real stellar system where this hypothesis cannot rule (at least aszero-order approximation) according to the stellar evolution theory.

9.2 Distribution Function

We introduce the most basic quantity in stellar systems, that is the fine-grained distribution function (DF) or phase-space density:

f (~x,~v, t) = limd6µ→0

d6m (~x,~v, t)

d3~x d3~v(9.1)

yielding the mass d6m (~x,~v, t) (or the number of objects) contained at timet within an infinitesimal phase-space volume d6µ = d3~x d3~v centered aroundany point (~x,~v) of the 6-dimensional phase-space of stellar motions (calledthe µ-space in statistical mechanics). Clearly, f ≥ 0, everywhere in phase-space. In the N-Body approximation the mass d6m (~x,~v, t) can be consideredproportional to the number of particles, i.e., stars or fluid elements (e.g., ofdark matter), within the volume d3~x d3~v. Furthermore, it is often convenientto introduce a coarse-grained distribution function:

F (~x,~v, t) =1

∆3~x∆3~v

∫

∆3~x∆3~v

f (~x,~v, t) d3~x d3~v (9.2)

which gives the average of the fine-grained distribution function f in a small,but not infinitesimal volume elements ∆3~x∆3~v around the phase-space points(~x,~v). Contrary to the fine-grained distribution f, the value of the coarse-grained distribution F depends on the particular choice of partitioning thephase-space in which the volume elements ∆3~x∆3~v are defined.

The distribution function can be used to derive several other useful quan-tities. For example, the spatial mass density ρ (~x, t) of the system is givenby the integral of DF over velocities, e.g., (in Cartesian coordinates):

ρ (~x, t) =

∫ ∞

−∞

∫ ∞

−∞

∫ ∞

−∞f (~x,~v, t) dvx dvy dvz (9.3)

The latter quantity, ρ (~x, t), can be used in turn to calculate the gravitationalpotential Φ (~x, t) via Poisson equation:

∇2Φ (~x, t) = 4πGρ (~x, t) (9.4)


The orbits of stars are given by the Hamiltonian:

H (~x, ~p, t) ≡ ~p 2

2+ Φ (~x, t) (9.5)

if the average mass particle is equal to unity, and ~p = ~v in Cartesian coordi-nates.

9.3 Stellar Dynamical Equilibria - Boltzmann

Equation

The basic equation governing the time evolution of the distribution function,DF, in collisionless stellar systems is:

df

dt=

∂f

∂t+ ~v

∂f

∂~x− ∂Φ

∂~x

∂f

∂~v= 0 (9.6)

otherwise called Boltzmann equation (or Vlasov equation in plasma physics).In Eq.(9.6), d~v

dt= −∂Φ

∂~x, to which the Euler equation reduces when pres-

sure with collisionless system vanishes. This equation states that the masscontained within any infinitesimal volume d6µ that travels in phase-spacealong the orbits corresponding to the potential Φ, determined by Eq.(9.4),is preserved. Furthermore, the measure of the volume d6µ is also preserved(Liouville theorem). Now, the morphological regularity and the commonlyobserved characteristics of most galaxies suggest that the majority of thesesystems are close to a state of statistical equilibrium. Thus, we often look forsteady-state solutions of Eq.(9.6) that do not have an explicit dependence of,DF, on time:

~v∂f

∂~x− ∂Φ

∂~x

∂f

∂~v= 0 (9.7)

It is relevant to underline as the stationary virial equation comes out fromthe stationary Vlasov equation (9.7) (Binney & Tremaine, 1987).

9.3.1 Jeans Theorem

Let us introduce the concept of constants and integrals of the motion. Acostant of motion in a given force field is any function C (~x,~v, t) that isconstant along any stellar orbit; that is, if the position and velocity along anorbit are given by ~x(t) and ~v(t) = d~x/ dt, then we have:

C [~x(t1), ~v(t1), t1] = C [~x(t2), ~v(t2), t2] (9.8)

9.4. INFORMATION THEORY AND STATISTICAL MECHANICS 73

for any t1 and t2.An integral of motion I (~x,~v) is any function only of the phase-space

coordinates (~x,~v) that is constant along any orbit, that is: ~v(t) = d~x/ dt,

I [~x(t1), ~v(t1)] = I [~x(t2), ~v(t2)]

d

dtI [~x(t), ~v(t)] = 0 (9.9)

Every integral is a constant of the motion but the converse is not true. TheEq.(9.9) becomes

dI

dt= ∇I · d~x

dt+

∂I

∂~v· d~v

dt= 0 (9.10)

which reads:

dI

dt= ~v · ∇I −∇Φ · ∂I

∂~v= 0 (9.11)

as soon as Euler Equation is used.Comparing this with Eq.(9.6), we see that the condition for I to be an

integral is identical with the condition for I to be a steady-state solutionof the collisionless Boltzmann (or Vlasov) equation. It follows that f isnecessarily a composite function of the phase space variables (~x,~v) throughone or more of the integral functions I1, I2, ... . That is:

f (~x,~v) ≡ f [I1 (~x,~v) , I2 (~x,~v) , ...] (9.12)

The last result is known as Jeans theorem of stellar dynamics (Jeans, 1915).In its complete form it reads as follows: ’Any steady-state solution of thecollisionless Boltzmann (or Vlasov) equation depends on the phase-space co-ordinates only through integrals of motion in the galactic potential, and anyfunction of the integrals yields a steady-state solution of the collisionlessBoltzmann equation’ (Binney & Tremaine, 1987, Chapt. 4).

9.4 Information Theory and Statistical Me-

chanics

Information theory provides a constructive criterion for setting up probabil-ity distributions on the basis of partial knowledge, and leads to a type ofstatistical inference which is called the maximum-entropy estimate (Jaynes,1957). Some of the elementary properties of maximum-entropy inference aredefined and estabilished in the following way.


The quantity x is able of assuming the discrete values xi (i = 1, 2, ..., n).We do not know the corresponding probabilities pi to obtain xi. All we knowis the expectation value of the function f(x):

〈f(x)〉 =n∑

i=1

pif(xi) (9.13)

We wish to know the probabilities pi that allow us to obtain the expectationvalue of a generic g(x). At first glance the problem seems insoluble becausethe given information is insufficient to determine the probabilities pi. TheEq.(9.13) and the normalization condition:

n∑

i=1

pi = 1 (9.14)

would have to be supplemented by (n − 2) additional conditions before pi

and then 〈g(x)〉 could be found (Jaynes, 1957).The great advance provided by information theory lies in the discovery

that there is a unique, unambiguous criterion for the ’amount of uncertainty ’represented by a discrete probability distribution, which agrees with our in-tuitive notions that a broad distribution represents more uncertainty thandoes a sharply peaked one, and satisfies all other conditions which make itreasonable.

Claude Shannon in 1948 proved that the quantity which is positive, whichincreases with increasing uncertainty, and is additive for independent sourcesof uncertainty, is

Q (p1, p2, ..., pn) = −K∑

i

pi ln pi (9.15)

where K is a positive constant. In other words, the Shannon’s theorem statesthat, if pi are a set of mutally exclusive probabilities, then the function Q isa unique function, which, when maximised, gives the most likely distributionof the pi for a given set of constraints.

Since this is just the expression for entropy as found in statistical mechan-ics, it will be called the entropy of the probability distribution pi. Henceforthwe can consider the terms ’entropy’ and ’uncertainty’ as synonymous. Infact, the correspondence with the law of entropy increase is striking andleads directely to the definition of the Gibbs entropy as:

S = −k∑

i

pi ln pi (9.16)

9.4. INFORMATION THEORY AND STATISTICAL MECHANICS 75

where now k is the Boltzmann constant. So the maximum of this functioncorresponds to the most probable distribution, the equilibrium state of thesystem. Notice that Eq.(9.16) provides a very powerful definition of theentropy of the system, which can be used even if it has not attained an equi-librium state. Notice also how this definition quantifies the relation betweenentropy, disorder and information (Longair, 1984). This last is defined asfollows:

information = k∑

i

pi ln pi (9.17)

Obviously the higher is the information the lower is the disorder of the sys-tem.

