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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

  • 2

  • 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 Einsteins Equations 92.1 General Relativity as Geometric Theory . . . . . . . . . . . . 92.2 On Einsteins equations by analogy . . . . . . . . . . . . . . . 112.3 On the meaning of Einsteins 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 CDM 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, relax, ff . . . . . . . . . . . . . . . . 60

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

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

  • CONTENTS iii

    8.4 Statistical Mechanics:Boltzmanns Equation for a collisionless particle system . . . . 62

    8.5 Analogy stars-pendulums in phase space . . . . . . . . . . . . 638.6 Movement of phase points in -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 Shus Criticism . . . . . . . . . . . . . . . . . . . . . . . . . . 81

    9.6.1 Shus approach . . . . . . . . . . . . . . . . . . . . . . 829.6.2 Incomplete Relaxation . . . . . . . . . . . . . . . . . . 87

    9.7 Mass Segregation . . . . . . . . . . . . . . . . . . . . . . . . . 879.8 Kull, Treumann & Bohringers Criticism . . . . . . . . . . . . 89

    9.8.1 Kull, Treumann & Bohringers approach . . . . . . . . 899.9 Nakamuras Criticism . . . . . . . . . . . . . . . . . . . . . . . 93

    9.9.1 Nakamuras 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

    xx

    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) =x

    yiA(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(x1 + dx1, x2 + dx2, .....xn + dxn) (1.7)

    Due to the differentation law we have:

    dyi =yi

    xdx; (i, = 1, 2, ....n) (1.8)

    which yields:

    Bi(y) =yi

    xA(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) =x

    yiA(, x) (1.10)

    which means (by convention);1

    Bi =x

    yiA (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) =x

    yix

    yjA(, ; 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 =x

    yix

    yjA (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

    xA(; x) (1.14)

    which by convention means:

    Bi =yi

    xA (contravariant law) (1.15)

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

    Bij =yi

    xyj

    xA (1.16)

    In the case of a tensor which transform according to:

    Bji (y) =x

    yiyj

    xA(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: