
Citation 
 Permanent Link:
 https://ufdc.ufl.edu/UFE0021102/00001
Material Information
 Title:
 Inner Caustics of Cold Dark Matter Halos
 Creator:
 Natarajan, Aravind
 Place of Publication:
 [Gainesville, Fla.]
Florida
 Publisher:
 University of Florida
 Publication Date:
 2007
 Language:
 english
 Physical Description:
 1 online resource (106 p.)
Thesis/Dissertation Information
 Degree:
 Doctorate ( Ph.D.)
 Degree Grantor:
 University of Florida
 Degree Disciplines:
 Physics
 Committee Chair:
 Sikivie, Pierre
 Committee Members:
 Dufty, James W.
Fry, James N. Woodard, Richard P. Sarajedini, Vicki L.
 Graduation Date:
 8/11/2007
Subjects
 Subjects / Keywords:
 Angular momentum ( jstor )
Caustic networks ( jstor ) Dark matter ( jstor ) Disasters ( jstor ) Galaxies ( jstor ) Galaxy rotation curves ( jstor ) Milky Way Galaxy ( jstor ) Symmetry ( jstor ) Velocity ( jstor ) Velocity distribution ( jstor ) Physics  Dissertations, Academic  UF catastrophe, caustic, dark
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) government publication (state, provincial, terriorial, dependent) ( marcgt ) borndigital ( sobekcm ) Electronic Thesis or Dissertation Physics thesis, Ph.D.
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. ( en )
 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.
 Thesis:
 Thesis (Ph.D.)University of Florida, 2007.
 Local:
 Adviser: Sikivie, Pierre.
 Statement of Responsibility:
 by Aravind Natarajan.
Record Information
 Source Institution:
 UFRGP
 Rights Management:
 Copyright Natarajan, Aravind. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 660256272 ( OCLC )
 Classification:
 LD1780 2007 ( lcc )

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Full Text 
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 rearranged 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. (B48) is replaced by J((, R), which includes the effect
of the descending rotation curve. The factor d2M/dl dt may be extracted from the
selfsimilar infall model [92, 93]
d2M 2
Sf o* (B50)
dQdt 17(F
where is the velocity of the particles in the ith caustic ring, and the dimensionless
coefficients fi characterize the density in the ith inandout flow. Rewriting equation (B49),
and using ~ bi [94]
S(r) Vrot [1 + fJ((, R)], (B51)
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 (B52)
2
p' is expected to be of order a. Fig. B3 shows a qualitative sketch of I(r) and J(r).
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
(nonclumpy) 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
(10o7( 3 3 Mt n2 b_.
1.8 x 10 c n ([ax) ( 10yr) (28)
v MeD bmin 10l0yr pc3
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 ~ 10f (M) (29)
\ A2f/(
I
Figure A3. Cusp catastrophe. (a) Each line represents a value of 0. (b) Bifurcation set.
 . 2
^ _0
(a) (b)
Figure A4. Swallowtail catastrophe. (a) Each line represents a value of 0. (b) Bifurcation
set.
* \.
'.. \
t, ^
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 (Al)
do
This is the parabola shown in Fig A2(a).
Setting the second derivative to zero gives us the :<,l;,ii, n:/; set
2f 0 {0 = 0}. (A2)
002
The projection of the singularity set onto the space of control variables {r} gives us the
bifurcation set
{r = 0}. (A3)
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 (A5)
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.
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 falloff 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 51. Emission measure averaged over a solid angle AQ = 105 sr, for three sets of
caustic parameters.
72 74
11oc
oo
cooo
looo
72 1
the caustic, it is on a circular orbit of radius r given by
12 2= r (432)
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)] (433)
Angular momentum conservation implies
r rf [1+ fJ (rf(r))] (434)
Fig. 43 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 (435)
and therefore
df(rf, a) = d(r) 1 + 2f J(rf) + frfr (rf) (436)
Assuming that the initial star density has no significant structure of its own, we have
df(r, a) d [ + 2fJ(r) + fr (r) (437)
dr
which becomes very large near r = a for r > a since
dJ(r) 1 (438)
(438)
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. 43.
i
S ..................... .
58 60 62 64 66 68 70 72 74
1 (degrees)
(a)
64 66 68 70 72 74
1 (degrees)
Figure 52. 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
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 NBody 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
Fig. Bi 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 (B3)
4 us
The caustic in cross section is the locus of points with D2 = 0. To derive the caustic
structure, let us use Eq. B3 to express Eq. B 1 as
p a (T 1)2 27 (ba )2
p 64 q
z = baroT (B4)
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 (B6)
2 1 +k 8/2 t 1< u< (
2 32 1< <9
where ( is given by
= +8 a (B7)
P
Figure B2 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. B2.
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).
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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 selfsimilar 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
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 106 cm2 S1 sr1 (59)
and NT,by(E1, E2) given by
NE,b2 (E, E)= dE 7 (510)
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 (E2.7) faster than the annihilation signal (E1.5), 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 105 sr.
Figures 52(a) and 52(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 falloff 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
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. 51(a)), the cutoff density was set equal to 400 GeV/cm3 everywhere, while for
the plots with p = 0.1 kpc (Figs 51(b) and 51(c)), the cutoff 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 105
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 51(b).
Table 54. 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
107 1549.8 2177.6
106 1048.3 1283.3
105 469.9 718.1
104 269.6 239.4
103 115.4 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
Taylor series in the variable r close to the outer turnaround radius ro
rro= c( To)2 + (24)
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
subvirial and presumably collisionless. The continuous infall of these cold collisionless
stars produces a series of caustics. The infalloutfall process repeats until the stars are
thermalized and lose their coldness. The result is a series of arcs (they are not complete
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 2sphere 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 selfsimilar 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 substructure. 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 'microcaustics' or 'micropancakes'
to distinguish them from the galactic caustics, which are the topic of this work.
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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
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The equilibrium surface and the singularity set are specified by
5 +g03 +02 +y+ = 0 (A22)
and the equation
504 +3g02 +2x0+ y 0 (A23)
respectively. To obtain the bifurcation set, let us consider a 2dimensional 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 (A24)
Differentiating equations A24 with respect to 0,
d 2003 + 6g0 + 2x
dO
d = 2004 + 6g2 + 2x 0d (A25)
dO dO
Both derivatives vanish when
+ + 0 (A26)
10 10
which has three real roots if
X2 + g3 < 0 (A27)
and one real root otherwise. The number of real roots gives us the number of cusps. From
equations A24, 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
2dimensional cross section. With x 0, Eq. A24 and A25 become
y 5 04 +3g02
z = 4 05 +2g3
d = 20 0 + 6g9
dO
S20 04 + 6g02 (A28)
dO
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 coordinates to the space of final coordinates 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] (31)
where Y is the Eulerian coordinate of the particle, j is the Lagrangian coordinate and
K(t, q is the peculiar gravitational potential, possibly due to nearby halos (Tidal Torque
Theory). Equation 31 implies the velocity field
(t, H(t (t) db (32)
a dt)jVqjq[1afr(32
For points close to the tricusp, in the z = 0 plane, the dark matter density is given by
I when R < 0
1R
GeV (f/102) (V,ot/220kms1)2
d(R, 0) a 0.34 Pkpc Pkpc ( 1 + when 0 < R < 1
cm pkpc Pkpc 1R )~
1 when R > 1
R1
(55)
(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/102) (Vrt/220kms1)2 1 (56)
d(R, Z) 0.17 (56)
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 ~ 104 pc (Chapter 2). f is set equal to 2 x 102 [25].
The cutoff 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 ~ 61/2 near the fold), but we will use the density close to the fold surface to
set the density cutoff.
