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MASSACHUSETTS INSTMITE A Tale of Two Particles OF TECHNOLOGY by AU6 15 2014 Katelin Schutz LIBRARIES 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. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part in any medium now known or hereafter created.

Signature redacted Author ...... Department of Physics Signature redacted May 9, 2014 Certified by...... A. David Kaiser Germeshausen Professor of the History of Science Senior Lecturer, Department of Physics Signature redacted Thesis Supervisor Certified by...... Tracy Slatyer Assistant Professor of Physics Signature redacted Thesis Supervisor Accepted by...... 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 self- interacting 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 newly- forming 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-set- buddy 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

1 Introduction 11 1.1 Inflationary Cosmology ...... 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 1.2 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

2 Multifield Inflation after Planck: Isocurvature Modes from Non- minimal Couplings 36 2.1 Introduction ...... 3 7 2.2 M odel ...... 3 9 2.2.1 Einstein-Frame Potential...... 4 0 2.2.2 Coupling Constants ...... 4 2 2.2.3 Dynamics and Transfer Functions ...... 4 5 2.3 Trajectories of Interest ...... 5 0 2.3.1 Geometry of the Potential ...... 5 0 2.3.2 Linearized Dynamics ...... 5 3 2.4 Results ...... 59 2.4.1 Local curvature of the potential ...... 5 9 2.4.2 Global structure of the potential...... 62

9 2.4.3 Initial Conditions ...... 64 2.4.4 CMB observables ...... 66 2.5 Conclusions ...... 71

3 Self-Scattering for Dark Matter with an Excited State 79 3.1 Introduction ...... 80 3.2 Dark Matter with Inelastic Scattering ...... 83 3.2.1 A Simple Model ...... 83 3.2.2 Approximate Wavefunctions ...... 84 3.3 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 3.4.1 Parameter Regimes of Phenomenological Interest ...... 99 3.5 Conclusions ...... 101

4 Conclusions 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 con- tinue 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. Chap- ter 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 small- scale 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 early- time behavior of the universe is simply determined by extrapolating the expansion backwards in time.

The theoretical framework for an expanding universe comes naturally from Ein- stein'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 + r2 d02 + 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 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,, (1.3) 2

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 k 3 - - (1.4)

and the trace of this tensor equation gives the Friedmann acceleration equation (1.5)3p).(p+1-= d 4irG 3 a 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 observ-

able 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 Nu-

cleosynthesis 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 in- tersected according to the hot Big Bang model. However, we observe those causally- disconnected 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

3H2 Pc = 8rG (1.6)

If we define Q = p/pc, we find that

Q_-1 3k Q 87rGpa 2 (1.7)

Conservation of stress-energy dictates that

p = -3H(p + p), (1.8) so assuming an equation of state

p = wp (1.9)

16 gives the time-dependence of p:

p(t) oc a(t)-(w+l) (1.10) so

oc a(t)3w+1. (.1

Relativistic matter (such as photons) has w = 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 , (1.12) which is the expansion governing a de Sitter spacetime. This exponential expansion 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, g*6oaflo + V()). (1.15)

Recalling that To = p and Tij = gijp, we find that

= #2 + V(0), p = 52- V(g). (1.16) p 2 2

Suppose that we are in a regime where 02

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,

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) = 1 f d3xf Vd 3k 3 f Vd k'- , ' (1..9, V I I (2r)3J (27)3 (

= (2ir d3k Jdk (k)I e*' (2_r where in the last line we have used the fact that

(2r)3 dar et-Ir = ( i). (1.20)

We therefore define the power spectrum as

P(k) a (k) (1.21) where we only have dependence on k =k because we invoke isotropy. One key property of the power spectrum is its functional dependence on the wavenumber, k. The spectral index of density perturbations, n, is defined as2

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].

