A Tale of Two Particles
MASSACHUSETTS INSTMITE
OF TECHNOLOGY
by
AU6 15 2014
Katelin Schutz
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Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
Bachelor of Science in Physics
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2014
@ Katelin Schutz, MMXIV. All rights reserved.
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Author
..................................
Department of Physics
May 9, 2014
..
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Certified by.
...........................
A.
David Kaiser
Germeshausen Professor of the History of Science
Senior Lecturer, Department of Physics
Thesis Supervisor
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.............................
Tracy Slatyer
Assistant Professor of Physics
Thesis Supervisor
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.....................
Nergis Mavalvala
Senior Thesis Coordinator
A Tale of Two Particles
by
Katelin Schutz
Submitted to the Department of Physics
on May 9, 2014, in partial fulfillment of the
requirements for the degree of
Bachelor of Science in Physics
Abstract
It was the earliest of times, it was the latest of times, it was the age of inflation, it
was the age of collapse, it was the epoch of perturbation growth, it was the epoch
of perturbation damping, it was the CMB of light, it was the dwarf galaxy of darkness, it was the largest of cosmic scales, it was the smallest of Milky Way subhalos,
we had multiple nonminimally coupled inflatons before us, we had inelastically selfinteracting dark matter before us, we were all going direct to the Planck scale, we
were all going direct the other way. Motivated by apparent discrepancies between the
standard theory and observation, we analyze two astrophysical systems in the context
of new particle physics. Taking a phenomenological approach, we calculate observable
consequences of novel particle models during two different stages in the development
of our universe. First, we explore the possibility that nonminimally coupled multifield inflation can generate a large primordial isocurvature fraction and account for
the "low-multipole anomaly" in the Cosmic Microwave Background. Second, we consider the effects of dark matter that inelastically self-interacts to determine the effect
on the structure and abundance of Milky Way satellites and dwarf galaxies. The disparity of time and energy scales examined in this thesis serves to highlight the range
of ways to use observables in the sky as a probe of new particle physics that may be
elusive at current experiments on the ground.
Thesis Supervisor: David Kaiser
Title: Germeshausen Professor of the History of Science
Senior Lecturer, Department of Physics
Thesis Supervisor: Tracy Slatyer
Title: Assistant Professor of Physics
2
Acknowledgments
They say that luck favors the prepared. While being an undergraduate at MIT
made me work harder than I ever could have imagined, I also recognize that I was
astonishingly lucky, particularly with regard to the people who helped me get to
where I am today. In fact, my biggest motivation for writing a senior thesis was this
acknowledgement section, since I am not required to write a thesis for my degree.
So for all of you reading this, pay attention to this section because the science that
follows is really just the icing on the cake, the victory lap, the symbol that I entered
MIT as a kid and have emerged as a real scientist. My ability to complete this work
is the direct product of the ample guidance from the people I am about to mention.
First and foremost, I must thank my family. My mother tells me that when I was
small, she used to think to herself "God, please don't let me mess this up." I think
it's clear now that she and my father have done quite the opposite by raising me the
way they did. Growing up, I always had the support that I needed to do whatever
I was interested in doing. I had a huge library of books and I spent my summers at
nerd camp learning number theory and philosophy. When I was bored by the pace
of my 6th grade math class and began quickly losing interest in science, my parents
fought the school so that I could take algebra and physics with the 8th graders. My
parents also forced me to do things that normal kids do, like playing team sports.
Though I resisted at the time, looking back on those experiences I am so grateful that
I had a proper childhood and that I learned basic skills like how to work in a team
and how to deal with other people who have different experiences from my own. In
particular, when my dad had me play on a boys' little league baseball team with my
brother and his friends, I had to learn grit in overcoming their teasing (one of the
most clever and observant remarks I often heard was that I "throw like a girl.") I
believe that the reason I was able to stay interested in math and science, even though
those subjects are "for boys," was because I was already used to being "the girl,"
and that I had learned ways of dealing with that. And most importantly, my parents
gave me a loving home and plenty of happy memories. Genetics aside, I would not
3
have gotten to where I am without my parents and the sacrifices that they made in
order to be good parents.
Before coming to MIT, there were many people who made me who I am, and I
want to briefly mention them here. There was John Lawrence, who taught me the
value of my own weirdness and showed me how to succeed in spite of real adversity.
I didn't know him for long, thanks to a long-fought battle with cancer, but his role
as my teacher and coach at the critical age of thirteen had a profound and lasting
influence on my values in life. I would like to thank Chuck Fujita for forcing me to
use my brain instead of reaching for my TI-89. I acknowledge coach Mancuso, who
taught me the true meaning of hard work. I also wish to thank Ray Perez for all the
life lessons learned and for helping me establish my personal motto: Sic Volvo. I want
to thank Rhonda Brown and Eliza Coyle for being like second mothers to me. And
finally, I wish to thank Gabby Perez and Lucy Coyle; though three of us have such
different talents and passions, their trailblazing in other fields continually inspires me
to be a more complete, well-rounded person.
When I arrived at MIT, the first friend I made turned out to be the most important friend of my life so far. I was a part of the first-ever PhysPOP, a freshman
pre-orientation program for people interested in physics. I found my peers in the
FPOP quite discouraging; many of them behaved as many MIT freshmen behave
(particularly freshmen who went to prestigious high schools) and were eager to show
how much they knew already. Feeling alienated by this, I befriended the PhysPOP
counselors, and in particular I befriended Adrian Liu. It is because of Adrian that I
decided to major in physics and he was the one who first got me interested in cosmology. After a talk he gave during PhysPOP, I was so fascinated with his research that
I sought out a UROP with Adrian's thesis advisor, Max Tegmark, who was working
on building a new kind of radio telescope. I really must acknowledge Max because he
gave me a chance to work in his lab; I have no idea what he saw in me, as I had no
clue what a Fourier transform even was at the time. Yet he chose to take me on as
a UROP student, and during my time in that group I learned a staggering amount
of physics, particularly from Adrian's summer cosmology boot camp. I ultimately
4
left the group because I was interested in more theoretical endeavors, but I stayed in
touch with the group, particularly with Adrian who went on to co-teach me quantum
mechanics. Though Adrian began a postdoc in California after my sophomore year,
he has been my closest friend, my co-conspirator, my moral compass, my shoulder to
cry on, my biggest cheerleader, and the love of my life. I learn new things from him
every day, and I hold him responsible for a significant portion of my education as a
physicist and as a person.
I also want to thank the people that made MIT home for me these past four
years, namely B Entry: Chris Kelly, Jamal Elkhader, Molly Kozminsky, Mary Knapp,
Andy Liang, Kirsten Hessler, Jan Sontag, Michael Pearce, Jon Allen, Anne Kim, Nick
Arango, Clarissa Towle, Nick Mohr, Lauren Wright, the list of crazy kids goes on.
But of all these people I especially must acknowledge Emily Nardoni, who was both
my suitemate and my physics sherpa. I followed her in a path that she forged for
herself, always a year ahead of me. It is because of her that I took Alan Guth's
Early Universe course, which solidified my decision to be a cosmologist. As a strong,
beautiful woman doing theoretical physics, I always had her as a role model and I
always had someone who I could vent to or ask for advice. Seeing the sheer amount
of work she put into becoming a physicist was a huge inspiration to me and made me
feel that the struggle would be worth it in the end.
Next, I wish to acknowledge my academic advisor, Allan Adams, to whom I
partially owe my career. He has been such a role model for me that if someone were
to ask me what I want to be when I grow up, I would probably reply "Allan Adams."
Time and time again I have gone to him when I was struggling most, and every time
he has picked me up off the metaphorical floor and set me back on the right path.
Sometimes, when I did not believe in myself, his encouraging emails were the only
thing that made me keep trying to follow my dream of becoming a physicist. When
I struggled with test anxiety, he had me practice taking tests under time pressure in
his office until I had the confidence to do well. When I had personal issues, he offered
to meet me at Flour and talk about it. In summary, he has gone out of his way to be
the most supportive advisor I could ever ask for, and I honestly do not know whether
5
I would be going to graduate school if he had not stepped in when he did.
I must also thank my UROP advisors, who helped me grow to love doing research. As I already mentioned, working for Max Tegmark perturbed my academic
trajectory in the direction of cosmology, which seems obvious in retrospect because
of his infectious enthusiasm for the subject. I also want to thank the Omniscopers
for fun times spent in lab and in West Forks, namely Eben Kunz, Nevada Sanchez,
Jon Losh, Ashley Perko, Hrant Gharibyan, Victor Buza, Josh Dillon, Yan Zhu, Andy
Lutomirski, and Shana Tribiano.
Next, in my sophomore year Alan Guth agreed to give me a spot in his newlyforming undergraduate group. That opportunity was one of the most important in
my life, and he casually offered me the spot with no questions asked. This highlights
one of my favorite things about him: despite his fame and the high quality of his
work, he agreed to take me on as a sophomore with relatively little experience. He is
definitely the most humble person I have ever met, and I am sure the thought never
crossed his mind that I might be unqualified at that stage in my physics education. I
will never forget how when he won the $3M Fundamental Physics Prize, his reaction
was not to plan an extravagant celebration with the canonical champagne but rather
he proposed that he order extra cheese on the pizza for our group meeting; that is
one of my fondest memories of working in the group, and I will always admire his
superlative humility. He has never been in this for the glory, but rather is a true
scholar of the physics, and I think that is the best possible motivation.
Having David Kaiser as my primary UROP advisor was a real stroke of luck. He
never tried to push me past my breaking point, but rather let me explore topics at
my own pace and with my own self-motivation. I think this internal drive that I
learned while working with him will be incredibly useful for my future in academia. I
also recognize that he went out of his way to be supportive of me. He offered to read
papers that I spent hours writing for classes and he took a general interest in my work
outside the context of the UROP. He was very positive and praised hard work, and
he would tell me on good research days that I should order celebratory-takeout-sushi.
He also promoted our publication to research colleagues and highlighted my work to
6
the professors who would be writing my letters of recommendation, which I'm sure
played a huge role in getting accepted to graduate school and getting fellowships.
Finally, I want to thank my most recent UROP advisor, Tracy Slatyer. Not only is
she a brilliant rising superstar in theoretical physics, but she has also been a fantastic
role model to me. As a female theorist (I believe she is only the second that MIT has
ever hired as faculty) she has been a great influence on the Center for Theoretical
Physics and her arrival tripled the number of female theory UROPs. She is also an
incredibly encouraging and nice human being, and personally it has given me hope
that I too can become a physics professor some day without sacrificing certain aspects
of my identity.
Now, I want to briefly thank the people at MIT who contributed to my experience
here, but whose roles do not have as obvious a location in the narrative structure of
this acknowledgements section. I want to thank my sophomore-year-problem-setbuddy and Junior Lab partner Olivia Mello, as well as all the friends who toiled with
me late at night solving quantum mechanics problems. I want to mention the beautiful
ladies of Sigma Kappa, whose support helped get me through the rough times, and
whose accomplishments in other areas continually inspire me; in particular, I'd like to
acknowledge Tonia Tsinman, Danielle Gorman, Vicky Gong, Angela Chu, Jean Xin,
Jackie Sly, Sarah Bindman, Ranna Zhou, Caro Roque, Danielle Chow, and Carolina
Lopez-Trevino. I want to acknowledge one of the smartest people I know and my
partner-in-snark, Ravi Charan, for useful conversations and insights. I want to thank
some of the amazing, strong women whom I have looked to as role models: Gabriella
Martini, Yan Zhu, Sarah Geller, and Chanda Prescod-Weinstein. I am grateful to
Dan Roberts and Jenny Schloss, whose advice helped me win the Hertz Fellowship
and secure funding for the next five years of graduate school. I must mention that
the MIT Physics Department would not be the wonderful and friendly place that it
is without Nancy Savioli, Cathy Modica, Krishna Rajagopal, and Ed Bertschinger.
Additionally I wish to thank some of the great professors I've had here: Peter Fisher,
Scott Hughes, Barton Zwiebach, Nergis Mavalvala, Iain Stewart, Jesse Thaler, Rob
Simcoe, and Bob Jaffe. Many of the ideas in this thesis were discussed at great
7
length with my friend and co-author, Evangelos Sfakianakis. I also wish to thank
people whose helpful conversations contributed to this work: Bruce Bassett, Jolyon
Bloomfield, Rhys Borchert, Xingang Chen, Stephen Face, Doug Finkbeiner, Illan
Halpern, Mark Hertzberg, Carter Huffman, Edward Mazenc, Neil Weiner, and many
others.
8
Contents
Introduction
13
1.1.1
Problems with the Standard Hot Big Bang
. . . . . . . . . .
13
1.1.2
The Inflationary Resolution
. . . . . . . .
. . . . . . . . . .
18
1.1.3
Other Inflationary Signatures
. . . . . . .
. . . . . . . . . .
21
Dark Matter . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . .
25
1.2.1
Proof for the Existence of Dark Matter . .
. . . . . . . . . .
25
1.2.2
Dark Matter Particle Physics
. . . . . . .
. . . . . . . . . .
29
1.2.3
Cosmological Structure Formation . . . . .
. . . . . . . . . .
34
.
.
.
.
.
. . . . . . . . . .
.
1.2
Inflationary Cosmology . . . . . . . . . . . . . . .
.
1.1
11
.
1
2 Multifield Inflation after Planck: Isocurvature Modes from Nonminimal Couplings
2.1
36
Introduction ...................
2.2 M odel ...................
. . . . . . . . . . . . . . . .
39
2.2.1
Einstein-Frame Potential.....
. . . . . . . . . . . . . . . .
40
2.2.2
Coupling Constants ........
. . . . . . . . . . . . . . . .
42
2.2.3
Dynamics and Transfer Functions . . . . . . . . . . . . . . . .
45
. . . . . . . . . . . . . . . .
50
2.3.1
Geometry of the Potential . . .
. . . . . . . . . . . . . . . .
50
2.3.2
Linearized Dynamics . . . . . .
. . . . . . . . . . . . . . . .
53
Results . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
59
2.4.1
Local curvature of the potential
. . . . . . . . . . . . . . . .
59
2.4.2
Global structure of the potential. . . . . . . . . . . . . . . . .
62
.
.
Trajectories of Interest ...........
.
2.4
37
.
2.3
. . . . . . . . . . . . . . . .
9
64
2.4.4
CMB observables . . . . . . . . . . . . . . . . . . .
66
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .
71
.
.
Initial Conditions . . . . . . . . . . . . . . . . . . .
.
2.5
2.4.3
3 Self-Scattering for Dark Matter with an Excited State
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . .
80
3.2
Dark Matter with Inelastic Scattering . . . . . . . . . . . .
. . . .
83
3.2.1
A Simple Model . . . . . . . . . . . . . . . . . . . .
. . . .
83
3.2.2
Approximate Wavefunctions . . . . . . . . . . . . .
. . . .
84
The Scattering Cross Sections . . . . . . . . . . . . . . . .
. . . .
88
3.3.1
Semi-Analytic Results . . . . . . . . . . . . . . . .
. . . .
88
3.3.2
Features and Limits of the Scattering Cross Sections
. . . .
90
3.4 Applications to Dark Matter Haloes . . . . . . . . . . . . .
. . . .
99
.
. . . .
99
3.5
.
.
.
.
.
Parameter Regimes of Phenomenological Interest
.
3.4.1
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .
.
3.3
.
3.1
.
79
4 Conclusions
. . . . 101
121
10
Chapter 1
Introduction
Particle physics has long been a data-driven endeavor. However, in recent years
we have reached a stage where bigger and better experiments come with a price tag of
millions (if not billions) of dollars, where international collaborations have hundreds
(if not thousands) of scientists pouring countless hours into accounting for every
possible systematic source of uncertainty. There is no doubt that these experiments
will probe and constrain many existing theories, and potentially discover a great deal
of exciting new physics. However, new particle theories are cheap, and constraining
the huge expanse of model-space is quite a daunting task if high-energy experiments
are the only source of such constraints.
The famed cosmologist Yakov Zel'dovich once referred to our universe as "the
poor man's [particle] accelerator" [1]. Along those lines, it will be necessary to continue making use of the full range of cosmic observables and to take advantage of
the wealth of new cosmological data to guide the next generation of high energy experiments. Indeed, new particle physics can have significant effects on major events
in the cosmological timeline; it turns out to be very difficult to make a universe like
ours, and there are all kinds of ways that things can go wrong. Really, cosmology is
the ultimate mystery, one that demands no less than a thoroughly consistent story,
and where tiny changes to the theory have implications on cosmic scales. In this way,
cosmology provides additional leverage onto figuring out what our universe is made
of and what underlying physics governs our existence.
11
There are gaping holes in our understanding of the universe, and plugging those
holes demands new particle physics beyond the Standard Model. For instance, we do
not know what the majority of our universe is made of; many independent observations confirm that ordinary baryonic matter (the stuff that makes up planets, stars
and galaxies) only accounts for roughly 5% of the energy content of our universe,
with dark matter making up roughly 25% and dark energy making up around 70%.
Furthermore, we do not fully understand how cosmological structures (such as galaxies and clusters of galaxies) formed in the way that they did, and whether unknown
particle physics played a role in their development. Additionally, the recent BICEP2
experiment provided strong indication of cosmic inflation as the leading paradigm for
the early universe. However, while the general mechanism is well-confirmed, we still
do not know what kind of particle was responsible for our early universe's exponential
expansion.
With such clear footholds onto physics beyond the Standard Model, present-day
cosmologists have a unique opportunity to synthesize a broad base of knowledge
from astrophysics and particle physics, and that is the overarching theme of this
thesis. In particular, the thesis will highlight how we can use astrophysics as a probe
of fundamental physics on the largest observable scales as well as on the smallest
cosmological scales.
The remainder of this chapter will give brief, self-contained introductions to our
basic understanding of cosmic inflation, dark matter, and structure formation. Chapter 2 will explore the possibility that certain large-scale anomalies in the Cosmic
Microwave Background can be explained by primordial perturbations from one class
of inflationary models. Chapter 3 reviews problems with our understanding of smallscale structure formation and their possible solution by dark matter self-interaction,
and develops an original semi-analytic model where the dark matter scattering can
be inelastic. Concluding remarks follow in Chapter 4.
12
1.1
Inflationary Cosmology
In this section, we attempt to provide a brief summary of inflationary cosmology [2],
which is by no means intended to be comprehensive. The objective is to introduce the
features and key observables of inflationary models, as well as some of the concepts
that will be necessary in later chapters.
1.1.1
Problems with the Standard Hot Big Bang
The conventional Big Bang model was originally developed as a theory to explain
the fact that the universe seems to be expanding. In the Big Bang theory, the earlytime behavior of the universe is simply determined by extrapolating the expansion
backwards in time.
The theoretical framework for an expanding universe comes naturally from Einstein's theory of General Relativity. We generally make two assumptions about our
universe: that there is no special point in space or time, and that there is no special
direction. The most general metric for a spacetime with those features (homogeneity
and isotropy) is the Friedmann-Robertson-Walker (FRW) metric,
ds 2 = g,,dx"dx" = -dt
2
+ a(t)2
(
r2 + r 2 d0 2 + r2 sin
9d)2
(1.1)
where a(t) is the expansion scale factor of the spacetime, as depicted in Figure 1-1.
,40(
*Notcb..O
Figure 1-1: An illustration of the changing scale factor for an expanding universe.
Here "notches" correspond to comoving coordinates. The comoving coordinates are
unaffected by the expansion, but the physical distances change by a factor of a(t).
Image source: Alan Guth.
13
=
1+
(p
3p).
(1.5)
The parameter k can be chosen to be either +1, -1, or 0 because we can always rescale
the radial cobrdinate, r, and those values of k determine whether the the geometry
is spatially closed, open, or flat as depicted in Figure 1-2.
Positive Curvature
Negative Curvature
Flat Curvature
Figure 1-2: From left to right, k = +1, k = -1, and k = 0 universes.
In a homogeneous and isotropic spacetime, the stress-energy tensor must be that
of a perfect fluid,
TL = (p + P)UIUV + pg1,,
(1.2)
where p is the pressure, p is the energy density, and u" is the fluid 4-velocityl. In
General Relativity, the Einstein Field Equations relate the stress-energy tensor to the
spacetime metric by
R14 - -Rglv = 87rGT,,
2
(1.3)
where R,,,, is the spacetime Ricci tensor and R is the spacetime Ricci scalar. The 00
component of this tensor equation gives the Friedmann equation
(&'2
87rGp
3 -
-
k
(1.4)
and the trace of this tensor equation gives the Friedmann acceleration equation
d
a
4irG
3
3
The term on the left hand side of Equation 1.4, &/ais often referred to as the Hubble
parameter, H and is responsible for the famous Hubble expansion law, v = Hr, which
'Note that in this thesis, we set c = h = kB = 1. In this chapter, we include factors of G but in
later chapters we will work in Planckian units and set G = 1.
14
was the original inspiration for the Big Bang theory.
Extrapolating the Friedmann expansion backward in time leads to certain observable predictions such as the existence of the Cosmic Microwave Background (CMB)
and primordial nucleosynthesis. The discovery of the CMB as a Big Bang relic and
the observational confirmation of elemental abundances as predicted by Big Bang Nucleosynthesis made the Big Bang the most successful theory in its day. However, to
quote Alan Guth [3], "the classic Big Bang theory says nothing about what banged,
why it banged, or what happened before it banged." In fact, the term "Big Bang"
was intended to be pejorative, as it was coined by Fred Hoyle, a proponent of the
"steady state" theory. There are other major problems with the conventional Big
Bang theory, which we describe further below.