Fig. 51 shows the emission measure averaged over a solid angle 105 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 51(a), 51(b) and 51(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.
nonluminous 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
particles at the location of the (inner) caustic (few x 100 km/s) [50]
200 x 3 x 107 100 GeV
6a 0 kpc
500 mx,
S 104 pc 00GeV (210)
V x
which is much smaller than 1 pc. We conclude that caustics are very small scale
(subparsec) structures and are therefore difficult to resolve with cosmological simulations.
Another difficulty with Nbody 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 selfsimilar infall model [92, 93], of the first and
The quartic B24 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 B27, 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 [ cos1 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
(B31)
(B25)
(B26)
(B27)
R2 0
/13 3PUi 2Q = 0
(B28)
3Z 2
(3Z)
4
(R+
Y1 
(B29)
(B30)
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.
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 (A34)
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 ] (A35)
Differentiating with respect to ,
dy 22 [sin + sin 2]
S2X2 [cos 2 cos ] (A36)
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. A37 has two solutions when y > x2 and no solution otherwise. If 0 = x,
Eq. A37 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. A6 shows the bifurcation set.
52 Annihilation flux for the three different sets of caustic parameters for mr = 50
GeV, compared with the EGRET measured diffuse background. . . 73
Ai The 04 singularity. .................. ... .......... 86
A2 Fold catastrophe: Equilibrium surface and bifurcation set. . .... 86
A3 Cusp catastrophe. .................. ... ........... 87
A4 Swallowtail catastrophe. ............... ........... 87
A5 Butterfly catastrophe. ............... ............ .. 88
A6 Elliptic umbilic catastrophe: Bifurcation set. ................. .. 88
A7 Hyperbolic umbilic catastrophe: Bifurcation set. ............... 88
Bl Dark matter trajectories forming a tricusp ring (in cross section). . ... 99
B2 Tricusp caustic in pz cross section. .................. .... 99
B3 Modified rotation curve. .................. .......... 99
Figure B1. Dark matter trajectories forming a tricusp ring (in cross section).
Figure B1. Dark matter trajectories ftruing a tricusp ring (in cross section).
Figure B2. Tricusp caustic in pz cross section.
B
Figure B3. Modified rotation curve. (a) The function I (b) The function J
aP
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Infall of a cold collisionless shell: Figures 37 and 38 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. 316. For Fig. 37, we chose c3 = 0 and gl = 0.0333. We
refer to this as Case 1. Fig. 38 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. 37(a) (Case 1) or 38(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. 37(c) (Case 1) and Fig. 38(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 37(g) (Case 1) and 38(g) (Case 2). The shell then takes the form shown in figures
37(h) (Case 1) and 38(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. 37(j) disappears through the sequence
of figures 37(k) 37(o). The disappearance of the loop implies the existence of a cusp
caustic in the z = 0 plane as well. In Fig. 37(p) the shell has regained an approximately
spherical form and is expanding to its original size.
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, (49)
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) (411)
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) (412)
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 + (413)
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 (414)
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.
(a) (b) (c) (d)
111 1012 002 .0112 ~ In' 0j I1
(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 37. Case 1: Infall of a shell. Irrotational flow with axial symmetry, Igl < g*
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 (37)
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 (38)
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. 38 implies a force per unit mass
t tan r (39)
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 104 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
P
(a)
Z
'0
Z o[
Figure 311. Caustic in cross section, for increasing c3.
Figure 312. Dark matter flows in xy cross section. (a) gi
z
U: \\\ r
?~ ~
,,,
\h,
r,
0Ot1
O04
92 (b) 91g / 9g2
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
nonaxially 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 coordinates 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(7To) sa2 (B1)
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] (B2)
oi:
(a)
DDx
x
(b)
Figure 31. Infall of a cold collisionless shell: Antisymmetric M.
z 0[
(c)
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
LIST OF FIGURES
Figure page
21 Dark matter trajectories forming a cusp catastrophe. ............. 27
22 Effect of averaging the density over a finite volume. ........... .. .. 28
31 Infall of a cold collisionless shell: Antisymmetric M. . . 43
32 Dark matter flows forming an inner caustic. ................ . 44
33 Tricusp ring with nonzero gl and g2. .................. .. 44
34 Effect of a random perturbation. . ............... . ... 45
35 Tricusp ring: NFW potential. .................. ....... 45
36 Tricusp ring: Nonspherically symmetric gravitational potential. . ... 45
37 Case 1: Infall of a shell. Irrotational flow with axial symmetry, gi < g, .. 46
38 Case 2: Infall of a shell. Irrotational flow with axial symmetry, Igll > g .. 47
39 Cross section of the inner caustic produced by an irrotational axially symmetric
velocity field (Case 1: Igll < g,). ............... ..... .. 48
310 Cross section of the inner caustic produced by an irrotational axially symmetric
velocity field (Case 2: g > g). ....... ........... ... ... 48
311 Caustic in cross section, for increasing c3. .................. ... 49
312 Dark matter flows in xy cross section. ............... .... 49
313 Gradient type caustic without axial symmetry. ..... . ... 50
314 Formation of the hyperbolic umbilic catastrophe. ............. 51
315 Caustic in cross section, for different (gi, g2) ................ 52
316 z = 0 sections of the caustic, showing the transformation from gradient type to
curl type ..................... .. ...... ......... .53
317 Transformation of a gradient type caustic to a curl type caustic. . ... 53
41 Rotation curve close to a caustic ring. ............. ... 64
42 Smooth potential V(r). ............... ............ 64
43 Adiabatic deformation of star orbits. .............. .... 65
51 Emission measure averaged over a solid angle Af = 105 sr, for three sets of
caustic parameters ................ ............. .. 72
(a)
Figure A5. Butterfly catastrophe. (a) Each line represents a value of 0. (b) Bifurcation
Figure A6.
Elliptic umbilic catastrophe: Bifurcation set.
i I
(b)
Hyperbolic umbilic catastrophe: Bifurcation set.
3
3
5 1
3
3
1
Figure A7.
[44] S.D. Hunter, D.L. Bertsch, J.R. Catelli, T.M. Dame, S.W. Digel, B.L. Dingus, et al.,
Ap. J. 481, 205 (1997)
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E.M. Lifshitz, 3rd edition, ButterworthHeinemann, Oxford
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Astron. Astrophys. 421, L29 (2004)
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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,... (429)
= 171, 17, 4, parsec.
(for a flat rotation curve) is of the
form
S0 rot 2r2
(430)
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)
(431)
V,(r) is plotted in Fig. 42 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 selfsimilar infall model
[ t2
[92, 93], a oc t39 Consider a particle of specific angular momentum 1. In the absence of
(427)
(428)
0001
Figure 32.
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 33. Tricusp ring with nonzero gi and g2.
N)'
I 1
_~t
rcc .
f"
.040
Y
... .....
Now let us consider the infall for Case 2. The loop which is present near the z = 0
plane in Fig. 38(j) disappears through a more complicated sequence of figures 38(k) 
38(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. 39(a) for the case (c3,91) (0, 0.0333)
and in Fig. 310(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. 39(a) and 310(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. 39(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 310. 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. 316) and we have cusps on the x axis and cusp/butterfly caustics on the
z axis, depending on the magnitude of gl.
Fig. 310(b) shows the dark matter flows in the vicinity of the butterfly caustic, for
g91 > gl,. Fig. 310(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. 311 shows the
transformation. We start with an irrotational velocity field (c3 = 0) in 311(a) and increase
C3 until the rotational component dominates the velocity field, in 311(d). The caustic line
for x < 0 and
d = (A12)
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 = () (A13)
For arbitrary (x, y), the density may be obtained by solving the cubic Eq. A7. 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 (A15)
and the singularity set
403 +2x0+ + y 0 (A16)
To sketch the bifurcation set, let us consider a 2 dimensional cross section with x fixed.
y = 403 + 20
z 304 + 02 (A17)
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. A17, we
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 [1214], 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 nonthermal 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
nonthermal (cold).