2Note 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 k4 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,"

d n 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, At r = -- (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

Vrot M (1.26) r

25 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 1 -- Clemens et al (1985) cc 100- 0 cc Keplerian Prediction

50-

0 1 2 3 4 5 6 7 8 Radius (kpc)

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 New- tonian 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 inter- acted 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 a = 0.025 WMAP 7-year data -

~6000 :

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;

1] Wavelength1000 AISO [h-p1 10 1

1000

100-

0Cosmk, Microwave Background 0 *SDSS galaxies *Cluster abundance C: 10 r 10 n Weak lensing ALyman Alpha Forest

0.001 0.01 0.1 1 10 Wavenumber k [h/Mpc]

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

nx, oc e~mx/T, (1.27) where nr is the number density of dark matter particles and mx is their mass. There- fore, 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

-3d~x =_U)(2 2 a d ni - n) (1.28) 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 2 ( nc \2 4da d = (ov)n, - 2feJ (1.29)

dne a (av) nc, ( n 2 da nK 1e H n2

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 -3

0 4 4 0310 1 01

...::: 4 .67 8 9 10 2* C 2 1 -50

10"10 4C6 8 1 20 3d0

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 pre- dictive 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 Planck experiment 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 two- field 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 quan- tizing 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 Planck results, 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 fNL- Nonminimal 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 field- space 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 < Mp. Hence we will neglect such terms here.

39 2.2.1 Einstein-Frame Potential

We perform a conformal transformation to the Einstein frame by rescaling the spacetime metric tensor, AV(X) = Q 2(x) gA,(x), (2.3) where the conformal factor Q 2(x) is related to the nonminimal coupling function via the relation

Q2 W 2 f (b(x)). (2.4) 1 MPi This transformation yields the action in the Einstein frame,

s = dcxV [L R - 0g9jjg, 418,0 - V(01)1, (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 = [--3i+ - f,3fj], (2.6) 2f f and the potential is also stretched so that it becomes

M4 ___ 4__ AO 4 V(, X) - (f ( , = 2 [4 + I252 + X - (2.7)

The form of the nonminimal coupling function is set by the requirements of renormalization [31, 32],

f(0, X) = [M2 + 42 + XX21, (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 shown here are Ax = 0.75 A, g = A, x = 1.2 0, with (4 >> 1 and A > 0. and asymptotically flat along either direction in field space, I = , ,

Mao)i A, M ( + M) (01+ ()2)] (2.9) 4 Q2 [1\ 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 - essentially, how quickly the fields

41 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 Mpi, (2.11) 2

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 v under renormalization-

group flow, these values correspond to A, ~ 0(10-1) and & ~, 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, j ~ 103, our analysis is unhindered by any potential breakdown of unitarity. 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 cut-off scale should be in the vicinity of M,1, Mpi/V , or Mpj/4 [56]. Even if one adopted the most stringent of these suggested cut-off scales, Mp/4 - 10- M,1 , 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. (The unitarity cut-off scale in multifield models in which the nonminimal couplings 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, - 0.1 requires g/A4 , ~ x/- ~ 1 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 gx2/167r2 [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 _ gUJyK =0, (2.12) where 04/= g" , 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:

2 2 t 2 ds = -(1 + 2A)dt + 2a(t)(c6 B)dx dt + a(t) [(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:

2 H 1 [1gI J + V(p)] p1 2 (2.16)

Using the covariant formalism of [48], we find the equations of motion for the perturbations,

DtQ2 + 3HDtQ'+ [2> MI 1 as (a0 0i QJ = 0, (2.17) where Q, is the gauge-invariant 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 - ~ZLMJ (2.19)

46 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,

4 W, (2.23) where w = 1w11. 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 -wM0102 _ 3 d a3&2 Q, a2 P dt ( (2.25) = 2- (w Q,) - 2 ++ dt o-H Q.,,

47 Q, + 3HQ, + T-+ M, +, = 4M21 W ,T (2.26) where T is the gauge-invariant Bardeen potential [29],

+ a2 H -)-,(2.27) and where M,, and M, are the adiabatic and entropic projections of the mass- squared 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 mass- squared of

2 = M,3 +3W 2 . (2.29)

In the usual fashion [29], we may construct the gauge-invariant curvature perturbation, RC ( +)-q (2.30) (P + A) where p and p are the background-order density and pressure and 6q is the energy- density 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

TRs(thc, t) = ft dt' a(t') H(t') TsS(thc, t)(

Tss(thc, t) = exp dt' 8(t') H(t') , 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 4(2.35)

where e, t,, and q,, are given by

M7oP MV (2.36)

The first two quantities function like the familiar slow-roll parameters from single- field 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 (thc, t] (2.37)

Ps(k) = Piz(khc) T2s(thc, t).