The Horizon Problem
One well-known problem with Big Bang cosmology is that the uniformity of our
universe would require extreme fine-tuning. For instance, the CMB is uniform to 1
part in 10,0000, and yet at the time of the CMB emission the different patches on the
sky were not in causal contact with each other. Given our knowledge of cosmological
Our present universe
the hot
ang
Figure 1-3: An artist's illustration of the horizon problem. At the time of the emission of the CMB, the past light cones of different CMB patches could not have intersected according to the hot Big Bang model. However, we observe those causallydisconnected regions to be uniform to 1 part in 100000. How could this be? This plot
was reproduced from [4].
15
parameters, a quick calculation shows that the physical size of our observable universe
at the time of CMB emission was 0(100) times greater than the distance that light
could have traveled since the Big Bang [2]. Put another way, if one looks at one
part of the CMB and then looks in the opposite direction, then light would have had
to travel 0(100) times faster in order for those pieces to have ever exchanged any
information with each other since the Big Bang. Therefore, the observed uniformity
of our universe can not be the result of any equilibration process.
Of course, if we postulate that the universe was originally incredibly uniform,
then we can achieve a universe that ends up being this uniform. However, there is
no theoretical reason why the Big Bang singularity should be perfectly uniform, and
such a scenario would have to be "tailor-made" to fit our own universe. The necessity
of fine-tuning the initial perturbations is called the horizon problem.
The Flatness Problem
Another well-known problem is that our universe is spatially flat to within 1%, which
is dynamically unstable, which we demonstrate as follows. If we refer to Equation
1.4, we find that k = 0 at the critical energy density
3H 2
Pc =
8rG
(1.6)
If we define Q = p/pc, we find that
Q_-1
3k
Q
87rGpa 2
(1.7)
p = -3H(p + p),
(1.8)
p = wp
(1.9)
Conservation of stress-energy dictates that
so assuming an equation of state
16
gives the time-dependence of p:
p(t) oc a(t)-(w+l)
(1.10)
so
oc a(t)3w+1.
Relativistic matter (such as photons) has w
=
(.1
j and nonrelativistic matter has w
=
0
(as the perfect fluid description of nonrelativistic matter is that of pressureless dust.)
Therefore, a homogeneous and isotropic universe with a perfect fluid description must
have a fractional curvature that grows linearly at the very least. Another quick calculation using well-measured cosmological parameters shows that in order for the
universe to be flat to within 1% today, at 1 second after the Big Bang (when the
temperature of the universe was at the well-understood energy scale of the electron
mass), I - 11 < 10-1 8 [2].
Even if we are skeptical of our precision measurements of Q = 1 (since historically,
astronomers before the 1990s thought that the universe was open with Q ~ 0.3), the
mere existence of our universe as we know it today points to incredibly fine-tuned
initial conditions, due to the instability of slight deviations from flatness. If our early
universe was ever-so-slightly more closed, then the universe would have collapsed
before structure formation and the evolution of life could occur. Conversely, if our
early universe was ever-so-slightly more open, then the universe would have expanded
too quickly for structures to coalesce, and again life as we know it could not have
evolved [2].
There is no physical law that forbids our universe from being exactly flat from the
very beginning. However, there is no apparent reason for such extreme fine-tuning,
and in principle the curvature could take on any value in the conventional Big Bang.
The necessity of fine tuning the initial energy density of our universe is known as the
flatness problem.
17
The Monopole Problem
The final problem of Big Bang cosmology is slightly more subtle, but was the original motivation for inflation, so it deserves at least a cursory mention. Essentially,
the simplest Grand Unified Theories (GUTs) are constructed so that spontaneous
symmetry breaking gives masses of 1016 GeV to SU(5) gauge bosons that unify the
electroweak and strong interactions. This mass scale is chosen so that the coupling
strengths of the strong, weak, and electromagnetic forces are unified in the Minimal
Supersymmetric Standard Model. At the GUT scale, magnetic monopoles are generically produced by the Kibble mechanism in large abundance as topological defects
in the Higgs fields. One can calculate the relic monopole energy density, and one
finds that magnetic monopoles would give Q ~ 100, which is unacceptably large and
would cause the universe to end in a fraction of a second [2]. Therefore, if any of
the machinery of theoretical particle physics is to be taken seriously, then we have
another serious problem for the conventional Big Bang.
1.1.2
The Inflationary Resolution
If the universe were full of some type of matter with constant energy density then we
would be able to solve all of the aforementioned problems, as we will now show.
The Flatness Problem
For matter with constant energy density, Equations 1.8 and 1.9 require w = -1. The
equation of state of w = -1
causes Q -+ 1 by Equation 1.11. So with this kind of
matter, the geometry is driven to being flat, so inflation says that we should expect
to see k = 0. In other words, regardless of the initial geometry (which could, in
principle, be anything), the expansion of the universe makes any patch (such as our
own observable universe) look locally flat [2].
18
The Horizon Problem
If we plug in a constant energy density po into Equation 1.4 and set k = 0 (as
previously discussed), then the solution is
a(t) oc
,
which is the expansion governing a de Sitter spacetime.
(1.12)
This exponential expan-
sion bolsters the claim that k -+ 0, since the scale factor that drives k to zero gets
exponentially larger. Furthermore, this exponential expansion means that any inhomogeneities prior to inflation are exponentially suppressed by the expansion. This is
smoothing is known as the "cosmological no-hair theorem," and shows how a uniform
de Sitter metric can arise without any fine-tuning of the initial conditions [2].
Furthermore, in an inflationary scenario, the entire observed universe comes from
a single region that coherently underwent inflation, meaning that it was causally
connected to itself prior to inflation and had time to equilibrate. In fact, the inhomogeneities we observe in the CMB do not come from the Big Bang initial conditions
but rather from quantum perturbations in the field responsible for inflation. These
quantum fluctuations in the field get stretched by the expansion into the seeds of
large scale structures that eventually developed into galaxies and galaxy clusters.
The Monopole Problem
In the standard scenario, the Kibble Mechanism produces such a great abundance
of monopoles that the universe would collapse back on itself. However, if inflation
happened after or during the production of the monopoles, then they were diluted to
such an extent as to make their contribution to the density of the universe negligible.
During inflation, the volume of the universe increased by a factor of at least O(1080),
which would explain why we do not see a universe filled with monopoles. As long as
inflation occurs below the GUT scale, we avoid this problem entirely [2].
19
The Inflaton
Fortunately, we can easily realize a state with constant energy density using quantum
field theory, using a scalar field that we will refer to as the inflaton. We first saw
experimental evidence for a scalar field at the LHC when the discovery of the Higgs
boson was announced [5]. For a number of subtle reasons, the inflaton is probably
not the Higgs, though determining whether they can be the same particle is an active
area of research.
The action for a homogeneous and isotropic scalar field 0 in a background metric
with potential energy V(#) is
s =
d v/t-x
g vO" ka# - V(#) .
(1.13)
If we plug in the FRW metric and vary the action with respect to the field q, we
arrive at the Euler-Lagrange equation, also known as the equation of motion for the
field 0:
q + 3H4 + V'(0) = 0.
(1.14)
This is analogous to the equation for a harmonic oscillator, with the strength of the
"drag" term set by the Hubble parameter. Physically this term is akin to a friction
term, but instead of converting an object's kinetic energy to heat, the field's kinetic
energy is put towards the expansion of the universe.
If we instead vary the action with respect to the spacetime metric, we find the
Einstein Field Equations from Equation 1.3 with
T, =
aJ$0,V
-
g*6oaflo + V()).
g,
(1.15)
Recalling that To = p and Tij = gijp, we find that
p = 2 #2 + V(0), p = 2 52- V(g).
(1.16)
Suppose that we are in a regime where 02 <V(q)- this physically corresponds
20
to a situation where the field is in a false vacuum state, meaning that the field is in
a semistable configuration away from its potential minimum, as shown in Figure 1-4.
Such a situation would (to good approximation) give w = -1, exactly as we desire,
V
Figure 1-4: An example of a potential which supports inflation in the asymptotically
flat region, which is a false vacuum. Here, the red dot indicates the field's value and
is completely analogous to the position of a classical ball slowly rolling down this
potential to the minimum. Since inflation is caused by a quantum field, we formulate
the quantum effects as perturbations about the classical trajectory.
and the consequences of that can solve the problems with the traditional Big Bang
theory. This general mechanism is known as inflation and (to date) it has been the
most successful theoretical paradigm for understanding the early universe.
1.1.3
Other Inflationary Signatures
There are numerous potentials and models that support an inflationary period in the
early universe, and it would be impossible to provide an exhaustive list of all the novel
observables corresponding to those models. However, there are a few salient observables that are generically predicted by inflation in a relatively model-independent way.
In this subsection, we discuss the key observables and determine a means to calculate
them in the context of the most basic single-field, slow-roll inflationary models.
21
The Power Spectrum
Though inflation smooths the initial pre-inflationary inhomogeneities (because of the
cosmological no-hair theorem) it turns out to also source the perturbations in the
density field of the post-inflationary universe. We cannot escape the fact that inflation is driven by a quantum field, and therefore there will be deviations from the
trajectory determined by the classical equations of motion. Perturbations in the field
will cause spatially varying amounts of inflation to occur, and the extent to which
the spacetime has been stretched will dilute the matter to varying degrees. Therefore
inflation generically predicts density perturbations, which will eventually form large
scale structures, such as galaxy clusters.
The two point correlation function is the easiest way to statistically analyze density
perturbations, 6(Y), in the early universe. With a density field defined by
p(s) = Po(1 + 6())
(1.17)
the correlation function () is the spatial correlation for two density features separated by f,
~()
=
(
+()6(z+f)),
(1.18)
where angular brackets denote a spatial average. If we assume that the perturbations
are gaussian (which they would generically tend to be due to the central limit theorem)
then we can completely specify the statistics of the perturbations by looking at the
two-point function because we can Wick expand any other statistical measure (such
as the three-point function). Additionally, if we assume that the perturbations are
homogeneous and isotropic, we can completely specify their statistics using a single
function, the power spectrum. Following [6], the correlation function in a volume V
22
can be written using its Fourier transform,
~
d3x j(g) j(- + r)
f d3xf
= 1
V
=
I
(2ir
(2_r
I
3 ,
Vd 3 k f Vd k'-
(27)3
(2r)3J
'
(1..9,
i).
(1.20)
(
=
d3k Jdk (k)I e*'
where in the last line we have used the fact that
(2r)3
= (
dar et-Ir
We therefore define the power spectrum as
P(k) a
where we only have dependence on k
(k)
(1.21)
because we invoke isotropy.
=k
One key property of the power spectrum is its functional dependence on the
wavenumber, k. The spectral index of density perturbations, n, is defined as 2
d In P(k)
dlnk,(1.22)
so if the power spectrum is a power law in k, then it is
P(k) = A, k".
(1.23)
For example, a scale-invariant spectrum is defined as having n, = 1 so that perturbations of all wavelengths have the same amplitude when they re-enter the horizon
[6]. Inflationary models generically predict a nearly scale-invariant spectrum with
n ,< 1, which concords with the reported spectral index from the recent Planck
satellite, n, = 0.9603
0.0073 [7].
2
Note that we are using the powers spectrum of density perturbations rather than the power
spectrum of perturbations to the gravitational potential, <. Poisson's equation gives an extra factor
of k 4 in that case, which will give a different definition for the spectral index.
23
If we believe that the power spectrum deviates from a power law in k, we can
further define the "running of the spectral index,"
dn
dln'
d ln k'
(1.24)
as well as a "running of the running of the spectral index," and so on. Observational
constraints on the presence of a nonzero running are fairly strong [7], though some
new evidence (which we will explain when we discuss gravitational waves produced
by inflation) suggests a higher running than previously thought.
Finally, it is possible that our assumption of gaussian density perturbations was
incorrect, and the effects of primordial nongaussianity have been a popular topic of
study in recent years. However, nongaussianity is highly constrained by Planck [8],
which tends to favor the simplest inflationary models.
Gravitational Waves
The exponential expansion of spacetime by a factor of roughly 1030 at a time 10-36
seconds after the Big Bang was an unimaginably violent process. Therefore, it seems
only natural that inflation should cause large perturbations in the fabric of spacetime
itself. Such a phenomenon is a unique prediction of inflation and has not yet been
accounted for by any of the competing models of the early universe. Therefore, when
the BICEP2 team announced the discovery of gravitational waves (tensor modes) in
the CMB [9], it was seen as the "smoking gun" for inflation.
Inflationary models generically predict a power law tensor mode power spectrum.
The BICEP2 experiment found that after subtracting foregrounds (such as dust), the
tensor-to-scalar ratio,
r
=
At
--
(1.25)
had a value of r = 0.16 at 5.9 a significance [9]. Interestingly, such a high value of
r is in tension with the measurements of other experiments (such as Planck) and it
remains to be seen whether those measured values will shift over time. However, such
tensions in the data can instantly be alleviated if a nonzero running, a, is invoked [9].
24
Therefore, at present there is a great deal of excitement in the field regarding both
the synthesis of these disparate datasets as well as attempting to find a unique model
of inflation that yields all the measured observables.
1.2
Dark Matter
The verification of the existence of dark matter is arguably the greatest triumph
of modem astronomy. Dark matter was proposed independently by Jan Oort and
Fritz Zwicky in the 1930s [10, 11], and since then we have gone on to corroborate
its presence time and time again. However, despite the fact that its existence has
been known for nearly a century, and despite its great abundance in our universe,
we embarrassingly still do not know what kind of particle makes up the dark matter.
In this section, we review some of the evidence that points to the existence of dark
matter, we describe one of the leading candidates and point out some problems with
the paradigm for how we treat cold dark matter.
1.2.1
Proof for the Existence of Dark Matter
Dark matter has been shown to exist in order to be consistent with observables on
length scales ranging from that of our own provincial galaxy all the way up to that of
our observable universe. We have so many independent measures of this that there is
very little room for other possibilities, such as modifications to Newtonian dynamics.
In this subsection, we will lay out some of the evidence for dark matter on various
scales.
Galaxies
Within our own Milky Way galaxy, the brightness pattern we observe suggests that
most of the mass should be concentrated at a central bulge, with a less-massive disk
that orbits about the bulge [12]. If we assume the contribution from the mass of the
disk to be negligible, then Newtonian dynamics indicates that the orbital velocity is
M
Vrot
r
25
(1.26)
In essence, we expect that at larger galactocentric radii, the galaxy should be rotating
at a slower rate than for small radii. However, as shown in Figure 1-5, our Milky
Galactic Rotation Curve
250 -k
-z
-o
150-
- My Experiment
Clemens et al (1985)
1 -cc 1000
cc
Keplerian Prediction
50-
0
1
2
3
4
5
Radius (kpc)
6
7
8
Figure 1-5: The rotation curve for the Milky Way galaxy, which the author measured
as part of a Junior Lab experiment. Also shown is the rotation curve as measured
by bona fide radio astronomers [13]. This curve as measured by the author was
determined by measuring the doppler shift of hydrogen in the interstellar medium at
different radii using the 21 cm hyperfine transition line.
Way has a fairly constant rotation speed even out to large radii. This means that
unless we abandon Newtonian dynamics, the assumption that we made about the
mass distribution in the Milky Way must be wrong.
In fact, to give a constant
rotation speed, one can show using Gauss' Law of gravity that one needs a spherical
density profile that goes like p oc r-2 . Ordinary baryonic matter cannot account for
this density profile, even if we are excessively generous with our mass-to-light ratios,
which indicates the presence of dark matter.
If we look outside of our own galaxy, we find similar rotation curves in other
galaxies where ordinary matter cannot give the right density profile. The ubiquity of
this situation makes galactic rotation curves an even stronger piece of evidence for
dark matter [14].
26
Galaxy Clusters
Galaxy clusters were one of the first motivations for dark matter, as Fritz Zwicky
invoked dark matter to account for the exceedingly high virial velocities within the
clusters. Specifically, he determined that even with excessively generous mass-to-light
ratios, the Coma Cluster would fly apart if not held together by an unseen source of
gravitational binding energy [11]. This turns out to be generically true for clusters,
which strengthens the case for dark matter modulo any modifications to Newtonian
dynamics [15].
Another way of measuring the mass of clusters that is independent of the Newtonian dynamics within the cluster is gravitationallensing from clusters. According
to Einstein's theory of general relativity, matter curves spacetime and that curvature
can distort light rays in a manner that is similar to the way that lenses bend light.
Therefore, by measuring the lensing of background light sources, we can infer the
mass of the corresponding gravitational lens that is causing the light to be bent. If
we try to "weigh" galaxy clusters in this manner, we find that there is much more
mass than can be accounted for by ordinary luminous matter [16].
The metaphorical "nail in the coffin" for modified theories of Newtonian dynamics
was the Bullet Cluster, as shown in Figure 1-6. In the image, we see two clusters
merging in a field of other background clusters. Shown in pink is data from the
Chandra X-ray telescope- the individual stars and galaxies within the clusters are
collisionless, so they fly right through each other, but the gas in the intergalactic
medium is highly collisional so it gets stuck and heats up which causes it to radiate
away its kinetic energy as X-rays [17]. Also shown in blue is the inferred mass from
gravitational lensing of the background clusters. Clearly the heated gas shown in pink,
which comprises the majority of the baryonic matter within clusters, cannot account
for the inferred lensing mass shown in blue. This result rules out modifications to
Newtonian dynamics at a reported confidence of 8o- [17]. Other such clusters have
been discovered that corroborate this result, such as the Train Wreck Cluster and the
Musket Ball Cluster [18, 19].
27
Figure 1-6: A composite Hubble Space Telescope image of the bullet cluster, with
observations from Chandra in pink and the mass inferred from lensing in blue. Image
source: NASA.
Baryon Acoustic Oscillations
The presence of dark matter even left its imprint on the largest scales in our observable universe. Before the CMB was emitted, the universe was a dense plasma,
opaque to light. This primordial plasma was comprised of photons and baryons, and
it underwent oscillations between periods of gravitational self-attraction and periods
of self-repulsion from radiation pressure. However, the primordial plasma only interacted with dark matter via gravity (and dark matter, by its very definition, does not
feel radiation pressure from photons) so the presence of dark matter had a significant
effect on the amplitude and scale of these oscillations in the primordial plasma, which
were imprinted on the CMB as shown in the angular power spectrum of Figure 1-7.
If we try to argue that the extra matter in galaxies and clusters is made of ordinary
baryons, then we face strong disagreement with the CMB [20].
After the CMB was emitted and overdense regions started to collapse, the imprint
left from the baryon acoustic oscillations affected the formation of large scale structures. Thus, galaxy surveys can also measure the baryon acoustic oscillations, albeit
at a very different distance scale. The results from galaxy surveys corroborate the
28
8000
Oa =
0.100
..................--
b =
0.075
--------
Ob =
0.048
=
0.025
a
-
WMAP 7-year data
~6000
:
0
2000
10
100
1000
Multipole moment 1
Figure 1-7: Baryon acoustic oscillations in the CMB angular power spectrum from the
interplay between gravitational attraction and radiation pressure. The amplitude and
size of the oscillations is highly sensitive to the amount of baryonic matter, Ob. Since
we know from inflation that Qtt = 1, and since the baryons only seem to comprise
around 5% of the energy content of the universe, there must be some non-baryonic
component of the energy density which prevents our universe from having an open
geometry. Dark matter turns out to account for roughly 25% of the energy budget
and dark energy accounts for the remaining 70%. This plot was reproduced from [20].
results from the CMB, and thus it is necessary for dark matter to exist in order to
explain the pattern of large scale structures that we observe [21].
1.2.2
Dark Matter Particle Physics
Now that we have established that the dominant source of matter in our universe is
non-baryonic, it remains to determine what dark matter could be made of. This is
one of the most obvious footholds we have onto physics beyond the Standard Model,
and as such it is imperative that we understand its particle properties. Understanding
the particle nature of dark matter is one of the major programs of current particle
physics research, and several experimental efforts are underway to try to detect dark
matter particles. In this subsection, we will discuss several dark matter candidates.
29
A Poor Candidate: Neutrinos
We have not yet shown that the existence of dark matter necessitates physics beyond
the Standard Model because of one known particle that fits all the criteria of dark
matter so far: the neutrino. Neutrinos are "dark" in the sense that they do not interact via electromagnetism, and neutrino oscillation experiments have demonstrated
that neutrinos have a nonzero rest mass. However, neutrinos cannot be the dark
matter, and the same can be said for any "hot" (relativistic) dark matter candidate;
[h-p1
AISO
Wavelength
1000
1]
10
1
1000
100-
0
C:
10 r
10
0Cosmk, Microwave Background
*SDSS galaxies
*Cluster abundance
n Weak lensing
ALyman Alpha Forest
0.001
0.01
0.1
Wavenumber k [h/Mpc]
1
10
Figure 1-8: The matter power spectrum, which is the Fourier decomposition of the
density field. The solid line marks the predictions of the standard model of cosmology,
whereas the dotted line shows the modification of replacing 7% of the dark matter
with 1 eV neutrinos. This plot shows that the contribution of these massive neutrinos
to the dark matter suppresses power on the galaxy-sized scales by roughly a factor of
two. This plot was reproduced from [22].
when structures were forming in the early universe, relativistic particles would have
free-streamed through the overdensitites that would have become galaxies, and those
overdensities could have never coalesced into galaxies. Put another way, lightweight
particles move at relativistic speeds that are much greater than the escape velocities
30
of newly-forming galaxies. Therefore, hot dark matter suppresses the formation of
structure and the mere existence of galaxies rules out neutrinos as being the dominant
component of dark matter [6].
Cold Dark Matter
We have established that the dark matter must be nonrelativistic, or "cold" in order
to clump and form cosmological structures. There are two main ways of achieving
this: either the dark matter particle must be very heavy, or it must have some way
of staying cold.
Heavy cold dark matter is easily achieved. One rather exotic possibility is that
dark matter is made of small primordial black holes; the parameter space for this is
extremely constrained, but there are some narrow ranges of parameters that escape
observational constraints [23]. Another more "vanilla" explanation is that dark matter
is just a heavy particle, and heavy particles generically abound in extensions of the
Standard Model [24].
One additional kind of cold dark matter is the axion, a particle which was proposed to explain why CP violation is exceedingly small in quantum chromodynamics.
Though axions are very light particles (with a mass of 1 eV or less) they are produced
nonthermally as a condensate and are decoupled from the Standard Model thermal
bath, and thus they remain cold. We will not discuss axions further, but they merit
a cursory mention as they are among the top candidates for dark matter [24].
The WIMP
One kind of proposed heavy dark matter is the weakly interacting massive particle
(WIMP). WIMPs are predicted by certain supersymmetric extensions of the Standard
Model, and are similar to neutrinos (both only interact via the weak force and gravity)
except for the fact that WIMPs are far more massive. Of all the cold dark matter
candidates, the WIMP is the most popular because of a phenomenon known as the
"WIMP miracle."
31
It would be reasonable to assume that heavy dark matter particles X can annihilate
to lighter particles that couple to the primordial plasma. If the dark matter remained
in equilibrium indefinitely, then it would all annihilate because of the Boltzmann
suppression
(1.27)
oc e~mx/T,
nx,
where nr is the number density of dark matter particles and mx is their mass. Therefore, unless the dark matter is totally sterile, there must be some means for falling
out of equilibrium in a process known as freeze out. Following [6], we can use the
Boltzmann equation to solve for the abundance over time
-3 d~x
a
2
=_U)(2
ni
d
-
(1.28)
n)
where n (ov) is the reaction rate for any general process with particle number density
n, interaction cross section a, and particle velocity v. Thus, the left hand side of
Equation 1.28 describes how the number density dilutes with the expansion of the
universe and the right hand side describes the interaction rate. This differential
equation is a Riccati equation and does not have an analytic solution, but we can
take some limits to understand the freeze out process. To simplify, we define the
comoving number density ne = nya3 , which simplifies the equation to
d nc
)
t= (ov) (n ,2 - n
dne da
4da d
dne a
-
= (ov)n,
(av) nc,
da nK
\2
nc
(
2
(1.29)
2feJ
(
1e H
n
n2
2
The ratio on the right hand side is a comparison of timescales, the equilibrium scattering rate n
e
(ov)
and the Hubble parameter H, which gives the expansion rate
of the universe. For early times, the scattering rate is much higher than the Hubble
expansion rate, and the solution to the differential equation is ne = nc,
e.
At late
times, the scattering rate is much lower than the Hubble rate and the solution is
32
n, = const. The transition between these behaviors is the freezeout process, and the
freezeout is sensitive to the particle physics of (uv). If we ask what o- would have to
be in order to give the right dark matter abundance, the answer is a weak scale cross
section! This is known as the "WIMP miracle" because it could have been absolutely
anything, and somehow it aligned with a previously-known scale in particle physics.
The WIMP miracle is one of the main reasons why WIMPs are the most popular
dark matter candidate, alongside the fact that WIMPs satisfy the other criteria of
being cold, fairly collisionless, and well-motivated by particle theory.
There has been a huge push in recent years to attempt to directly discover dark
matter in a detector or to produce it in a collider. However, those efforts are still
underway which motivates the further study of WIMPs and other dark matter candidates by looking at their astrophysical and cosmological signatures.
10 3910
0
0310
-3
4
1
01
...
:::4
2
4
.678910
2*C
1
-50
10"10
4C6
8
1
20
3d 0
WIMP mass [GeV/c2]
Figure 1-9: The latest direct detection results as reported by the CDMS collaboration.
The plot shows the 2c- confidence level upper bounds on the dark matter mass and on
its cross section with nucleons. The WIMP is either hiding somewhere in the allowed
regions of parameter space or we need to substantially re-evaluate our WIMP dark
matter models. This plot was reproduced from [25].
33
1.2.3
Cosmological Structure Formation
The way that structures (such as galaxies) formed in our universe is one of the aspects
of cosmology that is most difficult to model. The process of gravitational collapse of
structures is such a messy, nonlinear process with many factors that can affect the
outcome, and yet we see a very similar hierarchy of structures in different parts of our
universe. Evolving the nearly-smooth universe from the time of the emission of the
CMB up through the formation of galaxies is an exquisitely complicated endeavor, and
understanding this evolution requires huge simulations done on supercomputers, such
Figure 1-10: The results of a recently-developed hydrodynamic simulation which
starts 12 million years after the Big Bang and simulates 13 billion years of the evolution of structure. The left side of this picture is data from the Hubble Space Telescope,
and the right side of this picture is a simulated image from the code. This picture
was reproduced from [26].
as the one depicted in Figure 1-10. Because we do not yet have high-precision predictive power about galaxy formation and the like, there is a great degree of freedom
for exploring the consequences of novel physics on the development of cosmological
structures.
The standard procedure for doing simulations of structure formation is to populate
a universe with seed perturbations and evolve those perturbations accordingly. The
initial perturbations, as previously mentioned, typically come from the inflationary
power spectrum.
The evolution of perturbations is typically carried out using a
paradigm known as ACDM, where the A indicates the presence of a cosmological
34
constant (also known as dark energy) and where CDM denotes cold dark matter. In
particular, simulations typically assume that the dark matter is collisionless, which
is a good approximation for a "vanilla" WIMP because the weak scale cross section
is so small that the dark matter self-scattering is negligible. The growth of structure
tends to be hierarchical, meaning that small structures collapse first and then larger
structures form from mergers. ACDM works rather well for predicting the statistics
of large scale structure, but has problems with predicting smaller scale structures, like
for Milky Way subhalos as described in Chapter 3. It could be that our simulations
are not good enough yet (for example, it might be important to include the effects of
baryonic matter), or there could be some exotic dark sector physics that affects the
growth of structure.
35
Chapter 2
Multifield Inflation after Planck:
Isocurvature Modes from
Nonminimal Couplings
Recent measurements by the Planckexperiment of the power spectrum of temperature
anisotropies in the cosmic microwave background radiation (CMB) reveal a deficit of
power in low multipoles compared to the predictions from best-fit ACDM cosmology.
If the low-e anomaly persists after additional observations and analysis, it might
be explained by the presence of primordial isocurvature perturbations in addition
to the usual adiabatic spectrum, and hence may provide the first robust evidence
that early-universe inflation involved more than one scalar field. In this paper we
explore the production of isocurvature perturbations in nonminimally coupled twofield inflation. We find that this class of models readily produces enough power in
the isocurvature modes to account for the Planck low-i anomaly, while also providing
excellent agreement with the other Planck results.
36
2.1
Introduction
Inflation is a leading cosmological paradigm for the early universe, consistent with the
myriad of observable quantities that have been measured in the era of precision cosmology [27, 28, 29]. However, a persistent challenge has been to reconcile successful
inflationary scenarios with well-motivated models of high-energy physics. Realistic
models of high-energy physics, such as those inspired by supersymmetry or string
theory, routinely include multiple scalar fields at high energies [30]. Generically, each
scalar field should include a nonminimal coupling to the spacetime Ricci curvature
scalar, since nonminimal couplings arise as renormalization counterterms when quantizing scalax fields in curved spacetime [31, 32, 33, 34]. The nonminimal couplings
typically increase with energy-scale under renormalization-group flow [33], and hence
should be large at the energy-scales of interest for inflation. We therefore study a
class of inflationary models that includes multiple scalar fields with large nonminimal
couplings.
It is well known that the predicted perturbation spectra from single-field models
with nonminimal couplings produce a close fit to observations. Following conformal
transformation to the Einstein frame, in which the gravitational portion of the action
assumes canonical Einstein-Hilbert form, the effective potential for the scalar field is
stretched by the conformal factor to be concave rather than convex [35, 36], precisely
the form of inflationary potential most favored by the latest results from the Planck
experiment [37].
The most pronounced difference between multifield inflation and single-field inflation is the presence of more than one type of primordial quantum fluctuation that can
evolve and grow. The added degrees of freedom may lead to observable departures
from the predictions of single-field models, including the production and amplification
of isocurvature modes during inflation [38, 39, 40, 41, 42, 43, 44, 45].
Unlike adiabatic perturbations, which are fluctuations in the energy density, isocurvature perturbations arise from spatially varying fluctuations in the local equation
of state, or from relative velocities between various species of matter. When isocur-
37
vature modes are produced primordially and stretched beyond the Hubble radius,
causality prevents the redistribution of energy density on super-horizon scales. When
the perturbations later cross back within the Hubble radius, isocurvature modes create
pressure gradients that can push energy density around, sourcing curvature perturbations that contribute to large-scale anisotropies in the cosmic microwave background
radiation (CMB). (See, e.g., [46, 37].)
The recent measurements of CMB anisotropies by Planck favor a combination of
adiabatic and isocurvature perturbations in order to improve the fit at low multipoles
(i ~ 20 - 40) compared to the predictions from the simple, best-fit ACDM model in
which primordial perturbations are exclusively adiabatic. The best fit to the present
data arises from models with a modest contribution from isocurvature modes, whose
primordial power spectrum Ps(k) is either scale-invariant or slightly blue-tilted, while
the dominant adiabatic contribution, RP(k), is slightly red-tilted [37]. The low-i
anomaly thus might provide the first robust empirical evidence that early-universe
inflation involved more than one scalar field.
Well-known multifield models that produce isocurvature perturbations, such as
axion and curvaton models, are constrained by the Planck results and do not improve
the fit compared to the purely adiabatic ACDM model [37]. As we demonstrate here,
on the other hand, the general class of multifield models with nonminimal couplings
can readily produce isocurvature perturbations of the sort that could account for the
low-i anomaly in the Planck data, while also producing excellent agreement with the
other spectral observables measured or constrained by the Planckresults, such as the
spectral index n,, the tensor-to-scalar ratio r, the running of the spectral index a,
and the amplitude of primordial non-Gaussianity fNLNonminimal couplings in multifield models induce a curved field-space manifold
in the Einstein frame [47], and hence one must employ a covariant formalism for
this class of models. Here we make use of the covariant formalism developed in
[48], which builds on pioneering work in [39, 44]. In Section 2.2 we review the most
relevant features of our class of models, including the formal machinery required to
study the evolution of primordial isocurvature perturbations. In Section 2.3 we focus
38
on a regime of parameter space that is promising in the light of the Planck data,
and for which analytic approxmations are both tractable and in close agreement with
numerical simulations. In Section 2.4 we compare the predictions from this class of
models to the recent Planck findings. Concluding remarks follow in Section 2.5.
2.2
Model
We consider two nonminimally coupled scalar fields 0b'e {#, X}. We work in 3+1
spacetime dimensions with the spacetime metric signature (-, +, +, +). We express
our results in terms of the reduced Planck mass, M, = (87rG)-1
2
= 2.43 x 1018 GeV.
Greek letters (p, v) denote spacetime 4-vector indices, lower-case Roman letters (i,
j) denote spacetime 3-vector indices, and capital Roman letters (I, J) denote fieldspace indices. We indicate Jordan-frame quantities with a tilde, while Einstein-frame
quantities will be sans tilde. Subscripted commas indicate ordinary partial derivatives
and subscripted semicolons denote covariant derivatives with respect to the spacetime
coordinates.
We begin with the action in the Jordan frame, in which the fields' nonminimal
couplings remain explicit:
=J
- [f(#)A
"-
-
(2.1)
where R is the spacetime Ricci scalar, f(01) is the nonminimal coupling function, and
Orj is the Jordan-frame field space metric. We set
gjj
= 6.j, which gives canonical
kinetic terms in the Jordan frame. We take the Jordan-frame potential, V(k'), to
have a generic, renormalizable polynomial form with an interaction term:
V(#, X) = 4
+
2,2
+- -,4
(2.2)
with dimensionless coupling constants A, and g. As discussed in [48], the inflationary
dynamics in this class of models are relatively insensitive to the presence of mass
terms, m 2#
2
or m2X 2 , for realistic values of the masses that satisfy mk, mx
Hence we will neglect such terms here.
39
< Mp.
2.2.1
Einstein-Frame Potential
We perform a conformal transformation to the Einstein frame by rescaling the spacetime metric tensor,
(2.3)
AV(X) = Q 2(x) gA,(x),
where the conformal factor Q 2(x) is related to the nonminimal coupling function via
the relation
Q2 W
2 f
M Pi1
(b(x)).
(2.4)
This transformation yields the action in the Einstein frame,
s
=
dcxV
R-
[L
0g9jjg,
- V(01)1,
418,0
(2.5)
where all the terms sans tilde are stretched by the conformal factor. For instance,
the conformal transformation to the Einstein frame induces a nontrivial field-space
metric [47]
91g =
(2.6)
3i+ - f,3fj],
[--
f
2f
and the potential is also stretched so that it becomes
M4
V(, X)
-
(f
___
( ,
4__
AO
2 [4
=
+ I252 +
4
X
-
(2.7)
The form of the nonminimal coupling function is set by the requirements of renormalization [31, 32],
f(0, X)
=
42 + XX21,
[M2 +
(2.8)
where O and x are dimensionless couplings and M is some mass scale such that when
the fields settle into their vacuum expectation values,
f -+
M21/2. Here we assume
that any nonzero vacuum expectation values for 0 and X are much smaller than the
.
Planck scale, and hence we may take M = Mo1
The conformal stretching of the potential in the Einstein frame makes it concave
40
Figure 2-1: Potential in the Einstein frame, V(#5) in Eq. (2.7).
The parameters
and asymptotically flat along either direction in field space, I
Mao)
i A,
( + M)
4
M
(01+
()2)]
Q2 [1\
=
,
,
shown here are Ax = 0.75 A, g = A, x = 1.2 0, with (4 >> 1 and A > 0.
(2.9)
0'
(no sum on I). For non-symmetric couplings, in which AO =, Ax and/or (4 =
x, the
potential in the Einstein frame will develop ridges and valleys, as shown in Fig. 2-1.
Crucially, V > 0 even in the valleys (for g >
-VA9Ax),
and hence the system will
inflate (albeit at varying rates) whether the fields ride along a ridge or roll within a
valley, until the fields reach the global minimum of the potential at q = x = 0.
Across a wide range of couplings and initial conditions, the models in this class
obey a single-field attractor [451. If the fields happen to begin evolving along the top
of a ridge, they will eventually fall into a neighboring valley. Motion in field space
transverse to the valley will quickly damp away (thanks to Hubble drag), and the
fields' evolution will include almost no further turning in field space. Within that
single-field attractor, predictions for n,
r, a, and fNL all fall squarely within the
most-favored regions of the latest Planck measurements [45].
The fields' approach to the attractor behavior -
41
essentially, how quickly the fields
roll off a ridge and into a valley - depends on the local curvature of the potential
near the top of a ridge. Consider, for example, the case in which the direction X = 0
corresponds to a ridge. To first order, the curvature of the potential in the vicinity
of X = 0 is proportional to (g(. - Ao~x) [48]. As we develop in detail below, a
convenient combination with which to characterize the local curvature near the top
of such a ridge is
-
_
g )
(2.10)
As shown in Fig. 2-2, models in this class produce excellent agreement with the
latest measurements of n, from Planck across a wide range of parameters, where n, =
1 + dln'PR/dln k. Strong curvature near the top of the ridge corresponds to K > 1:
in that regime, the fields quickly roll off the ridge, settle into a valley of the potential,
and evolve along the single-field attractor for the duration of inflation, as analyzed in
[45]. More complicated field dynamics occur for intermediate values, 0.1 < K < 4, for
which multifield dynamics pull n, far out of agreement with empirical observations.
The models again produce excellent agreement with the Planck measurements of n,
in the regime of weak curvature, 0 <
K
< 0.1.
As we develop below, other observables of interest, such as r, a, and fNL, likewise show excellent fit with the latest observations. In addition, the regime of weak
curvature,
K
<
1, is particularly promising for producing primordial isocurvature
perturbations with characteristics that could explain the low-e anomaly in the recent
Planck measurements. Hence for the remainder of this paper we focus on the regime
K
< 1, a region that is amenable to analytic as well as numerical analysis.
2.2.2
Coupling Constants
The dynamics of this class of models depend upon combinations of dimensionless
coupling constants like
K
defined in Eq. (2.10) and others that we introduce below.
The phenomena analyzed here would therefore hold for various values of A, and Cr,
such that combinations like r were unchanged. Nonetheless, it is helpful to consider
reasonable ranges for the couplings on their own.
42
20.5
0.0
0.01
0.1
0
100
K
Figure 2-2: The spectral index n, (red), as given in Eq. (2.61), for different values
of K, which characterizes the local curvature of the potential near the top of a ridge.
Also shown are the la (thin, light blue) and 2o- (thick, dark blue) bounds on n, from
the Planck measurements. The couplings shown here correspond to 60 = 6x = 10',
A = 10-2, and Ax = g, fixed for a given value of r. from Eq. (2.10). The fields' initial
conditions are = 0.3, q 0 = 0, Xo = 10,
ko = 0, in units of Mpl.
The present upper bound on the tensor-to-scalar ratio, r < 0.12, constrains the
energy-scale during inflation to satisfy H(thc)/Mpl < 3.7 x 10-
5
[37], where H(th,)
is the Hubble parameter at the time during inflation when observationally relevant
perturbations first crossed outside the Hubble radius. During inflation the dominant
contribution to H will come from the value of the potential along the direction in
which the fields slowly evolve. Thus we may use the results from Planck and Eq.
(2.9) to set a basic scale for the ratios of couplings, AI/6f. For example, if the fields
evolve predominantly along the direction X
-
0, then during slow roll the Hubble
parameter will be
H ~- F
2
Mpi,
(2.11)
and hence the constraint from Planck requires AO/ j < 1.6 x i0-.
We adopt a scale for the self-couplings A, by considering a particularly elegant
member of this class of models. In Higgs inflation [36], the self-coupling AO is fixed
by measurements of the Higgs mass near the electroweak symmetry-breaking scale,
43
~ 0.1, corresponding to mH ce 125 GeV [49, 50]. Under renormalization-group
flow, AO will fall to the range 0 < AO < 0.01 at the inflationary energy scale [54]. Eq.
(2.11) with A = 0.01 requires (4
>
780 at inflationary energy scales to give the correct
amplitude of density perturbations.
For our general class of models, we therefore
-
consider couplings at the inflationary energy scale of order A,, g ~ 0(10-2) and j
0(103). Taking into account the running of both A, and
group flow, these values correspond to A, ~ 0(10-1) and
v
&
under renormalization-
~, 0(102) at low energies
[54].
We consider these to be reasonable ranges for the couplings. Though one might
prefer dimensionless coupling constants to be 0(1) in any "natural" scenario, the
ranges chosen here correspond to low-energy couplings that are no more fine-tuned
than the fine-structure constant, aEM ~ 1/137.
Indeed, our choices are relatively
conservative. For the case of Higgs inflation, the running of AO is particularly sensitive
to the mass of the top quark. Assuming a value for mi,
at the low end of the present
2- bound yields AO ~ 10' rather than 10-2 at high energies, which in turn requires
>
>4
80 at the inflationary energy scale rather than
O
780 [551. Nonetheless,
for illustrative purposes, we use A,, g ~ 10-2 and j ~. 101 for the remainder of our
analysis.
We further note that despite such large nonminimal couplings,
analysis is unhindered by any potential breakdown of unitarity.
~ 103, our
j
The energy scale
at which unitarity might be violated for Higgs inflation has occasioned a great deal
of heated debate in the literature, with conflicting claims that the renormalization
, or Mpj/4 [56]. Even if one
adopted the most stringent of these suggested cut-off scales, Mp/4
-
10-
M,1
,
cut-off scale should be in the vicinity of M, 1, Mpi/V
the relevant dynamics for our analysis would still occur at energy scales well below
the cut-off, given the constraint H(thc)
<
3.7 x 10-5 Mp.
scale in multifield models in which the nonminimal couplings
(The unitarity cut-off
r
are not all equal to
each other has been considered in [57], which likewise identify regimes of parameter
space in which Aff remains well above the energy scales and field values relevant to
inflation.) Moreover, models like Higgs inflation can easily be "unitarized" with the
44
addition of a single heavy scalar field [58], and hence all of the following analysis could
be considered the low-energy dynamics of a self-consistent effective field theory. The
methods developed here may be applied to a wide class of models, including those
studied in [66, 67, 68, 691.
Finally, we note that for couplings A,, g ~, 10-2 and
v
~ 103 at high energies, the
regime of weak curvature for the potential, r. < 0.1, requires that the couplings be
close but not identical to each other. In particular, r,
1
-
0.1 requires g/A4 ,~
x/-
~
O(10-5). Such small differences are exactly what one would expect if the effective
couplings at high energies arose from some softly broken symmetry. For example, the
field X could couple to some scalar cold dark matter (CDM) candidate (perhaps a
supersymmetric partner) or to a neutrino, precisely the kinds of couplings that would
be required if the primordial isocurvature perturbations were to survive to late times
and get imprinted in the CMB [46]. In that case, corrections to the P functions for
the renormalization-group flow of the couplings AX and x would appear of the form
gx2/167r 2 [33, 59], where gx is the coupling of X to the new field. For reasonable values
of gx ~ 10-1 - 10-2, such additional terms could easily account for the small but
non-zero differences among couplings at the inflationary energy scale.
2.2.3
Dynamics and Transfer Functions
When we vary the Einstein-frame action with respect to the fields 01, we get the
equations of motion, which may be written
001 + IK
where 04/= g"
_
gUJyK =0,
(2.12)
, and IK is the field-space Christoffel symbol.