Another approach is to carry out Nbody 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
x
(a)
(b)
(e)
(h)
A
/ k
(k)
LI z
S" x
(m) (n)
Figure 315. Caustic in cross section, for different (i, 92).
52
(c)
(f)
Ni
(U)
4
LI,
I
(d)
(g)
I
(j)
p
~~ ,
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 (315)
( )2 a2 a3
where al, a2 and a3 are dimensionless numbers. Fig. 36 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. 33 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. 35 is symmetric about the z axis when cl = c2 = 0
and g, = g2. Then
3 = gi sin(20) 8 + c3 sin0 0 (316)
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.
0.2 0 c
(a)
0.2
0.4
100
75
50
25
0
1U2
75
0
25
0
( pa
(b)
parse
(d)
Figure 22. 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 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
antisymmetric 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
viceversa.
C'! lpter 4 and C'! lpter 5 deal with the .i r ,!i, ii 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 selfsimilar 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
(k)
Figure 38. Case 2: Infall of a shell. Irrotational flow with axial symmetry, Ig l > g*
4
Lj
4
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
obtain the equations
d = 1202 + 2x
dO
dz dy
z 1203 + 20x 0 (A18)
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 (A20)
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 A4(a) represents a real root of 0. The
yz cross section of the bifurcation set with x < 0 is shown in Fig. A4(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
i/
z\~~,,.` 
Figure 39. Cross section of the inner caustic produced by
symmetric velocity field (Case 1: Igil < g,).
.
N
an irrotational axially
(a) (b) (c)
Figure 310. 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
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
(33)
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 nonzero 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 coordinates 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 (35)
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,!iics as a postdoctoral fellow at
Bielefeld University in Germany.
A.2.4 Hyperbolic Umbilic: Corank = 2, Codimension = 3
The unfolding is
03 +03 +x0 y z (A38)
The equilibrium surface is given by the equations
3 02 +x y 0
3 2 +X = 0 (A39)
The singularity set is defined by the condition
60 x
0 0 (A40)
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. A39, we find that y = 0 and z is positive. Similarly, putting = 0
in Eq. A39, 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. A39
.3
/ 3 02+ 6
360
z +x0 (A41)
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. A41 with respect to 0,
dy x3
60
dO 36 02
dz 6x4
d +x (A42)
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
4.3.4 Density Enhancement: Noncircular 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 (439)
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 (440)
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 selfsimilar 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
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, SungSoo, KC, Ian,
N ii',. and Emre. I thank the Institute for Fundamental Theory, University of Florida for
providing partial financial support.
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 nonradial, particularly in the inner regions of the halo. Fig. 21
shows an example of nonradial 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. 21 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
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) (312)
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 (313)
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. 35 shows the result of two simulations plotted on the same figure. The larger
caustic ring is obtained using the density profile of Eq. 313, while the smaller caustic
ring is obtained using the density profile of Eq. 38 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.
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, B2. 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 coordinates
R pa
P
Z (B8)
q
ao 0:
When a = 0, Eq. B 1 becomes
z 0
1
pa u(T To)2 (B9)
2
Solving for D2 and using Eq. B3, we obtain
D2 = 2bp R(R) forR > 0 (B10)
Summing the two solutions for ID211, 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
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)] (41)
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 (42)
+1/2 for r > a +p
and H(r) is given by
1 ra
H(r) tanh(r ) (43)
2 po
po is expected to be of order a. Fig. 41 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)]. (44)
r
From Eq. 44 and Fig. 41, 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 coordinates [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 (46)
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
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
coordinates 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 coordinates (x, y, z), the density of particles
d is [94]
d(t, x) d Nldd (Ai(t, X)) 1 (22)
Sdada2da3 del () ta:)
The sum over d1 is required because the mapping from (r, 0, Q) space to (x, y, z) space is
rn, ,ivI,,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
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: Noncircular 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 ........................................
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
(420)
The first order part of Eq. 418 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) (42
( dr" ) Or
We choose the form of the potential term UI(ro, o) as
i au1
2uwoio (ro, Po)
ro 00c
2)
(423)
Ui(ro, co) = Ui(ro) cos my
(424)
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. 423 to obtain
Ul, cos mwt
ro71 = 2 onl 
ro cot
(425)
where the constant of integration is absorbed by a redefinition of rl. Eliminating L from
Eq. 422, we have
r + 02 32" rI1
( ar2 +
Or
2 wo U, t
cos mwot
fOo \
(421)
(426)
roi1
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 41. Rotation curve close to a caustic ring (R = (r a)/p).
V,/2!u,,L
10
Figure 42. Smooth potential V,(r) (R
 R 2
(r a)/p).
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 51. N,/m2 in units of 104 GeV2 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 52. N,/m2 in units of 104 GeV2 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 53. N,/m2 in units of 104 GeV2 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 coordinates (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.
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
(B42)
C'!i ,iI.iig variables
ca
r70
(B43)
and using Eq. B3, we have
4G d2M dA
ab dQdt [R
(T 1)2 + (A2
1)2 + (A2]2 + 4A2T2
where R and ( are
R pa
R
su
b2
Using Eq. B39, 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.
(B44)
27 (p 2
16 q
(B45)
(B46)
(B47)
(B48)
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[90] A. Shirokov and E. Bertschinger, astroph/0505087
(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 zaxis. When g2 gl, it
is axially symmetric about the xaxis.
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 zaxis 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. 35 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
nonstationary 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)
In C'!I pter 4, we assumed q t 1 kpc and f = 4.6 x 102 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) (B38)
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, (B40)
J (pp) +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 [ (B41)
c a dldtj [p pI(a, )]12 B2 4,1T
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
MPEG1 Audio L vIr 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 Ai 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 Alt i .1!i1. 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.
with S and < EM > defined as
S(E1, E2) = dE bi (E) 2m
S i d2 M~x
< EM > (0, AE) = dQ EM(o0,) (52)
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(). (53)
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 1027 m3 S 1
Q 2
Qxh2
3 x 1026 cm3 s 1 (54)
The quantity dN,(E,)/dx for the dominant channels may be approximated by the form
[9, 11, 34] dN,/dx = a eb/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
Fig. 313 shows the inner caustic in 3 dimensions for the initial velocity field of Eq.
317 with ( = 0.005 and g = 0.05. The inner tube has a diamond shaped cross section
which is evident in Fig. 313(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 313(c) and 313(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. 313(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.
314(a) 314(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. 314(e) 
314(h) for the hyperbolic umbilic on the positive x side. The arc and the cusp approach
each other until they overlap (314(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. 314(i).
Let us mention that the reason there is no hyperbolic umbilic catastrophe on the y
axis of Fig. 314(a) 314(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
where we assumed Mn = f 1024 gm cm3 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 NBody Simulations
The existence and relevance of caustics in real galactic halos is sometimes disputed on
the basis of Nbody 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 21.
Figure 22(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 22(b), 22(c) and 22(d) show the effect of averaging the density over a
cube of side = 0.01 pc, 0.1 pc and 1 pc respectively. A cutoff density of 100 GeV cm3
was assumed. In Fig. 22(b), we see that the averaged density follows the true density
faithfully, but the density does not quite reach the cutoff value. Fig. 22(c) no longer
shows a sharp increase in density at the location of the caustic and Fig. 22(d) misses the
caustic completely.
The primordial velocity dispersion of WIMPs of mass mT is 6v = 3x 107 100 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
S2007 Aravind Natarajan
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 nonaxisymmetric 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
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)] (4M)
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 (417)
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 (419)
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 (25)
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 2sphere 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 (26)
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 (27)
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 nonzero angular momentum, xo / 0. Equations 26 and 27 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. 25 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].