It is now evident how to solve the problem; in making inferences onthe basis of partial information we must use that probability distributionwhich has maximum entropy subject to whatever is known. To maximize(9.15) subject to constraints (9.13) and (9.14), one introduces Lagrangianmultipliers λ, µ, in the usual way to obtain the result:

pi = e−λ−µf(xi) (9.18)

The constant λ, µ are determined by substituting into (9.13) and (9.14). Theresult may be written in the form:

〈f(x)〉 = − ∂

∂µln Z(µ) (9.19)

λ = ln Z(µ) (9.20)

where

Z (µ) =∑

i

e−µf(xi) (9.21)

will be called the partition function.This may be generalized to any number of functions fr(x). Given the

averages:

〈fr(x)〉 =∑

i

pifr(xi) (9.22)

form the partition function:

Z (λ1, ..., λm) =∑

i

exp − [λ1f1 (xi) + ... + λmfm (xi)] (9.23)


Then the maximum-entropy probability distribution is given by:

pi = exp − [λ0 + λ1f1 (xi) + ... + λmfm (xi)] (9.24)

in which the constants are determined from:

〈fr(x)〉 = − ∂

∂λrln Z (9.25)

λ0 = ln Z (9.26)

9.5 Lynden-Bell

In his seminal paper, Lynden-Bell (1967) argued that the violently chang-ing gravitational field of a newly formed galaxy leads to a redistribution ofenergies between stars and provides a mechanism analogous to a relaxationin a gas. The logical space for a treatment of this kind is the phase-space(µ-space) in which the phase-space volume is conserved and the followingrestrictions hold:

1. The total number of elements of phase (see later) which have any givenphase density, f (~x,~v, t), is conserved, see, Eq.(9.1).

2. The total energy is conserved.

3. As a corollary of item (1) overlapping between two generic elements ofphase cannot occur: if otherwise, the phase-space density would changein the overlapping region.

This last assumption is equivalent to admit an exclusion principle forthe phase-elements, and the fact that the phase-elements are distinguishableleads to introduce a fourth type of statistics besides the classical one ofMaxwell-Boltzmann and the two quantistic statistics, as follows:

Indistinguishable particles Distinguishable particlesNo exclusion I Einstein-Bose II Maxwell-BoltzmannExclusion III Fermi-Dirac IV Lynden-Bell

We will present now a simplified version of the main steps in the derivationof Lynden-Bell’s statistics, as well reviewed by Efthymiopoulos et al. (2006).

He considers a compact µ-space (i.e., the escapes are negligible), andimplements a coarse - graining process by dividing the µ-space in an enormous

9.5. LYNDEN-BELL 77

number of I macrocells of equal volume (the squares labelled by index i =1, 2, ..., I in Fig.9.1). He further divides each macrocell into a very largenumber, ν, of microcells each of the same volume that may or may notbe occupied by elements of the Liouville phase flow of the stars moving inµ-space. In Fig.9.1 these phase-elements (that is the microcells filled byparticles) are shown by dark squares within each macrocell.

I MACROCELLS OF EQUAL VOLUME (e.g., I=10)

ν MICROCELLS OF EQUAL VOLUME WITHIN EACH MACROCELL (e.g., ν=20)

ni OCCUPATION NUMBER OF THE iTH MACROCELL (e.g., n1=3)

N TOTAL NUMBER OF PHASE-ELEMENTS (e.g., N=29)

--------> MACROCELL

--------> PHASE-ELEMENT

--------> MICROCELL

ni -----> MACROSTATE

Figure 9.1: Lynden-Bell’s µ-space partition inside a macrostate.

He adopts the equal a priori probability assumption, namely he assumesthat each element of phase flow has equal a priori probability to be foundin any of I macrocells of Fig.9.1 (complete mixing). As the system evolvesin time, each phase element travels in phase space by respecting this as-sumption. We should note that, because of phase mixing, the form of thephase-elements also changes in time. However, this deformation does notchange the volume amount of an element. We can thus proceed in count-ing the number of phase-elements in each macrocell by keeping the simpleschematic picture of Fig.9.1.

He denotes ni the occupation number of the ith macrocell, i.e., the num-ber of fluid elements inside this macrocell at any fixed time t. The set ofnumbers ni = (n1, n2, ..., nI), called a macrostate, can thus be viewed as adiscretized realization of the coarse-grained distribution function of the sys-tem at the time t. In other words, that means the F (~x,~v, t) of Eq.(9.2) isdefined as a discrete function on the ith macrocell in this way:

Fi (~x,~v, t) =1

∆6µi

∫

∆6µi

fd3~xd3~v =nin

∆6µi

(9.27)


where n is the number of particles inside a phase-element.He calculates the number W (ni) of all possible microscopic configura-

tions that correspond to a given macrostate, and defines a Boltzmann en-tropy, S = ln W , for this particular macrostate. If we express by N =

∑Ii=1 ni

the total number of phase elements and by ν the (constant) number of mi-crocells within each macrocell, the combinatorial calculation of W readilyyields:

W (ni) =N !

n1!n2!...nI !

I∏

i=1

ν!

(ν − ni)!= N !

I∏

i=1

Pν (ni) (9.28)

where Pν(ni) is a binomial distribution, i.e., the probability of finding ni

phase-elements on a total of ν within the i-th macrocell.He finally seeks to determine a statistical equilibrium state as the most

probable macrostate, i.e., the one which maximizes S under the constraintsimposed by all preserved quantities of the phase flow. Besides mass conser-vation N =

∑Ii=1 ni, is also assumed total energy conservation of the system

E =∑I

i=1 niǫi (where ǫi is the average energy of particles in the ith macro-cell), and moreover the conservation of other quantities as the total angularmomentum (if spherical symmetry is preserved during the collapse) or anyother ’third integral’ of motion may be taken into account. In the simplestcase of mass and energy conservation, we maximize S by including the massand energy constraints as Lagrange multipliers λ1, λ2 in the maximizationprocess, namely:

δ ln W − λ1δN − λ2δE = 0 (9.29)

Applying Stirling’s formula for large numbers lnN ! ≈ N ln N −N , Eq.(9.27)becomes:

Fi =ηni

ν|S=max =

η

exp (λ1 + λ2ǫi) + 1(9.30)

where the value of the phase-space density inside each moving phase-spaceelement, η = n

d6µ= f (~x,~v, t), is taken as constant (in Fig.9.1, η turns to be

proportional to the ’darkness’ of phase-element). The Eq.(9.30) is Lynden-Bell formula for the value Fi of the coarse-grained distribution function withinthe ith macrocell at statistical equilibrium. Following the conventions ofthermodynamics, we interpret λ2 as an inverse temperature constant λ2 ≡β ∝ 1/T and λ1 in terms of an effective ’chemical potential’ µ = −λ1/β (or’Fermi energy’). We thus rewrite Eq.(9.30) in a familiar form reminiscent ofFermi-Dirac statistics:

Fi =η

exp [β (ǫi − µ)] + 1=

η exp [−β (ǫi − µ)]

1 + exp [−β (ǫi − µ)](9.31)

9.5. LYNDEN-BELL 79

The effective chemical potential in Eq.(9.31) has the same dimensions of theenergy per unit mass, ǫi.

At any rate, in the so-called non-degenerate limit:

Fi << η, (i.e., exp [β (ǫi − µ)] >> 1)

Eq.(9.31) tends to a Maxwell-Boltzmann distribution (that is, the final stateapproaches the isothermal model) in the following way:

Fi = η exp [−β (ǫi − µ)] = A exp (−βǫi) (9.32)

where A = η exp(βµ).The above exposition of Lynden-Bell theory is simplified in many aspects.

In particular:

• the expression given for the constraint of the total energy is not precise.One should calculate the energy self-consistently by the gravitationalinteraction of the masses contained in each phase-element. However,the final result turns out to be the same once this more precise calcu-lation is performed;

• all phase-elements in the above derivation are assumed to have thesame value of the phase space density η, i.e., the same ’darkness’ inFig.9.1.

A more general distribution function was derived by Lynden-Bell whenthe phase-elements of Fig.9.1 can be grouped into J groups of distinct dark-ness j = 1, ..., J (see, Fig.9.2).The final formula, derived also by the standard combinatorial calculation,reads:

Fi =J∑

j=1

Fj (9.33)

=

J∑

j=1

ηjexp [−βj (ǫi − µj)]

1 +∑J

j=1 exp [−βj (ǫi − µj)](9.34)

that is, it depends on a set of J pairs of Lagrange multipliers βj , µj. This morerealistic formula links the initial conditions of system formation, parametrizedby the values of ηj (conserved during the relaxation) to the final distributionfunction.

