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Inner Caustics of Cold Dark Matter Halos

Permanent Link: http://ufdc.ufl.edu/UFE0021102/00001

Material Information

Title: Inner Caustics of Cold Dark Matter Halos
Physical Description: 1 online resource (106 p.)
Language: english
Creator: Natarajan, Aravind
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: catastrophe, caustic, dark
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We investigate the caustics that occur when there is a continuous infall of collisionless matter from all directions in a galactic halo, in the limit of negligible velocity dispersion. We show that cold infall will necessarily produce a caustic without the need for simplifying assumptions. We discuss the complications that exist in real galactic halos and argue that caustics still exist in real galaxies. There are two kinds of caustics - outer and inner. The outer caustics are thin spherical shells surrounding galaxies. Inner caustics have a more complicated structure that depends on the spatial angular momentum distribution of the dark matter. We provide a detailed analysis of the structure of inner caustics for different initial conditions. The presence of dark matter caustics can have astrophysical effects. We explore a possible connection between the presence of a dark matter caustic and the location of the Monoceros Ring of stars. We show that there exist two mechanisms that can increase the baryonic density in the neighborhood of a dark matter caustic: One is the action of viscous torques on the baryonic material, while the other is the adiabatic deformation of star orbits as the caustic expands in radius. Finally, we investigate the possibility that caustics may be detected by observing the gamma rays that result from particle annihilation in caustics. We show that if the dark matter is composed of SUSY neutralinos, the annihilation flux has a distinct signature.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Aravind Natarajan.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Sikivie, Pierre.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021102:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021102/00001

Material Information

Title: Inner Caustics of Cold Dark Matter Halos
Physical Description: 1 online resource (106 p.)
Language: english
Creator: Natarajan, Aravind
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: catastrophe, caustic, dark
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We investigate the caustics that occur when there is a continuous infall of collisionless matter from all directions in a galactic halo, in the limit of negligible velocity dispersion. We show that cold infall will necessarily produce a caustic without the need for simplifying assumptions. We discuss the complications that exist in real galactic halos and argue that caustics still exist in real galaxies. There are two kinds of caustics - outer and inner. The outer caustics are thin spherical shells surrounding galaxies. Inner caustics have a more complicated structure that depends on the spatial angular momentum distribution of the dark matter. We provide a detailed analysis of the structure of inner caustics for different initial conditions. The presence of dark matter caustics can have astrophysical effects. We explore a possible connection between the presence of a dark matter caustic and the location of the Monoceros Ring of stars. We show that there exist two mechanisms that can increase the baryonic density in the neighborhood of a dark matter caustic: One is the action of viscous torques on the baryonic material, while the other is the adiabatic deformation of star orbits as the caustic expands in radius. Finally, we investigate the possibility that caustics may be detected by observing the gamma rays that result from particle annihilation in caustics. We show that if the dark matter is composed of SUSY neutralinos, the annihilation flux has a distinct signature.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Aravind Natarajan.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Sikivie, Pierre.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021102:00001


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1f0056ae2ccec00aaf7af7ead7fc7af9
5eff620051a81260d72854d4aa1b563726045084







INNER CAUSTICS OF COLD DARK MATTER HALOS


By

ARAVIND NATARAJAN



















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007

































S2007 Aravind Natarajan









ACKNOWLEDGMENTS

I would like to thank my parents for their love and support. I thank my Ph.D

supervisor Professor Pierre Sikivie and the other member of my supervisory committee,

Professors Richard Woodard, James Fry, James Dufty and Vicki Sarajedini. I am

especially grateful to Richard Woodard for his concern and advice on many occasions.

You will definitely be missed! It is my pleasure to thank Sush for all the great times

we had together. I also acknowledge my fellow physics nerds Jian, Sung-Soo, KC, Ian,

N ii',. and Emre. I thank the Institute for Fundamental Theory, University of Florida for

providing partial financial support.









TABLE OF CONTENTS
page

ACKNOW LEDGMENTS ................................. 3

LIST OF TABLES ....................... ............. 6

LIST OF FIGURES .................................... 7

ABSTRACT . . . . . . . . . . 9

CHAPTER

1 INTRODUCTION ................................ 10

2 THE FORMATION OF CAUSTICS ................... .... 15

2.1 Outer Caustics ...................... ......... 17
2.2 Inner Caustics ...................... .......... 19
2.3 Existence of Inner Caustics ................... ....... 19
2.4 Possible Complications that Affect the Existence of Caustics . . 21
2.4.1 Existence of a Large Number of Flows ............ .. 21
2.4.2 Presence of Small Scale Structure .................. .. 22
2.4.3 Gravitational Scattering by Inhomogeneities ..... . 23
2.5 Difficulty of Resolving Caustics in N-Body Simulations . .... 24
2.6 Possible Evidence for the Existence of Dark Matter Caustics . ... 25
2.6.1 Rises in the Rotation Curves of Spiral Galaxies . . ... 25
2.6.2 Rotation Curve of the Milky Way Galaxy . . 26
2.6.3 Triangular Feature in the IRAS Map ... . . 26
2.7 Discussion . ............... ............ .. 26

3 THE STRUCTURE OF INNER CAUSTICS ......... . .. 29

3.1 Linear Initial Velocity Field Approximation ... . . 29
3.2 Simulation . ............... ........... .31
3.3 Tricusp Ring .................. .......... .. 32
3.3.1 Axially Symmetric Case .................. .... .. 33
3.3.2 Perturbing the Initial Velocity Field ... . . 34
3.3.2.1 Effect of the gradient terms g1 and 92 ........... 34
3.3.2.2 Effect of a random perturbation . . 34
3.3.2.3 Effect of radial velocities .................. .. 34
3.3.3 Modifying the Gravitational Potential ................ .. 35
3.3.3.1 NFW profile .................. ....... .. 35
3.3.3.2 Breaking spherical symmetry ............... .. 36
3.4 General Structure of Inner Caustics .................. ..... 36
3.4.1 Axially Symmetric Case .................. .... .. 36
3.4.2 Infall without Axial Symmetry ...... .......... .... 39
3.5 Discussion ............... .............. .. 42









4 A POSSIBLE CONNECTION BETWEEN A DARK MATTER CAUSTIC
AND THE MONOCEROS RING OF STARS .. ...............

4.1 Angular Velocity of Gas in Circular Orbits Close to a Caustic Ring ...
4.2 Angular Momentum Transport by Viscous Torque .............
4.3 Effect on Star Orbits . . . . . . . .
4.3.1 Orbit Stability ...........................
4.3.2 Resonances ................... ...........
4.3.3 Density Enhancement: Circular Orbits .. .............
4.3.4 Density Enhancement: Non-circular Orbits .............
4.4 D discussion ..... . . . . . . .

5 WIMP ANNIHILATION IN INNER CAUSTICS .. ..............

5.1 A nnihilation Flux . . . . . . . .
5.1.1 Estim ating S . . . . . . . .
5.1.2 Estimating < EM > .. ......................
5.2 Comparing the Signal with the Background .. ..............
5.3 D discussion . . . . . . . . .

6 CO N CLU SIO N S . . . . . . . . .

APPENDIX

A CATASTROPHE THEORY .. ........................

A.1 Fold and Cusp Catastrophes . ...................
A.1.1 Fold: Corank = 1, Codimension = 1 ..................
A.1.2 Cusp: Corank = 1, Codimension = 2 ................
A.2 Higher Order Catastrophes . ....................
A.2.1 Swallowtail: Corank = 1, Codimension = 3 ..............
A.2.2 Butterfly: Corank = 1, Codimension = 4 ...............
A.2.3 Elliptic Umbilic: Corank = 2, Codimension =3 ...........
A.2.4 Hyperbolic Umbilic: Corank = 2, Codimension =3 .........

B PROPERTIES OF THE TRICUSP CAUSTIC RING ......


B.1 Density Near a Tricusp Caustic Ring .. ........
B.1.1 Case 1: Points in the z = 0 Plane .. ......
B.1.2 General Case: z : 0 .............
B.1.3 Solving the Quartic Equation to Obtain the Real
B.2 Mass Contained in the Ring .. ............
B.3 Effect on Surrounding B i .............


Roots of T


REFERENCES ................. ................. ....

BIOGRAPHICAL SKETCH ........................................









LIST OF TABLES


Table page

5-1 N,/m' in units of 10-4 GeV-2 for mx 50 GeV ................ .. 68

5-2 N,/m2 in units of 10-4 GeV-2 for m = 100 GeV ................ ..68

5-3 N,/m2 in units of 10-4 GeV-2 for mx = 200 GeV ................ ..68

5-4 Peaks of < EM > for (a 8.0, p =0.1,q 0.2) kpc .............. ..70









LIST OF FIGURES


Figure page

2-1 Dark matter trajectories forming a cusp catastrophe. ............. 27

2-2 Effect of averaging the density over a finite volume. ........... .. .. 28

3-1 Infall of a cold collisionless shell: Antisymmetric M. . . 43

3-2 Dark matter flows forming an inner caustic. ................ . 44

3-3 Tricusp ring with non-zero gl and g2. .................. .. 44

3-4 Effect of a random perturbation. . ............... . ... 45

3-5 Tricusp ring: NFW potential. .................. ....... 45

3-6 Tricusp ring: Non-spherically symmetric gravitational potential. . ... 45

3-7 Case 1: Infall of a shell. Irrotational flow with axial symmetry, gi < g, .. 46

3-8 Case 2: Infall of a shell. Irrotational flow with axial symmetry, Igll > g .. 47

3-9 Cross section of the inner caustic produced by an irrotational axially symmetric
velocity field (Case 1: Igll < g,). ............... ..... .. 48

3-10 Cross section of the inner caustic produced by an irrotational axially symmetric
velocity field (Case 2: g > g). ....... ........... ... ... 48

3-11 Caustic in cross section, for increasing c3. .................. ... 49

3-12 Dark matter flows in xy cross section. ............... .... 49

3-13 Gradient type caustic without axial symmetry. ..... . ... 50

3-14 Formation of the hyperbolic umbilic catastrophe. ............. 51

3-15 Caustic in cross section, for different (gi, g2) ................ 52

3-16 z = 0 sections of the caustic, showing the transformation from gradient type to
curl type ..................... .. ...... ......... .53

3-17 Transformation of a gradient type caustic to a curl type caustic. . ... 53

4-1 Rotation curve close to a caustic ring. ............. ... 64

4-2 Smooth potential V(r). ............... ............ 64

4-3 Adiabatic deformation of star orbits. .............. .... 65

5-1 Emission measure averaged over a solid angle Af = 10-5 sr, for three sets of
caustic parameters ................ ............. .. 72









5-2 Annihilation flux for the three different sets of caustic parameters for mr = 50
GeV, compared with the EGRET measured diffuse background. . . 73

A-i The 04 singularity. .................. ... .......... 86

A-2 Fold catastrophe: Equilibrium surface and bifurcation set. . .... 86

A-3 Cusp catastrophe. .................. ... ........... 87

A-4 Swallowtail catastrophe. ............... ........... 87

A-5 Butterfly catastrophe. ............... ............ .. 88

A-6 Elliptic umbilic catastrophe: Bifurcation set. ................. .. 88

A-7 Hyperbolic umbilic catastrophe: Bifurcation set. ............... 88

B-l Dark matter trajectories forming a tricusp ring (in cross section). . ... 99

B-2 Tricusp caustic in pz cross section. .................. .... 99

B-3 Modified rotation curve. .................. .......... 99









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

INNER CAUSTICS OF COLD DARK MATTER HALOS

By

Aravind Natarajan

August 2007

C(!i ': Pierre Sikivie
Major: Physics

We investigate the caustics that occur when there is a continuous infall of collisionless

matter from all directions in a galactic halo, in the limit of negligible velocity dispersion.

We show that cold infall will necessarily produce a caustic without the need for simplifying

assumptions. We discuss the complications that exist in real galactic halos and argue that

caustics still exist in real galaxies. There are two kinds of caustics outer and inner. The

outer caustics are thin spherical shells surrounding galaxies. Inner caustics have a more

complicated structure that depends on the spatial angular momentum distribution of the

dark matter. We provide a detailed analysis of the structure of inner caustics for different

initial conditions. The presence of dark matter caustics can have astrophysical effects.

We explore a possible connection between the presence of a dark matter caustic and the

location of the Monoceros Ring of stars. We show that there exist two mechanisms that

can increase the baryonic density in the neighborhood of a dark matter caustic: One is

the action of viscous torques on the baryonic material, while the other is the adiabatic

deformation of star orbits as the caustic expands in radius. Finally, we investigate the

possibility that caustics may be detected by observing the gamma rays that result from

particle annihilation in caustics. We show that if the dark matter is composed of SUSY

neutralinos, the annihilation flux has a distinct signature.









CHAPTER 1
INTRODUCTION

There are compelling reasons to believe that most of the matter in the Universe is

in a non-luminous form commonly known as "dark 1 il Il i the composition of which

remains a mystery. The study of dark matter and how it influences the formation and

evolution of galaxies is a central topic being pursued by both theorists and experimentalists.

Theoretical work involves studying the phenomenology of possible dark matter particles

that arise in theories that go beyond the standard model of particle physics, as well as

detailed computer simulations that test theories of structure formation. Experimental

work involves designing and calibrating detectors, improving background rejection,

developing efficient means of data analysis, etc.

The first indication of the existence of dark matter or i:::--:ig 1 i--' was in 1933,

when F. Zwicky [113] measured the radial velocities of seven galaxies in the Coma cluster.

The measured velocities differed from the mean radial velocity of the cluster by a large

amount (about 700 km/s RMS deviation). The luminous mass of the Coma cluster is

insufficient to account for such a large velocity dispersion. Zwicky therefore inferred the

existence of dark matter, whose presence is felt only gravitationally. With the currently

accepted value of the Hubble parameter, there is about 50 times as much matter in the

Coma cluster, as inferred from the cluster luminosity. A similar conclusion was reached

from Babcock's work [7] on the rotation velocities of M31 and by J. Oort's measurements

[71] of rotation and surface brightness of the galaxy NGC 3115.

Further evidence for dark matter in galaxies came in the 1970s with the study of the

rotation curves of spiral galaxies by V.C. Rubin and W.K. Ford [83] and by Roberts and

Whitehurst [82]. The measurements showed that the rotation velocity remains constant

with distance from the center even well outside the luminous disk. If the luminous matter

accounted for all the matter in the galaxy, one would expect to see a Keplerian fall-off of

rotation velocity. The observations contradict this and provide support for the existence of









non-luminous matter. Ostriker and Peebles [72] concluded from their study of the stability

of galactic disks that dark matter is present in the form of a i ,!i," in galaxies. [84, 104]

give a historical account.

Let us consider our nearest large galaxy M31 in the constellation Andromeda. The

Milky Way and M31 are separated by 0.73 Mpc and are approaching each other with a

line of sight velocity of 119 km/s. For the purpose of obtaining an approximate estimate

of the mass of these two galaxies, let us assume that they are point like. Further, let

us assume zero angular momentum. Although the actual physics is more complicated

(galaxies are not point masses), these assumptions allow us to make a rough calculation of

the mass in our galaxy. Assuming Newtonian gravity, the distance of separation r obeys

the equation
G (Mi + 11-.)
G(Ar1 +3(1 1)


It is easy to derive parametric expressions for the distance of separation r and the time t:


r(() 1ro[1 +cos(]
2
3
t(() ( 1 [( + sin (1 2)

using which we obtain for the velocity

r(t) sin (( +sin ()
t (1 +cos )2

Setting v = -119 km/s when t = 1.4 x 1010 years and r = 0.73 Mpc, we find ( 1.53 and

so M1 + i11. = 2.5 x 1012 M.. Assuming that the Milky Way galaxy and M31 have equal

masses, the mass of our galaxy is t 1012 Me (the acceleration of the Universe makes this

an underestimate). Since the mass in luminous matter in the Milky Way is t 1011 Me, we

find from our simple calculation that 90 of the mass in our galaxy is dark.

In recent years improved measurements of the properties of galaxies, as well as

theoretical input have made a very strong case for the reality of dark matter. If the









evolution of density perturbations into galaxies and clusters is governed by Einstein's

theory of general relativity, then the observed large scale structure requires the existence

of dark matter. Also, the matter must be 'cold', meaning that the primordial velocity

dispersion of the dark matter particles is very small. Observations of the gravitational

lensing of distant objects by an intervening cluster can be used to estimate the mass of

the cluster. The inferred mass once again, is far in excess of what would be expected if all

the mass were luminous. Current results from gravitational lensing are used to probe dark

matter substructure in our galaxy.

Perhaps the most accurate estimate of the amount of dark matter in the Universe

was obtained by the Wilkinson Microwave Anisotropy Probe (WMAP) satellite [98, 99].

The WMAP experiment has measured the temperature power spectrum of the cosmic

microwave background radiation to high precision. The results again point to the existence

of dark matter and are in agreement with the results of other experiments. Recently,

observations of 1E 0657-558 (the "bullet (l
matter [26]. This event is a merger of two clusters of galaxies. The intra-cluster gas

is collisional and heats up, emitting X-rays. If a collisionless component is present, it

would simply pass through and become separated from the gas. X-ray measurements of

the cluster are used to track the gas, while weak gravitational lensing tracks the total

mass. The observations show that the center of mass does not coincide with the luminous

matter.

Dark matter cosmology assumes the correctness of general relativity. General

relativity is consistent with solar system tests like the precession of planetary orbit

perihelia, or the bending of starlight by the sun's gravitational field [107]. The theory has

been tested to extremely high precision in the case of the Hulse-Taylor binary pulsar.

There exist cosmological theories that do not include dark matter (except perhaps

the massive neutrinos of the standard model). In these theories, gravity is not described

by Newton's law (or general relativity), but by a modification of it. The first theory that









included a modification of Newtonian gravity was proposed by M. Milgrom [59] in 1981.

More recently, a relativistic theory of modified gravity consistent with the tests of general

relativity, was proposed by J. Bekenstein [8]. However, in this work we will assume the

validity of Newtonian gravity and the existence of cold dark matter.

The most commonly cited cold dark matter candidates are the axion and the Weakly

Interacting Massive Particle (WIMP). These particles have the distinction that they

were proposed for purely particle physics reasons, though they are excellent dark matter

candidates. The axion solves the strong CP problem of QCD, while WIMPs arise in

supersymmetric extensions of the standard model. Axions are being searched for by the

Axion Dark Matter eXperiment (ADMX) [6, 33], while WIMP detection experiments

include DAMA/NaI [12-14], DAMA/LIBRA [15], CDMS [22], XENON [4], EDELWEISS

[85], DRIFT [97], Zeplin [3], CRESST [1] and others.
A central problem of structure formation studies is the question of how cold dark

matter is distributed in the halos of galaxies. The simplest halo model is the isothermal

model, which assumes that the dark matter particles form a self gravitating, thermalized

sphere. Some of the predictions of this model have been confirmed, notably the flatness of

rotation curves and the existence of core radii. However, it is not possible for all the dark

matter to be thermalized [91]. Even if dark matter had thermalized in the past by violent

relaxation [51], galaxy formation is an ongoing process. The late infall of dark matter

will cause non-thermal streams, as we show in Chapter 2. Since there is no evidence for

violent relaxation occurring tod iw, particles that have fallen into the inner regions of the

halo (which contain the most substructure) relatively recently may be expected to be

non-thermal (cold).

Another approach is to carry out N-body simulations of the formation of galactic

halos, on supercomputers. This approach is powerful since it requires no assumptions

of symmetries or special initial conditions, and presumably gives the correct results if

N is large enough. However, current simulations only have t 109 particles. If the dark









matter is composed of WIMPs, the number of particles in a Milky Way size galactic halo

is a 10'j- \ /100 GeV t 1067. If the dark matter is composed of axions, the number of

particles 10'-\ /10-5 eV W 10s3. Also, the mass per particle in simulations is very large

(several million solar masses) which can introduce spurious effects.

In this work, we will make use of the fact that cold dark matter particles exist on

a thin, three dimensional hypersurface in phase space. The phase space properties of

collisionless systems imply the existence of high density structures in physical space, called

caustics. In ('!i ipter 2, we show that cold infall of dark matter produces caustics and

provide possible evidence for the existence of caustics. We also examine the relevance of

caustics in real galaxies. In ('!i ipter 3, we provide a detailed description of the structure of

inner caustics, for different initial conditions.

('! ipter 4 and ('! ipter 5 deal with the .,-I i .1,-i-i 1 effects of dark matter caustics.

In C'!i ipter 4, we explore the possibility of a connection between a dark matter caustic

and the Monoceros Ring of stars. We show that there exist two mechanisms by which the

presence of a caustic could increase the density of baryonic matter in its neighborhood,

possibly explaining the formation of the Ring.

In C'!i ipter 5, we investigate the possibility that caustics may be detected by

indirect means (i.e., by observing gamma rays from the annihilation of particles in the

caustic). We compute the annihilation flux from a nearby caustic, assuming that the dark

matter is made up of supersymmetric WIMPs and compare the signal with the expected

background.









CHAPTER 2
THE FORMATION OF CAUSTICS

Dark matter caustics are regions of high density and form when the dark matter

particles are collisionless and have low velocity dispersion. Dark matter particles are

almost collisionless by definition (they have no electromagnetic or strong charge). The

primordial velocity dispersion of the leading dark matter candidates is expected to be very

small. The primordial velocity dispersions for axions(6va) and WIMPs(6vw) have been

estimated [94] to be


6va = 3.10-17 (10-5eV (t)2/3


6V, = 10- c /2 to (2-1)

where to is the present age of the universe and c is the speed of light.

Consider the continuous infall of dark matter particles with negligible velocity

dispersion from all directions in a galactic halo. Since the dark matter particles have

negligible velocity dispersion, these particles exist on a thin three dimensional hypersurface

in phase space, called the phase space sheet (the thickness of the phase space sheet is the

velocity dispersion) [94]. For collisionless matter, the density in phase space following the

trajectory of a particle is conserved by Liouville's theorem. This ensures that the phase

space sheet is continuous. Also, the phase space trajectories do not self-intersect. These

conditions constrain the topology of the phase space sheet irrespective of the details of the

galaxy formation process.

In order to obtain the density of dark matter particles in physical space, we must

make a mapping from phase space to physical space. Caustics are regions where this

mapping is singular. The singularities that occur in mappings are known as catastrophes

and have been well studied in the context of Catastrophe Theory (Appendix A). In

general, caustics are made up of sections of the elementary catastrophes [101].









Caustics are well known in the context of optics. Everydy,- occurrences of light

caustics include the heart shaped pattern that forms in a tea cup, the pattern of light

that forms on the bottom of a swimming pool, a rainbow, the brilliant points of light

that occur when sunlight reflects off a body of water, etc. Caustics form with light

because light is almost collisionless. Detailed work on optical caustics can be found in

[54, 68, 69, 103, 106]. In the context of cosmology, the study of caustics (or pancakes) was

pioneered by Y.B. Zeldovich, S.F. Shandarin, A.G. Doroshkevich, V.I. Arnold and many

others. [24, 30, 31, 48, 58, 88, 89].

Let us return to the phase space description of cold dark matter. If we may neglect

the thickness of the phase space sheet, we can assign to each particle, a three parameter

label d(al, a2, a3) which identifies the particle [94]. As an example, let us consider a

sphere with a conveniently chosen radius such that each particle in the flow passes through

the sphere once. Then, we may label the particles of the flow by the three parameters

(r, 0, Q) where r is the time when the particle crossed the reference sphere and 0, Q are the
co-ordinates of the point where the particle crossed the reference sphere. In our example,

the particle which crossed the sphere at time 7r at location 0, Q will be labeled by (7r, 0, Q)

at all times t. In terms of the physical space co-ordinates (x, y, z), the density of particles

d is [94]

d(t, x) d Nldd (Ai(t, X)) 1 (2-2)
Sdada2da3 del (|) ta:)
The sum over d1 is required because the mapping from (r, 0, Q) space to (x, y, z) space is

rn, ,iv-I,,-one. Caustics are locations where the Jacobian factor Idet (ax/ad) I 0. In the

limit of zero velocity dispersion, these are points of infinite density.

The particle trajectories in a galactic halo are characterized by two turnaround

radii (i.e., there are two points at which the radial velocity vanishes). Caustics occur at

both turnaround radii. The caustics that occur at the outer turnaround radii are called

'outer caustics'. They are topological spheres surrounding galaxies and typically occur on

scales of 100's of kpc, for a Milky Way size galaxy. The caustics that form at the inner









turnaround radii are called inner caustics. The inner caustics have a more complicated

structure and typically occur at distances ~ 10's of kpc. The importance of inner caustics

was first emphasized by P. Sikivie [94].

Fillmore and Goldreich [35] and Bertschinger [16] showed that the evolution of dark

matter halos is self-similar if the primordial overdensity has the profile


T- -( (2 3)

where i. is the mass inside a sphere of initial radius ri, at time ti, long before halo

formation started, and 611 is the excess mass due to the overdensity. .1, and e are

adjustable parameters. Self-similarity means that the phase space distribution of dark

matter is time-independent after rescaling all distances by a characteristic length scale

R(t), and all velocities by R(t)/t where t is the time since the Big Bang. The self-similar

model was modified to include the effect of non-zero angular momentum by Sikivie,

Tkatchev and Wang [92, 93]. With this modification, the locations of the inner caustics

are predicted by the theory.

2.1 Outer Caustics

As cold dark matter particles fall in and out of the gravitational potential well of the

galaxy, the phase space diagram acquires a number of 'folds', the projections of which

form catastrophes. Let us consider the simple case of isotropic infall, wherein particles

follow radial orbits. While dark matter in realistic halos is expected to carry angular

momentum, the infall is approximately isotropic in the outer regions of the halo (i.e.,

at distances that are large compared to the point of closest approach of the particles).

Close to the outer turnaround radius, we can therefore assume isotropy. Following our

convention, we label each particle by T, the time when it passed through the reference

sphere. 0, 0 are not relevant for isotropic infall. Similarly, the physical space density is

a function of only r = x2+ y2 + z2, when isotropy is assumed. The projection from

7 space to r space is singular when Or/ar = 0. Let us expand the radial position r in a









Taylor series in the variable r close to the outer turnaround radius ro


r-ro= -c( To)2 + (2-4)


where a is a constant and the particle labeled 7 To is at r ro (ar/ar = 0 when

7 To). The caustic is the locus of points that satisfy Or/r = 0 (i.e., the spherical

surface r = ro). The dark matter density is proportional to Or/7r ~ (ro r)-1/2 for r < ro

and is equal to 0 for r > ro. This is an example of a fold catastrophe (Appendix A).

There are caustics at each of the (n + 1) turnaround radii [16, 35] (There is no caustic

at the first turnaround radius because there is only an inflow of dark matter and no

outflow). This is best seen in the phase space diagram [94]. The projection of phase space

onto physical space is singular whenever the phase space sheet is tangent to velocity space

(i.e., at the turnaround radii). Thus in real galactic halos, we expect a series of outer

caustics which have the form of thin spherical shells.

