© 2017 nilanjan banik -...
TRANSCRIPT
ROLE OF AXIONS IN STRUCTURE FORMATION IN THE UNIVERSE
By
NILANJAN BANIK
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2017
© 2017 Nilanjan Banik
Dedicated to Mita (ma), Narayan (baba) and Yahaira (wife).
ACKNOWLEDGMENTS
The completion of this dissertation has been possible due to the generous support,
encouragement and guidance by several people. First and foremost, I would like to thank my
advisor, Pierre Sikivie for his constant guidance, support and encouragement to finish my
research. Thank you for being so patient with me, and for keeping me up on my toes and
pushing me to achieve my goals. It has been a privilege working with you. I will miss asking
you questions during our meetings and your patient, meticulous and insightful answers. I would
like to thank my collaborators and friends in the Physics Department especially Adam, whose
everlasting positive attitude and encouragement will be greatly missed.
I would like to thank my family for their unwavering support. Thanks to my parents, Ma
and Baba for tolerating my tantrums since childhood and for being so patient with me. Thank
you for your constant encouragement and understanding. Thanks to my lovely wife Yahaira
for being so patient and understanding when I worked late hours. Thanks for putting up with
all the papers and books occupying the coffee table, dining table and the couch. Thanks for
single-handedly taking care of our pets and the house beside working and attending school
for a whole year when I was in Fermilab. Thanks to my in-laws Jose and Sonia, for being so
supportive and helpful. You were there whenever we needed you. Last but not least, thanks to
Calculus, Tensor, Mozart and Einstein for keeping company whenever I worked from home.
4
TABLE OF CONTENTSpage
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
CHAPTER
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1 Historical Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1.1 Galaxy Rotation Curves . . . . . . . . . . . . . . . . . . . . . . . . . 111.1.2 Gravitational Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . 131.1.3 Cosmic Microwave Background Observations . . . . . . . . . . . . . . 141.1.4 Bullet Cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2 Dark Matter Candidates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 BOSE-EINSTEIN CONDENSATION OF COLD DARK MATTER AXIONS . . . . . 17
2.1 QCD Axions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Production of Cold Axions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3 Axion-Axion Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.1 Particle Kinetic Regime . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.2 Condensed Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Axion BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.5 Axions Are Distinguishable from Other Dark Matter Candidates . . . . . . . . 28
3 CDM EVOLUTION IN THE LINEAR REGIME USING THE SCHRÖDINGER-POISSONEQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.1 Wavefunction Description of Linear Perturbations . . . . . . . . . . . . . . . 313.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4 VORTICES IN AXION BEC DARK MATTER . . . . . . . . . . . . . . . . . . . . 46
4.1 Statistical Mechanics of Rotating Systems . . . . . . . . . . . . . . . . . . . 484.1.1 Temperature, Chemical Potential and Angular Velocity . . . . . . . . . 494.1.2 The Self-Gravitating Isothermal Sphere Revisited . . . . . . . . . . . . 504.1.3 The Rotating Bose-Einstein Condensate . . . . . . . . . . . . . . . . . 53
4.1.3.1 Bose-Einstein condensation . . . . . . . . . . . . . . . . . . 534.1.3.2 Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.1.3.3 Superfluid 4He . . . . . . . . . . . . . . . . . . . . . . . . . 564.1.3.4 Quasi-collisionless particles . . . . . . . . . . . . . . . . . . 58
5
4.1.4 Thermalization and Vortex Formation . . . . . . . . . . . . . . . . . . 594.2 Axions, Baryons and WIMPs . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2.1 Axions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.2.1.1 Axion thermalization . . . . . . . . . . . . . . . . . . . . . 624.2.1.2 Caustic rings . . . . . . . . . . . . . . . . . . . . . . . . . 644.2.1.3 n× (z × n) . . . . . . . . . . . . . . . . . . . . . . . . . . 654.2.1.4 R(t)2
t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2.1.5 jmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.2.2 Baryons and WIMPs . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3 Comparison with Observations . . . . . . . . . . . . . . . . . . . . . . . . . 734.3.1 Baryonic Angular Momentum Distribution . . . . . . . . . . . . . . . 74
4.3.1.1 If the dark matter is all WIMPs . . . . . . . . . . . . . . . . 744.3.1.2 If the dark matter is axions, at least in part . . . . . . . . . 76
4.3.2 Enhanced Caustic Rings . . . . . . . . . . . . . . . . . . . . . . . . . 784.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5 RECOMBINATION ERA MAGNETIC FIELDS FROM AXION DARK MATTER . . 85
5.1 Origin of Magnetic Fields in the Universe . . . . . . . . . . . . . . . . . . . . 855.2 The Axion Bose-Einstein Condensate . . . . . . . . . . . . . . . . . . . . . . 875.3 Vorticity from Tidal Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.4 Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6 EVOLUTION OF VELOCITY DISPERSION ALONG COLD COLLISIONLESS FLOWS 92
6.1 Laboratory Detection of Dark Matter . . . . . . . . . . . . . . . . . . . . . . 926.2 A Cold Flow in One Dimension . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.2.1 Zero Velocity Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . 956.2.2 Small velocity dispersion . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.3 An axisymmetric cold flow in two dimensions . . . . . . . . . . . . . . . . . . 976.3.1 Zero velocity dispersion . . . . . . . . . . . . . . . . . . . . . . . . . 986.3.2 Small Velocity Dispersion . . . . . . . . . . . . . . . . . . . . . . . . 99
6.4 Velocity Dispersion Near a Caustic . . . . . . . . . . . . . . . . . . . . . . . 1036.4.1 Velocity Dispersion Near a Fold Caustic . . . . . . . . . . . . . . . . . 1036.4.2 Velocity Dispersion Near a Cusp Caustic . . . . . . . . . . . . . . . . 105
6.5 Applications to the Big Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.5.1 Upper Limit on the Big Flow Energy Dispersion . . . . . . . . . . . . 1096.5.2 Velocity Dispersion Ellipse of the Big Flow . . . . . . . . . . . . . . . 111
6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7 NEW ASTROPHYSICAL BOUNDS ON ULTRALIGHT AXIONLIKE PARTICLES . 114
7.1 Ultralight Axionlike Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . 1147.2 Bose-Einstein Condensation of ULALPs . . . . . . . . . . . . . . . . . . . . . 1167.3 Bound from ULALP Infall . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6
7.4 Bound from the Sharpness of the Nearby Caustic Ring . . . . . . . . . . . . . 1247.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
8 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7
LIST OF FIGURESFigure page
1-1 Galactic rotation curve of the galaxy NGC 6503 from [5] based on data from [6]showing the contribution from the disk (dashed lines) and gas (dotted line). . . . . 12
1-2 Left panel shows the HST image of the galaxy cluster CL0024. The blue arcs are aresult of strong gravitational lensing . . . . . . . . . . . . . . . . . . . . . . . . . 13
1-3 CMB anisotropy map as seen by Planck showing the temperature fluctuations atthe last surface of scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1-4 CMB power spectrum of temperature fluctuations in the cosmic microwave backgroundfrom Planck data at different angular scales on the sky . . . . . . . . . . . . . . . 14
1-5 A composite image of the Bullet Cluster showing the collision of galaxy clusters. . . 15
4-1 Reproduction of Fig. 4 in the article The angular momentum content of dwarf galax-ies: new challenges for the theory of galaxy formation by F.C. van den Bosch et al.[49]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4-2 Specific angular momentum distributions if the dark matter is axions, for variousvalues of the parameter υ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4-3 The ratio of maximum to average angular momentum if the dark matter is axions,as a function of the parameter υ. . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4-4 Images of the region around galactic coordinates (80, 0) from the IRAS 12 µm andPlanck 857 GHz observations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6-1 The curve with the cusp is the location of the fold caustic in the flow described byEqs. (6–51). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6-2 Relative position of the Sun and the 5th caustic ring in the Caustic Ring Model ofthe Milky Way halo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7-1 Panoramic view of the Milky Way Galactic plane from the IRAS experiment in the12 µm band. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
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Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy
ROLE OF AXIONS IN STRUCTURE FORMATION IN THE UNIVERSE
By
Nilanjan Banik
August 2017
Chair: Pierre SikivieMajor: Physics
One of the most profound puzzles of modern day science is to understand the nature of
dark matter. Axions that arise as a result of the solution to the CP problem in the theory of
strong interactions are one of the leading dark matter candidates. A remarkable property that
sets axions apart from other dark matter candidates is that they thermalize via gravitational
self-interactions to form a Bose-Einstein Condensate (BEC). Axion BEC behaves differently
from other dark matter candidates over time scales longer than their thermalization time.
This dissertation is a compilation of my published works, in which we have explored novel
ways axion dark matter can affect structure formation in the Universe. After a brief historical
review on dark matter we review in detail the Bose-Einstein condensation of axion dark
matter. We show that density perturbations in the early Universe can be given a wavefunction
description. We then show that the evolution of axion dark matter can only be described by
quantum field theory on time scales longer than their thermalization time. Next, we show
that imparting angular momentum to a system of axion BEC gives rise to vortices and these
vortices can combine to form bigger vortices. Axion BEC vorticity will give rise to vorticity in
baryons due to thermalization. We show that during recombination, the baryon vorticity will
seed primordial magnetic fields.
We then present a technique for obtaining the leading behavior of the velocity dispersion
near caustics. The results are used to derive an upper limit on the energy dispersion of the
local flow of dark matter from the sharpness of the nearby caustic, and a prediction for the
9
dispersions in its velocity components is obtained. Ultralight axionlike particles (ULALPs) have
been predicted in many String Theory based extensions of Standard Model and is a popular
dark matter candidate. We show that like QCD axions, ULALPs form a BEC. We then derive
lower mass bounds on ULALPs based on heating effect of infalling ULALPs on galactic disk
stars and the thickness of the nearby caustic ring.
10
CHAPTER 1INTRODUCTION
1.1 Historical Development
Dark matter is a form of non-baryonic matter that is as of today undetected in the
laboratory but is considered to be a key component in driving structure formation in the
Universe. The Standard Model of Cosmology, which is a result of gathering an enormous
amount of data on our Universe, tells us that 26.7% [1] of the total energy density and ∼
85% of the total mass in the Universe is due to dark matter. The first evidence of dark
matter was obtained by Fritz Zwicky in 1933 [2]. While studying the Coma Cluster, Zwicky
observed that the galaxies were moving much too fast for them to be gravitationally bounded
by the observed mass content of the cluster. He inferred the presence of some undetected
mass which he called “dunkle Materie” (German for dark matter), that was providing the
additional gravitational potential to hold the fast galaxies to the cluster. In 1970, Rubin and
Ford observed [3] that stars in disk galaxies were also moving with unexpectedly large speeds
around their galactic centers. These two discoveries along with several others (see [4] for a
detailed history) have led to a general consensus in the scientific community that dark matter
exists.
Today, the quest for understanding the nature of dark matter is one of the vanguards of
modern day science. We have overwhelming data suggesting its existence. Below I mention
four of the most compelling evidences:
1.1.1 Galaxy Rotation Curves
Disk and spiral galaxies have most of the visible mass (in the form of stars, dust and gas)
concentrated near the galactic center. Therefore, the rotation velocity of the stars is expected
to decrease as we go away from the galactic center if all the matter is visible matter. Instead,
we observe that the rotation velocity remains largely constant up to very large radii. This
indicates that there is more mass in the galaxies than that is observed. Stellar kinematic data
implies that the galaxies are enveloped inside enormous dark matter halos and that more than
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95% of the mass of the galaxies is due to dark matter. Fig. 1-1 shows the rotation velocity
as a function of the radial distance for the galaxy NGC 6503. The baryonic matter (gas and
disk) cannot alone explain the galactic rotation curve. Addition of a dark matter halo allows a
proper fit to the data.
Figure 1-1. Galactic rotation curve of the galaxy NGC 6503 from [5] based on data from [6]showing the contribution from the disk (dashed lines) and gas (dotted line). Thedark matter halo contribution is represented by the dashed-dotted line that isrequired to match the data. The data was obtained with the Westerbork SynthesisRadio Telescope.
12
1.1.2 Gravitational Lensing
According to the general theory of relativity, mass bends (or lenses) light. Therefore, a
distant light emitting source will appear distorted (lensed) if there is some mass on the path
of the incoming light. Such distortions can be studied to understand the mass distribution
between the light source and us, even if the intervening matter can not be directly seen.
Depending on how much the light bends, the image of the distant source may appear visibly
distorted; in some cases multiple images of the same source may appear (strong lensing), or
lensing may appear as a very small consistent distortion which can be detected only when a
large number of sources are statistically analyzed (weak lensing). Weak lensing studies from
the Sloan Digital Sky Survey (SDSS) have discovered that galaxies, including our own, are
larger and more massive than accounted for by the visible disk and gas content. An example
of how gravitational lensing can be used to predict the mass distribution is shown in Fig. 1-2.
The left panel shows the image of the galaxy cluster CL0024 as seen by the Hubble Space
Telescope (HST). The blue arcs are a result of strong lensing of the background galaxies
due to the foreground galaxy cluster. On the right, a computer reconstruction of the mass
distribution of the foreground cluster inferred by lensing observations made by [7]. The peaks
represent the individual galaxies and the smooth background component is the dark matter
contained in clusters in between the galaxies, that is not accounted for by the luminous mass
of the galaxies.
13
Figure 1-2. Left panel shows the HST image of the galaxy cluster CL0024. The blue arcs are aresult of strong gravitational lensing of the background galaxies by the foregroundgalaxy cluster. The right panel shows a computer reconstruction of the lens from[7]. The smooth background is the dark matter halos around the galaxies.
1.1.3 Cosmic Microwave Background Observations
Big Bang cosmology tells us that the universe started hot and dense and then expanded
and cooled. In the hot, dense conditions of the early universe, baryons existed as charged
particles and photons were tightly coupled to them via Thomson scattering. As the Universe
expanded and cooled the charged particles combined to form neutral atoms thereby becoming
transparent to the photons. These free streaming photons which traveled through the universe
as it expanded and cooled, make up the cosmic microwave background (CMB) we see today.
Figure 1-3 shows the CMB map from Planck, the colors represent the temperature fluctuations.
These fluctuations are the imprint of the acoustic oscillations in the baryon-photon fluid at
the time of recombination when the photons decoupled from baryons at the surface of last
scattering.
Figure 1-4 shows the CMB temperature power spectrum. The peaks are sensitive to the
total matter density and the total baryon density; the height of the second peak tells us that
∼ 5% of the total energy density is baryons and that of the second and third peaks tells us
that ∼ 26% of total energy density is dark matter.
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Figure 1-3. CMB anisotropy map as seen by Planck showing the temperature fluctuations atthe last surface of scattering. Blue color indicates cold spots and red colorindicates hot spots. Image credits: ESA and the Planck Collaboration.
1.1.4 Bullet Cluster
One of the most striking evidences for the existence of dark matter is the composite
image of the Bullet cluster shown in Figure 1-5. The figure shows the collision of two galaxy
clusters. The individual galaxies are seen in the optical image data (from NASA/STScI;
Magellan/U.Arizona/D.Clowe et al.[157]), red hues shows the cloud of hot X-ray emitting gas
in the clusters (from NASA/CXC/CfA/ [158]), the blue hues are from gravitational lensing
data showing the mass distribution in the galaxy clusters (from NASA/STScI; ESO WFI;
Magellan/U.Arizona/ D.Clowe et al. [157]). The combined mass of all the galaxies is far less
than the mass of the X-ray emitting hot gas clouds. Yet, most of the mass is concentrated
away from the gas clouds. The accepted explanation is that when the galaxy clusters collided,
the galaxies which basically don’t collide, passed through, as did the collisionless dark matter
halos around them. The intracluster gas however collided, which is indicated by the bullet
shaped shock front formed on the right.
15
Figure 1-4. CMB power spectrum of temperature fluctuations in the cosmic microwavebackground from Planck data at different angular scales on the sky. Red dots witherror bars are the Planck data. The green curve represents the standard model ofcosmology, ΛCDM.
1.2 Dark Matter Candidates
Over the years, several dark matter candidates have been proposed. The dark matter must
be non-baryonic, cold and collisionless. Non-baryonic means that the dark matter is not made
of ordinary atoms and molecules. Cold means that the primordial velocity dispersion of the
dark matter particles is sufficiently small, less than about 10−8 c today, so that it may be set
equal to zero as far as the formation of large scale structure and galactic halos is concerned.
By primordial velocity dispersion we mean the velocity dispersion that the dark matter particles
have even in the absence of density perturbations. An upper limit of order 10−8c on the
primordial velocity dispersion follows from the requirement that free streaming of the dark
matter particles does not erase density perturbations on the smallest scales on which they
are observed. Collisionless means that the dark matter particles have, in first approximation,
16
Figure 1-5. A composite image of the Bullet Cluster showing the collision of galaxy clusters.Red regions indicate the X-ray emitting hot gas (from NASA/CXC/CfA/ [158]),blue regions indicate the mass distribution (from NASA/STScI; ESO WFI;Magellan/U.Arizona/ D.Clowe et al. [157]).
only gravitational interactions. Particles with the required properties are referred to as ‘cold
dark matter’ (CDM). The leading CDM candidates are weakly interacting massive particles
(WIMPs), axions, and sterile neutrinos. WIMPs are motivated by supersymmetric extensions of
the Standard Model. Their mass is typically 100 GeV, and their primordial velocity dispersion
of order 10−12c. Axions are motivated by the Peccei-Quinn solution of the strong CP problem,
the puzzle within the Standard Model why the strong interactions are P and CP invariant. The
axion mass is thought to be of order 10−5 eV/c2 and the axion primordial velocity dispersion
of order 10−17c. Sterile neutrinos have mass of order a few keV/c2 and primordial velocity
17
dispersion of order 10−8c, at the limit of what is allowed. For this reason, sterile neutrinos are
sometimes called “warm dark matter”. Axions and WIMPs are definitely cold dark matter.
1.3 Overview
The rest of the dissertation will be focused on axion dark matter, how they behave
differently from other dark matter candidates, and how we can exploit these differences
to predict their astrophysical signatures. In Chapter 2 we review the recent work on axion
Bose-Einstein Condensation and show that axions thermalize via gravitational self-interactions
and form a BEC when the background photon temperature was around 500 eV. In Chapter 3,
we demonstrate that the wavefunction description of CDM is equivalent to the fluid description
and emphasize that the evolution of axion dark matter on time scales longer than their
thermalization time can only be described by quantum field equations and not by classical
field equations as is commonly assumed in the literature. In Chapter 4, we show that vortices
appear on imparting angular momentum to a system of axion BEC and these vortices combine
to give rise to bigger vortices. A dark matter halo made of axion BEC will have a ‘big vortex’
along the rotational axis. The presence of this vortex modifies the specific angular momentum
distribution of baryons and solves the “galactic angular momentum problem”. In Chapter
5, we show that the vortices in axion BEC can impart vorticity in baryons which can seed
primordial magnetic fields during the recombination period. In Chapter 6, we predict the upper
limit on the energy dispersion of the ‘Big Flow’ from the sharpness of its nearby caustic, and
also predict the dispersions in its velocity components. Finally in Chapter 7, we derive novel
constraints to put lower mass limits on ultralight axionlike particles (ULALPs).
18
CHAPTER 2BOSE-EINSTEIN CONDENSATION OF COLD DARK MATTER AXIONS
In this chapter, our main goal is to show that CDM axions form a Bose-Einstein
Condensate (BEC). We will first describe how QCD axions solve the strong CP problem,
followed by how cold axions are produced in the early Universe. Next, we will present how
axions self-interact and finally we will show that the cold axions thermalize via gravitational self
interactions to form a BEC.
2.1 QCD Axions
The theory of strong interactions, called ”quantum chromodynamics” or QCD for short,
has in its Lagrangian density a ”θ-term” [9, 10, 11, 12]
Lθ = θg2s
32π2GaµνGa
µν (2–1)
where θ is an angle between 0 and 2π, gs is the coupling constant for strong interactions,
and Gaµν is the gluon field tensor. The θ-term is a 4-divergence and therefore has no effects
in perturbation theory. However, it can be shown to have non-perturbative effects, and these
are important at low energies/long distances. Since Lθ is P and CP odd, QCD violates those
discrete symmetries when θ = 0. The strong interactions are observed to be P and CP
symmetric, and therefore θ must be small. The experimental upper bound on the electric
dipole moment of the neutron implies θ . 0.7 × 10−11 [13, 14]. In the Standard Model of
particle physics there is no reason for θ to be small; it is expected to be of order one. That θ is
less than 10−11 is a puzzle, referred to as the strong CP problem.
Peccei and Quinn proposed [15, 16] solving the strong CP problem by introducing a
global U(1)PQ symmetry which is spontaneously broken. When some conditions are met,
This chapter is a reproduction of [8] which appears as a chapter in the book “UniversalThemes of Bose-Einstein Condensation”, N.P. Proukakis, D.W. Snoke and P.B. LIttlewood(Eds.) (Cambridge University Press, 2017). Reproduced with permission from CambridgeUniversity Press.
19
the parameter θ is promoted to a dynamical field ϕ(x)fa
, where fa is the energy scale at which
U(1)PQ is spontaneously broken, and ϕ(x) the associated Nambu-Goldstone boson field. The
theory now depends on the expectation value of ϕ(x). The latter minimizes the QCD effective
potential. It can be shown that the minima of the QCD effective potential occur where θ = 0
[17]. The strong CP problem is thus solved if there is a Peccei-Quinn symmetry.
Axions are the quanta of the field ϕ(x) [18, 19]. Axions acquire mass due to the
non-perturbative effects that make QCD depend on θ. The axion mass is given by
ma ≃ 10−6eV
(1012 GeV
fa
)(2–2)
when the temperature is zero.
2.2 Production of Cold Axions
The equation of motion for ϕ(x) is
DµDµϕ(x) + V ′
a(ϕ(x)) = 0 (2–3)
where V ′a is the derivative of the effective potential with respect to the axion field and Dµ
is the covariant derivative with respect to space-time coordinates. The effective potential may
be written as
Va = ma(t)2f 2
a
[1− cos
(ϕ(x)
fa
)]. (2–4)
The axion mass is temperature and hence time-dependent. It reaches its zero-temperature
value, Eq. (2–2), at temperatures well below 1 GeV. At temperatures much larger than 1 GeV,
ma is practically zero. The axion field starts to oscillate [20, 21, 22] at a time t1 after the Big
Bang given by
m(t1) · t1 = 1 . (2–5)
Throughout we use units in which ~ = c = 1. t1 is approximately 2 × 10−7s(
fa1012 GeV
)1/3.The temperature of the primordial plasma at that time is T1 ≃ 1 GeV
(1012 GeV
fa
)1/6. The ϕ(x)
20
oscillations describe a population of axions called “of vacuum realignment”. Their momenta are
of order t−11 at time t1, and are red-shifted by the expansion of the universe after t1:
δp(t) ∼ 1
t1
R(t1)
R(t)(2–6)
where R(t) is the scale factor. As a result the axions are non-relativistic soon after t1, and
today they are extremely cold. The fact that they are naturally abundant, weakly coupled
and very cold, and that they solve the strong CP problem as well, makes axions an attractive
candidate for the dark matter of the universe.
The number of axions produced depends on various circumstances, in particular whether
inflation occurred before or after the phase transition in which U(1)PQ is spontaneously broken,
hereafter called the PQ phase transition [23]. If inflation occurs after, it homogenizes the
axion field within the observable universe. The initial value of the axion field may then be
accidentally close to the CP conserving value in which case the cold axion population from
vacuum realignment is suppressed. If inflation occurs before the PQ phase transition, there
is always a vacuum realignment contribution (because the axion field has random unrelated
values in different QCD horizons) and there are additional contributions from axion string
decay and axion domain wall decay. The number density of cold axions is
n(t) ≃ 4× 1047
cm3X
(fa
1012GeV
)5/3(R(t1)
R(t)
)3
(2–7)
where X is a fudge factor. If inflation occurs before PQ phase transition, X is of order 2 or
20 depending on whose estimate of the string decay contribution one believes. If inflation
occurs after the PQ phase transition, X is of order 12sin2 α1, where α1 = ϕ(t1)/fa is the initial
misalignment angle.
Cold axions are effectively stable because their lifetime is vastly longer than the age of the
universe. The number of axions is effectively conserved. The phase-space density of cold axions
21
implied by Eqs. (2–6) and (2–7) is [24]
N ∼ n(2π)3
4π3(mδv)3
∼ 1061X
(fa
1012 GeV
)8/3
. (2–8)
N is the average occupation number of those axion states that are occupied. Because
their phase-space density is huge and their number is conserved, cold axions may form a
Bose-Einstein condensate (BEC). The remaining necessary and sufficient condition for the
axions to form a BEC is that they thermalize. Assuming thermal equilibrium, the critical
temperature is [24, 25]
Tc(t) =
(π2n(t)
ζ(3)
)1/3
≃ 300 GeV X1/3
(fa
1012 GeV
)5/9R(t1)
R(t). (2–9)
The critical temperature is enormous because the cosmic axion density is so very high. The
formula given in Eq. (2–9) differs from the one for atoms because, in thermal equilibrium, most
of the non-condensate axions would be relativistic.
The question is whether the axions thermalize. This is not at all obvious since axions
are extremely weakly coupled. Note that for Bose-Einstein condensation to occur, it is not
necessary that full thermal equilibrium be reached. It is sufficient that the rate of condensation
into the lowest energy available state be larger than the inverse age of the universe. Whether
this happens is the issue which we address next.
2.3 Axion-Axion Interactions
Axions interact by λϕ4 self-interactions and by gravitational self-interactions. In this
section we discuss these two processes in detail and calculate the corresponding relaxation rates
[25]. Let us introduce a cubic box of volume V = L3, with periodic boundary conditions at the
surface. The axion field and its canonically conjugate field π(x, t) may be written as
ϕ(x, t) =∑n
(an(t)Φn(x) + a†n(t)Φ∗n(x)) (2–10)
π(x, t) =∑n
(−iωn)(an(t)Φn(x)− a†n(t)Φ∗n(x)) (2–11)
22
inside the box, where
Φn(x) =eipn·x√2ωnV
, (2–12)
pn = 2πn/L, n = (n1, n2, n3) where n1, n2, n3 are integers, and ωn =√p2n +m2. The
creation and annihilation operators satisfy canonical equal-time commutation relations
[an(t), a†n′(t)] = δn,n′ , [an(t), an′(t)] = 0. (2–13)
The Hamiltonian, including λϕ4 self-interactions, is
H =∑n
ωna†nan +
∑n1,n2,n3,n4
1
4Λn3n4
s n1,n2a†n1
a†n2an3an4 (2–14)
where
Λn3n4
s n1,n2=
−λ4m2V
δn1+n2,n3+n4 . (2–15)
The Kronecker-delta ensures 3-momentum conservation. When deriving Eq. (2–14), axion
number violating terms such as aaaa, a†a†a†a†, a†aaa, a†a†a†a are neglected. Indeed, in
lowest order they allow only processes that are forbidden by energy-momentum conservation.