In the non-degenerate limit, the Eq.(9.33) represents a superposition ofnearly Boltzmann distributions, which means that each group of phase-elements is characterized by its own Maxwellian distribution of velocities


I MACROCELLS OF EQUAL VOLUME (e.g., I=10)

ν MICROCELLS OF EQUAL VOLUME WITHIN EACH MACROCELL (e.g., ν=20)

ni OCCUPATION NUMBER OF THE iTH MACROCELL (e.g., n1=3)


-------> MACROCELL

-------> PHASE-ELEMENT OF j-TYPE

(e.g., J=6)

-------> MICROCELL

nij -----> MACROSTATE

Figure 9.2: Lynden-Bell’s µ-space partition inside a macrostate in case of Jgroups of hase-elements.

which yields a different velocity dispersion in each group, related to the valueof ηj. In fact, if Fj << ηj then each factor exp [−βj (ǫi − µj)] has to be smalland we obtain the non-degenerate approximation:

Fi =

J∑

j=1

Aj exp (−βjǫi) (9.35)

where Aj = ηj exp (βjµj) is determined from the condition∫

Fj d6µ = Mj

( d6µ = d3~x d3~v and the integration is over all phase-space) and βj = βηj

η=

βηjM

∑

j ηjMjwhere β is in turn determined from the total energy condition.

The result shows the correct coarse-grained distribution function to bea superposition of Maxwellian components whose velocity dispersion are in-versely proportional to the phase-space density of the component at star

mixture[

〈v2〉j ∝ 1/ηj

]

. Indeed for the velocity distribution we obtain Fi ∼exp [−βjv

2i ] wich means βj ∝ 1

〈v2〉j∝ ηj . This poses the following problem:

how to express the overall distribution of velocities in the galaxy by a singleMaxwellian function (see, e.g., Shu (1978) and the debate Shu (1987)-Madsen(1987)).

9.6. SHU’S CRITICISM 81

9.5.1 Incomplete Relaxation

In real systems the relaxation is not complete (Lynden-Bell, 1967). In fact,when applied to a spherically symmetric stellar system, Eq.(9.35) substitutedinto Poisson’s equation leads to the Lane-Emden equation for an isothermalsphere. It is well known that unbounded isothermal spheres of extendedspatial structure have infinite masses (e.g., Chandrasekhar, 1939), whereasisolated isothermal spheres of finite mass have zero spatial structure. There-fore, unfortunately, the resulting equilibrium state of the violent relaxationhas infinite total mass. Thus real stellar systems tend towards the equilib-rium state during violent relaxation but can not attain it: the gravitationalpotential variations die away before the relaxation process is complete.

The suggested cure by Lynden-Bell for this is to introduce a cut-off inphase space so that complete violent relaxation takes place in a limited regiononly. While physically reasonable, the need for such a cure raises doubtsabout the usefulness of the theories and introduces the quest for a theory ofincomplete relaxation (Madsen, 1987). Nevertheless a statistical treatmentof incomplete violent relaxation is still lacking.

9.6 Shu’s Criticism

On reconsidering the foundations of Lynden-Bell’s statistical mechanical ofviolent relaxation in collisionless stellar system, Shu (1978) argues that Lyn-den Bell’s formulation in terms of a continuum description introduces unnec-essary complications. He considers a more conventional formulation in termsof particles. In fact, he stresses the point that stars are truly particles andnot infinitesimals part of a continuum. In his opinion it must be possible toformulate a statistical description on the basis of a particulate description.

He then finds the exclusion principle discovered by Lynden-Bell to bequantitatively important only at phase densities where two-body encountersare no longer negligible. Since the dynamical basis for the exclusion principlevanishes in such cases, the conclusion is that Lynden-Bell’s statistics alwaysreduces in practice to Maxwell-Boltzmann statistics when applied to stellarsystems.

The last loose end underlined by Shu concerns Lynden-Bell’s conclu-sion that the general cases of his new statistics leads to ’a superpositionof Maxwellian components whose velocity dispersions are inversely propor-tional to the phase space density of the star component’. This difficultiesmay vanish in the particulate description for a collisionless stellar system aslong as stars of different masses are initially well mixed in phase space.


9.6.1 Shu’s approach

The velocity dispersion problem has been first considered by Shu (1978) us-ing a method based on the particulate nature of the system. Shu dividesthe six-dimensional phase-space (µ-space) into an enormous number of fixedmicrocells of quite arbitrary volume g. Each microcell is occupied by 0 or 1particle (star of given mass, DM particle, black hole, etc.), in fact, the mi-crocell volume g is chosen small enough that each microcell may be host ofone star at most but large enough that two stars at adjacent microcells un-dergo no two-body encounter which would cause they travel, in a dynamicalcrossing time (the time scale associated with violent relaxation, Lynden-Bell1967) to a microcell different from its counterpart in absence of two-bodyencounter.

Shu makes a coarse-grained mesh of macrocells by grouping microcellsinto larger units of volume ωi = νig, where νi is the number of microcells inthe ith macrocell (its dimension may now be different for different macro-cells). The set ωi, i = 1, ..., I, constitutes a coarse mesh of µ-space, andthe set ni defines a given macrostate, where ni is the occupation number,i.e., the number of particles within ωi (see, Fig.9.3 for details).

I MACROCELLS OF VOLUME ωi=νig (e.g., I=11)

νi MICROCELLS OF EQUAL VOLUME g WITHIN EACH MACROCELL (e.g., ν1=20)

ni OCCUPATION NUMBER OF THE ith MACROCELL (e.g., n1=5)

N TOTAL NUMBER OF PARTICLES (e.g., N=51)

--------> MACROCELL

--------> PARTICLE

--------> MICROCELL

ni -----> MACROSTATE

Figure 9.3: Shu’s µ-space partition inside a macrostate.

Similarity and differences with Lynden-Bell’s approach are manifest: nidefines for both the macrostate but the meaning is different. In Lynden-Bellni is the number of phase-elements, in Shu the number of particles.

Since we deal with isolated systems (or with systems confined inside re-flecting walls), the set of occupation numbers ni satisfy the macroscopic


constraints:

I∑

i=1

mni = Nm (9.36)

I∑

i=1

mni

(

1

2|~vi|2 +

1

2Φi

)

= E (9.37)

where

Φi ≡ Φ (~xi, t) ≡ −I∑

l=1,i6=l

Gmnl

|~xi − ~xl|(9.38)

is the macroscopic gravitational field, m is the mass of the single species ofparticle, ~vi the velocity corresponding to the center of macrocell i.

The number W (ni) of all possible microscopic configurations in µ-spacethat correspond to a given macrostate ni defines a finite volume in 6N -dimensional Γ-space:

W (ni) =

[

N !

n1! · · · nI !

] [

ν1! · · · νI !

(ν1 − n1)! · · · (νI − nI)!

]

[

m3NgN]

(9.39)

= N ! · Pν1,...,νI(n1, ..., nI) ·

[

m3NgN]

where Pν1,...,νI(n1, ..., nI) is a multinomial distribution, i.e., the probability of

finding n1 particles on a total of ν1 within the first macrocell,..., nI particleson a total of νI within the I-th macrocell.

The macrostate occupying the largest volume W under the constraints ofconservation of mass, M, and energy, E, is the most probable state in whichthe system ends up, according to classical statistical mechanics. Entropy ismeasured by, ln W , so maximizing entropy under conservation of M and Ecorresponds to varying the following quantity:

lnW (ni) − αI∑

i=1

mni − βI∑

i=1

mni

(

1

2|~vi|2 +

1

2Φi

)

(9.40)

with respect to ni in order to find its extremum. The parameters α andβ are Lagrange multipliers introduced to remove the constraints on the in-dipendence of the ni variations. The most probable state according to thestatistical mechanical procedure described above, including the assumptionof complete mixing in phase space, turns out to be:

ni =νi

1 + exp [βm (ǫi − µ)](9.41)


where ǫi = 12|~vi|2 + Φi is the energy per unit mass in cell i and the chemical

potential µ ≡ −α/β. Shu called this distribution the Lynden-Bell distribu-tion for one type of particle. Disregarding to a normalization it resemblesthe Fermi-Dirac distribution, as might be expected due to the Pauli exclusionprinciple, but it is logically different since the particles are distinguishable.

In the nondegenerate limit (ni << νi) the Lynden-Bell distribution is sim-ply the Maxwell-Boltzmann distribution,

ni ≈ νi exp [−βm (ǫi − µ)] (9.42)

Up to now we confined the discussion to N stars of a single mass m.Generalizing to a collisionless stellar system with Nj particles of mass mj

the conserved quantities are now:

I∑

i=1

mjnij = Njmj (9.43)

J∑

j=1

I∑

i=1

mjnij

(

1

2|~vi|2 +

1

2Φi

)

= E (9.44)

Here the macrostate (see, Fig.9.4) is defined by the collection nij where iruns from 1 to I (the total number of macrocells) and j runs from 1 to J(the number of particles types, where J is much less than N (J << N), thetotal number of particles).

The general most probable state, assuming that all type of particles obeythe exclusion principle (that is at most one particle is allowed in each micro-cell) and assuming complete mixing in phase-space, is

nij = νiexp [−βmj (ǫi − µj)]

1 +∑

j exp [−βmj (ǫi − µj)](9.45)

where µj is the chemical potential of the j th particle type.As we can see, in the non-degenerate limit, nij << νi (i.e., exp [−βmj (ǫi − µj)]<<1), the most probable distribution is a sum of Maxwellians with the sameinverse ’temperature’ β:

nij = νi exp [−βmj (ǫi − µj)] (9.46)

But the mean kinetic energy of one particle of type j in the isothermal fluidis

1

2mj

⟨

v2⟩

j=

3

2kT ∼ kβ−1 (9.47)


I MACROCELLS OF VOLUME ωi=νig (e.g., I=11)

νi MICROCELLS OF EQUAL VOLUME g WITHIN EACH MACROCELL (e.g., ν1=20)

nij NUMBER OF PARTICLES WITH MASS mj WITHIN THE ith MACROCELL

Nj TOTAL NUMBER OF PARTICLES OF MASS mj

--------> MACROCELL

--------> PARTICLE OF MASS mj

--------> MICROCELL


Figure 9.4: Shu’s µ-space partition inside a macrostate in case of J groupsof phase-elements.

then the square of the velocity dispersion has to be inversely proportional

to mass[

〈v2〉j ∝ 1/mj

]

. This implies a dramatic consequence: the mass

segregation, that is the heaviest objects are more centrally concentrated inrespect to the lightest ones which go toward the border (see, Section 9.7).

Shu argued that mass segregation can be avoided if the initial conditionis well mixed, i.e., the mass composition of each macrocell is the same. For acollisionless stellar system, stars of all masses have nearly identical motionsif they have nearly identical µ-space locations. So the simplest assumptionhe made is that the mass function was initially well mixed. In Shu’s opinionthe motion equation assures that the mass distribution function has alwaysthe same form troughout µ-space, of this kind:

nij =Nj

Nni (9.48)

where ni is the total number of stars in the ith macrocell: ni =∑J

j=1 nij . Inother words, that means: owing to the fixed ratio of the total number of starswith mass mj over the total number N of the particles, the number of starsof given mass mj within the ith macrocell simply scales down proportionallyto the total number of stars within the same macrocell, i.e., the relativecomposition of each macrocell is the same.


Defining the coarse-grained mass density distribution function, F , as:

F (~x,~v, t) ∆3~x∆3~v = mass of stars at time t within ∆3~x∆3~v (9.49)

centered on (~x,~v)

we recover it as:

Fi (~xi, ~vi, t) ωi =J∑

j=1

mjnij = mni (9.50)

which is the analogous of Lynden-Bell’s course-grained number density Fi,Eq.(9.27), as soon as we remember that now: η = 1/g, gνi = ωi, m =∑

jmjnij

niis the average mass of the macrocell and (~xi, ~vi) is the center of the

ith macrocell.We may write the coarse-grained entropy as:

S = − k

m

∑

i

Fi lnFi∆6µ (9.51)

which, generally speaking, is the Gibbs entropy. The Eq.(9.51) becomes theBoltzmann entropy as soon as k is the Boltzmann constant. MaximizingW subject to the constraints (9.43), (9.44) and (9.48) is now equivalent tomaximizing S subject to the constraints:

∑

i

Fi∆6µ = constant (9.52)

∑

i

(

1

2|~v|2 +

1

2Φ

)

Fi∆6µ = constant (9.53)

The latter mathematical task is easy and the result is a single Maxwellian,

Fi = A exp(

−ǫi/σ20

)

(9.54)

where A and σ0 are constants and ǫ is the energy per unit mass of a star:

ǫi =1

2|~vi|2 + Φi (9.55)

It should be noted that the assumption that the mass distribution func-tion is well mixed in µ-space, Eq.(9.48), translates the collisionless problemwith J mass species to an equivalent N -body problem with a single massm. In particular, the equilibrium distribution function (9.54) is a singleMaxwellian with a uniform velocity dispersion σ0, the same for all the massspecies. In this way the mass segregation is completely avoided.

9.7. MASS SEGREGATION 87

9.6.2 Incomplete Relaxation

Finally, also Shu as Lynden-Bell (1967) explicitly recognized in his work theprimary difficulties (e.g., the predictions of systems with infinite masses) asso-ciated with the assumption of complete relaxation. Violent relaxation, beingrestricted in action both in space and time, must necessarily be incomplete.His own approach was to consider additional macroscopic constraints (e.g.,Shu, 1969) and another time cut-off procedures have to be applied to thestatistical mechanical results to obtain total masses which are not formallydivergent.

9.7 Mass Segregation

We consider a volume V in a system with a sample of particles of mass Mand a sample of particles of minor mass m.

If there is equipartition of the energy it means that

Ekinetic =1

2MV

2=

1

2mv2 (9.56)

where v is the typical velocity of the particles of mass m and V is the typicalvelocity of the particles of mass M . So, it is possible to assign, in the sameway as in a gas of molecules in equilibrium, a proper temperature T , such as

Ekinetic =3

2kT (9.57)

At the time t = t, we take into account the subsample of the m particles;since the negligible cross section, it will appear pratically decoupled fromthe sample of the M ’s. If the sample of the m’s is in virial equilibrium (see,Section 9.3) in the V volume, such as

2Tm = Ωm (9.58)

with

Tm =∑

j12mjv

2j

Ωm =∑

j mjφj

(9.59)

we wonder if, in the same volume, could the particles with mass M be inequilibrium too, i.e., if it is

2TM = ΩM (9.60)


with

TM =∑

i12Miv

2i

ΩM =∑

i Miφi

(9.61)

Since the stars move in the field generated by the entire galaxy, if it wasbe choosen a sufficiently small volume, we have

φj = φi ≃ cost. = φ (9.62)

as the particles M and m fill on average the same place. Moreover, since

V2

i ≃ cost. = V2

v2j ≃ cost. = v2 (9.63)

we can remove from the expressions of T and Ω the sign of sum.Therefore, knowing that

12mv2 = −1

2mφ

12mv2 = −1

2MV

2 (9.64)

we wonder if also the subsample of the Ms is in Viriual Equilibrium, i.e.,12MV

2= −1

2Mφ.

From (9.64) we infer φ = −Mm

V2

that gives

−ΩM = −Mφ =M

m· MV

2=

M

m2TM (9.65)

and becouse of the hypothesis M > m it results

|ΩM | > 2TM (9.66)

so the M particles are not in Virial Equilibrium in the V volume. Moreover,on the strenght of a kinetic energy too lower in respect to the potential energyin that position, the particles of mass M , beeing not supported by T , collapseto the center giving place to the mass segregation.

The construction can be inverted; if per hypothesis the M masses are inequilibrium, so

12MV

2= −1

2Mφ

12MV

2= −1

2mv2

(9.67)

from which we have φ = −mM

v2 that gives

−Ωm = −mφ =m

M· mv2 =

m

M2Tm (9.68)

and since m < M it results

|Ωm| < 2Tm (9.69)

Again the virial equilibrium is violated and the mass segregation arises be-cause the smaller masses go toward the border (see, Fig.9.5).

9.8. KULL, TREUMANN & BOHRINGER’S CRITICISM 89

--->

--->

--->

--->

--->

--->

--->

--->

Figure 9.5: Mass segregation: stars of mass M go to the center and stars ofmass m go toward the border.