In real galactic halos, the velocity dispersion of dark matter particles is small, but

not zero. In this case, the phase space sheet will have a finite thickness and the resulting

caustics will be spread over a distance 6r that depends on the magnitude of the velocity

dispersion. One must then average the density over a region of size ~ 6r, which renders

the density finite.

One of the best examples of outer caustics on galactic scales is the occurrence of shells

[18, 41, 53] around some giant elliptical galaxies which reside in rich clusters where the

merger probability is significant. These shells are caustics in the distribution of starlight.

They form when a dwarf galaxy falls into the gravitational potential of a giant galaxy

and is assimilated by it. The stars of the dwarf galaxy have a velocity dispersion that is

small compared to the virial velocity dispersion of the giant galaxy. The stars are therefore

sub-virial and presumably collisionless. The continuous infall of these cold collisionless

stars produces a series of caustics. The infall-outfall process repeats until the stars are

thermalized and lose their coldness. The result is a series of arcs (they are not complete









circles since the infall is not from all directions). This is strikingly evident in the case of

the galaxy NGC 3923 [18]. The occurrence of caustics with stars -ii--. -1- that the same

could occur with dark matter.

2.2 Inner Caustics

The caustics we have discussed so far are outer caustics, which we said, have the

topology of spheres. Let us now look at inner caustics. In the C'i plter 3, we will give a

detailed description of the structure of inner caustics, showing the catastrophes that form

for given initial conditions.

To see the formation of inner caustics, let us consider the infall of a cold flow of dark

matter particles. If the infall is spherically symmetric, the particle trajectories are radial

and the infall produces a singularity at the center. If instead, the dark matter particles

possess some distribution of angular momentum with respect to the halo center, the

particle trajectories are non-radial, particularly in the inner regions of the halo. Fig. 2-1

shows an example of non-radial infall, in xy cross section. The caustic is the envelope of

the family of dark matter trajectories (i.e., it is the locus of points tangent to the family

of trajectories). The dark matter density is enhanced along the envelope. The two curves

in Fig. 2-1 are fold catastrophes and their intersection is a cusp catastrophe. The caustic

curve and the density fall off are worked out in Appendix A.

2.3 Existence of Inner Caustics

We now provide a mathematical proof that the continuous infall of dark matter with

negligible velocity dispersion necessarily produces a caustic [63].

In accordance with our formalism, let us consider a cold collisionless flow of particles

and label the particles by three parameters (rT, 0, 0). Let us choose a reference sphere at

some convenient radius, such that each particle passes through the sphere once. r is the

time when the particle crossed the reference sphere and (0, Q) specify the location where

the particle crossed the reference sphere. To obtain the density is real space, one must









make a mapping from (7r, 0, 4) space to (x, y, z) space. The mapping is singular if

a(x,y,z) az a9z ax\
D = det x) = 0 (2-5)
O{T,0(p) OT (0 9O000

Consider the infall of a shell of dark matter particles with an arbitrary angular

momentum distribution. Since the tangential velocity vector on a 2-sphere must vanish

at two points at least, the angular momentum field has a maximum (otherwise it is

zero everywhere). Let us consider the point of closest approach of the particles with the

most angular momentum. Let us label these particles by (0o, 90). At the point of closest

approach, we have
dr 8.
r 0 (2-6)
a7 r 7T
for some (To, 0o, Qo) because the radial velocity vanishes at the point of closest approach.

Similarly, since the angular momentum does not vary to lowest order about (o, o00),


90 r 800=
ar a 0
=r 0 (2-7)
a4 r a4

for (To, 0o, 0) at the point of closest approach.

Define 0o = (To, 0o, Qo). Note that xYo = x(o) = 0 only in the limit of spherical

symmetry (i.e., when the inner caustic has collapsed to a singular point). For the more

general case of non-zero angular momentum, xo / 0. Equations 2-6 and 2-7 imply that

ax((ao)/o0, ax(do)/a94 and aS(yo0)/OT are all perpendicular to xo. Hence those three

vectors are linearly dependent which makes the Jacobian determinant Eq. 2-5 vanish.

Thus, we have shown that the point of closest approach of the particles with the most

angular momentum lies on a caustic. In general, the caustic is a surface.

So far, we have given a general description of caustics. Let us now consider various

complications that exist in real galactic halos [62].









2.4 Possible Complications that Affect the Existence of Caustics

2.4.1 Existence of a Large Number of Flows

The continuous infall of dark matter in a galactic halo results in a large, but discrete

set of velocities, which we refer to as 'flows'. The number of flows that exist at any given

location is the number of v--~i that dark matter particles can reach that location. Very

far from the galactic center but at distances smaller than the first turnaround radius (-iv

1 Mpc), there can be only one flow due to dark matter falling into the gravitational well

of the halo. At somewhat smaller distances, there are three flows that correspond to 3

v--,V- in which dark matter particles can reach that location (i) by falling in for the first

time, (ii) by falling from the opposite side and reaching the position under consideration

and (iii) by falling inward after reaching second turnaround. Similarly, at slightly smaller

distances, there are 5 flows, then 7 flows and so on. The number of flows at our location is

estimated to be of order 100.

After several infall times, the successive turnaround radii will be close to each other

because the potential does not change significantly during an infall time. Thus, the phase

space sheet is tightly wound in the inner regions of phase space. For an observer with

limited velocity resolution, it is possible that the velocity distribution appears to take

the form of a continuum, even though the microscopic structure is discrete, an effect

called 'phase mixing'. This effect is further exacerbated if the dark matter particles have

a significant velocity scatter. However, phase mixing occurs in phase space, not physical

space. This means that it is the inner regions of phase space that are smeared. In physical

space, the velocity distribution appears as a set of discrete flows superimposed over a

seemingly thermal continuum. This is best seen by drawing a vertical line through the

center of the phase space diagram [94] and counting the number of velocities. The result is

a large number of discrete values. The smaller velocity flows are very closely spaced, while

the larger velocity flows (the particles that populate the outer regions of the phase space









diagram) are well separated. High velocity, cold discrete flows can still be observed in the

inner regions of the halo.

It is important to note that the presence of angular momentum cannot prevent

particles from reaching the inner regions of the halo. This is because the angular

momentum field on a 2-sphere must vanish at two points at least. The particles

originating from these two locations follow radial orbits and pass through the center

of the halo. Thus for a continuous distribution of angular momentum, there are ahl--,v-

some particles that reach the inner regions. In the self-similar infall model with angular

momentum [92, 93], each flow contributes a few percent to the local dark matter density

at the sun's location. So, the contribution of a single cold flow to the dark matter density

at an arbitrary location may be quite small. However, the cold flow forms a caustic and

close to the caustic, the dark matter density can be large. Near a caustic, the contribution

of a single flow to the dark matter density can be as large as all the other flows put

together.

2.4.2 Presence of Small Scale Structure

In CDM cosmology, the spectrum of primordial density perturbations has power on all

scales. This implies that dark matter falling onto the galaxy may have clustered on smaller

scales, resulting in clumps. Indeed the hierarchical structure formation theory predicts

the existence of clumps, or sub-structure. The dark matter particles bound to a clump

fall in and out of the clump potential, forming a set of caustics, which are interesting in

their own right. These caustics are sometimes called 'micro-caustics' or 'micro-pancakes'

to distinguish them from the galactic caustics, which are the topic of this work.

The subflows belonging to the clump behave as a velocity dispersion from the point of

view of an observer. Since the clumps contain subflows, clumpy infall has a larger velocity

dispersion than smooth infall. For the individual flows to be resolved, it is important

that the phase space sheet does not become so thick that individual l~V, r- touch each

other. It is estimated that in a galaxy like ours, the velocity dispersion should be smaller









than about 30 km/s for some of the flows to be resolvable. It is unlikely that the infalling
clumps have such a large velocity dispersion. In any case, we expect a part of the smooth

(non-clumpy) component of the dark matter to have a small velocity dispersion.
2.4.3 Gravitational Scattering by Inhomogeneities

When a dark matter flow passes by a clump of matter, either baryonic or dark, the
particles of the flow are scattered by the gravitational potential of the clump. Baryonic
clumps include stars, globular clusters and giant molecular clouds. Consider the flow
of particles passing through a region populated by objects of mass M and density n.
Gravitational scattering causes each particle in the flow to have a random walk in velocity
space which results in a diffusion of the flow over a cone of angle AO [91]
/ bmax 4G2M2
(AO)2 dt b 2 nv 27b db
rbmin b2v
(10o-7( 3 3 Mt n2 b_.
1.8 x 10 ---c n ([ax) ( 10yr) (2-8)
v MeD bmin 10l0yr pc-3

where v is the velocity of the flow and t is the time over which it encountered the
inhomogeneities. For Giant Molecular Clouds, bmax and bmin are estimated to be ~ 1
kpc and 20 pc respectively. The result is that the flows of particles that spent most of
their past in the central parts of the galaxy could well have been washed out by scattering.
However, as we have emphasized, there are particles in the halo that have fallen into
the inner regions of the halo only a few times in the past. These particles which originate
from the outer regions of phase space are not scattered enough to lose their coldness. For
example, let us assume that a fraction f of the mass is in the form of clumps of mass M.
Then for a flow of particles with velocity v = 400 km/s, passing through the inner parts of
the galaxy (-v < 20 kpc) once, the effect of scattering is


(AO)2 ~ 10-f (M) (2-9)
\ A2f/(









where we assumed Mn = f 10-24 gm cm-3 and ln(bmx/bmin) ~ 4. Unless a significant

fraction of the mass is in clumps of mass 1010 Me or more, the effect of gravitational

scattering is not significant for at least some of the flows.

2.5 Difficulty of Resolving Caustics in N-Body Simulations

The existence and relevance of caustics in real galactic halos is sometimes disputed on

the basis of N-body cosmological simulations. However, the arguments that we have given

-,i--.- -1 that discrete flows and caustics must be present if the simulations have sufficient

resolution. Present dwi cosmological simulations have < 109 particles in them. While this

is a large number for most purposes, the claimed spatial resolution that may be obtained

is only of order ~ kpc (the actual spatial resolution may be smaller because of spurious

gravitational collisions). This means that the caustic density will be averaged over ~ 1 kpc

which would smear out the caustic to the extent that it is no longer relevant. To see the

effect of averaging the caustic density over a finite region, let us consider a cusp caustic

such as that shown in Figure 2-1.

Figure 2-2(a) shows the dark matter density close to one of the fold lines (averaged

over an infinitesimal volume) as a function of distance to the caustic along the line of

sight. Figures 2-2(b), 2-2(c) and 2-2(d) show the effect of averaging the density over a

cube of side = 0.01 pc, 0.1 pc and 1 pc respectively. A cut-off density of 100 GeV cm-3

was assumed. In Fig. 2-2(b), we see that the averaged density follows the true density

faithfully, but the density does not quite reach the cut-off value. Fig. 2-2(c) no longer

shows a sharp increase in density at the location of the caustic and Fig. 2-2(d) misses the

caustic completely.

The primordial velocity dispersion of WIMPs of mass mT is 6v = 3x 10-7 100 GeV/mx

km/s [94]. If the velocity dispersion is no larger than the primordial value, the resulting

caustics are spread over a region [95] 6a a 6v x a/v where a is the outer turnaround radius

(few hundred kpc) of the flow of particles forming the caustic and v is the speed of the









particles at the location of the (inner) caustic (few x 100 km/s) [50]

200 x 3 x 10-7 100 GeV
6a 0- kpc
500 mx,

S 10-4 pc 00GeV (2-10)
V x

which is much smaller than 1 pc. We conclude that caustics are very small scale

(sub-parsec) structures and are therefore difficult to resolve with cosmological simulations.

Another difficulty with N-body simulations is the large particle masses involved.

Since the mass of each particle can be several million solar masses, gravitational scattering

between two 'point' masses is not negligible. Close encounters of these spuriously massive

particles lead to large scattering angles [73], while gravitational scattering between two

WIMPs or two axions is completely negligible.

Note that when sufficient care is taken, it may be possible to resolve discrete flows

and caustics in cosmological simulations. The simulations of Stiff and Widrow [100] show

the existence of discrete flows in velocity space while the simulations of Bertschinger and

Shirokov [90] -,t'-: -1 the existence of caustics in physical space.

2.6 Possible Evidence for the Existence of Dark Matter Caustics

We have already mentioned that the existence of shells around elliptical galaxies

-ti -'- -i; that caustics of dark matter form in galactic halos. Here, we give some possible

observational evidence for the existence of dark matter caustics.

2.6.1 Rises in the Rotation Curves of Spiral Galaxies

If dark matter caustics exist in the galactic plane, they will perturb the gravitational

potential of the halo. As a result, the rotation velocity acquires a bump (i.e., a drop,

a sharp rise and a drop at the location of the caustic). W. Kinney and P. Sikivie have

combined the rotation curves (after rescaling them according to their mean rotation

velocities) of 32 well extended spiral galaxies [47]. The combined rotation curve shows

peaks at the locations expected in the self-similar infall model [92, 93], of the first and









second caustics. Thus while the bumps in the rotation curves of individual spiral galaxies

may have other explanations, the correlation of the bumps (specially at large distances,

outside the baryonic disk) is a good indication that they are due to caustics.

2.6.2 Rotation Curve of the Milky Way Galaxy

Binney and Dehnen [17] have conjectured that the anomalous behavior of the outer

Milky Way rotation curve could be explained if a ring of matter existed at ~13.6 kpc

which is close to the expected location of a caustic in the self-similar infall model of the

Milky Way [92, 93]. The rotation curve of the Milky Way galaxy also shows rises at the

expected locations of the caustics [96].

2.6.3 Triangular Feature in the IRAS Map

Caustics may reveal their presence by accreting ', ,i-!-; which could be visible. A

triangular feature in the galactic plane was identified by P. Sikivie in the IRAS map of the

galaxy [96]. The triangular feature is correctly oriented with respect to the galactic center.

Moreover, it coincides with the location of a rise in the rotation curve, strengthening the

hypothesis that the triangular feature is the imprint of a caustic on baryonic matter. As

will be shown in ('! Ilpter 3 (and Appendix A), the elliptic umbilic catastrophe has a cross

section that resembles a triangle.

2.7 Discussion

We have proved that the continuous infall of dark matter with low velocity dispersion

from all directions in a galactic halo produces caustics. There are two kinds of caustics -

outer and inner. The outer caustics are topological spheres surrounding galaxies, while the

inner caustics have a more complicated geometry.

We find that particles that have fallen into the central regions of the halo only a

few times in the past are not scattered significantly by inhomogeneities. These particles

may be expected to form caustics in physical space. The velocity space distribution

consists of discrete peaks in addition to the thermal continuum. The discrete flows

have distinct signatures in dark matter detectors. Each flow produces a peak in the









spectrum of microwave photons from axion to photon conversion in cavity detectors of

dark matter axions and a plateau in the recoil energy spectrum of nuclei struck by WIMPs

in WIMP detectors. As a result of the orbital motion of the Earth around the Sun, each

of these spectral features has a distinct annual modulation that depends on the flow

velocity. Caustics are also relevant to dark matter indirect searches [10, 43, 64, 78] and

gravitational lensing experiments [25, 36, 42, 70].








Figure, '-t -














1 0 1 2 3

Figure 2-1. Dark matter trajectories forming a cusp catastrophe.
Figure 2-1. Dark matter trajectories farming a cusp catastrophe.



























































-0.2 0 c



(a)


0.2


0.4


100



75



50



25



0









1U2



75



0



25



0


( pa


(b)


parse



(d)


Figure 2-2. Effect of averaging the density over a finite volume.


S= 0.1 paec

















-0.2 0 0,2 0.4
parse1c


6= par~


0.2


0.4









CHAPTER 3
THE STRUCTURE OF INNER CAUSTICS

In this chapter, we investigate the catastrophes that form when there is a steady

infall of cold collisionless matter from all directions in a galactic halo. We show that the

catastrophes that occur depend on the initial angular momentum distribution. Since the

inner caustics are made up of sections of the elementary catastrophes, the geometry of the

inner caustics depends on the initial angular momentum distribution. A brief discussion

of catastrophe theory is provided in Appendix A. Books on catastrophe theory include

[5, 23, 38, 79, 86].

We simulate a single cold flow of dark matter falling in a fixed gravitational potential.

The equations of motion are solved numerically and the Jacobian of the map from the

space of initial co-ordinates to the space of final co-ordinates is computed at every location

[63]. The zeroes of the Jacobian form the caustic surface. In Chapter 2, we showed that

the infall of a cold flow necessarily produces singular points. Here, we give a detailed

description of the caustics.

3.1 Linear Initial Velocity Field Approximation

In zeroth order of perturbation theory at an early epoch, the flow is given by Hubble's

law (i.e., iJ(t, r = H(t)r). To first order, the particle trajectories are given by the

Zeldovich approximation [112],


0 at,q)=a(t) [c- b(t)Vq4)iJ] (3-1)

where Y is the Eulerian co-ordinate of the particle, j is the Lagrangian co-ordinate and

K(t, q is the peculiar gravitational potential, possibly due to nearby halos (Tidal Torque

Theory). Equation 3-1 implies the velocity field


(t, H(t (t) db (3-2)
a dt)jVqjq[1afr(32









Following White [108], we may choose q= 0 at a minimum of K and expand K in a Taylor

series with the result that


U(f) = M F


(3-3)


where M is a symmetric matrix.

Let us consider a more general form for M. We write M as the sum of three parts

M = S + A + T where S is a symmetric traceless piece, A is an antisymmetric piece

and T is the trace. The trace of M does not affect the angular momentum and is not

important in determining the structure of inner caustics. We may therefore ignore T. In

the context of the Zeldovich approximation and Tidal Torque Theory, the flow is purely

irrotational with the result that A = 0. However, since there may be other means of

angular momentum transfer, we will work with the general matrix M which could include

a non-zero A. We study the inner caustics with purely rotational flow (M = A), purely

irrotational flow (M = S) and a mixture of rotational and irrotational terms (M = S + A).

We will see that the relative magnitudes of S and A determine the catastrophes that form

and hence, the geometry of the inner caustics.

Let us choose co-ordinates that diagonalize the symmetric matrix S. We have


91
M =R C3


C2

-Cl


(3 4)


-C2 C1 -gl 92

gl and g2 parametrize the symmetric part of M which yields the gradient part of i while

cl, C2 and 03 parametrize the antisymmetric part of M which yields the curl part of U. In

terms of these 5 parameters, the component of the initial velocity field tangent to the

turnaround sphere are

v6(9,Q) = ff-


ve (0, )


(g2 gl) sin 8 sin cos cos 0(cl cos C + ca sin Q) + C3 sin 0

sin cos [g,(1 + cos2 ) + 2(1 + sin2 )] co sin + cacos (3-5)









(The radial component is neglected since it does not contribute to the angular momentum

and does not influence the structure of the inner caustic).

Symmetries of the initial velocity field: Almost .li- i, we will take the

gravitational potential to be spherically symmetric, in which case the symmetry properties

of the initial velocity field are those of the subsequent evolution as well. In the irrotational

case (cl = C2 = C3 = 0), the initial velocity distribution is reflection symmetric about the

x = 0, y = 0 and z = 0 planes. Moreover it is axially symmetric when two of the three

eigenvalues (gi, g2, and g3 = -g g2) are equal. Most often we will chose the axes such

that gl < g2 < g3. The parameter space is then gi < 0 and gi < g2 < -g91. When g = g2,

the initial velocity distribution is axially symmetric about the z-axis. When g2 -gl, it

is axially symmetric about the x-axis.

In the case of pure rotation (gi = g2 = 0), we may choose axes such that t= cz. The

initial velocity distribution is alv--,v- axially symmetric in this case. When gi, g2, C1, C2

and C3 are all different from zero, the initial velocity distribution has no symmetry. When

axial symmetry about the z-axis is imposed, Cl = c2 = 0 and gl = g2.

3.2 Simulation

We simulate a single flow of zero velocity dispersion falling in and out of a time

independent gravitational potential 4(r), which is specified below. The initial conditions

are Eq. 3-5 plus v, = 0. We solve the equations of motion numerically and obtain the

trajectory (7r, 0, 4) of the particle that originated at the position (0, 0) on the turnaround

sphere at time r. Since neither the potential 4 nor the initial conditions are time varying,

the simulated flows are stationary (i.e., x(t; -, 0, 4) = x(t 7, 0, 0)). The simulation of

non-stationary flows would be straightforward but considerably more memory intensive

and time consuming, without being more revealing of the structure of inner caustics.

Using the equations of motion




z = z (r,, ) (36)









we compute the Jacobian from (r, 0, 0) space to (x, y, z) space

a(x, y, z)
D(TO --T xyz )
ax ax ax
90 9a 9aT
ao ag ag (3-7)
8z az az


The inner caustic is the locus of points with D = 0. In the limit of zero velocity

dispersion, these are infinite density points.

Unless stated otherwise, K is the gravitational potential produced by the density

profile given by

P(r) = ro (3-8)
47G(r2 + a2)
which implies an .-in',,l' ically flat rotation curve with rotation velocity Vrot. a is the

core radius. The density profile Eq. 3-8 implies a force per unit mass


t tan- r (3-9)
dt2 r r a/

The five parameters gi, g2, ca, C2 and c3 when expressed in units of vrot, are related to,

and are of order the dimensionless angular momentum parameter j of the self similar

infall model described in [92, 93]. This sets the overall scale for the values g1 .. c3 we are

interested in, and which are used in our simulations. Note that it is the relative values of

these five parameters that are relevant as far as the structure of caustics is concerned.

We use R, the radius of the turnaround sphere as the unit of distance and Vrot as

the unit of velocity. Since we simulate a single cold flow in a fixed potential, the particle

resolution is not a critical issue. We choose a resolution of 1 particle per degree interval in

0 and 0 and a time step of 10-4 in units of R/Vrot.

3.3 Tricusp Ring

In [94], it was shown by analytic means that when the initial velocity distribution

is dominated by net rotation, the inner caustic has the appearance of a ring whose cross

section is a tricusp. Here, we confirm this result using our simulations. We also study









the effect of various perturbations on the structure of the caustic. The properties of the

tricusp caustic ring are described in Appendix B.

3.3.1 Axially Symmetric Case

Let us consider the simple case where the initial velocity distribution is a linear

function of the co-ordinates and is purely rotational (i.e., the matrix M is purely

anti-symmetric). Since there is no symmetric component, we can choose co-ordinates

such that

S= C3 sin 0 (3-10)

(the antisymmetric part of a matrix is a vector, and we have aligned one axis with the

vector, which implies the existence of axial symmetry about that axis). Fig. 3-1 shows

the infall of a single shell in xz cross section, at successive times for c3 = -0.1. As the

shell falls in, deviations from spherical symmetry appear due to the presence of angular

momentum. The particles at the poles have zero angular momentum and fall in faster

than the particles near the equator. The shell takes on the form of Fig. 3-1(b). In Fig.

3-1(c), the particles which originated at the poles have crossed the z = 0 plane. The shell

has turned itself inside-out in Fig. 3-1(e), forming a crease. Finally the shell increases in

size and regains an approximately spherical shape.

The inner caustic occurs at and near the location of the crease in Fig. 3-1. Fig. 3-2(a)

shows the flows near the crease in pz cross section where p 2+ y2. The figure

shows that the pz plane is divided into two regions one region with four dark matter

trajectories passing through each point (inside) and the other with two dark matter

trajectories passing through each point(outside). The dark matter density is infinite at

the boundary between the two regions (when the velocity dispersion is zero). Fig. 3-2(b)

shows the caustic in three dimensions.









3.3.2 Perturbing the Initial Velocity Field

3.3.2.1 Effect of the gradient terms gi and g2

Let us perturb the initial velocity field by adding the gradient terms gl and g2. As

an example, we choose c3 = -0.1,gl = -0.033, g2 = 0.0267. Thus c31 is large compared

to gi or g2, and we may regard the presence of the gradient terms as perturbations to

the rotational flow. Fig. 3-3 shows the inner caustic obtained from the infall. It is again

a tricusp ring, but the cross section is Q dependent. In this case, the tricusp shrinks to

a point at 4 places along the ring. In the vicinity of such a point, the catastrophe is the

elliptic umbilic (D-4). In our example, the caustic is made up of four elliptic umbilic

catastrophe sections joined back-to-back.

3.3.2.2 Effect of a random perturbation

We have previously asserted that the assumption of symmetry is not required for the

formation of caustics. We now introduce a random perturbation to the initial velocity

distribution of Eq. 3-10. Fig. 3-4 shows the inner caustic. We note that the inner caustic

is still a tricusp ring, although it is deformed from what is was in Fig. 3-2(b)

3.3.2.3 Effect of radial velocities

We added radial velocities to the previously discussed axially symmetric velocity

distribution. We find that the radial velocity components result in only relatively small

changes to the dimensions of the tricusp ring. For the initial velocity field


S=C3 sin 8 +r) (3-11)


with c3 = -0.1, the tricusp ring radius was decreased by 0 -' compared to what it was

for the original initial velocity distribution ( = c3 sin 0 < ) and the transverse dimensions

of the tricusp were reduced by 11 and 11' in the directions perpendicular and parallel

to the plane of the ring.

The inner caustics are determined by the distribution of distances r,~i of closest

approach to the galactic center of the infalling particles. The distance of closest approach









is determined by angular momentum conservation: f = rin v,mx where is the specific

angular momentum and v,mx is the speed at the moment of closest approach, which can

be determined by energy conservation.


1 1
2 v x (v + v + v2) + (R) (rmin) (3-12)
22

The main contribution to ,max is from the gravitational potential energy released

while the particle falls in. The initial velocity components provide only corrections to

Vmax which are second order in v6, ',. and v,. Since does not depend on v, at all, radial
velocities produce only second order corrections to the distances of closest approach.

3.3.3 Modifying the Gravitational Potential

3.3.3.1 NFW profile

We carried out simulations of the infall of collisionless particles in the gravitational

potential produced by the density profile of N i, 11 ro, Frenk and White [66]


p(r) Ps 2 (3-13)
1 + -

The scale length r8 was chosen to be 25 kpc. ps was determined by requiring that the

rotational velocity at galactocentric distance re = 8.5 kpc is 220 km/s. The acceleration

of a particle orbiting in the potential produced by the NFW density profile is then

(220 km/s)2 2 ln(l + x)
d(r) = k j r (3 14)
Tr r2 ln(1 + Z) l

where x = r/rs and xe re/rv.

Fig. 3-5 shows the result of two simulations plotted on the same figure. The larger

caustic ring is obtained using the density profile of Eq. 3-13, while the smaller caustic

ring is obtained using the density profile of Eq. 3-8 with Vrot = 220 km/s and a = 4.84

kpc. In both cases, the turnaround radius R = 174 kpc and the initial velocity field

f = 0.2 sin 0 Q. The caustic is a tricusp ring in each case, but with different dimensions.