In higher orders they do lead to axion number violating processes but only on times scales that
are vastly longer than the age of the universe.
The gravitational self-interactions of the axion fluid are described by Newtonian gravity
since we only consider interactions on sub-horizon scales. The interaction Hamiltonian is
Hg = −G2
∫d3x d3x′
ρ(x, t)ρ(x′, t)
|x− x′|(2–16)
where ρ(x, t) = 12(π2 +m2ϕ2) is the axion energy density. In terms of creation and annihilation
operators [25]
Hg =∑
n1,n2,n3,n4
1
4Λn3n4
g n1,n2a†n1
a†n2an3an4 (2–17)
where
Λn3n4
g n1,n2= −4πGm2
Vδn1+n2,n3+n4
(1
|pn1 − pn3 |2+
1
|pn1 − pn4 |2
). (2–18)
Hg must be added to the RHS of Eq. (2–14).
23
In summary, we have found that the axion fluid is described by a set of coupled quantum
harmonic oscillators. We now estimate the resulting relaxation rates. There are two different
regimes of relaxation depending on the relative values of the relaxation rate Γ and the energy
dispersion δω. The condition Γ << δω defines the “particle kinetic regime”, whereas Γ >> δω
defines the “condensed regime”. Most physical systems relax in the particle kinetic regime.
Axions on the other hand relax in the condensed regime.
2.3.1 Particle Kinetic Regime
When Γ << δω, the rate of change of the occupation numbers Ni (i = 1, 2, ..M) of M
coupled oscillators is given by
Nl =M∑
i,j,k=1
1
2|Λkl
ij |2[NiNj(Nl+1)(Nk+1)−(Ni+1)(Nj+1)NlNk]2πδ(Ωlkij ) + O(Λ3) (2–19)
where Ωklij = ωk + ωl − ωi − ωj, and the Λkl
ij are the relevant couplings, such as are given
in Eqs. (2–15) and (2–18) for axions. If we substitute the couplings due to λϕ4 interactions,
Eq. (2–15), and replace the sums over modes by integrals over momenta, we obtain [26, 25]
N1 =1
2ω1
∫d3p2
(2π)32ω2
d3p3(2π)32ω3
d3p4(2π)32ω4
λ2(2π)4δ4(p1 + p2 − p3 − p4)
× 1
2[(N1 + 1)(N2 + 1)N3N4 −N1N2(N3 + 1)(N4 + 1)] , (2–20)
where N1 ≡ Np1 and so forth. When the states are not highly occupied (N . 1), Eq. (2–20)
implies the standard formula for the relaxation rate
Γ ∼ NN
∼ nσδv (2–21)
where σ = λ2/64πm2 is the scattering cross-section due to λϕ4 interactions, n is the particle
density and δv is the velocity dispersion. On the other hand, when the states are highly
occupied (N >> 1), Eq. (2–20) implies
Γ ∼ nσδvN . (2–22)
24
The relaxation rate is enhanced by the degeneracy factor, which is huge (N ∼ 1061) in
the axion case. The process of Bose-Einstein condensation occurs as a result of scatterings
a(p1) + a(p2) ↔ a(p3) + a(p4) in which N1, N2 and N3 are of order the large degeneracy
factor N whereas N4 << N . Eq. (2–20) implies that, as a result of such scatterings, the
occupation number of the lowest available energy state grows exponentially with the rate given
in Eq. (2–22) [26, 24, 27].
In contrast to λϕ4 interactions, gravitational interactions are long-range. The cross-section
for gravitational scattering is infinite due to the contribution from very small angle (forward)
scattering. But forward scattering does not contribute to relaxation, whereas scattering
through large angles does contribute. (The issue does not arise in the case of λϕ4 interactions,
for which there is no peak in the differential cross-section for forward scattering and scattering
is generically through large angles.) The upshot is that Eqs. (2–21) and (2–22) are still valid
for estimating the relaxation rate by gravitational interactions in the particle kinetic regime
provided one uses for σ the cross-section for large angle scattering. That cross-section is finite
and equals
σg ∼4G2m2
(δv)4(2–23)
in order of magnitude.
2.3.2 Condensed Regime
When Γ >> δω, one cannot use Eq. (2–19) because the derivation of that equation
involves an averaging over time that is valid only when Γ << δω. Instead we will use the
equations
ial(t) = ωlal(t) +M∑i,j,k
1
2Λij
kla†kaiaj (2–24)
which follow directly from the Hamiltonian, Eq. (2–14). It is convenient to define cl(t) ≡
al(t)eiωlt, in terms of which Eq. (2–24) becomes
cl(t) = −iM∑i=1
1
2Λij
klc†kcicje
iΩklij t (2–25)
25
where Ωklij ≡ ωk + ωl − ωi − ωj, as before. Further, because the occupation numbers of the
occupied states are huge, we write cl as a sum of a classical part Cl and a quantum part dl
cl(t) = Cl(t) + dl(t) . (2–26)
The Cl are c-number functions of order√Nl describing the bulk of the axion fluid. They
satisfy the equations of motion
Cl(t) = −iM∑
i,j,k=1
1
2Λij
klC∗kCiCje
iΩklij t. (2–27)
The dl and d†l are annihilation and creation operators satisfying canonical commutation
relations. Quantum statistics plays the essential role in determining the outcome of relaxation
to be the Bose-Einstein distribution. However, we may use classical physics to estimate the
rate of relaxation. The relaxation rate is the inverse time scale over which Cl(t) changes by an
amount of order Cl(t).
The sum in Eq. (2–27) is dominated by those states that are highly occupied. Let K be
the number of such states. Using the fact that in the condensed regime Ωklij t << 1, we may
rewrite Eq. (2–27) as
Cl(t) ∼ −iK∑
i,j,k=1
1
2Λij
klC∗kCiCj . (2–28)
If we substitute Eq. (2–15) for λϕ4 interactions, we get
Cp1(t) ∼ iλ
4m2V
∑p2,p3
1
2C∗
p2Cp3Cp4 (2–29)
where p4 = p1 + p2 − p3 and the sum is restricted to the highly occupied states. The sum is
similar to a random walk with each step of order ∼ N 3/2 and the number of steps of order K2.
Hence
Cp ∼λ
4m2VKN 3/2 ∼ λ
4m2VNN 3/2 (2–30)
26
where we used K ∼ N/N . Since Cl ∼√N , the relaxation rate due to λϕ4 interactions in the
condensed regime is [24, 25]
Γλ ∼ 1
4nλm−2 (2–31)
where n = N/V is the number density of the particles in highly occupied states. Likewise the
relaxation rate for gravitational scattering is found to be
Γg ∼ 4πGnm2ℓ2 (2–32)
where ℓ = 1/δp is the correlation length of the particles.
The expressions estimating the relaxation rates in the condensed regime, Eqs. (2–31) and
(2–32), are very different from the expression, Eq. (2–22), in the particle kinetic regime. In
particular, in the condensed regime, the relaxation rate is first order in the coupling, whereas
it is second order in the particle kinetic regime. But the expressions are compatible. At the
boundary between the two regimes, where δω ∼ Γ, the two estimates agree. At that boundary,
up to factors of order 2 or so,
δvN ∼ δvn
(δp)3∼ n
m2δω∼ n
m2Γ. (2–33)
Substituting this into Eq. (2–22) yields Eq. (2–31). Similarly for the relaxation rate due to
gravitational self-interactions.
2.4 Axion BEC
For a system of particles to form a BEC, four conditions must be satisfied:
1. the particles must be identical bosons,
2. their number must be conserved,
3. they must be degenerate, i.e. the average occupation number N of the states that theyoccupy should be order 1 or larger,
4. they must thermalize.
When the four conditions are satisfied, a macroscopically large fraction of the particles go
to the lowest energy available state. It may be useful to clarify the notion of lowest energy
27
available state [36]. Thermalization involves interactions. By lowest energy available state
we mean the lowest energy state that can be reached by the thermalizing interactions. In
general the system has states of yet lower energy. For example, and at the risk of stating the
obvious, when a beaker of superfluid 4He is sitting on a table, the condensed atoms are in
their lowest energy available state. This is not their absolute lowest energy state since the
energy of the condensed atoms can be lowered by placing the beaker on the floor. In the case
of atoms, it is relatively clear what state the atoms condense into when BEC occurs. The
case of axions is more confusing because the thermalizing interactions, both gravity and the
λϕ4 self-interactions, are attractive and therefore cause the system to be unstable. When the
system is unstable, the restriction to the lowest energy available state is especially crucial.
We saw in the first two sections that, for cold dark matter axions, the first three
conditions for BEC are manifestly satisfied. In this section we show that the fourth condition
is satisfied as well [24, 25]. Cold axions will thermalize if their relaxation time τ is shorter than
the age t of the universe, or equivalently if their relaxation rate Γ ≡ 1/τ is greater than the
Hubble expansion rate H ∼ 1/t.
The cold axion energy dispersion is
δω(t) =(δp(t))2
m(t). (2–34)
In view of Eqs. (2–5) and (2–6), δω(t1) ∼ 1/t1. If axions thermalize at time t1, we have
Γ(t1) > 1/t1 and therefore the thermalization is in the condensed regime or at the border
between the particle kinetic and condensed regimes. After time t1, δω(t) < 1/t since m(t)
increases sharply for a period after t1 whereas (δp(t))2 ∝ R(t)−2 ∝ 1/t, since R(t) ∝√t in the
radiation dominated era. So after t1, axions can only thermalize in the condensed regime.
To see whether the axions thermalize by λϕ4 self-interactions at time t1, we may use
either Eq. (2–31) or (2–22). Both estimates yield Γλ(t1) ∼ H(t1) indicating that the axions
thermalize at time t1 by λϕ4 self-interactions but only barely so. After t1 we must use Eq. (2–
31). It informs us that Γλ(t)/H(t) ∝ R(t)−3t ∝ t−12 , i.e. that even if axions thermalize at
28
time t1 they stop doing so shortly thereafter. Nothing much changes as a result of this brief
epoch of thermalization since in either case, whether it occurs or not, the correlation length
ℓ(t) ≡ 1/δp(t) ∼ t1R(t)/R(t1).
To see whether the axions thermalize by gravitational self-interactions we use Eq. (2–32).
It implies
Γg(t)/H(t) ∼ 8πGnm2ℓ2t ∼ 5 · 10−7 R(t1)
R(t)
t
t1X
(fa
1012GeV
) 23
(2–35)
once the axion mass has reached its zero temperature value, shortly after t1. Gravitational
self-interactions are too slow to cause thermalization of cold axions near the QCD phase
transition but, because Γg/H ∝ R−1(t)t ∝ R(t), they do cause the cold axions to thermalize
later on. The RHS of Eq. (2–35) reaches one at a time tBEC when the photon temperature is
of order
TBEC ∼ 500 eV X
(fa
1012 GeV
) 12
. (2–36)
The axions thermalize then and form a BEC as a result of their gravitational self-interactions.
The whole idea may seem far-fetched because we are used to think that gravitational
interactions among particles are negligible. The axion case is special, however, because
almost all particles are in a small number of states with very long de Broglie wavelength, and
gravity is long range.
Systems dominated by gravitational self-interactions are inherently unstable. In this
regard the axion BEC differs from the BECs that occur in superfluid 4He and dilute gases.
The axion fluid is subject to the Jeans gravitational instability and this is so whether the axion
fluid is a BEC or not [24]. The Jeans instability causes density perturbations to grow at a
rate of order the Hubble rate H(t), i.e. on a time scale of order the age of the universe at
the moment under consideration. Each mode of the axion fluid is Jeans unstable. We showed
however that, after tBEC, the thermalization rate is faster than the Hubble rate. The rate
at which quanta of the axion field jump between modes is faster than the rate at which the
Jeans instability develops. So the modes are essentially frozen on the time scale over which the
axions thermalize.
29
Finally, we comment on a misapprehension that appears in the literature. The axions do
not condense in the lowest momentum mode p = 0. Condensation into the p = 0 state would
mean that the fluid becomes homogeneous and at rest. Of course this is not what happens in
the axion case since the axion fluid is Jeans unstable. Despite a common misconception, it is
not a rule of BEC that the particles condense into the p = 0 state. The rule instead is that
they condense to the lowest energy available state, as defined earlier. Only in empty space, and
only if the total linear momentum and the total angular momentum of the particles are zero,
is the lowest energy state a state of zero momentum. It should be obvious that the particles
do not condense in the p = 0 state if they are moving or rotating. Nonetheless, Bose-Einstein
condensation occurs.
2.5 Axions Are Distinguishable from Other Dark Matter Candidates
For a long time, it was thought that axions and the other proposed forms of cold dark
matter behave in the same way on astronomical scales and are therefore indistinguishable
by observation. Axion BEC changed that. On time scales longer than their thermalization
time scale τ , axions almost all go to the lowest energy state available to them. The other
dark matter candidates, such as weakly interacting masssive particles (WIMPs) and sterile
neutrinos, do not do this. It was shown in Ref. [24] that, on all scales of observational interest,
density perturbations in axion BEC behave in exactly the same way as those in ordinary
cold dark matter provided the density perturbations are within the horizon and in the linear
regime. On the other hand, when density perturbations enter the horizon, or in second order of
perturbation theory, axions generally behave differently from ordinary cold dark matter because
the axions rethermalize so that the state most axions are in tracks the lowest energy available
state.
A distinction between axions and the other forms of cold dark matter arises in second
order of perturbation theory, in the context of the tidal torquing of galactic halos. Tidal
torquing is the mechanism by which galaxies acquire angular momentum. Before they fall
onto a galactic halo, the axions thermalize sufficiently fast that the axions that are about to
30
fall into a particular galactic gravitational potential well go to their lowest energy available
state consistent with the total angular momentum they acquired from nearby protogalaxies
through tidal torquing [36]. That state is a state of net overall rotation, more precisely a state
of rigid rotation on the turnaround sphere. In contrast, ordinary cold dark matter falls into a
galactic gravitational potential well with an irrotational velocity field [28]. The inner caustics
are different in the two cases. In the case of net overall rotation, the inner caustics are rings
[29] whose cross-section is a section of the elliptic umbilic D−4 catastrophe [30], called caustic
rings for short. If the velocity field of the infalling particles is irrotational, the inner caustics
have a ‘tent-like’ structure which is described in detail in ref. [28] and which is quite distinct
from caustic rings. Evidence was found for caustic rings. A summary of the evidence is given
in ref. [31]. Furthermore, it was shown in ref. [32] that the assumption that the dark matter
is axions explains not only the existence of caustic rings but also their detailed properties, in
particular the pattern of caustic ring radii and their overall size.
Vortices appear in the axion BEC as it is spun up by tidal torquing. The vortices in
the axion BEC are attractive, unlike those in superfluid 4He and dilute gases. Hence a large
fraction of the vortices in the axion BEC join into a single big vortex along the rotation axis of
the galaxy [36]. Baryons and ordinary cold dark matter particles that may be present, such as
WIMPs and/or sterile neurtinos, are entrained by the axion BEC and acquire the same velocity
distribution. The resulting baryonic angular momentum distribution gives a good qualitative
fit [36] to the angular momentum distributions observed in dwarf galaxies [33]. This resolves a
long-standing problem with ordinary cold dark matter called the ”galactic angular momentum
problem” [34, 35]. A minimum fraction of cold dark matter must be axions to explain the data.
That fraction is of order 35% [36]. A detailed account of this will be presented in Chapter. 4.
31
CHAPTER 3CDM EVOLUTION IN THE LINEAR REGIME USING THE SCHRÖDINGER-POISSON
EQUATIONS
In the previous chapter we established that axions form a BEC when the photon
temperature is around 500 eV. In this chapter, we will obtain the solutions of the Schrödinger
-Poisson equations that describe the homogeneous expanding Friedmann matter-dominated
universe and density perturbations therein. Each solution corresponds to a possible quantum
-mechanical state of dark matter axions. However, it should be kept in mind that the
gravitational and other self-interactions of the axions cause them to jump between those
states. In the second section of this chapter, we discuss the relationships between descriptions
of cold dark matter in terms of a pressureless fluid, in terms of a wavefunction, of a classical
scalar field, and a quantum scalar field. We identify the regimes where the various descriptions
coincide and where they differ.
Cold collisionless dark matter may be described in the linear regime of the evolution
of density perturbations as a pressureless fluid. This is the description of cold dark matter
in calculations of the cosmic microwave background anisotropies [38]. Since cold dark
matter plays an important role in this context and the calculations agree very well with
the observations, the pressureless fluid description has high credibility. To obtain the cosmic
microwave background anisotropies a full general relativistic treatment [39] is necessary
because the relevant evolution occurs in part on length scales of order the horizon. However,
on length scales much less than the horizon (i.e. for wavevectors much larger than the Hubble
rate) dark matter density perturbations are correctly described by Newtonian gravity. Linear
Newtonian perturbation theory [42] is simple, well understood and agrees with the general
relativisitic description on length scales much less than the horizon, where many of the
This chapter is a reproduction of [37], with permission from The American PhysicalSociety.
32
interesting phenomena in large scale structure formation occur. It is therefore a very useful
tool.
L. Widrow and N. Kaiser [43] pointed out that, on scales much less than the horizon, cold
collisionless dark matter can be described by a wavefunction satisfying the Schrödinger-Poisson
equations. As is discussed below, the wavefunction description is in many ways more powerful
than the pressureless fluid description. It allows the introduction of velocity dispersion whereas
the fluid description allows none. It can be used to describe multi-streaming and caustics in the
non-linear regime, whereas the fluid description breaks down in that regime. Indeed, Widrow
and Kaiser carried out numerical simulations of structure formation using a wavefunction
satisfying the Schrödinger equation, in lieu of N bodies satisfying Newton’s force law equation.
Several such simulations have been carried out [44].
In the following section, we reproduce the results of Newtonian linear perturbation theory
using a wavefunction solving the Schrödinger-Poisson equations. As far as we are aware,
this had not been done before although related work, also using the Schrödinger equation to
analyze the growth of density perturbations in the early universe, can be found in refs. [45]
and [46]. One of our motivations is to show that the formalism does indeed work as expected.
However, our main motivation is to prepare the ground for an in-depth study of the dynamical
evolution of axion dark matter.
3.1 Wavefunction Description of Linear Perturbations
Consider a fluid composed of a huge number N of particles that are all in the same
quantum-mechanical state. The wavefunction ψ(r, t) for the state satisfies the Schrödinger
equation
i∂tψ(r, t) =(− 1
2m∇2 +mΨ(r, t)
)ψ(r, t) , (3–1)
where m is the particle mass and Ψ(r, t) is the Newtonian gravitational potential. In this
section, we set ~ = c = 1. The density of the fluid is
n(r, t) = Nψ∗(r, t)ψ(r, t) . (3–2)
33
Let us assume that the only kind of matter present is the fluid of particles. The gravitational
potential is then given by the Poisson equation
∇2Ψ = 4πGmn(r, t) . (3–3)
The fluid density satisfies the continuity equation
∂tn+ ∇ · j = 0 , (3–4)
where
j =N
2m(ψ∗∇ψ − ψ∇ψ∗) . (3–5)
The fluid velocity v(r, t) is defined by j(r, t) ≡ n(r, t)v(r, t). If we write ψ(r, t) =√n(r, t)eiβ(r,t), then
v =1
m∇β . (3–6)
The velocity field satisfies the Euler-like equation
∂tv + (v · ∇)v = −∇Ψ− ∇q (3–7)
where
q = − 1
2m2
∇2√n√n
. (3–8)
q is commonly referred to as “quantum pressure”. Eqs. (3–4) and (3–7) follow from Eq. (3–1).
We want to use Eqs. (3–1) and (3–3) to describe the evolution of density perturbations
in an otherwise homogeneous Friedmann universe. The universe may be open or closed, or in
between. However, because our description uses Newtonian gravity, the cosmological constant
is set equal to zero.
The wavefunction describing the homogeneous universe is
ψ0(r, t) =√n0(t)e
i 12mH(t)r2 , (3–9)
34
where H(t) is the Lemaître-Hubble expansion rate. Indeed Eqs. (3–6) and (3–9) imply
v = Hr . (3–10)
The imaginary part of the Schrödinger equation is satisfied provided
∂tn0 + 3Hn0 = 0 (3–11)
and its real part is satisfied provided
Ψ0 = −1
2(∂tH +H2)r2 . (3–12)
The Poisson equation then implies the acceleration equation
∂tH +H2 = −4πG
3mn0(t) . (3–13)
The continuity and acceleration equations may be combined as usual to yield the Friedmann
equation
H(t)2 +K
a(t)2=
8πG
3mn0(t) (3–14)
where K = −1, 0,+1 depending on whether the universe is open, critical or closed, and a(t) is
the scale factor defined by H(t) = aa.
We now consider perturbations about this background:
ψ(r, t) = ψ0(r, t) + ψ1(r, t) . (3–15)
The perturbation is Fourier transformed in terms of comoving wavevector k as follows:
ψ1(r, t) = ψ0(r, t)
∫d3k ψ1(k, t)e
i k·ra(t) . (3–16)
Likewise the perturbation to the gravitational potential
Ψ1(r, t) =
∫d3k Ψ1(k, t)e
i k·ra(t) . (3–17)
35
The Schrödinger-Poisson equations expanded to linear order in the perturbations imply
i∂tψ1 = − 1
2m∇2ψ1 +m(Ψ0ψ1 +Ψ1ψ0) , (3–18)
and
∇2Ψ1 = 4πGm(ψ∗0ψ1 + ψ0ψ
∗1) . (3–19)
It is useful to introduce the functions
δ(k, t) ≡ ψ1(k, t) + ψ∗1(−k, t) , (3–20)
η(k, t) ≡ ψ1(k, t)− ψ∗1(−k, t) , (3–21)
in terms of which Eqs. (3–18) and (3–19) become
i∂tδ(k, t) − k2
2ma2(t)η(k, t) = 0 , (3–22)
i∂tη(k, t) +
(8πGm2n0(t)
k2a2(t)− k2
2ma2(t)
)δ(k, t) = 0 . (3–23)
These can be combined into one, second order differential equation for δ(k, t):
∂2t δ(k, t) +4
3t∂tδ(k, t)− 4πGρδ(k, t) +
k4
4m2a4(t)δ(k, t) = 0 , (3–24)
where ρ = mn0. The Fourier components of the perturbation to the wavefunction are given by
ψ1(k, t) =1
2δ(k, t) + i
ma(t)2
k2∂tδ(k, t) (3–25)
in terms of the solutions to Eq. (3–24).
The perturbation to the number density is
n1(r, t) = |ψ0(r, t)|2∫d3k
(ψ1(k, t) + ψ∗
1(−k, t))ei
k·ra(t) , (3–26)
and so the density contrast
δ(r, t) =n1(r, t)
n0(r, t)=
∫d3k δ(k, t)ei
k·ra(t) . (3–27)
36
By expanding
ψ(r, t) =√n0(t) + n1(r, t)e
i
(β0(r,t)+β1(r,t)
), (3–28)
one finds that
β1(r, t) =1
2i
∫d3k η(k, t)ei
k·ra(t) . (3–29)
Hence Eq. (3–22) implies
v(k, t) =ia(t)k
k · k∂tδ(k, t) , (3–30)
which is the same relationship between the velocity perturbation and the density contrast as in
the standard description of cold dark matter in terms of a pressureless fluid. Eq. (3–24) is also
the standard second order differential equation governing the evolution of the density contrast,
except for the last term. It arises due to quantum pressure in Eq. (3–7) and produces a Jeans
length [40, 41]
ℓJ = (16πGρm2)−14 = 1.02 · 1014cm
(10−5eV
m
) 12
(10−29g/cm3
ρ
) 14
. (3–31)
For k > a(t)ℓJ
, the Fourier components of the density perturbations oscillate in time. For
k << a(t)ℓJ
, the most general solution of Eq. (3–24) is
δ(k, t) = A(k)
(t
t0
)2/3
+B(k)
(t0t
), (3–32)
where A(k) and B(k) are the amplitudes of the growing and decaying modes, respectively. On
distance scales much larger than the Jeans length, the wavefunction description coincides in all
respects with the fluid description.
Let us mention briefly that the wavefunction can also describe rotational modes, provided
vortices are introduced. In a region where ∇ × v = 0, the vortices have the direction of ∇ × v
and have density (number of vortices per unit area) m2π|∇ × v|. By Kelvin’s theorem, the
vortices must move with the fluid. Therefore, in an expanding universe, the density of vortices
decreases as a(t)−2. Hence v ∝ a(t)−1 for rotational modes, which is again the usual result.
The vortices present in axion dark matter are discussed in Chapter 4.
37
3.2 Discussion
We saw in the previous section that density perturbations in the early universe may
be described by a wavefunction which solves the Schrödinger-Poisson equations and that
on length scales large compared to the Jeans length, Eq. (3–31), the resulting description
coincides with that in terms of a pressureless fluid. It is our purpose in the present section to
place this result in a wider physical context.
First let us state that, although it appears that the wavefunction description had not been
explicitly given before, it is no surprise that it exists since the Schrödinger equation implies
the continuity equation and the Euler-like equation (3–7). These two equations are the basic
equations describing a fluid. The only difference is the quantum pressure term in Eq. (3–7)
but that term is unimportant on distance scales large compared to the de Broglie wavelength.
The Jeans length of Eq. (3–31) can be viewed as the de Broglie wavelength of the minimum
energy state in a region of density ρ. Indeed in such a region, the gravitational potential is
Ψ = 2π3Gρr2 and hence the energy of a trial wavefunction of width b is of order
E(b) ∼ 1
2mb2+
2π
3Gρmb2 . (3–33)
E(b) reaches its minimum for b ∼ (4π3Gρm2)−
14 ∼ ℓJ .
However the mathematical equivalence of the two descriptions hides important physical
differences. This is perhaps best illustrated by an example. Consider the wavefunction
ψ(r, t) = A(eik·r + e−ik·r
)e−iωt . (3–34)
where A is a constant and ω = k·k2m
. It solves the Schrödinger equation for a free particle. The
fluid with N particles in the state of wavefunction ψ(x, t) has two flows, both with density
n1 = n2 = N |A|2, and with velocities v1 = km
and v2 = − km
. On the other hand, Eqs. (3–2)
and (3–6) map ψ(r, t) onto a fluid whose density is n(r) = 4|A|2 cos2(k · r) and whose velocity
v = 0. The two descriptions are mathematically equivalent in the sense that n(r, t) and v(r, t)
satisfy Eqs. (3–2) and (3–7) because ψ(r, t) satisfies Eq. (3–1). But the two descriptions are
38
not physically equivalent. They are physically equivalent only if we average over distances large
compared to the wavelength 2πk
and if we ignore the velocity dispersion ∆v = km
. The spatial
averaging is justified in the limit k → ∞. Ignoring the velocity dispersion is justified in the
limit, km
→ 0. The two limits are compatible only if m → ∞. This indicates that the physical
differences between the wavefunction and fluid description disappear completely only in the
limit where the dark matter particle is very heavy.
The fluid description never allows velocity dispersion since the velocity field v(r, t) has a
single value at every point. In contrast, the wavefunction description allows velocity dispersion
and multi-streaming. The wavefunction description is richer therefore. It can describe
everything that a fluid describes but the reverse is not true. Whether either description is
correct depends on the situation at hand.
Consider a dark matter particle with properties typical of a WIMP candidate: m ∼ 100
GeV, density today n0 ∼ 10−6/cm3, and primordial velocity dispersion today δv0 ∼ 10−12.
The de Broglie wavelength associated with the primordial velocity dispersion is of order 10−3
cm, much smaller than the average interparticle distance of order 1 m. The particles are highly
non-degenerate therefore. Provided that their primordial velocity dispersion is in fact irrelevant
to whatever phenomenon is under study (free streaming would be an exception since it is a
direct result of primordial velocity dispersion), the particles can be described as a pressureless
fluid. They can also be described by a wavefunction. The wavefunction description will in
almost all respects be equivalent to the pressureless fluid description but, unlike the latter, it
allows the inclusion of velocity dispersion and its associated effects. It is also applicable to the
non-linear regime, after shell crossing, when the fluid description in terms of its single velocity
field v breaks down. The wavefunction description is more powerful because it packs more
information. The wavefunction varies on a length scale of order 10−3 cm in the example given.
The fluid description is far coarser.
Next consider a dark matter candidate typical of axions or axion-like particles: spin
zero, m ∼ 10−5 eV, density today n0 ∼ 109/cm3, and primordial velocity dispersion today
39
δv0 ∼ 10−17. The de Broglie wavelength associated with the primordial velocity dispersion is of
order 1018 cm. The axion fluid is highly degenerate. The average occupation number of those
states that are occupied is huge, of order 1061. This suggests that axion dark matter is well
described by a classical scalar field φ(r, t). The remainder of this section considers whether this
is so.
A classical scalar field satisfies the Klein-Gordon equation
− c2Dµ∂µφ+ ω20φ+
λ
3!φ3 = 0 (3–35)
where Dµ is the covariant derivative of general relativity. We allow the presence of a
self-interaction Lφ4 = − λ4!φ4 in the action density. First, let us emphasize that the classical
field theory, Eq. (3–35), has no notion of axion. The axion is the quantum of the quantized
scalar field which we call Φ(r, t). There is no more notion of axion in Eq. (3–35) than there is
a notion of photon in Maxwell’s equations. Also there is no notion of mass since the mass m
is the energy of an axion, divided by c2. Henceforth, for the sake of clarity, we no longer set
~ and c equal to one. ω0 in Eq. (3–35) is not the axion mass but the oscillation frequency of
small perturbations in the classical scalar field in the infinite wavelength limit.
In the Newtonian limit of general relativity, the metric is g00 = −c2 − 2Ψ, g0i = 0,
gij = δij. Eq. (3–35) becomes then:
(∂2t − c2∇2 + ω20)φ+
λ
3!φ3 − (
2
c2Ψ∂2t + ∇Ψ · ∇+
1
c2∂tΨ∂t)φ = 0 . (3–36)
We obtain the non-relativistic limit of this equation by setting
φ(r, t) =√2 Re[e−iω0tϕ(r, t)] (3–37)
and neglecting Ψ versus c2, ∂tϕ versus ω0ϕ, ∂tΨ versus ω0Ψ, and dropping terms proportional
to e2iω0t and e−2iω0t which indeed oscillate so fast as to effectively average to zero. Eq. (3–36)
becomes then
i∂tϕ = − c2
2ω0
∇2ϕ+λ
8ω0
|ϕ|2ϕ+ω0
c2Ψϕ . (3–38)
40
To obtain the Schrödinger equation
i~∂tψ = − ~2
2m∇2ψ + V (r, t)ψ , (3–39)
substitute
ϕ(r, t) =
√~ω0
ψ(r, t) . (3–40)
in Eq. (3–38) and set m = ~ω0
c2. The potential energy in Eq. (3–39) is given, in the sense of
mean field theory, by
V (r, t) = mΨ(r, t) +~4λ
8m2c4|ψ(r, t)|2 . (3–41)
The Newtonian limit of Einstein’s equation is the Poisson equation, Eq. (3–3). The non-linear
version of Schrödinger’s equation obtained by substituting Eq. (3–41) into Eq. (3–39) is
commonly called the Gross-Pitaevskii equation.
So the Schrödinger equation describes the dynamics of a classical scalar field in the
non-relativistic limit. This result is not new of course. We reproduced it here to prepare the
ground for the actual question we want to discuss, namely whether dark matter axions (or
axion-like particles) are described by the Schrödinger-Poisson equations. Clearly, if axions
are described by a classical scalar field, the answer is yes as we just saw. But the axion is a
quantum field. It may behave like a classical field some of the time or perhaps even all the
time, but this is something that has to be proved. It cannot be merely assumed.
Inside a cubic box of volume V = L3 with periodic boundary conditions, the quantum
axion field may be expanded (see for example ref. [25])
Φ(r, t) =∑n
√~
2ωnV[an(t)e
i~ pn·r + a†n(t)e
− i~ pn·r] (3–42)
where n = (n1, n2, n3) with nk (k = 1, 2, 3) integers, pn = 2π~Ln, ω = c
~
√p · p+ c2m2. The
an and a†n are annihilation and creation operators satisfying canonical equal-time commutation
relations:
[an(t), a†n′(t)] = δn,n′ , [an(t), an′(t)] = 0 . (3–43)
41
The classical field limit is the limit where ~ → 0 with ~N held fixed, where N is the
quantum occupation number of the state described by a particular solution of the classical field
equations. The Hamiltonian for the quantum field Φ(r, t) which satisfies Eqs. (3–38) and (3–3)
in the classical field limit is [25]
H =∑n
~ωn a†nan +
∑n1,n2,n3,n4
1
4~Λn3,n4
n1,n2a†n1
a†n2an3an4 (3–44)
where Λn3,n4
n1,n2is the sum of two terms:
Λn3,n4
n1,n2= Λ n3,n4
s n1,n2+ Λ n3,n4
g n1,n2. (3–45)
The first term
Λ n3,n4
s n1,n2= +
λ~3
4m2c4Vδn1+n2,n3+n4 (3–46)
is due to the λΦ4 type self-interactions. The second term
Λ n3,n4
g n1,n2= −4πGm2~
Vδn1+n2,n3+n4
(1
|pn1 − pn3 |2+
1
|pn1 − pn4|2
)(3–47)
is due to the gravitational self-interactions. The Heisenberg equations of motion are
ian1 = −1
~[H, an1 ] = ωn1an1 +
1
2
∑n2,n3,n4
Λn3,n4
n1,n2a†n2
an3an4 . (3–48)
The classical field may be likewise expanded
ϕ(r, t) =∑n
√~
2ωnV[An(t)e
i~ pn·r + A†
n(t)e− i
~ pn·r] . (3–49)
The Fourier components An(t) satisfy
iAn1 = ωn1An1 +1
2
∑n2,n3,n4
Λn3,n4
n1,n2A∗
n2An3An4 . (3–50)
Eqs. (3–48) and (3–50) look similar but, as we will see, their physical implications are different
because the an(t) are operators whereas the An(t) are c-numbers.
42
Let us define the operator
Nn(t) = a†n(t) an(t) , (3–51)
i.e. the occupation number at time t of the state labeled n. It was shown in ref. [25] that the
Hamiltonian of Eq. (3–44) implies the following operator evolution equation
Ni =∑
k,i,j=1
1
2|Λkl
ij |2 [NiNj(Nl + 1)(Nk + 1)−NlNk(Ni + 1)(Nj + 1)] 2πδ(ωi+ωj−ωk−ωl) .
(3–52)
To remove unnecessary clutter, we replaced nj by j. The derivation of Eq. (3–52) assumes
only that the energy dispersion δω of the highly occupied states is much larger than the
relaxation rate Γ = 1τ. τ is the relaxation time, defined as the time scale over which the
distribution Nj changes completely. When δω >> Γ, the system is said to be in the
“particle kinetic” regime. The same derivation that yields Eq. (3–52) but applied to the
classical counterparts
Nn(t) = A∗n(t)An(t) (3–53)
yields
Ni =∑
k,i,j=1
1
2|Λkl
ij |2 [NiNjNl +NiNjNk −NlNkNi −NkNlNj] 2πδ(ωi + ωj − ωk − ωl) ,
(3–54)
again in the particle kinetic regime. Eqs. (3–52) and (3–54) are clearly different, and they
imply different outcomes.
Consider the process i + j → k + l where two quanta, initially in states i and j move
to states k and l. Assuming Λijkl = 0, this process always occurs in the quantum theory when
the initial states are occupied. In the classical theory, the corresponding process occurs only
if, in addition, at least one of the final states is occupied. In particular the scattering of two
waves does not happen in the classical theory of Eq. (3–38) if the only waves present are
the two waves in the initial state. The quantum theory behaves differently because the final
state oscillators have zero point oscillations. Incidentally, this observation shows that the
43
oft repeated statement that the quantum and classical theories differ only by loop effects is
incorrect.
After a sufficiently long time, the classical and quantum systems thermalize and reach an
equilibrium distribution. The time scale of thermalization of the classical system is of the same
order of magnitude as that of the quantum system [25] but the outcomes of thermalization are
different. In the quantum case, Eq. (3–52) implies that the equilibrium distribution Nj is
such that
NiNj(Nl + 1)(Nk + 1)−NlNk(Ni + 1)(Nj + 1) = 0 (3–55)
for every quartet of states such that ωi + ωj = ωk + ωl. Let us call ϵ = ~ω, and rewrite
Eq. (3–55) as
(1 +1
Ni
)(1 +1
Nj
) = (1 +1
Nk
)(1 +1
Nl
) (3–56)
whenever ϵi + ϵj = ϵk + ϵl. Eq. (3–56) is solved by
ϵi = C ln
[1 +
1
Ni
](3–57)
where C is a constant. Upon identifying C = kBT , this is seen to be the Bose-Einstein
distribution
Ni =1
eϵi
kBT − 1. (3–58)
On the other hand Eq. (3–54) implies that the equilibrium distribution Nj for the classical
case satisfies
(Ni +Nj)NkNl = NiNj(Nk +Nl) (3–59)
whenever ϵi + ϵj = ϵk + ϵl. Eq. (3–59), which may be rewritten
1
Ni
+1
Nj
=1
Nk
+1
Nl
, (3–60)
is solved by1
Ni
= C ′ϵi . (3–61)
44
Upon identifying C ′ = 1kBT
, we have
Niϵi = kBT (3–62)
which is indeed the standard result for classical oscillators at temperature T : each oscillator
has energy kBT on average.
We conclude that axion dark matter is not described by a classical field when it
thermalizes. Interactions are seen to have a dual role. They determine the behaviour of
the classical field as described by Eq. (3–38), or Eqs. (3–50) and (3–54) which follow directly
from Eq. (3–38). Eq. (3–38) has a set of solutions which we may label ϕα(r, t). The solutions
describe the states that the axions may occupy in the quantum theory. In the quantum theory,
however, the interactions have the additional role of causing transitions between the various
states ϕα(r, t). Indeed if there were no such transitions the outcomes of thermalization in the
quantum and classical theories would be the same. We just saw that they are not.
Let τ be the time scale over which the distribution Nα of the axions, over the states
described by the classical solutions ϕα(r, t), changes completely. We call τ the relaxation
or thermalization time scale. Note that full thermalization only happens generally on a time
scale much longer than τ . (The time scale for “full” thermalization depends on the degree
of thermalization required and is therefore less robustly defined than τ .) On time scales
short compared to τ , axion dark matter behaves as a classical field because only relatively
few transitions take place between the states described by the classical solutions ϕα(r, t).
On time scales long compared to τ , the axions are not described by a classical field because
their distribution over those states changes completely. On time scales long compared to τ
dark matter axions form a Bose-Einstein condensate (BEC) since they are highly degenerate
and their number is conserved. The time scale for BEC formation is the relaxation time τ
[26, 25, 27]. Axion BEC means that almost all axions go the lowest energy state available. The
question is then: does τ ever become shorter than the age of the universe t at that moment?
As was shown in Chapter 2 that τ ∼ t during the QCD phase transition at a temperature of
order 1 GeV when cold dark matter axions are first produced. The axions thermalize briefly
45
then as a result of their λΦ4 interactions. This brief period of thermalization has no known
observational consequences. However, when the photon temperature reaches of order 500 eV,
cold dark matter axions thermalize anew as a result of their gravitational self-interactions and
this does have observational consequences as mentioned earlier.
3.3 Summary
We derived solutions of the coupled Schrödinger and Poisson equations. The solutions
describe the homogeneous expanding matter-dominated universe and density perturbations
therein. The description is identical with that obtained by treating the dark matter as a
pressureless fluid, on all scales much larger than the de Broglie wavelength of the wavefunction.
In a number of respects, the wavefunction description is simpler and hence superior. It has
fewer degrees of freedom since a wavefunction is two real fields whereas a fluid is described by
four real fields, the density and the three components of velocity. The mathematics is simpler
as well. Even though it has only two components, the wavefunction can describe rotational
modes. On the other hand, the meaning of the wavefunction is less intuitively obvious.
We considered whether the wavefunction and fluid descriptions are equivalent. They are
equivalent in the sense that the density and velocity fields, given by Eqs. (3–2) and (3–6),
satisfy the fluid equations if the wavefunction satisfies the Schrödinger equation. However,
the wavefunction and the fluid describe objects which in general are not physically the same.
In particular the wavefunction description allows velocity dispersion and multi-streaming
whereas the fluid description does not. The physical distinction between the two descriptions
disappears completely only in the limit where the mass of the dark matter particle goes to
infinity. Because WIMPs are relatively heavy, the wavefunction and fluid descriptions of WIMP
dark matter are equivalent whenever velocity dispersion does not play a role. The wavefunction
description has the advantage that it can describe phenomena associated with velocity
dispersion, such as free-streaming, and that it can be used not only in the linear regime but
also in the non-linear regime, after shell crossing.
46
We asked whether the Schrödinger-Poisson equations correctly describe axion dark matter.
The answer is yes on time scales short compared to the relaxation time scale τ , and no on
time scales long compared to τ . Whenever τ is shorter than the age of the universe t, axion
dark matter is not correctly described by the Poisson-Schrödinger equations. Indeed axions
move towards a Bose-Einstein distribution on the time scale τ whereas the Schrödinger-Poisson
equations would predict that they move towards a Boltzmann distribution. Interactions, such
as gravity or λΦ4 interactions, play a dual role. On the one hand the interaction influences the
evolution of the classical field. The solutions of the classical field equation, which is equivalent
to the Schrödinger equation in the non-relativistic limit, describe the quantum states that the
axions may occupy. But the interaction has the additional role of causing the axions to jump
between those states. On time scales large compared to τ , the distribution of quanta over the
states described by the classical field changes completely. It was shown in refs.[24, 25] that, as
a result of axion gravitational self-interactions, τ becomes and remains shorter than the age
of the universe after the photon temperature has reached approximately 500 eV. Therefore the
commonly made assumption that axion dark matter is adequately described by classical field
equations at all times is incorrect.
47
CHAPTER 4VORTICES IN AXION BEC DARK MATTER
Its main purpose is to give a detailed description of the behavior of axion dark matter
before it falls into the gravitational potential well of a galaxy. In particular we want to
investigate the appearance and evolution of vortices in the rotating axion BEC, and ask
whether they have implications for observation. We obtain two results that may be somewhat
surprising. The first is that, unlike the vortices in superfluid 4He and in BECs of dilute gases,
the vortices in axion BEC attract each other. The reason for the difference in behavior is that
atoms have short range repulsive interactions whereas axions do not. The vortices in the axion
BEC join each other producing vortices of ever increasing size. When two vortices join, their
radii (not their cross-sectional areas) are added. We expect a huge vortex to form along the
rotation axis of the galaxy as the outcome of the joining of numerous smaller vortices. We call
it the ‘big vortex’. The presence of a big vortex implies that the infall is not isotropic as has
been assumed in the past [47, 29, 30, 31]. The axions fall in preferentially along the equatorial
plane. Caustic rings are enhanced as a result because the density of the flows that produce
the caustic rings is larger. We propose this as explanation of the fact that the rises in the
Milky Way rotation curve attributed to caustic rings [48] are typically a factor five larger than
predicted assuming that the infall is isotropic, a puzzle for which no compelling explanation
had been given in the past.
The second perhaps surprising finding is that baryons are entrained by the axion BEC
and acquire the same velocity distribution as the axion BEC. The underlying reason for this is
that the interactions through which axions thermalize are gravitational and gravity is universal.
The condition for the baryons to acquire net overall rotation by thermal contact with the
axion BEC is the same as the condition for the axions to acquire net overall rotation by
This chapter is a reproduction of [36], with permission from The American PhysicalSociety.
48
thermalizing among themselves. When baryons and axions are in thermal equilibrium, their
velocity fields are the same since otherwise entropy can be generated by energy-momentum
exchanging interactions between them. We expect that a big vortex forms in the baryon fluid
as well, although one of lesser size than the big vortex in the axion fluid. The resulting angular
momentum distribution of baryons agrees qualitatively with that observed by van den Bosch
et al. in dwarf galaxies [49]. In contrast, if the dark matter is WIMPs or sterile neutrinos,
the predicted angular momentum distribution of baryons in galaxies differs markedly from
the observed distribution, a discrepancy known as the ‘galactic angular momentum problem’
[34, 35]. At present, the most widely accepted solution to this problem is that gas outflows
driven by supernova explosions preferentially remove low angular momentum baryons from
galaxies [50].
We consider the possibility that the dark matter is a mixture of axions and another form
of cold dark matter. For the purposes of our discussion WIMPs and sterile neutrinos behave
in the same way. So we call the other form of cold dark matter WIMPs for the sake of brevity.
There is a minimum fraction of dark matter, of order 3%, that must be axions for the axions
to thermalize by gravitational interactions before they fall into galactic gravitational potential
wells. When they thermalize, the axions acquire net overall rotation and entrain the baryons
and WIMPs with them. The baryons and WIMPs acquire the same velocity distribution as the
axions before falling onto galactic halos. The WIMPs therefore produce the same caustic rings,
and at the same locations, as the axions. However, to account for the typical size of the rises
in the Milky Way rotation curve attributed to caustic rings, we find that the fraction of dark
matter in axions must be of order 37% or more.
To investigate the issues of interest here we generalize the statistical mechanics of many
body systems in thermal equilibrium to the case when the system is rotating and total angular
momentum is conserved. When total angular momentum is conserved, a system of identical
particles in thermal equilibrium is characterized by an angular frequency ω in addition to its
temperature T and its chemical potential µ. Broadly speaking, the state of thermal equilibrium
49
of such systems is one of rigid rotation with angular frequency ω. Incidentally, we find that
there is no satisfactory generalization of the isothermal sphere model of galactic halos with
ω = 0. This is a serious flaw of that model since galactic halos acquire angular momentum
from tidal torquing. For a Bose-Einstein condensate, we derive the state that most particles
condense into when ω = 0. In superfluid 4He this state is one of rigid rotation except for a
regular array of vortices embedded in the fluid. For a BEC of collisionless particles contained in
a cylindrical volume the state is one of quasi-rigid rotation with all the particles as far removed
from the axis of rotation as allowed by the Heisenberg uncertainty principle. For an axion BEC
about to fall into a galactic gravitational well the state is one in which each spherical shell
rotates rigidly but the rotational frequency varies with the shell’s radius r as r−2. (The motion
is similar to that of water draining through a hole in a sink). If the axions were to equilibrate
fully, they would all move very close to the equatorial plane. Because their thermalization rate
is not much larger than the Hubble rate, we expect that the axions start to move towards the
equator but there is not enough time for the axions to get all localized there. The motion of
the axions toward the equatorial plane is another way to see why the big vortex forms.
At first, we generalize the rules of statistical mechanics to the case where the many body
system conserves angular momentum. We derive the rule that determines the state that a
rotating BEC condenses into. We verify that the rule is consistent with the known behaviour
of superfluid 4He and derive the expected behavior of a rotating BEC of quasi-collisionless
particles. Next, we obtain the expected behavior of an axion BEC about to fall into a galactic
gravitational potential well, and the response of baryons and WIMPS to the presence of the
axion BEC. We then show that the axion BEC provides a solution to the galactic angular
problem and derive a minimum fraction of dark matter in axions (37%) from the typical size of
rises in the Milky Way rotation curve attributed to caustic rings.
4.1 Statistical Mechanics of Rotating Systems
In this section we discuss theoretical issues related to the statistical mechanics of rotating
many-body systems. First we generalize the well-known equilibrium Bose-Einstein, Fermi-Dirac
50
and Maxwell-Boltzmann distributions to the case where angular momentum is conserved.
Systems which conserve angular momentum are characterized by an angular velocity, in
addition to their temperature and chemical potential. Next we discuss the self-gravitating
isothermal sphere [55] as a model for galactic halos. We show that the model is a reasonably
good description only when the angular momentum is zero. Next we discuss rotating
Bose-Einstein condensates (BEC) and obtain the rule that determines the state which particles
condense into. We analyze the properties of the vortices that must be present in any rotating
Bose-Einstein condensate, and discuss the contrasting behaviors of vortices in superfluid4He and in a fluid of quasi-collisionless particles. Finally, we identify rethermalization as the
mechanism by which vortices appear in a BEC after it has been given angular momentum.
4.1.1 Temperature, Chemical Potential and Angular Velocity
A standard textbook result gives the average occupation number Ni of particle state i
in a system composed of a huge number of identical particles at temperature T and chemical
potential µ:
Ni =1
e1T(ϵi−µ) − σ
(4–1)
where ϵi is the energy of particle state i, and σ = 0, +1 or -1. If the particles are distinguishable,
one must take σ = 0 and the distribution is called Maxwell-Boltzmann. If the particles are
bosons, σ = + 1 and the distribution is called Bose-Einstein. If the particles are fermions,
σ = −1 and the distribution is called Fermi-Dirac.
To obtain Eq. (4–1), one considers a system with given total energy E =∑
i niϵi and
given total number of particles N =∑
i ni. The Ni are the values of the ni which maximize
the entropy [56, 57]. One may repeat this exercise in the case of a system that conserves total
angular momentum L =∑
i nili about some axis, say z. Maximizing the entropy for given
total energy E, total number of particles N and total angular momentum L, one finds:
Ni =1
e1T(ϵi−µ−ωli) − σ
(4–2)
51
where ω is an angular velocity. The system at equilibrium is characterized by T , µ and ω. If
the total number of particles is not conserved, one must set µ = 0. Likewise if total angular
momentum is not conserved one must set ω = 0.
4.1.2 The Self-Gravitating Isothermal Sphere Revisited
Consider a huge number of self-gravitating identical classical particles. (Although
identical, they are distinguishable by arbitrarily unobtrusive labels.) A particle state is given by
its location (r, v) in phase-space. According to Eq. (4–2), the particle density in phase-space is
given at thermal equilibrium by
N (r, v) = N0 e−m
T[ 12v·v+Φ(r)−ωz·(r×v)] (4–3)
where m is the particle mass, N0 ≡ eµT , and Φ(r) is the gravitational potential. Newtonian
gravity is assumed. The gravitational potential satisfies the Poisson equation:
∇2Φ(r) = 4πGmn(r) (4–4)
where
n(r) =
∫d3v N (r, v) (4–5)
is the physical space density.
When ω = 0, Eqs. (4–3) and (4–5) imply
n(r) = n0 e− 3
<v2>Φ(r) (4–6)
where < v2 >= 3Tm
is the velocity dispersion of the particles and n0 = N0(2π<v2>
3)32 .
Combining Eqs. (4–4) and (4–6) one obtains
∇2Φ(r) = 4πGmn0 e− 3
<v2>Φ(r) . (4–7)
52
This equation permits a spherically symmetric ansatz, Φ(r) = Φ(r). It can then be readily
solved by numerical integration. The solutions have the form
n(r) = n0 d(r/s) (4–8)
where
s =
(< v2 >
12πGmn0
) 12
(4–9)
and d(x) is a unique function with the limiting behaviors: d(x) → 1 as x → 0 and d(x) →
2/x2 as x → ∞. A plot of the function d(x) is shown, for example, in Fig. 1 of ref. [58] or
Fig. 4.7 of ref. [59]. The function d(x) is often approximated by 22+x2 for convenience. The
phase-space distribution
N (r, v) = N0 e− 3
<v2>v·v d(r/s) (4–10)
is called an ‘isothermal sphere’ [55].
The isothermal sphere is often used as a model for galactic halos [60, 61]. As such it
has many attractive properties. First, the isothermal sphere model is very predictive since it
gives the full phase-space distribution in terms of just two parameters, < v2 > and s. Second,
these two parameters are directly related to observable properties of a galaxy: < v2 > is
related to the galactic rotation velocity at large radii vrot by vrot =√
23< v2 > and s is related
to the galactic halo core radius a by a =√2s. Third, since n(r) ∝ 1/r2 for large r, the
isothermal model predicts galactic rotation curves to be flat at large r. This is consistent with
observation. Fourth, since n(r) ≃ n0 for small r, galactic halos have inner cores where the
density is constant. This is also consistent with observation. Fifth, the model is based on a
simple physical principle, namely thermalization.
For all its virtues, we do not believe the isothermal model to be a good description of
galactic halos. The reason is that present day galactic halos, such as that of the Milky Way,
are unlikely to be in thermal equilibrium. If for some unexplained reason the Milky Way halo
were in thermal equilibrium today, it would soon leave thermal equilibrium because it accretes
surrounding dark matter. The infalling dark matter particles only thermalize on time scales
53
that are much longer than the age of the universe [62, 63]. The flows of infalling dark matter
produce peaks in the velocity distribution. A large fraction of the halo, over 90% in the model
of ref. [31], is in cold flows. This disagrees with the smooth Maxwell-Boltzmann distribution,
Eq. (4–10), of the isothermal model. The presence of infall flows, with high density contrast in
phase-space, has been confirmed by cosmological N-body simulations [64].
Here we point to another flaw of the isothermal sphere as a model of galactic halos.
Galactic halos acquire angular momentum from tidal torquing. If they are in thermal
equilibrium, as the isothermal model supposes, the phase-space distribution must be given
by Eq. (4–3) with ω = 0. However, Eq. (4–3) is an unacceptably poor description of galactic
halos as soon as ω = 0. Indeed, Eq. (4–3) may be rewritten
N (r, v) = N0 e−m
T[ 12(v−ω×r)2+Φ(r)− 1
2(ω×r)2] (4–11)
where ω = ωz. Compared to the ω = 0 case, the velocity distribution is locally boosted by the
rigid rotation velocity ω × r. The physical space density is
n(r) = n0e−m
T[Φ(r)− 1
2(ω×r)2] . (4–12)
Substituting this into Eq. (4–4), one obtains
∇2Φ(r) = 4πGmn0 emT[−Φ(r)+ 1
2ω2ρ2] . (4–13)
where (ρ, z, ϕ) are cylindrical coordinates. Eq. (4–13) does not have any solutions for which
the density n(r) goes to zero for large ρ. Indeed d(r) ∝ em2T
ω2ρ2 at large ρ unless Φ → 12ω2ρ2
there. But this implies, through Eq. (4–4), that the density goes to the constant value ω2
2πGm
at large ρ. The particles at large ρ have huge bulk motion with average velocity ω × r. Thus
the rotating isothermal sphere is an object of infinite extent in a state of rigid rotation. This is
certainly inconsistent with the properties of galactic halos.