We further expand each scalar field to first order in perturbations about its classical background value,
#(XIA)
-
WI(t) + 6#o(xp)
(2.13)
and we consider scalar perturbations to the spacetime metric (which we assume to
45
be a spatially flat Friedmann-Robertson-Walker metric) to first order:
ds2 = -(1 + 2A)dt 2 + 2a(t)(c6 B)dx t dt + a(t) 2 [(1 - 21P)6 3 + 2a18O EJdx'dx9, (2.14)
where a(t) is the scale factor and A, B, i and E are the scalar gauge degrees of
freedom.
Under this expansion, the full equations of motion separate into background and
first-order equations. The background equations are given by
Vti" + 3HO, + gIJVj = 0,
(2.15)
where VAI = OJAI+FIJKAK for an arbitrary vector, A,, on the field-space manifold;
DVAI = ObVjAI is a directional derivative; and H = i/a is the Hubble parameter.
The 00 and Oi components of the background-order Einstein equations yield:
H2
1
p1
[1gI
2
J + V(p)]
(2.16)
Using the covariant formalism of [48], we find the equations of motion for the perturbations,
DtQ2
where
+ 3HDtQ'+ [2>
Q, is the gauge-invariant
MI
1 as
(a0
0i
QJ = 0,
(2.17)
Mukhanov-Sasaki variable
QI - Q1 +
(2.18)
,)
and Q, is a covariant fluctuation vector that reduces to 6
1
to first order in the
fluctuations. Additionally, MI' is the effective mass-squared matrix given by
gIKDJK
46
-
~ZLMJ
(2.19)
where 'LMJ is the field-space Riemann tensor.
The degrees of freedom of the system may be decomposed into adiabatic and
entropic (or isocurvature) by introducing the magnitude of the background fields'
velocity vector,
&
=
(2.20)
|
with which we may define the unit vector
(2.21)
which points along the fields' motion. Another important dynamical quantity is the
turn-rate of the background fields, given by
W = Vt&I,
(2.22)
with which we may construct another important unit vector,
where w =
1w11.
(2.23)
W,
4
The vector SI points perpendicular to the fields' motion, SM&, = 0.
The unit vectors &I and S^, effectively act like projection vectors, with which we may
decompose any vector into adiabatic and entropic components. In particular, we may
decompose the vector of fluctuations QI,
(2.24)
in terms of which Eq. (2.17) separates into two equations of motion:
Q
y,+3HQ,+
-+M
= 2- (w Q,) - 2
dt
2_
-wM010
a2
++
o-H
47
3
P
Q.,,
(
d a3&2
dt
Q,
(2.25)
Q, + 3HQ, +
= 4M21 W
M, +,
T-+
where T is the gauge-invariant Bardeen potential
+ a2 H
,T
(2.26)
[29],
-)-,(2.27)
and where M,, and M, are the adiabatic and entropic projections of the masssquared matrix, M' from (2.19). More explicitly,
00
(2.28)
M 8 .9=
As Eqs.
(2.25) and (2.26) make clear, the entropy perturbations will source the
adiabatic perturbations but not the other way around, contingent on the turn-rate w
being nonzero. We also note that the entropy perturbations have an effective masssquared of
2 = M,3 +3W
2
.
(2.29)
In the usual fashion [29], we may construct the gauge-invariant curvature perturbation,
RC
( +)-q
(P + A)
(2.30)
where p and p are the background-order density and pressure and 6q is the energydensity flux of the perturbed fluid. In terms of our projected perturbations, we find
[48]
H
R= -Q.
(2.31)
Analogously, we may define a normalized entropy (or isocurvature) perturbation as
[29, 39, 40, 42, 44, 48]
H
S a-Q,9.
(2.32)
In the long-wavelength limit, the coupled perturbations obey relations of the form
48
[29, 39, 40, 42, 44, 48]:
7c ~ aHS
(2.33)
S~ 8HS,
which allows us to write the transfer functions as
t)
=
ft dt' a(t') H(t') TsS(thc, t)(
Tss(thc, t) = exp
dt' 8(t') H(t')
,
TRs(thc,
where thc is the time when a fiducial scale of interest first crosses the Hubble radius
during inflation, khe
=
a(thc)H(th,).
We find [48]
2w
W
where e, t,,
4(2.35)
and q,, are given by
H
(2.36)
M7o
P MV
The first two quantities function like the familiar slow-roll parameters from singlefield inflation: 7,, = 1 marks the end of the fields' slow-roll evolution, after which
& ~ H&, while E = 1 marks the end of inflation (d = 0 for e = 1). The third quantity,
7,s, is related to the effective mass of the isocurvature perturbations, and need not
remain small during inflation.
Using the transfer functions, we may relate the power spectra at
the
to spectra at
later times. In the regime of interest, for late times and long wavelengths, we have
P-(k) = PR(khc) [1 + T
Ps(k) = Piz(khc)
(thc, t]
T2s(thc, t).
49
(2.37)
Ultimately, we may use Ts
and Tss to calculate the isocurvature fraction,
Ps
T
2____
+ S
T2S +'T2
Sp=
AS s +PpS
j.g
(2.38)
which may be compared to recent observables reported by the Planck collaboration.
An example of the fields' trajectory of interest is shown in Fig. 2-3. As shown
in Fig. 2-4, while the fields evolve near the top of the ridge, the isocurvature modes
are tachyonic, p2 < 0, leading to the rapid amplification of isocurvature modes.
When the turn-rate is nonzero, w 0 0, the growth of
the adiabatic perturbations,
Q,.
Q,
can transfer power to
If Ts grows too large from this transfer, then
predictions for observable quantities such as n, can get pulled out of agreement with
present observations, as shown in the intermediate region of Fig. 2-2 and developed
in more detail in Section 2.4. On the other hand, growth of Q. is strongly suppressed
when fields evolve in a valley, since p,/H2 > 1. In order to produce an appropriate
fraction of isocurvature perturbations while also keeping observables such as n, close
to their measured values, one therefore needs field trajectories that stay on a ridge for
a significant number of e-folds and have only a modest turn-rate so as not to transfer
too much power to the adiabatic modes. This may be accomplished in the regime of
weak curvature, r
2.3
2.3.1
<
1.
Trajectories of Interest
Geometry of the Potential
As just noted, significant growth of isocurvature perturbations occurs when A < 0,
when the fields begin near the top of a ridge. If the fields start in a valley, or if the
curvature near the top of the ridge is large enough (r > 1) so that the fields rapidly
fall into a valley, then the system quickly relaxes to the single-field attractor found in
[451, for which flio -+ 0. To understand the implications for quantities such as 3 i.,
it is therefore important to understand the geomtery of the potential. This may be
50
V
Figure 2-3: The fields' trajectory (red) superimposed upon the effective potential in
the Einstein frame, V, with couplings O = 1000, x = 1000.015, AO = Ax = g = 0.01,
and initial conditions 0 = 0.35, Xo = 8.1 x 10-4, #0 = *o = 0, in units of Mpi.
accomplished by working with the field-space coordinates r and 9, defined via
(2.39)
q5=rcos9 , y = rsin9.
+
2
(The parameter 9 was labeled y in [53].) Inflation in these models occurs for 00k
xX
2
> M2 [48]. That limit corresponds to taking r
-+
oo, for which the potential
becomes
MA1 2g cos2 sin2 + A cos 9+ Ax sin
(4 cos2 9 + x sin2 )2
4
We further note that for our choice of potential in Eq. (2.7), V(#,
symmetries,
#
-+
-0 and X
to only one quarter of the
#
-+
-X.
(2.40)
x) has two discrete
This means that we may restrict our attention
- X plane. We choose
#
> 0 and x > 0 without loss of
generality.
The extrema (ridges and valleys) are those places where VO = 0, which formally
51
1.0
s2 /
2
(w/H)x10 3
0.8.
0.6
0.4
0.2
0.0
-0.2-
~-~
-0
40
20
80
60
N,
Figure 2-4: The mass of the isocurvature modes, p /H 2 (blue, solid), and the turn
rate, (w/H) x 103 (red, dotted), versus e-folds from the end of inflation, N., for the
trajectory shown in Fig. 2-3. Note that while the fields ride along the ridge, the
isocurvature modes are tachyonic, p2 < 0, leading to an amplification of isocurvature
perturbations. The mass p2 becomes large and positive once the fields roll off the
ridge, suppressing further growth of isocurvature modes.
has three solutions for 0 < 0 < 7r/2 and r -+ oo:
01
= 0
02=,
93
2
A
= cos[
A
,
(2.41)
Ao + Ax
where we have defined the convenient combinations
-
Ax aAX
In order for
93
(2.42)
- gx.
to be a real angle (between 0 and 7r/2), the argument of the inverse
cosine in Eq. (2.41) must be real and bounded by 0 and 1. If Ax and A0 have the
same sign, both conditions are automatically satisfied. If Ax and A4 have different
signs then the argument may be either imaginary or larger than 1, in which case there
is no real solution
93.
for Ax > A4 , then
93
If both Ax and A , have the same sign, the limiting cases are:
-+ 0, and for Ax < A then
52
93 -+
7r/2.
In each quarter of the
#-x plane, we therefore have either two or three extrema,
as
shown in Fig. 2-5. Because of the mean-value theorem, two ridges must be separated
by a valley and vice versa. If AX and A 4 have opposite signs, there are only two
extrema, one valley and one ridge. This was the case for the parameters studied in
[48]. If A4 and Ax have the same sign, then there is a third extremum (either two
ridges and one valley or two valleys and one ridge) within each quarter plane. In the
case of two ridges, their asymptotic heights are
Vr-00(01) =
4 2
>i
(2.43)
(4
0
X
Vr.(+oo(02 )=
and the valley lies along the direction
03.
In the limit r -+ oo, the curvature of the
potential at each of these extrema is given by
=
VYOO 1=03 =
~
4
A M
=
2AA 4 (A 4 + AX) 2 M4
xA~
(AA (
~
i
(2.44)
'
K01e=o
.OAM
A.M
, Yeele=r/2
In this section we have ignored the curvature of the field-space manifold, since for
large field values the manifold is close to flat [48], and hence ordinary and covariant
derivatives nearly coincide. We demonstrate in Appendix B that the classification of
local curvature introduced here holds generally for the dynamics relevant to inflation,
even when one takes into account the nontrivial field-space manifold.
2.3.2
Linearized Dynamics
In this section we will examine trajectories for which w is small but nonzero: small
enough so that the isocurvature perturbations do no transfer all their energy away
to the adiabatic modes, but large enough so that genuine multifield effects (such as
i, # 0) persist rather than relaxing to effectively single-field evolution.
We focus on situations in which inflation begins near the top of a ridge of the
53
V 00
41
0.5
1.0
1.5
2.0
2.5
3.0
Figure 2-5: The asymptotic value r -+ oo for three potentials with AX= -0.001
(blue dashed), AX = 0 (red solid), and AX = 0.001 (yellow dotted), as a function of
the angle 0 = arctan(X/0). For all three cases, AO = 0.0015, 4 =
'x = 1000, and
AO= 0.01.
potential, with
40
large and both X0 and X O small. Trajectories for which the fields
remain near the top of the ridge for a substantial number of e-folds will produce
a significant amplification of isocurvature modes, since p2 < 0 near the top of the
ridge and hence the isocurvature perturbations grow via tachyonic instability. Prom
a model-building perspective it is easy to motivate such initial conditions by postulating a waterfall transition, similar to hybrid inflation scenarios [65], that pins the
X field exactly on the ridge. Anything from a small tilt of the potential to quantum
fluctuations would then nudge the field off-center.
With X0 small, sufficient inflation requires
plished with sub-Planckian field values given
000
>> M2, which is easily accom-
O > 1. We set the scale for Xo by
imagining that X begins exactly on top of the ridge. In the regime of weak curvature,
r. < 1, quantum fluctuations will be of order
(X) 2= H- 2
27r
where we take X-
2
=> XrM =
H
N/2-i
(245
2.5
(X2 to be a classical estimator of the excursion of the field
54
away from the ridge. The constraint from Planck that H/M,1 < 3.7 x 10inflation then allows us to estimate Xr
during
(A
~ 10-' Mp, at the start of inflation.
Gaussian wavepacket for X will then spread as vx/, where N is the number of e-folds
of inflation.) This sets a reasonable scale for Xo; we examine the dynamics of the
system as we vary Xo around X-.
We may now expand the full background dynamics in the limit of small
x.
K,
X, and
The equation of motion for 0, given by Eq. (2.15), does not include any terms
linear in X or - , so the evolution of 0 in this limit reduces to the single-field equation
of motion, which reduces to
VAOM3
sR ~
-
(2.46)
3V3-b2
in the slow-roll limit [53]. To first approximation, the 0 field rolls slowly along the
top of the ridge. Upon using Eq. (2.11), we may integrate Eq. (2.46) to yield
M
3
(2.47)
N.
where N. is the number of e-folds from the end of inflation, and we have used 4(t)
>
*
4(tend). The slow-roll parameters may then be evaluated to lowest order in X and
and take the form [45]
3
7ary, ~
--
1
/
1 - ---
3
(2.48)
4N*/
-N*
Expanding the equation of motion for the X field and considering (4,6x
> 1 we
find the linearized equation of motion
i+ 3Hj -O
55
2
X -e 0,
4M2)
(2.49)
which has the simple solution
X(t) c~- X0 exp [-2
3H
9H2 AOM21
2 +-q )N(t)
ftt
where we again used Eq. (2.11) for H, and N(t)
since the start of inflation. If we assume that A 4 M2/,
1
<
to A4/A4
(2.50)
,
Hdt' is the number of e-folds
<
9H2/4, which is equivalent
3/16, then we may Taylor expand the square root in the exponent of
X(t). This is equivalent to dropping the
term from the equation of motion. In this
limit the solution becomes
X(t) ~ XoeKN(t),
where
K
is defined in Eq. (2.10). Upon using the definition of A4, in Eq. (2.42), we
now recognize
the limit
(2.51)
K
K
= 4AO/A4 . Our approximation of neglecting J thus corresponds to
< 3/4.
When applying our set of approximations to the isocurvature mass in Eq. (2.29),
we find that the M,, term dominates w 2 /H 2 , and the behavior of M,. in turn is
dominated by DJDKV rather than the term involving R KL. Since we are projecting
the mass-squared matrix orthogonal to the fields' motion, and since we are starting on
a ridge along the
# direction,
the derivative of V that matters most to the dynamics
of the system in this limit is DxxV evaluated at small X. To second order in X, we
find
2
X V#
+
P(1+6)
2A(1 + 60)
]
[3(1 +64)Ax + (1-e)(1+6 %)A
, +6(1
+
D V=[4
+ 0(1+ 6(.)#4 04
- e)(1+6
,
4
)A - Aoe,
(2.52)
where we have used A4 and Ax as given in Eq. (2.42) and also introduced
=
56
1
-
X
(2.53)
These terms each illuminate an aspect of the geometry of the potential: as we found
in Eq. (2.44), AO and Ax are proportional to the curvature of the potential along the
q and x axes respectively, and e is the ellipticity of the potential for large field values.
Intuition coming from these geometric quantities motivates us to use them as a basis
for determining the dynamics in our simulations. The approximations hold well for
the first several e-folds of inflation, before the fields fall off the ridge of the potential.
Based on our linearized approximation we may expand all kinematical quantities
in power series of Xo and 1/N.. We refer to the intermediate quantities in Appendix B
and report here the important quantities that characterize the generation and transfer
of isocurvature perturbations. To lowest order in x and X, the parameter 7, defined
in Eq. (2.36) takes the form
3
-K -34Nt,
77a~9
(
+
2e
3)
-
showing that to lowest order in 1/N., 7,.
+
3
8*
(2.54)
(3)
< 0 and hence the isocurvature modes
begin with a tachyonic mass. The quantities a and 6 from Eq. (2.35) to first order
are
r.xoexp[(Nt - N.)]
V2OMpj
[3r
Q~x+
E
(2.55)
1
_ 9]
*+ T
4 +2
8'
where Nwt is the total number of e-folds of inflation. These expansions allow us to
approximate the transfer function Tss of Eq. (2.34),
Tss ~
N
-exp
4~
~2
[ (N1c - N,)
3
(3 -
1
1
N)
(2.56)
where Nhc is the number of e-folds before the end of inflation at which Hubble crossing
occurs for the fiducial scale of interest. We may then use a semi-analytic form for
TRs by putting Eq. (2.56) into Eq. (2.34). This approximation is depicted in Fig.
2-6.
57
4.
-
- Exact
-- Analyticl
3
N
1
0
0
10
20
30
N.
40
0.035
50
60
-Exact
0 Semi-Analytic
0.030
0.025
0.020
0.015
0.010
0.005
0.000
0
10
20
30
40
50
60
Figure 2-6: The evolution of Tss (top) and TRs (bottom) using the exact and approximated expressions, for . = 4A4/A0 = 0.06, 4Ax/Ax = -0.06 and e = -1.5 x 10-5,
with #0 = 0.35 MP1, Xo = 8.1 x 10-4 Mpl, and q 0 =)o = 0. We take Nhc = 60
and plot Tss and Tps against N., the number of e-folds before the end of inflation.
The approximation works particularly well at early times and matches the qualitative
behavior of the exact numerical solution at late times.
Our analytic approximation for Tss vanishes identically in the limit N, -+ 0 (at
the end of inflation), though it gives an excellent indication of the general shape of
Tss for the duration of inflation. We further note that Tss is independent of Xo to
lowest order, while Tps oc a oc ryo and hence remains small in the limit we are
considering. Thus for small r., we expect
#8j
58
to be fairly insensitive to changes in Xo.
Results
2.4
We want to examine how the isocurvature fraction
#i.
varies as we change the shape of
the potential. We are particularly interested in the dependence of 6i. on x, since the
leading-order contribution to the isocurvature fraction from the shape of the potential
is proportional to n. Guided by our approximations, we simulated trajectories across
1400 potentials and we show the results in Figures 2-7 - 2-10. The simulations were
done using zero initial velocities for k and X, and were performed using both Matlab
and Mathematica, as a consistency check. We compare analytical approximations in
certain regimes with our numerical findings.
As expected, we find that there is an interesting competition between the degree
to which the isocurvature mass is tachyonic and the propensity of the fields to fall off
the ridge. More explicitly, for small
K
we expect the fields to stay on the ridge for
most of inflation with a small turn rate that transfers little power to the adiabatic
modes. Therefore, in the small-K limit, Tps remains small while Tss (and hence
#i..)
increases exponentially with increasing
show that
#6,
vs.
K
K.
Indeed, all the numerical simulations
increases linearly on a semilog scale for small
K.
However, in
the small-K limit, the tachyonic isocurvature mass is also small, so ,6 i. remains fairly
small in that regime. Meanwhile, for large
K
we expect the fields to have a larger
tachyonic mass while near the top of the ridge, but to roll off the ridge (and transfer
significant power to the adiabatic modes) earlier in the evolution of the system. There
should be an intermediate regime of K in which the isocurvature mass is fairly large
(and tachyonic) and the fields do not fall off the ridge too early. Indeed, a ubiquitous
feature of our numerical simulations is that 8i. is always maximized around
K
, 0.1,
regardless of the other parameters of the potential.
2.4.1
Local curvature of the potential
In Fig. 2-7, we examine the variation of 6i. as we change Xo and
#8i, has no dependence
on Xo for small
K.
Increasing
K
K.
As expected,
breaks the Xo degeneracy: the
closer the fields start to the top of the ridge, the more time the fields remain near the
59
top before rolling off the ridge and transferring power to the adiabatic modes. Just
as expected, for the smallest value of Xo, we see the largest isocurvature fraction.
Even for relatively large Xo, there is still a nontrivial contribution of isocurvature
modes to the perturbation spectrum. Therefore, our model generically yields a large
isocurvature fraction with little fine-tuning of the initial field values in the regime
<
1
0.1
-Xo = 10-4
.-- xo=10 3
Xo= 10-2
-.-.
0.01
00.001
10-
-
10-1
10-6
0.00
0.10
0.05
0.20
0.15
K
Figure 2-7: The isocurvature fraction for different values of Xo (in units of Mpl) as a
function of the curvature of the ridge, r.. All of the trajectories began at 0 = 0.3 M, 1
which yields Net~ = 65.7. For these potentials, O = 1000, AO = 0.01, e = 0, and
AX = 0. The trajectories that begin closest to the top of the ridge have the largest
values of #irj, with some regions of parameter space nearly saturating Oiso = 1.
,
«1.
We may calculate
#ij
3
for the limiting case of zero curvature, r.
-+
0, the vicinity
in which the curves in Fig. 2-7 become degenerate. Taking the limit r.
-+
0 means
essentially reverting to a Higgs-like case, a fully SO(2) symmetric potential with no
turning of the trajectory in field space [53]. As expected, our approximate expression
in Eq. (2.56) for Tps
-+
0 in the limit r. -+ 0, and hence we need only consider Tss.
As noted above, our approximate expression for Tss in Eq. (2.56) vanishes in
the limit N
-+
0. Eq. (2.56) was derived for the regime in which our approximate
expressions for the slow-roll parameters e and q, in Eq. (2.48) are reasonably accurate. Clearly the expressions in Eq. (2.48) will cease to be accurate near the end
of inflation. Indeed, taking the expressions at face value, we would expect slow roll
to end (Iq,, = 1) at N, = 1/2, and inflation to end (e = 1) at N, = 2/Vs, rather
60
10
-- Tss
'
0.1
TRS
0.001
10
10-7
10-9
0.00
0.05
0.10
0.15
0.20
K
Figure 2-8: Contributions of TRs and Tss to 8i.,. The parameters used are O=
0.3 Mpi, Xo = 10- 3 MPI, O = 10 3 , AO = 0.01, e = 0 and Ax = 0. For small K, Oi., is
dominated by Tss; for larger K, Tps becomes more important and ultimately reduces
than at N. = 0. Thus we might expect Eq. (2.48) to be reliable until around N, ~_1
which matches the behavior we found in a previous numerical study [45]. Hence we
will evaluate our analytic approximation for Tss in Eq. (2.56) between Nhc
N, ~ 1, rather than all the way to N.,
0. In the limit .
-+
0 and e
-+
=
60 and
0 and using
N, = 1, Eq. (2.56) yields
Tss ~
upon taking Nhc
> N..
hence #I.O ~ 2.9 x 10
#iso
= 2.3 x 10
5
5
For
Nhc =
(2.57)
-- exp [-9/81,
Nhc
60, we therefore find Tss ~ 5.4 x 10
3
, and
. This value may be compared with the exact numerical value,
. Despite the severity of our approximations, our analytic expression
provides an excellent guide to the behavior of the system in the limit of small
As we increase
K,
K.
the fields roll off the ridge correspondingly earlier in their evolu-
tion. The nonzero turn-rate causes a significant transfer of power from the isocurvature modes to the adiabatic modes. As Tps grows larger, it lowers the overall value
of fl.,O. See Fig. 2-8.