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. B3,
2 27 Z2
R= (T 1) (B19)
64 T2
where T = 7/T. We need to solve the quartic equation
T4 2 T3 + ( RZ2 = 0. (B20)
64
The density d(R, Z) is given by
p(RZ 7 GeV (f/1O2) (Vot/220kms1)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. B20. The number of real roots is
given by the sign of the discriminant S
S= 144FR2 R+ 128F2 (R+ +4R2 (R + 16 R 27R42563
(B22)
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. B20 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 rewrite Eq. B20 in term of T1 by making the substitution T = T + thereby
eliminating the cubic term:
Ti4 (R ~+1 2 RT 0 (B24)
( t/
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'\ /105 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,ii 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 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
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 21. Dark matter trajectories forming a cusp catastrophe.
Figure 21. Dark matter trajectories farming a cusp catastrophe.
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 = 3.1017 (105eV (t)2/3
6V, = 10 c /2 to (21)
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 selfintersect. 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].
CHAPTER 1
INTRODUCTION
There are compelling reasons to believe that most of the matter in the Universe is
in a nonluminous 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 falloff of
rotation velocity. The observations contradict this and provide support for the existence of
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 0657558 (the "bullet (l
matter [26]. This event is a merger of two clusters of galaxies. The intracluster gas
is collisional and heats up, emitting Xrays. If a collisionless component is present, it
would simply pass through and become separated from the gas. Xray 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 HulseTaylor 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
which gives us
a = 2( R) for R < 1 (B13)
Solving for ID21 and summing the two solutions for ID2 1, we find
1 1 for R < 1 (B14)
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 + (B15)
D21 bp t R (1 2/5
The physical space density is given by
1 dM cos(a)
d(p, z) = (a,) ) (B16)
p d~dt D2 (a, )
where the sum is over D2 1. We use the selfsimilar infall model with angular momentum
[92, 93] to estimate the mass infall rate
dM fV V2Qt ( 17)
d t fv 4r (B17)
dQdt 47G
where f is a selfsimilar 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
1R
d(RO) 0. 34GeV (f/102) (V,/220kms1)2 ) o < l (B18)
cm3 pkpc Pkpc 1R 0 <
1 R>1
p1
pkp~c and pkpc are distances measured in kpc.
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. 33 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 (D4). In our example, the caustic is made up of four elliptic umbilic
catastrophe sections joined backtoback.
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. 310. Fig. 34 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. 32(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) (311)
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
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 [9294], 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 selfsimilar 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 selfsimilar 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.
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. 315 show respectively the z 0, y = 0
and x = 0 sections of the inner caustic produced by the initial velocity field of Eq. 317
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. 315 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. 317. Fig. 316 shows
the z 0 cross sections of the inner caustic during such a transition. In Fig. 316(a),
the initial velocity field is irrotational and we see a circle and diamond, as before. In
Fig. 316(b), the diamond is skewed because of the rotation in the z 0 plane introduced
by C3 / 0. Fig. 316(c) shows the case c3 = As c3 is increased further, the diamond
transforms into two swallowtail (A4) catastrophes joined back to back (Fig. 316(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
Z22 02
0 //
02 002
002 'r OZ
(a) (b)
006 006
Z o Z o
006 006 y
403 0 03 003 0 003
x y
(c) (d)
Figure 313. 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.
outside the circle. Finally, the swallowtails pinch off to form the inner circle of the z = 0
section of the tricusp ring. Fig. 317 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 nonzero 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.
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 i1 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
LIST OF TABLES
Table page
51 N,/m' in units of 104 GeV2 for mx 50 GeV ................ .. 68
52 N,/m2 in units of 104 GeV2 for m = 100 GeV ................ ..68
53 N,/m2 in units of 104 GeV2 for mx = 200 GeV ................ ..68
54 Peaks of < EM > for (a 8.0, p =0.1,q 0.2) kpc .............. ..70
points since neither derivatives in Eq. A42 vanish. To determine the different regions
of the bifurcation set, let us consider points on the line y = z with x = 1. From Eq.
A39, we have 0 = 0 or 0 + 0 = 1/3. If 0 = Eq. A39 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 A7 shows the bifurcation set. It has three distinct regions.
/
//
\ / ^
V
(a) (b)
Figure Ai. The 04 singularity. (a) 04 (b) 04 + x02 (c) 04 + x02 + y0
r
(a) (b)
Figure A2. Fold catastrophe: Equilibrium surface and bifurcation set.
(a)
Figure 316.
(b)
(c)
z = 0 sections of the caustic, showing the transformation from gradient type
to curl type.
(c)
/
(d)
oo. Y
x
Figure 317. Transformation of a gradient type caustic to a curl type caustic.
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 selfsimilar 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. Selfsimilarity means that the phase space distribution of dark
matter is timeindependent 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 selfsimilar
model was modified to include the effect of nonzero 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
.0.005 .. ......... ..
tl(X)5 '
Figure 34. Effect of a random perturbation.
0. ^..**
0.01
Figure 35. Tricusp ring: NFW potential.
0.02
0.0.02 
Sa
Figure 36. Tricusp ring: Nonspherically symmetric gravitational potential.
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 (A29)
5
which meets the z axis at z = 4 ( g)5/2 2g ( 3)3/2
z 0 0 0, 02 (A30)
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 A5(a) shows the family of 0 curves. Each
curve corresponds to a real root of the quintic Eq. A22.
Consider Eq. A22 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. A5(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 + (A32)
3
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.