As was discussed by Lynden-Bell and Wood [65], isothermal spheres and self-gravitating
systems in general are unstable because their specific heat is negative, i.e. they get hotter
54
when energy is extracted from them. The instability implies a gravo-thermal catastrophe on
a time scale which may be inconsistent with the age of galactic halos and thus cause further
difficulties in using isothermal spheres as models for galactic halos. The instability of rotating
isothermal spheres is discussed in ref. [66].
4.1.3 The Rotating Bose-Einstein Condensate
4.1.3.1 Bose-Einstein condensation
Bose-Einstein condensation occurs when the following four conditions are satisfied: 1) the
system is composed of a huge number of identical bosons, 2) the bosons are highly degenerate,
i.e. their average quantum state occupation number is larger than some critical value of order
one, 3) the number of bosons is conserved, and 4) the system is in thermal equilibrium. When
the four conditions are satisfied a fraction of order one of all the bosons are in the same state.
Let us recall a simple argument [67] why Bose-Einstein condensation occurs. We first set
ω = 0. For identical bosons, Eq. (4–2) states
Ni =1
e1T(ϵi−µ) − 1
. (4–14)
Let i = 0 be the ground state, i.e. the particle state with lowest energy. It is necessary that the
chemical potential remain smaller than the ground state energy ϵ0 at all times since Eq. (4–14)
does not make sense for ϵi < µ. The total number of particles N(T, µ) =∑
i Ni is an
increasing function of µ for fixed T since each Ni has that property. Let us imagine that the
total number N of particles is increased while T is held fixed. The chemical potential increases
till it reaches ϵ0. At that point the total number of particles in excited (i > 0) states has its
maximum value
Nex(T, µ = ϵ0) =∑i>0
1
e1T(ϵi−ϵ0) − 1
. (4–15)
In three spatial dimensions, Nex(T, µ = ϵ0) is finite [67]. Consider what happens when, at a
fixed temperature T , N is made larger than Nex(T, µ = ϵ0). The only possible system response
is for the extra N −Nex(T, µ = ϵ0) particles to go to the ground state. Indeed the occupation
55
number N0 of that state becomes arbitrarily large as µ approaches ϵ0 from below. This is the
phenomenon of Bose-Einstein condensation.
We may repeat the above argument for ω = 0. In this case the chemical potential µ must
remain smaller than the smallest ηi ≡ ϵi − ωli. The existence of a minimum ηi is guaranteed
because the particle energy ϵi always contains a piece that is quadratic in li, namely the kinetic
energy associated with motion in the ϕ direction. Let
η0 ≡ mini[ϵi − ωli] (4–16)
and i = 0 the particle state that minimizes η. (We are relabeling the states compared to the
ω = 0 case.) The largest possible number of particles in states i = 0 at a given temperature T
is
Nex(T, µ = η0) =∑i =0
1
e1T(ηi−η0) − 1
. (4–17)
If the total number N of particles is larger than Nex(T, µ = η0) , the extra N −Nex(T, µ = η0)
particles go to the i = 0 state.
4.1.3.2 Vortices
It may appear at first that the requirement ∇×v = 0, implied by Eq. (3–6), disagrees with
the principle just stated, since a fluid of particles may have a rotational velocity field whereas
the wave description allows apparently only irrotational flow. However, that appearance is
deceiving because Eq. (3–6) is valid only where Ψ = 0. Indeed β is not well defined where
Ψ = 0. The following example is instructive
Ψ(ρ, ϕ, t) = AJl(kρ) eilϕ e−i k2
2mt (4–18)
where Jl is the Bessel function of index l. This wavefunction solves the Schrödinger equation
with Φ = V = 0. It describes an axially symmetric flow with energy k2
2mand z-component
of angular momentum l per particle. The fluid is clearly rotating since Eq.(3–6) implies the
56
velocity field
v =l
mρϕ . (4–19)
The curl of that velocity field vanishes everywhere except on the z-axis: ∇ × v = 2πlmzδ2(x, y)
where x and y are Cartesian coordinates in the plane perpendicular to z. Ψ vanishes on the
z-axis if l = 0. Furthermore, since the Bessel function Jl(s) ≃ 0 for s << l [70], the density
implied by Eq. (4–18) is tiny for ρ much less than the classical turnaround radius ρc = l/k.
Thus, on length scales large compared to k−1, the fluid motion described by the wavefunction
(4–18) is the same as the motion of a fluid composed of classical particles, in agreement with
the principle stated at the end of the previous paragraph.
That the motion of a fluid of cold dark matter particles can be described by a wavefunction
was emphasized by Widrow and Kaiser some twenty years ago [43]. Ordinary (non BEC)
cold dark matter particles have irrotational flow because their rotational modes have been
suppressed by the expansion of the universe and, when density perturbations start to grow,
Eq.(3–7) implies ∇ × v = 0 at all times if ∇ × v = 0 initially [28]. The previous paragraph
emphasizes that a fluid of cold dark matter particles can be described by a wavefunction
whether or not the velocity field is irrotational. The only requirement is that the wavefunction
vanish on a set of lines, called vortices, if the velocity field is rotational.
Each vortex carries an integer number of units of angular momentum, l units in the
example of Eq. (4–18). The following rule applies. Let C[Γ] be the circulation of the velocity
field along a closed path Γ
C[Γ] ≡∮Γ
dr · v(r, t) . (4–20)
If the fluid is represented by a wave, Eq. (3–6) implies
C[Γ] = ∆β
m≡ 2π
ml[Γ] (4–21)
where ∆β is the change of the phase β when going around Γ. Since the wavefunction is single
valued, l[Γ] ≡ 12π∆β is an integer. The surface enclosed by a closed path with non-zero
circulation must be traversed by vortices whose units of angular momentum add up to l[Γ].
57
Another rule is that as long as the fluid is described by a wavefunction Ψ(r, t), vortices cannot
appear spontaneously in the fluid. They can only move about. Indeed, consider an arbitrary
closed path Γ. The total number of vortices encircled by Γ, counting a vortex with angular
momentum l as l vortices, is determined by the constraint of Eq. (4–21). The only way the
RHS of that equation can change is by having the wavefunction Ψ vanish somewhere on Γ
and letting a vortex cross that curve. Finally, a third rule: vortices must follow the motion of
the fluid. This is a corollary of Kelvin’s theorem which states that, in gradient flow - i.e. if
the RHS of Euler’s equation is a gradient, as is the case for Eq. (3–7) - the circulation of the
velocity field along a closed path that moves with the fluid is constant in time.
4.1.3.3 Superfluid 4He
Let us see how the above considerations apply first to the case of superfluid 4He, and next
to the case of a collisionless fluid. Of course, we have nothing new to say about superfluid4He, which we discuss merely to build confidence in the general approach described so far.
We found that, when Bose-Einstein condensation occurs, a macroscopically large number of
particles, say N , condense into the state with lowest η = ϵ − ωl. We have, in terms of the
quantities defined earlier,
ϵ =
∫d3r Ψ∗(− 1
2m∇2 +mΦ + V )Ψ
=1
N
∫d3r n [
m
2v · v +m(q + Φ) + V ] (4–22)
and
l =
∫d3r Ψ∗z · (r × 1
i∇)Ψ =
∫d3r Ψ∗1
i
∂
∂ϕΨ
=1
N
∫d3r n mz · (r × v) . (4–23)
He atoms have an interparticle potential Vp that describes forces that are strongly repulsive at
short range and weakly attractive at long range. In the liquid state, the average interatomic
distance is of order the atom size. Thus the density of superfluid 4He has an approximately
58
constant value n0. In that case
ϵ ≃ n0
N
∫d3r
m
2v · v , l ≃ n0
N
∫d3r mz · (r × v) (4–24)
and therefore
η ≃ n0m
2N
∫d3r [(vz)
2 + (vρ)2 + (vϕ − ωρ)2 − ω2ρ2] . (4–25)
η is minimized when v = ωz × r. So, when superfluid 4He carries angular momentum, it
is in a state of rigid rotation when viewed on length scales large compared to the de Broglie
wavelength. Because rigid rotation implies that the velocity field has non-zero curl, vortices are
present. The vortices are parallel to the z-axis and their number density per unit area is
m
2πz · (∇ × v) =
mω
π(4–26)
according to Eqs. (4–20) and (4–21). The transverse size of a vortex is determined by
balancing the competing effects of −∇q and − 1m∇V in Eq. (3–7). −∇q tends to increase the
transverse size of the vortex whereas − 1m∇V tends to decrease it assuming, as is the case in
superfluid 4He, that the interparticle interactions are repulsive at short distances. The outcome
determines the transverse size of a vortex to be a characteristic length ξ, called the ‘healing
length’ [67]. For the interparticle potential
Vp(x− x′) = U0δ3(x− x′) (4–27)
the healing length is
ξ =1√
2mn0U0
. (4–28)
The transverse size of a l-vortex, i.e. a vortex that carries l units of angular momentum, is
lξ. Indeed the behavior at short distances to the vortex center is the same as in Eq. (4–18)
with k replaced by ξ−1. The cross-sectional area of a l-vortex is therefore of order πξ2l2. Also
its energy per unit length [67] is proportional to l2. The vortices repel each other because a
l-vortex has more energy per unit length than l 1-vortices. The lowest energy configuration for
given angular momentum per unit area is a triangular lattice of parallel 1-vortices [71]. Such
59
triangular vortex arrays were observed in superfluid 4He [72] and in Bose-Einstein condensed
gases [73].
4.1.3.4 Quasi-collisionless particles
We finally arrive at the object of our interest, a Bose-Einstein condensate of quasi-collisionless
particles. The particles cannot be exactly collisionless since they must thermalize to form a
BEC and they can only thermalize if they interact. However, the interaction by which the
particles thermalize can be arbitrarily weak since the thermalization may, in principle, occur on
an arbitrarily long time scale. In that limit we may set V = 0 in Eq. (3–1). For the sake of
definiteness we set the gravitational field Φ = 0 as well. We have then
η =
∫d3r Ψ∗(− 1
2m∇2 − ω
i
∂
∂ϕ)Ψ
=1
N
∫d3r n [
m
2v · v +mq −mωz · (r × v)] . (4–29)
If one approximates the Bose-Einstein condensate as a fluid of classical particles (setting q = 0
and taking n(r) and v(r) to be independent variables), the state of lowest η is one of rigid
rotation with angular velocity ωz and all particles placed as far from the z-axis as possible. For
the reasons stated earlier, this is a good approximation only on length scales large compared to
the BEC de Broglie wavelength. To obtain the exact BEC state, one must solve the eigenvalue
problem
(− 1
2m∇2 − ω
1
i
∂
∂ϕ)Ψi = ηiΨi . (4–30)
The BEC state is then Ψ0 such that η0 = mini ηi. Since by assumption the system conserves
angular momentum, the operators − 12m
∇2 and 1i
∂∂ϕ
are simultaneously diagonalizable. Thus
(i = kl)
Ψkl(z, ρ, ϕ) = Akl(z, ρ)eilϕ , ηkl = ϵkl − ωl (4–31)
and
− 1
2m(∂2
∂z2+
1
ρ
∂
∂ρρ∂
∂ρ− l2
ρ2)Akl = ϵklAkl . (4–32)
60
Let us consider the particular example of a BEC contained in a cylinder of radius R and height
h. In this case, the operators − 12m
∇2, 1i
∂∂ϕ
and(1i
∂∂z
)2 are simultaneously diagonalized by
Ψlpn = eilϕ sin(πp
hz)Jl(xln
ρ
R) (4–33)
where l = 0,±1,±2, ... , p = 1, 2, 3, ... , n = 1, 2, 3, ... , and xln is the nth root of Jl(x) with
xl1 < xl2 < xl3 < ... . Since
ϵlpn =1
2m[(πp
h)2 + (
xlnR
)2] (4–34)
ηlpn is minimized by setting p = n = 1 and l = l0 where l0 minimizes 12m
(xl1
R)2 − ωl. For large
l, the first zero of Jl [70]
xl1 ≃ l + 1.85575 l13 +O(l−
13 ) . (4–35)
Hence
l0 = mR2ω[1− 2.47433 (mR2ω)−23 +O(mR2ω)−
43 ] . (4–36)
When the BEC is approximated as a fluid of classical particles, the BEC state is rigid rotation
with all the particles located at ρ = R, not necessarily in a uniform way. In the actual BEC
state the particles are, for large l, uniformly located just inside the ρ = R surface, in a film of
thickness δρ ∼ R l−23 .
Unlike the case of superfluid 4He, vortices in a collisionless BEC attract each other. Indeed
the lowest energy state for given total angular momentum l is a single l-vortex with transverse
size as large as possible. We may imagine turning off the interparticle repulsion in superfluid4He placed in a cylindrical container. Starting with a triangular array of l parallel 1-vortices
but progressively decreasing U0, the vortices grow in transverse size till they join into a single
l-vortex and all matter is uniformly concentrated near the ρ = R surface.
4.1.4 Thermalization and Vortex Formation
We emphasized that vortices cannot appear spontaneously in a fluid that is described
by a (single) wavefunction Ψ. The Gross-Pitaevskii equation can only describe the motion of
vortices, not their appearance. How then do the vortices appear? The vortices appear when
61
the bosons move between different particle states, some of which have vortices and some of
which don’t. When angular momentum is given to a BEC that is free of vortices, it will at first
remain free of vortices even though it carries angular momentum. The vortices only appear
when the BEC rethermalizes and the particles go to the new lowest energy state consistent
with the angular momentum the BEC received.
Consider, for example, a BEC of spin zero particles in a cylindrical volume. The
wavefunctions of the particle states are given by Eq. (4–33). The Hamiltonian is the sum
of free and interacting parts: H = H0 +H1. The free Hamiltonian is:
H0 =∑lpn
ϵlpna†lpnalpn (4–37)
where alpn and a†lpn are annihilation and creation operators satisfying canonical commutation
relations and generating a Fock space in the usual fashion. We assume that the interaction has
the general form
H1 =∑
i,i′,i′′,i′′′
1
4Λi i′
i′′ i′′′ a†i′′′ a
†i′′ ai′ ai (4–38)
where i ≡ lpn, i′ ≡ l′p′n′ and so forth, so that the total number of particles
N =∑lpn
a†lpn alpn (4–39)
is conserved. In addition we require that
Λi i′
i′′ i′′′ = 0 unless l + l′ = l′′ + l′′′ (4–40)
so that the total angular momentum
L =∑lpn
l a†lpn alpn (4–41)
is conserved as well. The interaction H1 causes the system to thermalize on some time scale
τ = 1Γ. Ref. [25] estimates the thermalization rate Γ of cold dark matter axions through their
62
λϕ4 and gravitational self-interactions. The relevant thing for our discussion here is only that
there is a finite time scale τ = 1Γ
over which the system thermalizes.
Let us suppose that N particles are in thermal equilibrium in the cylinder with ω = 0
and temperature T well below the critical temperature for Bose-Einstein condensation. A
macroscopically large number N0 of particles are in the ground state (l, p, n)0 = (0, 1, 1), which
we label i = 0 for short. The remaining N − N0 particles are in excited (i = 0) states. The
vorticity of each state equals its l quantum number. The N0 particles in the ground state form
a fluid with zero vorticity. Many excited states carry vorticity but their occupation numbers are
small compared to N0. The particles in excited states merely constitute a gas at temperature
T . Let us suppose that the fluid is then given some angular momentum. This can be done, for
example, by having a large mass M which gravitationally attracts the particles in the cylinder
go by, producing a time-dependent potential energy Vext(r, t). We assume for the sake of
definiteness that the mass M passes by the cylinder on a time scale τM which is much shorter
than the thermal relaxation time scale τ . While the mass M passes by, each ϕ particle stays
in whatever state it was in to start with since the interaction H1 that allows particles to jump
between states is, by assumption, too feeble to have any effect on the τM time scale. The
wavefunction of each state satisfies the time-dependent Schrödinger equation:
i∂tΨi(r, t) = [− 1
2m∇2 + Vext(r, t)]Ψi(r, t) (4–42)
with the initial condition Ψi(r, t = −∞) = Ψi(r). Although each Ψi(r, t) changes in
time, for the reasons given earlier, its vorticity does not. Therefore, just after the mass
M has passed, the macroscopic fluid described by Ψ0(r, t) has no vorticity although it
generally has angular momentum. After a time of order τ , the N particles acquire a thermal
distribution, Eq. (4–2) with σ = +1, consistent with the total number of particles N , the
angular momentum L acquired from the passing mass and total energy E including some
energy acquired from the passing mass. Assuming the temperature is still below the critical
temperature for Bose-Einstein condensation, a macroscopically large number of particles are in
63
the state (l, p, n)′0 = (l′0, 1, 1) with l′0 given by Eq. (4–36). That state describes a fluid which
carries a single vortex with l′0 units of angular momentum.
4.2 Axions, Baryons and WIMPs
In this section we apply the considerations of Section 4.1 to dark matter axions when
they are about to fall into a galactic gravitational potential well. We also discuss the behavior,
in the presence of dark matter axions, of baryons and of a possible ordinary cold dark matter
component made of weakly interacting massive particles (WIMPs) and/or sterile neutrinos.
First we discuss the axions by themselves, ignoring the other particles.
4.2.1 Axions
Axions behave differently from ordinary cold dark matter particles, such as WIMPs or
sterile neutrinos, on time scales long compared to their thermalization time scale τ ≡ 1Γ
because on time scales long compared to τ the axions form a BEC and almost all axions go to
their lowest energy available state [24, 25]. Ordinary cold dark matter particles do not do this.
Axions behave in the same way as ordinary cold dark matter on time scales short
compared to their thermalization time scale τ [24]. So, to make a distinction between axions
and ordinary cold dark matter it is necessary to observe the dark matter on time scales long
compared to τ . The critical question is then: what is the thermalization time scale τ?
4.2.1.1 Axion thermalization
The relaxation rate of axions through gravitational self-interactions is of order [24, 25, 53]
Γ ∼ 4πGnm2ℓ2 (4–43)
where n and m are their density and mass, and ℓ ≡ 1δp
their correlation length. δp is their
momentum dispersion. A heuristic derivation of Eq. (4–43) is as follows. If the axions have
density n and correlation length ℓ, they produce gravitational fields of order g ∼ 4πGnmℓ.
Those fields completely change the typical momentum δp of axions in a time δpgm
. Γ is
the inverse of that time. To estimate the axion relaxation time today, let us substitute
nm ≃ 0.23 · 10−29 gr/cc (the average dark matter density today), m ≃ 10−5 eV (a typical mass
64
for dark matter axions) and ℓ ≃ 2 · 10−7sec GeV10−4eV
= 0.6 · 1017cm (the horizon during the QCD
phase transition, stretched by the universe’s expansion until today). This yields a relaxation
time τ of order 105 years, much shorter than the present age of the universe. So dark matter
axions formed a BEC a long time ago already. It is found in refs. [24, 25] that the axions first
thermalize and form a BEC when the photon temperature is approximately 500 eV(
fa1012 GeV
) 12
where fa is the axion decay constant. After the axions form a BEC their correlation length ℓ
increases until it is of order the horizon since the BEC size is limited only by causality.
It may seem surprising that axions thermalize as a result of their gravitational self-interactions
since gravitational interactions among particles are usually negligible. Dark matter axions are
an exception because the axions occupy in huge numbers a small number of states (the typical
quantum state occupation number is 1061) and those states have enormous correlation lengths,
as was just discussed.
It has been claimed [24, 32, 25] that the dark matter is axions, at least in part, because
axions explain the occurrence of caustic rings of dark matter in galactic halos. For the
explanation to succeed it is necessary that the axions that are about to fall onto a galaxy
thermalize sufficiently fast that they almost all go to the lowest energy available state
consistent with the angular momentum they acquired from neighboring protogalaxies by
tidal torquing [74]. Heuristically, the condition is [25]
4πGnm2ℓ > p = mv (4–44)
where v is the acceleration necessary for the axions to remain in the lowest energy state as the
tidal torque is applied. Here ℓ must be taken to be of order the size of the system, i.e. some
fraction of the distance between neighboring protogalaxies. It was found in ref. [25] that the
inequality (4–44) is satisfied by a factor of order 30 - i.e. that its LHS is of order 30 times
larger than its RHS - independently of the system size.
65
4.2.1.2 Caustic rings
The evidence for caustic rings is summarized in ref. [31]. It is accounted for if the angular
momentum distribution of the dark matter particles on the turnaround sphere of a galaxy is
given by
l(n, t) = m jmax n× (z × n)R(t)2
t(4–45)
where t is time since the Big Bang, R(t) is the radius of the turnaround sphere, z the galactic
rotation axis, n the unit vector pointing to an arbitrary point on the turnaround sphere, and
jmax a dimensionless parameter that characterizes the amount of angular momentum the
particular galaxy has. The turnaround sphere is defined as the locus of particles which have
zero radial velocity with respect to the galactic center for the first time, their outward Hubble
flow having just been arrested by the gravitational pull of the galaxy. Eq. (4–45) states that
the particles on the turnaround sphere rotate rigidly with angular velocity vector ω = jmax
tz.
The time-dependence of the turnaround radius is predicted by the self-similar infall model [75]
to be R(t) ∝ t23+ 2
9ϵ . The parameter ϵ is related to the slope of the evolved power spectrum
of density perturbations on galaxy scales [76]. This implies that ϵ is in the range 0.25 to 0.35
[47]. The evidence for caustic rings is consistent with that particular range of values of ϵ.
Each property of the angular momentum distribution given in Eq. (4–45) maps onto
an observable property of the inner caustics of galactic halos: the rigid rotation implied by
the factor n × (z × n) causes the inner caustics to be rings of the type described in refs.
[29, 30, 28], the value of jmax determines their overall size, and the R(t)2
ttime dependence
causes, in the stated ϵ range, the caustic radii an to be proportional to 1/n (n = 1, 2, 3...).
The prediction for the caustic radii is
an ≃ 40 kpc
n
(vrot
220 km/s
) (jmax
0.18
)(4–46)
where vrot is the galactic rotation velocity. To account for the evidence for caustic rings, axions
must explain Eq. (4–45) in all its aspects. We now show, elaborating the arguments originally
given in ref.[32], that axions do in fact account for each factor on the RHS of Eq. (4–45).
66
4.2.1.3 n× (z × n)
Consider a comoving spherical volume of radius S(t) centered on a protogalaxy. At early
times S(t) = a(t)S where a(t) is the cosmological scale factor. At later times S(t) deviates
from Hubble flow as a result of the gravitational pull of the protogalactic overdensity. At
some point it reaches its maximum value. At that moment it equals the galactic turnaround
radius. S is taken to be of order but smaller than the distance to the nearest protogalaxy of
comparable size, say one third of that distance. In the absence of angular momentum, the
axions have a purely radial motion described by a wavefuntion Ψ(r, t) = U(r, t) where r is the
radial coordinate relative to the center of the sphere. When angular momentum is included
the radial motion is modified at small radii by the introduction of an angular momentum
barrier. This modification of the radial motion is relatively unimportant and we neglect it. The
wavefunctions of the states that the axions occupy are thus taken to be
Ψl,p(r, θ, ϕ, t) = U(r, t)Al,p(θ, ϕ) (4–47)
where θ and ϕ are the usual spherical angular coordinates (0 ≤ θ ≤ π, 0 ≤ ϕ < 2π),
and l and p are quantum numbers. l is as before the eigenvalue of the z-component of
angular momentum. The z-direction is the direction of the total angular momentum acquired
inside the sphere as a result of tidal torquing. We will see below that that direction is time
independent. p is an additional quantum number, associated with motion in θ. We normalize
U(r, t) and the various A(θ, ϕ) such that∫ S(t)
0
r2 dr |U(r, t)|2 =∫ π
0
sin θ dθ
∫ 2π
0
dϕ |A(θ, ϕ)|2 = 1 . (4–48)
We suppress the quantum numbers l and p henceforth.
According to Eq. (4–44) the axions thermalize on a time scale τ that is short compared to
the age of the universe. Hence we expect most axions to keep moving to the state of lowest
η = ϵ − ω(t)l . The angular frequency ω is time dependent since the angular momentum is
growing by tidal torquing and the moment of inertia is increasing due to the expansion of the
67
volume under consideration. We have
ϵ =
∫r<S(t)
d3x Ψ∗[− 1
2m∇2 +mΦ(r, t)]Ψ
=
∫ S(t)
0
r2dr U∗[− 1
2mr2∂
∂rr2∂
∂r+ mΦ(r, t)]U
+1
2I(t)
∫dΩ A∗[− 1
sin θ
∂
∂θsin θ
∂
∂θ− 1
sin2 θ
∂2
∂ϕ2]A (4–49)
where1
I(t)=
1
m
∫ S(t)
0
dr |U(r, t)|2 . (4–50)
I(t) is similar to a moment of inertia but differs from the usual definition because the volume
to which it refers is not rotating like a rigid body in three dimensions. Eq. (4–47) implies
instead that each spherical shell of that volume rotates with the same angular momentum
distribution. The associated angular velocities vary with shell radius r as r−2. The inner
shells rotate faster than the outer shells because all shells have the same angular momentum
distribution.
We take the gravitational potential Φ to be spherically symmetric. The first term on the
RHS of Eq. (4–49) is then independent of the angular variables and irrelevant to what follows.
We will ignore it henceforth. Since
l =
∫dΩ A∗ 1
i
∂
∂ϕA (4–51)
we have
η =
∫dΩ A∗[
1
2I(t)(− 1
sin θ
∂
∂θsin θ
∂
∂θ− 1
sin2 θ
∂2
∂ϕ2)− ω(t)
1
i
∂
∂ϕ]A
=
∫dΩ A∗[
1
2I(t)
(− 1
sin θ
∂
∂θsin θ
∂
∂θ+
1
sin2 θ(1
i
∂
∂ϕ− ω(t)I(t) sin2 θ)2
)− 1
2ω2(t)I(t) sin2 θ]A
=
∫dΩ [
1
2I(t)|dAdθ
|2 − 1
2ω2(t)I(t) sin2 θ|A|2
+1
2I(t) sin2 θ|(1i
∂
∂ϕ− ω(t)I(t) sin2 θ)A|2] . (4–52)
68
The ϕ dependence of A that minimizes η is
A(θ, ϕ, t) = Θ(θ, t)eiω(t)I(t) sin2 θ ϕ . (4–53)
However that exact ϕ dependence is not allowed because the wavefunction must be
single-valued. Instead we have
A(θ, ϕ, t) ≃ Θ(θ, t)eiω(t)I(t) sin2 θ ϕ (4–54)
by which we mean that A(θ, ϕ, t) is as given in Eq. (4–53) except for the insertion of small
defects (vortices) that allow A to be single-valued. The vortices are discussed below.