49 Ultimately, we may use Ts and Tss to calculate the isocurvature fraction,

Ps T 2____ AS s +PpS Sp=j.g T2S +'T2+ S (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 Q, can transfer power to the adiabatic perturbations, Q,. 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 < 1.

2.3 Trajectories of Interest

2.3.1 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 3i., 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

q5=rcos9 , y = rsin9. (2.39)

2 (The parameter 9 was labeled y in [53].) Inflation in these models occurs for 00k +

2 the potential xX > M2 [48]. That limit corresponds to taking r -+ oo, for which becomes MA1 2g cos2 sin2 + A cos 9+ Ax sin (2.40) 4 (4 cos2 9 + x sin2 )2

We further note that for our choice of potential in Eq. (2.7), V(#, x) has two discrete symmetries, # -+ -0 and X -+ -X. This means that we may restrict our attention to only one quarter of the # - 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 2 2 s / 0.8. (w/H)x10 3

0.6

0.4

0.2

0.0 -- ~-~

-0.2- 0 20 40 60 80 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 = cos[ A A , (2.41) 2 Ao + Ax where we have defined the convenient combinations

- (2.42) Ax aAX - gx.

In order for 93 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. If both Ax and A , have the same sign, the limiting cases are: for Ax > A4 , then 93 -+ 0, and for Ax < A then 93 -+ 7r/2.

52 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 0 (2.43) (4 Vr.(+oo(02 )= X 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 .OAM 4 A.M A M K01e=o = ~ , Yeele=r/2 = ~ i

2 2AA4 (A4 + AX) M4 (2.44) VYOO 1=03 = (AA ( xA~ '

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 V00

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 postu- lating 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 000 >> M2, which is easily accomplished with sub-Planckian field values given 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

2 H2 H (X) = - => XrM = (245 27r N/2-i 2.5 where we take X- 2 (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 10- during inflation then allows us to estimate Xr ~ 10-' Mp, at the start of inflation. (A

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 K, X, and x. 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 ~ - 3V3-b2 (2.46) 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

N. (2.47) M 3 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]

1 / 3 (2.48) 7ary, ~ -- 1 - --- -N* 4N*/

Expanding the equation of motion for the X field and considering (4,6x > 1 we find the linearized equation of motion

4M2) i+ 3Hj -O 2 X -e 0, (2.49)

55 which has the simple solution

3H 9H2 AOM21 X(t) c~- X0 exp [-2 2 +-q )N(t) , (2.50) where we again used Eq. (2.11) for H, and N(t) ftt Hdt' is the number of e-folds 2 since the start of inflation. If we assume that A4 M2/,1 < 9H /4, which is equivalent to A4/A4 < 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), (2.51) where K is defined in Eq. (2.10). Upon using the definition of A4, in Eq. (2.42), we now recognize K = 4AO/A4 . Our approximation of neglecting J thus corresponds to the limit K < 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/H2, 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

] D V=[- + 2A(1 + 60) 2 04 X V# + 0(1+ 6(.)#4

4 + P(1+6) [3(1 +64)Ax + (1-e)(1+6 %)A , +6(1 - e)(1+6 4, )A - Aoe, (2.52) where we have used A4 and Ax as given in Eq. (2.42) and also introduced

= 1 - X (2.53)

56 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 ( 2e 3 77a~9 -K -3 + + (3) (2.54) 4Nt, 3) 8* showing that to lowest order in 1/N., 7,. - < 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 (2.55) [3r E 1 _ 9] Q~x+ 4 +2 *+ T 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),

N 4~ ~2 3 1 1 Tss ~ -exp [ (N1c - N,) (3 - 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

0 0 10 20 30 40 50 60 N.

0.035 -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 to be fairly insensitive to changes in Xo.