61
2.4.2
Global structure of the potential
The previous discussion considered the behavior for AX = 0. As shown in Fig. 2-5,
the global structure of the potential will change if Ax = 0. In the limit r. < 1, the
fields never roll far from the top of the ridge along the X = 0 direction, and therefore
the shape of the potential along the X direction has no bearing on
#is.
However,
large , breaks the degeneracy in AX because the fields will roll off the original ridge
and probe features of the potential along the X direction. See Fig. 2-9.
0.1
.-
-
'
s...----
=-0.001
AX = 0
10-10
io- 13
0.00
0.05
0.10
0.15
0.20
K
Figure 2-9: The isocurvature fraction for different values of AX as a function of the
curvature of the ridge, K. All of the trajectories began at 0 = 0.3 Mp1 and Xo =
10- 4 Mp1 , yielding Net~ = 65.7. For these potentials, O = 1000, AO = 0.01, and
e = 0. Potentials with AX < 0 yield the largest 8ir,
peaks, though in those cases
A#is falls fastest in the large-K limit due to sensitive changes in curvature along the
trajectory. Meanwhile, potentials with positive AX suppress the maximum value of
once K > 0.1 and local curvature becomes important.
In the case AX = 0, the fields roll off the ridge and eventually land on a plain,
where the isocurvature perturbations are minimally suppressed, since A
~ 0. For
AX > 0, there is a ridge along the X direction as well as along X = 0, which means
that there must be a valley at some intermediate angle in field space. When the
fields roll off the original ridge, they reach the valley in which p2 > 0, and hence the
isocurvature modes are more strongly suppressed than in the AX = 0 case.
Interesting behavior may occur for the case AX < 0. There exists a range of
62
K
for which the isocurvature perturbations are more strongly amplified than a naive
estimate would suggest, thanks to the late-time behavior of 7, ~ (DxxV)/V. If the
second derivative decreases more slowly than the potential itself, then the isocurvature
modes may be amplified for a short time as the fields roll down the ridge. This added
contribution is sufficient to increase Pi. compared to the cases in which AX > 0.
However, the effect becomes subdominant as the curvature of the original ridge, r,
is increased. For larger r, the fields spend more time in the valley, in which the
isocurvature modes are strongly suppressed.
In Figure 2-10, we isolate effects of e and n on
small (which implies that
K
#
3
.j. From Eq. (2.52), when A, is
is small), e sets the scale of the isocurvature mass. Positive
e makes the isocurvature mass-squared more negative near
K
= 0, which increases the
power in isocurvature modes. Conversely, negative e makes the isocurvature masssquared less negative near r = 0, which decreases the power in isocurvature modes.
In geometrical terms, in the limit AO = Ax = 0, equipotential surfaces are ellipses
with eccenticity V- for e > 0 and
#8i. exactly
v/(--
1) for e < 0. In this limit we may calculate
as we did for the case of e = 0.
#
The other effect of changing e is that it elongates the potential in either the
or X direction. This deformation of the potential either enhances or decreases the
degree to which the fields can turn, which in turn will affect the large-K behavior. In
particular, for e > 0 the potential is elongated along the
#
direction, which means
that when the fields roll off the ridge, they immediately start turning and transferring
power to the adiabatic modes. Conversely, for e < 0 the potential is elongated along
the x direction, so once the fields fall off the ridge, they travel farther before they
start turning. Therefore, in the large-K limit, P/.' falls off more quickly for e > 0 than
for e < 0.
We may use our analytic expression for Tss in Eq. (2.56) for the case in which
r -+ 0 with e =A 0. We find the value of 3, ~= TsS changes by a factor of 11 when
we vary e t 1/2, while our numerical solutions in Fig. 2-10 vary by a factor of 21.
Given the severity of some of our analytic approximations, this close match again
seems reassuring.
63
-
e= 1/2
,
0.1
e=-1/2
0.001
10-1
10-7
0.00
0.05
0.10
0.15
0.20
K
Figure 2-10: The isocurvature fraction for different values of e as a function of the
curvature of the ridge, is. All of the trajectories began at Xo = 10-3 Mp1 and #o =
0.3 Mpi, with N 0tt = 65.7. For these potentials, O = 1000, AO = 0.01, and Ax = 0.
Here we see the competition between e setting the scale of the isocurvature mass and
affecting the amount of turning in field-space.
2.4.3
Initial Conditions
The quantity
i,, varies with the fields' initial conditions as well as with the param-
eters of the potential. Given the form of TRs and Tss in Eq. (2.34), we see that the
value of A,3. depends only on the behavior of the fields between Nhc and the end of
inflation. This means that if we were to change 40 and Xo in such a way that the
fields followed the same trajectory following Nhc, the resulting values for %3 j0 would
be identical.
We have seen in Eq. (2.47) that we may use
#
as our inflationary clock,
6,02/M 2
~
4N*/3, where N, = Ntot - N(t) is the number of e-folds before the end of inflation.
We have also seen, in Eq.
(2.51), that for small , we may approximate X(t) er
Xo exp[t'N(t)]. If we impose that two such trajectories cross Nhc with the same value
A(logXo) = KAN
=
--
3
4
#
of X, then we find
r, A
M2i
.
(2.58)
We tested the approximation in Eq. (2.58) by numerically simulating over 15,000
trajectories in the same potential with different initial conditions.
64
The numerical
Logo(pliso)
-4.0
-0.5
-4.5
-0.75
-1
-5.0
-2
-5.5
-6.30
0.32
0.34
0.36
0.38
#0
Figure 2-11: Numerical simulations of &3 j. for various initial conditions (in units of
Mpi). All trajectories shown here were for a potential with r = 4AO/AO = 0.116,
4Ax/Ax = -160.12, and e = -2.9 x 10-. Also shown are our analytic predictions
for contours of constant #i, derived from Eq. (2.58) and represented by dark, solid
lines. From top right to bottom left, the contours have #3 jo = 0.071, 0.307, 0.054,
0.183, and 0.355.
results are shown in Fig. 2-11, along with our analytic predictions, from Eq. (2.58),
that contours of constant Bi.O should appear parabolic in the semilog graph.
As
shown in Fig. 2-11, our analytic approximation matches the full numerical results
remarkably well. We also note from Fig. 2-11 that for a given value of Xo, if we
increase <0o (thereby increasing the total duration of inflation, N 0tt), we will decrease
Aiso, behavior that is consistent with our approximate expressions for Tps and Tss in
Eq. (2.56).
65
V
Figure 2-12: Two trajectories from Fig. 2-11 that lie along the #ij = 0.183 line, for
#0 = 0.3 Mpl and #o = 0.365 Mpl. The dots mark the fields' initial values. The two
trajectories eventually become indistinguishable, and hence produce identical values
of
2.4.4
CMB observables
Recent analyses of the Planck data for low multipoles suggests an improvement of fit
between data and underlying model if one includes a substantial fraction of primordial isocurvature modes,
#i.
-
0(0.1). The best fits are obtained for isocurvature
perturbations with a slightly blue spectral tilt, nr = 1 + d ln Ps/dIn k ;> 1.0 [37]. In
the previous sections we have demonstrated that our general class of models readily produces 6io
-
0(0.1) in the regime , ,< 0.1. The spectral tilt, nI, for these
perturbations goes as [40, 44]
n, = 1 - 2e + 277,,,
66
(2.59)
where e and 7, are evaluated at Hubble-crossing, Nc. Given our expressions in Eqs.
(2.48) and (2.54), we then find
3
ns1-2r-
3
2N.
2e
+
3
4N*2
(1+6).
(2.60)
For trajectories that produce a nonzero fraction of isocurvature modes, the isocurva-
ture perturbations are tachyonic at the time of Hubble-crossing, with 7, Oc .M,,
2
< 0. Hence in general we find n, will be slightly red-tilted, n,
1. However, in
the regime of weak curvature, x < 1, we may find n, ~ 1. In particular, in the limit
K
-+ 0 and e -+ 0, then n, -+ 1- 3/(4N*) ~ 1- O(10-4), effectively indistinguishable
from a flat, scale-invariant spectrum. In general for r < 0.02, we therefore expect
ni > n,, where n, ~ 0.96 is the spectral index for adiabatic perturbations. In that
regime, the isocurvature perturbations would have a bluer spectrum than the adiabatic modes, albeit not a genuinely blue spectrum. An important test of our models
will therefore be if future observations and analysis require n, > 1 in order to address the present low-I anomaly in the Planckmeasurements of the CMB temperature
anisotropies.
Beyond 8i. and n., there are other important quantities that we need to address,
and that can be used to distinguish between similar models: the spectral index for
the adiabatic modes, n,, and its running, a = dn,/d In k; the tensor-to-scalar ratio, r;
and the amplitude of primordial non-Gaussianity,
of large curvature,
K
>
fNL.
As shown in [45], in the limit
1, the system quickly relaxes to the single-field attractor for
which 0.960 < n. 5 0.967, a ~ Q(10-4), 0.0033 < r < 0.0048, and IfNL< 1- (The
ranges for n, and r come from considering Nh, = 50 - 60.) Because the single-field
attractor evolution occurs when the fields rapidly roll off a ridge and remain in a
valley, in which p2 > 0, the models generically predict
< 1 in the limit
<i.
K
> 1 as
well. Here we examine how these observables evolve in the limit of weak curvature,
K
< 1, for which, as we have seen, the models may produce substantial pi" ~ 0(0.1).
67
0.98
0.97
0.96
2
0.95
0.94
0.93
O.9&00
0.05
0.10
0.15
0.20
K
Figure 2-13: The spectral index n, for different values of the local curvature n. The
parameters used are #0 = 0.3 Mpl, X0 = 10-3 Mpi, 0 = 1000, A0 = 0.01, e = 0 and
AX = 0. Comparing this with Fig. 2-7 we see that the peak in the i, curve occurs
within the Planck allowed region.
Let us start with the spectral index, n. If isocurvature modes grow and transfer
substantial power to the adiabatic modes before the end of inflation, then they may
affect the value of n,. In particular, we have [40, 44, 48]
n. =
ns(thc)
+
1[-a(thc)
3
-
(thc)TRs]
sin(2A),
(2-61)
where
n,(thc)
=
1 - 6 + 271,
(2.62)
and a and 3 are given in Eq. (2.35). The angle A is defined via
cosA =
Ts
.
(2.63)
-/1+ T
The turn rate a = 2w/H is small at the moment when perturbations exit the
Hubble radius, and the trigonometric factor obeys -1 < sin(2A) < 1. We also have
#6
~
+ O(N;') at early times, from Eq.
(2.55).
Hence we see that Ts
must
be significant in order to cause a substantial change in n, compared to the value at
Hubble crossing, n,(thc)- Yet we found in Fig. 2-8 that Tps grows large after
#is,
has
reached its maximum value. We therefore expect n. to be equal to its value in the
single-field attractor for n
< 0.1.
68
This is indeed what we find when we study the exact numerical evolution of n,
over a wide range of K, as in Fig. 2-2, as well as in the regime of weak curvature,
K
<
1, as shown in Fig. 2-13. For
K
< 0.1 and using NhC = 60, we find n, well
within the present bounds from the Planck measurements: n, = 0.9603
[37]. Moreover, because the regime
K
0.0073
< 0.1 corresponds to TRs < 1, the analysis
of the running of the spectral index, a, remains unchanged from [45], and we again
find a ~- -2/N.
a = -0.0134
O(10-4), easily consistent with the constraints from Planck,
'
0.0090 [37].
0.002
0.001
5 x10-
5x10-4
0.00
0.05
0.10
0.15
0.20
K
Figure 2-14: The tensor-to-scalar ratio as a function of the local curvature parameter
K. The parameters used are # 0 = 0.3 Mp1 , Xo = 10-3 Mp1 , 0 = 1000, A = 0.01, e = 0
and AX = 0.
Another important observational tool for distinguishing between inflation models
is the value of the tensor-to-scalar ratio, r. Although the current constraints are at
the 10-1 level, future experiments may be able to lower the sensitivity by one or two
orders of magnitude, making exact predictions potentially testable. For our models
the value of r is given by [45]
r=
We see that once TRs
-
1
1+'T2
(2.64)
0(1), the value of r decreases, as is depicted in Fig. 2-14.
One possible means to break the degeneracy between this family of models, apart
from / 3 i, is the correlation between r and n,. In the limit of vanishing Tps, both n,
and r revert to their single-field values, though they both vary in calculable ways as
Tps grows to be 0(1). See Fig. 2-15.
69
0.00305
0.00300
0.00295
S0.00290'
0.00285.
0.00275.
.
0.960
ns
0.955
.
.
. . .
0.965
.
0.00280'
Figure 2-15: The correlation between r and n, could theoretically break the degeneracy between our models. The parameters used for this plot are #0 = 0.3 Mpi,
Xo = 10-3 M, 1, O= 1000, AO = 0.01, e = 0 and Ax = 0, with 0 < , < 0.1.
We studied the behavior of
fNL in our family of models in detail in [48].
There we
found that substantial fNL required a large value of T&S by the end of inflation. In this
paper we have found that TRS remains small in the regime of weak curvature, r < 0.1.
Using the methods described in detail in [48], we have evaluated fNL numerically for
the broad class of potentials and trajectories described above, in the limit of weak
curvature (r < 1), and we find IfNLI< 0(1) for the entire range of parameters and
initial conditions, fully consistent with the latest bounds from Planck [62].
Thus we have found that there exists a range of parameter space in which multifield dynamics remain nontrivial, producing ,
- 0 (0.1), even as the other impor-
tant observable quantities remain well within the most-favored region of the latest
observations from Planck.
70
2.5
Conclusions
Previous work has demonstrated that multifield inflation with nonminimal couplings
provides close agreement with a number of spectral observables measured by the
[45] (see also [67]). In the limit of strong curvature of the effective
potential in the Einstein frame, > 1, the single-field attractor for this class of models
Planckcollaboration
pins the predicted value of the spectral index, n,, to within 1a of the present best-fit
observational value, while also keeping the tensor-to-scalar ratio, r, well below the
present upper bounds. In the limit of r
>
1, these models also generically predict no
observable running of the spectral index, and (in the absence of severe fine-tuning of
initial conditions [48]) no observable non-Gaussianity,
Ial, IfNLI
<
1- In the limit of
the single-field attractor, however, these models also predict no observable multifield
effects, such as amplification of primordial isocurvature modes, hence Pi.
limit
K
-
0 in the
> 1.
In this paper, we have demonstrated that the same class of models can produce
significant isocurvature modes, Pi,. ~ 0(0.1), in the limit of weak curvature of the
Einstein-frame potential,
K
< 0.1. In that limit, these models again predict values for
n,, a, r, and fNL squarely within the present best-fit bounds, while also providing a
plausible explanation for the observed anomaly at low multipoles in recent measurements of CMB temperature anisotropies [371. These models predict non-negligible
isocurvature fractions across a wide range of initial field values, with a dependence
of / 3 i. on couplings that admits an analytic, intuitive, geometric interpretation. Our
geometric approach provides an analytically tractable method in excellent agreement
with numerical simulations, which could be applied to other multifield models in
which the effective potential is "lumpy."
The mechanism for generating
#i.
0.1 that we have investigated in this paper
is based on the idea that a symmetry among the fields' bare couplings A,, g, and
r
is
softly broken. Such soft breaking would result from a coupling of one of the fields (say,
X) to either a CDM scalar field or to a neutrino species; some such coupling would
be required in order for the primordial isocurvature perturbations to survive to the
71
era of photon decoupling, so that the primordial perturbations could be impressed in
the CMB [46]. Hence whatever couplings might have enabled primordial isocurvature
modes to modify the usual predictions from the simple, purely adiabatic ACDM model
might also have generated weak but nonzero curvature in the effective potential,
. < 1. If the couplings A 1 , g, and C, were not subject to a (softly broken) symmetry,
or if the fields' initial conditions were not such that the fields began near the top of
a ridge in the potential, then the predictions from this class of models would revert
to the single-field attractor results analyzed in detail in [45).
Inflation in this class of models ends with the fields oscillating around the global
minimum of the potential. Preheating in such models offers additional interesting
phenomena [64], and further analysis is required to understand how the primordial
perturbations analyzed here might be affected by preheating dynamics. In particular, preheating in multifield models -
under certain conditions -
can amplify per-
turbations on cosmologically interesting length scales [70]. Thus the behavior of
isocurvature modes during preheating [71] requires careful study, to confirm whether
preheating effects in the family of models considered here could affect any of the predictions for observable quantities calculated in this paper. We are presently studying
effects of preheating in this family of models.
Finally, expected improvements in observable constraints on the tensor-to-scalar
ratio, as well as additional data on the low-e portion of the CMB power spectrum,
could further test this general class of models and perhaps distinguish among members
of the class.
72
Appendix A: Approximated Dynamical Quantities
In this appendix, we present results for dynamical quantities under our approxima-
tions that 0, x > 1,
p42
> M21, and Xo < Mpl.
First we expand quantities associated with field-space curvature, starting with the
field-space metric, gjj, using the definition from Eq. (2.6). We arrive at the following
expressions:
6M 2
Gx
=
6M
x
(2.65)
x
M21
We also find
6M21
g9ox = Gx_
~x'~ x
gxx
~
(2.66)
02
M21
PI
Next we expand the field-space Christoffel symbols, F,
-
-
xx
and find
(2.67)
xXX
rx
100
00-~
,,
_
xX (260
73
-
6x
The nonzero components of the field-space Riemann curvature tensor become
=OX
1X
'?RXXo
6
= -RX
(2.68)
e
We also expand dynamical quantities, beginning with the fields' velocity:
6 ~
and the turn rate w in the
V2oM4
(2.69)
4 and X directions:
Wo ,e 0
342 (2MpA+X - V
+O)
(2.70)
Appendix B: Covariant Formalism and Potential
Topography
We have defined the character of the maxima and minima of the potential using the
(normal) partial derivative at asymptotically large field values, where the manifold
is asymptotically flat, hence the normal and covariant derivatives asymptote to the
same value. By keeping the next to leading order term in the series expansion, we
can test the validity of this approach for characterizing the nature of the extrema.
We take as an example the potential parameters used in Fig. 2-3, specifically
S= 1000,
X = 999.985, A = 0.01, AX = 0.01, g = 0.01. The ridge of the potential
occurs at X = 0.
74
The asymptotic value of the second partial derivative is
-M A
_
ExxPx= -+P32
-M I x 1.5 - 10-(
Let us look at the partial second derivative for x
Kxx1x=O = MP44
1q
2
[-A.O
,O
[g
(MP2]
2+
O(M21
+0
(2.71)
#:
=
0 and finite
[
0.015 ('#2 + 10M2 1 j.
.1)
We see that the two terms can be comparable. In particular, the second derivative
changes sign at
Yxx1x=O =
0
=>
(2.72)
4,002, ~ 667M 1
which is a field value larger than the one we used for our calculation. In order to get
70 efolds of inflation, (q# 2 ~ 100MP , significantly smaller than the transition value.
For
#
< O1t, the second derivative is positive, meaning there is a transition where the
local maximum becomes a local minimum. This means that if one was to take our
Einstein frame potential as a phenomenological model without considering the field
space metric, even at large field values, where slow roll inflation occurs, the results
would be qualitatively different.
Let us now focus our attention on the covariant derivative, keeping in mind that
in a curved manifold it is a much more accurate indicator of the underlying dynamics.
(2.73)
'DxxV = 1 xx _ r-p v. - rhx.
Looking at the extra terms and keeping the lowest order terms we have Yx = 0 by
symmetry, V4 ~ A/((q# 3 ), and FP
= 4g(1 + 6x)k/C ~x/( 0#).
We will now expand the covariant derivative term in 1/# and also in
4
and x.
This way we will make sure that there is no transition in the behavior of the extremum
for varying field values, that is to say the character of the extremum will be conserved
term by term in the expansion (we only show this for the first couple of terms, but
75
the trend is evident). We find
DxxV =
A M+
4
2A4 -
(
A , 4(1+(2.74)
2_6
+
M-)
62(60(02)2
(1---+...
(1+2e)-
3A,,+
6
0 (1+e)+...]+.
3660
We have written the covariant derivative using the geometrically intuitive combinations of parameters, which was done in the main text in a more general setting
(x 4 0). It is worthwhile to note that we did not write the closed form solution
for DxxV (which is straightforward to calculate using the Christoffel symbols, given
explicitly in [48]), since this power series expansion is both more useful and more
geometrically transparent, since it is easy to see the order at which each effect is first
introduced.
We see that once we take out the (1/6,k 2) behavior there remains a multiple series
expansion as follows
" Series in (1/6402)
* Each term of the above series is expanded in inverse powers of (.
For the example of Fig. 2-3 the relevant quantity that defines to lowest order in
6, and (x all terms of the series is A 4 = 0.015.