After Eq. (4–54) is satisfied, we have
η ≃∫dΩ [
1
2I(t)|dΘdθ
|2 − 1
2ω2(t)I(t) sin2 θ|Θ|2] . (4–55)
η is further minimized by having Θ peaked at θ = π2, i.e. at the equator. The width of the
peak is of order δθ ∼ 1√ωI
. Eq. (4–54) shows that L ≡ ωI is the angular momentum per
particle in the galactic plane. A typical value is
L ∼ (500km
s)(10 kpc) m ≃ 2.6 · 1019
( m
10−5 eV
). (4–56)
Therefore in their state of lowest η the axions are almost all within a very small angular
distance, of order 10−10 radians, from the galactic plane.
The state just described is the state most axions would be in after a sufficiently long
period of thermalization. Because the thermalization criterion of Eq. (4–44) is only satisfied
by a factor of order 30 we expect that, although the axions start to move towards the equator,
there is not enough time for all the axions to get localized there. We expect the system to
behave as follows. As the axions acquire angular momentum they go to a state, described
by Eqs. (4–47) and (4–54), in which each spherical shell rotates rigidly with angular velocity
69
proportional to r−2 where r is the shell radius. The axion velocity field is
v ≃ vrr + vθθ +1
mrL(t) sin θϕ . (4–57)
The ≃ sign indicates that the LHS and RHS equal each other except for the presence of
vortices. The vortices have direction and density per unit surface given by [see Eqs. (4–20) and
(4–21)]m
2π∇ × v ≃ ∇ ×
(L(t)
mrsin θϕ
)=L(t)
πr2cos θ r . (4–58)
They point in the radial direction, and are more dense near the poles than near the equator.
The total vortex number penetrating the northern hemisphere is∫ 2π
0
dϕ
∫ π2
0
sin θ dθ r2m
2π∇ × v = L . (4–59)
As discussed in Section 4.1, axion vortices attract each other. When two vortices combine,
their diameters are added. (Two vortices of equal diameter, and hence of equal cross-sectional
area, combine into a vortex with four times that cross-sectional area). Assuming that a fraction
of order one of all the vortices combine with one another, a huge vortex appears along the z
axis. We will refer to it as the ‘big vortex’. The intersection of the big vortex with the galactic
plane is a circle whose radius a′ ≃ L′
kwhere k is the momentum of axions in the equatorial
plane at their closest approach to the galactic center and L′ is the angular momentum carried
by the big vortex. The distance of closest approach to the galactic center of axions in the
equatorial plane is a = Lk. a is also the radius of the caustic ring made by the axions as they
fall through the galaxy for the first time. Because of incomplete thermalization, we expect that
some fraction of the vortices have not joined the big vortex, implying that L′ < L and hence
a′ < a.
The factor n × (z × n) in the angular momentum distribution on the turnaround sphere,
Eq. (4–45), is thus accounted for by the fact that the axions on the turnaround sphere rotate
rigidly. After turnaround there is not enough time for further thermalization and whatever
further thermalization may occur would not make an appreciable difference. Thus, after
70
turnaround, the axions fall in and out of the galaxy like ordinary cold collisionless particles
but they do so with net overall rotation whereas ordinary cold dark matter falls in with an
irrotational velocity field [28].
Why and how the angular momentum distribution of Eq. (4–45) yields caustic rings is
explained in refs. [30, 31]. However those papers, written before the discovery of Bose-Einstein
condensation of dark matter axions [24], assume that the infall is isotropic, i.e. that the mass
falling onto the galaxy per unit time and unit solid angle dMdΩdt
does not depend on θ (nor on
ϕ). The above discussion suggests that this assumption should be modified since a big vortex
is now expected along the z axis, implying that the infall rate is suppressed near θ = 0 and
π. In the self-similar infall model the total mass M of the halo grows as t 23ϵ [75]. Therefore
dMdΩdt
= M6πϵt
in the isotropic case. We replace this with
dM
dΩdt(θ, t) = Nυ(sin θ)
υ M
6πϵt(4–60)
where υ (lower case upsilon, not to be confused with v the magnitude of velocity) is a
parameter describing the size of the big vortex. The normalization factor
Nυ =Γ(υ + 2)
2υ(Γ(υ2+ 1))2
(4–61)
is such that the total infall rate, integrated over solid angle, remains the same as before.
The new model for the infall rate, Eq. (4–60), does not change the prediction that the inner
caustics are rings nor the prediction, Eq. (4–46), for the caustic ring radii [36]. It does however
imply (for large υ) that the caustic rings are more prominent than in the isotropic infall case
since the axions fall in preferentially along the galactic plane. This will be discussed in Section
4.3.
4.2.1.4 R(t)2
t
We now show, repeating the argument of ref. [32], that the R(t)2
ttime dependence on
the RHS of Eq. (4–45) follows from tidal torque theory in linear order of perturbation theory.
Consider again the comoving sphere of radius S(t) introduced above. The total gravitational
71
torque applied to the volume V (t) of the sphere is
τ(t) =
∫V (t)
d3r δρ(r, t) r × (−∇ϕ(r, t)) (4–62)
where δρ(r, t) = ρ(r, t) − ρ0(t) is the density perturbation. ρ0(t) is the unperturbed density.
In first order of perturbation theory, the gravitational potential does not depend on time
when expressed in terms of comoving coordinates, i.e. ϕ(r = a(t)x, t) = ϕ(x). Moreover
δ(r, t) ≡ δρ(r,t)ρ0(t)
has the form δ(r = a(t)x, t) = a(t)δ(x). Hence
τ(t) = ρ0(t)a(t)4
∫V
d3x δ(x) x× (−∇xϕ(x)) . (4–63)
Eq. (4–63) shows that the direction of the torque is time independent. Hence the rotation axis
is time independent, as in Eq. (4–45). Furthermore, since ρ0(t) ∝ a(t)−3, τ(t) ∝ a(t) ∝ t23 and
hence the angular momentum increases with time proportionally to t 53 . Since R(t) ∝ t23+ 2
9ϵ ,
tidal torque theory predicts the time dependence of Eq. (4–45) provided ϵ = 0.33. This
value of ϵ is in the range, 0.25 < ϵ < 0.35, predicted by the evolved spectrum of density
perturbations and supported by the evidence for caustic rings. So the time dependence in
Eq. (4–45) is accounted for.
4.2.1.5 jmax
Here we compare the average value of jmax implied by the evidence for caustic rings
with the amount of angular momentum expected from tidal torquing. The amount of angular
momentum acquired by a galaxy through tidal torquing can be reliably estimated by numerical
simulation because it does not depend on any small feature of the initial mass configuration, so
that the resolution of present simulations is not an issue in this case. The amount of galactic
angular momentum is usually given in terms of the dimensionless quantity [77]
λ ≡ L|E| 12GM 5
2
, (4–64)
where L is the angular momentum of the galaxy, M its mass and E its net mechanical (kinetic
plus gravitational potential) energy. λ was found in numerical simulations to have a broad
72
distribution with median value 0.05 [78]. λ may also be estimated from observations of the
luminous matter by making some assumptions, in particular the assumption that the angular
momentum per unit mass of the disk and the halo are equal. Using such methods, Hernandez
et al. [79] derived the λ distribution of a large sample of spiral galaxies from the Sloan Digital
Sky Survey and found it to be consistent with the expectations from numerical simulations.
On the other hand, in the caustic ring model, the dimensionless measure of galactic
angular momentum is jmax. The evidence for caustic rings implies that the jmax distribution is
peaked at jmax ≃ 0.18. In case of isotropic infall (υ = 0), the relationship between jmax and λ
is [32]
λ =
√6
5− 3ϵ
8
10 + 3ϵ
1
πjmax . (4–65)
For ϵ = 0.33, Eq. (4–65) implies λ/jmax = 0.283. Hence there is excellent agreement between
jmax ≃ 0.18 and λ ∼ 0.05 when υ = 0. That the agreement is so good is likely somewhat
fortuitous since neither the λ nor the jmax distribution is very well established. Also, because
the extent of galactic halos is not uniformly agreed upon, there is some ambiguity in the
definition of the quantities, L, E and M, that enter λ. In deriving Eq. (4–65), the halo was
taken to extend all the way to the turnaround radius [32]. In obtaining the λ distribution from
numerical simulations, a definition of a galactic halo convenient in numerical simulations is
used [78]. The two definitions are not clearly equivalent. All together the agreement between λ
and jmax is significant only within some factor of order one, perhaps as large as 2.
For υ = 0, the relationship between λ and jmax is more difficult to derive because of the
lack of spherical symmetry. A calculation implies that the RHS of Eq. (4–65) is multiplied by a
factor of order 1+υ/21+υ/3
. If υ is much larger than one, the RHS of Eq. (4–65) is multiplied by 3/2
so that λjmax
≃ 0.426 for ϵ = 0.33. The values λ ∼ 0.05 and jmax ≃ 0.18 are then consistent at
the 50% level only.
4.2.2 Baryons and WIMPs
The gravitational forces produced by the axion BEC act not only on the axions themselves
but also on all other particles present. In particular, the axion BEC interacts gravitationally
73
with baryons, and with WIMPs if WIMPs are present. The condition for baryons/WIMPs to
acquire net overall rotation by gravitational interaction with the axion BEC is heuristically
4πGnmm′ℓ > m′ v , (4–66)
where m′ is the baryon/WIMP mass. The accelerations v necessary to acquire net overall
rotation are the same for baryons/WIMPs as for axions. Since m′ cancels out of the inequality
(4–66), conditions (4–66) and (4–44) are the same. It was found in ref. [25] that, if the dark
matter is entirely axions, the inequality (4–44) is satisfied by a factor of order 30. Therefore
conditions (4–44)and (4–66) are equivalent to
n m & 1
30ρDM (4–67)
where ρDM is the total cold dark matter density. If the axion fraction of cold dark matter
is larger than of order 3%, we expect the axion BEC to acquire net overall rotation and to
entrain the baryons and WIMPs along.
The baryons and WIMPs do not form a BEC but, by being in thermal contact with
the axion BEC, they behave in very much the same way. Indeed, thermal contact between
baryons/WIMPs and axions implies that they have the same temperature T and the same
angular velocity ω, as was discussed in subsection 4.1.1. The temperature of axions is certainly
smaller than the typical kinetic energy of axions in a galactic halo because axions with larger
kinetic energy would escape. Since the typical halo velocity is v ∼ 10−3c,
T . 1
2mv2 ≃ 6 · 10−8 K
( m
10−5 eV
). (4–68)
The velocity dispersion of the baryons and WIMPs, being much heavier than the axions but
at the same temperature, is tiny: less than 30 cm/s if m′ ≥ 1 GeV and m < 10−3 eV. The
temperature of baryons and WIMPs is thus effectively zero. Baryons and WIMPs are therefore
in their own state of lowest available η = ϵ − ωl, with the same ω as the axions. That
state may be derived by the same methods as we used for the axions in Section 4.1.1. The
74
outcome is that the baryons/WIMPs are in a state of rigid rotation (again not in the three
dimensional sense, but in the sense that each spherical shell rotates rigidly with angular velocity
proportional to r−2 where r is the shell’s radius) with the same velocity field, Eq. (4–57), as
the axion BEC. The underlying reason for this outcome is simple. If the baryons and WIMPs
were not locally at rest with respect to the axion BEC, entropy could be generated by bringing
them to such a state. Furthermore, as was the case for the axions, the lowest η state is one
where all the baryons and WIMPs are near the equator. We assume again, as we did for axions,
that the baryons/WIMPs start to move towards the equator but that there is not enough
time for them to all get there. Thus the baryon/WIMP infall rate is taken to have the same
functional dependence on t and θ as for axions
dM ′
dΩdt= Nυ′(sin θ)υ
′ M ′
6πϵt(4–69)
but we allow a different value υ′ for the υ parameter. Indeed we expect that the axions move
to the equator first and that the baryons/WIMPs follow them there. Since the baryons/WIMPs
are locally at rest with respect to the axion BEC, their angular momentum distribution on the
turnaround sphere is the same as in Eq. (4–45) but with m replaced by m′:
l′(n, t) = m′ jmax n× (z × n)R(t)2
t. (4–70)
Since the WIMPs fall in with the same initial velocity distribution as the axions, they move
in the same way after falling onto the galaxy and produce the same caustic structures. The
baryons also fall in the same way initially, but being collisionfull, separate from the axions and
WIMPs after shell crossing starts.
4.3 Comparison with Observations
Here we discuss two observations that appear related to the physics described in Section
4.2. The first is a measurement of the angular momentum distribution of baryonic matter in
dwarf galaxies by van den Bosch et al. [49]. The second is the typical size of the observed rises
75
in the Milky Way rotation curve, compared to the prediction from caustic rings of dark matter
[48].
4.3.1 Baryonic Angular Momentum Distribution
In the first part of this subsection we recount a discrepancy between the observed and
predicted angular momentum distributions of baryons in galaxies if the dark matter is ordinary
cold dark matter, such as WIMPs. This discrepancy is commonly referred to as the ’galactic
angular momentum problem’. In the second part we show how the discrepancy is resolved if
the dark matter is axions. Note that the galactic angular problem has potential solutions that
are more widely accepted by the community than the solution, exploiting the special properties
of axion dark matter, that we propose here. At present, the most widely accepted solution is
that gas outflows driven by supernova explosions preferentially remove low angular momentum
baryons from galaxies [50].
4.3.1.1 If the dark matter is all WIMPs
Since we assume in this subsection that none of the dark matter is axions, the considerations
of Section 4.2 do not apply.
By the principle of equivalence, tidal torquing gives the same amount and the same
distribution of specific angular momentum (i.e. angular momentum per unit mass) to baryons
and to dark matter before they fall onto galactic halos. Let us assume, to start with, that
the individual angular momentum of each particle is conserved from its turnaround till today.
In that case the observed amount and distribution of baryonic specific angular momentum is
the same as predicted for dark matter by numerical simulations. It was mentioned already
in subsection 4.2.1.5 that the amount of specific angular momentum observed in the
baryonic components of disk galaxies is consistent with the amount expected from numerical
simulations, lending support to the hypothesis that the angular momentum of each particle is
conserved. However, the observed specific angular momentum distribution of baryons in disk
galaxies differs markedly from that predicted by numerical simulations for WIMP dark matter.
The predicted distribution has many more particles with low specific angular momentum than
76
the observed distribution and a compensating (to keep the average the same) population of
particles with much higher specific angular momentum. The simulations predict too high a
concentration of baryons at the centers of galaxies.
At first it may appear that the solution to this discrepancy is simply to abandon the
notion that the angular momentum of individual particles is conserved after they have fallen
onto the galaxy. However, when the processes that allow angular momentum exchange are
modeled, it is found that they aggravate the discrepancy rather than resolve it. Frictional
forces among baryons have the general effect of removing angular momentum from baryons
that have little angular momentum and transferring it to baryons that have a lot. Dynamical
friction of dark matter on clumps of baryonic matter has the general effect of transferring
angular momentum from the baryons to the dark matter. Both processes tend to concentrate
baryons at galactic centers even more, aggravating the discrepancy [34]. The discrepancy
is commonly referred to as the ‘galactic angular momentum problem’. See ref. [35] for a
review. As mentioned already, the problem may have a solution other than the one we propose
here. At present the most widely accepted proposal is that gas outflows driven by supernova
explosions preferentially remove low angular momentum baryons from the galaxies [50].
The problem is thrown into sharp relief by comparing the universal angular momentum
distribution obtained by Bullock et al. from numerical simulations [80] with the observed
angular momentum distribution of baryons in dwarf galaxies [49]. Bullock et al. found that the
specific angular momentum distributions of the galaxies in simulations are all well fitted by a
single two parameter function:
dM
dl=
µMvl0(l0 + l)2
for 0 ≤ l ≤ lmax =l0
µ− 1(4–71)
where µ > 1, and Mv is the halo’s virial mass. Each galaxy has its own value of µ and lmax.
The distribution of log10(µ−1) values for the galaxies in the simulations is nearly Gaussian with
average -0.6 and standard deviation 0.4, implying that 90% of halos have 0.06 < µ − 1 < 1.0.
The median µ value is 1.25. The ratio of the average specific angular momentum lav to the
77
maximum specific angular momentum is given in terms of the parameter µ by
lavlmax
= (µ− 1)
[µ ln
(µ
µ− 1
)− 1
]. (4–72)
The broad distribution of µ values implies a correspondingly broad distribution of lavlmax
values
with average near 0.25 [49].
van den Bosch et al. [49] derived the baryonic angular momentum distribution of fourteen
dwarf galaxies from observations by Swaters [81]. The distributions are shown in Fig. 4-1
which is a reproduction of the relevant figure in ref. [49].
The prediction of Eq. (4–71) with µ = 1.25, the median value, is shown as a solid line in
each panel. The observed distributions are markedly different from the prediction of Eq. (4–
71). Perhaps the most striking difference is that Eq. (4–71) predicts dMdl
to be maximum at
l = 0 whereas the observed distributions appear to go to zero at l = 0 and have their maxima
around l = lav. Another striking difference, pointed out in ref. [49], is that the observed
values of lavlmax
are strongly peaked near 0.375. This is apparent from the fact that many of the
distributions in Fig. 4-1 end at lmax ≃ 2.6 lav. As mentioned, the numerical simulations predictlavlmax
to have a broad distribution with median value 0.25. If this were so, the distributions in
Fig. 4-1 would end at a wide variety of lmax
lavvalues and half of these values would be larger
than 4.
4.3.1.2 If the dark matter is axions, at least in part
As described in Section 4.2, the angular momentum distributions of the baryons and
WIMPs are modified by gravitational interactions with the axion BEC. The outcome is Eq. (4–
70) for the angular momentum distribution on the turnaround sphere and Eq. (4–69) for the
infall rate. We assume that the angular momentum of each particle is conserved after it crosses
the turnaround sphere. Eq. (4–70) implies for the angular momentum distribution on the
turnaround sphere at time t
l(θ, t) = z · l(n, t) = lmax (sin θ)2(t
t0
) 53
(4–73)
78
Figure 4-1. Reproduction of Fig. 4 in the article The angular momentum content of dwarfgalaxies: new challenges for the theory of galaxy formation by F.C. van den Boschet al. [49]. The shaded areas indicate the specific angular momentum distributionsof baryons in fourteen dwarf galaxies. In terms of the quantities defined in the text,s = l
lavand p(s) ∝ dM
dl(l). The solid curve is the distribution, Eq. (4–71), predicted
by numerical simulations of galaxy formation with ordinary cold dark matter, for µ= 1.25.
79
where t0 is the present age of the universe and
lmax = m jmaxR2
0
t0(4–74)
is the angular momentum of particles falling in along the galactic plane today. R0 is the
present turnaround radius. We are removing the primes from l′, m′, M ′ and so forth to avoid
cluttering the equations unnecessarily. To obtain Eq. (4–73) from Eq. (4–70) we used the fact
that l ∝ t53 ; see subsection 4.2.1.4. The angular momentum distribution today is
dM
dl(l) =
∫dΩ
∫ t0
0
dtdM
dΩdt(θ, t) δ(l − l(θ, t)) . (4–75)
Substituting Eqs. (4–69) and (4–73) and carrying out the t integration, one finds for ϵ = 1/3
dM
dl(l) =
6
5
M0
lmax
(l
lmax
) 15
Iυ
(l
lmax
)(4–76)
where
Iυ(r) = Nυ
∫ √1−r
0
dx
(1− x2)65−υ
2
. (4–77)
The predicted angular momentum distributions are shown in Fig. 4-2 below for υ = 0, 0.5, 1.0,
1.5, 2.0, 3.0, 5.0, 10., 50. and 100.
For υ = 0, dMdl(l) has a sharp cusp at l = 0 with dM
dl(0) = 3 M0
lmax. However, as soon as
υ > 0, dMdl
∝ l15 near l = 0. For υ in the range 0.5 to 2.0, dM
dl(l) is qualitatively similar to the
angular momentum distributions found by van den Bosch et al. in dwarf galaxies.
The average angular momentum is
lav =1
M0
∫ lmax
0
dl ldM
dl(l) = lmax
3
11
√π
2υ(υ
2+ 1)
Γ(υ + 2)
Γ(υ2+ 1) Γ(υ
2+ 5
2)
. (4–78)
lmax/lav, plotted in Fig. 4-3, decreases monotonically with υ, from 2.75 at υ = 0 to 1.83 at
υ = ∞. In the range 0.5 < υ < 2.0, lmax/lav ranges from 2.56 to 2.29.
So we find that in the axion case 1) the lmax/lav distribution is sharply peaked, like the
observed distribution, and 2) it is peaked at roughly the same value (2.6) as the observed
distribution.
80
4.3.2 Enhanced Caustic Rings
The observational evidence in support of the caustic ring halo model is summarized in
ref. [31]. A large part of that evidence is based on the existence of statistically significant
correlations between bumps in galactic rotation curves, consistent with the assumption that
some of the bumps are caused by caustic rings of dark matter and that the caustic ring radii
obey Eq. (4–46) [82, 48]. Additional evidence is provided by the fact that the bumps in the
high resolution inner rotation curve of the Milky Way published in ref. [83] are kinky, i.e. they
start with an upward kink and end with a downward kink [48]. The kinks are explained by
the fact that the dark matter density diverges at caustic surfaces [30]. Yet more evidence is
provided by the existence of a triangular shape in the IRAS (Infrared Astronomical Satellite)
and Planck map of the galactic plane in one of the two tangent directions to the nearest
caustic ring (n = 5). The position of the triangular shape coincides in galactic longitude with
the position of the rise in the rotation curve associated with that caustic ring. The triangular
shape is explained as the imprint of the gravitational field of the caustic ring on dust and gas
in the galactic disk.
There is however a puzzle with the interpretation of the evidence: the effects attributed
to caustic rings are too large compared to theoretical expectation. Specifically, the bumps
in the Milky Way rotation curve are on average a factor 5 larger [48] than expected in the
caustic ring model if the infall is isotropic [29, 30, 31] and if the bumps are due solely to the
caustic rings themselves. The sizes of the bumps are not actually predicted precisely by the
caustic ring model. They are given as a product of two factors, one of which is predicted by
the model. The other factor depends on details that the model (in its present state) does not
predict and which fluctuate from one caustic ring to the next. Nonetheless, this second factor
is expected to be generally of order one. There is no reason why its average should be five. See
refs. [30, 48] for details.
To account for this discrepancy, ref. [48] proposed that the rises in rotation curves are
amplified by gas that is gravitationally attracted to, and accreted onto, the caustic rings. The
81
square of the velocity dispersion of gas in the galactic disk is sufficiently small compared to
the gravitational potential ripples caused by caustic rings that a large amplification factor is
plausible. However, in this proposal it is hard to understand the kinkiness of the bumps in the
Milky Way rotation curve since the gas distribution would follow the gravitational potential of
the caustic rings, which is much smoother than the density of the caustic rings.
The physics discussed in Section 4.2 suggests a simpler and more compelling explanation,
to wit that the infalling axion BEC has a big vortex along the galactic symmetry axis and
hence that the infall is not isotropic. The caustic rings are enhanced because more dark matter
falls in near the galactic plane. If the infall rate is given by Eq. (4–60), the density of the flow
producing the caustic ring is increased by the factor Nυ given in Eq. (4–61). For Nυ to be of
order five, υ must be of order forty.
There is a restriction on how large υ can be because a caustic ring is partly erased if
the infalling dark matter is too concentrated near the galactic plane. Using the description
in ref. [30], one readily finds that caustic rings are formed in the flow of particles whose
declination α ≡ π2− θ at their last turnaround is less in magnitude than αm = 1
2
√usτ0 ≃
√p2a
where p is the width of the caustic ring. See ref. [30] for definitions of u, s, τ0. Some of the
caustics may be partly erased but not n = 5 since, as mentioned above, its full cross-section
appears in IRAS and Planck maps of the galactic plane as shown in Fig .4-4, (also see
http://www.phys.ufl.edu/~sikivie/triangle/index.htm).
For that ring, p = 0.018a and hence αm = 5.5. Requiring (cosαm)υ > 0.5 allows υ
as large as 150 and hence an enhancement by the factor Nυ as large as 10. The rise in the
Milky Way rotation curve associated with the n = 5 ring is a factor three larger [48] than
expected in the isotropic model (υ = 0), and therefore consistent with the appearance of the
full cross-section of the caustic ring in the aforementioned IRAS map.
If the existence of a big vortex is the correct explanation for the enhanced effect of caustic
rings of dark matter on galactic rotation curves, we may derive a lower limit on the fraction
Xa of dark matter that is axions. Let XW = 1 −Xa be the dark matter fraction in WIMPs or
82
some other form of ordinary cold dark matter. We need
Xa Nυ + XW Nυ′ ≃ 5 (4–79)
to account for the average strength of the rises in the Milky Way rotation curve. On the other
hand Nυ . 10, otherwise the corners of the triangular feature in the IRAS map get erased.
Also υ′ . 5, otherwise the angular momentum distribution of baryonic matter becomes too
dissimilar to the distributions observed by van den Bosch et al.; compare Figs. 4-2 and 4-3.
Eq. (4–61) implies then Nυ′ . 2. Combining all this, one obtains
Xa &3
8, (4–80)
i.e. a lower limit of approximately 37.5% on the axion dark matter fraction.
4.4 Summary
The goal of this chapter was to increase our understanding of the behavior of axion BEC
dark matter before it falls into the gravitational potential well of a galaxy. In particular we
wanted to see how axion BEC vortices appear and evolve, and whether they have implications
for observation.
In Section 4.1, we discussed the properties of rotating many body systems in thermal
equilibrium. We showed that the widely used self-gravitating isothermal sphere model is an
unacceptably poor description of galactic halos as soon as angular momentum is introduced.
We showed that the vortices that appear in a BEC of quasi-collisionless particles, such as
an axion BEC, attract each other, in contrast to the repulsive behavior of the vortices in
superfluid 4He and dilute gases. We showed that vortices in any BEC appear only as part of
the process of rethermalization after the BEC has been given angular momentum. Neither the
thermalization of a BEC nor the appearance of its vortices is described by the Gross-Pitaevskii
equation. That equation describes the behavior of the BEC, including the motion of its
vortices, only after it has formed.
83
In Section 4.2, we used the results of Section 4.1 to try and predict the behavior of axion
BEC dark matter before it falls into the gravitational potential well of a galaxy. As angular
momentum is acquired by the axion BEC through tidal torquing, the axions go to a state
where all spherical shells rotate rigidly about a common axis, with angular velocity proportional
to r−2 where r is the radius of the shell. The resulting angular momentum distribution on
the turnaround sphere, Eq. (4–45), is precisely and in all respects that which accounts for the
evidence for caustic rings of dark matter. Because axion BEC vortices are attractive, we expect
that most join into one big vortex. The radius of this big vortex is smaller than but of order
the radius of the first caustic ring made by the axion BEC as it falls in and out of the galaxy.
We modified the caustic ring model of galactic halos to include the presence of the big vortex.