58 2.4 Results

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 K. Indeed, all the numerical simulations show that #6, vs. K increases linearly on a semilog scale for small K. However, in the small-K limit, the tachyonic isocurvature mass is also small, so ,6i. 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 K. As expected,

#8i, has no dependence on Xo for small K. Increasing K 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. < 1 -Xo = 10-4 3 0.1 -.-. .-- xo=10 Xo= 10-2 0.01

00.001

10- -

10-1

10-6 0.00 0.05 0.10 0.15 0.20 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.

3 We may calculate #ij 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 ac-

curate. 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

10-7

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 = 60 and

N, ~ 1, rather than all the way to N., 0. In the limit . -+ 0 and e -+ 0 and using

N, = 1, Eq. (2.56) yields

Tss ~ -- exp [-9/81, (2.57) Nhc

3 upon taking Nhc > N.. For Nhc = 60, we therefore find Tss ~ 5.4 x 10 , and hence #I.O ~ 2.9 x 10 5 . This value may be compared with the exact numerical value, #iso = 2.3 x 10 5 . Despite the severity of our approximations, our analytic expression provides an excellent guide to the behavior of the system in the limit of small K.

As we increase K, the fields roll off the ridge correspondingly earlier in their evolution. 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 .- - =-0.001 ' s ...---- 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 = 4 10- 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 K

62 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 #3 .j. From Eq. (2.52), when A, is small (which implies that K 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 mass- squared 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 v/(-- 1) for e < 0. In this limit we may calculate #8i. exactly 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 N0tt = 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 parameters 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

3 fields followed the same trajectory following Nhc, the resulting values for %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 of X, then we find 3 # A(logXo) = KAN = -- r, A . (2.58) 4 M2i

We tested the approximation in Eq. (2.58) by numerically simulating over 15,000 trajectories in the same potential with different initial conditions. The numerical

64 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 3 Figure 2-11: Numerical simulations of & 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

3 lines. From top right to bottom left, the contours have # 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,,, (2.59)

66 where e and 7, are evaluated at Hubble-crossing, Nc. Given our expressions in Eqs. (2.48) and (2.54), we then find 3 2e ns1-2r- 3 + (1+6). (2.60) 2N. 3 4N*2

For trajectories that produce a nonzero fraction of isocurvature modes, the isocurvature 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 Planck measurements 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, fNL. As shown in [45], in the limit of large curvature, K > 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 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]

3 n. = ns(thc) + 1[-a(thc) - (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 0.0073

[37]. Moreover, because the regime K < 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. ' O(10-4), easily consistent with the constraints from Planck, a = -0.0134 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 = 1 (2.64) 1+'T2

We see that once TRs - 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. 0.00280' 0.00275...... 0.955 0.960 0.965 ns

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 important 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 Planck collaboration [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 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. - 0 in the limit K > 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 perturbations 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 approximations 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: 6M2

Gx = x 6M x (2.65)

M21

We also find

6M2 1 g9ox = Gx_~x'~ x (2.66)

gxx ~ 02 M21 PI Next we expand the field-space Christoffel symbols, F, and find

xx - - (2.67) rx xXX 00-~ 100

- 6x ,, _ xX (260

73 The nonzero components of the field-space Riemann curvature tensor become

=OX '?RXXo 6 (2.68) 1X = -RX e

We also expand dynamical quantities, beginning with the fields' velocity:

V2oM4 6 ~ (2.69) and the turn rate w in the 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 _ -M I x 1.5 - 10-( ExxPx= -+P32 (2.71)

Let us look at the partial second derivative for x = 0 and finite #:

[-A.O ,O 2+ Kxx1x=O = 4 2 [g (MP2] [ 0.015 ('#2 + 10M2 j. MP41q O(M21 +0 .1) 1

We see that the two terms can be comparable. In particular, the second derivative changes sign at

Yxx1x=O = 0 => 4,002, ~ 667M 1 (2.72) 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.