9.8 Kull, Treumann & Bohringer’s Criticism

In the present Section we deal with another issue which is related to themain crucial point of two previous approaches of violent relaxation: con-tinuum and particulate. As we have seen, since the first investigation byLynden-Bell (1967), one of the most important flaws of the theory has beenconsidered to be the fact that despite the collisionless nature of violent relax-ation, the statistical mechanics approach predicts a thermalized final state(Madsen, 1987; Shu, 1987), i.e., in which each phase-space element of dif-ferent densities has a square velocity dispersion inversely proportional to itsdensity (in Lynden-Bell: 〈v2〉j ∝ 1/ηj; in Shu 〈v2〉j ∝ 1/mj). The final re-sult, tendentially present also in Lynden-Bell (1967), is the mass sgregation.In order to solve this problem, Shu (1978) applied to his particle approachvery stringent assumptions on the initial mean distribution function.

In their work of 1997, Kull, Treumann & Bohringer re-examine the sta-tistical mechanics of violent relaxation in terms of phase-space elements ofdifferent densities. Starting from the consideration that Lynden-Bell’s sta-tistical approach uses objects which are phase-elements of constant volumebut different mass, they conclude that the mass independence of the finalresult is not fully represented. Obviously, considering phase-space elementsof different volume but constant mass, they remove the presence of differentvelocity dispersions linked to different mass elements.

9.8.1 Kull, Treumann & Bohringer’s approach

The authors (hereafter, KTB) attempt to derive the coarse-grained phase-space distribution, F (~x,~v, t), of a system final state consisting of phase-space


elements of J different phase-space densities, ηj, subject to violent relaxationmechanism. The densities ηj are obtained from the initial fine-grained phase-space density, f(~x,~v, t) . To determine the final state, they apply the samemaximum entropy principle used by Lynden-Bell (1967): as usual the statethat maximizes the entropy under the constraint of conserved total energyand mass is considered the most probable final state attained by the system.

To apply statistical mechanics, the µ-space is divided into a large numberof microcells. ηj is the phase-space density of a phase-space element (i.e.,a microcell occupied by mass of j -type) and gj is its volume, so that themass associated with the phase element is ηjgj. A microstate may then bedescribed by the set of gj occupied microcells each of them has density ηj .

With respect to the continuum limit, the volumes gj of the phase-spaceelements are arbitrary and have no direct physical significance. Instead,physical significance is attributed to the mass ηjgj (or the correspondingmass differences) of the phase-space elements in µ-space. In this respect,it becomes natural to incorporate the universal mass independence of finalmotion, after the violent relaxation, into the statistical mechanics pictureby introducing a constraint on the volumes gj . Indeed, the authors take asconstant the phase-element mass, it means:

ηjgj = m = const. (9.70)

Therefore, in contrast to Lynden-Bell (1967), where the phase-elements havedifferent mass but constant volume, the phase-elements considered here differin volume and have constant mass.

According to Eq.(9.70), the volume gj of a phase-element of density ηj

and mass m, is given by

gj = m/ηj (9.71)

In order to simplify the discussion without introducing further restrictions,KTB assume that the different phase-space volumes gj are all multiples ofsome smallest elementary volume g, so that the volume gj occupied by aphase-element of density ηj becomes,

gj = cjg (9.72)

where cj is the factor by which gj is larger than the elementary volume, g.In other words, the microcell gj is made up of cj microcells of elementaryvolume g.

In conclusion the µ-space is assumed to be divided into a large numberof elementary microcells each one of volume g. At the macroscopic level,

9.8. KULL, TREUMANN & BOHRINGER’S CRITICISM 91

these microcells are grouped into macrocells containing a large number, ν, ofmicrocells. The corresponding volume of the macrocell is νg. Suppose thatthere is a macrostate in which the ith macrocell contains nij phase-elementsof different densities ηj (see, Fig.9.6). Because of the collisionless natureof the interaction in the violent relaxation process described by the Vlasovequation, there is no cohabitation of microcells which form a phase-element(Lynden-Bell, 1967).

I MACROCELLS OF EQUAL VOLUME νg (e.g., I=16)

ν ELEMENTARY MICROCELLS WITHIN EACH MACROCELL (e.g., ν=200)

nij PHASE-ELEMENTS OF DENSITIY ηj WITHIN THE ith MACROCELL

cj NUMBER OF ELEMENTARY MICROCELLS WITHIN A jth MICROCELL


--------> jth MICROCELL OF VOLUME gj=cjg

--------> jth PHASE-ELEMENT OF MASS m=ηjgj

--------> ELEMENTARY MICROCELL OF EQUAL VOLUME g

--------> MACROCELL


Figure 9.6: KTB’s µ-space partition inside a macrostate.

Under the assumption that the phase-space is resolved to the scale of thevolume of the smallest microcell, g, TKB find the number of ways of assigningnij microcells of volume gj without cohabitation to the ith macrocell, andthen the total number of ways of assigning the

∑

j nij phase-space elementsto the ith macrocell, w (nij).

The total number of microstates W (nij) corresponding to a given set ofoccupation numbers nij is found by multiplying w (nij) and taking into ac-count the number of ways of splitting the total of N distinguishable elementsinto groups nij. This yelds:

W (nij) =∏

j

N !∏

i nij !

∏

i

ν!(

ν −∑

j cjnij

)

!(9.73)

The macroscopic constraints to which the system is submitted are the Jconstraints related to conservation of the total number Nj (or total mass Mj)


of phase-space elements of densities ηj; that is:

∑

i

ηjgjnij = ηjgjNj = Mj (9.74)

The energy constraint reads:

∑

j

∑

i

ηjgjnij

(

1

2|~vi|2 +

1

2Φi

)

= E (9.75)

where the gravitational potential is defined by:

Φi = Φ (xi) = −∑

j

∑

l=1,l 6=i

Gηjgjnl

|xi − xl|(9.76)

The most probable state is found by the standard procedure of maximiz-ing, ln W , subject to the constraints of constant total energy and constantmasses. Introducing the Lagrangian multipliers αj and β, which are relatedrespectively to the constraints (9.74) and (9.75), the expression to be maxi-mized is:

ln [W (nij)] − αj

∑

j

∑

i

ηjgjnij − β∑

j

∑

i

ηjgjnij

(

1

2|~vi|2 +

1

2Φi

)

(9.77)

Defining µj = −αj/β and taking into account Eq.(9.71), the most prob-able occupation numbers nij become:

nij =ν exp [−βm (ǫi − µj)]

∑

j cj exp [−βm (ǫi − µj)] + 1(9.78)

where ǫi = 12|~vi|2 + Φi stands for the total energy per unit mass related to

the ith macrocell.The coarse-grained phase-space distribution F is defined as the sum of

the J phase-space distributions where:

Fj (~x,~v) ≈ Fj (~xi, ~vi) =cjnijηj

ν(9.79)

which is the analogous of Eq.(9.27) by considering that inside a phase-elementthere are cj elementary microcells. It is thus given as:

Fi (~x,~v) ≈ Fi (~xi, ~vi) =∑

j

Fj (~xi, ~vi) =∑

j

cjnijηj

ν(9.80)

9.9. NAKAMURA’S CRITICISM 93

Substituting nij from Eq.(9.80), the coarse-grained phase-space distributionF becomes finally

Fi (~x,~v) =∑

j

cjηj exp −βm [ǫ (~v, ~x) − µj ]∑

j cj exp −βm [ǫ (~v, ~x) − µj] + 1(9.81)

where ǫi (~v, ~x) ≈ ǫi (~vi, ~xi) = 12|~vi|2 + Φi is the total energy per unit mass.

In the non-degenerate limit (Fj << ηj , ∀j) the coarse-grained phase-spacedistribution (9.81) becomes a sum of Maxwellians. But the Maxwellians areall characterized by the same temperature β−1. The fact that all the phase-elements have the same mass, m, allows to translate the same temperatureinto the same velocity dispersion σ2

o = β−1. The coarse-grained phase-spacedistribution in the non-degenerate limit is:

Fi (~v, ~x) =∑

j

cjηj exp −βm [ǫi (~v, ~x) − µj] (9.82)

In conclusion, in the non-degenerate limit, the final state of the violent re-laxation process defined by equation (9.82) is a superposition of Maxwellianscharacterized by a common velocity dispersion that is equivalent to an equipar-tition of energy per unit mass without regarding to the mass species. As aconsequence, there is no mass segregation in systems which have undergonea violent relaxation, as one expects from the typical energy variation per unitmass due to this mechanism. That is

e∗ = Φ (9.83)

e∗ beeing the energy per unit star mass. This is in agreement with the com-mon velocity dispersion of the different components observed, for instance,in clusters of galaxies (see, e.g., Lubin & Bahcall 1993).