Whereas the previous version of that model assumed that the infall is isotropic, the new version
assumes that the infall rate is given by Eq. (4–60) where υ parametrizes the size of the big
vortex.
The rate at which baryons and WIMPs reach thermal equilibrium with the axion BEC
was found to be qualitatively the same as the rate at which axions reach thermal equilibrium
among themselves. That thermalization rate is larger than the Hubble rate provided the
axion dark matter fraction is more than of order 3%. Because baryons and WIMPs are much
heavier than axions, the temperature of baryons and WIMPs is effectively zero when they are in
thermal contact with the axions. In that case, baryons and WIMPs acquire the same velocity
distribution as the axion BEC before falling onto galactic halos, and WIMPs produce the same
caustic rings as axions do, and at the same locations. We expect the baryons and WIMPs to
produce their own big vortex although with radius smaller than the radius of the big vortex in
axions. The specific angular momentum distribution of baryons and WIMPs on the turnaround
sphere is the same as for axions. The infall rate is also the same but with a smaller value υ′ of
the parameter υ.
In Section 4.4, we compared the specific angular momentum distribution predicted for
baryons, when the axion dark matter fraction is more than of order 3%, with the specific
84
angular momentum distribution of baryons observed in dwarf galaxies. They are qualitatively
similar for 0.5 . υ′ . 2. Moreover, in this range the ratio lmax/lav of maximum to average
specific angular momentum is predicted to be near 2.4. This is in qualitative agreement with
the observed distributions since most of the latter have lmax/lav ≃ 2.6. In contrast, if the dark
matter is all WIMPs, the specific angular momentum distribution differs markedly from the
observed distributions; see Fig. 4-1. Furthermore, lmax/lav is predicted in the WIMP case to
vary from galaxy to galaxy, the median value being 4 and the 90% range from 2.6 to 8.1. The
ability of axion dark matter to qualitatively explain the observed angular momentum of baryons
in dwarf galaxies is further evidence that at least part of the dark matter is axions.
The appearance of a large vortex in the axion BEC provides a plausible solution to a past
puzzle, namely that the rises in galactic rotation curves attributed to caustic rings of dark
matter are typically a factor 5 larger than expected when the dark matter infall is assumed
to be isotropic. The presence of a big vortex implies that more dark matter falls in along the
galactic plane and hence that the density of the flows producing the caustic rings is increased.
If all the dark matter is axions, the factor 5 enhancement is accounted for if υ is of order 40.
If the dark matter is partly axions and partly WIMPs, with the axion fraction more than of
order 3%, axions and WIMPs co-produce the caustic rings. If one requires υ′ < 5 to have
an acceptable fit between the predicted and observed specific angular momentum distribution
of baryons in dwarf galaxies, the caustic ring enhancement of a factor of five can only be
accounted for if at least 37% of the dark matter is axions.
85
0.0 0.5 1.0 1.5 2.0 2.5 3.0
l
lav
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
dM
d l
lav
Mo
Υ=0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
l
lav
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
dM
d l
lav
Mo
Υ=0.5
0.0 0.5 1.0 1.5 2.0 2.5 3.0
l
lav
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
dM
d l
lav
Mo
Υ=1
0.0 0.5 1.0 1.5 2.0 2.5 3.0
l
lav
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
dM
d l
lav
Mo
Υ=1.5
0.0 0.5 1.0 1.5 2.0 2.5 3.0
l
lav
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
dM
d l
lav
Mo
Υ=2
0.0 0.5 1.0 1.5 2.0 2.5 3.0
l
lav
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
dM
d l
lav
Mo
Υ=3
0.0 0.5 1.0 1.5 2.0 2.5 3.0
l
lav
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
dM
d l
lav
Mo
Υ=5
0.0 0.5 1.0 1.5 2.0 2.5 3.0
l
lav
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
dM
d l
lav
Mo
Υ=10
0.0 0.5 1.0 1.5 2.0 2.5 3.0
l
lav
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
dM
d l
lav
Mo
Υ=50
0.0 0.5 1.0 1.5 2.0 2.5 3.0
l
lav
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
dM
d l
lav
Mo
Υ=100
Figure 4-2. Specific angular momentum distributions if the dark matter is axions, for variousvalues of the parameter υ.
86
0 5 10 15Υ
1
2
3
4
lmax
lav
Figure 4-3. The ratio of maximum to average angular momentum if the dark matter is axions,as a function of the parameter υ. For the fourteen dwarf galaxies observed by vanden Bosch et al. this ratio is narrowly peaked near 2.6. In numerical simulations ofgalaxy formation with ordinary cold dark matter, this ratio is predicted to vary fromgalaxy to galaxy over the range 2.6 to 8.1 (90 %CL), with median value 4.0.
Figure 4-4. Images of the region around galactic coordinates (80, 0) from the IRAS 12 µm andPlanck 857 GHz observations. The triangular feature, indicative of a tricusp causticring, is prominent in both data sets.
87
CHAPTER 5RECOMBINATION ERA MAGNETIC FIELDS FROM AXION DARK MATTER
In Chapter 4 we showed how vortices appear in axion BEC infalling onto a galactic
potential well. In this chapter, we introduce a new mechanism for generating magnetic fields
in the recombination era. This Harrison-like mechanism utilizes vorticity in baryons that is
sourced through the Bose-Einstein condensate of axions via gravitational interactions. The
magnetic fields generated are on galactic scales ∼ 10 kpc and have a magnitude of the order
of B ∼ 10−23G today. The field has a greater magnitude than those generated from other
mechanisms relying on second order perturbation theory, and is sufficient to provide a seed for
battery mechanisms.
5.1 Origin of Magnetic Fields in the Universe
One of the most pressing problems in modern astronomy is determining the origin
of magnetic fields in the Universe. Magnetic fields are observed on all scales, from small
scales, such as inside our own solar system, to the largest bound structures, galaxy clusters
[85, 86, 87]. In fact, recent observations have even detected an intergalactic magnetic field
existing in the void regions of the cosmic large scale structure, with magnitude BIGMF ∼
10−18 − 10−15G (e.g., Refs. [88, 89]). Despite their prevalence, there is a large amount of
uncertainty in how the first magnetic fields were created.
Setting aside the possibility that magnetic fields were present as initial conditions which
is incredibly unappealing, we are yet to fully understand the origin of the first fields. These
primordial seeds need only be small as there are several ways in which they can then amplified
by astrophysical processes; for example, adiabatic contraction or turbulent shock flows during
structure formation (see, e.g. Ref. [90] for a review). The seeds are required to have an
amplitude in the range of 10−30 − 10−20G, with the specific magnitude depending on the
This chapter is a reproduction of [84], with permission from The American PhysicalSociety.
88
details of the dynamo model. There has been much work over the years addressing the
generation of the primordial seed field.
Harrison [91, 92] was one of the first to attempt to explain the origin of the seed magnetic
field generated by vorticity in a rotating protogalaxy prior to decoupling. This provided a
seed field of the order of B ∼ 10−19G, which is large enough to source galactic dynamo
mechanisms which enhance this initial seed field to currently observed magnitudes. However,
since (at linear order in perturbation theory) vorticity decays [93], there is no way to support
the vorticity in a post-recombination universe, and so this mechanism of magnetic field
generation was criticized [94]. Later, Mishustin & Ruzmaĭkin [95] investigated the generation
of magnetic fields in the post-recombination plasma, finding fields with magnitude of around
10−17G (evaluated at z ∼ 100). Similarly to Harrison, this work required the existence of
primordial vorticity. Another recent piece of work used vorticity from the texture scenario of
large scale structure formation [96]. However, the resulting field is too weak to act as the
primordial seed.
Other mechanisms for generating magnetic fields in the very early universe have been
studied comprehensively in the literature. These roughly include fields generated during
inflation, with a breaking of the conformal invariance of electromagnetism [97], during
phase-transitions [98, 99] or during (p)reheating [100, 101]. These all have issues, and
sustaining magnetic fields in the very early universe proves to be difficult.
One interesting method for generating magnetic fields around recombination builds upon
the ideas of Harrison, and uses second order cosmological perturbation theory [102, 103, 104].
While vorticity decays at linear order in perturbation theory, there are source terms at second
order – these look very much like the baroclinic term in the Biermann battery – that allow
for vorticity generation [105, 106, 107]. It is therefore possible that this vorticity comes
hand-in-hand with a magnetic field. There have been numerous works to this end [108, 109,
110, 111, 112, 113], that all obtain fields with roughly the same magnitude of 10−26 − 10−23G
at recombination.
89
5.2 The Axion Bose-Einstein Condensate
In Chapter 2 it was shown that axions form a re-thermalizing BEC through gravitational
self-interactions when the photon temperature was around 500 eV [24, 25, 8]. The axion BEC
interacts gravitationally with baryons with a relaxation rate [25]
ΓG ∼ 4πGnmmbℓ
∆pb, (5–1)
where n and m are the number density and mass of axions respectively, mb is the mass of
baryons which is of order 1 GeV, ℓ ∼ 1H
is the correlation length of the axion BEC, ∆pb ∼√3mbT is the momentum dispersion of the baryons, where T is the photon temperature.
For T < 1 keV the dominant interaction between photons and baryons is Compton
scattering. The relaxation rate for an electron to gain or lose energy by Compton scattering off
photons is known from standard cosmology to be [114]
Γe ∼ 9× 10−21 s−1Ωbh2
(T
Tγ0
)4
(5–2)
where Tγ0 is the present day photon temperature and Ωbh2 is the present day physical baryon
density parameter. It can be shown that both the ratios ΓG/H and ΓG/Γe are greater than
one around matter radiation equality and keeps increasing thereafter. Therefore the baryons
thermalize with the axion BEC.
5.3 Vorticity from Tidal Torque
In the standard picture of structure formation baryons collapse onto dark matter
overdensities. Tidal torque from nearby inhomogeneities imparts angular momentum onto
such protogalaxies. It was shown in Ref. [36] that when a system of axion BEC acquires
angular momentum, the axions thermalize and most of them go to a state with minimum
ηi = ϵi − ξ(t)li, where ϵi and li are the energy and angular momentum respectively of the
ith state and ξ(t) is the angular velocity of the system which grows with time as angular
momentum grows. This lowest η state has non-zero vorticity and therefore the axions acquire a
net rotational velocity field.
90
The baryons being in thermal contact with the axion BEC are dragged along with the
axion flow and acquire the same rotational velocity field as the axions. The baryons therefore
acquire vorticity from tidal torquing as a result of thermalization with the axion BEC. It should
be noted that in general tidal torque on collisionless particles cannot generate rotational flow
[115]. Before shell crossing the baryons behave like collisionless particles since dissipative
processes and shocks are absent. Therefore if dark matter is made of only WIMPs then there is
no vorticity.
5.4 Magnetic Fields
We will now show how vorticity in the baryon fluid can generate a magnetic field. We
follow an approach similar to that of Ref. [95]. As recombination begins at redshift zr ∼ 1500,
the photon free streaming length grows rapidly such that, on galactic scales, it can be treated
as a homogeneous radiation background. The charged particles moving in this radiation
background experience a Thomson drag force, [116] FT = 4σTργ3c
v, where σT is the Thomson
cross-section, ργ is the energy density of photons and v is the velocity of the charged particle
relative to the background radiation. Because electrons are much lighter than protons, their
acceleration due to this force is much greater than that of protons. Neglecting the effects from
neutral species, the equations of motion for electrons and protons with velocities ve and vp
respectively are
dvedt
= − e
me
(E +
vec× B
)− 4σTργ ve
3cme
+vp − veτep
+ agrav , (5–3)
dvpdt
=e
mp
(E +
vpc× B
)− 4σTργ vp
3cmp
− vp − veτep
+ agrav , (5–4)
where τep is the characteristic time for momentum exchange via Coulomb scattering, me and
mp are the masses of the electron and proton, respectively, e is the charge of the electron,
and agrav is the acceleration due to gravitational interactions with the axion BEC. Since we
are interested in showing how the vorticity in baryons sourced by the axion BEC can generate
91
magnetic fields, we have neglected the electron and proton pressure terms in the above
equations.
The current density is defined as J = ene(vp − ve), where ne ≃ np, by local charge
neutrality. The Thomson drag term becomes negligibly small after z ∼ 900 when the timescale
of Thomson scattering is greater than the Hubble time. We have assumed that initially there is
no magnetic field and neglected the back reaction from the generated magnetic field.
On taking the difference of Eqs. (5–3) and (5–4) and neglecting the Thomson drag term
for protons, followed by the curl, we arrive at an equation for the vorticity of electron fluid, ωe,
d
dt∇ ×
(J
ene
)=
e
me
∇ × E +4σTργ3mec
ωe − 2∇ ×
(Je
meσ
), (5–5)
where σ is the conductivity of the background medium. The LHS of Eq. (5–5) is negligibly
small compared to the first term on the RHS on galactic scales [85]. The last term on the RHS
of Eq. (5–5) is proportional to the magnetic diffusion term which can be neglected because of
the high conductivity of the background medium. We are therefore left with
e
me
(∇ × E
)= −4σTργ
3mecωe . (5–6)
On invoking the Maxwell equation
1
c
∂B
∂t= −∇ × E , (5–7)
Eq. (5–6), becomes∂B
∂t=
4σTργ3e
ωe . (5–8)
Of course, these calculations are performed in a static universe, therefore we must transform to
the expanding universe in which we live. On doing so, Eq. (5–6) becomes
1
a2∂(a2B)
∂t=
4σTργ0a−4
3eωe(t) , (5–9)
where a(t) is the scale factor and a subscript zero denotes the present-day value of a quantity.
92
Let us consider a galaxy sized (∼ 10 kpc) spherical overdensity of axion BEC onto which
baryons are falling. Tidal torque imparts the same specific angular momentum to the infalling
matter. Thermalization with the axion BEC results in vorticity in the baryons which is of the
order ω ∼ L/MR2, where L is the total angular momentum, M is the total mass of the
infalling baryons and R is the size of the protogalaxy. Following Peebles [117], the angular
momentum of a protogalaxy grows as t5/3 in the linear regime, which implies that the vorticity
grows as t1/3. At z ∼ 10 the protogalaxies reach their turnaround radius after which they begin
to collapse and separate from the background. We denote this redshift by z∗ in the following.
From this time onwards the evolution is complicated to handle analytically as non linear effects
play a significant role. We make an estimate by considering that the angular momentum of the
protogalaxy is conserved per comoving volume after they separated from the background, so
the vorticity decays like t−4/3.
To summarize in terms of redshift we have
ω(z) =
ω0 (1 + z∗)
5/2(1 + z)−1/2 , z∗ < z < zr
ω0(1 + z)2 , 0 ≤ z ≤ z∗ ,
(5–10)
where ω0 is the present day value of the vorticity which, for our galaxy, is ω0 ∼ 10−15 s−1.
Expressing Eq. (5–9) in terms of redshift and using the above expression for vorticity we get an
equation which can be integrated from the beginning of recombination upto z ∼ 900 when the
battery shuts down. For zr > z > 900, we have
B(z)
z2∼ 10−22 G
( z∗10
)5/2 ( ω0
10−15 s−1
)ln(zrz
). (5–11)
The magnetic field grows up to z ∼ 900 when it has magnitude B ∼ 10−17 G. After this
time it is frozen into the residual free charges and decays with the expansion of the universe.
The magnetic field today has a magnitude of B0 ∼ 10−23 G on scales of order 10 kpc.
93
5.5 Discussion
In this chapter, we have investigated the generation of magnetic fields from vorticity in
the recombination era. We have used a Harrison-like mechanism, with the novelty lying in the
fact that the vorticity is not assumed, but rather is inherent in the Bose-Einstein condensate of
axions. This provides a natural source of vorticity which is present only for axion dark matter.
The magnetic field sourced by this vorticity has a magnitude of B ∼ 10−17G peaking at
redshift z = 900, on scales of 10 kpc whose value today is of order 10−23 G. The magnetic
field generated through this process acts as a seed for astrophysical amplification mechanisms
through the later stages of galaxy formation. There are several different dynamic mechanisms
which can amplify seeds by upwards of ten orders of magnitude [118, 90], and result in the
observed fields of the order of a few microGauss at redshift less than one.
Furthermore, the magnetic field generated from axion dark matter is larger in magnitude
that those created by mechanisms relying on higher order fluctuations within the standard
ΛCDM cosmological model. Therefore, this allows for less effective amplification mechanisms
to enhance the primordial seed to the observable size.
94
CHAPTER 6EVOLUTION OF VELOCITY DISPERSION ALONG COLD COLLISIONLESS FLOWS
The infall of cold dark matter onto a galaxy produces cold collisionless flows and
caustics in its halo. If a signal is found in the cavity detector of dark matter axions, the flows
will be readily apparent as peaks in the energy spectrum of photons from axion conversion,
allowing the densities, velocity vectors and velocity dispersions of the flows to be determined.
In this chapter, we discuss the evolution of velocity dispersion along cold collisionless flows in
one and two dimensions. A technique is presented for obtaining the leading behavior of the
velocity dispersion near caustics. The results are used to derive an upper limit on the energy
dispersion of the Big Flow from the sharpness of its nearby caustic, and a prediction for the
dispersions in its velocity components.
6.1 Laboratory Detection of Dark Matter
One approach to the identification of dark matter is to attempt to detect dark matter
particles in the laboratory. WIMP dark matter can be searched for on Earth by looking for the
recoil of nuclei that have been struck by a WIMP [119]. Axion dark matter can be searched
for by looking for the conversion of axions to photons in an electromagnetic cavity permeated
by a strong magnetic field [120, 121]. The spectrum of photons produced in the cavity is
directly related to the axion energy spectrum in the laboratory since energy is conserved in the
conversion process:
hν = Ea = mac2 +
1
2mav
2 (6–1)
where ν is the frequency of a photon produced by axion to photon conversion, Ea the energy
of the axion that converted into it, ma the axion mass and v the velocity of the axion in the
rest frame of the cavity. Since the velocity dispersion of halo axions is of order 10−3c, the
width of the axion signal is of order 10−6ν. If, for example, the axion signal occurs at 1 GHz,
This chapter is a reproduction of [156], with permission from The American PhysicalSociety.
95
its width is of order 1 kHz. On the other hand, the resolution with which the signal can be
spectrum analyzed is the inverse of the time over which it is observed [122]. If the signal
is observed for 100 seconds, for example, the achievable resolution is 0.01 Hz. Thus, under
the example given, the kinetic energy spectrum of halo axions is resolved into 105 bins. The
Earth’s rotation changes the velocities relative to the laboratory frame by amounts of order
1 m/s in 100 s, and therefore introduces Döppler shifts of order δDν ∼ (300 km/s)(1 m/s)
ν/c2 ≃ 0.3 · 10−11ν, which is 0.003 Hz in the example given. If the data taking runs are much
longer than 100 s, the resolution is limited by the Döppler shifts. However, for a given velocity
vector in the rest frame of the Galaxy, the Döppler shifts may be removed and in that case the
resolution can be much higher than 0.01 Hz.
Narrow peaks in the velocity spectrum of dark matter on Earth are expected because a
galactic halo grows continuously by accreting the dark matter that surrounds it. The infalling
dark matter produces a set of flows in the halo since the dark matter particles oscillate back
and forth many times in the galactic gravitational potential well before they are thermalized
by gravitational scattering off inhomogeneities in the galaxy [62]. The flows are cold and
collisionless and therefore produce caustics [29, 30, 28, 63]. Caustics are surfaces in physical
space where the density is very high. At the location of a caustic, a flow “folds back” in
phase space. Each flow has a local density and velocity vector, and produces a peak with the
corresponding properties in the energy spectrum of photons from axion conversion in a cavity
detector. All three components of a flow’s velocity vector can be determined by observing the
peak’s frequency and Döppler’s shift as a function of time of day and time of year [123]. Thus
an axion dark matter signal would be a rich source of information on the formation history of
the Milky Way halo. Moreover, the flows are relevant to the search itself, before a signal is
found, because in a high resolution analysis a narrow prominent peak may have higher signal to
noise than the full signal.
Motivated by these considerations, the self-similar infall model of galactic halo formation
[75] was used to predict the densities and velocity vectors of the Milky Way flows [47, 31].
96
The original model was generalized to include angular momentum for the infalling particles.
The flow properties near Earth are sensitive to the dark matter angular momentum distribution
because the angular momentum distribution determines the structure and location of the
halo’s inner caustics [28]. If the dark matter particles fall in with net overall rotation, the
inner caustics are rings. The self-similar infall model predicts the radii of the caustic rings.
The caustic rings produce bumps in the galactic rotation curve at those radii. Mainly from
the study of galactic rotation curves, but also from other data, evidence was found for
caustic rings of dark matter at the locations predicted by the self-similar infall model. The
evidence is summarized in ref.[31]. The model of the Milky Way halo that results from fitting
the self-similar infall model to the data is called the “Caustic Ring Model”. It is a detailed
description of the full phase-space structure of the Milky Way halo [31].
The caustic ring model predicts that the local velocity distribution on Earth is dominated
by a single flow, dubbed the “Big Flow” [48]. The reason for this is our proximity to a cusp in
the 5th caustic ring in the Milky Way. Up to a two-fold ambiguity, the Big Flow has a known
velocity vector; see Section 6.5. Its density on Earth is estimated to be of order 1.5 · 10−24
gr/cm3, i.e. a factor three larger than typical estimates (0.5 · 10−24 gr/cm3) found in the
literature for the total dark matter density on Earth. The existence of the Big Flow provides
strong additional motivation for high resolution analysis of the output of the cavity detector,
since it produces a prominent narrow peak in the energy spectrum. It is desirable to have an
estimate of the width of that peak since this determines the signal to noise ratio of a high
resolution search for it. The width of the peak is the energy dispersion of the Big Flow. One
of our main goals is to place an upper limit on the energy dispersion of the Big Flow from the
observed sharpness of the 5th caustic ring. More generally we want to study the evolution of
velocity dispersion along cold collision flows, the relation between velocity dispersion and the
distance scale over which caustics are smoothed, and the behavior of velocity dispersion very
close to a caustic. Our results may be relevant to other cold collisionless flows, in particular
97
the streams of stars that result from the tidal disruption of galactic satellites, such as the
Sagittarius Stream [124].
In Section 6.2, we study the evolution of velocity dispersion along a cold collisionless
flow in one dimension. In Section 6.3, we do the same for an axially symmetric flow in two
dimensions. In Section 6.4.1 we present a technique for obtaining the leading behavior of
velocity dispersion near caustics. We apply it to fold caustics and cusp caustics. In Section
6.5, we use our results to derive an upper limit on the energy dispersion of the Big Flow from
the sharpness of the 5th caustic ring and make a prediction for the relative dispersions of its
velocity components.
6.2 A Cold Flow in One Dimension
In this section, we study how velocity dispersion changes along a cold collisionless flow
in one dimension. We consider a large number of particles moving in a time-independent
potential V (x) and forming a stationary flow. We first discuss the case where the velocity
dispersion vanishes and then the case where the velocity dispersion is small.
6.2.1 Zero Velocity Dispersion
In the case of zero velocity dispersion, all particles have the same energy
E =m
2v2 + V (x) . (6–2)
Hence their velocity at location x is
v(x) = ±√
2
m(E − V (x)) . (6–3)
For the sake of definiteness, we assume that the particles are bound to a minimum of V (x),
going back and forth between x1 and x2, defined by E = V (x1) = V (x2). For x1 < x < x2
there are two flows, one with v(x) > 0 and one with v(x) < 0. There are no flows at x < x1
or x > x2. Since the overall flow is stationary (∂n∂t
= 0), the continuity equation
∂n
∂t+
∂
∂x(nv) = 0 , (6–4)
98
implies that the densities of both the left- and right-moving flows equal
n(x) =J√
2m(E − V (x))
, (6–5)
where J is a constant. J is the flux (number of particles per unit time) in the left- and the
right-moving flows.
There are two caustics, one at x1 and the other at x2. The caustics are simple fold (A2)
catastrophes. For x near x1
V (x) = E + (x− x1)dV
dx(x1) +O(x− x1)
2 (6–6)
with dVdx(x1) < 0. Hence
n(x) =J√
− 2m
dVdx(x1)
1√x− x1
(6–7)
as x approaches x1 from above. Likewise
n(x) =J√
+ 2m
dVdx(x2)
1√x2 − x
(6–8)
when x approaches x2 from below.
6.2.2 Small velocity dispersion
Next we consider the same flow but with a small energy dispersion δE. We assume that
the energy distribution dNdE
(E) is narrowly peaked about its average E. δE is defined as usual
by
δE =< (E − E)2 >12 . (6–9)
Brackets indicate averaging over the energy distribution. At location x, the two flows have
average velocity
v(x) ≃ v(x, E) = ±√
1
2m(E − V (x)) (6–10)
and velocity dispersion
δv(x) =< (v(x,E)− v(x))2 >12≃<
((E − E)
∂v
∂E(x, E)
)2
>12=
δE
m|v(x, E)|. (6–11)
99
We assumed that dNdE
(E) goes to zero rapidly as |E − E| increases, as is the case e.g. for a
Gaussian distribution. Eq. (6–10) is exact then in the limit δE → 0. Also Eq. (6–11) is exact
in that limit provided that, in addition, ∂E∂v|x = mv(x, E) does not vanish, i.e. that one is not
close to a caustic.
The density of each of the flows is given by Eq. (6–5), with E replaced by E. The
phase-space density of the flow is therefore
N =n(x)
mδv(x)≃ J
δE. (6–12)
It is independent of x, as required by Liouville’s theorem.
Velocity dispersion smooths the caustics. Indeed, when δE = 0, the caustics are spread
over a thickness (j = 1, 2)
δxj =δE∣∣∣dVdx (xj)∣∣∣ . (6–13)
The density reaches at the caustics a maximum value
nmax,j ∼√
m
2∣∣∣dVdx (xj)∣∣∣
J√δxj
=
√m
2δEJ . (6–14)
Also the velocity dispersion reaches a maximum value
δvmax,j ∼√δE
2m. (6–15)
It is interesting that nmax and δvmax are the same at one caustic as at the other. Eq. (6–15)
follows from Eq. (6–14) and Liouville’s theorem. It also follows from the fact that v = 0 at the
caustics, so that δE ∼ m2(δvmax)
2 there.
6.3 An axisymmetric cold flow in two dimensions
In this section we study a stationary, axisymmetric, cold, collisionless flow of particles
moving in two dimensions in an axisymmetric time-independent potential V (r). Again, we
discuss first the flow with vanishing velocity dispersion, and then the flow with a small velocity
dispersion.
100
6.3.1 Zero velocity dispersion
Consider a flow of particles moving in a plane, under the influence of a potential V (r).
(r, ϕ) are polar coordinates in the plane. All particles have the same energy E and the same
angular momentum L. Hence the velocity field
v(r, ϕ) = vr(r)r + vϕ(r)ϕ (6–16)
with
vϕ(r) =L
mrand vr(r) = ±
√2
m(E − Veff(r)) (6–17)
where
Veff = V (r) +L2
2mr2. (6–18)
Provided L = 0 the particles have a non-zero distance of closest approach a: E = Veff(a). Let
us assume they also have a turnaround radius R, with E = Veff(R) and R > a. There are two
flows for a < r < R. We call them the ”in” (vr < 0) and ”out” (vr > 0) flows. There are no
flows for r < a or r > R.