'DxxV = 1 xx _ r-p v. - rhx. (2.73)

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 4M+ ( 2A4 - (1---+... 2_6 A, 4(1+(2.74) + M-) 3A,,+ (1+2e)- 0 (1+e)+...]+. 62(60(02)2 6 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 A4 = 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 02 > M21 and the RZJKL term is multiplied by two factors of the fields' velocity. If in addition we take X = X = 0, then M,., becomes

+44M2 M, s = (i + VXXV. (2.75)

Using the double series expansion of Eq. (2.74) the entropic mass-squared becomes

M.,V+4M2-A4M, = +[AO - --1 6 (1 --+..[ e 1 +V C 2 6. +2 J(2 .7 6 )

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

6 = M41PP O M 12#2 + M+P1404 (2.77)

The calculation of 77,, is now a straightforward exercise giving

e-.0 -1 -4A, +M2 [4A., 2E+0 ( 1 7799 V AO C02 AO 3 CO M4 1 + i (1-e)+O(-)+ () (2.78) (C002)2 g,(CO02)3 3 2e] 9 [ ~- + -2-- +(1 -_ 4N. 3 +16 N,2 ) 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 - 1 + . 3 = -2E - 7,9 + tlo, : K + +

From Eq. (2.34), we see that Tss depends on the integral

Nhe It #3Hdt'= J dN'. (2.79) th. N

Plugging in the expression for P from Eq. (2.79)

Nh #dN' = K(Nhc - N.) - c ln -N*- - 2 (2.80)

where

Ci = 1 - (2.81) 4 2 9 3e C2 = - 3. (2.82) 8 8

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 2 sections, c-/M - 0.1 - 1 cm /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 long- range 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 non- Abelian (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 nearly- degenerate 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 self-

82 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 Dark Matter with Inelastic Scattering

3.2.1 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 E 2j ) CO -- (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 1 A= -2+e 2 26 + r 2 'r %/ -. 71-7 FF3.4) VE ev=1+(4e-'O*")/(r2C ( There is a transition in the behavior of the eigenvalues and eigenvectors when .- c+r ~ d. In the regime where > , the eigenvalues and eigenvectors r 2r 2 can be approximated as

e2 er 1 TF1 A+ + _a (2 (3.5) 2 2 r d 2

When 9 < :, the eigenvalues and eigenvectors can be approximated as 2 2 2 7 e (\('

A+ 2 f2 e-2cr'' - e-2co 0 1 (36 A V ~-,+e+ 2 -~ V 26,+,-2 00 r~,r ', 1 0

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 large- r 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 potential- dominated 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) - 1/r for small r V(r) > e/2, V(r) > E2, e = 0

WKB Approximation for intermediate r V(r) , , e/2, E/26

V(r) ~ e-1' for large r V(r) < E/2

Neglect small-r repulsed eigenstate e, Pp

Substantially Enhanced Cross Section efV, EO, es $ , 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 r we approximate e~rM =mxk2) lag the Yukawa potential as an exponential 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 V(r) - 1/r -WKB region - V(r) ~ VO ew 101 (potential dominated) W(exactly solvable 2-state system)

10- i -2 ......

1r=1 r=r! 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 large- distance 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 e1 , 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 The Scattering Cross Sections

3.3.1 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 2a2m2) to obtain the physical cross sections, since we initially rescaled r by amxc/h) and the final results are:

2te cosh sinh ((" p

aground-+ ground = i + * 2 cosh sinh -2i1A (3.7)

88 cosh sin

Uexcited- d excited 1o T)j (3.8) escoh2x~ -. )cosh WC") -incos2 2 2r cos W sinh sinh ( ) U'ground-+ excited =- 2 (CA.E)( I (E"+-A) A o(2) (3.9)

0 excited-+ ground " 2 2ir c) s (A (3.10) 2 CA cosh ('(,~'v) (cosh ( 14(''E) - cos(2W)) where we have defined eA = /e -! J, and P and Vo are parameters in the exponential potential,

1 1+ ,4 = (3.11) 2 2 E4rM rM

with rM chosen so that eEorM = max ! . The terms . and r, come from matching the WKB wavefunction onto the wavefunction for the exponential potential, and are defined by,

P, ~ ~ ~ e +iC+A 2 + ( ) ((3.12) rA r (i+ r ("A~ + 1 ) r (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 (3.13) with r, chosen such that V0 e-'' >> 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