By inspection of the terms, we can see that for our choice of parameters the first
term defines the behavior of the covariant derivative, which is also the asymptotic
value of the normal second derivative that we used to characterize the character of
the extremum. In the case when A4 = 0 the ellipticity term e is dominant. Even if
AO = e = 0 then the dominant term comes at an even higher order and is proportional
.
to A4 )
In other words, the character of the extremum is conserved if one considers the
covariant derivatives. For asymptotically large field values the two coincide, since the
curvature vanishes. It is thus not only quantitatively but also qualitatively essential
to use our covariant formalism for the study of these models, even at large field values
where the curvature of the manifold is small.
76
Now that the character of the maximum is clear we can proceed to calculating all
r7,,.
We neglect the term in M., that is proportional to RIZKL, since the curvature
of the field-space manifold is subdominant for
> M21 and the RZJKL term is
02
multiplied by two factors of the fields' velocity. If in addition we take X = X = 0,
then M,., becomes
M2
+44
M, s
VXXV.
(i +
=
(2.75)
Using the double series expansion of Eq. (2.74) the entropic mass-squared becomes
1
e
6.
+V C 2
+2
J(2
.76
)
[
1 (1 --+..
= +[AO - --6
M.,V+4M2-A4M,
To find the generalized slow roll parameter 7, we need to divide by the potential,
which again can be expanded in a power series for X -+ 0 as
= M41PPO
M 612#2
+ M+
P1404
(2.77)
The calculation of 77,, is now a straightforward exercise giving
-1
e-.0
-4A, +M2
7799 V
AO
[4A.,
C02
AO
2E+0
3
M4
~- +
3
4N.
(C002)2
-2--
i (1-e)+O(-)+
9
2e]
3
+(1
+16 N,2
1
CO
1
g,(CO02)3
()
(2.78)
[
-_
)
+
(
where we used the slow-roll solution for # from Eq. (2.47), identifying it as the
inflationary clock and the definition n = 4AO/Ao. By setting n = e = 0 we see that
even in the fully symmetric case the isocurvature mass is small but positive.
In the limit of X -+ 0 there is no turning (w = 0), and hence TRS = 0. In order to
77
calculate Tss we need
3 = -2E - 7,9 +
.
3 -1 +
tlo,
+
: K+
From Eq. (2.34), we see that Tss depends on the integral
Nhe
It
#3Hdt'=
dN'.
J
th.
(2.79)
N
Plugging in the expression for P from Eq. (2.79)
Nh
#dN' = K(Nhc - N.) - c ln
-N*-
-
(2.80)
2
where
Ci = 1 C2 =
9
8
4
3e
3.
8
(2.81)
2
(2.82)
Of course there is the ambiguity of stopping the integration one e-fold before the end
of inflation. If one plots P vs. N. and does a rough integration of the volume under
the curve, one finds this area giving an extra contribution
fl' #dN
~ -1. This is a
change, but not a severe one. We will neglect it for now, keeping in mind that there is
an 0(1) multiplicative factor missing from the correct result. However since 8 varies
over a few orders of magnitude, we can consider this factor a small price to pay for
such a simple analytical result.
78
Chapter 3
Self-Scattering for Dark Matter
with an Excited State
Self-interacting dark matter scenarios have recently attracted much attention as a
possible means to alleviate the tension between N-body simulations and observations
of the dark matter distribution on Galactic scales. The presence of internal structure for the dark matter - for example, a nearly-degenerate state in the spectrum
that could decay, or be collisionally excited or de-excited - has also been proposed
as a possible means to address these discrepancies. We analyze a simple model of
dark matter self-scattering including a nearly-degenerate excited state, and develop
an accurate analytic approximation for the elastic and inelastic s-wave cross sections,
which is valid provided the particle velocity is low (this condition is also required for
the s-wave to dominate over higher partial waves) and the conditions for a substantial self-interaction cross section are satisfied. This approximation may be useful in
incorporating inelastic self-scattering into N-body simulations, in order to study the
quantitative impact of nearly-degenerate states in the dark matter spectrum.
79
3.1
Introduction
The verification of the existence of dark matter on wildly disparate scales is one of
the greatest triumphs of modern astrophysics. Despite the fact that it is five times
more abundant than ordinary baryonic matter, there is no known particle that can
serve as a good dark matter candidate. Thus the dark sector remains one of the most
promising potential windows onto physics beyond the Standard Model. While there
are many efforts underway to probe the particle nature of dark matter through its
interactions with the Standard Model, to date dark matter has only been detected
by its gravitational interactions.
The distribution of dark matter on the sky can be inferred from measurements,
and may provide insight into its non-gravitational interactions as well. The formation
of structures in our universe is highly sensitive to dark sector physics. In particular,
the approach of treating cold dark (CDM) matter as effectively collisionless- such
as in the case of the weakly interacting massive particle (WIMP)- has been very
successful at explaining large scale phenomena such as the bullet cluster and baryon
acoustic oscillations. However, there are some discrepancies between collisionless
cold dark matter (CCDM) simulations and observations, particularly at small scales.
The disagreement between CCDM simulations and observation is most apparent in
dwarf galaxies, which are dominated by dark matter and thus make relatively pristine
(uncontaminated by messy baryonic physics) laboratories for studying the interplay
between the particle properties of dark matter and the structure of dark matter halos.
One such disparity, the "core-cusp problem" [72], is that CCDM simulations have
shown that virtualized CDM halos give rise to a prominent central cusp in the density
profile, usually characterized by the Einasto or Navarro-Frenk-White (NFW) profiles.
However, many observed dwarf galaxies have stable orbiting globular clusters, which
indicates a central core rather than a cusp, and their inferred rotation curves also
imply a flat core rather than a cusp (for example, [73, 74]). Another issue, the "missing
satellites problem," is that bottom-up CCDM structure formation simulations predict
an order of magnitude more dwarf galaxies than have been observed [75, 76, 77, 78].
One might wonder whether we may be unable to see Milky Way satellites simply
80
because they have so few stars and are too dim, which would arise from a deficit
of baryons. However, as was pointed out in 2011 and dubbed the "too big to fail
problem"
[79], the dwarf galaxies
we have observed are far less dense than many of
those formed in simulations. The densest subhalos predicted by simulation should
generally be dense enough to attract the necessary baryons such that star formation
would render those galaxies visible, and yet we do not observe these rather dense
subhalos either in the Milky Way or in the Local Group [80].
Mechanisms for resolving these discrepancies via the baryonic matter have been
proposed, and are an active field of research [81, 82, 83, 84, 85, 86]; however, they
might also be clues to dark matter microphysics. One interesting possible remedy
for these problems is that dark matter could be self-interacting via some novel dark
sector interaction [87]. Indeed, simulations have shown that self-interacting dark
matter can alleviate some of the tension between theory and observation with regard
to the problems listed above [88, 89, 90, 91].
Consistency with existing limits on dark matter self-interaction from large scale
structures (which have much greater virial velocities than dwarf galaxies) is most
easily achieved if the scattering cross-section has a velocity dependence, growing
larger at low velocities. (In the absence of velocity dependence, a small range of cross
sections, c-/M
-
0.1 - 1 cm 2 /g, can still solve the small-scale structure problems
while evading constraints from larger scales.) Fortunately, it is not difficult to realize
a velocity-dependent cross-section: if the dark matter is interacting via some longrange dark force, then at low velocities, there is a non-perturbative enhancement to
the scattering rate.
From the perspective of non-relativistic quantum mechanics, the perturbative expansion for calculating the scattering cross-section breaks down because the wavefunctions are substantially deformed by the presence of the potential; in terms of
Feynman diagrams, the result is dominated by an infinite series of ladder diagrams
(see e.g. [92] for a discussion in terms of the Bethe-Salpeter formalism). Instead,
one can formulate this interaction as a nonrelativistic quantum mechanical scattering
problem.
81
Additionally, if dark matter is charged under some new dark gauge symmetry
which is broken, then the states in the dark matter multiplet can naturally acquire
a small mass splitting, regardless of whether the gauge group is Abelian or nonAbelian (e.g. [93, 94, 95, 96]). Furthermore, if the dark matter is a Majorana fermion
or a real scalar, its vector couplings must be off-diagonal (in other words, there is
no vertex consisting of the vector and two identical scalars or Majorana fermions)
since it cannot carry conserved charge. In models of this type, dark matter can only
scatter elastically through the vector mediator by virtually exciting some slightly more
massive state, and its tree-level scatterings are purely inelastic. Such "inelastic dark
matter" models can have interesting implications for direct detection experiments (e.g.
[97, 98, 99]). More relevantly for the discrepancies described above, the presence of a
nearly-degenerate excited state in the dark matter spectrum could lead to interesting
kinematic modifications to the dark matter self-scattering. Exothermic scatterings
from this excited state could deliver velocity "kicks" to dark matter particles that are
comparable or larger than the escape velocity of the systems in question, especially
in the slow-moving environs of dwarf galaxies, and in this way dilute dense cusps
[100]. (Velocity kicks from late-time decays of a metastable excited state have been
considered in e.g. [101, 102, 103, 104, 105, 106].)
Previous work in simulating self-interacting dark matter, and understanding its
effects analytically, has focused on the case where only a single dark matter state
participates in the interaction. Even in situations where the scattering is purely
elastic (i.e. there is no transition to a state of different mass) the presence of a nearlydegenerate state in the spectrum can significantly modify the resonance structure of
the scattering cross sections. However, the addition of a second state adds at least one
additional parameter to the problem (the mass splitting), making numerical analysis
computationally expensive and, in parts of parameter space, unstable. One main
goal in this chapter is to work out an analytic approximation for the cross section
of dark matter scattering via an off-diagonal Yukawa interaction in the presence
of an excited state. This corresponds to the case where the dark gauge group is
U(1) and provides a simple and illustrative toy model for inelastic dark matter self82
scattering more generally. As we will show, our expressions give good agreement with
numerically solving the Schr6dinger equation and they are also relatively simple and
highly intuitive. We identify regions of parameter space with particular relevance to
dwarf-galaxy-sized halos, and consequently to the discrepancies described above.
The chapter is organized as follows. The numerical quirks of solving the Schrbdinger
equation in our model motivate a series of analytic approximations, as described in
Section 3.2. We derive the approximate scattering cross sections, which we then further discuss and examine in several regimes of interest in Section 3.3. In Section 3.3,
we also numerically verify the validity of our approximations for the scattering cross
sections and display the resonances inherent in a system with an excited state. We
discuss the plausibility of this mechanism for explaining the observations of Milky
Way satellites' internal structure and dynamics in Section 3.4. Concluding remarks
follow in Section 3.5.
3.2
3.2.1
Dark Matter with Inelastic Scattering
A Simple Model
We consider the case of a Yukawa-like interaction coupling two states with some small
mass splitting, J. We follow the phenomenological model discussed in [93, 107, 108],
where the dark matter is a pseudo-Dirac fermion charged under a dark U(1) gauge
group. At high energies, where the U(1) symmetry is unbroken, the dark matter is
a charged Dirac fermion; at low energies, a small Majorana mass splits the Dirac
fermion into two nearly-degenerate Majorana states.
The potential matrix coupling the 111) and 122) two-body states (corresponding to
both particles being in the ground state or both particles being in the excited state)
will be
0
- hca*
--hcae-or
2Jc2
83
where a is the coupling between the dark matter and the mediator, mo is the mass
of the mediator, 6 is the mass splitting between the ground and excited states, and
the first row corresponds to the ground state 111).
Note that the interaction between the ground state, 11) and the excited state, 12)
is purely off-diagonal. This is a natural consequence of taking the force carrier to be a
vector, as the mass eigenstates are 45* rotations of the high-energy gauge eigenstates,
and do not carry a conserved charge. As a result, two particles initially in the same
state (ground or excited) can only scatter into the two-body states where they are
both in the ground state, or both in the excited state. If the initial state is 112),
i.e. one particle is in the ground state and the other in the excited state, then their
scattering decouples from the other two-body states and is elastic, with the final state
being 112) or 121). This case can be treated by the existing methods in the literature
(e.g. [109]).
3.2.2
Approximate Wavefunctions
We define the dimensionless parameters:
e,
=
-
7
2j ) CO
E
--
(3.2)
so that rescaling r by amxc/h gives the s-wave Schrodinger equation:
e-e~E2
_
2
Note here that mX is the mass of the dark matter, and v is the individual velocity
of either of the dark matter particles in the center-of-mass frame (half the relative
velocity). The wave function o(r) is the physical wavefunction rescaled by r. The
eigenvalues A and eigenvectors ?P of that matrix are
84
26 +
r 2 'r
%/
71-7
-.
VE
F1F3.4)
ev=1+(4e-'O*")/(r2C
(
A= -2+e 2
There is a transition in the behavior of the eigenvalues and eigenvectors when
.- c+r ~ d. In the regime where
> 2 , the eigenvalues and eigenvectors can be
2r
r
approximated as
A+
<
When 9
2
2
e2
+ _a
2
er
r
1
d 2
TF1
(2
(3.5)
:, the eigenvalues and eigenvectors can be approximated as
27
f2
A+
2
A V~-,+e+
(\('
e2
e-2cr''
2
r~,r
-~
V
e-2co
26,+,-2
',
0
1
1
0
(36
00
Because the Yukawa potential has very different small-r and large-r behavior,
the diagonalization of this matrix is roughly independent of r within those regimes,
provided that e2/2 < CO (if e/2 > e, then as r gets larger, the diagonalization
of the matrix changes before the behavior of the potential changes.) Therefore, if
q4'(r) = A 0 (r), then 0i(r)o is an approximate solution to the matrix Schrodinger
equation in those regimes. However, in the transition region where *--
d, the
eigenvectors will vary as a function of r, so we will match the small-r and larger wavefunctions using a WKB approximation to acquire the wavefunction in this
region.
A more specific (albeit schematic) outline of how the wavefunctions are derived is
as follows. For r < 1, we approximate the Yukawa potential as V(r) ~ 1/r, assume
the potential term dominates (since e2 and e2 are assumed to be small), solve the
resulting Schr6dinger equation exactly, and propagate that solution outward using
a WKB approximation. The validity of the WKB approximation in the potentialdominated regime requires that
/V'(r)/V(r)j = Iv/ eEr/ 2 (e,,+ 1/r)
< 1, which is
just equivalent to requiring that the spatial variation of the local DeBroglie wavelength
85
V(r)
-
V(r) > e/2, V(r) > E2, e = 0
1/r for small r
WKB Approximation for intermediate r
V(r) ,
,
e/2, E/2
6
V(r) < E/2
V(r) ~ e-1' for large r
Neglect small-r repulsed eigenstate
Substantially Enhanced Cross Section
e,
efV, EO,
es
p
P
$
,
e= 0
Table 3.1: A summary of the various approximations and assumptions used for deriving the wavefunction in different regimes. Despite all of these restrictions, the various
regimes of validity overlap a great deal, making these approximations useful in large
swatches of parameter space.
is sufficiently gradual. For r > 1, the condition for validity of the approximation is
that eO//V(r) < 1, so the approximation breaks down when V(r) ~
C2.
Where
the WKB approximation breaks down (at V(r) $ e2) or where the diagonalization
approximation fails (at V(r) $ e/2), we match the WKB solution to the large-r
solution. Therefore, in this work, we will choose the matching radius rM such that
e- M = max
,1f.
For large
lag r we approximate the Yukawa potential as an exe~rM =mxk2)
ponential potential V(r) ~ Voe-Ar (on the condition that the matrix Schrbdinger
equation is approximately diagonal), which can be solved exactly. We summarize the
different regimes in Fig. 3-1. The large-r solution involves hypergeometric functions
which can only be analytically matched at rM by using an asymptotic expansion; however, if e, >; p, then there is an exponentially suppressed term with an exponentially
large prefactor and the exact details of the resulting phase shift will be dependent on
the matching procedure. In order to extract cross sections in a way that is insensitive
to the matching procedure, we choose to work in the e, < M regime, where it is safe
to ignore this term.
86
101
V(r) - 1/r
(potential dominated)
-WKB
- V(r) ~ VO ew
solvable
region
W(exactly
2-state system)
10-
.
......................
i -2
r=r!
1r=1
0.1
1.0
10.0
100.0
Figure 3-1: Example of the different r-regimes and matching points for a sample
parameter set (eV = 0.1, ej = 0.02, co = 0.05), following [107]. The plot shows the
exact Yukawa potential (solid black line) and the approximate potentials we employ,
in their regimes of validity. In the r < rM region where the eigenstates are decoupled,
for r < 1 the potential dominates the kinetic energy and mass splitting and is well
approximated by V(r) ~ 1/r (red dotted line), whereas for 1 < r < rM the WKB
approximation is employed to obtain an approximate wavefunction. At r > rM the
WKB approximation may break down, but there V(r) ~ Voe-l" (dashed blue line).
Note that for this entire calculation, we neglected higher-order partial waves, as
our method does not generalize straightforwardly to that case [107]. In general, the
s-wave only dominates for e, < EO [109]; however, since eo ~ M (to match the largedistance behavior of the potential), the regime where the higher partial waves can
be neglected is identical to the regime where our approximation is well-controlled, as
discussed above and in Appendix A. While the e, > eO regime may be interesting, the
techniques of this chapter are not applicable there anyway (with a caveat about the
s-wave in the case where the mass splitting is substantial and the transition from large
r to small r is adiabatic, as discussed in Appendix D.) The main point here is that we
will generically ignore the regime where c, > M because we are dominated by higher
partial waves and our approximation for the s-wave usually breaks down anyway. In
the language of [109], our results are an extension of their analytic approximation in
the "resonant regime" to the case with two interacting states; the "classical regime"
in this two-state case is beyond the scope of this work.
We also require that e 1 , eS, and co are all less than 1 in order to see substantial
enhancement to the s-wave cross section.
If c, > 1, the kinetic energy is large
compared to the potential energy; if co > 1, the range of the interaction is short;
87
in both cases the presence of the potential does not significantly deform the wavefunction. If e > 1, the mass splitting is large compared to the Bohr potential energy
(~
a2mX, leading to a suppression of virtual excitations.) Since the potential is purely
off-diagonal, this suppresses the elastic scattering cross section as well.
For completeness, we include a full mathematical description of the approximated
wavefunctions in Appendix A, including the WKB matching between the small-r and
large-r wavefunctions. A more in-depth derivation of these wavefunctions (including
more extensive discussion of the regimes of validity) can be found in [107].
3.3
3.3.1
The Scattering Cross Sections
Semi-Analytic Results
We will apply the solution to the matrix Schr6dinger equation (Eq. 3.3) derived
in [107] to extract the scattering cross sections. In this case, we use the regular
boundary conditions, 0+ (0) = 0_(0) = 0 (recall
4
is defined in terms of ?/, the
radial wavefunction rescaled by r, and the wavefunctions should be regular at the
origin). Note this is not the same as the approach used in [107], where the Sommerfeld
enhancement to annihilation was extracted from irregular solutions with 0 (0) = 0.
Additionally we will impose one of two sets of boundary conditions: the radially
ingoing particles will either be purely in the ground state or purely in the excited state.
We include a derivation of the dimensionless transfer cross sections in Appendix A
(these must be multiplied by h2 /(c 2 a2 m2) to obtain the physical cross sections, since
we initially rescaled r by amxc/h) and the final results are:
2te
aground-+ ground =
i
+
*
2
cosh
sinh (("
cosh
sinh
p
-2i1A
(3.7)
88
sin
cosh
1o T)j
d excited
Uexcited-
(3.8)
U'ground-+
0
excited
excited-+ ground "
(CA.E)(
2
=-
WC")
sinh
-incos2
(
(E"+-A)
I
)
escoh2x~ -. )cosh
2r cos 2 W sinh
A
o(2)
(3.9)
(3.10)
s
(A
2ir c)
2
CA cosh 2 ('(,~'v) (cosh (
14(''E) - cos(2W))
/e -! J, and P and Vo are parameters in the exponential
where we have defined eA =
potential,
2
2
(3.11)
=
,4
1+
1
E4rM
with rM chosen so that
rM
= max
eEorM
. The terms
!
. and r,
come from
matching the WKB wavefunction onto the wavefunction for the exponential potential,
and are defined by,
(
~
)
((3.12)
+ 1) r
r ("A~
r (i+
rA
e +iC+A 2
~
+
P,~
(v+
EA +
).
Finally, W is a phase that comes from extending the WKB solution to the matching
region,
iW
T
_dr'+f A dr+ 2i V -,-,/2
with r, chosen such that V0 e-''
>>
(3.13)
v, C. We emphasize again that these cross
sections only hold in the regimes described in Table I. Since we are only computing
the s-wave piece of the scattering amplitude, which is angle-independent, the viscosity
and transfer cross sections are trivially related to a.
The astute reader will notice the following salient feature of the scattering cross
89
sections: the elastic and inelastic cross sections are the same whether the system starts
in the ground state or the excited state, modulo a swap of e, with e, (assuming that
E& is the same in both cases, which requires the system to be above threshold.) This
reflects the identical interactions of the ground and excited states: swapping E, ++ e,
simply corresponds to relabeling the states. The result also agrees with our intuition
from quantum mechanical scattering off a 1D step potential: the transmission and
reflection amplitudes are the same for "downhill" and "uphill" scattering when the
particle's energy is greater than the potential barrier, and the same is true in this
system above the mass-splitting threshold.
There are a few other interesting features (such as resonances, cutoffs, and extrema) of the amplitudes that are slightly more subtle, which we will expand on in
the following subsection. We will also show evidence supporting the claim of our
approximations' validity in the regime of interest.
3.3.2
Features and Limits of the Scattering Cross Sections
Here we explore features of the approximate cross sections and depict their behavior
alongside the exact cross sections, which we determined by numerically solving the
matrix Schrddinger equation using Mathematica. Numerically solving for the scattering amplitudes proved computationally expensive in certain regions of parameter
space, which further motivates the use of these approximations.