Since the flow is stationary and axisymmetric, the continuity equation implies that the
density of particles of both the in and out flows is
n(r) =J
r|vr(r)|(6–19)
where J is a constant. J is the number of particles per unit time and per radian. There are
simple fold caustics at r = a and r = R. When r approaches a from above
n(r) =J
a√
− 2m
dVeff
dr(a)
1√r − a
, (6–20)
whereas
n(r) =J
R√
+ 2m
dVeff
dr(R)
1√R− r
(6–21)
when r approaches R from below.
101
6.3.2 Small Velocity Dispersion
We now consider the same flow as in the previous subsection but with a Gaussian
distribution of energy and angular momentum of the form
d2N
dEdL=
N
2πσEσLe− 1
2(E−E
σE)2− 1
2(L−L
σL)2
. (6–22)
For this distribution δE = σE, δL = σL, and
< (E − E)(L− L) > = 0 . (6–23)
The most general Gaussian would allow < (E − E)(L − L) >= 0. However our main interest
is the evolution of the velocity dispersion of flows of cold dark matter particles falling onto a
galactic halo and sloshing back and forth thereafter. We now argue that Eq. (6–23) is a good
approximation for that case.
The primordial velocity dispersion δvp of a flow of cold dark matter particles is negligibly
small. By primordial velocity dispersion, we mean the velocity dispersion that the particles
have in the absence of structure formation. δvp ∼ 10−17 is typical of axions, δvp ∼ 10−12 for
WIMPs, and δvp ∼ 10−8 for sterile neutrinos. The main contributions to the velocity dispersion
of a flow of cold dark matter particles falling onto a galaxy are instead from gravitational
scattering off inhomogeneities in the galaxy (such as globular clusters and molecular clouds)
and from the growth by gravitational instability of small scale density perturbations in the flow
itself. When a process produces a velocity dispersion δv, the associated energy dispersion is
of order δE ∼ mvδv where v is the velocity of the flow in the galactic reference frame when
the process occurs, and the associated angular momentum dispersion is of order δL ∼ mDδv
where D is the distance from the galactic center where the process occurs. So, δE and δL are
not independent quantities but related by
δL = δEDav
vav(6–24)
102
where vav is an average flow velocity, say 300 km/s for our Milky Way galaxy, and Dav is an
average distance from the galactic center where the flow acquired velocity dispersion. We may
only give a rough guess for the order of magnitude of Dav, perhaps 100 kpc for our galaxy.
In view of Eq. (6–24) we define an overall flow velocity dispersion σv: δL = mDavσv and
δE = mvavσv.
Furthermore, in the limit where the galaxy has no angular momentum (L = 0), there
is no preference for the many events that produce velocity dispersion to increase or decrease
< (E− E)(L− L) > since this quantity is odd under L→ −L. Therefore < (E− E)(L− L) >
is proportional to L. Disk galaxies have angular momentum but, relative to their size and
typical velocities, that angular momentum is small. Indeed all dimensionless measures of
galactic angular momentum have values much less than one. One such measure is the galactic
spin parameter [116]
λ =L|E| 12GM 5
2
(6–25)
where L is the angular momentum of the galaxy, E its net mechanical (kinetic plus gravitational)
energy and M its mass. G is Newton’s gravitational constant. A typical value is λ ∼ 0.05.
Another dimensionless measure of galactic angular momentum is the ratio a/R of caustic ring
radius a to turnaround radius R for the flows of dark matter particles in the halo. A typical
value is a/R ∼ 0.1 [31]. Finally, the dimensionless number that controls the amount of galactic
angular momentum in the Caustic Ring Model is jmax. A typical value is jmax ∼ 0.2. Since
< (E − E)(L − L) > is proportional to galactic angular momentum, and galactic angular
momentum is of order 0.1 in dimensionless units, < (E− E)(L− L) > is suppressed relative to
δE δL by a similar factor of order 0.1. To simplify our calculations, we set < (E−E)(L−L) >
= 0 as a first approximation.
Provided δE and δL are sufficiently small, we may within the support of the d2NdEdL
distribution express small deviations dE = E − E and dL = L − L of the particle energy and
angular momentum from its average values as linear functions of small deviations dvr = vr − vr
and dvϕ = vϕ − vϕ of the velocity components from their average values at a given position
103
(r, ϕ). Henceforth we set m = 1 to avoid cluttering the equations unnecessarily. Eqs. (6–17)
imply
dE = vrdvr + vϕdvϕ and dL = rdvϕ . (6–26)
Therefore the exponent in Eq. (6–22) may be rewritten using
(dE dL)
1σ2E
0
0 1σ2L
dE
dL
= (dvr dvϕ)
v2rσ2E
vrvϕσ2E
vrvϕσ2E
v2ϕσ2E+ r2
σ2L
dvr
dvϕ
.
(6–27)
We rotate dvr
dvϕ
=
cos θ sin θ
− sin θ cos θ
dv1
dv2
(6–28)
so as to diagonalize the 2x2 matrix on the RHS of Eq. (6–27). Provided
tan 2θ =2vrvϕ
v2ϕ +(
σE
σLr)2
− v2r
(6–29)
we have (dE
σE
)2
+
(dL
σL
)2
=
(dv1σ1
)2
+
(dv2σ2
)2
(6–30)
where1
(σ 12)2
=1
2
(v2r + v2ϕσ2E
+r2
σ2L
)∓
√1
4
(v2r + v2ϕσ2E
+r2
σ2L
)2
− r2v2rσ2Eσ
2L
. (6–31)
This implies
σ1 σ2 =σE σLr|vr(r)|
. (6–32)
At a given location, the velocity distribution is
d2N
dvrdvϕ=
d2N
dEdL| det
(∂(E,L)
∂(vr, vϕ)
)| = r|vr(r)|
d2N
dEdL. (6–33)
We have
(δvr)2 = cos2 θ(σ1)
2 + sin2 θ(σ2)2
(δvϕ)2 = sin2 θ(σ1)
2 + cos2 θ(σ2)2
104
< dvr dvϕ > = (σ22 − σ2
1) sin θ cos θ . (6–34)
Liouville’s theorem is satisfied since the phase space density
N =n(r)
σ1σ2=n(r)r|vr(r)|
σEσL=
J
σEσL(6–35)
does not depend on r.
At the caustics, θ → 0 since vr → 0. σ2 becomes δvϕ and remains finite:
σ2 →1√(
LrcσE
)2+(
rcσL
)2 ≡ δvϕ(rc) (6–36)
with rc = a or R. σ1 becomes δvr and is large. According to Eqs. (6–31) and (6–19), δvr and
n(r) become infinite. However, those equations cannot be used right at the caustics since they
neglect second order terms in Eqs. (6–26), and this is inaccurate when vr → 0.
We may use Eq. (6–35) to estimate the maximum density and velocity dispersion at the
caustics since that equation follows directly from Liouville’s theorem. The divergence at the
caustics is cut off by the fact that vr ∼ δvr there. Through Eqs. (6–32) and (6–19), this
implies
δvr,max|rc ∼√
σEσLrcδvϕ(rc)
(6–37)
and
nmax|rc ∼ J
√δvϕ(rc)
rcσEσL. (6–38)
At the inner caustic
δvϕ(a) =1√(
LaσE
)2+(
aσL
)2 =1√(
vϕ(a)
vavσv
)2+(
aDavσv
)2 ≃ σvvavvϕ(a)
(6–39)
in the limit of small angular momentum (a << R) since vϕ(a) = La
is of order vav whereas Dav
is of order R and therefore much larger than a. Hence
δvr,max|a ∼√
1
aDavvϕ(a)σv =
1
a
√LσL (6–40)
105
and
nmax|a ∼J√LσL
. (6–41)
Eqs. (6–40) and (6–41) show that the properties of the inner caustic are controlled by the
angular momentum distribution in the small angular momentum limit.
In contrast, at the outer caustic,
δvϕ(R) =1√(
LRσE
)2+(
RσL
)2 =1√(
avϕ(a)
Rvavσv
)2+(
RDavσv
)2 ≃ σvDav
R(6–42)
since a << R in the limit of small angular momentum. Hence
δvr,max|R ∼√vavσv =
√δE (6–43)
and
nmax ∼J
R√δE
. (6–44)
The properties of the outer caustic are controlled by the energy distribution in the small
angular momentum limit.
6.4 Velocity Dispersion Near a Caustic
The local velocity dispersion becomes large when a caustic is approached since the
physical space density becomes large but the phase-space density remains constant. In this
section we present a technique for deriving the leading behavior of the velocity dispersion near
caustics. We apply it to the case of fold caustics and of cusp caustics.
6.4.1 Velocity Dispersion Near a Fold Caustic
The fold caustic is described by the simplest (A2) of catastrophes. It is the only caustic
possible in one dimension. In one dimension it occurs at a point. In two dimensions it occurs
on a line, and in three dimensions it occurs on a surface, with the dimensions parallel to the
line or surface playing spectator roles only.
Consider a flow of particles moving in one dimension in the potential V (x) = gx
where g is a constant acceleration. We set the mass of the particles equal to one, as in the
106
previous section. We assume the flow to be stationary for simplicity. This is not an essential
assumption. In the limit of zero velocity dispersion, the flow at a given time is described by the
map
x(τ) = x0 −1
2gτ 2 (6–45)
which gives the position of the particles as a function of their age. We define the age τ of a
particle as minus the time at which it passes through its maximum x value. The velocity is
v(τ) =∂x
∂τ= −gτ . (6–46)
At position y there are two flows if y < x0, and no flow if y > x0. For y < x0, the particles at
position y have ages τ = ±√
2g(x0 − y). The density of each of the two flows is
n(y) ≡ dN
dy(y) =
dN
dτ|dτdy
| = dN
dτ
1√2g(x0 − y)
. (6–47)
N is number of particles, as before. The fold caustic is located at y = x0.
We now allow the particles to have a distribution dNdx0
of x0 values, with average x0 and a
small dispersion δx0. Since the particles at a given location y have a distribution of x0 values,
they also have a distribution of ages. Small deviations dx0 and dτ from the average x0 and the
average age at a given location are related by
0 = dx = dx0 − gτdτ . (6–48)
Hence the dispersion in ages at a given location is
δτ =√< (dτ)2 > =
√<
(dx0gτ
)2
> =1
g|τ |δx0 . (6–49)
Brackets indicate averages over the distribution dNdx0
. Small deviations in velocity are related to
small deviations in age by dv = −gdτ . Hence the velocity dispersion at position y is
δv(y) = gδτ =δx0|τ |
= δx0
√g
2(x0 − y). (6–50)
107
The phase space density N = n/δv has no singularity at the caustic, in accordance with
Liouville’s theorem. Although everything we found here was already obtained in Section 6.2,
the present approach is more efficient is obtaining the characteristic behavior of the velocity
dispersion right at the caustic. The technique can be readily applied to more complicated
cases, such as the cusp caustic which we discuss next.
6.4.2 Velocity Dispersion Near a Cusp Caustic
The cusp caustic is described by the A3 catastrophe, the next simplest after the fold
catastrophe. It can only exist in two dimensions or higher. In two dimensions, it occurs at a
point. In three dimensions, it occurs along a line with the dimension parallel to the line playing
a spectator role only.
A cusp caustic appears in the map
z = z0 + bατ
x = x0 − cτ − 1
2sα2 (6–51)
giving the positions (x, z) of particles in a flow from right to left with the particles on the right
and top moving downwards and the particles on the right and bottom moving upwards. See
Fig. 6-1.
For simplicity, we take the flow to be stationary, and the acceleration to vanish
everywhere. The particles are labeled by (α, τ) where τ is an age parameter, defined as
minus the time a particle crosses the z = z0 axis. α labels the different trajectories. The
velocities are
vz =∂z
∂τ= bα
vx =∂x
∂τ= −c . (6–52)
The density of the flow is
d(x, z) ≡ d2N
dxdz=
d2N
dαdτ
1
|D(α, τ)|(6–53)
108
Figure 6-1. The curve with the cusp is the location of the fold caustic in the flow described byEqs. (6–51). The arrows indicate local velocity vectors. There is one flow at everypoint to the right of the curve, and three flows at every point to its left.
where
D(α, τ) = det
(∂(x, z)
∂(α, τ)
)= b(−sα2 + cτ) . (6–54)
Caustics occur where D(α, τ) = 0, i.e. where the map is singular. The equation D(α, τ) = 0
defines the curve
x = x0 −3
2
(sc2
b2
) 13
|z − z0|23 , (6–55)
shown by the solid line in Fig. 1. There are three flows at every location to the left of the
curve whereas there is only one flow in the region to its right. The number of flows at a
location (x, z) is the number of (α, τ) values that solve Eqs. (6–51). The curve is the location
of a fold caustic. The cusp caustic is the special point (x0, z0). When the curve is traversed
starting on the side with three flows, two of the flows disappear. At any point other than
(x0, z0), the density and the velocity dispersion of those two flows behave as described in the
previous subsection, i.e. the density diverges as 1√h
where h is the distance to the curve and
the dispersion in the velocity component perpendicular to the curve also diverges as 1√h. The
dispersion in the velocity component parallel to the curve remains finite.
109
In this subsection, we are interested in the behavior of the density and velocity dispersion
at the cusp. A complete answer can be given by obtaining the inverse of the map in Eqs.
(6–51). This involves solving a third order polynomial equation, and inserting the resulting
functions α(x, z) and τ(x, z) into the RHS of Eqs. (6–54) and (6–53). Here we content
ourselves with the behavior of the density and the velocity dispersion ellipse when the cusp is
approached from particular directions. If the cusp is approached along the z = z0 axis from the
right (x > x0), the single flow there has α = 0 and τ = −(x− x0)/c. The density of that flow
is
d(x, z0) =d2N
dαdτ
1
b|x− x0|. (6–56)
If the cusp is approached along the x = x0 axis, from the top or from the bottom, the single
flow there has τ = − s2cα2 and z − z0 = − bs
2cα3. The density is
d(x0, z) =d2N
dαdτ
1
3
(2
bsc2
) 13 1
|z − z0|23
. (6–57)
If the cusp is approached along the z = z0 axis from the left (x < x0) there are three
flows: i) α = 0 and τ = (x0 − x)/c, ii) τ = 0 and α =√
2s(x0 − x), and iii) τ = 0 and
α = −√
2s(x0 − x). Flow i) has the same density as given in Eq. (6–56) whereas flows ii) and
iii) each have half the density given in Eq. (6–56).
To obtain the velocity dispersions we consider a set of maps, as in Eq. (6–51), but with
a distribution d5Ndz0 db dx0 dc ds
of the constants z0, b, x0, c and s that appear there. We assume
that the distribution is narrowly peaked around average values of these constants. We consider
small deviations dz0 ... ds of the constants from their average values plus small deviations dα
and dτ in the flow parameters such that
dz = dz0 + dbατ + bdατ + bαdτ = 0
dx = dx0 − dcτ − cdτ − 1
2dsα2 − sαdα = 0 . (6–58)
110
Eqs. (6–58) imply that, at a given physical point, the deviations in the flow parameters are
given in terms of the deviations in the constants by bτ bα
sα c
dα
dτ
=
−dz0 − ατdb
dx0 − τdc− 12α2ds
. (6–59)
When approaching the cusp, α → 0 and τ → 0, the RHS of Eq. (6–59) goes to(
−dz0dx0
), and
therefore dα
dτ
=1
b(cτ − sα2)
c − bα
−sα bτ
−dz0
dx0
. (6–60)
Likewise, when approaching the cusp, the deviations in the velocity components are
dvz = bdα = − cdz0cτ − sα2
dvx = −dc . (6–61)
Hence
δvz =√< (dvz)2 > =
cδz0|cτ − sα2|
, δvx = δc . (6–62)
If dz0 and dc are correlated
< dvz dvx >=c
cτ − sα2< dz0 dc > . (6–63)
For each of the flows at the cusp, the dispersion in the velocity component parallel to the
axis (x) of the cusp remains finite whereas the dispersion in the component of velocity in the
direction perpendicular (z) to the axis of the cusp diverges. This might have been expected
since the flow folds in the direction perpendicular to the axis of the cusp. The phase space
density remains finite since the divergence of the physical density is canceled by the divergence
of the velocity dispersion.
6.5 Applications to the Big Flow
In the Caustic Ring Model of the Milky Way halo, the dark matter density on Earth is
dominated by a single cold flow, dubbed the ‘Big Flow’, because of our proximity to a cusp
111
in the 5th caustic ring in our galaxy. In the Caustic Ring Model, there are two flows on Earth
associated with the 5th caustic ring. Their velocity vectors are [48] [31]
v5± ≃ (505 ϕ± 120 r) km/s (6–64)
where ϕ is the unit vector in the direction of Galactic rotation and r the unit vector in the
radially outward direction. The density of the Big Flow on Earth is estimated to be 1.5 · 10−24
gr/cm3. The density of the other flow associated with the 5th caustic ring, hereafter called
the ‘Little Flow’, is estimated to be 0.15 · 10−24 gr/cm3. It is not known whether the Big
Flow has velocity v5− and the Little Flow has velocity v5+, or vice-versa. The existence of the
Big Flow provides strong additional motivation for high resolution analysis of the output of a
cavity detector of dark matter axions, since it produces a prominent narrow peak in the energy
spectrum. The width of the peak determines the signal to noise ratio of a high resolution
search for it. The width of the peak is the energy dispersion of the Big Flow. Here we place an
upper limit on the energy dispersion of the Big Flow from the observed sharpness of the 5th
caustic ring. The same upper limit applies to the Little Flow.
The best lower limit on the sharpness of the 5th caustic ring is obtained by considering a
triangular feature in the Infrared Astronomical Satellite (IRAS) and Planck map of the Galactic
plane [48]. The feature is in a direction tangent to the 5th caustic ring. The gravitational
fields of caustic rings of dark matter leave imprints upon the spatial distribution of ordinary
matter. Looking tangentially to a caustic ring, from a vantage point in the plane of the ring,
one may have the good fortune of recognizing the tricusp shape [30] of the cross-section of
a caustic ring. The maps are shown in Fig. 4-4. The triangular feature is correctly oriented
with respect to the galactic plane and the galactic center. Its position is consistent within
measurement errors with the position of the sharp rise in the Galactic rotation curve due to
the 5th caustic ring. If the velocity dispersion of the flow forming the 5th caustic ring were
large, the triangular feature in the IRAS map would be blurred. The sharpness of the triangular
112
feature in the IRAS map implies that the 5th caustic ring is spread over a distance less than
approximately 10 pc.
6.5.1 Upper Limit on the Big Flow Energy Dispersion
The particles forming a caustic ring fall in and out of the galaxy near the galactic plane.
Let E and L be respectively the energy and angular momentum of the particles that form the
5th caustic ring and are in the galactic plane. We have
E =1
2vr(r)
2 +L2
2r2+ V (r) (6–65)
where vr(r) is the radial velocity of the particles at galactocentric radius r and V (r) is the
gravitational potential in which they move. We will use
V (r) = v2rot ln(r) (6–66)
consistent with a flat galactic rotation curve, with rotation velocity vrot. For the Milky Way,
vrot ≃ 220 km/s. The inner radius a of a caustic ring is the distance of closest approach to the
galactic center of the particles in the galactic plane. Therefore
E =L2
2a2+ V (a) . (6–67)
Small deviations dE, dL and da from the average values of E, L and a are related by
dE =LdL
a2− L2
a3da+
dV
dr(a)da . (6–68)
In view of Eq. (6–66) this may be rewritten
da =a
v2rot − vϕ(a)2(dE − LdL
a2) (6–69)
where vϕ(r) = L/r is the velocity in the direction of galactic rotation of the particles that are
in the galactic plane. The spread δa in caustic ring radius is therefore given by
(δa)2 ≡< (da)2 >=a2
(v2rot − vϕ(a)2)2[(δE)2 +
L2
a4(δL)2 − 2
L
a2< dE dL >] (6–70)
113
where, as in Section 6.3, brackets indicate averaging over the d2NdEdL
distribution of the dark
matter particles, δE ≡√< (dE)2 > and δL ≡
√< (dL)2 >. The second term in the square
brackets on the RHS of Eq. (6–70) dominates over the first term since
a2δE
LδL=
avavvϕ(a)Dav
(6–71)
and vav is of order vϕ(a) whereas Dav is much larger than a, as was discussed in Section 6.3.
The second term also dominates over the third term since
< dE dL > ∼ δE δLa
R<< δE δL . (6–72)
Hence
δa ≃ σvvϕ(a)Dav
v2ϕ(a)− v2rot≃ σv
426 km/sDav (6–73)
where we used vϕ(a) = 520 km/s [31]. It remains to estimate Dav, the average distance from
the galactic center where the processes took place by which the flow presently constituting the
5th caustic ring acquired velocity dispersion. Of course it is hard to give a precise value. The
present turnaround radius of the flow constituting the 5th caustic ring is R5 = 121 kpc [31].
As a rough estimate, we set Dav ∼ R5/2. With δa . 10 pc, this yields
σv . 71 m/s . (6–74)
In ref. [48], an upper limit on σv was estimated using δa ∼ Rvσv with R the turnaround radius
and v the velocity of the flow at the caustic. This yielded σv . 53 m/s. Although far more
work went into justifying Eqs. (6–73) and (6–74), the two estimates are qualitatively consistent
since Dav ∼ R and v = vϕ(a). The difference between the two estimates may be taken as a
measure of the uncertainty on the bound. At any rate, the bound on σv from the sharpness
of caustic rings is extraordinarily severe in view of the commonly made assertion that the dark
matter falling onto a galaxy is in clumps with velocity dispersion of order 10 km/s.
114
To obtain an upper limit on the energy dispersion δE = mvavσv of the flow forming the
5th caustic ring we set vav ∼ 300 km/s. This yields
δE
m. 2.4 · 10−10 . (6–75)
If the axion frequency is 1 GHz, as in the example given in the Introduction, the upper limit on
the widths of the peaks associated with the Big Flow and the Little Flow in the cavity detector
of dark matter axions is of order 0.24 Hz. Let us emphasize that there is nothing to suggest
that the upper bound is saturated.
6.5.2 Velocity Dispersion Ellipse of the Big Flow
In the Caustic Ring Model of the Milky Way halo, the Big Flow has a large density on
Earth as a result of our proximity to a cusp in the 5th caustic ring of dark matter. The inner
radius of the 5th caustic ring, derived from a rise in the Milky Way rotation curve and from
the triangular feature in the IRAS map of the Galactic plane, is a = 8.31 kpc whereas its outer
radius is a+ p = 8.44 kpc [48]. See Fig. 6-2.
These values assume that our own distance to the Galactic center, which sets the scale,
is 8.5 kpc. Note that the values of the inner and outer radii are at the tangent point of our
line of sight to the 5th caustic ring. If axial symmetry is assumed, as in the Caustic Ring
Model, we are just outside the tricusp cross-section of the 5th caustic ring, near the cusp at
the outer radius, which we call henceforth the ‘outer cusp’. In that case there are two flows on
Earth associated with the 5th caustic ring, the Big Flow and the Little Flow. The uncertainty
on the density of the Big Flow is large since it is sensitive to our distance to the outer cusp.
It should be noted that although axial symmetry may be a very good approximation for the
overall structure of the Milky Way halo, it may not be a good approximation in predicting the
position of the Sun relative to the nearby caustic ring. This is because caustic rings should
not be expected to be exactly circular nor exactly centered on the galactic coordinate system.
Axial symmetry could be sufficiently broken that we are located inside the tricusp instead of
just outside. If we are located inside the tricusp, there are four flows on Earth associated with
115
Figure 6-2. Relative position of the Sun and the 5th caustic ring in the Caustic Ring Model ofthe Milky Way halo. The Sun’s position is indicated by the dot. The x-axis isparallel to the Galactic plane. The tricusp shape is the cross-section of the 5thcaustic ring. The model assumes axial symmetry. Because of axial symmetrybreaking, the size of the tricusp and the position of the Sun relative to it may differfrom what the figure shows.
the 5th caustic ring. If we are inside the tricusp and near the outer cusp three of the flows
are Big Flows and the fourth is the Little Flow. The Little Flow is the same as before. It does
not participate in the outer cusp singularity. The Big Flows do participate in the outer cusp
singularity. The densities, velocities and velocity dispersions of the Big Flows are given by the
equations in Section 6.4.1 as a function of position relative to the cusp. The Big Flows all have
comparatively large dispersions in the component of velocity perpendicular to the symmetry
axis of the cusp, i.e. their velocity ellipses are elongated in the direction perpendicular to the
Galactic plane.
116
6.6 Summary
Motivated by the prospect that dark matter may some day be detected on Earth, we set
out to predict properties of the velocity dispersion ellipsoid of the Big Flow. The Big Flow
dominates the local dark matter distribution in the Caustic Ring Model of the Milky Way halo
due to our proximity to the 5th caustic ring of dark matter in our galaxy. We analyzed a cold
collisionless stationary flow in one dimension and derived how the velocity dispersion changes
along such a flow. In one dimension the problem is simple because energy conservation and
Liouville’s theorem control the outcome. To make headway in two dimensions we assumed
that the potential in which the particles fall is axially symmetric as well as time-independent
so that both energy and angular momentum are conserved. We derive the evolution of the
velocity dispersion ellipse along a stationary axially symmetric flow under those assumptions.
The local velocity dispersion always becomes large when approaching a caustic because the
density becomes large but the phase space density is constant. We introduced a technique
for obtaining the leading behavior of the velocity dispersion near caustics, and applied the
technique to fold and cusp caustics. Finally we used our results to obtain an upper limit on the
energy dispersion of the Big Flow from the observed sharpness of the 5th caustic ring and a
prediction for the dispersion in its velocity components.
117
CHAPTER 7NEW ASTROPHYSICAL BOUNDS ON ULTRALIGHT AXIONLIKE PARTICLES
Motivated by tension between the predictions of ordinary cold dark matter (CDM)
and observations at galactic scales, ultralight axionlike particles (ULALPs) with mass of the
order 10−22 eV have been proposed as an alternative CDM candidate. We consider cold and
collisionless ULALPs produced in the early Universe by the vacuum realignment mechanism
and constituting most of CDM. The ULALP fluid is commonly described by classical field
equations. However, we show that, like QCD axions, the ULALPs thermalize by gravitational
self-interactions and form a Bose-Einstein condensate, a quantum phenomenon. ULALPs, like
QCD axions, explain the observational evidence for caustic rings of dark matter because they
thermalize and go to the lowest energy state available to them. This is one of rigid rotation
on the turnaround sphere. By studying the heating effect of infalling ULALPs on galactic disk
stars and the thickness of the nearby caustic ring as observed from a triangular feature in the
IRAS map of our galactic disk, we obtain lower-mass bounds on the ULALP mass of order
10−23 and 10−20 eV, respectively.