9.9 Nakamura’s Criticism

The reason which motivate Nakamura’s review work (2000) is essentially theattempt to solve the velocity dispersion problem risen in the seminal work ofLynden-Bell.

In the Nakamura’s opinion in fact, neither Shu (1978) nor Kull, Treumann& Bohringer (1997), succeeded in a satisfactory way in the purpose.

As we have seen, in the Shu’s approach, based on the particulate natureof the system, the velocity dispersion depends on the mass of particle species,i.e., the equilibrium state exhibits mass segregation. Shu claims that mass


segregation can be avoided if the initial condition is well mixed, i.e., themass composition of each macrocell is the same, see, Eq.(9.74). This massdistribution function in a macrocell remains unchanged throughout the timeevolution.

But, as Nakamura remarks, if this assertion is true, trajectories of twoneighboring particles with different mass must be so close that the two par-ticles are always in the same cell throughout the relaxation process. Onthe other hand, trajectories of two neighboring particles with the same massmight be far apart at the equilibrium state because Shu’s theory allows per-mutation of any two particles in the final equilibrium state. Therefore, theinitial condition assumed by Shu (1978) may be overrestrictive. Further in-vestigation will be required to clarify the problem of mass segregation in thisparticulate approach.

In his review analysis Nakamura takes also into account as Lynden-Bellhave shown that a single Gaussian distribution is obtained when we usemicrocells with equal mass instead of dividing the phase space into microcellswith equal phase-space volume like in Lynden-Bell’s theory.

Though the theory by Kull et al. seems to have successfully solved theproblem, in Nakamura’s opinion, it still contains two basic defects: one con-ceptual and one methodological. The conceptual defect is about the basis ofthe equal mass microcells. Why do we have to use microcells with equalmass, not equal volume? Kull et al. argue that mass has more physicalsignificance than phase-space volume, however, this statement is subjective.One may consider, for instance, that energy is more fundamental and mayuse microcells with equal energy. Then one would end up with a distributioncompletely different from the Gaussian distribution.

The methodological defect is the size and shape of microcells. In Lynden-Bell (1967) theory the microcells can be identical hypercubes in six-dimensionalphase space; it is easy to fill the whole phase space with these hypercubeswith any combination. On the contrary, microcells with equal mass inevitablyhave different shapes and sizes, thus only limited combinations are allowed tofill the phase space without gaps (see, e.g., the white squares in Fig.9.6). Forinstance, the phase-space density at v → ∞ (v, velocity) is infinitesimallysmall, thus a cell with equal mass must be infinitely large there.

The purpose of the Nakamura work is to introduce a statistical theory ofa collisionless system based on the ’maximum information entropy principle’introduced by Jaynes (1957a, 1957b and 1983). The guiding line is that: ’theprobability distribution over microscopic states which has maximum entropysubject to whatever is known, provides the most unbiased representation ofour knowledge of the state of the system’ (Jaynes, 1957). Nakamura appliesthis principle to the relaxation of the coarse-grained distribution function to


show that the equilibrium state is a single Gaussian distribution in the non-degenerate limit. In the case of ordinary collisional gases, the entropy mustbe calculated from the probability of particle existence, i.e., the probability tofind a particle at a certain position in the phase space. It is possible to use thesame kind of probability for a collisionless system because the essential struc-ture of the problem must be the same. It will be shown (SubSection 9.9.2)that Lynden-Bell’s statistics are equivalent to calculate the entropy from theprobability of particle transition, i.e., the probability that a particle at a cer-tain location at the initial time moves to another location, at infinite time.This is the reason, in Nakamura’s thought, why Lynden-Bell’s statistic givesthe wrong answer.

9.9.1 Nakamura’s approach

Nakamura divides the µ-space into a set of i small cells of equal volume∆µ = ∆3~x∆3~v. The box-averaged distribution Fi (t) is defined as:

Fi (t) =

∫

∆µi

f (~x,~v, t) d3~xd3~v (9.84)

where f (~x,~v, t) is the true distribution (the fine-grained distribution) and∆µi = ∆µ indicates the volume integration over the ith cell. The box-averaged distribution Fi represents the probability of finding a particle inthe ith cell.

What author wishes to do is to calculate the box-averaged distribution inthe limit of t → ∞, starting with a given initial distribution at t = t0. Here,the phase-space volume is a conserved quantity in addition to the energyand mass in ordinary statistical physics. Thus, he must specify the initialdistribution to know the phase-space volume corresponding to it. Then, heintroduces the joint probability Pi,ξ as a probability to find a particle in theξth cell at t = t0 and find the same particle in the ith cell at t = ∞. Theinitial and equilibrium distribution is calculated from Pi,ξ as Fξ (t0) =

∑

i Pi,ξ

and Fi (∞) =∑

ξ Pi,ξ (see, e.g., the examples in Fig.9.7).The initial state of the fine-grained distribution f (~x,~v, t0) is assumed

to be so smooth that we can regard it as a constant within a cell, i.e.,f (~x,~v, t0) = Fξ (t0) /∆µ.

The maximum entropy principle gives the inference that probability Pi,ξ

is the one that maximizes the following information entropy

S = −∑

i

Fi (∞) ln Fi (∞) = −∑

i,ξ

Pi,ξ ln Pi,ξ (9.85)

under the constraints of energy and phase-space volume conservation.


iξ=2 ξ=3ξ=1

t=ooξ i=1 i=2 i=3 t=0

t=oot=0

Figure 9.7: Exemples of initial (top) and equilibrium (down) distribution:Fξ (t0) = P1,ξ + P2,ξ + P3,ξ and Fξ (∞) = Pi,1 + Pi,2 + Pi,3, respectively.

We introduce now the quantities:

σi,ξ =Pi,ξ

Fξ (t0)(9.86)

where Fξ (t0) may be known or not. In this second case, Fξ (t0), works simplyas a mathematical normalization factor for the joint probability Pi,ξ. In thefirst case it changes the meaning of the two probabilities Pi,ξ and σi,ξ (see,later). Then the equilibrium distribution can be expressed in the followingway:

F∞i =

∑

ξ

σi,ξF0ξ (9.87)

where we write Fξ (t0) = F 0ξ and Fi (∞) = F∞

i .The goal of this Section is to maximize the entropy S putting Eq.(9.87)

into Eq.(9.85) under the conservation of energy and phase-space volume.From the above equation the energy conservation may be written as:

∑

i,ξ

σi,ξF0ξ

(

1

2|~vi|2 + Φi

)

= constant (9.88)

where vi and Φi is the velocity and potential per mass at the ith cell. Notingthat the fine-grained distribution is uniform within a cell, we understand that


∆µ σi,ξ represents the fraction of the phase-space volume of the ξth cell thatwill go to the ith cell at t → ∞. The phase-space volume conservation isthen expressed as:

∑

ξ

∆µσi,ξ = ∆µ (9.89)

There is one additional condition:∑

i

σi,ξ = 1 (9.90)

to ensure∑

i,ξ Pi,ξ = 1.Using Lagrange’s method to find the maximum of the entropy S in

Eq.(9.85) under the constraints of equations (9.88), (9.89) and (9.90), thefunction to maximize is:

−∑

i,ξ

σi,ξF0ξ ln σi,ξF

0ξ − β

∑

i,ξ

σi,ξF0ξ

(

1

2|~vi|2 + Φi

)

− λi

∑

ξ

σi,ξ − δξ

∑

i

σi,ξ(9.91)

where β, λi and δξ are the Lagrange’s undetermined coefficients related tothe constraints. The result is:

σi,ξ = AξB1/F 0

ξ

i exp (−βǫi) (9.92)

with Aξ = exp(

−δξ/F0ξ − 1

)

, Bi = exp (−λi), and ǫi = v2i /2 + Φi. The

distribution at t → ∞ can be calculated from the above equation combinedwith Eq.(9.87) as:

F∞i =

∑

ξ

AξB1/F 0

ξ

i F 0ξ exp (−βǫi) (9.93)

The parameters Aξ, Bi, and β must be determined in order to satisfyequations (9.88), (9.89) and (9.90). Due to the difficulty to obtain explicitexpressions for these parameters, Nakamura examines, in what follows, onelimiting case, the non-degenerated limit.

There are cells in which the probability distribution is negligibly small(F 0

ξ ∼ 0) at the initial time. We assume these empty cells have a very small

probability F and calculate the limit of F → 0. Introducing a new parameter

B′

i = B1/Fi , the σi,ξ may be re-written as:

σi,ξ = limF→0

AξB′

iF/F 0

ξ exp (−βǫi)

=

A1 exp (−βǫi)(

F ξ0 6= F

)

A0B′

i exp (−βǫi)(

F ξ0 = F

) (9.94)


The equilibrium distribution of Eq.(9.93) becomes:

F∞i = exp (−βǫi) lim

F→0

∑

F ξ0 6=F

A1Fξ0 +

∑

F ξ0 6=F

A0B′

iF

= A1 exp (−βǫi)∑

F ξ0 6=F

F ξ0 (9.95)

This result means the equilibrium state is a single Gaussian distributionthat is proportional to exp (−βǫi). Indeed we obtain:

F∞i ∼ A1

[

exp(

−β |~vi|2)

· (−βΦi)]

(9.96)

where the Gaussian distribution of the velocity is characterized by a uniquemean square velocity dispersion β−1.

Before closing let us examine the condition under which the above cal-culation is valid. The σi,ξ of Eq.(9.94) when F ξ

0 = 0 may be cast in thisway:

∑

ξ

|σi,ξ]F ξ0 =0 = M0A0B

′

i exp (−βǫi)

= 1 − M1A1 exp (−βǫi) (9.97)

where M0 and M1 are the number of cells with F ξ0 = 0 and F ξ

0 6= 0, respec-tively. Since σ has to be a positive number and the exp (−βǫi) ranges inside1 ÷ 0, then the following inequality has to hold:

M1A1 < 1 (9.98)

Therefore the condition is:

F∞i = A1e

−βǫi

∑

ξ

F 0ξ < A1

∑

ξ

F 0ξ <

1

M1

∑

ξ

F 0ξ =

⟨

F 0ξ

⟩

(9.99)

where 〈...〉 denotes the average over non-zero cells.What means this result?

1. The mean number of particles per unit volume inside the ξ-cell at t = 0has to be equal to the number of particles per unit volume inside thei -cell at t = ∞.


2. Owing to item (1), that means there is not overlap inside each cell,then the exclusion principle does not play a role in this case. In otherwords we are in the non-degenerate limit.

3. But if no-overlap is synonimous of F∞i =

⟨

F 0ξ

⟩

without regarding to theindices (i, ξ) and to time (∞, 0), that from one side is what one expectsto occur (see, Lynden-Bell’s items (1) and (3)) by choosing the phase-space volume conservation, from the other side it proves, in Nakamurawords, ’as in this limit the conservation of phase-space volume doesnot play a role and the equilibrium state becomes the same as the oneobtained without it’. Even if this steatment may only intuitively beaccepted, in our opinion it appears not clearly demonstrated.

4. If the assumption of Nakamura related to a smooth fine-grained distri-bution is taken into account, the item 3. becomes exactly F∞

i = F 0ξ

owing to:

∑

ξ

∆µ σi,ξ = ∆µ

∑

ξ

Fξ(t0)

f(x,v, t0)σi,ξ = ∆µ

∑

ξ

Fξ(t0)

f(x,v, t0)

Pi,ξ

Fξ(t0)= ∆µ

∑

ξ

Pi,ξ

f(x,v, t0)= ∆µ (9.100)

1

f(x,v, t0)

∑

ξ

Pi,ξ = ∆µ

Fi(∞)

f(x,v, t0)= ∆µ

Fi(∞)

f(x,v, t0)= ∆µ =

Fξ(t0)

f(x,v, t0)

Fi(∞) = Fξ(t0)

It should be noted that this strong condition is a consequence of theNakamura assumption. Even if he claims the formulation of his statis-tics is completely general (see, the Section 4 of his paper), we maintainsome reserves on this crucial point.


9.9.2 To get Lynden-Bell statistics

In the previous Section the equilibrium distribution is obtained by maximiz-ing the entropy defined in Eq.(9.85). It is possible to think about anotherdefinition of entropy, i.e., entropy calculated from the probability of particletransition: σi,ξ = Pi,ξ/Fξ(t0), which means the probability that a particle inthe ξth cell at t = t0 goes to the ith cell at t → ∞.

It should be noted that the meaning of particles transition probabilityσi,ξ becomes different in respect to the joint probability Pi,ξ because now theterm Fξ(t0) gains a deep meaning transforming the joint probability into aconditional probability. It means a probability after we know the fact thatthe particle was in ξth cell.

Because of the probability for total ways of transition is expressed asP =

∑

ξ σi,ξ thus the entropy may be defined as

S ′ = −∑

i,ξ

(

∏

ξ

σi,ξ

)

ln

(

∏

ξ

σi,ξ

)

= −M∑

i,ξ

σi,ξ ln σi,ξ (9.101)

where M is the total number of cells.Maximizing S’ under the constraints of (9.88), (9.89) and (9.90), we ob-

tain

σi,ξ = AξBi exp(

−βF 0ξ ǫi

)

(9.102)

where Aξ and Bi are coefficients determined by the constraints. When Aξ isknown, we can express Bi as:

Bi =1

∑

ξ Aξ exp(

−βF 0ξ ǫi

) (9.103)

The equilibrium distribution calculated from Eq.(9.87) becomes:

F∞i =

∑

ξ

AξF0ξ exp

(

−βF 0ξ ǫi

)

∑

ξ Aξ exp(

−βF 0ξ ǫi

)

=∑

F 0ξ 6=0

F 0ξ exp

(

−βF 0ξ ǫi − µξ

)

1 +∑

F 0ξ 6=0 exp

(

−βF 0ξ ǫi − µξ

) (9.104)

where µξ is defined as

exp (−µξ) =Aξ

∑

F 0ξ =0 Aξ exp

(

−βF 0ξ ǫi

) (9.105)

9.10. INCONSISTENCY IN THEORIES OF VIOLENT RELAXATION101

From the comparison of Eq.(9.104) with Eq.(9.95) it is now manifestthe great difference by use of the joint probability instead of conditionalprobability. The terms F 0

ξ are outside of the exponentials in the first case,whereas they are rigidly located inside the exponentials in the second case.

The above expression is identical to the result of Lynden-Bell (1967) givenin his Appendix I. Therefore, we understand that Lynden-Bell’s theory isequivalent to applying the maximum entropy principle with respect to theprobability of particle transition, σi,ξ.

Then the question is : what is true? We can determine it when we omitthe constraint of phase-space volume conservation. Indeed the Eq.(9.104)differs from a single Gaussian owing to the presence of F 0

ξ . But F 0ξ are the

terms which link the joint probability, Pi,ξ, to the transition probability σi,ξ.Using Pi,ξ is equivalent to have no complete knowledge about the particledistribution F 0

ξ which in turn is equivalent, in Nakamura’s opinion, to disre-gard the volume conservation (item (3)). But the same does not occur for theLynden-Bell’s approach based on the σi,ξ. Indeed, in the Nakamura’s words:’Without this constraint (volume conservation) the theory must agree with thestatistical physics of an ordinary gas, and the equilibrium distribution mustbe a Gaussian distribution. It can be done in our theory by omitting equation(9.88), and the result gives a single Gaussian distribution. In Lynden-Bell’stheory, one can remove the phase-space volume conservation to calculate thenumber of combinations; the result fails to give a single Gaussian distribu-tion’.

9.10 Inconsistency in theories of violent re-

laxation

In a recent work (Arad & Lynden-Bell, 2005) re-examine to what extent thefundamental assumptions of statistical mechanics apply in violent relaxationmechanism. They concentrate on the well-known Lynden-Bell theory andthe more recent Nakamura’s theory. However, they do not try to estabil-ish what among the two above mentioned theories is more correct, as, intheir opinion, this is still an open question1 but instead they highlight aninconsistency present in both the theories. The inconsistency arises from thenon-transitive nature of these theories: a system that undergoes a violentrelaxation, relaxes and then, under an addition of energy, undergoes again anew violent relaxation and would settle in an equilibrium state that is differ-

1They do not consider Nakamura’s argument to be a proof for his correctness over theLynden-Bell’s theory.


ent from the one that is predicted if the system had gone directly from theinitial to the final state.