Analytic Approximation 4 10 --- Exact Numerics

1000

100

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

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 -1 0 -4 -3 -2 -1 0 Iog(E,) log(e,)

logo0) log(q)

7 7

5 5

-3 S 3

-3 1 -3 1

-4 -4 -4 -3 -2 -1 0 -4 -3 -2 -1 0 log(E,) 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 - - V27/e, meaning the resonances occur

at E ; . In the analysis of [109] the resonances occur at co = 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:

-2(3./14 Sr (1+ i- ) isin cosh (yr ) (V Uelastic =7- 1+ 0 .sJA (3.14) 72 r (1 - sinh (r-- iW) tj

Off-resonance, in the limit as e,, a+0, sinh ( - i) -+ -isin p, and the cross section scales as 1/pi2 . 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 - 0.005 0.010 0.050 0.100 0.500 1.000

"Inelastic" Scattering with Ej = 0 106

100

0.01

10- 4 0.005 0.010 0.050 0.100 0.500 1.000

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

2 2 27r cos W sinh ( irE) Uinelastic = (3.16) e (cosh (A) - cos(2tp))

94 Off-resonance, as c, -+ 0, this probability approaches

inelastic - 7r()cot . (3.17)

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

scattering in the ground state. Expanding -ground-ground to first order in E, yields

7r 7rE8 VoE'5 12 - rcot (O - + 2 In - + 2y Oground-+ground- 41n2 - 2b ~ r + (1) 0 r2p a is t u2p 2 (3.18) 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

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 - even small ones- changes the resonance positions, especially for small co.

95 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 (M. Uground-+excited . 2 11ecos s ot () (3-19) ,i (cosh (')-cos(2p))

2 47r 2 cos Wtanh (3.20 2 2p(3-20) ground A (excited-+( E6 CA 11 cosh JA - cos(2w))

96 2 1 227r cos /2 O-excited-+excited : (c2 + 2Csin o + cos Wsech Scosh (e)- cos(20) 2 (3.21) where for convenience, we have defined the subdominant 0(1) term

( 2-1-2,0o -+b - + ixrtanh -T' + ln ( , (3.22) 2M 2 21t 161t 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/(EgAp2) 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 excited-+groundVexc OC off-resonance, a near-resonance (3.23)

oexcited-+ excited OC off-resonance, near-resonance. (3.24)

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:

2 rx -+ r 1+ fr + ), (3.25) A 2p 2) ( ) 2

-r 1 sinh p + i 1 a(ground -+ ground) = - 1+ _) (3.26) V V2 Lsinh - i p

The term involving sinh's approaches 1 when W -+ 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 \

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 low- mass-splitting limits we have studied; it is identical to the expression obtained by first taking E, -+ 0 and then es -+ 0.

3.4 Applications to Dark Matter Haloes

3.4.1 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 a2mx (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 a2/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- 5m. 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 we will also require E < co, i.e. 23/a 2MX m2/a2m2 => Mis > V6h . The requirement that m4 < amx means that the higher a is above its lower bound of / 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 dominates the eigenvalues at small r so Ai 1/r. The s-wave solutions to the rescaled Schrodinger equation can be expressed in terms of Bessel functions:

0 (r) = A- /iJi(2V.) - Or#_(0),Y1(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)+ ef '''dr' + (V +(o) ~A+ +o' ') -~L -/f~2r VA 1/ 1. (r) (A + iir_(O)) efo'V-('')'+ (A_ - iro-_(0)) e-Eof -(2-') 2V/7r NF A- 4 2 (3.29) If the particle velocity is high enough (above threshold) such that there exists a radius r* in this regime where V(r*) = ,S then A+(r*) = 0 and we must perform a WKB approximation about the turning point. Linearizing the potential and matching the wavefunctions on either side using the connection formulae yields:

-1 [(1/ A +1 .A ~ i~\ e-2o* /,\(r)dr\ eTi fo (r)dr' A+ [2# 2 \ 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 radius. The potentials should match at rM so *~9 = Voe-M". We also require

7 that fr e- dr -= f r Vo e~'r dr, which comes from solving the Lippman-Schwinger 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