7.1 Ultralight Axionlike Particles
Ultralight axionlike particles (ULALPs) with a wide range of masses between 10−33 eV ≤
m ≤ 10−18 eV, the so-called “Axiverse”, [126, 127, 128], are predicted in string theory-based
extensions of the standard model. They are dark matter candidates as well, with properties
similar to the QCD axion but much lighter. If sufficiently light, they suppress structure
formation on small scales [129, 130, 131, 132, 133, 134, 135, 136] because they have a Jeans
length [40, 41],
ℓJ = (16πGρm2)−14 = 1.01× 1014 cm
(10−5 eV
m
) 12
(10−29 g/cm3
ρ
) 14
(7–1)
This chapter is a reproduction of [125], with permission from The American PhysicalSociety.
118
where ρ is the ULALP mass density.
Numerous bounds have been placed on the ULALP mass using observational data. V.
Lora et al. [137] found that the mass range 0.3 × 10−22 < m < 10−22 eV provides a best fit
to the properties of dwarf spheroidal galaxies. The cosmic microwave background anisotropy
observations require m > 10−24 eV [138]. Numerical simulations of structure formation with
ULALP dark matter have been carried out and found to give a good description of the core
properties of dwarf galaxies when m ∼ 10−22 eV [139, 140]. ULALP dark matter with mass
m ∼ 10−21 eV was found to alleviate the problems of excess small-scale structure that plague
ordinary cold dark matter [141]. Recent work finds that data on the Draco II and Triangulum II
dwarf galaxies are best fit by a ULALP with mass m ∼ 3.7− 5.6× 10−22 eV [142], while other
authors obtain m < 0.4 × 10−22 eV from Fornax and Sculptor data [143]. Because ULALPs
cause reionization to occur at a lower redshift, it has been argued that reconstruction of the
UV-luminosity function restricts m & 10−22 eV [144].
L. Hui et al. [145] have recently written a comprehensive overview of the ULALP
literature, including a discussion of the relaxation of ULALP dark matter in gravitationally
bound objects. Relaxation (also known as thermalization) is the key ingredient for Bose-Einstein
condensation to occur. Ref.[146] have studied the effects of Bose-Einstein condensate of
ultralight dark matter on the propagation of gravitational waves. We show in Section 7.3 how
the analysis of Ref. [145] relates to the earlier results in Refs. [24, 25, 147, 27].
A relic ULALP population can be produced thermally (generated from the radiation bath)
or nonthermally by the vacuum realignment mechanism. The former class of ULALPs behaves
like dark radiation. In this paper, we are solely interested in ULALPs generated by the vacuum
realignment mechanism, which behave like cold dark matter.
In Section 7.2, we show that ULALPs thermalize via gravitational self-interactions and
hence form a Bose-Einstein condensate, in a manner analogous to QCD axions. This is
something that the previous works have not taken into account. In Section 7.3, we estimate a
lower mass bound on ULALPs by requiring that the infalling ULALPs do not excessively heat
119
the galactic disk stars. In Section 7.4, we obtain a stronger bound from the sharpness of the
fifth caustic ring of the Milky Way galaxy which appears as a triangular feature in infrared
astronomical satellite (IRAS) and Planck maps. Section 7.5 provides a summary.
7.2 Bose-Einstein Condensation of ULALPs
In Chapter 2 we showed that QCD axions form a BEC because the following conditions
are satisfied: i) the system is composed of a huge number of identical bosons, ii) these
particles are highly degenerate, iii) their number is conserved, and iv) they thermalize.
Bose-Einstein condensation means that most of the particles go to the lowest energy state
available through the thermalizing interactions. We show below that the cold ULALP fluid
produced by the vacuum realignment mechanism satisfies all four conditions.
The ULALP is described by two parameters: its mass m, which sets the time when
the ULALP field begins to oscillate in the early Universe and its decay constant f , with
dimensions of energy, which sets the magnitude of its initial misalignment and the strength of
its interactions. Unlike the case of QCD axions, m and f are independent parameters. Also,
unlike the case of QCD axions, the ULALP mass is taken to be temperature independent.
The time when the ULALP field starts to oscillate is of order t1 ≡ 1/m. The ULALPs
produced by the vacuum realignment mechanism [20, 21, 22] have at that time number density
n(t1) ∼ mϕ21, where ϕ1 is the value of the field then. ϕ1 is of order the decay constant f .
After t1, the number of ULALPs is conserved. So their number density at later times is
n(t) ∼ mϕ21
(a(t1)
a(t)
)3
, (7–2)
where a(t) is the cosmic scale factor. By demanding that the ULALPs make up the majority of
CDM, we relate their initial number density to their mass,
n(t1) ∼ρc,0m
(t0teq
)2(teqt1
)3/2
, (7–3)
where t0 and ρc,0 are the age of the Universe and cold dark matter density today, respectively.
We assumed that t1 is in the radiation-dominated era, which requires m > 2 × 10−28 eV. We
120
have therefore
ϕ1 ∼√ρc,0t0
(mteq)14
∼ 3× 1017 GeV
(10−22 eV
m
) 14
. (7–4)
If inflation does not homogenize the ULALP field, the cold ULALPs produced by vacuum
realignment have momentum dispersion of order
δp(t) ∼ 1
t1
(a(t1)
a(t)
), (7–5)
and hence their average state occupation number is
N ∼ (2π)3
4π/3
n(t)
(δp)3∼ 5× 1098
(10−22 eV
m
) 52
. (7–6)
If inflation does homogenize the ULALP field, the momentum dispersion is smaller yet, and the
average quantum state occupation number is higher. Therefore, the ULALPs certainly form a
highly degenerate Bose gas.
The decay rate of ULALPs into two photons, in analogy with the QCD axion, is of order
Γaγγ ∼ 1
64π
(απ
)2m3
f 2=
1
2.5× 10110 sec
( m
10−22 eV
)3(1017 GeV
f
)2
(7–7)
where α is the fine structure constant. The ULALP is thus stable on the time scale of the
age of the Universe. More generally, because all ULALP number changing interactions are
suppressed by one or more powers of 1/f , the number of ULALPs is conserved on time
scales of order the age of the Universe. So, the third condition for ULALP Bose-Einstein
condensation is satisfied as well.
The fourth condition is that ULALPs thermalize on a time scale shorter than the age of
the universe. We call the time scale over which the momentum distribution of the ULALPs
changes completely as a result of their self-interactions the relaxation time τ . The relaxation
rate is Γ ≡ 1/τ . ULALPs certainly have gravitational self-interactions but perhaps also λϕ4
self-interactions. If thermalization occurs, it occurs in a regime where the energy dispersion
δω is smaller than the relaxation rate Γ. Indeed, since Eq. (7–5) gives an upper limit on the
121
momentum dispersion, we have
δω(t) ∼ (δp(t))2
2m<
1
2mt21
(a(t1)
a(t)
)2
= H for t < teq
=3
4
(teqt
) 13
H for t > teq (7–8)
and H < Γ is necessary for thermalization to occur. The condition δω < Γ defines the
“condensed regime”.
In the condensed regime, the relaxation rate due to λϕ4 self-interactions is of order
[24, 25]
Γλ ∼ |λ|n4m2
. (7–9)
For QCD axions, |λ| ∼ m2
f2 . If we assume the same holds true for ULALPs and furthermore
that ϕ1 ∼ f , we have
Γλ(t)/H(t) ∼ n(t1)
2f 2
(a(t1)
a(t)
)3
t ∼(t1t
) 12
for t < teq
∼√t1teqt
for t > teq . (7–10)
Thus we find that, like QCD axions, the ULALPs may briefly thermalize through their λϕ4
self-interactions when they are first produced by vacuum realignment in the early Universe but
that they will at any rate stop doing so shortly thereafter.
In the condensed regime, the relaxation rate due to gravitational self-interactions is of
order [24, 25]
Γg ∼ 4πGnm2ℓ2 (7–11)
where ℓ ≡ 1/δp is the correlation length. Since Eq. (7–5) gives an upper limit on the
momentum dispersion, and we assume that ULALPs constitute most of the dark matter, we
have
Γg(t)/H(t) ∼ 3H
2
n(t)m2
ρtot(t)
1
(δp(t))2& 3H
2m
ρ(t)
ρtot(t)
(a(t)
a(t1)
)2
122
&√
t
teqfor t < teq
&(t
teq
) 13
for t > teq , (7–12)
where ρ = nm. We used the Friedmann equation to relate the total density ρtot to the Hubble
constant. Equation (7–12) shows that, independently of the ULALP mass, the ULALP fluid
thermalizes by gravitational self-interactions at the time of equality, or earlier if the ULALP
field was homogenized by inflation.
It was shown in Ref. [25] that QCD axions that are about to fall into a galactic halo
thermalize sufficiently fast by gravitational self-interactions that they almost all go the lowest
energy state available to them consistent with the total angular momentum they acquired from
tidal torquing by galactic neighbors. That lowest energy state is one of rigid rotation on the
turnaround sphere. It was shown in Ref. [32] that this redistribution of angular momentum
among infalling dark matter axions explains precisely and in all respects the properties of
caustic rings of dark matter for which observational evidence had been found earlier. The
observational evidence for caustic rings of dark matter in the Milky Way and other isolated
disk galaxies is summarized in Ref. [31]. Furthermore, it was shown in Ref. [36] that this
redistribution of angular momentum solves the galactic angular momentum problem, which
is the tendency of cold dark matter, in numerical simulations of structure formation, to be
too concentrated at galactic centers. The fact that Bose-Einstein condensation of the dark
matter particles explains the observational evidence for caustic rings and solves the galactic
angular momentum problem and the fact that Bose-Einstein condensation is a property of dark
matter axions but not of the other dark matter candidates, constitute an argument that the
dark matter is axions, at least in part [148]. Although these studies [25, 32, 36] were motivated
primarily by QCD axions, they do not depend in an essential way on the axion mass and apply
equally well to any axionlike dark matter candidate produced by the vacuum realignment
mechanism, including ULALPs. To summarize, if ULALPs are the dark matter, they thermalize
by gravitational interactions and form a Bose-Einstein condensate at or before the time of
123
equality between matter and radiation. They thermalize sufficiently fast before falling onto
galactic halos to acquire quasirigid rotation on the turnaround sphere. They then form caustic
rings with properties that are in accord with observations. The observational evidence for
caustic rings therefore supports the hypothesis that the dark matter is ULALPs but also, as we
will see, constrains that hypothesis.
We conclude this section with a discussion of the results of Ref. [145] on ULALP
relaxation in gravitationally bound objects. We will show that the relaxation rate obtained
there is consistent with the earlier results on axion relaxation in Refs. [24, 25]. The relaxation
rate in the particle kinetic regime is [25]
Γ ∼ n σ δv N (7–13)
where n is the particle density, σ is the scattering cross section, δv is the velocity dispersion,
and N is the degeneracy (i.e.,the average occupation number of those particle states that are
occupied). The relevant cross-section for relaxation by gravitational interactions in the particle
kinetic regime is [25]
σg ∼4G2m2
(δv)4. (7–14)
The average quantum degeneracy is
N ∼ (2π)3n4π3(mδv)3
. (7–15)
Combining Eqs. (7–13 - 7–15) gives the relaxation rate by gravitational interactions in the
particle kinetic regime
Γg ∼ 24π2 (Gρ)2
m3(δv)6. (7–16)
For the special case of a gravitationally bound object, the crossing time is of order tcr ∼√
3π4Gρ
and the size of the object is of order r ∼ δvtcr/π. The relaxation time is then of order
τg =1
Γg
∼ 2
27r4m3(δv)2 . (7–17)
124
Ignoring the numerical prefactor, which is at any rate poorly known, this is the result given in
Ref. [145] for gravitationally bound systems. Hui et al. consider several systems that may relax
in the particle kinetic regime and find either that they do not relax or that relaxation has no
observable consequences. Whether the ULALP fluid relaxes or not is of paramount importance
in determining its behavior because it does not obey classical field equations when it relaxes.
We found earlier in this section, following the discussion in ref. [25], that the ULALP fluid does
relax before it falls into galactic halos. It was our purpose in this paragraph to show that the
general discussion of cosmic axion relaxation in Ref. [25] is consistent with both the relaxation
rate for gravitationally bound objects given in ref. [145] and our claim that the ULALP fluid
relaxes before falling into galactic halos.
7.3 Bound from ULALP Infall
Provided no thermalization is occurring [25, 37], ULALP dark matter is represented by a
wave function solving the Schrödinger-Poisson equations. The wave function may be written in
terms of a real amplitude A and a phase β:
Ψ(r, t) = A(r, t)eiβ(r,t) . (7–18)
In the linear regime of the growth of density perturbations, before multistreaming occurs,
Eq. (7–18) describes a flow of density,
n(r, t) = (A(r, t))2 (7–19)
and velocity,
v(r, t) =1
m∇β(r, t) . (7–20)
In particular, a homogeneous flow of density n and velocity v is described by
Ψ(r, t) =√n ei(p·r−ωt), (7–21)
where p = mv is the momentum and ω = p2/2m is the energy of each particle. An attractive
feature of the wave function description is that it readily accommodates multistreaming [43].
125
In a region with two homogeneous cold dark matter flows, with densities ni and velocities
vi (i = 1, 2), the wave function is
Ψ(r, t) =√n1 e
i(p1·r−ω1t) +√n2 e
i(p2·r−ω2t−δ) . (7–22)
The associated density is
n(r, t) = |Ψ(r, t)|2 = n1 + n2 + 2√n1n2 cos(∆p · r −∆ωt− δ) , (7–23)
where ∆p = p2 − p1 and ∆ω = ω2 − ω1. The interference term is important for ULALP
dark matter because the correlation length ℓ = 1|∆p| is large (since m is small and ∆v is
generally fixed by observation) and gravity is long range. The gravitational field sourced by the
interference term in Eq. (7–23) is
g = −8πG√n1n2 mℓ sin(∆p · r −∆ωt− δ) n (7–24)
where n = ℓ∆p. It is the gravitational fields associated with the interference terms in the
ULALP fluid density that cause the ULALP dark matter thermalization discussed in the
previous section [25].
An isolated galaxy such as our own Milky Way continually accretes surrounding dark
matter. There are on Earth two flows of dark matter falling in and out of the Galaxy for
the first time (n =1), two flows falling in and out for the second time (n = 2), two flows
falling in and out for the third time (n = 3), and so forth. The total number of flows on
Earth is of order the age of the Universe divided by the time it takes a particle initially at rest
at our galactocentric distance to fall through the Galaxy and reach the opposite side, i.e.,
1010 yr/108 yr = 100. The flows of particles that fell into the halo relatively recently (n . 10)
have not been thermalized as a result of gravitational scattering off inhomogeneities in the
Galaxy [149]. Those flows and their associated caustics are a robust prediction [28] of galactic
halo formation with cold dark matter.
126
For n . 10, the density of each flow is of order 2% or 3% [149, 47, 31] of the total
average dark matter density at our distance from the Galactic center; i.e., each flow has density
of order ρfl ∼ 10−26 gr/cm3. Since the typical difference in velocities between pairs of flows is
of order 10−3c, the coherence lengths associated with the interference terms are of order
ℓ ∼ 1
10−3m∼ 64 pc
(10−22 eV
m
)(7–25)
and their correlation times are of order
T ∼ 1
10−6m∼ 2× 105 year
(10−22 eV
m
). (7–26)
The gravitational fields sourced by the interference terms are of order
g ∼ 4πGρfl ℓ ∼ 0.5km
sec×Gyear
(10−22 eV
m
). (7–27)
A star in the galactic disk acquires from each pair of flows a velocity increment of order gT
after a coherence time T . The total velocity acquired by a star is the result of a random walk
in its velocity space, and of order
∆v ∼ gT
√t0T
√1
2N(N − 1) ∼ 0.5
km
sec
(10−22 eV
m
) 32
(7–28)
where t0 ∼ 1010 yr is the age of the Universe and N ∼ 20 is the number of flows with density
ρfl. Note that ∆v is not sharply sensitive to our assumption on the number of flows since ρfl
and g are inversely proportional to N , whereas the number of flow pairs is proportional to
N2. The thin disk of the Milky Way is made of stars with vertical velocity dispersion σz ∼ 20
km/s−1. We obtain a lower bound on the ULALP mass
m & 10−23 eV (7–29)
by requiring ∆v < σz. The result obtained here is broadly consistent with the discussion of the
thickening of the Galactic disk by ULALP dark matter in Ref. [145].
127
It was observed in Ref. [150] that ULALP dark matter has an oscillating pressure/tension
with angular frequency equal to twice the ULALP mass and that this oscillating pressure/tension,
being a source of gravity in general relativity, has measurable effects. The idea presented here
is similar but makes use of Newtonian gravity and multistreaming instead.
7.4 Bound from the Sharpness of the Nearby Caustic Ring
Cold, collisionless dark matter lies in six-dimensional phase space on a thin three-dimensional
hypersurface, the thickness of which is the velocity dispersion of the dark matter particles. We
refer to this hypersurface as the “phase space sheet”. As dark matter particles fall in and out
of a galactic gravitational potential well, their phase space sheet wraps up. Locations where
the phase space sheet folds back onto itself have large particle density in physical space [47],
[29], [30], [28]. Indeed, the phase space sheet is tangent to velocity space at the location of
such folds, and therefore the physical space density diverges there in the limit of vanishing
sheet thickness. For small velocity dispersion, the physical density at the location of a fold is
finite but very large. These locations of high density are called caustics. They are generically
two-dimensional surfaces in physical space. The kinds of caustics that appear are classified by
catastrophe theory.
It was shown [28] that galactic halos built up by the infall of cold collisionless dark matter
have two sets of caustics, inner and outer. There is one inner and one outer caustic for each
value of the integer n introduced in the previous section. The catastrophe structure of the
inner caustics depends on the angular momentum distribution of the infalling particles. If the
velocity field of the infalling particles is dominated by net overall rotation (∇ × v = 0), the
inner caustics are closed tubes of which the cross section is a section of the elliptic umbilic
(D−4) catastrophe [30], called caustic rings for short. Evidence for caustic rings of dark matter
was derived from a variety of observations. The evidence is summarized in Ref. [31]. The
evidence is explained in all its aspects if the dark matter is axions or axionlike particles such as
ULALPs [32].
128
The caustic ring model of the Milky Way halo was independently tested against
observations in Ref. [151]. Recently, Y. Huang et al. [152] found evidence for the second
and third (n = 2,3) caustic rings in our Galaxy from their measurement of the Milky Way
rotation curve out to very large (100 kpc) radii.
Caustic rings of dark matter produce sharp rises in galactic rotation curves [30]. Part of
the observational evidence for caustic rings is that sharp rises appear in the Milky Way rotation
curve [153] at radii consistent with the predictions for caustic ring radii by the self-similar
infall model [48]. Furthermore, there is a triangular feature in the IRAS map [154] of the
Galactic plane of which position on the sky matches the position of the sharp rise in the Milky
Way rotation attributed to the caustic ring (n = 5) nearest to us. The triangular feature
appears in the direction of galactic coordinates (l, b) = (80,0). It is seen also in the recent
Planck observations [1]; see Fig. 4-4. In Fig. 7-1 we show the panoramic view of the Milky
Way provided by the IRAS 12 µm observations. The triangular feature is clearly visible on the
left-hand side of the image. It can be understood as the imprint of the nearby caustic ring of
dark matter upon the gas and dust in the Galactic disk [48].
The sharpness of the triangular feature implies that its edges are not smoothed on
distances larger than approximately 10 pc. Because velocity dispersion smooths caustics, the
sharpness of the triangular feature implies an upper limit of order 50 m/s [48, 156] on the
velocity dispersion of the flow (n = 5) that produces the nearby caustic ring. As we show now,
it also implies a lower limit on the mass of ULALP dark matter, because the wave description
smooths caustics as well.
Consider the simple fold caustic that occurs at a caustic ring radius, where the particles
that fall in along the galactic plane reach their distance of closest approach to the galactic
center before falling back out. In the particle description, the particles satisfy the equation of
motion
md2r
dt2= −dVeff(r)
dr(7–30)
129
where r is the distance to the galactic center and Veff(r) is the effective potential for radial
motion, including the repulsive angular momentum barrier. In the neighborhood of the caustic,
the radial velocity of the particles is
vr(r) = ±√
− 2
m
dVeffdr
(a)(r − a) (7–31)
where a is the caustic ring radius. Conservation of the number of particles implies that the
density near the caustic is proportional to
n(r) ∝ 1
r2|vr(r)|. (7–32)
The density diverges therefore at r = a as 1/√r − a, which is the characteristic behavior of
a simple fold caustic. Spreading of the caustic due to velocity dispersion is discussed in Ref.
[156].
In the wave description, the behavior of the dark matter fluid is given by a wavefunction
Ψ(r) that satisfies the Schrödinger equation. At the location of the fold discussed in the
previous paragraph, the radial part of the wave function satisfies
− ~2
2m
1
r2d
drr2dΨ(r)
dr+ Veff(r)Ψ(r) = EΨ(r) (7–33)
with E = Veff(a). Near r = a, the wave function is proportional to an Airy function [155]
Ψ(r) = C1
rAi(γ(a− r)) (7–34)
with
γ =
[−2m
~2dVeffdr
(a)
] 13
. (7–35)
The wave function smooths the fold caustic over a distance scale γ−1. The sharpness of the
triangular feature in the IRAS map associated with the fifth caustic ring implies an upper
bound on γ−1 of order 10 pc. The dominant contribution to −dVeff
dr(a) is the centrifugal force
mvϕ(a)2/a where vϕ(a) ≃ 500 km/s is the velocity component of the flow producing the fifth
130
caustic ring in the direction of galactic rotation. We obtain the bound
m & 10−20 eV (7–36)
by requiring γ−1 . 10 pc. Finally we note that high- resolution observation of the nearby
caustic provides, in principle, a means to determine the ULALP mass since the gravitational
field in the plane of the ring near r = a has characteristic structure on the length scale
γ−1 ∝ m−2/3.
Figure 7-1. Panoramic view of the Milky Way Galactic plane from the IRAS experiment in the12 µm band.
7.5 Summary
In this chapter, we have explored ULALPs, an alternative cold dark matter candidate
discussed by many authors. We showed that ULALPs thermalize through gravitational
self-interactions and form a Bose-Einstein condensate in a manner analogous to QCD axions.
This was not taken into account in the previous literature.
We placed new constraints on the mass of ULALPs. First, we considered the heating
effect of infalling ULALPs on the disk stars of the Milky Way galaxy. We derived therefrom a
lower bound of the ULALP mass, which is comparable in strength to the previous bounds from
structure formation and cosmic microwave background data.
Our tightest bound comes from taking into account the Bose-Einstein condensate nature
of ULALPs. In a way analogous to the QCD axion, Bose-Einstein condensation enables the
ULALP dark matter to acquire net overall rotation and hence form caustic rings with the
tricusp cross section. From the observed sharpness of the triangular feature due to the fifth
caustic ring in the Milky Way, we derive a bound on the spread of the ULALP wave function
and, in turn, on the ULALP mass. The bound we obtain, ma & 10−20 eV, disfavors a large
131
fraction of the ULALPs from being the dark matter. It is in tension with recent works that find
m ∼ 10−22 eV fits the data best.
132
CHAPTER 8CONCLUSIONS
There are overwhelming observational evidences that tell us that around 85% of the
matter in the Universe is as yet undetected and is referred to as dark matter. QCD axions
is one of the leading dark matter candidates. They are remarkably different from other dark
matter candidates because they thermalize by gravitational self-interactions and form a
Bose-Einstein condensate when the CMB temperature is of order 500 eV. On time scales
shorter than the thermalization time, axions behave like ordinary dark matter candidates
and their evolution can be described by classical field equations. However, on time scales of
order the thermalization time, the axions move towards a Bose-Einstein distribution and their
evolution can only be described by quantum field equations.
On imparting angular momentum to a system of axion BEC, vortices will appear. These
vortices can combine to give rise to bigger vortices. A dark matter halo composed of axion
BEC dark matter, around a disk galaxy such as our Milky Way, will have a big vortex along
the axis of rotation. The big vortex will modify the specific angular momentum distribution of
the infalling baryons and provide a better fit to observation. The vortices in axion BEC dark
matter can give rise to vorticity in baryons. During the recombination era, this baryon vorticity
will seed primordial magnetic field at the galactic scales. The resulting magnetic field intensity
is greater than those that are generated from second order perturbation theory and is sufficient
for amplification to the present day observed value by battery mechanism.
In Chapter 6, we obtained an upper limit on the energy dispersion of the Big Flow from
the observed sharpness of the 5th caustic ring and a prediction for the dispersion in its velocity
components. Finally in Chapter 7, we showed that ULALPs also form a BEC. From the heating
effect of infalling ULALPs on galactic disk stars and the thickness of the nearby caustic ring
as observed from a triangular feature in the IRAS map of our galactic disk, we obtained
lower-mass bounds on the ULALP mass of order 10−23 and 10−20 eV, respectively.
133
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BIOGRAPHICAL SKETCH
Nilanjan Banik was born to Mita and Narayan Banik in 1987 in Kolkata, India. He
grew up in Mumbai, India. From his early childhood he showed great interest in science and
mathematics. His father greatly encouraged him towards science and math by designing
homemade experiments, telling him about stars, planets and the Moon, going for walks to
the beach in Mumbai and discussing mathematics and encouraging him to ask challenging
questions. As a child, Nilanjan was interested in chemistry, so much so that he setup his own
chemistry laboratory in his seventh grade in the storage room of his house in Mumbai. He
conducted several experiments like titration, electrolysis, salt analysis etc.
Gradually, his interest shifted towards physics since he was more interested in understanding
how and why the chemical reactions were taking place. He started dabbling with quantum
physics and special relativity. His father taught him calculus, vector analysis and 3D geometry,
based on which he self-taught himself vector calculus. He joined Indian School of Mines in
2006 to pursue a bachelor’s in Mechanical Engineering and graduated with a 1st class. While
studying mechanical engineering, he realized that his true passion was theoretical physics and
that he wanted to pursue a career in it. Due to the dearth of theoretical physics faculty in his
engineering college, he spent his summer break of 2008 in Harish Chandra Research Institute,
Allahabad, India as a visiting student, studying about the accelerated expansion of the Universe
under L. Sriramkumar. In August 2010 he joined University of Florida to pursue a PhD in
Physics. He worked with Pierre Sikivie on axion dark matter and how they affect structure
formation in the Universe. While in Gainesville, Florida, he met his lovely wife Yahaira,
who greatly supported him in his life and career. In 2015, Nilanjan was awarded a Fermilab
Graduate Fellowship in Theoretical Physics, with this he spent a year in Fermilab working with
Scott Dodelson on galactic dynamics and Dark Energy survey. This greatly helped Nilanjan
in focusing towards data driven Cosmology as his future endeavor. Nilanjan lives happily with
his wife, their three dogs (Calculus, Tensor and Mozart) and one chinchilla named Einstein
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in Gainesville, Florida. In September 2017 he will move to the Netherlands to start a joint
postdoctoral position in GRAPPA, University of Amsterdam and University of Leiden.
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