The caustic ring produced by the NFW profile has a larger radius than that produced

by the isothermal profile because the NFW gravitational potential is shallower than the

isothermal one at the location of the caustic (rcaustic 16 kpc). Since = rin ,max is the

same, rin is larger in the NFW case because vmax is smaller.

3.3.3.2 Breaking spherical symmetry

We also simulated the infall of collisionless particles in a non spherically symmetric

gravitational potential. For the latter, we chose the triaxial form:



(r) ln 2 R 2 (3-15)

( )2 a2 a3

where al, a2 and a3 are dimensionless numbers. Fig. 3-6 shows the inner caustic for the

case where al = 0.95, a2 = 1.0 and a3 = 1.05, and the initial velocity field J = 0.2 sin 0 .

It is again a tricusp ring. Its axial symmetry is lost due to the absence of axial symmetry

in the potential. The tricusp ring still has reflection symmetry about the xy, yz and xz

planes. As in Fig. 3-3 the tricusp shrinks to a point four times along the ring.

3.4 General Structure of Inner Caustics

In this section, we describe the structure of inner caustics when the initial velocity

field is not dominated by a rotational component. We first discuss the axially symmetric

case, for irrotational flow. We show that adding a rotational component transforms a

'gradient type' caustic into a tricusp ring. Finally we work out the general case when the

initial field does not have axial symmetry.

3.4.1 Axially Symmetric Case

The initial velocity field of Eq. 3-5 is symmetric about the z axis when cl = c2 = 0

and g, = g2. Then

3 = gi sin(20) 8 + c3 sin0 0 (3-16)

We first simulate the flow and obtain the inner caustic in the irrotational case C3 = 0.

Next we see how the caustic is modified when c3 / 0.









Infall of a cold collisionless shell: Figures 3-7 and 3-8 show the infall of a cold

collisionless shell. The figures show the qualitative evolution of a shell whose initial

velocity field is given by Eq. 3-16. For Fig. 3-7, we chose c3 = 0 and gl = -0.0333. We

refer to this as Case 1. Fig. 3-8 shows the qualitative evolution of a shell with c3 = 0 and

gl = -0.0667, which will be referred to as Case 2.

Since c3 = 0 in both cases, each particle stays in the plane containing the z axis and

its initial position on the turnaround sphere. The figures therefore show the particles in

the y = 0 plane. The angular momentum vanishes at 0 = 0 and 0 = r/2 where 0 is the

polar coordinate of the particle at its initial position. Hence, the particles labeled 0 = 0

and 0 = 7/2 follow radial orbits. The angular momentum increases in magnitude from

0 = 7/2, reaches a maximum at 0 = 7/4 and returns to zero at 0 = 0. The sign of the

angular momentum does not change during this interval.

The shell starts out as shown in Fig. 3-7(a) (Case 1) or 3-8(a) (Case 2). As the shell

falls in, the particles at 0 / 0, r/2 move towards the poles. These particles feel an angular

momentum barrier and fall in more slowly than the particles at 0 0. This results in the

formation of a loop in Fig. 3-7(c) (Case 1) and Fig. 3-8(c) (Case 2). The formation of the

loop implies a cusp caustic on the z axis. The particles labeled 0 = 0 and 0 = 7 have

crossed the z = 0 plane and the particles labeled 0 = r/2 have crossed the x = 0 plane in

figures 3-7(g) (Case 1) and 3-8(g) (Case 2). The shell then takes the form shown in figures

3-7(h) (Case 1) and 3-8(h) (Case 2).

The further evolution depends on the magnitude of the angular momentum (i.e., on

the value of Igi ) and is different for the two cases. Consider the infall for Case 1. The

loop that is present near the z = 0 plane in Fig. 3-7(j) disappears through the sequence

of figures 3-7(k) 3-7(o). The disappearance of the loop implies the existence of a cusp

caustic in the z = 0 plane as well. In Fig. 3-7(p) the shell has regained an approximately

spherical form and is expanding to its original size.









Now let us consider the infall for Case 2. The loop which is present near the z = 0

plane in Fig. 3-8(j) disappears through a more complicated sequence of figures 3-8(k) -

3-8(o). The particles near 0 = r/2 cross the z = 0 plane before the sphere turns itself

inside out. This crossover produces additional structure and a more complicated caustic.

The critical value of g1 \, below which the qualitative evolution is that of Case 1 and above

which that of Case 2 is gl, 0.05.

Caustic structure: The butterfly catastrophe. The inner caustic is a surface of

revolution whose cross section is shown in Fig. 3-9(a) for the case (c3,91) (0, -0.0333)

and in Fig. 3-10(a) for (c3,g1) = (0, -0.0667).

On the z axis, there is a caustic line. Caustic lines are not generic. The caustic

line in Figs. 3-9(a) and 3-10(a) occur only because the initial velocity field is axially

symmetric and irrotational. We will see below that when axial symmetry is broken or

when a rotational component is added, the line becomes a caustic tube.

Fig. 3-9(b) shows the dark matter flows in the vicinity of the cusp for Ig1 < gl,.

There are four flows everywhere inside the caustic and two flows everywhere outside.

Note the occurrence of the butterfly configuration when g91| > gl, in Fig 3-10. Since

the magnitude of |gi| determines whether or not the butterfly configuration occurs, |gi|

may be termed the "butterfly factor" [86].

If gl is chosen positive instead of negative, the behavior at the poles and the equator

is reversed (Eq. 3-16) and we have cusps on the x axis and cusp/butterfly caustics on the

z axis, depending on the magnitude of gl.

Fig. 3-10(b) shows the dark matter flows in the vicinity of the butterfly caustic, for

|g91 > gl,. Fig. 3-10(c) shows the number of flows in each region of the butterfly.

Adding a rotational component: Here we show, in the axially symmetric case, the

effect of adding a rotational component to the initial velocity field. Fig. 3-11 shows the

transformation. We start with an irrotational velocity field (c3 = 0) in 3-11(a) and increase

C3 until the rotational component dominates the velocity field, in 3-11(d). The caustic line









on the z axis changes to a tube of circular cross section. The radius of this tube increases

until the gradient type caustic becomes indistinguishable from a tricusp ring.

3.4.2 Infall without Axial Symmetry

We start off by discussing the flows and caustics resulting from irrotational initial

velocity fields. With =` 0, Eqs. (3-5) become


i = sin0 sin(20) + sin(20)(( sin2 +g) 0 (3 17)

where ( = (g2 91) and g g1 + 9g2. is a measure of z axial symmetry breaking in

the irrotational case. We first let ( < g. Next, we explore all of (g, g2) parameter space.

Finally, we add a rotational component by letting c3 / 0.

Irrotational infall without axial symmetry: We saw in the case of axially

symmetric infall of an irrotational flow that there is a caustic line on the z axis.

Fig. 3-12(a) shows the trajectories of the particles in the z = 0 plane for such a case.

The orbits are radial. Indeed all particles have zero angular momentum with respect to

the z axis when the initial velocity field is irrotational and axially symmetric. Because

all trajectories intersect the z axis, there is a pile up of particles on that axis and hence a

caustic line.

Fig. 3-12(b) shows the trajectories of the particles in the z = 0 plane for the initial

velocity field of Eq. (3-17) with = 0.01 and g = -0.05. The particles do have angular

momentum with respect to the z axis now. The caustic line on the z axis spreads into a

tube whose cross section is the diamond shaped envelope of particle trajectories shown

in Fig. 3-12(b). That envelope has four cusps. The flows and caustic have reflection

symmetry about the xy, xz and yz planes because the initial velocity field has those

symmetries. Fig. 3-12(b) shows four flows inside the diamond-shaped caustic and two

flows outside. The infall of dark matter particles from regions above and below the z = 0

plane will add two more flows at each point, which are not shown in Fig. 3-12 for clarity.









Fig. 3-13 shows the inner caustic in 3 dimensions for the initial velocity field of Eq.

3-17 with ( = -0.005 and g = -0.05. The inner tube has a diamond shaped cross section

which is evident in Fig. 3-13(b), which shows a succession of constant z sections. Near the

z = 0 plane, there are six flows inside the diamond, four flows in the other regions inside

the caustic tent, and two outside. Figures 3-13(c) and 3-13(d) show y = 0 and x = 0

sections of the caustic.

Hyperbolic umbilic catastrophe: Let us look more closely at the two regions

(top and bottom) in Fig. 3-13(c) where the inner surface reaches and traverses the outer

surface. We will show the existence of a higher order structure at the boundary. Figs.

3-14(a) 3-14(d) show z = constant sections of the inner caustic in such a region. As z| is

increased, the two cusps on the y axis simply pass through the outer surface, whereas the

cusps on the x axis traverse the outer surface by forming with the latter, two ii;/,, i,. ../.

umbilic (D+4) catastrophes, one on the positive x side and one on the negative x side.

The sequence through which this happens is shown in greater detail in Figs. 3-14(e) -

3-14(h) for the hyperbolic umbilic on the positive x side. The arc and the cusp approach

each other until they overlap (3-14(g)), forming a corner. The cusp is transferred from one

section to the other as the two sections pass through. This behaviour is characteristic of

the hyperbolic umbilic catastrophe. There are four hyperbolic umbilics embedded in the

caustic two (x > 0 and x < 0) at the top (z > 0) and two at the bottom (z < 0). The

hyperbolic umbilic at z > 0 and x < 0 is shown in three dimensions in Fig. 3-14(i).

Let us mention that the reason there is no hyperbolic umbilic catastrophe on the y

axis of Fig. 3-14(a) 3-14(d) is because the particles forming the inner and outer surfaces

here originate from different patches of the initial turnaround sphere. The particles

forming the two caustic surfaces near the hyperbolic umbilic originate from the same patch

of the initial turnaround sphere.

(g9, g2) landscape: Here we describe the inner caustic in the irrotational case for g,
and g2 far from those values where the flow is axially symmetric. Recall that the flow is









symmetric about the z axis when gi g2 (g3 = -2g), about the y axis when g2 = -2gl

(g3 gi), and about the x axis when g2 -- 9g (g3 g2). In terms of and g, these
conditions for axial symmetry are =- 0, g = 0, and = -g, respectively.

The first, second and third columns of Fig. 3-15 show respectively the z 0, y = 0

and x = 0 sections of the inner caustic produced by the initial velocity field of Eq. 3-17

for various values of (9g, g2). The ratio g2/gi decreases uniformly from 1 (top row) to

-1/2 (bottom row). Note that the third column describes a sequence which is that of the

first column in reverse, and that the first half of the sequence in the second column is the

reverse of the sequence in its second half, with x and z axes interchanged.

In the first row, the caustic is axially symmetric about the z axis. There is a caustic

line seen in the xz and yz cross sections. The line appears as a point in the xy cross

section. In the second row, axial symmetry is broken and the familiar diamond shaped

curve appears. The hyperbolic umbilics are apparent in the xz cross section at the

points where the two curves meet. In the third row, the smooth curve passes through the

diamond seen in xy cross section. This curves turns into a caustic line in the fifth row, as

seen in the xy and xz cross sections and appearing as a point in the yz cross section.

The plots of Fig. 3-15 are reminiscent of caustics seen in gravitational lensing theory

[19, 76, 77], and in the analysis of the stability of ships using catastrophe theory [79, 111].

Adding a rotational component: The swallowtail catastrophe. Here we add

a rotational component (C3 / 0) to the initial velocity field of Eq. 3-17. Fig. 3-16 shows

the z 0 cross sections of the inner caustic during such a transition. In Fig. 3-16(a),

the initial velocity field is irrotational and we see a circle and diamond, as before. In

Fig. 3-16(b), the diamond is skewed because of the rotation in the z 0 plane introduced

by C3 / 0. Fig. 3-16(c) shows the case c3 = As c3 is increased further, the diamond

transforms into two swallowtail (A4) catastrophes joined back to back (Fig. 3-16(d)).

There are two flows in the central region formed by the swallowtails, six flows in the

cusped region of each swallowtail, four in the other regions inside the circle and two









outside the circle. Finally, the swallowtails pinch off to form the inner circle of the z = 0

section of the tricusp ring. Fig. 3-17 shows the transition in three dimensions.

3.5 Discussion

We discussed the structure of inner caustics for different initial velocity distributions.

We simulated the flow of cold collisionless particles falling in and out of a fixed gravitational

potential. We restricted ourselves to initial velocity fields of the form U = MI where x is

the initial position and M is a 3 x 3 real traceless matrix. The matrix M can be split into

two parts as M = S + A where S is a symmetric part and A is an antisymmetric part.

Tidal Torque Theory / Zeldovich approximation predicts A = 0. However, we have studied

the more general case of non-zero A, as well as the irrotational case A = 0.

When the initial velocity distribution is dominated by a rotational component, the

inner caustic has the appearance of a ring, whose cross section is a tricusp. When the

initial velocity distribution depends on the azimuthal angle Q, the cross section of the

caustic varies along the ring. In the neighborhood of a point where the tricusp dimensions

have shrunk to zero, the catastrophe is the elliptic umbilic.

We showed that the caustic is stable under perturbations both in the initial velocity

field and in the gravitational potential. This stability is not a surprise since it is well

known that catastrophes are stable to perturbations.

We simulated the infall of an irrotational flow, with and without axial symmetry.

For axially symmetric infall, the inner caustic structure consists of cusp and butterfly

caustics. When axial symmetry is broken, hyperbolic umbilic catastrophes occur. When

a rotational component is added, swallowtail catastrophes appear as the gradient type

caustics smoothly transform into curl type caustics.
































oi:






(a)







DDx


x

(b)


Figure 3-1. Infall of a cold collisionless shell: Antisymmetric M.


z 0[


(c)













0001


Figure 3-2.


0.002
Z 0
-0.002


A

(//


(


Dark matter flows forming an inner caustic. (a) Dark matter flows near the
inner caustic, in cross section. (b) The axially symmetric tricusp ring in three
dimensions.


0.0)5


-0.005


111:22';:,


Figure 3-3. Tricusp ring with non-zero gi and g2.


N)-'


I 1


_~-t-

rcc .
f--"


.040
-----Y


... .....





















.0.005 .. .-......-.. ..
tl-(X)5 '














Figure 3-4. Effect of a random perturbation.








-0.--------- -^-----.----.--**
0.01















Figure 3-5. Tricusp ring: NFW potential.





0.02









0.-0.02 --



Sa



Figure 3-6. Tricusp ring: Non-spherically symmetric gravitational potential.

























(a) (b) (c) (d)








111 1012 002 .0112 ~ In' 0j I-1


(e) (f) (g) (h)

--I U -




1, 0 1
0I I III IN 001 i Iij I I1 02



(i) W) (k) (1)











(m) (n) (o) (p)

Figure 3-7. Case 1: Infall of a shell. Irrotational flow with axial symmetry, Igl < g*


















(k)


Figure 3-8. Case 2: Infall of a shell. Irrotational flow with axial symmetry, Ig l > g*


4


Lj


4


















i/











z---\~--~----,---,.` ------


Figure 3-9. Cross section of the inner caustic produced by
symmetric velocity field (Case 1: Igil < g,).


.

N


an irrotational axially


(a) (b) (c)

Figure 3-10. Cross section of the inner caustic produced by an irrotational axially
symmetric velocity field (Case 2: Igll > g,). The number of dark matter
trajectories at each point is indicated.


0.05 I


Z oF


-0.05 F


2

4


4

6 2
4
4

2


j







i


oOu4


004


0/J6





























P

(a)


Z


'0


Z o[


Figure 3-11. Caustic in cross section, for increasing c3.


Figure 3-12. Dark matter flows in xy cross section. (a) gi


z


U: \\\ r










?~ ~
,,,


\h,
r,


0Ot1


O04


92 (b) 91g / 9g2





























Z22 02









0 //
02 002


002 'r OZ





(a) (b)




006 006







Z o Z o







006 -006 y


4-03 0 03 -003 0 003
x y

(c) (d)

Figure 3-13. Gradient type caustic without axial symmetry. (a) The caustic in 3
dimensions. (b) A succession of constant z sections. (c) and (d) Cross
sectional view.



























y +


V
Y


(b)





o> *
|4











(f)
t" <. *
'*- -'



-r ^


1) M.lli tl'..;



(f)


IIy--


x

(c)


+":


: '.





twp ow~ os


Figure 3-14. Formation of the hyperbolic umbilic catastrophe.


> y



V 1


(a)



4

4

*
I,

( e
r

P


I,
i
oult ,) o a r) 11 ) POi)


()


















x

(a)


(b)










(e)










(h)

A






/ k

(k)


LI z




S" x
(m) (n)

Figure 3-15. Caustic in cross section, for different (i, 92).

52


(c)











(f)




Ni





(U)




-4


LI,


I
(d)


(g)


I
(j)


p


~~ ,
























(a)


Figure 3-16.


(b)


(c)


z = 0 sections of the caustic, showing the transformation from gradient type
to curl type.


(c)


/







(d)
oo. Y
x


Figure 3-17. Transformation of a gradient type caustic to a curl type caustic.









CHAPTER 4
A POSSIBLE CONNECTION BETWEEN A DARK MATTER CAUSTIC AND THE
MONOCEROS RING OF STARS

In recent years, the Sloan Digital Sky Survey (SDSS), the Two Micron All Sky Survey

(2MASS) and the Isaac Newton Telescope Wide Field Camera (INT WFC) have revealed

the existence of a stream of stars at a galactocentric distance ~ 18 kpc, now commonly

known as the Monoceros ring. This stream of stars was first discovered by Newberg

et.al. [67] using SDSS data. Subsequent work by several authors [27, 29, 45, 55, 56, 81,

109] has confirmed the overdensity of stars and uncovered new details.

The stars of the Monoceros ring are observed over galactic longitude 1200 < 1 < 2700

and galactic latitude Ibl < 350 [57]. Assuming it is a complete circle, the total mass in the

Ring is estimated to be in the range 2 x 107 5 x 10s Me in [109] and 2 x 108 109 M.

in [45]. The scale height of the Ring stars in the direction perpendicular to the Galactic

plane is estimated to be 1.6 0.5 kpc in [109], 0.75 0.04 kpc in [45] and 1.3 0.4

kpc in [81]. The scale height in the direction parallel to the plane is also of order kpc.

The stars in the Ring move with a speed of approximately 220 km/s in the direction of

galactic rotation [29, 110]. Their velocity dispersion along the line of sight is small. It was

estimated to be between 20 and 30 km/s in [109].

Rings are in fact, quite common in spiral galaxies [20]. They are usually caused by

some non-axisymmetric component associated with the baryonic disk, such as the presence

of a bar, which induces resonances in the disk. The torque exerted by the bar on the gas

changes sign across a resonance, causing gas to accumulate at the Lindblad resonances

in the form of rings [20]. However, such rings are located within the disk itself. It is not

clear what kind of perturbation would create a ring of stars so far from the galactic center.

Another possibility is that the Monoceros ring is a detection of the galactic warp [61].

Conn et al. [28] argue against this possibility since the stars of the ring are observed

equally on both sides of the galactic plane, while the warp would be preferentially seen









either above or below the plane. Also, the ring stars are located within 20 kpc, while the

distance to the warp is expected to be larger than 30 kpc [28].

One of the widely accepted views is that the Monoceros ring of stars was formed by

the tidal disruption of a satellite galaxy of the Milky Way. Penarrubia et al. [75] have

constructed a theoretical model to explain the observations based on the tidal disruption

of a satellite galaxy which was initially close to the galactic plane. The simulations of

Helmi et. al. [40] find that rings do form due to the tidal disruption of satellites, but such

rings may not be long lived. Here we explore a different proposal altogether, namely that

the Monoceros Ring of stars formed as a result of the gravitational forces exerted by the

second caustic ring of dark matter in the Milky Way [65].

Caustic rings of dark matter had been predicted [92-94], prior to the discovery of

the Monoceros Ring, to lie in the Galactic plane at radii given by the approximate law 40

kpc/n where n = 1,2, 3 *. Since the Monoceros Ring is located near the second (n = 2)

caustic ring of dark matter, it is natural to ask whether the former is a consequence

of the latter. If the answer is yes, the position of the Monoceros Ring in the Galactic

plane and its 20 kpc radius are immediately accounted for. As was mentioned already,

the self-similar infall model predicts that the radius a2 of the second caustic ring of

dark matter is approximately 20 kpc in our galaxy. The transverse sizes p and q are not

predicted by the self-similar infall model. However, the expectation for p and q is that

they are of order 1 kpc for the n = 2 ring. So the transverse sizes of the second caustic

ring of dark matter are of order the transverse sizes of the Monoceros Ring. Moreover, for

q = 1 kpc, the dark matter mass contained in the n = 2 ring is a 6 x 108 Me (Appendix

B). This is of order the total observed mass in the Monoceros Ring.









4.1 Angular Velocity of Gas in Circular Orbits Close to a Caustic Ring

The caustic ring exerts a gravitational force on matter close to the ring. The net

gravitational force close to the caustic is given by g(rl = -g(r)r

2
g(r) v [1 + 2fJ(r)] (4-1)

The function J(r) = I(r) + H(r) is as defined in Appendix B (for = 1)

-1/2 for r < a ,

I -1/2 + for a < r < a +p (4-2)

+1/2 for r > a +p

and H(r) is given by
1 r-a
H(r) tanh(r- ) (4-3)
2 po
po is expected to be of order a. Fig. 4-1 shows the rotation velocity close to a caustic ring.

The angular velocity of gas in the vicinity of the caustic is given by


Vt [1 + fJ(r)]. (4-4)
r

From Eq. 4-4 and Fig. 4-1, we note that the angular velocity has a minimum at r = a.

4.2 Angular Momentum Transport by Viscous Torque

The (r, 0) component of the viscous stress tensor in cylindrical co-ordinates [49] is

\9Vr, =Vp V -5)
[r Or r(45)

where v is the kinematic viscosity. Neglecting the dependence of v, on 0 and setting

v6 = Q r, this becomes

I6, p v (r) ]

p= r a (4-6)
Sis the component of the rce pe unit area crossing the r constant surface (a
Il6,, is the 0 component of the force per unit area crossing the r = constant surface (a









cylinder with radius r). To find FO, we integrate over the area.


FO = (27r)vr dz p

S27avrj2 (4 7)
Or

where a(t, r) is the surface density. The viscous torque is therefore


T 27ravr 3 (4 8)
Or

Consider an annulus at position r, of thickness 6r. The annulus contains a mass

6m = a(27rr6r). The net torque on the annulus equals the rate of change of angular

momentum of the matter in the annulus. [52, 80]

dL d( 6ml)
dt dt
= 6m + r 1
[at Or
S(27r6ra)v, (4-9)
Or

where 1 is the specific angular momentum = angular momentum per unit mass = Q r2, L

is the angular momentum and v, is the radial velocity. Axial symmetry and stationarity of

the potential are assumed. Let us define the two dimensionless quantities

r d9R
Al rQ
R r
r2 02Q
A2 (4 10)


Using I = Q r2, we have
dL
d=6r 27j ver2Q (A1 + 2) (4-11)
dt
Consider an annulus within the caustic ring (i.e., in the region a < r < a + p).

The matter inside radius r has smaller Q than the matter in the annulus and tends to

slow down the matter in the annulus. The matter outside r + 6r has larger Q than the

matter in the annulus and tends to speed up the matter in the annulus. The total torque









is -r(r) + r(r + 6r) = 6r dr and we have


d Var = -a Vr2o (A, + 2) (4-12)
dr Or

and we therefore have

2 v A2 7J' 37 32 2
r20A1 + -- + + r2aAl vr 1 +-
Av A 2 2

S3+ + + v vr 1 + (4-13)
r A, a A,

where the primes represent derivatives with respect to position.

Close to r = a, for r > a,

af
A1
2 /p(r a)
fa2

A2 3
2 a (4-14)
A1 2 (r a)

and so

Vr + v +- (4 15)
2(r a) a
If at an early time, a is constant and v' is negligible, the flow of gas will be in the

negative radial direction (i.e., towards r = a). This will make a' < 0 which will make v,

more negative. A similar calculation in the region r < a, shows that the gas velocity is in

the negative radial direction. However, the magnitude of the flow velocity is larger in the

region a < r < a + p, especially when r is close to a. The movement of gas in the region

a < r < a +p towards r = a may have physical consequences and could possibly ]'1 iv a role

in star formation. The viscous transport process will be halted by the back reaction of the

gas, which we have neglected in our analysis.









4.3 Effect on Star Orbits


4.3.1 Orbit Stability

In a gravitational field f(rf = -g(r)r, the angular frequency squared, of small

oscillations about a circular orbit of radius r is

1 d
K' W t d [r g(r)] (4-M)
2 3 dr

The orbit is stable if K2 > 0 (then K is the epii' L- frequency). In the neighborhood of a

caustic ring, we have

2(r) -2( + 2f J + fr (4-17)

For a < r < a + p, all the terms are positive, while for r > a + p, the third term is

dJ/dr sech2 However, since we expect po to be large, of order a, the sum

of the three terms is still positive, implying that circular orbits are stable. Deviations

from circular symmetry can induce instabilities through the phenomenon of Lindblad

resonances.

4.3.2 Resonances

Let U(r, 90) be the gravitational potential close to a caustic ring which is nearly

circular. The equations of motion close to the ring are [18]

2 U
r = 2 r au--
Or
( 2) au (4 18)


where w = -h. Consider a small perturbation:

U(r, y) = Uo(r) + Ui(r, p)
r = ro + rl
S= Po + 1 (4-19)









where the first order terms are due to the deviation from circular symmetry of the caustic

ring. We may expand U(r, o) into zeroth and first order parts -


auo(ro)
U(r, <) = Uo(ro) + UI(ro, Or


(4-20)


The first order part of Eq. 4-18 reads


r1 + 2 u(7o0) 01
12 Oo2o0r)
ro47 + 2 jor1 + 2 foi + -
ro


au1
2 oiro1 (ro, Io)
r
-2 uon a(ro, Po)
to 9y


For small departures from circular orbits, we may neglect the terms involving ro and

io when they are multiplied by first order quantities. We then have


rl+ (2 (0) ) r1 2M0oro01 l(ro, o) (4-2
( dr" ) Or


We choose the form of the potential term UI(ro, o) as


i au1
-2uwoio (ro, Po)
ro 00c


2)


(4-23)


Ui(ro, co) = Ui(ro) cos my


(4-24)


where p = wo t and m is an integer.