(11 + 4 V es~ -2i1g -+- 0+=(3.31) 2 2 ErM rM

The wavefunctions for an exponential potential can be solved for exactly in terms of OF3 hypergeometric functions [1111 as follows:

Voe~Pr 2 Pc -[) 1 i3 i(+e EA) 1+ i(e, + EA) 4p2 it 2 2p 2 2y 4y2

VF- [-iCEI 1 i(E + EA ) 1 i(E - 1 (V oe~ r) 21 + C2 42 Ap2 oF3 1 p'2 2p '22 22M4L p

+ (C3Voe 4M2 4p2

x oF 3 i(E -ea) 3 i(e + ,E +e, 2] 3 2 21+2 2M A42

Voe )~ 3 i(e, + EA) 3 i(E - EA) i }(Voe 2]) C4 2 - -- + 4p2 2 2p '2 2M -3 4,uAp (3.32)

103 (VoePr C1 ( 1e) " 02 + '2

i(E, -EA) 3 i(e + ) iE Voe -,, \ 21 xoF3 3 +,p + ,1+- [{3 2 2p p 4p~2 j

3 i(E +EA) 3 i(e, - EA) 1- , )2] + C2 (Vc oF3 ' (1 ie-)2 + 4 2-- 2p '2 2p p) \41P2 ) E2+ 1 ')2) -C3 (V42 ) +1(2

1 2pi(e- C') 1 x oF3 1 + + i(e + ( 4p2V p 2 2p ) )2]

(E2(+1 ~ Voe-" ~- ie)2) C4 4p2 J +t (2 2/pz + 62p) 1 x oF 1 1- + i(e 1 eC) 4pp( 2 1 [1 2 2pi ' 2 2p I (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

0= 1A (Ej-e fo VA/Crdr' + F,e -for V\IE&)dr'). (3-34)

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 2 A (3 35 A 21 2pp Irv r + 1-r (i62 -S& + ( +

1) (r++ F& r 1 + r ( + A 2pL 2 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

V el" = - -01+2 e (l f /4jdr' 2i F+ (2r / e A (C1Fr + C 2P* + C 3 7r, + C 4q* *)

x C1 ~ +C 2F*,e? -C I ae~ -C 4?,*PTe)

F= eJ'$-A'''dr'- e~ (2r) 3 /2

x (Cive + C2F*e~J - C 3 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) M A- E+ _ Ae+ f ' 2~ - 6i

F+ = (v +( ) - ) e2i i

E - (A.. + iirk...(O)) (3.37

F... =( )/ (A.. - ilr q_(O)) e~o(~/Dr and similarly, above threshold, equating (3.34) with (3.30) gives

106 iA e 2 ffo

+ 1 iA+

x [e-f;'m W'\ V )dr + e2 I~d-f.~Tfr(/~~)r + ' ( X + (0 ) - ,,~ 1iA+ ( iAd+ r

x [e-f;m(vI\+V+) dr -2f t VA+dr _ e2 fo* 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 large- r 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)

JO rm1M 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

E+ _ A+ee= 0

F+ = i( 2 )/ 2 [c1rV+c 2 *U+C3 77A+c4?* r*

EA= e- = ( [Cre + C r*'e - C 7 ae~ - C ele* -V '7F (27r) 3/2 1 e' 2 3 4 7 F = A= 2 3 2 C1e + C2 r*e - C3 i ei - C4 7,* F*e- , 7r (2ir) / 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 v3 fi)) j =+

F+ iAj [e 2for V-+d r- 2fort Vl/\+dr- forJ ( VT+-/) dr + e for (V--\-Vl7)dr]

t (+N- V/)r] - [e-f;m(v1' V'q7)dr2 for VId -2 fo T+dr +fom(,A

+C4 l'A] = (2 )3 /2 [C1rV+C2r*+C 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 + C4 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' +C 2 (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 ikz fx(g)e Ar e Air _ e-ikr fx(ggeikr ~=N(r (e0 ) + fy()eik'r N 2ik 0 + 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:

dP =I12 Vin = IN12 hkin

dPout 2 2 Vot = IN2 ) hk (3.47) r 2dQdt r2 m du kin

Equating the top row of (3.45) with ibx gives

N A= , B Nfx(0) do elastic 2 1 1A1 2 (3.48) 2 2 => i x = 4k 1B|

= >O elastic = k AB21-

Similarly, equating the bottom row of (3.45) gives

C = Nfy(O) 2 doineiastic _ k' k' |C1 dfl = k 4k04)12-k'|| JAI (3.49) 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 + (3.50) E2 4p2 4A2 and the inelastic scattering cross section is