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Iwouldliketothankmyparentsfortheirloveandsupport.IthankmyPh.DsupervisorProfessorPierreSikivieandtheothermemberofmysupervisorycommittee,ProfessorsRichardWoodard,JamesFry,JamesDuftyandVickiSarajedini.IamespeciallygratefultoRichardWoodardforhisconcernandadviceonmanyoccasions.Youwilldenitelybemissed!ItismypleasuretothankSushforallthegreattimeswehadtogether.IalsoacknowledgemyfellowphysicsnerdsJian,SungSoo,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 2.5DicultyofResolvingCausticsinNBodySimulations ........... 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:NoncircularOrbits .............. 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 51N=m2inunitsof104GeV2form=50GeV ................. 68 52N=m2inunitsof104GeV2form=100GeV ................. 68 53N=m2inunitsof104GeV2form=200GeV ................. 68 54Peaksoffor(a=8:0;p=0:1;q=0:2)kpc ............... 70 6
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Figure page 21Darkmattertrajectoriesformingacuspcatastrophe. ............... 27 22Eectofaveragingthedensityoveranitevolume. ............... 28 31Infallofacoldcollisionlessshell:AntisymmetricM. ............... 43 32Darkmatterowsforminganinnercaustic. .................... 44 33Tricuspringwithnonzerog1andg2. ....................... 44 34Eectofarandomperturbation. .......................... 45 35Tricuspring:NFWpotential. ............................ 45 36Tricuspring:Nonsphericallysymmetricgravitationalpotential. ........ 45 37Case1:Infallofashell.Irrotationalowwithaxialsymmetry,jg1jg 47 39Crosssectionoftheinnercausticproducedbyanirrotationalaxiallysymmetricvelocityeld(Case1:jg1jg). ........................... 48 311Causticincrosssection,forincreasingc3. ..................... 49 312Darkmatterowsinxycrosssection. ........................ 49 313Gradienttypecausticwithoutaxialsymmetry. ................... 50 314Formationofthehyperbolicumbiliccatastrophe. ................. 51 315Causticincrosssection,fordierent(g1;g2). ................... 52 316z=0sectionsofthecaustic,showingthetransformationfromgradienttypetocurltype. ....................................... 53 317Transformationofagradienttypecaustictoacurltypecaustic. ........ 53 41Rotationcurveclosetoacausticring. ....................... 64 42SmoothpotentialVc(r). ............................... 64 43Adiabaticdeformationofstarorbits. ........................ 65 51Emissionmeasureaveragedoverasolidangle=105sr,forthreesetsofcausticparameters. ................................. 72 7
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........ 73 A1The4singularity. .................................. 86 A2Foldcatastrophe:Equilibriumsurfaceandbifurcationset. ............ 86 A3Cuspcatastrophe. ................................... 87 A4Swallowtailcatastrophe. ............................... 87 A5Butterycatastrophe. ................................ 88 A6Ellipticumbiliccatastrophe:Bifurcationset. ................... 88 A7Hyperbolicumbiliccatastrophe:Bifurcationset. ................. 88 B1Darkmattertrajectoriesformingatricuspring(incrosssection). ........ 99 B2Tricuspcausticinzcrosssection. ......................... 99 B3Modiedrotationcurve. ............................... 99 8
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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,onewouldexpecttoseeaKeplerianfalloofrotationvelocity.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,observationsof1E0657558(the\bulletcluster")haveprovideddirectevidencefordarkmatter[ 26 ].Thiseventisamergeroftwoclustersofgalaxies.Theintraclustergasiscollisionalandheatsup,emittingXrays.Ifacollisionlesscomponentispresent,itwouldsimplypassthroughandbecomeseparatedfromthegas.Xraymeasurementsoftheclusterareusedtotrackthegas,whileweakgravitationallensingtracksthetotalmass.Theobservationsshowthatthecenterofmassdoesnotcoincidewiththeluminousmatter.Darkmattercosmologyassumesthecorrectnessofgeneralrelativity.Generalrelativityisconsistentwithsolarsystemtestsliketheprecessionofplanetaryorbitperihelia,orthebendingofstarlightbythesun'sgravitationaleld[ 107 ].ThetheoryhasbeentestedtoextremelyhighprecisioninthecaseoftheHulseTaylorbinarypulsar.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.Thelateinfallofdarkmatterwillcausenonthermalstreams,asweshowin Chapter2 .Sincethereisnoevidenceforviolentrelaxationoccurringtoday,particlesthathavefallenintotheinnerregionsofthehalo(whichcontainthemostsubstructure)relativelyrecentlymaybeexpectedtobenonthermal(cold).AnotherapproachistocarryoutNbodysimulationsoftheformationofgalactichalos,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,thephasespacetrajectoriesdonotselfintersect.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;arethecoordinatesofthepointwheretheparticlecrossedthereferencesphere.Inourexample,theparticlewhichcrossedthesphereattimeatlocation;willbelabeledby(;;)atalltimest.Intermsofthephysicalspacecoordinates(x;y;z),thedensityofparticlesdis[ 94 ] d1d2d3(~i(t;~x))1 @~~i(t;~x)(2{2)Thesumover~iisrequiredbecausethemappingfrom(;;)spaceto(x;y;z)spaceismanytoone.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 ]showedthattheevolutionofdarkmatterhalosisselfsimilariftheprimordialoverdensityhastheprole 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.Thestarsarethereforesubvirialandpresumablycollisionless.Thecontinuousinfallofthesecoldcollisionlessstarsproducesaseriesofcaustics.Theinfalloutfallprocessrepeatsuntilthestarsarethermalizedandlosetheircoldness.Theresultisaseriesofarcs(theyarenotcomplete 18
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18 ].Theoccurrenceofcausticswithstarssuggeststhatthesamecouldoccurwithdarkmatter. Chapter3 ,wewillgiveadetaileddescriptionofthestructureofinnercaustics,showingthecatastrophesthatformforgiveninitialconditions.Toseetheformationofinnercaustics,letusconsidertheinfallofacoldowofdarkmatterparticles.Iftheinfallissphericallysymmetric,theparticletrajectoriesareradialandtheinfallproducesasingularityatthecenter.Ifinstead,thedarkmatterparticlespossesssomedistributionofangularmomentumwithrespecttothehalocenter,theparticletrajectoriesarenonradial,particularlyintheinnerregionsofthehalo.Fig. 21 showsanexampleofnonradialinfall,inxycrosssection.Thecausticistheenvelopeofthefamilyofdarkmattertrajectories(i.e.,itisthelocusofpointstangenttothefamilyoftrajectories).Thedarkmatterdensityisenhancedalongtheenvelope.ThetwocurvesinFig. 21 arefoldcatastrophesandtheirintersectionisacuspcatastrophe.Thecausticcurveandthedensityfalloareworkedoutin AppendixA 63 ].Inaccordancewithourformalism,letusconsideracoldcollisionlessowofparticlesandlabeltheparticlesbythreeparameters(;;).Letuschooseareferencesphereatsomeconvenientradius,suchthateachparticlepassesthroughthesphereonce.isthetimewhentheparticlecrossedthereferencesphereand(;)specifythelocationwheretheparticlecrossedthereferencesphere.Toobtainthedensityisrealspace,onemust 19