Since we have neglected the time variation of w and ro when they multiply first order

terms, we can integrate Eq. 4-23 to obtain


Ul, cos mwt
ro71 = -2 onl -
ro cot


(4-25)


where the constant of integration is absorbed by a redefinition of rl. Eliminating L from

Eq. 4-22, we have


r + 02 32" rI1
( ar2 +


Or


2 wo U, t
cos mwot
fOo \


(4-21)


(4-26)


roi1









Large oscillations occur at resonance, that is when


2 2Uo
m2 0 Or2 + 3 W


Using the relations


J(r) 1 r a
2 p
Uo r2
Vrot [1 + 2 f J(r)]

LLo [t + f J(r0


we have the resonance condition for r a < a,

a2 1
r a = m
4p (_- 1)2

With a = 18kpc, f = 0.046 and p =1 kpc, we find r a

4.3.3 Density Enhancement: Circular Orbits

In the absence of the caustic, the effective potential


= 2, 3,... (4-29)


= 171, 17, 4, parsec.



(for a flat rotation curve) is of the


form


S0 rot 2r2


(4-30)


where 1 is the specific angular momentum and ro is a constant. When the caustic is

present, the effective potential is modified to Vff(r) = Vff,o(r) + Vc(r) where


,(r) -2 2fV -r J(r)


(4-31)


V,(r) is plotted in Fig. 4-2 with a = 20 kpc, p = 1 kpc and po = 5 kpc. The effective

potential is smooth even though its second derivative diverges at r = a and r = a + p.

The caustic ring radius increases with time. According to the self-similar infall model
[ t2
[92, 93], a oc t39 Consider a particle of specific angular momentum 1. In the absence of


(4-27)


(4-28)









the caustic, it is on a circular orbit of radius r given by


12 2= r (4-32)

or it is oscillating about a circular orbit of that radius. In the presence of a caustic of

radius a, the particle is on or oscillating about a circular orbit of radius rf(r, a) given by

12 = rf2Vo [1 + 2fJ(rf)] (4-33)

Angular momentum conservation implies

r rf [1+ fJ (rf(r))] (4-34)

Fig. 4-3 shows rf(r, a) as a function of a. Each line in that figure corresponds to a

different value of r. Let d(r) be the density of stars in the absence of the caustic, and

df(r, a) their density in the presence of a caustic with radius a. Assuming that all stars
remain on circular orbits, conservation of the number of stars implies

rf df(rf, a) Arf = rd(r, a) Ar (4-35)

and therefore
df(rf, a) = d(r) 1 + 2f J(rf) + frfr (rf) (4-36)

Assuming that the initial star density has no significant structure of its own, we have

df(r, a) d [ + 2fJ(r) + fr (r) (4-37)
dr

which becomes very large near r = a for r > a since

dJ(r) 1 (438)
(4-38)
dr 2 Vp(rr- a)

Thus in the limit that all stars are on circular orbits, the star density adopts the same

divergent profile as the dark matter density, at r = a. The increase in density is clearly

seen in Fig. 4-3.









4.3.4 Density Enhancement: Non-circular Orbits

The observed radial velocity dispersion Av of stars in the Monoceros Ring is of order

20 km/s. This implies that the stars do not move on circular orbits, but oscillate in the

radial direction with typical amplitude


Ar kms 20k 1 Vrot= 1.3 kpc (4-39)
w s 5 /2 20kpc

The density profile of stars in the neighborhood of a caustic ring will therefore be averaged

over the length scale Ar. The sharp features at r = a and r = a + p will be smoothed out.

However, there will be a relative overdensity

Ad a d af kpc (440)
-- ~ af | -), ~ -- ^1 (4-40)
d dr p p

which is an order 1C( 1' enhancement in star density.

4.4 Discussion

As was described in C'! plter 2 and ('!Ci pter 3, the continuous infall of cold collisionless

matter from all directions in a galactic halo results in the formation of inner and

outer caustics. The inner caustics are rings in the galactic plane provided the angular

momentum distribution of the infalling dark matter is characterized by net overall

rotation. Assuming self-similar infall, the radii of the caustic rings of dark matter in our

galaxy were predicted to be 40 kpc / n with n = 1, 2, 3 ... Because the Monoceros Ring

of stars is located near the second caustic ring of dark matter we looked for processes

by which the latter may cause the former. We have identified two such processes which

may help in explaining the formation of the ring of stars. The first is the flow of gas in

the disk towards the sharp angular velocity minimum located at the caustic ring radius,

possibly increasing the rate of star formation there. To the extent that this process is

responsible for the formation of the Monoceros Ring, the Ring stars are predicted to be

younger than average. [57] find that the stars are bluer, and hence younger than average

stars. The second process is the adiabatic deformation of star orbits in the neighborhood










of the caustic ring. As the spatial dependence of the gravitational field of a caustic ring

is known, it is straightforward to obtain the map of initial to final orbits for disk stars.

The resulting enhancement of disk star density at the location of the second caustic ring

is of order 1011' Because of uncertainties in the caustic parameters and in the velocity

distribution of the disk stars, the strength of the enhancement can only be estimated to

within a factor of two or so. The self similar infall model of galactic halo formation is

expected to describe the halos of all isolated spiral galaxies. The caustic rings of dark

matter in exterior galaxies may also be revealed by the baryonic matter they attract. Our

analysis is relevant to those cases as well.


10 1?


Figure 4-1. Rotation curve close to a caustic ring (R = (r a)/p).


V,/2!u,,L


-10


Figure 4-2. Smooth potential V,(r) (R


- R 2


(r a)/p).

































7-
7,


-----~----- \ 7






/\ \ /








/J


\ ~//x I


Figure 4-3. Adiabatic deformation of star orbits. The enhancement in density is
proportional to the slope dr/drf which is infinite at rf = a.




















65









CHAPTER 5
WIMP ANNIHILATION IN INNER CAUSTICS

In recent years there has been a lot of interest in dark matter detection. Dark matter

detection experiments are of two kinds (i) Direct and (ii) Indirect detection experiments.

Direct detection experiments are sensitive to dark matter particles interacting with target

nuclei. Indirect detection experiments look for standard model particles that result from

dark matter particle annihilation. Here we investigate the flux of gamma ray photons

produced by WIMP annihilation in dark matter inner caustics. Since caustics are regions

of high density, one may expect caustics to be relevant to dark matter searches. We show

that if the dark matter is the SUSY neutralino, the annihilation of neutralinos in caustic

rings produces a distinct signature, which in principle may be detected [64].

We estimate the number of photons produced in different energy bands, when WIMPs

annihilate into standard model particles. We then compute the expected gamma ray

annihilation flux and compare this with the expected diffuse gamma ray background.

Previous work on particle annihilation in caustics includes [10, 43, 60, 78]. Caustics

and their associated cold flows are also relevant to direct detection experiments [2, 37, 39,

50, 87, 105].

5.1 Annihilation Flux

In the minimal supersymmetric extension of the standard model (i\SSM), a good

candidate for the WIMP is the lightest neutralino which is a linear combination of the

supersymmetric partners of the neutral electroweak gauge bosons and the neutral Higgs

bosons [46]. The characteristics of the annihilation flux depend both on the composition

of the WIMP and its mass mX. The line emission signal (XX -- y XX --i Z) is loop

suppressed and is therefore smaller than the continuum signal. The continuum flux

(number of photons received with energies ranging from El to E2 per unit detector area,

per unit solid angle, per unit time) is given by [21, 78, 102]

< EM >
) (El, E2, 0, QA) S(E1, E2) x (0, O, AQ) (5 1)
4r









with S and < EM > defined as


S(E1, E2) = dE bi (E) 2m
S i d2 M~x

< EM > (0, AE) = dQ EM(o0,) (5-2)


dNi/dE is the number of photons produced per annihilation channel per unit energy, bi

is the branching fraction of channel i and < av > is the thermally averaged cross section

times the relative velocity. The factor of 2 in the denominator accounts for the fact that

two WIMPs disappear per annihilation. The quantity EM(0, Q) is called the emission

measure and is the dark matter density squared, integrated along the line of sight


EM(0, ) = dx p2(). (5-3)
Jlos

and < EM > (0, Q, AQ) represents the emission measure in the direction (0, Q) averaged

over a cone of angular extent AQ. We note that S depends solely on the particle physics

while < EM > depends solely on the dark matter distribution.

5.1.1 Estimating S

Assuming that all the dark matter is composed of neutralinos, the quantity < av > is

constrained by the known dark matter abundance [46]

3 x 10-27 m3 S -1
Q 2
Qxh2
3 x 10-26 cm3 s -1 (5-4)


The quantity dN,(E,)/dx for the dominant channels may be approximated by the form

[9, 11, 34] dN,/dx = a e-b/x where x is the dimensionless quantity E,/mx and (a, b) are

constants for a given annihilation channel. The values of (a, b) for the important channels

are given in [34]. Using these values, we may calculate the number of photons produced

per annihilation within a specified energy range. Let us consider four energy bands:

Energy Band I with photon energies from 30 MeV to 100 MeV, Band II with energies

from 100 MeV to 1 GeV, Band III with energies from 1 GeV to 10 GeV and Band IV









containing photon energies 10 GeV upto mx. The values of N./m2 are tabulated for the

different energy bands, for mx = 50,100, 200 GeV.

Table 5-1. N,/m2 in units of 10-4 GeV-2 for mx = 50 GeV
C'!h .i,. I I II III IV
WW, ZZ(0.73, 7.76) 106.8 85.2 18 0.52
bb(1, 10.7) 146 114.4 21.6 0.32
tt(1.1,15.1) 160 122.4 19.6 0.12
uu(0.95, 6.5) 139.2 111.6 25.2 0.96



Table 5-2. N,/m2 in units of 10-4 GeV-2 for mx = 100 GeV
C('1 .i1. I I II III IV
WW, ZZ(0.73, 7.76) 38 30.8 7.9 0.6
bb(1, 10.7) 51.9 41.8 10 0.5
tt(1.1, 15.1) 57 45.4 9.8 0.3
uu(0.95, 6.5) 49.4 40.3 10.7 1.0



Table 5-3. N,/m2 in units of 10-4 GeV-2 for mx = 200 GeV
C'!h .i1,. I I II III IV
WW, ZZ(0.73,7.76) 13.5 11.0 3.1 0.4
bb(1, 10.7) 18.4 15.0 4.1 0.4
tt(1.1, 15.1) 20.2 16.4 4.3 0.3
uu(0.95, 6.5) 17.5 14.4 4.2 0.6


5.1.2 Estimating < EM >

As we showed in C'!i plter 3, the inner caustic has the appearance of a ring when the

initial velocity distribution has a net rotational component. Here, we assume that this is

the case. It is easiest to calculate the emission measure from a tricusp caustic ring because

of the advantage that it can be treated analytically.

Let us consider cylindrical co-ordinates (p, z) where p = + y2. We assume axially

symmetric infall about the z axis and reflection symmetry about the z = 0 plane. We can

then obtain an analytic solution for the dark matter density at points close to the caustic.









For points close to the tricusp, in the z = 0 plane, the dark matter density is given by

I when R < 0
1-R
GeV (f/10-2) (V,ot/220kms-1)2
d(R, 0) a 0.34- Pkpc Pkpc ( 1 + when 0 < R < 1
cm pkpc Pkpc 1-R )~
1 when R > 1
R-1
(5-5)

(Appendix B) where pkpc and pkpc are distances measured in kpc. For points not in the

z = 0 plane, the density is obtained by computing the sum (Appendix B)

d(RZ) 7 0.1 Gev (f/10-2) (Vrt/220kms-1)2 1 (56)
d(R, Z) 0.17 (5-6)
cm3 Pkpc Pkpc |2T2 3T + ( R)

where T, are the real roots of the quartic

T4 2 T3 + (1 R) T2 2 0 (57)
64

At every point inside the caustic, there are four real roots, while outside, there are two.

The above formulae are only valid at points close to the caustic (distances of order p or q).

The emission measure is calculated by integrating the density squared along the line

of sight. Let b = r/2 0 be the galactic latitude. I is the galactic longitude chosen so

that the galactic center is located in the direction = 0, b = 0. We will assume that the

caustics are spread over a distance ~ 10-4 pc (Chapter 2). f is set equal to 2 x 10-2 [25].

The cut-off density close to the fold surface (near R = 0, Z = 0) is then A 2.15 x 103/a p

GeV/cm3. The density close to the cusp will be larger than this (since p ~ 61 near the

cusp, while p ~ 6-1/2 near the fold), but we will use the density close to the fold surface to

set the density cut-off.

Fig. 5-1 shows the emission measure averaged over a solid angle 10-5 sr for three

different sets of caustic parameters, as a function of longitude 1. b is set equal to 0 and

we assume that the caustic lies in the galactic plane. Figures 5-1(a), 5-1(b) and 5-1(c) are

plotted for (a,p, q) = (7.5, 0.5,0.5), (8.0, 0., 0.2) and (8.0, 0.1,0.5) respectively with all

distances measured in kpc. The earth's location is set equal to 8.5 kpc from the center.









We expect the signal to be strongest when the line of sight is tangent to the ring. From

the figures, we see that the emission measure is sensitive to the caustic geometry. The

prominent features are the pair of peaks, or 'hot spots' separated by a few degrees. The

first peak occurs when the line of sight is tangent to the fold surface (when p = a). The

second peak occurs when the line of sight is tangent to the cusp line (when p = a + p).

(In the limit p, q -i 0, the two peaks coincide [78]). For the case when p = 0.5 kpc

(Fig. 5-1(a)), the cut-off density was set equal to 400 GeV/cm3 everywhere, while for

the plots with p = 0.1 kpc (Figs 5-1(b) and 5-1(c)), the cut-off density was set equal to

800 GeV/cm3 everywhere. The magnitudes of < EM > for the hot spots depend on the

values of the caustic parameters and also on the averaging scale (here chosen to be 10-5

sr). Table 4 shows the annihilation flux for the two hot spots for different values of the

averaging scale AQ, for the case (a = 8.0,p = 0.1, q = 0.2) kpc. It is worth pointing out

that if the triangular feature in the IRAS map is interpreted as the imprint of the nearest

caustic on the surrounding gas as in [95], the implied caustic parameters are close to what

we have assumed for Fig 5-1(b).

Table 5-4. Peaks of < EM > for (a = 8.0,p 0.1, q = 0.2) kpc
AQ Fold peak Cusp peak
(sr) (GeV/cc)2 kpc (GeV/cc)2 kpc
10-7 1549.8 2177.6
10-6 1048.3 1283.3
10-5 469.9 718.1
10-4 269.6 239.4
10-3 115.4 52.0


5.2 Comparing the Signal with the Background

The annihilation flux from caustics is thus given by

< EM >
S= Sx
47
Ny < EM > _-
S110 rN (mn/100 GeV)2 (GeV/cc)2 kpc









Let us compare this flux with the expected background. The EGRET measured

background flux by( (E1, E2) from energies El to E2 is given by [11, 44]


)bg(Ei, E2, 8, NQ7,bg(El, E2) X NO(0, Q) x 10-6 cm-2 S-1 sr-1 (5-9)

and NT,by(E1, E2) given by


NE,b2 (E, E)= dE 7 (5-10)

The function No(0, p) is energy independent and follows the fitting form given in [11]. For

the four energy bands we have considered, NT,b = 198.8 for Band I (30 MeV 100 MeV),

28.9 for Band II (100 MeV 1 GeV), 0.58 for Band III (1 GeV 10 GeV) and 0.012 for

Band IV (above 10 GeV).

We now compare the caustic signal with the expected background. Since the gamma

ray background falls off with energy (E-2.7) faster than the annihilation signal (E-1.5), we

expect that the best chance for detection is at moderately high energies. At low energies,

the background flux overwhelms the signal, while at very high energies, the signal is weak.

We choose mx = 50 GeV since this choice gives the largest flux. For the quantity N,/m,

we use the average value for the band. The averaging scale AQ is set to 10-5 sr.

Figures 5-2(a) and 5-2(b) show the expected annihilation flux (number of photons per

square meter, per steradian, per year) as a function of angle I near the plane of the galaxy

(b = 0) for the three sets of caustic parameters we considered, for Energy Bands III and

IV respectively. This is contrasted with the expected diffuse gamma ray background. In

principle, the peaks in the signal and the sharp fall-off of flux are helpful in identifying the

annihilation signal, particularly for the more optimistic caustic parameters and for small

WIMP masses. For large WIMP masses, the annihilation signal is significantly smaller.

5.3 Discussion

We calculated the gamma ray annihilation signal from a nearby dark matter caustic

having the geometry of a ring with a tricusp cross section near the plane of the galaxy, in












different energy bands. For such a caustic, the annihilation signal has two peaks, separated

by a few degrees, depending on the size of the caustic. There is an abrupt fall-off of

flux after the second peak. Since the diffuse gamma ray background flux falls off with

energy faster than the signal, it is advantageous to look for the signal at moderately high

energies. We compared the expected annihilation flux with the expected diffuse gamma

ray background. The characteristics of the annihilation flux can in principle, be used to

discriminate between the signal and the background. In practice however, we expect this

to be a challenging task.


(a)

B 5 0 62 6' 63 os TO ra


(a)


(b)


(c)
(C)


Figure 5-1. Emission measure averaged over a solid angle AQ = 10-5 sr, for three sets of
caustic parameters.


72 74


11oc

-oo
cooo
looo


72 1

































-i









S ..................... .

58 60 62 64 66 68 70 72 74
1 (degrees)


(a)


64 66 68 70 72 74
1 (degrees)


Figure 5-2. Annihilation flux for the three different sets of caustic parameters for mx = 50

GeV, compared with the EGRET measured diffuse background.


8 x 106



6 x 106




4 x 106



2 x 106


16 x 104



12 x 104


4 x 104


58 60 62


I

i I
i i

iL"
g









CHAPTER 6
CONCLUSIONS

In this work, we investigated the structure and properties of cold dark matter

caustics. In C'!i pter 1, we provided an introduction to dark matter cosmology and

described the observational evidence for dark matter. In C'! lpter 2, we showed that the

continuous infall of dark matter particles with low velocity dispersion from all directions

in a galactic halo leads to the formation of caustics. We therefore expect caustics to exist

in galactic halos. There are two kinds of caustics: outer and inner. The outer caustics

are thin spherical shells surrounding galaxies, while the inner caustics have a more

complicated geometry. We gave possible observational evidence in favor of caustics.

In C'! lpter 3, we provided a detailed analysis of the structure of inner caustics. We

found that the catastrophes that form, and hence the geometry of the caustic, depends

on the spatial distribution of the dark matter angular momentum. We used the linear

velocity field approximation = MI with the matrix M made up of symmetric and/or

anti-symmetric parts. We found that when M is mostly antisymmetric (rotational or curl

flow), the caustics are made up of elliptic umbilic catastrophe sections, while a symmetric

M (irrotational or gradient flow) produces caustics with hyperbolic umbilic catastrophe

sections. We also showed the formation of the swallowtail and butterfly catastrophes. It

is possible to smoothly transform the gradient type caustics into curl type caustics and

vice-versa.

C'! lpter 4 and C'! lpter 5 deal with the .i- r ,!|i, -i-i 1 effects of dark matter caustics.

C'! lpter 4 explores a possible connection between a dark matter caustic and the

Monoceros Ring of stars. The existence of a dark matter caustic in the plane of the

galaxy at ~ 20 kpc was predicted by the self-similar infall model of Sikivie, Tkatchev and

Wang prior to the discovery of the Monoceros Ring. We found two mechanisms by which

a dark matter caustic can increase the star density in its neighborhood. The first is the









action of viscous torques, by which gas is transported to the location of the caustic. The

second is the adiabatic deformation of star orbits as the caustic increases in radius.

In ('!i ipter 5, we calculated the expected gamma ray annihilation flux from a nearby

dark matter caustic ring, assuming that the dark matter consists of SUSY neutralinos. We

compared this flux with the diffuse gamma ray background. The flux from the caustic has

a distinct signature which in principle, can be detected. However, we expect this to be a

challenging task.









APPENDIX A
CATASTROPHE THEORY

Catastrophes are abrupt changes that occur in a system when smooth changes are

made to variables governing the system. Water boils suddenly, ice melts, aircraft produce

sonic booms, ships that are stable when inclined by 0 capsize when inclined by 0 + 60, etc.

The term 'catastrophe' refers to the fact that an observable quantity changes suddenly.

The study of systems that exhibit sudden changes is called catastrophe theory.

The mathematician Hassler Whitney laid the foundation of singularity theory in

1955 with his study of the singularities that occur in mappings. Singularity theory was

extended to apply to observable phenomena by the topologist Ren6 Thom and later by

C(i i-1 11.'!h r Zeeman, Vladimir Arnold and others. The combination of singularity theory

with its practical applications forms catastrophe theory.

In a catastrophe theoretical analysis of a system, we start by making a list of the

variables that critically affect the behavior of the system. The variables are then divided

into two sets One set of variables can be controlled by the observer and is called the

control set. The other set cannot be directly controlled and is called the 'hidden set'

or 'state set'. We will use the term 'control variable' to refer to a variable that can be

controlled and the term 'state variable' to describe a variable that is not controlled by the

observer. The number of control variables is called the codimension of the catastrophe.

The number of state variables is called the corank.

Consider the familiar example of the heart shaped pattern that forms on the surface

of tea in a tea cup (you need some milk in the tea to make it reflective). This is a light

caustic. Imagine a light meter placed on the surface of the tea. The light meter can be

moved in two dimensions (x, y). Normally, small changes in (x, y) result in small changes

in the measured light intensity. However, when the light meter crosses the caustic, the

light intensity changes suddenly (i.e., a catastrophe occurs). The two variables we are free

to control are (x, y), which are therefore control variables. The caustic is the envelope of









the family of reflected light rays, parametrized by the angle of incidence 0. 0 is the state

variable. The catastrophe is the cusp.

Catastrophes are described by a 'potential' or 'generating function' f that involves

the control and state variables. Varying f with respect to the state variables (keeping the

control variables fixed) gives us the 'equilibrium surface'. The equilibrium surface is the

set of equations that describes the physical system. In the example of the light caustic, the

equilibrium surface is the set of reflected light rays.

Consider the function f(0) = 04. f(0) has a minimum at 0 = 0 (Fig. A-l(a)) ,

but it is not stable to perturbations. If the function is changed to f(0, x) = 04 + x2,

f has a maximum at 0 = 0 for negative x, as seen in Fig. A-l(b). The new function

f(0, x) is not stable either because the addition of the term 0y alters the form of f near
0 = 0. There is now neither a maximum, nor a minimum at 0 = 0, as seen in Fig.

A-l(c). However, the function f(0, x, y) = 04 + 02 + yO is stable because the addition

of a perturbing term cannot change the behavior of the function near 0 = 0. This is

because the function f(0, x, y) already includes all possible perturbing terms. (A cubic

term is irrelevant because a quartic can ah--iv-i be put into a form that does not involve

a cubic by a redefinition of 0. A constant term cannot change the nature of the critical

points of 0 and a term like a 05 introduces a new critical point at 0 = -4/5 a which

can be moved arbitrarily far from 0 = 0 by making a arbitrarily small). The function

f(0, x, y) = 04 + 02 + yO is the unfolding of the singularity 04 and is the generating
function of the cusp catastrophe. Coming back to the example of the light caustic, the

family of reflected light rays is obtained by setting Of(x = xo)/O0 = 0.

Here we give a brief description of some of the catastrophes and derive their

bifurcation sets [79, 86]. We restrict ourselves to the catastrophes encountered in ('! lpter

3, namely, the fold, the cusp, the swallowtail, the butterfly, the elliptic umbilic and the

hyperbolic umbilic.









A.1 Fold and Cusp Catastrophes.

A.1.1 Fold: Corank = 1, Codimension = 1

The fold is the simplest catastrophe. It involves 1 state variable 0 and 1 control

variable r. The unfolding is

f(0;r)= +rO

The physics of the system is described by the equilibrium surface

8f
a 0 =02 +r 0 (A-l)
do

This is the parabola shown in Fig A-2(a).

Setting the second derivative to zero gives us the -:<,l;,ii, n:/; set

2f 0 {0 = 0}. (A-2)
002

The projection of the singularity set onto the space of control variables {r} gives us the

bifurcation set

{r = 0}. (A-3)

The bifurcation set of the fold catastrophe divides control space {r} into two regions. The

region r < 0 has two solutions of 0. The region r > 0 has no solution.

The intensity of the observable quantity (or simply referred to as the density) is

proportional to the sum

dfC z 2 (A4)

in the region r < 0. The sum is taken over the different solutions of 0 in Eq. A- For the

fold catastrophe, there are two solutions of 0 for r < 0

0 = + r (A-5)


and so the density d = df/- for r < 0 where df is a constant. df is not predicted by

catastrophe theory since the theory is qualitative. The catastrophe occurs at the point

r 0.









The projection of a sphere onto a plane produces a fold catastrophe at the equator.

The rainbow caustic is a physical example of a fold catastrophe. In physical phenomena,

the density is never really infinite at r = 0, but much larger than the average density.

Fold catastrophes are points in 1 dimension, lines in 2 and surfaces in 3.

A.1.2 Cusp: Corank = 1, Codimension = 2

As discussed before, the cusp catastrophe has 1 state variable and two control

variables. The unfolding is
04 02
f(0;x,y) +x- +y0 (A-6)
4 2
The equilibrium surface is given by

8f
= 0 3 +X + y 0 (A-7)


The singularity set is
92f
= 0 = 302 + = 0. (A-8)
002

The bifurcation set is obtained by eliminating 0 from Eq. A-8 and Eq. A-7

y = 2 (-X)3/2 (A-9)
373

for x < 0. This is shown in Fig. A-3. The bifurcation set divides (x, y) space into two

regions one with 3 solutions of 0 and the other with 1 solution. The catastrophe occurs

when crossing the boundary between the two regions.

The density d is proportional to the sum

d o 1302 + -1 (A-10)

where the sum is over the real roots Oi, of Eq. A-7. For the special case y = 0, there are

three solutions for x < 0, namely 0 = 0, -x/3 and one solution 0 = 0 for x > 0.

Therefore, the density d
d1 1 1 2d1
d d + + =-(A-ll)
x 2x 2 x









for x < 0 and

d = (A-12)
x

for x > 0 where dc is a constant. For x = 0, there is one real solution 08 = -y and the

density d is
dc 1 2/3
d = () (A-13)

For arbitrary (x, y), the density may be obtained by solving the cubic Eq. A-7. The

proportionality constant dc must be determined by the physics of the system. Cusps are

points in 2 dimensional space and lines in 3.

We now discuss the higher order catastrophes, following the treatment of Saunders

[86].

A.2 Higher Order Catastrophes.

A.2.1 Swallowtail: Corank = 1, Codimension = 3

The swallowtail has 3 control variables x, y, z. There is 1 state variable 0. The

standard unfolding is
85 83 82
& 0+2+y z (A 14)

giving the equilibrium surface

04 +x2+ y+z 0 (A-15)

and the singularity set

403 +2x0+ + y 0 (A-16)

To sketch the bifurcation set, let us consider a 2 dimensional cross section with x fixed.