Oinelastic = jCI 4 *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/ Kts/ 2

Oelastic = 7 C4 q* ( 1,q12 (3.52) and the inelastic scattering cross section is

Oinelastic = I21 (3.53)

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 C3 = 0. We are free to set C1 = 1 up to some overall normalization. Dividing the E- equation by the F_ equation yields

e2ie rve +C e - C 7* F*e e +C_ =___&e2 (3.54)4 r~e A +C 2I'e- JA c4 q* r&e A

111 and combining this with (3.42) gives

-2 , sinh [1 + e 2 iw' C * r* e u + e 2e (e -A + e ~ (3.55) 2sinh(Ire) [1 + e 2 W](

Irwe IrA, \ Up ZC e, e& +eT ) -e P(e- +e~ A+ ) C2 E

So by (3.50) the elastic scattering cross section is

r v0 2 r cosh 21(EA+^, sinh 2A+ oeist V= 1+ 42 Lpv/cosh (*"2~,u sinh 21A -ip (3.56) and by (3.51), the inelastic scattering cross section is

27r cos2V sinh (2MEL) sinh (MA)

2 C= cosh ((e -e) (cosh ( 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 C2r*e 4 - 77 PAe~ -C4 A* FX (3.58) C2r*e -,q -rC,&F47*rae

and combining this with (3.42) gives

__we, v.V 2 C * * [ (ee + e ) - e e (e- + e (3.59)

-2 tr sinh (EA) [1 + e2'w] C2 =A*+e ? A e e (e- u+ e ~

112 So by (3.52), the elastic scattering cross section is

( - 2 A cosh 7(e4,+e, sinh (ea-e,) + ip) 'elastic = C2 1 jF el 42L~ cosh ( 2/At,~)) sinh ( 2(EAuv) - ip) (3.60) and by (3.53), the inelastic scattering cross section is

27r cos2 Wsinh () sinh (MA) 2 7r(e,.+c4) ( (3.61) O'inelastic = . CA cosh 21 (j~"cosh 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= df (1 - cos 0) d (3.62) f dQ 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, a repulsive and attractive potential. For the - state, let the scattering solution 0-(r) (for the coefficient of the / 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 J e (~ce2ig +e2i6+) e-k = (e ikr (eCtJ- e26+ ) + e-kr) = kr (ei2 + e i+(3.64)

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:

7r ee2i- + e 2i,2 o(ground -+ ground) = 1 - 2 2 (3.6

o(ground -+ excited) = le 2e (3.65 T2 2 1 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 =

2 -7 (11 - e2iJ + 11- e2ai+12) = o_- + o+, as it must - the total scattering rate cannot 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 low- velocity limit are given by:

J_- -[2-y + 0(1 + Vc) + 4'(1 - Vd)]ac,

-27 +4(1 +i/c) +/(1 - iv/c)]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 -- oo, are O(z) e ln(z) + iir(i cot(7rz) -1) [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:

o(ground -+ ground) = 7r [47 + 2 ln(c) + ir cot(-7r/C)] 2 c2 ,

3 2 2 cr(ground - excited) = 7r cot (7rx/2)c . (3.69)

(Note the prefactor 1/k2 has canceled out the factors of a in the phase shifts.)

115 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)+ i sin Wcosh V -2ie/p2 Uelastic - 2 1 + ._(3.71)

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 cot], (3.72)

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

2 2 27r cos W sinh ( WE- ( 2irEti) c(3.73) 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 ~, ' /27r/o, we see that they agree exactly if

K = 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 < E., requiring eu < 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

- Analytic Approximation 1 - 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, 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.

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