In fact, the main purpose of their paper is to demonstrate that theoriespredict a definite statistical equilibrium, do not predict the same final statewhen the system undergoes two violent relaxation sessions separated in time,as they do when the two sessions are treated as one. This may be called aninconsistency or at best a lack of transitivity.

In order to do this analysis they re-derive the Nakamura’s theory, whichis based on the information-theory approach, using a combinatorial approachthat enables them to compare it to Lynden-Bell’s theory. This derivation ispossible realizing the phase-space density distribution using N >> 1 elementsof equal mass m. As in Lynden-Bell theory, the authors assume that initiallythe system is made of a discrete set of density levels η1, η2, ... occupyingphase-space volumes V1, V2, .... Then the overall number of elements thatrealize a phase-space density ηJ is NJ = VJηJ/m.

As a second step, the authors test the transitivity of the theories anddemonstrate that both the statistical-mechanical theories of violent relax-ation by Lynden-Bell and Nakamura give a negative conclusion. This non-transitivity is a result of the phase mixing that occurs when the systemrelaxes. As the fine-grained phase-space density filaments become thinnerand thinner, the system is better described in terms of the coarse-grainedphase-space density. Any further relaxation of the system should be there-fore considered in terms of the coarse-grained phase-space density which, aswe have seen, would yield different results with respect to the predictionsbased on the initial fine-grained phase-space density. This is a worrying as-pect of these theories, as it is easy to imagine a scenario where part of thesystem mixes, then it fluctuates, and then mixes once again. The predictionsof the theory, based on the fine-grained density, will then provide a wrongresult.

Arad and Lynden-Bell believe that these difficulties and ambiguities inexactly applying the statistical mechanics to collisionless particles which obeyBoltzmann’s equation, teach us an important lesson. The non-transitivitythat they have shown is a sign that a kinetic description of violent relaxationis probably incomplete, as the equilibrium is dependent on the evolutionarypath of the system. Instead, what is probably needed is a dynamical approachto the problem. Indeed, most of the above difficulties are circumvented if,instead of aiming to derive a universal most probable state, we reduce ouraim to that of finding an appropriate and useful evolution equation for thecoarse-grained distribution function.

9.11. THE PHASE-SPACE STRUCTURE OF DARK MATTER HALOS103

9.11 The phase-space structure of Dark Mat-

ter halos

Recent developments have been performed in order to get new insight intothe phase-space structure of dark matter halos.

Dark-matter halos are the basic entities in which luminous galaxies formand live. Their gravitational potentials have a crucial role in determining thegalaxy properties. While many of the systematic features of halo structureand kinematics have been revealed by N-body simulations, the origin of thesefeatures is still not understood, despite the fact that they are governed bysimple Newtonian gravity.

The halo density profile ρ (r) is a typical example. It is found in thecold dark matter (CDM) simulations to have a robust non-power-law shape(originally, Navarro et al., 1997; Power et al., 2003 and Hayashi et al., 2004and references therein), with a log slope of -3 at large radii, varying graduallytoward -1 or even flatter at small radii. The slope shows only a weak sensitiv-ity to the cosmological model and to the initial fluctuation power spectrum(e.g., Colin et al., 2004 and Navarro et al., 2004), indicating that its origin isdue to a robust relaxation process rather than specific initial consitions. Inparticular, violent relaxation (Lynden-Bell, 1967) may be involved in shapingup the density profile, but we have no idea why this profile has to assumethe specific NFW (Navarro, Frenk & White) shape.

An interesting attempt to address the origin of the halo profile has beenmade by Taylor & Navarro (2001), who measured a poor-man phase-spacedensity profile by fTN (r) = ρ (r) /σ3 (r), and found that it displays an ap-proximate power-law behavior, fTN ∝ r−α with α = 1.87, over more thantwo decades in r. Using the Jeans equation, they showed that this power lawpermits a whole family of density profiles, and that a limiting case of thisfamily is a profile similar to NFW, but with an asymptotic inner slope of-0.75 as r → 0. This scale-free behavior of fTN (r) is intriguing, and it moti-vates further studies of halo structure by means of phase-space density (i.e.,Arad, Dekel & Klypin, 2004). Other studies have subsequently confirmedthat ρ/σ3 is a power law in radius, but estimates of the exponent differsomewhat from the Taylor-Navarro value: α = 1.95, 1.90 or 1.94 accordingto Rasia et al. (2004), Ascasibar et al. (2004) and Dehnen &McLaughlin(2005), respectively.

Hartwick (2007) by starting from the ’pseudo’ phase-space density Q (r) =ρ/σ3, presents a halo model which posses a constant Q core followed by aradial CDM-like power-law decrease in Q. The space density profile derivedfrom this model has a constant density core and falls off rapidly beyond.


Modelling dark matter halos with constant density cores is not new butit is usually done by parametrizing the density profile (see, e.g., Burkert,1995, and references therein). Here the goal is to follow the effects of afinite primordial ’pseudo’ phase-space density upper limit. Thus the constantdensity core results from a solution of the Jeans equation with a parametrizedphase-space density profile. In fact, simple analytical arguments suggestthat the effects of a primordial phase-space density bound should be seen inpresent structures even after many mergings (e.g., Dalcanton & Hogan, 2001).In the absence of cosmological simulations which include such a primordialbound, the author rely on the good agreement of predictions from his simplemodel with observations to argue that standard CDM simulations and hencethe NFW profile may not be giving a complete picture.

Finally, in a very recent paper, Vogelsberger et al., 2007 present a newand completely general technique for calculating the fine-grained phase-spacestructure of dark matter Galactic halo. Rather than improving simulationssimply by increasing the number of particles, they attach additional infor-mation to each particle, namely a phase-space distortion tensor which allowsthem to follow the evolution of the fine-grained phase-space distribution inthe immediate neighbourhood of the particle. They introduce the GeodesicDeviation Equation (GDE) as a general tool for calculating the evolutionof this distortion along any particle trajectory. The projection from phase-space to configuration-space yields the density of the particular CDM streamthat a selected particle is embedded in. This technique appears to be generaland powerful in order to analyse phase-space structures embedded in differ-ent potentials. More sophisticated cosmological simulations should lead to afuller understanding of the dark matter halo structures in the phase-space.

9.12 Conclusion

We have analyzed the thermodynamics of violent relaxation in collisionlesssystems from the point of view of the DF in µ-space. Different coarse-grainedDFs proposed by different authors have been taken into consideration.

It appears that for a collisionless gas of a unique mass specie there is notsignificant discrepancies among the different approaches which converge tothe same DF at the end of relaxation process.

The main problem is to avoid the non observed mass segregation in thecase of multi-species composition. On this topic the results are very differentand are depending on the shape and size one chooses for µ-space tiles.

Our run on the literature about this argument is started from the seminalpaper of Lynden-Bell (1967) and is closed by that of the same author (Arad

9.12. CONCLUSION 105

& Lynden-Bell, 2005). According to this last criticism we can deduce thefollowing focal points:

• the real relaxation is probably more complicated in respect to the idealtreatements until now considered. In them there is an inconsistency orat best a lack of transitivity;

• the fate of the system at the end of the violent relaxation is not defini-tively assigned;

• anyways it seems that if we take off the phase-space volume conserva-tion, a single Gaussian would characterize the proper DF of a collision-less gas with a mass mixture.

A number of authors have attempted to use entropy arguments to drawconclusions about the end state of collisionless relaxation (Tremaine et al.,1986; White and Narayan, 1987; Stiavelli, 1987; Stiavelli and Bertin, 1987;Spergel and Hernquist, 1992 and Soaker, 1996). In addition to the difficultyjust mentioned, these studies must deal with the fact that the equilibriumstate of a stellar system strongly depends on the initial conditions. If thecorrect final state is to be singled out by an entropy maximization, the de-pendence of the final state on the initial state must somehow be translatedinto a set of constraints. Inferring the final state then becomes equivalentto specifying these constraints. There is currently no clear understanding ofhow this can be done, and it is not even certain that the exercise would bephysically enlightening (Merritt, 1999). It should be underlined that all theconsiderations of the present review concern a single component of collision-less ingredients. But a real galaxy has its baryonic component embeddedinside a dark matter halo. Then the problem we are dealing with is to takeinto account a two-component system. In our knowledge an equivalent treat-ment as that before considered does not exist even if some efforts have beendone (see, e.g., Stiavelli & Sparke, 1991).

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