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@@~x @@~x @=0(2{5)Considertheinfallofashellofdarkmatterparticleswithanarbitraryangularmomentumdistribution.Sincethetangentialvelocityvectorona2spheremustvanishattwopointsatleast,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).Forthemoregeneralcaseofnonzeroangularmomentum,~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. 21 .Figure 22(a) showsthedarkmatterdensityclosetooneofthefoldlines(averagedoveraninnitesimalvolume)asafunctionofdistancetothecausticalongthelineofsight.Figures 22(b) 22(c) and 22(d) showtheeectofaveragingthedensityoveracubeofside=0.01pc,0.1pcand1pcrespectively.Acutodensityof100GeVcm3wasassumed.InFig. 22(b) ,weseethattheaverageddensityfollowsthetruedensityfaithfully,butthedensitydoesnotquitereachthecutovalue.Fig. 22(c) nolongershowsasharpincreaseindensityatthelocationofthecausticandFig. 22(d) missesthecausticcompletely.TheprimordialvelocitydispersionofWIMPsofmassmisv=3107p 94 ].Ifthevelocitydispersionisnolargerthantheprimordialvalue,theresultingcausticsarespreadoveraregion[ 95 ]ava=vwhereaistheouterturnaroundradius(fewhundredkpc)oftheowofparticlesformingthecausticandvisthespeedofthe 24
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50 ] whichismuchsmallerthan1pc.Weconcludethatcausticsareverysmallscale(subparsec)structuresandarethereforediculttoresolvewithcosmologicalsimulations.AnotherdicultywithNbodysimulationsisthelargeparticlemassesinvolved.Sincethemassofeachparticlecanbeseveralmillionsolarmasses,gravitationalscatteringbetweentwo`point'massesisnotnegligible.Closeencountersofthesespuriouslymassiveparticlesleadtolargescatteringangles[ 73 ],whilegravitationalscatteringbetweentwoWIMPsortwoaxionsiscompletelynegligible.Notethatwhensucientcareistaken,itmaybepossibletoresolvediscreteowsandcausticsincosmologicalsimulations.ThesimulationsofStiandWidrow[ 100 ]showtheexistenceofdiscreteowsinvelocityspacewhilethesimulationsofBertschingerandShirokov[ 90 ]suggesttheexistenceofcausticsinphysicalspace. 47 ].Thecombinedrotationcurveshowspeaksatthelocationsexpectedintheselfsimilarinfallmodel[ 92 93 ],oftherstand 25
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17 ]haveconjecturedthattheanomalousbehavioroftheouterMilkyWayrotationcurvecouldbeexplainedifaringofmatterexistedat13.6kpcwhichisclosetotheexpectedlocationofacausticintheselfsimilarinfallmodeloftheMilkyWay[ 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 ]. Figure21. Darkmattertrajectoriesformingacuspcatastrophe. 27
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(b) (c) (d)Figure22. Eectofaveragingthedensityoveranitevolume. 28
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AppendixA .Booksoncatastrophetheoryinclude[ 5 23 38 79 86 ].Wesimulateasinglecoldowofdarkmatterfallinginaxedgravitationalpotential.TheequationsofmotionaresolvednumericallyandtheJacobianofthemapfromthespaceofinitialcoordinatestothespaceofnalcoordinatesiscomputedateverylocation[ 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,theinitialvelocitydistributionisaxiallysymmetricaboutthezaxis.Wheng2=1 2g1,itisaxiallysymmetricaboutthexaxis.Inthecaseofpurerotation(g1=g2=0),wemaychooseaxessuchthat~c=c^z.Theinitialvelocitydistributionisalwaysaxiallysymmetricinthiscase.Wheng1,g2,c1,c2andc3arealldierentfromzero,theinitialvelocitydistributionhasnosymmetry.Whenaxialsymmetryaboutthezaxisisimposed,c1=c2=0andg1=g2. 3{5 plusvr=0.Wesolvetheequationsofmotionnumericallyandobtainthetrajectory~x(;;)oftheparticlethatoriginatedattheposition(;)ontheturnaroundsphereattime.Sinceneitherthepotentialnortheinitialconditionsaretimevarying,thesimulatedowsarestationary(i.e.,~x(t;;;)=~x(t;;)).Thesimulationofnonstationaryowswouldbestraightforwardbutconsiderablymorememoryintensiveandtimeconsuming,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 31 showstheinfallofasingleshellinxzcrosssection,atsuccessivetimesforc3=0:1.Astheshellfallsin,deviationsfromsphericalsymmetryappearduetothepresenceofangularmomentum.Theparticlesatthepoleshavezeroangularmomentumandfallinfasterthantheparticlesneartheequator.TheshelltakesontheformofFig. 31(b) .InFig. 31(c) ,theparticleswhichoriginatedatthepoleshavecrossedthez=0plane.TheshellhasturneditselfinsideoutinFig. 31(e) ,formingacrease.Finallytheshellincreasesinsizeandregainsanapproximatelysphericalshape.TheinnercausticoccursatandnearthelocationofthecreaseinFig. 31 .Fig. 32(a) showstheowsnearthecreaseinzcrosssectionwhere=p 32(b) showsthecausticinthreedimensions. 33
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3.3.2.1Eectofthegradienttermsg1andg2Letusperturbtheinitialvelocityeldbyaddingthegradienttermsg1andg2.Asanexample,wechoosec3=0:1;g1=0:033;g2=0:0267.Thusjc3jislargecomparedtog1org2,andwemayregardthepresenceofthegradienttermsasperturbationstotherotationalow.Fig. 33 showstheinnercausticobtainedfromtheinfall.Itisagainatricuspring,butthecrosssectionisdependent.Inthiscase,thetricuspshrinkstoapointat4placesalongthering.Inthevicinityofsuchapoint,thecatastropheistheellipticumbilic(D4).Inourexample,thecausticismadeupoffourellipticumbiliccatastrophesectionsjoinedbacktoback. 3{10 .Fig. 34 showstheinnercaustic.Wenotethattheinnercausticisstillatricuspring,althoughitisdeformedfromwhatiswasinFig. 32(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. 35 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. 36 showstheinnercausticforthecasewherea1=0:95,a2=1:0anda3=1:05,andtheinitialvelocityeld~v=0:2sin^.Itisagainatricuspring.Itsaxialsymmetryislostduetotheabsenceofaxialsymmetryinthepotential.Thetricuspringstillhasreectionsymmetryaboutthexy,yzandxzplanes.AsinFig. 33 thetricuspshrinkstoapointfourtimesalongthering. 3{5 issymmetricaboutthezaxiswhenc1=c2=0andg1=g2.Then 2g1sin(2)^+c3sin^(3{16)Werstsimulatetheowandobtaintheinnercausticintheirrotationalcasec3=0.Nextweseehowthecausticismodiedwhenc36=0. 36
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37 and 38 showtheinfallofacoldcollisionlessshell.TheguresshowthequalitativeevolutionofashellwhoseinitialvelocityeldisgivenbyEq. 3{16 .ForFig. 37 ,wechosec3=0andg1=0:0333.WerefertothisasCase1.Fig. 38 showsthequalitativeevolutionofashellwithc3=0andg1=0:0667,whichwillbereferredtoasCase2.Sincec3=0inbothcases,eachparticlestaysintheplanecontainingthezaxisanditsinitialpositionontheturnaroundsphere.Theguresthereforeshowtheparticlesinthey=0plane.Theangularmomentumvanishesat=0and==2whereisthepolarcoordinateoftheparticleatitsinitialposition.Hence,theparticleslabeled=0and==2followradialorbits.Theangularmomentumincreasesinmagnitudefrom==2,reachesamaximumat==4andreturnstozeroat=0.Thesignoftheangularmomentumdoesnotchangeduringthisinterval.TheshellstartsoutasshowninFig. 37(a) (Case1)or 38(a) (Case2).Astheshellfallsin,theparticlesat6=0;=2movetowardsthepoles.Theseparticlesfeelanangularmomentumbarrierandfallinmoreslowlythantheparticlesat=0.ThisresultsintheformationofaloopinFig. 37(c) (Case1)andFig. 38(c) (Case2).Theformationoftheloopimpliesacuspcausticonthezaxis.Theparticleslabeled=0and=havecrossedthez=0planeandtheparticleslabeled==2havecrossedthex=0planeingures 37(g) (Case1)and 38(g) (Case2).Theshellthentakestheformshowningures 37(h) (Case1)and 38(h) (Case2).Thefurtherevolutiondependsonthemagnitudeoftheangularmomentum(i.e.,onthevalueofjg1j)andisdierentforthetwocases.ConsidertheinfallforCase1.Theloopthatispresentnearthez=0planeinFig. 37(j) disappearsthroughthesequenceofgures 37(k) 37(o) .Thedisappearanceoftheloopimpliestheexistenceofacuspcausticinthez=0planeaswell.InFig. 37(p) theshellhasregainedanapproximatelysphericalformandisexpandingtoitsoriginalsize. 37