-y = 403 + 20
z 304 + 02 (A-17)


y is an odd function of 0 while z is an even function of 0. This means z is an even function

of y and there is reflection symmetry about the z axis. Differentiating Eq. A-17, we









obtain the equations

d = 1202 + 2x
dO
dz dy
z 1203 + 20x -0 (A-18)
dO dO

Both derivatives vanish when 02 = -x/6, which means that, for x < 0, there are two cusps

located at y = (2/3)3/2 X13/2,z -_ 2/12. For x > 0, there are no cusps.

Next, we determine the points where the curve crosses the axes, for x < 0. For z = 0

there are two crossings, while for y = 0, there is only one crossing at x2/4, which implies

that there is a point of self intersection on the z axis.

To determine the different regions, consider Eq. A 15 for z = 0 (i.e., points on the y

axis). Eq. A 15 simplifies to

04+ 02+y0 = (A 19)

which has solutions

2
02 [l 1 (A-20)

where = 4y/x2.

If x > 0, there are two real solutions for I < 0 and no solution for ( > 0. In this

regime, the swallowtail resembles a fold catastrophe. If x < 0, there are four real solutions

for 0 < K < 1, two real solutions for K < 0 and no real solution for ( > 1. Thus, there are

three distinct regions when x < 0. Each line in Fig A-4(a) represents a real root of 0. The

yz cross section of the bifurcation set with x < 0 is shown in Fig. A-4(b), indicating the

number of real roots in each of the three regions.

A.2.2 Butterfly: Corank = 1, Codimension = 4

The state space is 1 dimensional and the control space is 4 dimensional. The standard

unfolding is
06 04 03 02
+ g- + x- + y + z. (A21)
6 4 3 2









The equilibrium surface and the singularity set are specified by


5 +g03 +02 +y+ = 0 (A-22)


and the equation

504 +3g02 +2x0+ y 0 (A-23)

respectively. To obtain the bifurcation set, let us consider a 2-dimensional cross section

with g and x fixed. Writing y and z as functions of 0, we have [86]

-y = 504 + 3g02 + 2x0
z = 405 + 2g03 +x02 (A-24)

Differentiating equations A-24 with respect to 0,


d 2003 + 6g0 + 2x
dO
d = 2004 + 6g2 + 2x -0d (A-25)
dO dO

Both derivatives vanish when

+ + 0 (A-26)
10 10
which has three real roots if

X2 + g3 < 0 (A-27)

and one real root otherwise. The number of real roots gives us the number of cusps. From

equations A-24, we note that for the special case x 0, z is an odd function of 0 and

y is an even function of 0. So for x 0, y is an even function of z, implying reflection

symmetry about the y axis. Let us therefore set x = 0 to sketch the bifurcation set in

2-dimensional cross section. With x 0, Eq. A-24 and A-25 become

-y 5 04 +3g02
z = 4 05 +2g3
d = 20 0 + 6g9
dO
S20 04 + 6g02 (A-28)
dO









Both derivatives vanish for 0 0 or 0 = +- 0. For g > 0, there is only 1 cusp which

occurs at y = 0, z = 0. For g < 0, there are three cusps, which occurs at y = 0, z = 0 and
S z [4 (9)52 + 2g ( ) 3/2] The sign of g determines the number of cusps,

and hence the geometry of the bifurcation set. g is therefore called the 'butterfly factor'. x

is called the 'bias factor' because the yz cross section is reflection symmetric with x = 0.

To determine the points of intersection with the axes, we set

S= 0 =0 = 0, 2 3g (A-29)
5

which meets the z axis at z = 4 ( g)5/2 2g ( 3)3/2

z 0 0 0, 02 (A-30)
2

which meets the y axis at y = g2/4. Since both values of 0 give the same y, there is a

point of self intersection on the y axis. Fig A-5(a) shows the family of 0 curves. Each

curve corresponds to a real root of the quintic Eq. A-22.

Consider Eq. A-22 with x = 0, z = 0 (i.e., points on the y axis). The solutions of 0 are

0= 0
02 = -g9 1 (A 31)

For g > 0, there is one solution (0 0) if 4y/g2 > 0 and three solutions otherwise. This is

characteristic of the cusp catastrophe. For g < 0, there are five solutions if 0 < 4y/g2 < 1,

three solutions if 4y/g2 < 0 and one solution if 4y/g2 > 1. (In C'i plter 3, the butterfly we

encountered contained an extra flow that was not singular). The different regions of the yz

cross section of bifurcation set with g < 0 are shown in the Fig. A-5(b).
A.2.3 Elliptic Umbilic: Corank = 2, Codimension = 3

The elliptic umbilic is described by 2 state variables 0, Q and 3 control variables

x, y, z. We use the unfolding

o03 02 (02 + 2) yO + (A-32)
3









The equilibrium surface is given by the equations


02 +2 + 2x0- y = 0
-20 + 2xO. + z 0 (A 33)

The singularity set is obtained by setting the Hessian to zero,

2(8 + x) -2
22( x)+x 0 (A-34)
-20 -2(8 2)

that is, 02 +42 = 2.

To obtain the bifurcation set, let us consider x = constant surfaces. If x is held

constant, we can write 0 = x cos ( and = x sin (. Eq. A 33 becomes

y = x2 [cos 2 + 2 cos
z= x2 [sin 2- 2 sin ] (A-35)

Differentiating with respect to ,

dy 22 [sin + sin 2]

S2X2 [cos 2 cos ] (A-36)


Both derivatives vanish when = 0, 27r/3. Hence there are three cusps located at

(y, z) = (3x2, 0), (-3x2/2, 3v/3X2/2), (-3x2/2, -3V/3X2/2). dy/d = 0 for 7 = but
dz/dl is not. Therefore, dy/dz 0 for 0 which occurs at (-x2, 0). As x is decreased,

the cross section shrinks to the elliptic umbilic point ~ x.2 at (x = 0, y 0, z 0). To

determine the different regions of the bifurcation set, let us consider points on the y axis

(i.e., z = 0) and for fixed x. Eq. A 33 becomes

y = 02 2 + 2x0
S(x ) = 0 (A 37)

If = 0, Eq. A-37 has two solutions when y > -x2 and no solution otherwise. If 0 = x,

Eq. A-37 has two solutions when y < 3 x2. Thus the bifurcation set has two regions in

the region -x 2 < y < 3 x2, there are four solutions for (0, 4) while outside this region,

there are two. Fig. A-6 shows the bifurcation set.









A.2.4 Hyperbolic Umbilic: Corank = 2, Codimension = 3

The unfolding is

03 +03 +x0 y z (A-38)

The equilibrium surface is given by the equations

3 02 +x y 0
3 2 +X = 0 (A-39)

The singularity set is defined by the condition

60 x
0 0 (A-40)
x 60

that is x2 360 0.

We proceed by considering x = constant sections. If x = 0, then 0 = 0 or 0 = 0.

Putting 0 = 0 in Eq. A-39, we find that y = 0 and z is positive. Similarly, putting = 0

in Eq. A-39, we find z 0 and y is positive. Therefore, when x = 0, the bifurcation set is

contained in the positive region of the x and z axes.

When x / 0, we may write = x2/36 0 and so, from Eq. A-39
.3
-/ 3 02+ 6
360

z +x0 (A-41)
362 82

When 0 is close to zero, but negative, y is a large negative number, while z is a large

positive number. When 0 is close to zero, but positive, both y and z are positive and

large. The graph consists of two disjoint pieces.

Differentiating Eq. A-41 with respect to 0,

dy x3
60-
dO 36 02
dz 6x4
d +x (A-42)
dO 362 03

Both derivatives vanish when x = 60. Therefore, there is one cusp at (y = X2/4, z = 2/4).

If x > 0, the portion that corresponds to 0 < 0 is smooth and has no cusps and no critical









points since neither derivatives in Eq. A-42 vanish. To determine the different regions
of the bifurcation set, let us consider points on the line y = z with x = 1. From Eq.

A-39, we have 0 = 0 or 0 + 0 = 1/3. If 0 = Eq. A-39 has two real solutions of 0 if
y > -1/12. If 0 + ~ 1/3, 0 has two real roots if y > 1/4. The hyperbolic umbilic point is

at (x = 0, y 0, z = 0). Fig A-7 shows the bifurcation set. It has three distinct regions.



/

//
\ / ^


V


(a) (b)

Figure A-i. The 04 singularity. (a) 04 (b) 04 + x02 (c) 04 + x02 + y0


r
(a) (b)

Figure A-2. Fold catastrophe: Equilibrium surface and bifurcation set.



































I--


Figure A-3. Cusp catastrophe. (a) Each line represents a value of 0. (b) Bifurcation set.


- -. 2


^- _0


(a) (b)


Figure A-4. Swallowtail catastrophe. (a) Each line represents a value of 0. (b) Bifurcation

set.


*- \.
'-.. \
t, ^





























(a)


Figure A-5. Butterfly catastrophe. (a) Each line represents a value of 0. (b) Bifurcation


Figure A-6.


Elliptic umbilic catastrophe: Bifurcation set.


i I


(b)


Hyperbolic umbilic catastrophe: Bifurcation set.


3

3
5 1
3

3

1


Figure A-7.









APPENDIX B
PROPERTIES OF THE TRICUSP CAUSTIC RING

In C'! ipter 3, we showed that when the initial velocity field is dominated by a net

rotational component, the resulting inner caustic has the appearance of a ring, whose

cross section has three cusps. We called this a 'tricusp caustic ring'. In the limit of axial

symmetry about the z axis and reflection symmetry about the z = 0 plane, it is possible

to derive an analytic expression for the caustic structure, as well as to calculate the dark

matter density at points close to the ring. The properties of the caustic ring were first

described by P. Sikivie [94]. Here we give a brief description.

Let us assume an axially symmetric flow. We have shown in Chapter 3 that

non-axially symmetric flows also produce caustics. Hence, the assumption of axial

symmetry should be regarded as a simplifying feature and not as a necessity.

With the assumed axial symmetry, we may parametrize each particle in the flow in

terms of two variables r and a. a = r/2 0, where 0 is the polar angle of the position of

the particle when it crossed the reference sphere. (The particles which are confined to the

z = 0 plane have a = 0). We define r = 0 as the time when the particles just above the

z = 0 plane (the particles parametrized by a = 0 + 6a) cross this plane. The azimuthal

angle Q is not relevant due to the assumed axial symmetry. Let p + +y2 and z be

the cylindrical co-ordinates of physical space. We assume reflection symmetry about the

z = 0 plane. With these assumptions, the flow at points close to the caustic, in pz cross

section is obtained by performing a Taylor series expansion about (a = 0, r = 0) [94]

z(a,7) bar
1 1
p(a,r)-a -u(7-To) sa2 (B-1)
2 2

where b, u, s and To are constants and a is the caustic ring radius. The two dimensional

Jacobian determinant (x, y)/a(a, 7) is


D2(a, r) b [ur ( o) + sa22] (B-2)









Fig. B-i shows the dark matter flows forming a tricusp ring caustic (in pz cross

section), for continuous r and for discrete a. (each line represents a particular value of a

and each line is made up of many points, each point representing a particular value of T).

p and q are the horizontal and vertical extents of the tricusp respectively [94]

1 2
p = 0

v 5 bp
q (B-3)
4 us

The caustic in cross section is the locus of points with D2 = 0. To derive the caustic

structure, let us use Eq. B-3 to express Eq. B 1 as

p -a (T 1)2 27 (ba )2
p 64 q
z = baroT (B-4)

where T = 7/To. The condition D2 = 0 implies


T(T t1) + 0 (B5)

Eliminating T and a, the cross section is described by the curve

1 -3/2 o0 < < 1
2 3 27 27
z (B-6)
2 1 +k -8-/2 t 1< u< (
2 32- 1< <9

where ( is given by
= +8 a (B-7)
P
Figure B-2 shows the cross section. If the z axis is rescaled relative to the p axis so as

to make the tricusp equilateral, the tricusp has a Z3 symmetry [94] consisting of rotations

by multiples of 27/3 about the point of coordinates (pc, z,) = (a + p/4, 0). This point may

be called the center of the tricusp. It is indicated by a star in Fig. B-2.

When axial symmetry is not present, the cross section varies along the ring, as we

showed in C'!h p III. In general, there are points where the cross section shrinks to zero,

forming elliptic umbilic catastrophes (Appendix A).









B.1 Density Near a Tricusp Caustic Ring

Let us now calculate the dark matter density at points near the caustic using

equations B 1, B-2. We first compute the density in the z = 0 plane. We then move

on to the more general case.

B.1.1 Case 1: Points in the z = 0 Plane

Let us first consider the simple case when z = 0. From Eq. B 1, we see that the

condition z = 0 implies a = 0 or r = 0. For points (p, 0) with p > a + p, we have

a = 0, 7 / 0. For points (p, 0) with p < a, we have r = 0, a / 0. For points (p, 0) with

a < p < a + p, both a and 7 are zero. Define the dimensionless co-ordinates

R p-a
P
Z (B-8)
q

ao 0:

When a = 0, Eq. B 1 becomes

z 0
1
p-a -u(T -To)2 (B-9)
2
Solving for D2 and using Eq. B-3, we obtain

D2 = 2bp R(R) forR > 0 (B10)

Summing the two solutions for ID21-1, we find


ID21 2bpR vR 1 +
1 1
bpr RR > r 0 (B 11)

r= 0:
When 7 = 0, we have

z 0
p- a = -ur2 -a2 (B12)
2 2









which gives us

a = 2( R) for R < 1 (B-13)

Solving for ID21 and summing the two solutions for ID2 -1, we find

1- 1- for R < 1 (B-14)
SiD2 bp 1 R

In the region 0 < R < 1, we must sum over the four solutions. In this region (inside

the caustic, z = 0),

S 1 1 + (B-15)
|D21 bp t R (1 2/5
The physical space density is given by

1 dM cos(a)
d(p, z) =- (a,) ) (B-16)
p d~dt |D2 (a, )

where the sum is over D2 -1. We use the self-similar infall model with angular momentum

[92, 93] to estimate the mass infall rate

dM fV V2Qt ( 17)
d t fv 4r (B-17)
dQdt 47G

where f is a self-similar parameter which specifies the flow density and v is the speed of

the particles in the flow a b [94]. We may also approximate cos a t 1 for p, q < a. The

density d(R, 0) at points close to the tricusp caustic, for z = 0 is given by

I R<0
1-R
d(RO) 0. 34GeV (f/10-2) (V,/220kms-1)2 ) o < l (B18)
cm3 pkpc Pkpc 1-R 0 <
1 R>1
p-1

pkp~c and pkpc are distances measured in kpc.









B.1.2 General Case: z / 0

We now calculate the density near the caustic at points z / 0. Since z / 0, both a

and 7 are nonzero. Setting a = z/br and using Eq. B-3,

2 27 Z2
R= (T -1) (B-19)
64 T2

where T = 7/T-. We need to solve the quartic equation

T4 2 T3 + ( RZ2 = 0. (B-20)
64

The density d(R, Z) is given by

p(RZ 7 GeV (f/1O-2) (Vot/220kms-1)2 1 t
p(R, Z) a 0.17ck pT ( (B221)
cm3 Pkpc Pkpc 2T2 i + -

where the sum is over the real roots of the quartic Eq. B-20. The number of real roots is

given by the sign of the discriminant S

S= 144FR2 R+ -128F2 (R+ +4R2 (R + -16 R -27R4-2563

(B-22)

with F given by
S27 2R 1 (B 23)
64 4 16
S is positive for points inside the caustic and negative for points outside. Since there

must be at least two roots everywhere, Eq. B-20 has two real roots when S < 0 and four

real roots when S > 0.

B.1.3 Solving the Quartic Equation to Obtain the Real Roots of T

Let us re-write Eq. B-20 in term of T1 by making the substitution T = T + thereby

eliminating the cubic term:

Ti4 -(R ~+1 2 RT 0 (B-24)
( t/








The quartic B-24 can be expressed as the product of two quadratics


(T12 + eT1 + f) (T12


eT1 + h) = 0,


where h and f are given by


1
2
1
f 2
2


(Rt) ;]

K2 (


and e" solves the cubic

(e2)3 F- 272] .
(2)3 2 R+ (62 R2 + 2R 27 2

We can solve for T1 once we have solved the cubic B 27.


To solve B-27, we set y


e2 and make the redefinition y


y + j + + 1 ) to obtain


for yi,


where P and Q are given by


S- R 2)

- -( 3


One real root of y1 is given by


Q + Q2 3]
2 P cos [ cos-1 Q


S[Q_ V/Q2 P3]


when Q2 P3 > 0
when Q2 P3 < 0


When Q2 p3 < 0, there are three real roots, but we only need one real root. Putting


everything together and using e2


y 1 + (R + ) andT


1e
2-
1+e
2


T1 + {, we have the result


(l)2 -
L2


(B-31)


(B-25)


(B-26)


(B-27)


R2 0


/13 3PUi 2Q = 0


(B-28)


3Z 2


(3Z)
4


(R+


Y1 -


(B-29)


(B-30)









Once the real roots of T are obtained using B-31, we can compute the density using Eq.
B-21.
B.2 Mass Contained in the Ring.

With the assumptions of axial and reflection symmetry, we can use Eq. B-6 and Eq.
B-21 to compute the total mass in a caustic ring of radius a and cross section p, q. The
mass enclosed is

M = dp dzJ pdd(p,z)

S27a / dp dz d(p,z)
p1 Z Zm a
2a x 2pq dR dZ d(R, Z) (B-32)
R=-1/8 where Zmin and Zmax are specified by

1 /2_ 1 S 3 27 27 8 --
Zmin = (B-33)
0 O
1 2 8 3 2
Zmax i + 3 (B34)
2 3 27 27
S 1 + 8R and we have used p < a. Using Eq. B-21,

M z 5.57 x 107 Mo 2 2) ( V2 2k T (B-35)

where T is the integral

S dR2 dZ 1 (B36)
JR-1/8 J ZZmi 2T2 3T + (1 R)I

and Ti are the real roots of Eq. B-20. A numerical calculation gives T w 2.4 and so the
mass contained in the ring

M 1.34 x 108Me vrot (B-37)
kpc 10-2 220 km/s v









In C'!I pter 4, we assumed q t 1 kpc and f = 4.6 x 10-2 for the n = 2 ring which gives us

1i. ~6 x 108 Me.

B.3 Effect on Surrounding Baryons

If gas or other baryonic material in the plane of the galaxy at radius r moves on a

circular orbit with velocity v(r), then

=( F(r) (B-38)


gives the inward gravitational force per unit mass at r. Caustics exert a gravitational force

on the surrounding matter. The effect of this force is to perturb the circular velocities of

matter close to the caustic, producing bumps in the rotation curve. The perturbation to

the circular velocity v, is given by

2
Fc(r) =- Vrot Vc(r) (B39)
r

where Fc is the force exerted on the gas by the caustic. Let us consider a caustic with

transverse sizes p, q < a. Consider a point p in the plane of the galaxy and located very

close to the ring (i.e., |p al < a). In this limit, the ring may be approximated by a long,

straight tube. We may then integrate over the length of the tube. The gravitational force

F, of the caustic ring is given by [94]


Fc(p, z = 0) 2G dp dz d(p', z') p 2 p, (B-40)
J (p-p) +z2(p ')

To simplify this result, let us change variables (p', z') -+ (a, 'r). Since we are working in

the limit p, q < a, most of the contribution to the force comes from small values of (a, 'r).

In this limit, and assuming stationarity of the potential, we may ignore the dependence of

dM/dQdt on a and r. We find

F( 2G d2M P p_ (a,r ) +( )
F(p) dadr [ (B-41)
c a dldtj [p pI(a, )]12 B2 4,1T









In terms of a and 7,


2G d2M fdad
Fc(p) = d rd
a dGdt J


(p
[(p a)


a) u( 7o)2 + 1a2
u( T70)2 + .a2]2 + b2t272


(B-42)


C'!i ,iI.iig variables


c-a
r70


(B-43)


and using Eq. B-3, we have


4G d2M dA
ab dQdt [R


(T 1)2 + (A2
1)2 + (A2]2 + 4A2T2


where R and ( are


R p-a
R


su
b2


Using Eq. B-39, the perturbation to the rotation velocity is


47G d2M (I
b v,,ot dUdt


with I given by


1 dAdT.
I((, R) dAdT
27 J


R (T 1)2 + (A2
[R- (T- 1)2 + (A]2 + 4A2T2


I((, R) is constant for R > 1 and R < 0. For the special case of ( 1,


-1/2

I(1,R) = -1/2+ v

+1/2


for R < 0 ,

for 0 < R < 1


for R > 1.


The sudden change in I((, R) at R = 0 and R = 1 is due to the fact that the dark matter

density changes suddenly when crossing those points.


(B-44)


27 (p 2
16 q


(B-45)


(B-46)


(B-47)


(B-48)









To obtain the full rotation curve, we must include the perturbation v,(r) due to the

caustic. However, since the caustic is not added to the halo, but instead is made up of

dark matter re-arranged in the form of a ring, the rotation curve is flat on average. The

perturbation v, should therefore be added to a descending curve. The modified rotation

curve may be expressed as

v(r) r + 47 J((, R)j (B 49)
b V2 dadt
b IUMo(

The function I((, R) of eq. (B-48) is replaced by J((, R), which includes the effect

of the descending rotation curve. The factor d2M/dl dt may be extracted from the

self-similar infall model [92, 93]
d2M 2
Sf o* (B-50)
dQdt 17(F
where is the velocity of the particles in the ith caustic ring, and the dimensionless

co-efficients fi characterize the density in the ith in-and-out flow. Re-writing equation (B-49),

and using ~ bi [94]

S(r) Vrot [1 + fJ((, R)], (B-51)

where (r) is the rotation velocity near the ith caustic ring. We choose J to be of the form

J(r) (r) tanh r a (B-52)
2

p' is expected to be of order a. Fig. B-3 shows a qualitative sketch of I(r) and J(r).





























Figure B-1. Dark matter trajectories forming a tricusp ring (in cross section).
Figure B-1. Dark matter trajectories ftruing a tricusp ring (in cross section).


Figure B-2. Tricusp caustic in pz cross section.


B


Figure B-3. Modified rotation curve. (a) The function I (b) The function J


aP









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[102] P. Ullio, L. Bergstrom, J. Edsjo and C. Lacey, Phys. Rev. D66, 123502 (2002)

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BIOGRAPHICAL SKETCH

Aravind Natarajan was born in Trivandrum, India, in 1978. He completed his

undergraduate studies at Bangalore University, 1i Pii -i-; in electronics and communication

engineering. His undergraduate thesis work was on frequency selective processing of

MPEG-1 Audio L vI--r- bitstreams. He later became interested in physics and worked

for 8 months as a Research Assistant in the Devices Lab, Department of Electrical

Communication Engineering, Indian Institute of Science. This lead to his first publication,

in the Journal of Applied Physics. He then joined the Joint A-i in.. in v Program at the

Department of Physics, Indian Institute of Science, to study physics and astronomy and

enrolled at the University of Florida in August of 2002, as a graduate student.

Aravind successfully took several courses at the University of Florida including

Electricity and Magnetism, Quantum Mechanics, Quantum Field Theory, Standard Model

of Particle Physics, Statistical Mechanics, The Early Universe, Functional Integration,

Particle A-lt i .1!i--1.- and Dark Matter. He received his master's degree from the

University of Florida in August 2004. His graduate school GPA was 4.0.

Aravind received many academic awards during the course of his study. In the spring

of 2005, he received the J. Michael Harris Award, which is given to two theory students

each spring. In the spring of 2006, he received the Outstanding International Student

Award, given by the International Center. In fall 2006, he was awarded the Chuck Hooper

Memorial Award for distinction in research and teaching.

In summer 2004, Aravind began to work on his thesis, which involves a detailed study

of the properties of dark matter caustics and related .-lr i, !i-- i1. 1 effects. He attended the

"Santa Fe Cosmology Workshop" at Santa Fe, in the summer of 2004 and the summer of

2006 and the "Particles, Strings and Cosmology School" held in Japan, in the fall of 2006.

He has presented talks and posters at many conferences. He has also been a science fair

judge at the Kanapaha middle school in Gainesville.









Aravind's main ]..1.1i,- is photography, which he practices in his spare time. His

photographic subjects are mostly nature, landscapes, and wildlife. After graduation, he

intends to continue research in cosmology and ,-ar ,1l,!i-ics as a postdoctoral fellow at

Bielefeld University in Germany.