One particularly salient feature of the dimensionless cross sections is that the
"uphill" and "downhill" scattering amplitudes are identical above the mass-splitting
threshold, and that below threshold, the probability of any kind of inelastic scattering
goes to zero. As we can see from Figure 3-2, the dimensionless cross sections exactly
match far above the threshold, but near threshold the 1/E factor in the cross section
from the excited state to the ground state makes it far more likely that a particle will
de-excite. These cross sections correspond to different physical scenarios; an excited
state particle in a virialized halo will give a larger ground state velocity and thus
correspond to a larger resulting e, than for virialized ground state particles in the
halo. We further discuss the astrophenomenology of these situations in Section 3.4.
90
Inelastic Scattering from the Ground State
105
10 4
Analytic Approximation
--- Exact Numerics
1000
100
10
I.I I
0.002
0.005
0.010
0.020
0.050
0.100
Inelastic Scattering from the Excited State
106
105
104
b 1000
-
Analytic Approximation
--- Exact Numerics
100
10
0.01
0.002
0.005
0.010
0.020
0.050
0.100
Figure 3-2: A comparison of inelastic scattering from the ground state to the excited
state (left) and from the excited state to the ground state (right) with es = 0.01 and
co = 0.04. The top panels give the dimensionless cross section as a function of the
ground state particle's velocity E,, and the bottom panels give the dimensionless cross
section in terms of the excited state particle's velocity eA. While scattering from the
ground drops off to zero below threshold, scattering from the excited state goes as
1/e for small eA, which shows that it is much more likely for particles to de-excite
near threshold. For large E,, scattering from the ground and excited states is equally
as likely, since there is little energetic "overhead" to upscattering. We can see that
our approximations agree well with the exact numerical solution for E, < 6,, where
the contribution from the repulsed eigenstates can be ignored.
91
1og(o-)
log(o-)
09
0
7
7
5
5
-2
-2
3
3
3-1
-4
-3
-2
Iog(E,)
-1
0
-4
-3
-2
-1
0
log(e,)
logo0)
log(q)
0
7
7
5
5
-3
S
1
-3
-3
-4
-4
-3
-2
-1
3
-4
-4
0
log(E,)
1
-3
-2
-1
0
log(e,)
Figure 3-3: An analytic calculation of the elastic ground state cross section in the CO
vs. e, plane with e, = 0 (top left), c8 = 0.01 (top right), ES = 0.05 (bottom left), and
Ej = 0.1 (bottom right). This resonance structures in the top left panel are exactly
the same as depicted in Figure 1 of [109]. Note that to the lower right of the diagonal,
the approximation is no longer valid because of the repulsed eigenstates.
Another feature of our approximate cross sections is the shift in resonances and
anti-resonances for different mass splittings, as depicted in Figure 3-3. These certainly
warrant some further explanation, which motivates our expansion of our expressions
in a variety of limits.
92
The Degenerate Limit
In the limit where 6 -+ 0, expressions for the scattering cross section in a repulsive or
attractive potential have been previously derived [109]. In this limit, our "elastic" and
"inelastic" cross sections refer to scatterings between particular linear combinations
of the attracted and repulsed two-body eigenstates, and it is natural to switch to
the basis of (dark) charge eigenstates accordingly (which in this limit are also mass
eigenstates). In Appendix 3.5 we show in detail how this conversion is done, and find
good agreement with previous results [109]. In particular, we find that the resonance
positions occur when p = nir and n is an integer; as noted in [107], when 1 > c4 ,E
and e,, < p, W is well approximated by W - at E ;
.
V27/e, meaning the resonances
In the analysis of [109] the resonances occur at co =
occur
where the
parameter r, is chosen to be 1.6, in close agreement.
For completeness, we include the 6 -+ 0 expansion in the body of the text. The
elastic scattering cross section to first order in Ej (for both the ground and excited
state, since they are now degenerate) becomes:
Uelastic =7-
72
1+
cosh (yr
Sr (1+ i- )
isin
r (1 -
sinh (r--
)
-2(3./14
(V
.sJA
0
Off-resonance, in the limit as e,, a+0, sinh
iW)
(
- i)
(3.14)
tj
-+
-isin p, and the cross
section scales as 1/pi 2 . More precisely, a Taylor expansion yields:
Oelastic -+
17
cot W + 21n (L)+
2_Y
,
(3.15)
where -y is the Euler-Mascheroni constant. On-resonance, where sin V = 0, the elastic
scattering amplitude is simply 1, and the dimensionless cross section is accordingly
Oelastic
= I/E,.
93
"Elastic" Scattering with
E, =
0
105
1000
0.1
-
10
0.005
0.010
0.050 0.100
"Inelastic" Scattering with Ej
=
0.500
1.000
0.500
1.000
0
106
100
1
0.01
10- 4
0.005 0.010
0.050
0.100
Figure 3-4: Scattering with ej = 0 and c, = 0.01, where the red solid curve comes
from our analytic approximation and the blue dashed curve comes from numerically
solving the Schrddinger equation. The quotation marks in the plot titles serve as
a reminder that with Es = 0 there is no inelastic scattering because the states are
degenerate. Resonances occur as predicted by our Taylor expansions.
Meanwhile, for the inelastic case, setting e, = eA yields
27r cos 2 W sinh 2
( irE)
(3.16)
Uinelastic =
e (cosh (A)
94
- cos(2tp))
Off-resonance, as c, -+ 0, this probability approaches
(3.17)
.
7r
()cot
inelastic -
On-resonance, where cos(2 o) = 1, the inelastic scattering amplitude approaches 1,
and again
Uinelastic -+
7r/E 2
The Low-Velocity Limit
Now let us consider the case where E, -+ 0, without first setting es -+ 0.
In this
limit, scattering into the excited state is forbidden, so we will only examine elastic
-ground-ground to first order in E, yields
scattering in the ground state. Expanding
7r
-
rcot (O - + 2 In
-
- 41n2 - 2b0~
+ 2yOground-+ground
r2p
7rE8
r
12
+ (1)
VoE'5
a
2
(3.18)
is t u2p
where 00 is the digamma function and -y is the Euler-Mascheroni constant.
j
Elastic Scattering in the Small E, Limit
106
104
I
AI
100
1
0.01
0.01
0.02
0.05
0.10
0.20
0.50
1.00
Figure 3-5: Scattering from the ground state to the ground state with E, = 0.001.
Shown here are different scenarios with E, = 0 (solid, red), ej = 0.01 (dashed, blue),
and E6 = 0.03 (dotted, purple).
The presence of the mass splitting -
ones- changes the resonance positions, especially for small co.
95
even small
We see that the cross section does not vary with e, in this low-velocity limit, with
the cross section approaching the expected geometric size of ~ 7r/m2 once we convert
to dimensionful parameters (making the approximation A ~
=
mo/(amx), and
then multiplying the dimensionless cross section by 1/(amx) 2 as usual). Resonances
occur when p = (n + e6/2p)7r, and in the case where 6 = 0 (as mentioned previously)
the resonance positions are W = nir. Thus the presence of a mass splitting induces
a shift to the resonance positions at velocities below the threshold. The shift in the
resonances is the same as found for the case of annihilation [107].
Except for the shift in resonance positions, this cross section is very similar in
form to (3.15); in the limit as e6 -+ 0 (but e, < e5) they are identical, except that
(3.18) has a 4-y term rather than 2y (two extra -y's come from the 7O(l)). This is
a subdominant correction; generally larger contributions will arise from the cot and
log terms. So we see that for elastic scattering in the ground state, the effect of the
mass splitting is primarily just to shift the resonance positions; this contrasts with
the case of annihilation where switching on the mass splitting can lead to a generic
enhancement of the cross section by a factor of 2-4 at low velocities [107].
The Threshold (E, = e5) Limit
Scattering amplitudes involving the excited state will be suppressed by
EA
as eA
approaches zero from above, but the corresponding cross sections need not vanish.
The case where c, ~ c, corresponds, for particles initially in the excited state, to
very low physical velocities. We perform a Taylor expansion in small (but real and
positive)
EA,
finding for the cross sections:
47r 2eCACOS2 W tanh
Uground-+excited
.
2 11ecos
,i
(cosh
ot
s
(M.
()
(')-cos(2p))
47r 2 cos 2 Wtanh
2 2p(3-20)
ground
(excited-+
(
E6 CA 11 cosh
(3.20
A
96
JA
(3-19)
-
cos(2w))
2
O-excited-+excited
:
cos
1 227r
(c2 +
Scosh
(e)-
/2
2Csin o + cos Wsech
cos(20)
2
(3.21)
where for convenience, we have defined the subdominant 0(1) term
( 2-1-2,0o
-+b 2M
2
+ ixrtanh
-T' + ln (
21t
161t
,
(3.22)
which is a real quantity because the term with the hyperbolic tangent cancels out the
imaginary part of the digamma function.
We see that in all cases there is a potentially large enhancement corresponding to
the zero-J resonances, W = nir so cos(2#) = 1. The cross section does not actually
diverge at these pseudo-resonances, but scales as 1/(cosh(rej/tt) - 1), and so is large
when ES
< t.
The upscattering cross section vanishes as e,
-+ 0, as expected, as the phase
space for newly-excited particles shrinks to zero. The elastic scattering cross section
for the particles in the excited state scales parametrically as 1/p2, except close to
the resonances, where it instead scales as 1/el if Es < p. Both these behaviors
correspond to geometric cross sections, one governed by the range of the force and
one by the momentum transfer associated with virtual de-excitation to the ground
state. The downscattering cross section, perhaps most interesting for scenarios where
an abundant relic population of dark matter exists in the excited state, diverges as
1/EA, meaning that if vec = aEA is the physical velocity of the incoming particles in
the excited state, -veXC will approach a constant value at low velocities. For e
<
I,
the cross section scales as 1/(EgAp 2 ) away from the resonances, and 1/(eAe3) close to
the resonances. Inserting the dimensionful prefactors, the physical cross sections for
downscattering and elastic scattering in the excited state have the following scaling
behavior:
97
off-resonance,
excited-+groundVexc OC
near-resonance
a
(3.24)
near-resonance.
off-resonance,
oexcited-+ excited OC
(3.23)
We note that for particles initially in the excited state and slow-moving, inelastic
downscattering will generally dominate over elastic scattering (due to the 1/v scaling). The constant av for downscattering implies that the argument given in
[1001,
predicting a constant density core in dwarf galaxies as a direct result of a constant
ov for exothermic interactions, holds even at low velocities where the perturbative
approach used in that work is invalid. However, the scaling of the constant av with
the parameters of the model is quite different to the perturbative case. Note in particular that in regions of parameter space close to a resonance, large scattering cross
sections can be achieved even for large mo (provided mo < amx so our approximation
holds), depending only on the mass splitting and the dark matter mass rather than
the mediator mass.
The cross section for elastic scattering in the ground state does not have a simple
behavior close to threshold, since there is nothing special about c, ~ e6 from the
perspective of the ground state. Setting e& = 0 we obtain:
rx
-+
r
fr
1+
2
A
+
2p
),
(3.25)
2)
(
a(ground -+ ground) = -
1+
_)
sinh
-r 1
V
V2
The term involving sinh's approaches 1 when W
-+
Lsinh
)
p +i
-
2
1
(3.26)
ip
nir and -1 when W
-4
(n + 1)7r/2,
and these give rise to the characteristic resonances and anti-resonances in the low-v
limit. More explicitly, we can perform a Taylor expansion in the low-velocity limit
c, < P (here having already set e, = e6), obtaining:
98
Elastic Scattering at Threshold
4
10
100
\
1
0.01
0.02
0.05
0.10
0.20
0.50
1.00
Figure 3-6: Scattering at threshold, e =E, = 0.01, where the solid red line is from
the analytic approximation and the blue dashed line is from numerically solving the
Schr6dinger equation. Resonances occur as predicted.
a-(ground -+ ground) ~
7r cot(p) + 2 In
+ 47.
(3.27)
We see that this cross section has the same form as the other low-velocity and lowmass-splitting limits we have studied; it is identical to the expression obtained by
first taking E, -+ 0 and then es -+ 0.
3.4
3.4.1
Applications to Dark Matter Haloes
Parameter Regimes of Phenomenological Interest
On dwarf galaxy scales, an elastic scattering cross section of roughly o/mx > 0.1
cm 2 /g is required in order for dark matter self-scattering to have a significant impact
on the internal structure [891. This corresponds to particles in the core interacting
once on average over the age of the universe [100], and so is likely also a necessary
condition for exothermic downscattering to be relevant. We will thus use this cross
section as a benchmark.
99
As discussed in [100], requiring a significant relic population of particles in the
excited state at late times (that was not depleted by scatterings in the early universe) requires mx at the MeV scale or lighter. However, the excited state might
be populated non-thermally, in which case much heavier DM masses might also be
viable.
For the non-degeneracy of the excited state to have a significant impact on scattering in dwarf galaxies, the mass splitting should be significant compared to the
typical kinetic energy of the dark matter particles. Taking the typical velocity in
dwarf galaxies to be 10 km/s~ 3 x 10- 5 c [110], this implies 6 > 10-mx in order to
see differences from purely elastic scattering. If the mass splitting is radiatively generated, the natural scales are of order a2 mx (the Bohr radius of the dark matter if the
mediator mass can be neglected), amo or a 2mO (if the splitting is generated by loops
of the mediator) [93]; if the mass splitting is generated by some higher-dimension
operator as in [108], then its size depends on the heavy mass scale. Our requirements
that c6
5
1, co < 1 impose that 6 < a 2 m, and amo < a2m . So in order for our
approximation to be valid and the mass splitting to be interestingly large, we will
focus on the range a > 10' (or higher for larger 6: a > V/iMX), which will also
guarantee e, = v/a < 1 as required. For a vector mediator, this is in agreement with
broad expectations from the Standard Model, if the coupling is not fine-tuned to be
small.
In general, we will treat a as a free parameter within the range 10-'
a < 1,
since the constraints on it are rather model-dependent. There is an upper bound on
a from the requirement that annihilations not over-deplete the dark matter density
in the early universe, unless the observed relic density is non-thermally produced at
some later time or the dark matter is asymmetric. This upper bound is somewhat
model-dependent; focusing on TeV-scale DM in the same simplified model used in
this work, [108] found typical values for a (yielding the correct relic density) of a few
times 10-2. Annihilation of the dark matter to the force carriers has an annihilation
cross section scaling as a 2/m2, so lighter DM generally implies a smaller value of
a if the DM is indeed a thermal relic. In any case there is no corresponding lower
100
bound on a, as annihilation channels not involved in the self-scattering could prevent
over-closure of the universe even if a is very small.
There are few model-independent constraints on the mediator mass mo; the coupling of the force carrier to Standard Model particles is independent of its role
here of mediating dark matter scattering. For significant scattering we require that
M,
am , and for s-wave scattering to dominate and our approximation to be valid
we will generally require that e, < co, i.e. mo > mxv ~ 3 x 10- 5 m.
When we consider the exothermic scenario with a significant population of particles initially in the excited state, their scatterings have f, ~e6 in our notation,
assuming the kinetic energy of the excited-state particles (limited by the escape velocity of the dwarf) is small compared to the mass splitting. Thus for this scenario
m2/a2m2 => Mis > V6h . The require-
ment that m4 < amx means that the higher a is above its lower bound of
/
we will also require E < co, i.e. 23/a 2 MX
the more valid parameter space there will be for mo (although raising m , above the
upper bound should just send the scattering cross section toward zero.)
3.5
Conclusions
We have presented an analytic approximation for the scattering cross sections for dark
matter interacting via an off-diagonal dark Yukawa potential. Our approximate cross
sections make contact with previous work in the literature and provide interesting
new phenomenological features to be studied. The regime of validity of our approximations corresponds to the astrophysical regime of interest, and in those regimes the
dark matter self-interaction cross section can potentially be large enough to dissipate
the central cusps of dwarf halos. We hope that in future work, incorporating these
semi-analytic scattering cross sections into numerical simulations will allow the first
detailed studies of haloes containing inelastically scattering dark matter.
101
Appendix A: A Brief Review of Previous Results
Approximate Small-r Wavefunctions
For small r, we can approximate the Yukawa potential as 1/r, which therefore domi1/r. The s-wave solutions to the rescaled
nates the eigenvalues at small r so Ai
Schrodinger equation can be expressed in terms of Bessel functions:
0 (r) = A- /iJi(2V.) -
Or#_(0),Y 1(2/f)
(3.28)
0k+(r) = A+ fI1 (2 f) + 2 +(0) VK1 (2 V),
where A+ and A- are the coefficients of the repulsed and attracted eigenstates, respectively. Moving radially outward (but still within the regime of validity for the
V ~ 1/r approximation), the large-r asymptotics of the Bessel functions give
-+1)+
-~L
(r)
A- 4
'''dr'
+
(V
+(o)
~A+
-/
f~2
r VA
'
+o
')
1.
1/
2V/7r NF
ef
2
(A + iir_(O)) efo'V-('')'+
of
(A_ - iro-_(0)) e-E
-(2-')
(3.29)
If the particle velocity is high enough (above threshold) such that there exists a radius
,S then A+(r*) = 0 and we must perform a
r* in this regime where V(r*) =
WKB approximation about the turning point. Linearizing the potential and matching
the wavefunctions on either side using the connection formulae yields:
[(1/ A
-1
A+ [2#
+
+1
2
~
.A
i~\
\
e-2o* /,\(r)dr\
eTi
fo
(r)dr'
2
/- (~e2f,* VrA+r)dr' +2 (V+ (0) 2./- -
~ etfo \-\()drj'
(3.30)
In either case (above or below threshold), the small-r wavefunctions are in a form
that will match smoothly onto the WKB wavefunctions.
102
Approximate Large-r Wavefunctions
At large r, we can approximate the Yukawa potential as a purely exponential
potential of the form V eP'. We impose conditions on V and I by requiring that
the exponential potential mimic the Yukawa for r > rM, where rM is the matching
= Voe-M". We also require
radius. The potentials should match at rM so *~9
that fr e-
7
dr
r Vo e~'r dr, which comes from solving the Lippman-Schwinger
-= f
form of the Schr6dinger equation and requiring that the rescaled wavefunctions from
both potentials match to first order in the coupling constant a. The parameters p
and V are therefore given by
(1
-+2
2
es~
V
0+=(3.31)
4
+
1
ErM
-2i1g
rM
The wavefunctions for an exponential potential can be solved for exactly in terms
of OF3 hypergeometric functions [1111 as follows:
Pc -[)
1
4p2
+ C2
+
4M2
x oF3
C4
VF-
Ap2 oF3
42
Voe
2
4p2
2
it
[-iCEI
1
1+ i(e, + EA)
i(+e EA)
2p
i3
1
i(E
-
2
4y2
2y
2
i(E + EA ) 1
2p
2p '22
p'2
Voe~Pr
1
2M4L
2
(V oe~ r) 21
p
(C3Voe
4p2
3
2
)~
3
21+
2 2M A42 i(e +
i(E -ea)
,E+e,
2]
i(E - EA)
i(e, + EA) 3
3
- -+
2
2p
'2
2M
i
-3
}(Voe
2])
4,uAp
(3.32)
103
C1 ( 1e)
"
(VoePr
3 +,pi(E,
3
-EA)
[{3
(1
ie-)2 + 4
+
2
oF3
(V42
)
C4
x oF
1
1
[1
-
21
4p~2
j
p
i(e, - EA)
'2
2p
,
1p)
)2]
\41P2
')2)
1
+ i(e
(
+
2pi(e
C')
2
p
4p2
3
Voe -,, \
+1(2
1+
~ Voe-"
1
J
~
(E2(+1
ie)2)
+t (2
2/pz
-
x oF3
2p
iE
,1+-
3 i(E +EA)
2-2p
E2+
-C3
)
i(e +
)
C2 (Vc
'2
12
2p
)
xoF3
+
+
'
02
+ 62p) 1
+ i(e 1 eC)
2pi
' 2
2p
I
4p2V )2]
4pp(2
1
(3.33)
where we have defined eA = V/e - ej. The wavefunctions are expressed in terms of
four linearly-independent solutions, corresponding to ingoing or outgoing particles in
the ground or excited states. In particular,
* the C1 term represents an ingoing wave in the ground state,
" the C2 term represents an outgoing wave in the ground state,
" the C3 term represents an ingoing wave in the excited state,
* the C4 term represents an outgoing wave in the excited state.
104
WKB Approximation for the Intermediate-r Wavefunctions
To match the large-r wavefunctions to the small-r wavefunctions, one can use the
WKB approximation to propagate the known wavefunctions of the exponential potential into the transition region. We write the large-r WKB solutions as
(3-34)
1A (Ej-e fo VA/Crdr' + F,e -for V\IE&)dr').
0=
where ). are the eigenvalues of the matrix Schr6dinger equation with an exponential
potential rather than a Yukawa.
In order to match the WKB solution with the exact solution for the large-r exponential potential, we define the following convenient quantities:
+
1) (r+
2
A
Irv
r
F&
r 1+
+
A
21
1-r (i62 -S&
r
A
+
2pp
(
2pL
2
(3 35
(
+
1
+
2p
2)
Then, deriving expressions for the WKB coefficients E and F to match onto the
exponential wavefunctions is a matter of using the asymptotic behavior of the oF3
hypergeometric functions in the r -* -oo limit. We find that'
'Note that by setting E+ = 0, we are neglecting a term that is exponentially suppressed. However,
if e, > p, then that term has a cosine prefactor with a large imaginary component, and this will cause
the matching procedure to affect the apparent phase shifts. Such an artifact would give incorrect
cross sections, so we work in the regime where e,, < p.