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38(j) disappearsthroughamorecomplicatedsequenceofgures 38(k) 38(o) .Theparticlesnear==2crossthez=0planebeforethesphereturnsitselfinsideout.Thiscrossoverproducesadditionalstructureandamorecomplicatedcaustic.Thecriticalvalueofjg1j,belowwhichthequalitativeevolutionisthatofCase1andabovewhichthatofCase2isg1'0:05.Causticstructure:Thebutterycatastrophe.TheinnercausticisasurfaceofrevolutionwhosecrosssectionisshowninFig. 39(a) forthecase(c3;g1)=(0;0:0333)andinFig. 310(a) for(c3;g1)=(0;0:0667).Onthezaxis,thereisacausticline.Causticlinesarenotgeneric.ThecausticlineinFigs. 39(a) and 310(a) occuronlybecausetheinitialvelocityeldisaxiallysymmetricandirrotational.Wewillseebelowthatwhenaxialsymmetryisbrokenorwhenarotationalcomponentisadded,thelinebecomesacaustictube.Fig. 39(b) showsthedarkmatterowsinthevicinityofthecuspforjg1jg1inFig 310 .Sincethemagnitudeofjg1jdetermineswhetherornotthebutterycongurationoccurs,jg1jmaybetermedthe\butteryfactor"[ 86 ].Ifg1ischosenpositiveinsteadofnegative,thebehavioratthepolesandtheequatorisreversed(Eq. 3{16 )andwehavecuspsonthexaxisandcusp/butterycausticsonthezaxis,dependingonthemagnitudeofg1.Fig. 310(b) showsthedarkmatterowsinthevicinityofthebutterycaustic,forjg1j>g1.Fig. 310(c) showsthenumberofowsineachregionofthebuttery.Addingarotationalcomponent:Hereweshow,intheaxiallysymmetriccase,theeectofaddingarotationalcomponenttotheinitialvelocityeld.Fig. 311 showsthetransformation.Westartwithanirrotationalvelocityeld(c3=0)in 311(a) andincreasec3untiltherotationalcomponentdominatesthevelocityeld,in 311(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. 312(a) showsthetrajectoriesoftheparticlesinthez=0planeforsuchacase.Theorbitsareradial.Indeedallparticleshavezeroangularmomentumwithrespecttothezaxiswhentheinitialvelocityeldisirrotationalandaxiallysymmetric.Becausealltrajectoriesintersectthezaxis,thereisapileupofparticlesonthataxisandhenceacausticline.Fig. 312(b) showsthetrajectoriesoftheparticlesinthez=0planefortheinitialvelocityeldofEq.( 3{17 )with=0:01andg=0:05.Theparticlesdohaveangularmomentumwithrespecttothezaxisnow.ThecausticlineonthezaxisspreadsintoatubewhosecrosssectionisthediamondshapedenvelopeofparticletrajectoriesshowninFig. 312(b) .Thatenvelopehasfourcusps.Theowsandcaustichavereectionsymmetryaboutthexy,xzandyzplanesbecausetheinitialvelocityeldhasthosesymmetries.Fig. 312(b) showsfourowsinsidethediamondshapedcausticandtwoowsoutside.Theinfallofdarkmatterparticlesfromregionsaboveandbelowthez=0planewilladdtwomoreowsateachpoint,whicharenotshowninFig. 312 forclarity. 39
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313 showstheinnercausticin3dimensionsfortheinitialvelocityeldofEq. 3{17 with=0:005andg=0:05.TheinnertubehasadiamondshapedcrosssectionwhichisevidentinFig. 313(b) ,whichshowsasuccessionofconstantzsections.Nearthez=0plane,therearesixowsinsidethediamond,fourowsintheotherregionsinsidethecaustictent,andtwooutside.Figures 313(c) and 313(d) showy=0andx=0sectionsofthecaustic.Hyperbolicumbiliccatastrophe:Letuslookmorecloselyatthetworegions(topandbottom)inFig. 313(c) wheretheinnersurfacereachesandtraversestheoutersurface.Wewillshowtheexistenceofahigherorderstructureattheboundary.Figs. 314(a) 314(d) showz=constantsectionsoftheinnercausticinsucharegion.Asjzjisincreased,thetwocuspsontheyaxissimplypassthroughtheoutersurface,whereasthecuspsonthexaxistraversetheoutersurfacebyformingwiththelatter,twohyperbolicumbilic(D+4)catastrophes,oneonthepositivexsideandoneonthenegativexside.ThesequencethroughwhichthishappensisshowningreaterdetailinFigs. 314(e) 314(h) forthehyperbolicumbiliconthepositivexside.Thearcandthecuspapproacheachotheruntiltheyoverlap( 314(g) ),formingacorner.Thecuspistransferredfromonesectiontotheotherasthetwosectionspassthrough.Thisbehaviourischaracteristicofthehyperbolicumbiliccatastrophe.Therearefourhyperbolicumbilicsembeddedinthecaustictwo(x>0andx<0)atthetop(z>0)andtwoatthebottom(z<0).Thehyperbolicumbilicatz>0andx<0isshowninthreedimensionsinFig. 314(i) .LetusmentionthatthereasonthereisnohyperbolicumbiliccatastropheontheyaxisofFig. 314(a) 314(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. 315 showrespectivelythez=0,y=0andx=0sectionsoftheinnercausticproducedbytheinitialvelocityeldofEq. 3{17 forvariousvaluesof(g1;g2).Theratiog2=g1decreasesuniformlyfrom1(toprow)to1/2(bottomrow).Notethatthethirdcolumndescribesasequencewhichisthatoftherstcolumninreverse,andthatthersthalfofthesequenceinthesecondcolumnisthereverseofthesequenceinitssecondhalf,withxandzaxesinterchanged.Intherstrow,thecausticisaxiallysymmetricaboutthezaxis.Thereisacausticlineseeninthexzandyzcrosssections.Thelineappearsasapointinthexycrosssection.Inthesecondrow,axialsymmetryisbrokenandthefamiliardiamondshapedcurveappears.Thehyperbolicumbilicsareapparentinthexzcrosssectionatthepointswherethetwocurvesmeet.Inthethirdrow,thesmoothcurvepassesthroughthediamondseeninxycrosssection.Thiscurvesturnsintoacausticlineinthefthrow,asseeninthexyandxzcrosssectionsandappearingasapointintheyzcrosssection.TheplotsofFig. 315 arereminiscentofcausticsseeningravitationallensingtheory[ 19 76 77 ],andintheanalysisofthestabilityofshipsusingcatastrophetheory[ 79 111 ].Addingarotationalcomponent:Theswallowtailcatastrophe.Hereweaddarotationalcomponent(c36=0)totheinitialvelocityeldofEq. 3{17 .Fig. 316 showsthez=0crosssectionsoftheinnercausticduringsuchatransition.InFig. 316(a) ,theinitialvelocityeldisirrotationalandweseeacircleanddiamond,asbefore.InFig. 316(b) ,thediamondisskewedbecauseoftherotationinthez=0planeintroducedbyc36=0.Fig. 316(c) showsthecasec3=.Asc3isincreasedfurther,thediamondtransformsintotwoswallowtail(A4)catastrophesjoinedbacktoback(Fig. 316(d) ).Therearetwoowsinthecentralregionformedbytheswallowtails,sixowsinthecuspedregionofeachswallowtail,fourintheotherregionsinsidethecircleandtwo 41
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317 showsthetransitioninthreedimensions. 42
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(b) (c) (d) (e) (f)Figure31. Infallofacoldcollisionlessshell:AntisymmetricM. 43
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(b)Figure32. Darkmatterowsforminganinnercaustic.(a)Darkmatterowsneartheinnercaustic,incrosssection.(b)Theaxiallysymmetrictricuspringinthreedimensions. Tricuspringwithnonzerog1andg2. 44
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Eectofarandomperturbation. Tricuspring:NFWpotential. Tricuspring:Nonsphericallysymmetricgravitationalpotential. 45
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(b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p)Figure37. Case1:Infallofashell.Irrotationalowwithaxialsymmetry,jg1j
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(b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p)Figure38. 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,theselfsimilarinfallmodelpredictsthattheradiusa2ofthesecondcausticringofdarkmatterisapproximately20kpcinourgalaxy.Thetransversesizespandqarenotpredictedbytheselfsimilarinfallmodel.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. 41 showstherotationvelocityclosetoacausticring.Theangularvelocityofgasinthevicinityofthecausticisgivenby vrot 4{4 andFig. 41 ,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) 42 witha=20kpc,p=1kpcandp0=5kpc.Theeectivepotentialissmootheventhoughitssecondderivativedivergesatr=aandr=a+p.Thecausticringradiusincreaseswithtime.Accordingtotheselfsimilarinfallmodel[ 92 93 ],a/t2 3+2 9.Consideraparticleofspecicangularmomentuml.Intheabsenceof 61