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Iwouldliketothankmyparentsfortheirloveandsupport.IthankmyPh.DsupervisorProfessorPierreSikivieandtheothermemberofmysupervisorycommittee,ProfessorsRichardWoodard,JamesFry,JamesDuftyandVickiSarajedini.IamespeciallygratefultoRichardWoodardforhisconcernandadviceonmanyoccasions.Youwilldenitelybemissed!ItismypleasuretothankSushforallthegreattimeswehadtogether.IalsoacknowledgemyfellowphysicsnerdsJian,Sung-Soo,KC,Ian,NaveenandEmre.IthanktheInstituteforFundamentalTheory,UniversityofFloridaforprovidingpartialnancialsupport. 3

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page ACKNOWLEDGMENTS ................................. 3 LISTOFTABLES ..................................... 6 LISTOFFIGURES .................................... 7 ABSTRACT ........................................ 9 CHAPTER 1INTRODUCTION .................................. 10 2THEFORMATIONOFCAUSTICS ........................ 15 2.1OuterCaustics ................................. 17 2.2InnerCaustics .................................. 19 2.3ExistenceofInnerCaustics ........................... 19 2.4PossibleComplicationsthatAecttheExistenceofCaustics ........ 21 2.4.1ExistenceofaLargeNumberofFlows ................ 21 2.4.2PresenceofSmallScaleStructure ................... 22 2.4.3GravitationalScatteringbyInhomogeneities ............. 23 2.5DicultyofResolvingCausticsinN-BodySimulations ........... 24 2.6PossibleEvidencefortheExistenceofDarkMatterCaustics ........ 25 2.6.1RisesintheRotationCurvesofSpiralGalaxies ............ 25 2.6.2RotationCurveoftheMilkyWayGalaxy ............... 26 2.6.3TriangularFeatureintheIRASMap ................. 26 2.7Discussion .................................... 26 3THESTRUCTUREOFINNERCAUSTICS ................... 29 3.1LinearInitialVelocityFieldApproximation ................. 29 3.2Simulation .................................... 31 3.3TricuspRing .................................. 32 3.3.1AxiallySymmetricCase ........................ 33 3.3.2PerturbingtheInitialVelocityField .................. 34 3.3.2.1Eectofthegradienttermsg1andg2 34 3.3.2.2Eectofarandomperturbation ............... 34 3.3.2.3Eectofradialvelocities ................... 34 3.3.3ModifyingtheGravitationalPotential ................. 35 3.3.3.1NFWprole .......................... 35 3.3.3.2Breakingsphericalsymmetry ................ 36 3.4GeneralStructureofInnerCaustics ...................... 36 3.4.1AxiallySymmetricCase ........................ 36 3.4.2InfallwithoutAxialSymmetry ..................... 39 3.5Discussion ................................... 42 4

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................... 54 4.1AngularVelocityofGasinCircularOrbitsClosetoaCausticRing .... 56 4.2AngularMomentumTransportbyViscousTorque .............. 56 4.3EectonStarOrbits .............................. 59 4.3.1OrbitStability .............................. 59 4.3.2Resonances ................................ 59 4.3.3DensityEnhancement:CircularOrbits ................ 61 4.3.4DensityEnhancement:Non-circularOrbits .............. 63 4.4Discussion .................................... 63 5WIMPANNIHILATIONININNERCAUSTICS ................. 66 5.1AnnihilationFlux ................................ 66 5.1.1EstimatingS 67 5.1.2Estimating 68 5.2ComparingtheSignalwiththeBackground ................. 70 5.3Discussion .................................... 71 6CONCLUSIONS ................................... 74 APPENDIX ACATASTROPHETHEORY ............................. 76 A.1FoldandCuspCatastrophes. ......................... 78 A.1.1Fold:Corank=1,Codimension=1 .................. 78 A.1.2Cusp:Corank=1,Codimension=2 ................. 79 A.2HigherOrderCatastrophes. .......................... 80 A.2.1Swallowtail:Corank=1,Codimension=3 .............. 80 A.2.2Buttery:Corank=1,Codimension=4 ............... 81 A.2.3EllipticUmbilic:Corank=2,Codimension=3 ........... 83 A.2.4HyperbolicUmbilic:Corank=2,Codimension=3 ......... 85 BPROPERTIESOFTHETRICUSPCAUSTICRING .............. 89 B.1DensityNearaTricuspCausticRing ..................... 91 B.1.1Case1:Pointsinthez=0Plane ................... 91 B.1.2GeneralCase:z6=0 ........................... 93 B.1.3SolvingtheQuarticEquationtoObtaintheRealRootsofT 93 B.2MassContainedintheRing. .......................... 95 B.3EectonSurroundingBaryons ......................... 96 REFERENCES ....................................... 100 BIOGRAPHICALSKETCH ................................ 105 5

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Table page 5-1N=m2inunitsof104GeV2form=50GeV ................. 68 5-2N=m2inunitsof104GeV2form=100GeV ................. 68 5-3N=m2inunitsof104GeV2form=200GeV ................. 68 5-4Peaksoffor(a=8:0;p=0:1;q=0:2)kpc ............... 70 6

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Figure page 2-1Darkmattertrajectoriesformingacuspcatastrophe. ............... 27 2-2Eectofaveragingthedensityoveranitevolume. ............... 28 3-1Infallofacoldcollisionlessshell:AntisymmetricM. ............... 43 3-2Darkmatterowsforminganinnercaustic. .................... 44 3-3Tricuspringwithnon-zerog1andg2. ....................... 44 3-4Eectofarandomperturbation. .......................... 45 3-5Tricuspring:NFWpotential. ............................ 45 3-6Tricuspring:Non-sphericallysymmetricgravitationalpotential. ........ 45 3-7Case1:Infallofashell.Irrotationalowwithaxialsymmetry,jg1jg 47 3-9Crosssectionoftheinnercausticproducedbyanirrotationalaxiallysymmetricvelocityeld(Case1:jg1jg). ........................... 48 3-11Causticincrosssection,forincreasingc3. ..................... 49 3-12Darkmatterowsinxycrosssection. ........................ 49 3-13Gradienttypecausticwithoutaxialsymmetry. ................... 50 3-14Formationofthehyperbolicumbiliccatastrophe. ................. 51 3-15Causticincrosssection,fordierent(g1;g2). ................... 52 3-16z=0sectionsofthecaustic,showingthetransformationfromgradienttypetocurltype. ....................................... 53 3-17Transformationofagradienttypecaustictoacurltypecaustic. ........ 53 4-1Rotationcurveclosetoacausticring. ....................... 64 4-2SmoothpotentialVc(r). ............................... 64 4-3Adiabaticdeformationofstarorbits. ........................ 65 5-1Emissionmeasureaveragedoverasolidangle=105sr,forthreesetsofcausticparameters. ................................. 72 7

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........ 73 A-1The4singularity. .................................. 86 A-2Foldcatastrophe:Equilibriumsurfaceandbifurcationset. ............ 86 A-3Cuspcatastrophe. ................................... 87 A-4Swallowtailcatastrophe. ............................... 87 A-5Butterycatastrophe. ................................ 88 A-6Ellipticumbiliccatastrophe:Bifurcationset. ................... 88 A-7Hyperbolicumbiliccatastrophe:Bifurcationset. ................. 88 B-1Darkmattertrajectoriesformingatricuspring(incrosssection). ........ 99 B-2Tricuspcausticinzcrosssection. ......................... 99 B-3Modiedrotationcurve. ............................... 99 8

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9

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113 ]measuredtheradialvelocitiesofsevengalaxiesintheComacluster.Themeasuredvelocitiesdieredfromthemeanradialvelocityoftheclusterbyalargeamount(about700km/sRMSdeviation).TheluminousmassoftheComaclusterisinsucienttoaccountforsuchalargevelocitydispersion.Zwickythereforeinferredtheexistenceofdarkmatter,whosepresenceisfeltonlygravitationally.WiththecurrentlyacceptedvalueoftheHubbleparameter,thereisabout50timesasmuchmatterintheComacluster,asinferredfromtheclusterluminosity.AsimilarconclusionwasreachedfromBabcock'swork[ 7 ]ontherotationvelocitiesofM31andbyJ.Oort'smeasurements[ 71 ]ofrotationandsurfacebrightnessofthegalaxyNGC3115.Furtherevidencefordarkmatteringalaxiescameinthe1970swiththestudyoftherotationcurvesofspiralgalaxiesbyV.C.RubinandW.K.Ford[ 83 ]andbyRobertsandWhitehurst[ 82 ].Themeasurementsshowedthattherotationvelocityremainsconstantwithdistancefromthecenterevenwelloutsidetheluminousdisk.Iftheluminousmatteraccountedforallthematterinthegalaxy,onewouldexpecttoseeaKeplerianfall-oofrotationvelocity.Theobservationscontradictthisandprovidesupportfortheexistenceof 10

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72 ]concludedfromtheirstudyofthestabilityofgalacticdisksthatdarkmatterispresentintheformofa\halo"ingalaxies.[ 84 104 ]giveahistoricalaccount.LetusconsiderournearestlargegalaxyM31intheconstellationAndromeda.TheMilkyWayandM31areseparatedby0.73Mpcandareapproachingeachotherwithalineofsightvelocityof119km/s.Forthepurposeofobtaininganapproximateestimateofthemassofthesetwogalaxies,letusassumethattheyarepointlike.Further,letusassumezeroangularmomentum.Althoughtheactualphysicsismorecomplicated(galaxiesarenotpointmasses),theseassumptionsallowustomakearoughcalculationofthemassinourgalaxy.AssumingNewtoniangravity,thedistanceofseparationrobeystheequation r=G(M1+M2) 2r0[1+cos]t()=s (1{2) usingwhichweobtainforthevelocity (1+cos)2(1{3)Settingvr=119km/swhent=1:41010yearsandr=0:73Mpc,wend=1:53andsoM1+M22:51012M.AssumingthattheMilkyWaygalaxyandM31haveequalmasses,themassofourgalaxyis1012M(theaccelerationoftheUniversemakesthisanunderestimate).SincethemassinluminousmatterintheMilkyWayis1011M,wendfromoursimplecalculationthat90%ofthemassinourgalaxyisdark.Inrecentyearsimprovedmeasurementsofthepropertiesofgalaxies,aswellastheoreticalinputhavemadeaverystrongcasefortherealityofdarkmatter.Ifthe 11

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98 99 ].TheWMAPexperimenthasmeasuredthetemperaturepowerspectrumofthecosmicmicrowavebackgroundradiationtohighprecision.Theresultsagainpointtotheexistenceofdarkmatterandareinagreementwiththeresultsofotherexperiments.Recently,observationsof1E0657-558(the\bulletcluster")haveprovideddirectevidencefordarkmatter[ 26 ].Thiseventisamergeroftwoclustersofgalaxies.Theintra-clustergasiscollisionalandheatsup,emittingX-rays.Ifacollisionlesscomponentispresent,itwouldsimplypassthroughandbecomeseparatedfromthegas.X-raymeasurementsoftheclusterareusedtotrackthegas,whileweakgravitationallensingtracksthetotalmass.Theobservationsshowthatthecenterofmassdoesnotcoincidewiththeluminousmatter.Darkmattercosmologyassumesthecorrectnessofgeneralrelativity.Generalrelativityisconsistentwithsolarsystemtestsliketheprecessionofplanetaryorbitperihelia,orthebendingofstarlightbythesun'sgravitationaleld[ 107 ].ThetheoryhasbeentestedtoextremelyhighprecisioninthecaseoftheHulse-Taylorbinarypulsar.Thereexistcosmologicaltheoriesthatdonotincludedarkmatter(exceptperhapsthemassiveneutrinosofthestandardmodel).Inthesetheories,gravityisnotdescribedbyNewton'slaw(orgeneralrelativity),butbyamodicationofit.Thersttheorythat 12

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59 ]in1981.Morerecently,arelativistictheoryofmodiedgravityconsistentwiththetestsofgeneralrelativity,wasproposedbyJ.Bekenstein[ 8 ].However,inthisworkwewillassumethevalidityofNewtoniangravityandtheexistenceofcolddarkmatter.ThemostcommonlycitedcolddarkmattercandidatesaretheaxionandtheWeaklyInteractingMassiveParticle(WIMP).Theseparticleshavethedistinctionthattheywereproposedforpurelyparticlephysicsreasons,thoughtheyareexcellentdarkmattercandidates.TheaxionsolvesthestrongCPproblemofQCD,whileWIMPsariseinsupersymmetricextensionsofthestandardmodel.AxionsarebeingsearchedforbytheAxionDarkMattereXperiment(ADMX)[ 6 33 ],whileWIMPdetectionexperimentsincludeDAMA/NaI[ 12 { 14 ],DAMA/LIBRA[ 15 ],CDMS[ 22 ],XENON[ 4 ],EDELWEISS[ 85 ],DRIFT[ 97 ],Zeplin[ 3 ],CRESST[ 1 ]andothers.Acentralproblemofstructureformationstudiesisthequestionofhowcolddarkmatterisdistributedinthehalosofgalaxies.Thesimplesthalomodelistheisothermalmodel,whichassumesthatthedarkmatterparticlesformaselfgravitating,thermalizedsphere.Someofthepredictionsofthismodelhavebeenconrmed,notablytheatnessofrotationcurvesandtheexistenceofcoreradii.However,itisnotpossibleforallthedarkmattertobethermalized[ 91 ].Evenifdarkmatterhadthermalizedinthepastbyviolentrelaxation[ 51 ],galaxyformationisanongoingprocess.Thelateinfallofdarkmatterwillcausenon-thermalstreams,asweshowin Chapter2 .Sincethereisnoevidenceforviolentrelaxationoccurringtoday,particlesthathavefallenintotheinnerregionsofthehalo(whichcontainthemostsubstructure)relativelyrecentlymaybeexpectedtobenon-thermal(cold).AnotherapproachistocarryoutN-bodysimulationsoftheformationofgalactichalos,onsupercomputers.Thisapproachispowerfulsinceitrequiresnoassumptionsofsymmetriesorspecialinitialconditions,andpresumablygivesthecorrectresultsifNislargeenough.However,currentsimulationsonlyhave109particles.Ifthedark 13

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Chapter2 ,weshowthatcoldinfallofdarkmatterproducescausticsandprovidepossibleevidencefortheexistenceofcaustics.Wealsoexaminetherelevanceofcausticsinrealgalaxies.In Chapter3 ,weprovideadetaileddescriptionofthestructureofinnercaustics,fordierentinitialconditions. Chapter4 and Chapter5 dealwiththeastrophysicaleectsofdarkmattercaustics.In Chapter4 ,weexplorethepossibilityofaconnectionbetweenadarkmattercausticandtheMonocerosRingofstars.Weshowthatthereexisttwomechanismsbywhichthepresenceofacausticcouldincreasethedensityofbaryonicmatterinitsneighborhood,possiblyexplainingtheformationoftheRing.In Chapter5 ,weinvestigatethepossibilitythatcausticsmaybedetectedbyindirectmeans(i.e.,byobservinggammaraysfromtheannihilationofparticlesinthecaustic).Wecomputetheannihilationuxfromanearbycaustic,assumingthatthedarkmatterismadeupofsupersymmetricWIMPsandcomparethesignalwiththeexpectedbackground. 14

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94 ]tobe wheret0isthepresentageoftheuniverseandcisthespeedoflight.Considerthecontinuousinfallofdarkmatterparticleswithnegligiblevelocitydispersionfromalldirectionsinagalactichalo.Sincethedarkmatterparticleshavenegligiblevelocitydispersion,theseparticlesexistonathinthreedimensionalhypersurfaceinphasespace,calledthephasespacesheet(thethicknessofthephasespacesheetisthevelocitydispersion)[ 94 ].Forcollisionlessmatter,thedensityinphasespacefollowingthetrajectoryofaparticleisconservedbyLiouville'stheorem.Thisensuresthatthephasespacesheetiscontinuous.Also,thephasespacetrajectoriesdonotself-intersect.Theseconditionsconstrainthetopologyofthephasespacesheetirrespectiveofthedetailsofthegalaxyformationprocess.Inordertoobtainthedensityofdarkmatterparticlesinphysicalspace,wemustmakeamappingfromphasespacetophysicalspace.Causticsareregionswherethismappingissingular.ThesingularitiesthatoccurinmappingsareknownascatastrophesandhavebeenwellstudiedinthecontextofCatastropheTheory( AppendixA ).Ingeneral,causticsaremadeupofsectionsoftheelementarycatastrophes[ 101 ]. 15

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54 68 69 103 106 ].Inthecontextofcosmology,thestudyofcaustics(orpancakes)waspioneeredbyY.B.Zeldovich,S.F.Shandarin,A.G.Doroshkevich,V.I.Arnoldandmanyothers.[ 24 30 31 48 58 88 89 ].Letusreturntothephasespacedescriptionofcolddarkmatter.Ifwemayneglectthethicknessofthephasespacesheet,wecanassigntoeachparticle,athreeparameterlabel~(1;2;3)whichidentiestheparticle[ 94 ].Asanexample,letusconsideraspherewithaconvenientlychosenradiussuchthateachparticleintheowpassesthroughthesphereonce.Then,wemaylabeltheparticlesoftheowbythethreeparameters(;;)whereisthetimewhentheparticlecrossedthereferencesphereand;aretheco-ordinatesofthepointwheretheparticlecrossedthereferencesphere.Inourexample,theparticlewhichcrossedthesphereattimeatlocation;willbelabeledby(;;)atalltimest.Intermsofthephysicalspaceco-ordinates(x;y;z),thedensityofparticlesdis[ 94 ] d1d2d3(~i(t;~x))1 @~~i(t;~x)(2{2)Thesumover~iisrequiredbecausethemappingfrom(;;)spaceto(x;y;z)spaceismany-to-one.CausticsarelocationswheretheJacobianfactorjdet(@~x=@~)j=0.Inthelimitofzerovelocitydispersion,thesearepointsofinnitedensity.Theparticletrajectoriesinagalactichaloarecharacterizedbytwoturnaroundradii(i.e.,therearetwopointsatwhichtheradialvelocityvanishes).Causticsoccuratbothturnaroundradii.Thecausticsthatoccurattheouterturnaroundradiiarecalled`outercaustics'.Theyaretopologicalspheressurroundinggalaxiesandtypicallyoccuronscalesof100'sofkpc,foraMilkyWaysizegalaxy.Thecausticsthatformattheinner 16

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94 ].FillmoreandGoldreich[ 35 ]andBertschinger[ 16 ]showedthattheevolutionofdarkmatterhalosisself-similariftheprimordialoverdensityhastheprole 92 93 ].Withthismodication,thelocationsoftheinnercausticsarepredictedbythetheory. 17

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AppendixA ).Therearecausticsateachofthe(n+1)turnaroundradii[ 16 35 ](Thereisnocausticattherstturnaroundradiusbecausethereisonlyaninowofdarkmatterandnooutow).Thisisbestseeninthephasespacediagram[ 94 ].Theprojectionofphasespaceontophysicalspaceissingularwheneverthephasespacesheetistangenttovelocityspace(i.e.,attheturnaroundradii).Thusinrealgalactichalos,weexpectaseriesofoutercausticswhichhavetheformofthinsphericalshells.Inrealgalactichalos,thevelocitydispersionofdarkmatterparticlesissmall,butnotzero.Inthiscase,thephasespacesheetwillhaveanitethicknessandtheresultingcausticswillbespreadoveradistancerthatdependsonthemagnitudeofthevelocitydispersion.Onemustthenaveragethedensityoveraregionofsizer,whichrendersthedensitynite.Oneofthebestexamplesofoutercausticsongalacticscalesistheoccurrenceofshells[ 18 41 53 ]aroundsomegiantellipticalgalaxieswhichresideinrichclusterswherethemergerprobabilityissignicant.Theseshellsarecausticsinthedistributionofstarlight.Theyformwhenadwarfgalaxyfallsintothegravitationalpotentialofagiantgalaxyandisassimilatedbyit.Thestarsofthedwarfgalaxyhaveavelocitydispersionthatissmallcomparedtothevirialvelocitydispersionofthegiantgalaxy.Thestarsarethereforesub-virialandpresumablycollisionless.Thecontinuousinfallofthesecoldcollisionlessstarsproducesaseriesofcaustics.Theinfall-outfallprocessrepeatsuntilthestarsarethermalizedandlosetheircoldness.Theresultisaseriesofarcs(theyarenotcomplete 18

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18 ].Theoccurrenceofcausticswithstarssuggeststhatthesamecouldoccurwithdarkmatter. Chapter3 ,wewillgiveadetaileddescriptionofthestructureofinnercaustics,showingthecatastrophesthatformforgiveninitialconditions.Toseetheformationofinnercaustics,letusconsidertheinfallofacoldowofdarkmatterparticles.Iftheinfallissphericallysymmetric,theparticletrajectoriesareradialandtheinfallproducesasingularityatthecenter.Ifinstead,thedarkmatterparticlespossesssomedistributionofangularmomentumwithrespecttothehalocenter,theparticletrajectoriesarenon-radial,particularlyintheinnerregionsofthehalo.Fig. 2-1 showsanexampleofnon-radialinfall,inxycrosssection.Thecausticistheenvelopeofthefamilyofdarkmattertrajectories(i.e.,itisthelocusofpointstangenttothefamilyoftrajectories).Thedarkmatterdensityisenhancedalongtheenvelope.ThetwocurvesinFig. 2-1 arefoldcatastrophesandtheirintersectionisacuspcatastrophe.Thecausticcurveandthedensityfalloareworkedoutin AppendixA 63 ].Inaccordancewithourformalism,letusconsideracoldcollisionlessowofparticlesandlabeltheparticlesbythreeparameters(;;).Letuschooseareferencesphereatsomeconvenientradius,suchthateachparticlepassesthroughthesphereonce.isthetimewhentheparticlecrossedthereferencesphereand(;)specifythelocationwheretheparticlecrossedthereferencesphere.Toobtainthedensityisrealspace,onemust 19

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@@~x @@~x @=0(2{5)Considertheinfallofashellofdarkmatterparticleswithanarbitraryangularmomentumdistribution.Sincethetangentialvelocityvectorona2-spheremustvanishattwopointsatleast,theangularmomentumeldhasamaximum(otherwiseitiszeroeverywhere).Letusconsiderthepointofclosestapproachoftheparticleswiththemostangularmomentum.Letuslabeltheseparticlesby(0;0).Atthepointofclosestapproach,wehave @=~x r@~x @=0(2{6)forsome(0;0;0)becausetheradialvelocityvanishesatthepointofclosestapproach.Similarly,sincetheangularmomentumdoesnotvarytolowestorderabout(0;0), @=~x r@~x @=0@r @=~x r@~x @=0 (2{7) for(0;0;0)atthepointofclosestapproach.Dene~0=(0;0;0).Notethat~x0=~x(~0)=0onlyinthelimitofsphericalsymmetry(i.e.,whentheinnercaustichascollapsedtoasingularpoint).Forthemoregeneralcaseofnon-zeroangularmomentum,~x06=0.Equations 2{6 and 2{7 implythat@~x(~0)=@;@~x(~0)=@and@~x(~0)=@areallperpendicularto~x0.HencethosethreevectorsarelinearlydependentwhichmakestheJacobiandeterminantEq. 2{5 vanish.Thus,wehaveshownthatthepointofclosestapproachoftheparticleswiththemostangularmomentumliesonacaustic.Ingeneral,thecausticisasurface.Sofar,wehavegivenageneraldescriptionofcaustics.Letusnowconsidervariouscomplicationsthatexistinrealgalactichalos[ 62 ]. 20

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2.4.1ExistenceofaLargeNumberofFlowsThecontinuousinfallofdarkmatterinagalactichaloresultsinalarge,butdiscretesetofvelocities,whichwerefertoas`ows'.Thenumberofowsthatexistatanygivenlocationisthenumberofwaysthatdarkmatterparticlescanreachthatlocation.Veryfarfromthegalacticcenterbutatdistancessmallerthantherstturnaroundradius(say1Mpc),therecanbeonlyoneowduetodarkmatterfallingintothegravitationalwellofthehalo.Atsomewhatsmallerdistances,therearethreeowsthatcorrespondto3waysinwhichdarkmatterparticlescanreachthatlocation-(i)byfallinginforthersttime,(ii)byfallingfromtheoppositesideandreachingthepositionunderconsiderationand(iii)byfallinginwardafterreachingsecondturnaround.Similarly,atslightlysmallerdistances,thereare5ows,then7owsandsoon.Thenumberofowsatourlocationisestimatedtobeoforder100.Afterseveralinfalltimes,thesuccessiveturnaroundradiiwillbeclosetoeachotherbecausethepotentialdoesnotchangesignicantlyduringaninfalltime.Thus,thephasespacesheetistightlywoundintheinnerregionsofphasespace.Foranobserverwithlimitedvelocityresolution,itispossiblethatthevelocitydistributionappearstotaketheformofacontinuum,eventhoughthemicroscopicstructureisdiscrete,aneectcalled`phasemixing'.Thiseectisfurtherexacerbatedifthedarkmatterparticleshaveasignicantvelocityscatter.However,phasemixingoccursinphasespace,notphysicalspace.Thismeansthatitistheinnerregionsofphasespacethataresmeared.Inphysicalspace,thevelocitydistributionappearsasasetofdiscreteowssuperimposedoveraseeminglythermalcontinuum.Thisisbestseenbydrawingaverticallinethroughthecenterofthephasespacediagram[ 94 ]andcountingthenumberofvelocities.Theresultisalargenumberofdiscretevalues.Thesmallervelocityowsareverycloselyspaced,whilethelargervelocityows(theparticlesthatpopulatetheouterregionsofthephasespace 21

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92 93 ],eachowcontributesafewpercenttothelocaldarkmatterdensityatthesun'slocation.So,thecontributionofasinglecoldowtothedarkmatterdensityatanarbitrarylocationmaybequitesmall.However,thecoldowformsacausticandclosetothecaustic,thedarkmatterdensitycanbelarge.Nearacaustic,thecontributionofasingleowtothedarkmatterdensitycanbeaslargeasalltheotherowsputtogether. 22

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91 ] ()2=ZdtZbmaxbmin4G2M2 v3M M2lnbmax wherevisthevelocityoftheowandtisthetimeoverwhichitencounteredtheinhomogeneities.ForGiantMolecularClouds,bmaxandbminareestimatedtobe1kpcand20pcrespectively.Theresultisthattheowsofparticlesthatspentmostoftheirpastinthecentralpartsofthegalaxycouldwellhavebeenwashedoutbyscattering.However,aswehaveemphasized,thereareparticlesinthehalothathavefallenintotheinnerregionsofthehaloonlyafewtimesinthepast.Theseparticleswhichoriginatefromtheouterregionsofphasespacearenotscatteredenoughtolosetheircoldness.Forexample,letusassumethatafractionfofthemassisintheformofclumpsofmassM.Thenforaowofparticleswithvelocityv=400km/s,passingthroughtheinnerpartsofthegalaxy(say<20kpc)once,theeectofscatteringis ()21011fM M(2{9) 23

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31024gmcm3andln(bmax=bmin)4.Unlessasignicantfractionofthemassisinclumpsofmass1010Mormore,theeectofgravitationalscatteringisnotsignicantforatleastsomeoftheows. 2-1 .Figure 2-2(a) showsthedarkmatterdensityclosetooneofthefoldlines(averagedoveraninnitesimalvolume)asafunctionofdistancetothecausticalongthelineofsight.Figures 2-2(b) 2-2(c) and 2-2(d) showtheeectofaveragingthedensityoveracubeofside=0.01pc,0.1pcand1pcrespectively.Acut-odensityof100GeVcm3wasassumed.InFig. 2-2(b) ,weseethattheaverageddensityfollowsthetruedensityfaithfully,butthedensitydoesnotquitereachthecut-ovalue.Fig. 2-2(c) nolongershowsasharpincreaseindensityatthelocationofthecausticandFig. 2-2(d) missesthecausticcompletely.TheprimordialvelocitydispersionofWIMPsofmassmisv=3107p 94 ].Ifthevelocitydispersionisnolargerthantheprimordialvalue,theresultingcausticsarespreadoveraregion[ 95 ]ava=vwhereaistheouterturnaroundradius(fewhundredkpc)oftheowofparticlesformingthecausticandvisthespeedofthe 24

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50 ] whichismuchsmallerthan1pc.Weconcludethatcausticsareverysmallscale(sub-parsec)structuresandarethereforediculttoresolvewithcosmologicalsimulations.AnotherdicultywithN-bodysimulationsisthelargeparticlemassesinvolved.Sincethemassofeachparticlecanbeseveralmillionsolarmasses,gravitationalscatteringbetweentwo`point'massesisnotnegligible.Closeencountersofthesespuriouslymassiveparticlesleadtolargescatteringangles[ 73 ],whilegravitationalscatteringbetweentwoWIMPsortwoaxionsiscompletelynegligible.Notethatwhensucientcareistaken,itmaybepossibletoresolvediscreteowsandcausticsincosmologicalsimulations.ThesimulationsofStiandWidrow[ 100 ]showtheexistenceofdiscreteowsinvelocityspacewhilethesimulationsofBertschingerandShirokov[ 90 ]suggesttheexistenceofcausticsinphysicalspace. 47 ].Thecombinedrotationcurveshowspeaksatthelocationsexpectedintheself-similarinfallmodel[ 92 93 ],oftherstand 25