105
E+=0
F+
=
(2r-01+
/2
-
e (l f
eJ'$-A'''dr'-
F=
e
2i V el"
(C1Fr + C 2P* + C 3 7r, + C 4q* *)
A
+C 2 F*,e? -C I ae~
~
x C1
/4jdr'
-C 4 ?,*PTe)
e~
3 2
(2r) /
+ C 2F*e~J
x (Cive
C3 r
-
e
-
C42i*Pse
(3.36)
where r, is some radius chosen such that V 0 e-L'
>>
e2
In order to match the WKB solution with the small-r solution below threshold,
we equate (3.34) with (3.29), which gives
(3.36)
_
F+ =
E
+
Ae
f M' 2~
(v +(
-
F... =( )/
A
-
) e2i
) -
6i
-
E+
i
(A.. + iirk...(O))
(A..
-
ilr q_(O)) e~o(~/Dr
and similarly, above threshold, equating (3.34) with (3.30) gives
106
(3.37
e 2 ffo
iA
+
1
iA+
x [e-f;'m W'\
+ ' (
X + (0 ) -
1 ( iAd+
iA+
V )dr + e2
I~d-f.~Tfr(/~~)r
,,~
r
x [e-f;m(vI\+V+)dr -2f t
VA+dr _ e2fo* Xdr+frm(VA+ - V/+)dr]
(3.38)
where rf is the radius above threshold at which the eigenvalue of the repulsed eigenstate of the exponential potential passes through zero, defined by V e-W'
= e,
V,/27
We can then equate the coefficients from matching the WKB solution onto the
small-r solution with the coefficients from matching the WKB solution with the larger solutions. After imposing appropriate boundary conditions, we can then determine
the coefficients for the various physical solutions to the Schrodinger equation. For
instance, we can calculate the coefficients for the repulsed and attracted eigenstates
for the small-r solution. We can also determine the coefficients for the ingoing and
outgoing spherical waves in the ground or excited states for the large-r solution.
Appendix B: Extracting Scattering Amplitudes from
the Wavefunctions
Matching the Wavefunctions Using the Boundary Conditions
As mentioned in Section 3.3, we require that 0+(0) = 0-(0) = 0, since O(r) is the
radial wavefunction rescaled by r and the wavefunctions should be regular at the
107
origin. For utility, we then define the following useful phases:
i0
(3.39)
rm1M
JO
>
where r, is some radius chosen such that V e-'
C2,,
, as in (3.36). Then, for the
below-threshold case, equating (3.36) with (3.37) at the matching radius, rM gives
0
F+ = i(
EA=
2 )/ 2
[c1rV+c
2
*U+C3 77A+c4?* r*
e- = ( 3 2 [Cre
1 e'
(27r) /
+ C2 r*'e
2 3 /2 C1e
F = A7r= (2ir)
+ C2 r*e 7
-V '7F
- C 3 7 ae~
- C3 i
- C4
el
e*
ei - C4 7,* F*e-
,
E+ _ A+ee=
p
(3.40)
where we have defined eA = V/2 -ef and r,, and
rA
are defined in (3.35). Similarly,
above threshold, equating (3.36) with (3.38) gives
[
E,\=
f7 (e-f-m
d-2=0
25
1 (e-f;m(Vr\TV1+)dr +e2 fo t VA+ dr -2 fo*VAdr+for
F+ iAj
-
[e 2for
[e-f;m(v1'
V-+d r- 2fort Vl/\+dr- forJ
V'q7)dr2 fort
= (2 )3 / 2 [C1rV+C2r*+C
-2
VId
A
(
+C4
VT+-/) dr
fo
v3
fi))
j
=+
+ efor (V--\-Vl7)dr]
T+dr +fom(,A (+N- V/)r]
l'A]
(3.41)
Both above and below threshold, the E+ equation gives us A+ = 0, which is akin to
neglecting the contribution from the repulsed eigenstate at small radii. This indicates
that repulsive scattering occurs most significantly at large r, where the exponential
108
part of the Yukawa dominates the behavior of the wavefunction. This makes intuitive
sense since we are interested in the low-velocity limit, which means that incoming
particles must climb up to the classically disallowed region of a repulsive potential in
order to even reach the small-r region. Since A+ = 0, the F+ equation gives
Cir,, + C 2r*+ C3 1rA + C 4 hI* = 0,
(3.42)
both above and below threshold.
General Scattering Amplitudes
Once we have imposed the relevant boundary conditions on the large-r wavefunctions,
we can extract the scattering amplitudes by reading off the coefficients of the outgoing
solutions. Since the hypergeometric functions asymptote to 1 as r -+ oo, the large-r
ground and excited wavefunctions approach ingoing and outgoing spherical waves:
?01 =
C1--
e-'u' +C2 (e)
+p
4C2
(3.43)
2 = -C 3 77($-
-
/e~iE
C4,*
V0
es.
More generally, consider wavefunctions for nondegenerate states X and Y given
by
V)x = (A + B)eikr - Aeikr
y = Ceik'r
where the A terms represent the unscattered wavefunction in state X, the B term
represents the elastically scattered wavefunction in state X, and the C term represents the inelastically scattered wavefunction in state Y (hence the wavenumber k'
as distinct from k.) In this case, conservation of probability current dictates that
k 1A1 2 = k JA + B1 2 + k' ICj 2 . We can reformulate these wavefunctions (recall that all
wavefunctions used in this chapter are rescaled by r) in the context of a 3-dimensional
109
scattering problem as an ingoing cylindrical wave and a scattered outgoing spherical
wave:
e Air _ e-ikr
0
fx(g)e Ar
e ikz
~=N(r (e0 ) +
fy()eik'r
N
2ik
fx(ggeikr
+
fy()eWr'
(3.45)
where the second equality follows because for the s-wave,
etkz -+ jo(kr)Po (cos 0) = sin kr
(3.46)
kr
In general to get a differential cross section, we relate the incident probability flux
through an area to the scattered outgoing probability flux through a solid angle:
=I12 Vin = IN1 2 hkin
dP
dPout
2 Vot
=
IN2
2
r dQdt
)
2
hk
r2
du
(3.47)
m
kin
Equating the top row of (3.45) with ibx gives
N
B
,
A=
Nfx(0)
2
do elastic
x=
k' |C1 2
4)12-k'||
4k0 JAI
(3.49)
AB21
k
= >O elastic =
(3.48)
-
=>
i
1 1A1 2
2
4k21B|
Similarly, equating the bottom row of (3.45) gives
C = Nfy(O)
doineiastic _ k'
= kk
dfl
irk' IC| 2
=>Oineastic
k
1A1 2
110
If we apply the analogy to our wavefunctions for the case where we begin purely in
the ground state (which corresponds to setting C3 to zero), then the elastic scattering
cross section is
Oelatic =-
C2
E2
+
4p2
4A2
(3.50)
and the inelastic scattering cross section is
Oinelastic
IjC4 *2.
=
(3.51)
In the same vein, for the case where we begin purely in the excited state (which
corresponds to setting C1 to zero), then the elastic scattering cross section is
If-11A/
Oelastic
= 7
C4 q*
(
Kts/ 2
1,q12
(3.52)
I21
(3.53)
and the inelastic scattering cross section is
Oinelastic =
Deriving the Scattering cross sections for our Model
Incoming in the Ground State
We will impose boundary conditions such that the ingoing wave is purely in the
ground state, which implies that C 3 = 0. We are free to set C1 = 1 up to some overall
normalization. Dividing the E- equation by the F_ equation yields
ee2ie
rve
e (3.54)
- C4 7* F*e
+C
2
+C_ =___&e
r~e A +C2I'e-
111
JA
c 4 q* r&e A
and combining this with (3.42) gives
-2
C * r*
, sinh
[1 + e 2 iw'
e 2e (e -A + e ~
e u +
(3.55)
2sinh(Ire) [1 + e 2
C2
A+
)
e& +eT ) -e P(e- +e~
Irwe
IrA,
E
Up
\
W](
ZC
e,
So by (3.50) the elastic scattering cross section is
r cosh
v0 2
Lpv/cosh (*"2~,u sinh
r
oeist
= 1+ 42
V
21A
sinh
21(EA+^,
2A+
-ip
(3.56)
and by (3.51), the inelastic scattering cross section is
27r cos2V
C= cosh 2
sinh
(2MEL)
sinh (MA)
-e) (cosh (
((e
-
A(''+**
cos(29))
Incoming in the Excited State
We will impose boundary conditions such that the ingoing wave is purely in the
excited state, which implies that C1 = 0. We are free to set C3 = 1 up to some overall
normalization. Dividing the E- equation by the F. equation yields
e~e
(3.58)
-C4 A* FX
C 2r*e 4 - 77 PAe~
C2r*e
-,q
-rC,&F47*
rae
and combining this with (3.42) gives
v.V
__we,
C * * [ (ee
+e
) - e2 e (e-
-2 tr sinh (EA)
C2 =A*
+e ? Ae e (e- u+e ~
112
[1 + e 2'w]
+e
(3.59)
So by (3.52), the elastic scattering cross section is
(
'elastic =
C2
1
-
cosh
~
el 42L cosh
A
2
jF
7(e4,+e,
(
2/At,~))
sinh (ea-e,) + ip)
sinh (
2(EAuv) -
ip)
(3.60)
and by (3.53), the inelastic scattering cross section is
27r cos2 Wsinh
27r(e,.+c4) ( (3.61)
()
sinh (MA)
O'inelastic =
.
CA cosh
(j~"
cosh
21
JA("
cos(2 p))
Relation to the Transfer Cross Section
When considering the effects of DM scattering on structure formation, the physically
significant quantity is the transfer cross section, which determines the longitudinal
momentum transfer:
UT=
f
df (1 - cos 0) d
dQ
(3.62)
Since our differential cross sections are angle-independent, we can pull those out of
the integral. Since the cos 6 term is orthogonal to the sin0 term in the do Jacobian,
the remaining integral just gives
Gr
= 47r d,
(3.63)
which is the same as the cross section that we computed.
Appendix C: Comparing our 6 -+ 0 Limit with Pre-
vious Results
An analytic approximate form for the phase shift due to elastic scattering, for both
attractive and repulsive potentials, has previously been presented in the literature
[109]. We show here how to recover the analogous result in our approximation.
In the J -+ 0 limit, the potential matrix can be diagonalized, yielding the exact
113
eigenstate basis *+ = 7 (-1, 1), ?P- = 1 (1, 1) (as in Eq. 3.5). Since the potential
is now diagonal, scatterings from *+ to the 0- (and vice versa) do not occur: the
two eigenstates are decoupled. The ?P+ and ?P- eigenstates experience, respectively,
For the - state, let the scattering solution 0-(r) (for the coefficient of the
/
a repulsive and attractive potential.
eigenvector) have the asymptotic form 0-(r) = ei(kr+ 26 -) + e ikr. The phase shift 6.
characterizes the scattering amplitude, which is given by 11 - e 2
-12
. Likewise, for
the + state, let the phase shift be 6+.
Since the differential equation is linear, any linear combination of these solutions
(Aq_(r)40_ + Bo+(r)?P+) is also a solution. In particular, if we set A = B = 1/Vr
and A = -B = 1N/2, we obtain the two solutions:
e
=
(re~J_e
(e ikr (eCtJ-
e26+
(~c
e
J
) + e-kr)
e2ig +e2i6+)
+ e i+(3.64)
(ei2
kr
=
e-k
These correspond to the cases we studied above, where the particles are initially
purely in the ground or excited states. So we see that by calculating the phase shifts
for these initial conditions (given by the A, B and C coefficients in Eq. ??), we can
recover the values for 6_ and 6+, and vice versa. Our cross sections are given by:
o(ground
o(ground
ee2i- + e 2i,2
7r
-+
-+
ground) =
excited) =
1
2
-
le 2e
T2
2
2
1
(3.6
(3.65
and in this case, since 6 = 0, swapping the identifications of "ground" and "excited" states has no effect. Note that the sum of these cross sections gives aUot =
-7
(11
-
e2iJ 2
+ 11- e2ai+12)
= o_- + o+, as it must - the total scattering rate can-
not depend on the choice of basis.
In the limit where the phase shifts are small (which we will see is the case at low
114
velocities and away from resonances), we can expand:
u(ground
-+
ground) ; T2- 13.. + ,+32
u(ground
-+
excited) =
T
16- -
J+I2
The authors of [109] define a quantity a which corresponds to our e,, and a quantity
c=
1/(KEO),
where , is set to 1.6.
The phase shifts derived for the repulsive and attractive case by [109] in the lowvelocity limit are given by:
J_-
-27 +4(1 +i/c) +/(1 - iv/c)]ac,
-[2-y + 0(1 + Vc) + 4'(1 - Vd)]ac,
J+
[
where -y is the Euler-Mascheroni constant and O(z) is the digamma function. Note
these phase shifts become small in the low-velocity limit due to the scaling with a, as
claimed above.
The asymptotic expansions of the digamma function, as jzj
ln(z) + iir(i cot(7rz) -1)
--
oo, are O(z) e
[arg(z)I/7rj for z not a negative integer. Thus in the large-c
limit (corresponding to eo
<
1, which is necessary for our approximations to hold),
and neglecting terms of O(1/c) and higher, these phase shifts approach:
3. = - [27 + ln(c) + 7r cot(7rv/6)] ac,
5+ = - [27 + ln(c)] ac.
(3.68)
So the cross sections for ground-ground and ground-excited scattering should be set
by:
[47 + 2 ln(c) + ir cot(-7r/C)] 2 c2
-+
ground)
cr(ground
-
excited) = 7r3 cot 2 (7rx/2)c2 .
= 7r
,
o(ground
(Note the prefactor 1/k 2 has canceled out the factors of a in the phase shifts.)
115
(3.69)
We now take the same limits (first 6 -+ 0, and then v -+ 0) in our semi-analytic
approximation. Setting c, = e4 we obtain:
r,
V
r 1+
r'
+
= ir2
r (1 +
,
(3.70)
where the second equality comes from the gamma function identities r (1+z) = z r (z),
r(1 - z) r(z) = 7r/sin(7rz), and r(z)r(z + 1/2) = 21-2z VF(2z). So the elastic
scattering cross section to first order in e6 (for both the ground and excited state,
since they are now degenerate) becomes:
r (1)+
Uelastic
-
i sin Wcosh
V
-2ie/p2
._(3.71)
2 1 +
Off-resonance, in the limit as e, -+ 0, sinh
-
i
-+
-i sin V, and the scattering
amplitude approaches zero. More precisely, a Taylor expansion yields:
Oeiastic
+
[27+ 2In
( )+
7r
(3.72)
cot],
where -y is the Euler-Mascheroni constant. On-resonance, where sin W =0, the elastic
scattering amplitude is simply 1.
Meanwhile, for the inelastic case, setting e, = c, yields
27r cos2 W sinh 2
( 2irEti) c(3.73)
( WE-
.
Oinelastic =
1E.,
(cosh
Ag
- cos(2V))
Off-resonance, as ev -+ 0, this probability approaches
0
inelastic -
(r cot
)
.
(3.74)
On-resonance, where cos(2V) = 1, the inelastic scattering amplitude simply approaches 1. We see that these would agree precisely with the approximate forms
116
of the cross sections derived from the results of [109] if we made the replacements:
p -+ 1/c,
y -+ 7rV2,
ln(Vo/fp) -+ -y.
(3.75)
These replacements are parametrically correct - p ~ co ~ 1/c up to 0(1) factors, likewise V ~1p up to 0(1) corrections. The results are most sensitive to the
identification W
-+
r/,
since this sets the resonance positions: taking c = 1/(4),
and our approximate expression W ~,
K =
'
/27r/o, we see that they agree exactly if
r/2 ; 1.57. The value of K = 1.6 chosen by [109] therefore leads to percent-level
agreement in the resonance positions.
Perfect agreement between the two analyses should not be expected, since they
use different potentials (albeit with similar properties), but our approach agrees both
qualitatively and quantitatively with the results of [109] in the region of parameter
space where they can both be used.
Appendix D: Beyond the Regime of Validity
Higher Partial Waves
Inspection of Fig. 3-2 shows that as expected, our analytic approximation breaks
down when E, > E ~
(note that yI and co are generally equal up to a 0(1) factor).
Conveniently, e, < fo is also precisely the condition for s-wave scattering to dominate
over the higher partial waves. Consequently, while a more careful treatment of the
matching between the WKB and large-r regimes (see Fig. 3-1) might allow extension
of our approximation for the s-wave to the region with c, > co, at that point it would
be necessary to include the higher partial waves as well.
This can be easily seen by comparing the relevant length scales: for the ith partial
wave, the vacuum solution is proportional to the Bessel function je(evr), which peaks
when f ~, evr, i.e. r ~ f/e,. In order for scattering of the fth partial wave to be
significant, this peak must lie within the range of the potential, i.e. r < 1/e, and so
we must have f < ev/eo. If e, /e < 1, then only the s-wave term can penetrate the
potential far enough to experience significant scattering.
117
In the case where J is non-zero, this argument still holds - for particles in the
excited state, the asymptotic wave function is now je(Vj - E2), but since
E., requiring eu
<
E
<
eo is certainly sufficient to ensure that the potential cuts off at
smaller r than the peak of the higher-e wavefunctions. Since whenever particles in the
excited state are present and the scattering rate is significant, their downscatterings
will populate the ground state, we will generally consider e, < E to be both a
necessary and sufficient condition for our approximate solution to be useful.
The Adiabatic Regime
However, there is a regime where , , e but our approximate solution remains valid,
although the s-wave does not generally dominate scattering in this part of parameter
space, and so we caution that our s-wave result should not be used as a proxy for the
total scattering cross section.
By discarding the exponentially suppressed term discussed in Appendix A, we are
essentially neglecting scattering of the repulsed eigenstate at small distances, which
is valid when the range of the potential is relatively short and so the scattering
wavefunction for the repulsed eigenstate is peaked outside its range (note this is the
same reason we can ignore the higher partial waves in this regime).
There is another regime where this approximation is valid, for a different reason.
Suppose the system starts with both dark matter particles in the ground state (i.e. the
state of lowest energy). At short distances, the lowest-energy eigenstate is the one that
experiences an attractive potential (corresponding to the +- two-body state at high
energies). If the transition from long distances to short distances is adiabatic - i.e.
this transition occurs slowly relative to the scale associated with the splitting between
the eigenstates - then particles in the lowest-energy eigenstate at long distances will
find themselves entirely in the attracted eigenstate at short distances, in analogy to
the adiabatic theorem, and ignoring the repulsed eigenstate will be valid because it
will simply never be populated.
118
The splitting between the eigenstates corresponds to an energy scale of
q2
in
our dimensionless coordinates, and hence to a time scale of 1/6; the corresponding
distance scale, for an inward-moving wavepacket, would be ~ e,/e2.
The rotation
of the eigenstates, as described in Sec. 3.2.1, occurs when V(r) = e~r/r becomes
comparable to E2/2.
If the cause of this transition is the exponential cutoff, i.e.
r - 1/co, then the transition occurs over a radius Ar ~ 1/4; if E,
< E
then it occurs
.
when V(r) ~1/r and over a range Ar ~ 1/c
Elastic Scattering and Adiabaticity
104
100
1
-
Analytic Approximation
-
Exact Numerics
0.01
10
0.02
0.04
0.06
0.08
0.10
E,
Figure 3-7: A scan through e, with e = 0.03 and c, = 0.04. This demonstrates the
shift from the transition to small-r being adiabatic vs. nonadiabatic. We can see the
breakdown near 6 - eEO, which happens near e, ~ 0.06.
So the criterion for adiabaticity is e/Ej2
1/f, if o j> E2, or ,/
$
1/
oth-
erwise. In the first case, the transition is adiabatic for e,eo < ES; in the second case
adiabaticity always holds for e, < 1 (however, note that EO
< f2
is a regime where the
approximations we use are known to be less accurate [107]), which by the condition
60
,<
3j
implies foe,
.
We can summarize this by saying adiabaticity holds if and
only if eoe, < q , provided our other assumptions hold (that is, ev, e5, c4 5 1).
This mechanism is also responsible for the enhancement in the annihilation rate
noted in [107] for the case with a mass splitting, compared to the case where the
mass splitting is negligible relative to the kinetic energy and can be ignored; under
adiabatic conditions, the presence of the mass splitting causes particles initially in
119
the ground state to transition into a purely attracted state, rather than a equal linear
combination of attracted and repulsed states.
This argument cannot be applied to scattering from the excited state or into
the excited state, as if the excited state is populated then this implies the repulsed
eigenstate will also be populated and cannot be ignored. But provided we are only
interested in elastic scattering from the ground state (i.e. for the below-threshold case
C,, ,<e), we expect our results to be accurate (for the s-wave) even when e,, > E, in
.
the event that o, ,< E
120
Chapter 4
Conclusions
We have analyzed two contrasting models for particle physics beyond the Standard
Model by considering their phenomenological consequences in the sky. In particular,
motivated by observational astrophysics, we have studied the isocurvature modes
from nonminimally coupled multifield inflation and we have derived the cross section
for dark matter with an excited state that self-interacts via an off-diagonal Yukawa
potential.
These two topics correspond to physical scenarios that happen in two vastly different regimes. At one extreme, we have inflation, which occurred at 10-36 seconds after
the Big Bang at an extremely high energy scale that remains elusive to ground-based
colliders. Inflation happened on a length scale corresponding to a patch of space that
was once far smaller than an atom which then grew to be larger than our observable
universe. At the other extreme, we have considered dwarf galaxies at the present day.
In dwarf galaxies (which are the smallest cosmologically-relevant scales) the kinetic
energy of a 100 GeV particle is 100 eV, which is comparable to the binding energy of
Hydrogen.
Clearly these two systems represent different ends of a spectrum, and we have only
focused on these extreme examples. We have not even begun to describe any of the
additional countless ways that astrophysics can be exploited asa probe of fundamental
physics at intermediate scales. We hope this thesis leaves the reader with a sense of
the awesome prospects for astroparticle physics as a means to understand our cosmic
origins.
121
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