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43 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 43 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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Table51. ChannelIIIIIIIV Table52. ChannelIIIIIIIV Table53. ChannelIIIIIIIV Chapter3 ,theinnercaustichastheappearanceofaringwhentheinitialvelocitydistributionhasanetrotationalcomponent.Here,weassumethatthisisthecase.Itiseasiesttocalculatetheemissionmeasurefromatricuspcausticringbecauseoftheadvantagethatitcanbetreatedanalytically.Letusconsidercylindricalcoordinates(;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 ].Thecutodensityclosetothefoldsurface(nearR=0;Z=0)isthen2:15103=ap 51 showstheemissionmeasureaveragedoverasolidangle105srforthreedierentsetsofcausticparameters,asafunctionoflongitudel.bissetequalto0andweassumethatthecausticliesinthegalacticplane.Figures 51(a) 51(b) and 51(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. 51(a) ),thecutodensitywassetequalto400GeV/cm3everywhere,whilefortheplotswithp=0:1kpc(Figs 51(b) and 51(c) ),thecutodensitywassetequalto800GeV/cm3everywhere.Themagnitudesofforthehotspotsdependonthevaluesofthecausticparametersandalsoontheaveragingscale(herechosentobe105sr). Table4 showstheannihilationuxforthetwohotspotsfordierentvaluesoftheaveragingscale,forthecase(a=8:0;p=0:1;q=0:2)kpc.ItisworthpointingoutthatifthetriangularfeautureintheIRASmapisinterpretedastheimprintofthenearestcausticonthesurroundinggasasin[ 95 ],theimpliedcausticparametersareclosetowhatwehaveassumedforFig 51(b) Table54. 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(30MeV100MeV),28.9forBandII(100MeV1GeV),0.58forBandIII(1GeV10GeV)and0.012forBandIV(above10GeV).Wenowcomparethecausticsignalwiththeexpectedbackground.Sincethegammaraybackgroundfallsowithenergy(E2:7)fasterthantheannihilationsignal(E1:5),weexpectthatthebestchancefordetectionisatmoderatelyhighenergies.Atlowenergies,thebackgrounduxoverwhelmsthesignal,whileatveryhighenergies,thesignalisweak.Wechoosem=50GeVsincethischoicegivesthelargestux.ForthequantityN=m2,weusetheaveragevaluefortheband.Theaveragingscaleissetto105sr.Figures 52(a) and 52(b) showtheexpectedannihilationux(numberofphotonspersquaremeter,persteradian,peryear)asafunctionofanglelneartheplaneofthegalaxy(b=0)forthethreesetsofcausticparametersweconsidered,forEnergyBandsIIIandIVrespectively.Thisiscontrastedwiththeexpecteddiusegammaraybackground.Inprinciple,thepeaksinthesignalandthesharpfalloofuxarehelpfulinidentifyingtheannihilationsignal,particularlyforthemoreoptimisticcausticparametersandforsmallWIMPmasses.ForlargeWIMPmasses,theannihilationsignalissignicantlysmaller. 71
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(b) (c)Figure51. Emissionmeasureaveragedoverasolidangle=105sr,forthreesetsofcausticparameters. 72
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(b)Figure52. 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/orantisymmetricparts.WefoundthatwhenMismostlyantisymmetric(rotationalorcurlow),thecausticsaremadeupofellipticumbiliccatastrophesections,whileasymmetricM(irrotationalorgradientow)producescausticswithhyperbolicumbiliccatastrophesections.Wealsoshowedtheformationoftheswallowtailandbutterycatastrophes.Itispossibletosmoothlytransformthegradienttypecausticsintocurltypecausticsandviceversa. Chapter4 and Chapter5 dealwiththeastrophysicaleectsofdarkmattercaustics. Chapter4 exploresapossibleconnectionbetweenadarkmattercausticandtheMonocerosRingofstars.Theexistenceofadarkmattercausticintheplaneofthegalaxyat20kpcwaspredictedbytheselfsimilarinfallmodelofSikivie,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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A1(a) ),butitisnotstabletoperturbations.Ifthefunctionischangedtof(;x)=4+x2,fhasamaximumat=0fornegativex,asseeninFig. A1(b) .Thenewfunctionf(;x)isnotstableeitherbecausetheadditionofthetermyalterstheformoffnear=0.Thereisnowneitheramaximum,noraminimumat=0,asseeninFig. A1(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 A2(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 A3 .Thebifurcationsetdivides(x;y)spaceintotworegionsonewith3solutionsofandtheotherwith1solution.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 A4(a) representsarealrootof.Theyzcrosssectionofthebifurcationsetwithx<0isshowninFig. A4(b) ,indicatingthenumberofrealrootsineachofthethreeregions. 81
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54+3g2+2x+y=0(A{23)respectively.Toobtainthebifurcationset,letusconsidera2dimensionalcrosssectionwithgandxxed.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=0tosketchthebifurcationsetin2dimensionalcrosssection.Withx=0,Eq. A{24 and A{25 become d=203+6gdz d=204+6g2 82
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A5(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. A5(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.Thusthebifurcationsethastworegionsintheregionx2
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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 A7 showsthebifurcationset.Ithasthreedistinctregions. (b) (c)FigureA1. The4singularity.(a)4(b)4+x2(c)4+x2+y (b)FigureA2. Foldcatastrophe:Equilibriumsurfaceandbifurcationset. 86
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(b)FigureA3. Cuspcatastrophe.(a)Eachlinerepresentsavalueof.(b)Bifurcationset. (b)FigureA4. Swallowtailcatastrophe.(a)Eachlinerepresentsavalueof.(b)Bifurcationset. 87
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(b)FigureA5. Butterycatastrophe.(a)Eachlinerepresentsavalueof.(b)Bifurcationset. (b)FigureA6. Ellipticumbiliccatastrophe:Bifurcationset. (b)FigureA7. Hyperbolicumbiliccatastrophe:Bifurcationset. 88
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Chapter3 ,weshowedthatwhentheinitialvelocityeldisdominatedbyanetrotationalcomponent,theresultinginnercaustichastheappearanceofaring,whosecrosssectionhasthreecusps.Wecalledthisa`tricuspcausticring'.Inthelimitofaxialsymmetryaboutthezaxisandreectionsymmetryaboutthez=0plane,itispossibletoderiveananalyticexpressionforthecausticstructure,aswellastocalculatethedarkmatterdensityatpointsclosetothering.ThepropertiesofthecausticringwererstdescribedbyP.Sikivie[ 94 ].Herewegiveabriefdescription.Letusassumeanaxiallysymmetricow.Wehaveshownin Chapter3 thatnonaxiallysymmetricowsalsoproducecaustics.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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B1 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 B2 showsthecrosssection.Ifthezaxisisrescaledrelativetotheaxissoastomakethetricuspequilateral,thetricusphasaZ3symmetry[ 94 ]consistingofrotationsbymultiplesof2=3aboutthepointofcoordinates(c;zc)=(a+p=4;0).Thispointmaybecalledthecenterofthetricusp.ItisindicatedbyastarinFig. B2 .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=ddtmaybeextractedfromtheselfsimilarinfallmodel[ 92 93 ] ddti=fiviv2rot B{49 ),andusingvibi[ 94 ] 2tanhra p0:(B{52)p0isexpectedtobeofordera.Fig. B3 showsaqualitativesketchofI(r)andJ(r). 98
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Darkmattertrajectoriesformingatricuspring(incrosssection). FigureB2. Tricuspcausticinzcrosssection. (b)FigureB3. Modiedrotationcurve.(a)ThefunctionI(b)ThefunctionJ
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AravindNatarajanwasborninTrivandrum,India,in1978.HecompletedhisundergraduatestudiesatBangaloreUniversity,majoringinelectronicsandcommunicationengineering.HisundergraduatethesisworkwasonfrequencyselectiveprocessingofMPEG1AudioLayer1bitstreams.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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