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17 ]haveconjecturedthattheanomalousbehavioroftheouterMilkyWayrotationcurvecouldbeexplainedifaringofmatterexistedat13.6kpcwhichisclosetotheexpectedlocationofacausticintheself-similarinfallmodeloftheMilkyWay[ 92 93 ].TherotationcurveoftheMilkyWaygalaxyalsoshowsrisesattheexpectedlocationsofthecaustics[ 96 ]. 96 ].Thetriangularfeatureiscorrectlyorientedwithrespecttothegalacticcenter.Moreover,itcoincideswiththelocationofariseintherotationcurve,strengtheningthehypothesisthatthetriangularfeatureistheimprintofacausticonbaryonicmatter.Aswillbeshownin Chapter3 (and AppendixA ),theellipticumbiliccatastrophehasacrosssectionthatresemblesatriangle. 26

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10 43 64 78 ]andgravitationallensingexperiments[ 25 36 42 70 ]. Figure2-1. Darkmattertrajectoriesformingacuspcatastrophe. 27

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(b) (c) (d)Figure2-2. Eectofaveragingthedensityoveranitevolume. 28

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AppendixA .Booksoncatastrophetheoryinclude[ 5 23 38 79 86 ].Wesimulateasinglecoldowofdarkmatterfallinginaxedgravitationalpotential.TheequationsofmotionaresolvednumericallyandtheJacobianofthemapfromthespaceofinitialco-ordinatestothespaceofnalco-ordinatesiscomputedateverylocation[ 63 ].ThezeroesoftheJacobianformthecausticsurface.In Chapter2 ,weshowedthattheinfallofacoldownecessarilyproducessingularpoints.Here,wegiveadetaileddescriptionofthecaustics. 112 ], 3{1 impliesthevelocityeld dt~rq(~q)j~q=[1=a(t)]~r(3{2) 29

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108 ],wemaychoose~q=0ataminimumofandexpandinaTaylorserieswiththeresultthat 30

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2g1.Wheng1=g2,theinitialvelocitydistributionisaxiallysymmetricaboutthez-axis.Wheng2=1 2g1,itisaxiallysymmetricaboutthex-axis.Inthecaseofpurerotation(g1=g2=0),wemaychooseaxessuchthat~c=c^z.Theinitialvelocitydistributionisalwaysaxiallysymmetricinthiscase.Wheng1,g2,c1,c2andc3arealldierentfromzero,theinitialvelocitydistributionhasnosymmetry.Whenaxialsymmetryaboutthez-axisisimposed,c1=c2=0andg1=g2. 3{5 plusvr=0.Wesolvetheequationsofmotionnumericallyandobtainthetrajectory~x(;;)oftheparticlethatoriginatedattheposition(;)ontheturnaroundsphereattime.Sinceneitherthepotentialnortheinitialconditionsaretimevarying,thesimulatedowsarestationary(i.e.,~x(t;;;)=~x(t;;)).Thesimulationofnon-stationaryowswouldbestraightforwardbutconsiderablymorememoryintensiveandtimeconsuming,withoutbeingmorerevealingofthestructureofinnercaustics.Usingtheequationsofmotion (3{6) 31

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@@x @@x @@y @@y @@y @@z @@z @@z @ TheinnercausticisthelocusofpointswithD=0.Inthelimitofzerovelocitydispersion,theseareinnitedensitypoints.Unlessstatedotherwise,isthegravitationalpotentialproducedbythedensityprolegivenby 3{8 impliesaforceperunitmass dt2=v2rot rtan1r ai^r(3{9)Theveparametersg1;g2;c1;c2andc3whenexpressedinunitsofvrot,arerelatedto,andareoforderthedimensionlessangularmomentumparameterjoftheselfsimilarinfallmodeldescribedin[ 92 93 ].Thissetstheoverallscaleforthevaluesg1c3weareinterestedin,andwhichareusedinoursimulations.Notethatitistherelativevaluesoftheseveparametersthatarerelevantasfarasthestructureofcausticsisconcerned.WeuseR,theradiusoftheturnaroundsphereastheunitofdistanceandvrotastheunitofvelocity.Sincewesimulateasinglecoldowinaxedpotential,theparticleresolutionisnotacriticalissue.Wechoosearesolutionof1particleperdegreeintervalinandandatimestepof104inunitsofR=vrot. 94 ],itwasshownbyanalyticmeansthatwhentheinitialvelocitydistributionisdominatedbynetrotation,theinnercaustichastheappearanceofaringwhosecrosssectionisatricusp.Here,weconrmthisresultusingoursimulations.Wealsostudy 32

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AppendixB 3-1 showstheinfallofasingleshellinxzcrosssection,atsuccessivetimesforc3=0:1.Astheshellfallsin,deviationsfromsphericalsymmetryappearduetothepresenceofangularmomentum.Theparticlesatthepoleshavezeroangularmomentumandfallinfasterthantheparticlesneartheequator.TheshelltakesontheformofFig. 3-1(b) .InFig. 3-1(c) ,theparticleswhichoriginatedatthepoleshavecrossedthez=0plane.Theshellhasturneditselfinside-outinFig. 3-1(e) ,formingacrease.Finallytheshellincreasesinsizeandregainsanapproximatelysphericalshape.TheinnercausticoccursatandnearthelocationofthecreaseinFig. 3-1 .Fig. 3-2(a) showstheowsnearthecreaseinzcrosssectionwhere=p 3-2(b) showsthecausticinthreedimensions. 33

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3.3.2.1Eectofthegradienttermsg1andg2Letusperturbtheinitialvelocityeldbyaddingthegradienttermsg1andg2.Asanexample,wechoosec3=0:1;g1=0:033;g2=0:0267.Thusjc3jislargecomparedtog1org2,andwemayregardthepresenceofthegradienttermsasperturbationstotherotationalow.Fig. 3-3 showstheinnercausticobtainedfromtheinfall.Itisagainatricuspring,butthecrosssectionisdependent.Inthiscase,thetricuspshrinkstoapointat4placesalongthering.Inthevicinityofsuchapoint,thecatastropheistheellipticumbilic(D4).Inourexample,thecausticismadeupoffourellipticumbiliccatastrophesectionsjoinedback-to-back. 3{10 .Fig. 3-4 showstheinnercaustic.Wenotethattheinnercausticisstillatricuspring,althoughitisdeformedfromwhatiswasinFig. 3-2(b) 34

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1 2v2max=1 2(v2+v2+v2r)+(R)(rmin)(3{12)Themaincontributiontovmaxisfromthegravitationalpotentialenergyreleasedwhiletheparticlefallsin.Theinitialvelocitycomponentsprovideonlycorrectionstovmaxwhicharesecondorderinv;vandvr.Since`doesnotdependonvratall,radialvelocitiesproduceonlysecondordercorrectionstothedistancesofclosestapproach. 3.3.3.1NFWproleWecarriedoutsimulationsoftheinfallofcollisionlessparticlesinthegravitationalpotentialproducedbythedensityproleofNavarro,FrenkandWhite[ 66 ] r rsh1+r rsi2(3{13)Thescalelengthrswaschosentobe25kpc.swasdeterminedbyrequiringthattherotationalvelocityatgalactocentricdistancer=8.5kpcis220km/s.TheaccelerationofaparticleorbitinginthepotentialproducedbytheNFWdensityproleisthen wherex=r=rsandx=r=rs.Fig. 3-5 showstheresultoftwosimulationsplottedonthesamegure.ThelargercausticringisobtainedusingthedensityproleofEq. 3{13 ,whilethesmallercausticringisobtainedusingthedensityproleofEq. 3{8 withvrot=220km/sanda=4:84kpc.Inbothcases,theturnaroundradiusR=174kpcandtheinitialvelocityeld~v=0:2sin^.Thecausticisatricuspringineachcase,butwithdierentdimensions. 35

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(r)=v2rotln0BB@R x a12+y a22+z a321CCA(3{15)wherea1,a2anda3aredimensionlessnumbers.Fig. 3-6 showstheinnercausticforthecasewherea1=0:95,a2=1:0anda3=1:05,andtheinitialvelocityeld~v=0:2sin^.Itisagainatricuspring.Itsaxialsymmetryislostduetotheabsenceofaxialsymmetryinthepotential.Thetricuspringstillhasreectionsymmetryaboutthexy,yzandxzplanes.AsinFig. 3-3 thetricuspshrinkstoapointfourtimesalongthering. 3{5 issymmetricaboutthezaxiswhenc1=c2=0andg1=g2.Then 2g1sin(2)^+c3sin^(3{16)Werstsimulatetheowandobtaintheinnercausticintheirrotationalcasec3=0.Nextweseehowthecausticismodiedwhenc36=0. 36

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3-7 and 3-8 showtheinfallofacoldcollisionlessshell.TheguresshowthequalitativeevolutionofashellwhoseinitialvelocityeldisgivenbyEq. 3{16 .ForFig. 3-7 ,wechosec3=0andg1=0:0333.WerefertothisasCase1.Fig. 3-8 showsthequalitativeevolutionofashellwithc3=0andg1=0:0667,whichwillbereferredtoasCase2.Sincec3=0inbothcases,eachparticlestaysintheplanecontainingthezaxisanditsinitialpositionontheturnaroundsphere.Theguresthereforeshowtheparticlesinthey=0plane.Theangularmomentumvanishesat=0and==2whereisthepolarcoordinateoftheparticleatitsinitialposition.Hence,theparticleslabeled=0and==2followradialorbits.Theangularmomentumincreasesinmagnitudefrom==2,reachesamaximumat==4andreturnstozeroat=0.Thesignoftheangularmomentumdoesnotchangeduringthisinterval.TheshellstartsoutasshowninFig. 3-7(a) (Case1)or 3-8(a) (Case2).Astheshellfallsin,theparticlesat6=0;=2movetowardsthepoles.Theseparticlesfeelanangularmomentumbarrierandfallinmoreslowlythantheparticlesat=0.ThisresultsintheformationofaloopinFig. 3-7(c) (Case1)andFig. 3-8(c) (Case2).Theformationoftheloopimpliesacuspcausticonthezaxis.Theparticleslabeled=0and=havecrossedthez=0planeandtheparticleslabeled==2havecrossedthex=0planeingures 3-7(g) (Case1)and 3-8(g) (Case2).Theshellthentakestheformshowningures 3-7(h) (Case1)and 3-8(h) (Case2).Thefurtherevolutiondependsonthemagnitudeoftheangularmomentum(i.e.,onthevalueofjg1j)andisdierentforthetwocases.ConsidertheinfallforCase1.Theloopthatispresentnearthez=0planeinFig. 3-7(j) disappearsthroughthesequenceofgures 3-7(k) 3-7(o) .Thedisappearanceoftheloopimpliestheexistenceofacuspcausticinthez=0planeaswell.InFig. 3-7(p) theshellhasregainedanapproximatelysphericalformandisexpandingtoitsoriginalsize. 37

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3-8(j) disappearsthroughamorecomplicatedsequenceofgures 3-8(k) 3-8(o) .Theparticlesnear==2crossthez=0planebeforethesphereturnsitselfinsideout.Thiscrossoverproducesadditionalstructureandamorecomplicatedcaustic.Thecriticalvalueofjg1j,belowwhichthequalitativeevolutionisthatofCase1andabovewhichthatofCase2isg1'0:05.Causticstructure:Thebutterycatastrophe.TheinnercausticisasurfaceofrevolutionwhosecrosssectionisshowninFig. 3-9(a) forthecase(c3;g1)=(0;0:0333)andinFig. 3-10(a) for(c3;g1)=(0;0:0667).Onthezaxis,thereisacausticline.Causticlinesarenotgeneric.ThecausticlineinFigs. 3-9(a) and 3-10(a) occuronlybecausetheinitialvelocityeldisaxiallysymmetricandirrotational.Wewillseebelowthatwhenaxialsymmetryisbrokenorwhenarotationalcomponentisadded,thelinebecomesacaustictube.Fig. 3-9(b) showsthedarkmatterowsinthevicinityofthecuspforjg1jg1inFig 3-10 .Sincethemagnitudeofjg1jdetermineswhetherornotthebutterycongurationoccurs,jg1jmaybetermedthe\butteryfactor"[ 86 ].Ifg1ischosenpositiveinsteadofnegative,thebehavioratthepolesandtheequatorisreversed(Eq. 3{16 )andwehavecuspsonthexaxisandcusp/butterycausticsonthezaxis,dependingonthemagnitudeofg1.Fig. 3-10(b) showsthedarkmatterowsinthevicinityofthebutterycaustic,forjg1j>g1.Fig. 3-10(c) showsthenumberofowsineachregionofthebuttery.Addingarotationalcomponent:Hereweshow,intheaxiallysymmetriccase,theeectofaddingarotationalcomponenttotheinitialvelocityeld.Fig. 3-11 showsthetransformation.Westartwithanirrotationalvelocityeld(c3=0)in 3-11(a) andincreasec3untiltherotationalcomponentdominatesthevelocityeld,in 3-11(d) .Thecausticline 38

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3{5 )become 2(g2g1)andgg1+1 2g2.isameasureof^zaxialsymmetrybreakingintheirrotationalcase.Werstletg.Next,weexploreallof(g1;g2)parameterspace.Finally,weaddarotationalcomponentbylettingc36=0.Irrotationalinfallwithoutaxialsymmetry:Wesawinthecaseofaxiallysymmetricinfallofanirrotationalowthatthereisacausticlineonthezaxis.Fig. 3-12(a) showsthetrajectoriesoftheparticlesinthez=0planeforsuchacase.Theorbitsareradial.Indeedallparticleshavezeroangularmomentumwithrespecttothezaxiswhentheinitialvelocityeldisirrotationalandaxiallysymmetric.Becausealltrajectoriesintersectthezaxis,thereisapileupofparticlesonthataxisandhenceacausticline.Fig. 3-12(b) showsthetrajectoriesoftheparticlesinthez=0planefortheinitialvelocityeldofEq.( 3{17 )with=0:01andg=0:05.Theparticlesdohaveangularmomentumwithrespecttothezaxisnow.ThecausticlineonthezaxisspreadsintoatubewhosecrosssectionisthediamondshapedenvelopeofparticletrajectoriesshowninFig. 3-12(b) .Thatenvelopehasfourcusps.Theowsandcaustichavereectionsymmetryaboutthexy,xzandyzplanesbecausetheinitialvelocityeldhasthosesymmetries.Fig. 3-12(b) showsfourowsinsidethediamond-shapedcausticandtwoowsoutside.Theinfallofdarkmatterparticlesfromregionsaboveandbelowthez=0planewilladdtwomoreowsateachpoint,whicharenotshowninFig. 3-12 forclarity. 39

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3-13 showstheinnercausticin3dimensionsfortheinitialvelocityeldofEq. 3{17 with=0:005andg=0:05.TheinnertubehasadiamondshapedcrosssectionwhichisevidentinFig. 3-13(b) ,whichshowsasuccessionofconstantzsections.Nearthez=0plane,therearesixowsinsidethediamond,fourowsintheotherregionsinsidethecaustictent,andtwooutside.Figures 3-13(c) and 3-13(d) showy=0andx=0sectionsofthecaustic.Hyperbolicumbiliccatastrophe:Letuslookmorecloselyatthetworegions(topandbottom)inFig. 3-13(c) wheretheinnersurfacereachesandtraversestheoutersurface.Wewillshowtheexistenceofahigherorderstructureattheboundary.Figs. 3-14(a) 3-14(d) showz=constantsectionsoftheinnercausticinsucharegion.Asjzjisincreased,thetwocuspsontheyaxissimplypassthroughtheoutersurface,whereasthecuspsonthexaxistraversetheoutersurfacebyformingwiththelatter,twohyperbolicumbilic(D+4)catastrophes,oneonthepositivexsideandoneonthenegativexside.ThesequencethroughwhichthishappensisshowningreaterdetailinFigs. 3-14(e) 3-14(h) forthehyperbolicumbiliconthepositivexside.Thearcandthecuspapproacheachotheruntiltheyoverlap( 3-14(g) ),formingacorner.Thecuspistransferredfromonesectiontotheotherasthetwosectionspassthrough.Thisbehaviourischaracteristicofthehyperbolicumbiliccatastrophe.Therearefourhyperbolicumbilicsembeddedinthecaustic-two(x>0andx<0)atthetop(z>0)andtwoatthebottom(z<0).Thehyperbolicumbilicatz>0andx<0isshowninthreedimensionsinFig. 3-14(i) .LetusmentionthatthereasonthereisnohyperbolicumbiliccatastropheontheyaxisofFig. 3-14(a) 3-14(d) isbecausetheparticlesformingtheinnerandoutersurfaceshereoriginatefromdierentpatchesoftheinitialturnaroundsphere.Theparticlesformingthetwocausticsurfacesnearthehyperbolicumbilicoriginatefromthesamepatchoftheinitialturnaroundsphere.(g1;g2)landscape:Herewedescribetheinnercausticintheirrotationalcaseforg1andg2farfromthosevalueswheretheowisaxiallysymmetric.Recallthattheowis 40

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2g1(g3=g2).Intermsofandg,theseconditionsforaxialsymmetryare=0,g=0,and=g,respectively.Therst,secondandthirdcolumnsofFig. 3-15 showrespectivelythez=0,y=0andx=0sectionsoftheinnercausticproducedbytheinitialvelocityeldofEq. 3{17 forvariousvaluesof(g1;g2).Theratiog2=g1decreasesuniformlyfrom1(toprow)to-1/2(bottomrow).Notethatthethirdcolumndescribesasequencewhichisthatoftherstcolumninreverse,andthatthersthalfofthesequenceinthesecondcolumnisthereverseofthesequenceinitssecondhalf,withxandzaxesinterchanged.Intherstrow,thecausticisaxiallysymmetricaboutthezaxis.Thereisacausticlineseeninthexzandyzcrosssections.Thelineappearsasapointinthexycrosssection.Inthesecondrow,axialsymmetryisbrokenandthefamiliardiamondshapedcurveappears.Thehyperbolicumbilicsareapparentinthexzcrosssectionatthepointswherethetwocurvesmeet.Inthethirdrow,thesmoothcurvepassesthroughthediamondseeninxycrosssection.Thiscurvesturnsintoacausticlineinthefthrow,asseeninthexyandxzcrosssectionsandappearingasapointintheyzcrosssection.TheplotsofFig. 3-15 arereminiscentofcausticsseeningravitationallensingtheory[ 19 76 77 ],andintheanalysisofthestabilityofshipsusingcatastrophetheory[ 79 111 ].Addingarotationalcomponent:Theswallowtailcatastrophe.Hereweaddarotationalcomponent(c36=0)totheinitialvelocityeldofEq. 3{17 .Fig. 3-16 showsthez=0crosssectionsoftheinnercausticduringsuchatransition.InFig. 3-16(a) ,theinitialvelocityeldisirrotationalandweseeacircleanddiamond,asbefore.InFig. 3-16(b) ,thediamondisskewedbecauseoftherotationinthez=0planeintroducedbyc36=0.Fig. 3-16(c) showsthecasec3=.Asc3isincreasedfurther,thediamondtransformsintotwoswallowtail(A4)catastrophesjoinedbacktoback(Fig. 3-16(d) ).Therearetwoowsinthecentralregionformedbytheswallowtails,sixowsinthecuspedregionofeachswallowtail,fourintheotherregionsinsidethecircleandtwo 41

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3-17 showsthetransitioninthreedimensions. 42

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(b) (c) (d) (e) (f)Figure3-1. Infallofacoldcollisionlessshell:AntisymmetricM. 43

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(b)Figure3-2. Darkmatterowsforminganinnercaustic.(a)Darkmatterowsneartheinnercaustic,incrosssection.(b)Theaxiallysymmetrictricuspringinthreedimensions. Tricuspringwithnon-zerog1andg2. 44

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Eectofarandomperturbation. Tricuspring:NFWpotential. Tricuspring:Non-sphericallysymmetricgravitationalpotential. 45

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(b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p)Figure3-7. Case1:Infallofashell.Irrotationalowwithaxialsymmetry,jg1j
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(b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p)Figure3-8. Case2:Infallofashell.Irrotationalowwithaxialsymmetry,jg1j>g

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Crosssectionoftheinnercausticproducedbyanirrotationalaxiallysymmetricvelocityeld(Case1:jg1jg).Thenumberofdarkmattertrajectoriesateachpointisindicated. 48

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Causticincrosssection,forincreasingc3. Darkmatterowsinxycrosssection.(a)g1=g2(b)g16=g2. 49

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Gradienttypecausticwithoutaxialsymmetry.(a)Thecausticin3dimensions.(b)Asuccessionofconstantzsections.(c)and(d)Crosssectionalview. 50

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Formationofthehyperbolicumbiliccatastrophe. 51

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Causticincrosssection,fordierent(g1;g2). 52

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Transformationofagradienttypecaustictoacurltypecaustic. 53

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67 ]usingSDSSdata.Subsequentworkbyseveralauthors[ 27 29 45 55 56 81 109 ]hasconrmedtheoverdensityofstarsanduncoverednewdetails.ThestarsoftheMonocerosringareobservedovergalacticlongitude120
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28 ].OneofthewidelyacceptedviewsisthattheMonocerosringofstarswasformedbythetidaldisruptionofasatellitegalaxyoftheMilkyWay.Pe~narrubiaetal.[ 75 ]haveconstructedatheoreticalmodeltoexplaintheobservationsbasedonthetidaldisruptionofasatellitegalaxywhichwasinitiallyclosetothegalacticplane.ThesimulationsofHelmiet.al.[ 40 ]ndthatringsdoformduetothetidaldisruptionofsatellites,butsuchringsmaynotbelonglived.Hereweexploreadierentproposalaltogether,namelythattheMonocerosRingofstarsformedasaresultofthegravitationalforcesexertedbythesecondcausticringofdarkmatterintheMilkyWay[ 65 ].Causticringsofdarkmatterhadbeenpredicted[ 92 { 94 ],priortothediscoveryoftheMonocerosRing,tolieintheGalacticplaneatradiigivenbytheapproximatelaw40kpc=nwheren=1;2;3SincetheMonocerosRingislocatednearthesecond(n=2)causticringofdarkmatter,itisnaturaltoaskwhethertheformerisaconsequenceofthelatter.Iftheanswerisyes,thepositionoftheMonocerosRingintheGalacticplaneandits20kpcradiusareimmediatelyaccountedfor.Aswasmentionedalready,theself-similarinfallmodelpredictsthattheradiusa2ofthesecondcausticringofdarkmatterisapproximately20kpcinourgalaxy.Thetransversesizespandqarenotpredictedbytheself-similarinfallmodel.However,theexpectationforpandqisthattheyareoforder1kpcforthen=2ring.SothetransversesizesofthesecondcausticringofdarkmatterareoforderthetransversesizesoftheMonocerosRing.Moreover,forq=1kpc,thedarkmattermasscontainedinthen=2ringis6108M( AppendixB ).ThisisoforderthetotalobservedmassintheMonocerosRing. 55

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g(r)v2rot AppendixB (for=1) pforara+p+1=2forr>a+p andH(r)isgivenby 2tanhra p0(4{3)p0isexpectedtobeofordera.Fig. 4-1 showstherotationvelocityclosetoacausticring.Theangularvelocityofgasinthevicinityofthecausticisgivenby vrot 4{4 andFig. 4-1 ,wenotethattheangularvelocityhasaminimumatr=a. 49 ]is ;r=1 ;r=@ @r(r)=r@ ;risthecomponentoftheforceperunitareacrossingther=constantsurface(a 56

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where(t;r)isthesurfacedensity.Theviscoustorqueistherefore 52 80 ] dt=d(ml) @t+vr@ @rl=(2rr)vr@l @r wherelisthespecicangularmomentum=angularmomentumperunitmass=r2,Listheangularmomentumandvristheradialvelocity.Axialsymmetryandstationarityofthepotentialareassumed.Letusdenethetwodimensionlessquantities Usingl=r2,wehave dt=r2vrr2(A1+2)(4{11)Consideranannuluswithinthecausticring(i.e.,intheregiona
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drr3@ rA2 r=r2A1vr1+2 r3+A2 wheretheprimesrepresentderivativeswithrespecttoposition.Closetor=a,forr>a, (4{14) andso 58

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4.3.1OrbitStabilityInagravitationaleld~g(~r)=g(r)^r,theangularfrequencysquared,ofsmalloscillationsaboutacircularorbitofradiusris drr3g(r):(4{16)Theorbitisstableif2>0(thenistheepicyclefrequency).Intheneighborhoodofacausticring,wehave dr:(4{17)Foraa+p,thethirdtermisdJ=dr=1 2p0sech2ra p0.However,sinceweexpectp0tobelarge,ofordera,thesumofthethreetermsisstillpositive,implyingthatcircularorbitsarestable.DeviationsfromcircularsymmetrycaninduceinstabilitiesthroughthephenomenonofLindbladresonances. 18 ] r=!2r@U @rd dt(!r2)=@U @' where!=_'.Considerasmallperturbation: 59

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4{18 reads r1+@2U0 (4{21) Forsmalldeparturesfromcircularorbits,wemayneglectthetermsinvolving_r0and_!0whentheyaremultipliedbyrstorderquatities.Wethenhave r1+@2U0 (4{22) (4{23) WechoosetheformofthepotentialtermU1(r0;'0)as 4{23 toobtain r0!0t(4{25)wheretheconstantofintegrationisabsorbedbyaredenitionofr1.Eliminating_'fromEq. 4{22 ,wehave r1+@2U0 60

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2+r p@U0 (4{28) wehavetheresonanceconditionforraa, wherelisthespecicangularmomentumandr0isaconstant.Whenthecausticispresent,theeectivepotentialismodiedtoVeff(r)=Veff;0(r)+Vc(r)where rJ(r) (4{31) 4-2 witha=20kpc,p=1kpcandp0=5kpc.Theeectivepotentialissmootheventhoughitssecondderivativedivergesatr=aandr=a+p.Thecausticringradiusincreaseswithtime.Accordingtotheself-similarinfallmodel[ 92 93 ],a/t2 3+2 9.Consideraparticleofspecicangularmomentuml.Intheabsenceof 61

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4-3 showsrf(r;a)asafunctionofa.Eachlineinthatgurecorrespondstoadierentvalueofr.Letd(r)bethedensityofstarsintheabsenceofthecaustic,anddf(r;a)theirdensityinthepresenceofacausticwithradiusa.Assumingthatallstarsremainoncircularorbits,conservationofthenumberofstarsimplies dr(rf)(4{36)Assumingthattheinitialstardensityhasnosignicantstructureofitsown,wehave dr(r)(4{37)whichbecomesverylargenearr=aforr>asince 2p 4-3 62

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r=20km/s s1 d d'afdJ drja
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Rotationcurveclosetoacausticring(R=(ra)=p). SmoothpotentialVc(r)(R=(ra)=p). 64

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Adiabaticdeformationofstarorbits.Theenhancementindensityisproportionaltotheslopedr=drfwhichisinniteatrf=a. 65

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64 ].Weestimatethenumberofphotonsproducedindierentenergybands,whenWIMPsannihilateintostandardmodelparticles.Wethencomputetheexpectedgammarayannihilationuxandcomparethiswiththeexpecteddiusegammaraybackground.Previousworkonparticleannihilationincausticsincludes[ 10 43 60 78 ].Causticsandtheirassociatedcoldowsarealsorelevanttodirectdetectionexperiments[ 2 37 39 50 87 105 ]. 46 ].ThecharacteristicsoftheannihilationuxdependbothonthecompositionoftheWIMPanditsmassm.Thelineemissionsignal(!;!Z)isloopsuppressedandisthereforesmallerthanthecontinuumsignal.Thecontinuumux(numberofphotonsreceivedwithenergiesrangingfromE1toE2perunitdetectorarea,perunitsolidangle,perunittime)isgivenby[ 21 78 102 ] (E1;E2;;;)=S(E1;E2) 66

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ZdEM(;) (5{2) 46 ] ThequantitydN(E)=dxforthedominantchannelsmaybeapproximatedbytheform[ 9 11 34 ]dN=dx=aebx=xwherexisthedimensionlessquantityE=mand(a;b)areconstantsforagivenannihilationchannel.Thevaluesof(a;b)fortheimportantchannelsaregivenin[ 34 ].Usingthesevalues,wemaycalculatethenumberofphotonsproducedperannihilationwithinaspeciedenergyrange.Letusconsiderfourenergybands:EnergyBandIwithphotonenergiesfrom30MeVto100MeV,BandIIwithenergiesfrom100MeVto1GeV,BandIIIwithenergiesfrom1GeVto10GeVandBandIV 67

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Table5-1. ChannelIIIIIIIV Table5-2. ChannelIIIIIIIV Table5-3. ChannelIIIIIIIV Chapter3 ,theinnercaustichastheappearanceofaringwhentheinitialvelocitydistributionhasanetrotationalcomponent.Here,weassumethatthisisthecase.Itiseasiesttocalculatetheemissionmeasurefromatricuspcausticringbecauseoftheadvantagethatitcanbetreatedanalytically.Letusconsidercylindricalco-ordinates(;z)where=p 68

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cm3(f=102)(Vrot=220kms1)2 1RwhenR01 1R1+1 AppendixB )wherekpcandpkpcaredistancesmeasuredinkpc.Forpointsnotinthez=0plane,thedensityisobtainedbycomputingthesum( AppendixB ) cm3(f=102)(Vrot=220kms1)2 64Z2=0(5{7)Ateverypointinsidethecaustic,therearefourrealroots,whileoutside,therearetwo.Theaboveformulaeareonlyvalidatpointsclosetothecaustic(distancesoforderporq).Theemissionmeasureiscalculatedbyintegratingthedensitysquaredalongthelineofsight.Letb==2bethegalacticlatitude.listhegalacticlongitudechosensothatthegalacticcenterislocatedinthedirectionl=0;b=0.Wewillassumethatthecausticsarespreadoveradistance104pc( Chapter2 ).fissetequalto2102[ 25 ].Thecut-odensityclosetothefoldsurface(nearR=0;Z=0)isthen2:15103=ap 5-1 showstheemissionmeasureaveragedoverasolidangle105srforthreedierentsetsofcausticparameters,asafunctionoflongitudel.bissetequalto0andweassumethatthecausticliesinthegalacticplane.Figures 5-1(a) 5-1(b) and 5-1(c) areplottedfor(a;p;q)=(7:5;0:5;0:5);(8:0;0:1;0:2)and(8:0;0:1;0:5)respectivelywithalldistancesmeasuredinkpc.Theearth'slocationissetequalto8:5kpcfromthecenter. 69

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78 ]).Forthecasewhenp=0:5kpc(Fig. 5-1(a) ),thecut-odensitywassetequalto400GeV/cm3everywhere,whilefortheplotswithp=0:1kpc(Figs 5-1(b) and 5-1(c) ),thecut-odensitywassetequalto800GeV/cm3everywhere.Themagnitudesofforthehotspotsdependonthevaluesofthecausticparametersandalsoontheaveragingscale(herechosentobe105sr). Table4 showstheannihilationuxforthetwohotspotsfordierentvaluesoftheaveragingscale,forthecase(a=8:0;p=0:1;q=0:2)kpc.ItisworthpointingoutthatifthetriangularfeautureintheIRASmapisinterpretedastheimprintofthenearestcausticonthesurroundinggasasin[ 95 ],theimpliedcausticparametersareclosetowhatwehaveassumedforFig 5-1(b) Table5-4. Peaksoffor(a=8:0;p=0:1;q=0:2)kpc Foldpeak Cusppeak (sr) (GeV/cc)2kpc (GeV/cc)2kpc 107 2177.6 106 1283.3 105 718.1 104 239.4 103 52.0 =S 70

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11 44 ] bg(E1;E2;;)=N;bg(E1;E2)N0(;)106cm2s1sr1(5{9)andN;bg(E1;E2)givenby 11 ].Forthefourenergybandswehaveconsidered,N;bg=198.8forBandI(30MeV-100MeV),28.9forBandII(100MeV-1GeV),0.58forBandIII(1GeV-10GeV)and0.012forBandIV(above10GeV).Wenowcomparethecausticsignalwiththeexpectedbackground.Sincethegammaraybackgroundfallsowithenergy(E2:7)fasterthantheannihilationsignal(E1:5),weexpectthatthebestchancefordetectionisatmoderatelyhighenergies.Atlowenergies,thebackgrounduxoverwhelmsthesignal,whileatveryhighenergies,thesignalisweak.Wechoosem=50GeVsincethischoicegivesthelargestux.ForthequantityN=m2,weusetheaveragevaluefortheband.Theaveragingscaleissetto105sr.Figures 5-2(a) and 5-2(b) showtheexpectedannihilationux(numberofphotonspersquaremeter,persteradian,peryear)asafunctionofanglelneartheplaneofthegalaxy(b=0)forthethreesetsofcausticparametersweconsidered,forEnergyBandsIIIandIVrespectively.Thisiscontrastedwiththeexpecteddiusegammaraybackground.Inprinciple,thepeaksinthesignalandthesharpfall-oofuxarehelpfulinidentifyingtheannihilationsignal,particularlyforthemoreoptimisticcausticparametersandforsmallWIMPmasses.ForlargeWIMPmasses,theannihilationsignalissignicantlysmaller. 71

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(b) (c)Figure5-1. Emissionmeasureaveragedoverasolidangle=105sr,forthreesetsofcausticparameters. 72

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(b)Figure5-2. Annihilationuxforthethreedierentsetsofcausticparametersform=50GeV,comparedwiththeEGRETmeasureddiusebackground. 73

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Chapter1 ,weprovidedanintroductiontodarkmattercosmologyanddescribedtheobservationalevidencefordarkmatter.In Chapter2 ,weshowedthatthecontinuousinfallofdarkmatterparticleswithlowvelocitydispersionfromalldirectionsinagalactichaloleadstotheformationofcaustics.Wethereforeexpectcausticstoexistingalactichalos.Therearetwokindsofcaustics:outerandinner.Theoutercausticsarethinsphericalshellssurroundinggalaxies,whiletheinnercausticshaveamorecomplicatedgeometry.Wegavepossibleobservationalevidenceinfavorofcaustics.In Chapter3 ,weprovidedadetailedanalysisofthestructureofinnercaustics.Wefoundthatthecatastrophesthatform,andhencethegeometryofthecaustic,dependsonthespatialdistributionofthedarkmatterangularmomentum.Weusedthelinearvelocityeldapproximation~v=M~xwiththematrixMmadeupofsymmetricand/oranti-symmetricparts.WefoundthatwhenMismostlyantisymmetric(rotationalorcurlow),thecausticsaremadeupofellipticumbiliccatastrophesections,whileasymmetricM(irrotationalorgradientow)producescausticswithhyperbolicumbiliccatastrophesections.Wealsoshowedtheformationoftheswallowtailandbutterycatastrophes.Itispossibletosmoothlytransformthegradienttypecausticsintocurltypecausticsandvice-versa. Chapter4 and Chapter5 dealwiththeastrophysicaleectsofdarkmattercaustics. Chapter4 exploresapossibleconnectionbetweenadarkmattercausticandtheMonocerosRingofstars.Theexistenceofadarkmattercausticintheplaneofthegalaxyat20kpcwaspredictedbytheself-similarinfallmodelofSikivie,TkatchevandWangpriortothediscoveryoftheMonocerosRing.Wefoundtwomechanismsbywhichadarkmattercausticcanincreasethestardensityinitsneighborhood.Therstisthe 74

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Chapter5 ,wecalculatedtheexpectedgammarayannihilationuxfromanearbydarkmattercausticring,assumingthatthedarkmatterconsistsofSUSYneutralinos.Wecomparedthisuxwiththediusegammaraybackground.Theuxfromthecaustichasadistinctsignaturewhichinprinciple,canbedetected.However,weexpectthistobeachallengingtask. 75

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76

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A-1(a) ),butitisnotstabletoperturbations.Ifthefunctionischangedtof(;x)=4+x2,fhasamaximumat=0fornegativex,asseeninFig. A-1(b) .Thenewfunctionf(;x)isnotstableeitherbecausetheadditionofthetermyalterstheformoffnear=0.Thereisnowneitheramaximum,noraminimumat=0,asseeninFig. A-1(c) .However,thefunctionf(;x;y)=4+x2+yisstablebecausetheadditionofaperturbingtermcannotchangethebehaviorofthefunctionnear=0.Thisisbecausethefunctionf(;x;y)alreadyincludesallpossibleperturbingterms.(Acubictermisirrelevantbecauseaquarticcanalwaysbeputintoaformthatdoesnotinvolveacubicbyaredenitionof.Aconstanttermcannotchangethenatureofthecriticalpointsofandatermlikea5introducesanewcriticalpointat=4=5awhichcanbemovedarbitrarilyfarfrom=0bymakingaarbitrarilysmall).Thefunctionf(;x;y)=4+x2+yistheunfoldingofthesingularity4andisthegeneratingfunctionofthecuspcatastrophe.Comingbacktotheexampleofthelightcaustic,thefamilyofreectedlightraysisobtainedbysetting@f(x=x0)=@=0.Herewegiveabriefdescriptionofsomeofthecatastrophesandderivetheirbifurcationsets[ 79 86 ].Werestrictourselvestothecatastrophesencounteredin Chapter3 ,namely,thefold,thecusp,theswallowtail,thebuttery,theellipticumbilicandthehyperbolicumbilic. 77

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A.1.1Fold:Corank=1,Codimension=1Thefoldisthesimplestcatastrophe.Itinvolves1statevariableand1controlvariabler.Theunfoldingis @=0)2+r=0(A{1)ThisistheparabolashowninFig A-2(a) .Settingthesecondderivativetozerogivesusthesingularityset @2=0)f=0g:(A{2)Theprojectionofthesingularitysetontothespaceofcontrolvariablesfrggivesusthebifurcationset @21(A{4)intheregionr<0.ThesumistakenoverthedierentsolutionsofinEq. A{1 .Forthefoldcatastrophe,therearetwosolutionsofforr<0 r(A{5)andsothedensityd=df=p rforr<0wheredfisaconstant.dfisnotpredictedbycatastrophetheorysincethetheoryisqualitative.Thecatastropheoccursatthepointr=0. 78

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@=0)3+x+y=0(A{7)Thesingularitysetis @2=0)32+x=0:(A{8)ThebifurcationsetisobtainedbyeliminatingfromEq. A{8 andEq. A{7 3p A-3 .Thebifurcationsetdivides(x;y)spaceintotworegions-onewith3solutionsofandtheotherwith1solution.Thecatastropheoccurswhencrossingtheboundarybetweenthetworegions.Thedensitydisproportionaltothesum A{7 .Forthespecialcasey=0,therearethreesolutionsforx<0,namely=0;p d=dc1 2x+1 2x=2dc 79

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A{7 .Theproportionalityconstantdcmustbedeterminedbythephysicsofthesystem.Cuspsarepointsin2dimensionalspaceandlinesin3.Wenowdiscussthehigherordercatastrophes,followingthetreatmentofSaunders[ 86 ]. A.2.1Swallowtail:Corank=1,Codimension=3Theswallowtailhas3controlvariablesx;y;z.Thereis1statevariable.Thestandardunfoldingis 43+2x+y=0(A{16)Tosketchthebifurcationset,letusconsidera2dimensionalcrosssectionwithxxed. A{17 ,we 80

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d=122+2xdz d=123+2x=dy d Bothderivativesvanishwhen2=x=6,whichmeansthat,forx<0,therearetwocuspslocatedaty=(2=3)3=2jxj3=2;z=x2=12.Forx>0,therearenocusps.Next,wedeterminethepointswherethecurvecrossestheaxes,forx<0.Forz=0therearetwocrossings,whilefory=0,thereisonlyonecrossingatx2=4,whichimpliesthatthereisapointofselfintersectiononthezaxis.Todeterminethedierentregions,considerEq. A{15 forz=0(i.e.,pointsontheyaxis).Eq. A{15 simpliesto A-4(a) representsarealrootof.Theyzcrosssectionofthebifurcationsetwithx<0isshowninFig. A-4(b) ,indicatingthenumberofrealrootsineachofthethreeregions. 81

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54+3g2+2x+y=0(A{23)respectively.Toobtainthebifurcationset,letusconsidera2-dimensionalcrosssectionwithgandxxed.Writingyandzasfunctionsof,wehave[ 86 ] Dierentiatingequations A{24 withrespectto, d=203+6g+2xdz d=204+6g2+2x=dy d Bothderivativesvanishwhen 5g3<0(A{27)andonerealroototherwise.Thenumberofrealrootsgivesusthenumberofcusps.Fromequations A{24 ,wenotethatforthespecialcasex=0,zisanoddfunctionofandyisanevenfunctionof.Soforx=0,yisanevenfunctionofz,implyingreectionsymmetryabouttheyaxis.Letusthereforesetx=0tosketchthebifurcationsetin2-dimensionalcrosssection.Withx=0,Eq. A{24 and A{25 become d=203+6gdz d=204+6g2 82

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A-5(a) showsthefamilyofcurves.EachcurvecorrespondstoarealrootofthequinticEq. A{22 .ConsiderEq. A{22 withx=0;z=0(i.e.,pointsontheyaxis).Thesolutionsofare g2 Forg>0,thereisonesolution(=0)if4y=g2>0andthreesolutionsotherwise.Thisischaracteristicofthecuspcatastrophe.Forg<0,therearevesolutionsif0<4y=g2<1,threesolutionsif4y=g2<0andonesolutionif4y=g2>1.(In Chapter3 ,thebutteryweencounteredcontainedanextraowthatwasnotsingular).Thedierentregionsoftheyzcrosssectionofbifurcationsetwithg<0areshownintheFig. A-5(b) 1 332+x2+2y+z(A{32) 83

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(A{33) ThesingularitysetisobtainedbysettingtheHessiantozero, A{33 becomes (A{35) Dierentiatingwithrespectto, d=2x2[sin+sin2]dz d=2x2[cos2cos] (A{36) Bothderivativesvanishwhen=0;2=3.Hencetherearethreecuspslocatedat(y;z)=(3x2;0);(3x2=2;3p A{33 becomes (A{37) If=0,Eq. A{37 hastwosolutionswheny>x2andnosolutionotherwise.If=x,Eq. A{37 hastwosolutionswheny<3x2.Thusthebifurcationsethastworegions-intheregionx2
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32+xy=032+xz=0 (A{39) Thesingularitysetisdenedbythecondition A{39 ,wendthaty=0andzispositive.Similarly,putting=0inEq. A{39 ,wendz=0andyispositive.Therefore,whenx=0,thebifurcationsetiscontainedinthepositiveregionofthexandzaxes.Whenx6=0,wemaywrite=x2=36andso,fromEq. A{39 Whenisclosetozero,butnegative,yisalargenegativenumber,whilezisalargepositivenumber.Whenisclosetozero,butpositive,bothyandzarepositiveandlarge.Thegraphconsistsoftwodisjointpieces.DierentiatingEq. A{41 withrespectto, d=6x3 d=6x4 Bothderivativesvanishwhenx=6.Therefore,thereisonecuspat(y=x2=4;z=x2=4).Ifx>0,theportionthatcorrespondsto<0issmoothandhasnocuspsandnocritical 85

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A{42 vanish.Todeterminethedierentregionsofthebifurcationset,letusconsiderpointsontheliney=zwithx=1.FromEq. A{39 ,wehave=or+=1=3.If=,Eq. A{39 hastworealsolutionsofify>1=12.If+=1=3,hastworealrootsify>1=4.Thehyperbolicumbilicpointisat(x=0;y=0;z=0).Fig A-7 showsthebifurcationset.Ithasthreedistinctregions. (b) (c)FigureA-1. The4singularity.(a)4(b)4+x2(c)4+x2+y (b)FigureA-2. Foldcatastrophe:Equilibriumsurfaceandbifurcationset. 86

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(b)FigureA-3. Cuspcatastrophe.(a)Eachlinerepresentsavalueof.(b)Bifurcationset. (b)FigureA-4. Swallowtailcatastrophe.(a)Eachlinerepresentsavalueof.(b)Bifurcationset. 87

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(b)FigureA-5. Butterycatastrophe.(a)Eachlinerepresentsavalueof.(b)Bifurcationset. (b)FigureA-6. Ellipticumbiliccatastrophe:Bifurcationset. (b)FigureA-7. Hyperbolicumbiliccatastrophe:Bifurcationset. 88

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Chapter3 ,weshowedthatwhentheinitialvelocityeldisdominatedbyanetrotationalcomponent,theresultinginnercaustichastheappearanceofaring,whosecrosssectionhasthreecusps.Wecalledthisa`tricuspcausticring'.Inthelimitofaxialsymmetryaboutthezaxisandreectionsymmetryaboutthez=0plane,itispossibletoderiveananalyticexpressionforthecausticstructure,aswellastocalculatethedarkmatterdensityatpointsclosetothering.ThepropertiesofthecausticringwererstdescribedbyP.Sikivie[ 94 ].Herewegiveabriefdescription.Letusassumeanaxiallysymmetricow.Wehaveshownin Chapter3 thatnon-axiallysymmetricowsalsoproducecaustics.Hence,theassumptionofaxialsymmetryshouldberegardedasasimplifyingfeatureandnotasanecessity.Withtheassumedaxialsymmetry,wemayparametrizeeachparticleintheowintermsoftwovariablesand.==2,whereisthepolarangleofthepositionoftheparticlewhenitcrossedthereferencesphere.(Theparticleswhichareconnedtothez=0planehave=0).Wedene=0asthetimewhentheparticlesjustabovethez=0plane(theparticlesparametrizedby=0+)crossthisplane.Theazimuthalangleisnotrelevantduetotheassumedaxialsymmetry.Let=p 94 ] 2u(0)21 2s2 whereb;u;sand0areconstantsandaisthecausticringradius.ThetwodimensionalJacobiandeterminant@(x;y)=@(;)is 89

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B-1 showsthedarkmatterowsformingatricuspringcaustic(inzcrosssection),forcontinuousandfordiscrete.(eachlinerepresentsaparticularvalueofandeachlineismadeupofmanypoints,eachpointrepresentingaparticularvalueof).pandqarethehorizontalandverticalextentsofthetricusprespectively[ 94 ] 2u20q=p 4bp ThecausticincrosssectionisthelocusofpointswithD2=0.Toderivethecausticstructure,letususeEq. B{3 toexpressEq. B{1 as p=(T1)227 64b0 whereT==0.TheconditionD2=0implies 64b0 38 273=22 3+8 273=22 (B{6) whereisgivenby p:(B{7)Figure B-2 showsthecrosssection.Ifthezaxisisrescaledrelativetotheaxissoastomakethetricuspequilateral,thetricusphasaZ3symmetry[ 94 ]consistingofrotationsbymultiplesof2=3aboutthepointofcoordinates(c;zc)=(a+p=4;0).Thispointmaybecalledthecenterofthetricusp.ItisindicatedbyastarinFig. B-2 .Whenaxialsymmetryisnotpresent,thecrosssectionvariesalongthering,asweshowedinChapIII.Ingeneral,therearepointswherethecrosssectionshrinkstozero,formingellipticumbiliccatastrophes( AppendixA ). 90

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B{1 B{2 .Werstcomputethedensityinthez=0plane.Wethenmoveontothemoregeneralcase. B{1 ,weseethattheconditionz=0implies=0or=0.Forpoints(;0)with>a+p,wehave=0;6=0.Forpoints(;0)with
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1RforR1(B{14)Intheregion0R1,wemustsumoverthefoursolutions.Inthisregion(insidethecaustic,z=0), 1R1+1 ddt(;)cos() 92 93 ]toestimatethemassinfallrate ddt=fvv2rot 94 ].Wemayalsoapproximatecos1forp;qa.Thedensityd(R;0)atpointsclosetothetricuspcaustic,forz=0isgivenby cm3(f=102)(Vrot=220kms1)2 1RR01 1R1+1 (B{18) 92

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B{3 64Z2 64Z2=0:(B{20)Thedensityd(R;Z)isgivenby cm3(f=102)(Vrot=220kms1)2 B{20 .ThenumberofrealrootsisgivenbythesignofthediscriminantS S=144R2R+1 21282R+1 22+4R2R+1 2316R+1 2427R42563(B{22)withgivenby =27 64Z2+R 16:(B{23)Sispositiveforpointsinsidethecausticandnegativeforpointsoutside.Sincetheremustbeatleasttworootseverywhere,Eq. B{20 hastworealrootswhenS<0andfourrealrootswhenS>0. B{20 intermofT1bymakingthesubstitutionT=T1+1 2,therebyeliminatingthecubicterm: 2T12RT1=0(B{24) 93

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B{24 canbeexpressedastheproductoftwoquadratics 2e2R+1 2R ef=1 2e2R+1 2+R e ande2solvesthecubic 2e22+R2+2R+27 16Z2e2R2=0(B{27)WecansolveforT1oncewehavesolvedthecubic B{27 .Tosolve B{27 ,wesety=e2andmaketheredenitiony=y1+2 3R+1 2toobtainfory1, 323Z 333Z 2 Onerealrootofy1isgivenby 3cos1Q Pp (B{30) WhenQ2P30,therearethreerealroots,butweonlyneedonerealroot.Puttingeverythingtogetherandusinge2=y=y1+2 3R+1 2andT=T1+1 2,wehavetheresult e e 94

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B{31 ,wecancomputethedensityusingEq. B{21 B{6 andEq. B{21 tocomputethetotalmassinacausticringofradiusaandcrosssectionp;q.Themassenclosedis (B{32) whereZminandZmaxarespeciedby 2q 38 273=22 8R000R1 (B{33) 2r 3+8 273=22 B{21 B{20 .AnumericalcalculationgivesT2:4andsothemasscontainedinthering 95

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Chapter4 ,weassumedq1kpcandf=4:6102forthen=2ringwhichgivesusM26108M. 94 ] ad2M ddtZdd0(;) [0(;)]2+z02(;)(B{41) 96

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ad2M ddtZdd(a)1 2u(0)2+1 2s2 2u(0)2+1 2s22+b222(B{42)Changingvariables u0T= 0 andusingEq. B{3 ,wehave abd2M ddtZdAdTR(T1)2+A2 p=su b2=27 16p q2: UsingEq. B{39 ,theperturbationtotherotationvelocityis bvrotd2M ddtI(;R);(B{46)withIgivenby 2ZdAdTR(T1)2+A2 (B{48) ThesuddenchangeinI(;R)atR=0andR=1isduetothefactthatthedarkmatterdensitychangessuddenlywhencrossingthosepoints. 97

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bv2rotd2M ddtJ(;R):(B{49)ThefunctionI(;R)ofeq.( B{48 )isreplacedbyJ(;R),whichincludestheeectofthedescendingrotationcurve.Thefactord2M=ddtmaybeextractedfromtheself-similarinfallmodel[ 92 93 ] ddti=fiviv2rot B{49 ),andusingvibi[ 94 ] 2tanhra p0:(B{52)p0isexpectedtobeofordera.Fig. B-3 showsaqualitativesketchofI(r)andJ(r). 98

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Darkmattertrajectoriesformingatricuspring(incrosssection). FigureB-2. Tricuspcausticinzcrosssection. (b)FigureB-3. Modiedrotationcurve.(a)ThefunctionI(b)ThefunctionJ

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AravindNatarajanwasborninTrivandrum,India,in1978.HecompletedhisundergraduatestudiesatBangaloreUniversity,majoringinelectronicsandcommunicationengineering.HisundergraduatethesisworkwasonfrequencyselectiveprocessingofMPEG-1AudioLayer-1bitstreams.Helaterbecameinterestedinphysicsandworkedfor8monthsasaResearchAssistantintheDevicesLab,DepartmentofElectricalCommunicationEngineering,IndianInstituteofScience.Thisleadtohisrstpublication,intheJournalofAppliedPhysics.HethenjoinedtheJointAstronomyProgramattheDepartmentofPhysics,IndianInstituteofScience,tostudyphysicsandastronomyandenrolledattheUniversityofFloridainAugustof2002,asagraduatestudent.AravindsuccessfullytookseveralcoursesattheUniversityofFloridaincludingElectricityandMagnetism,QuantumMechanics,QuantumFieldTheory,StandardModelofParticlePhysics,StatisticalMechanics,TheEarlyUniverse,FunctionalIntegration,ParticleAstrophysics,andDarkMatter.Hereceivedhismaster'sdegreefromtheUniversityofFloridainAugust2004.HisgraduateschoolGPAwas4.0.Aravindreceivedmanyacademicawardsduringthecourseofhisstudy.Inthespringof2005,hereceivedtheJ.MichaelHarrisAward,whichisgiventotwotheorystudentseachspring.Inthespringof2006,hereceivedtheOutstandingInternationalStudentAward,givenbytheInternationalCenter.Infall2006,hewasawardedtheChuckHooperMemorialAwardfordistinctioninresearchandteaching.Insummer2004,Aravindbegantoworkonhisthesis,whichinvolvesadetailedstudyofthepropertiesofdarkmattercausticsandrelatedastrophysicaleects.Heattendedthe\SantaFeCosmologyWorkshop"atSantaFe,inthesummerof2004andthesummerof2006andthe\Particles,StringsandCosmologySchool"heldinJapan,inthefallof2006.Hehaspresentedtalksandpostersatmanyconferences.HehasalsobeenasciencefairjudgeattheKanapahamiddleschoolinGainesville. 105

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