# Apply to join IQIM!

Editor’s Note: Dr. Chandni Usha is an IQIM postdoctoral scholar working with Prof. Eisenstein. We asked her to describe her experience as an IQIM fellow.

Just another day at work!

When I look back at how I ended up here, I find myself in a couple of metastable states. Every state pushed me to newer avenues of knowledge. Interestingly, growing up I never really knew what it was like to be a scientist. I had not watched any of those sci-fi movies or related TV series as a kid. No outreach program ever reached me in my years of schooling! My first career choice was to be a lawyer. But a casual comment by a friend that lawyers are ‘liars’ was strong enough to change my mind. Strangely enough, now the quest is for the truth, in a lab down at the sub-basement of one of the world’s best research institutes.

I did my masters in Physics at this beautiful place called the Indian Institute of Science in Bangalore. I realized that I like doing things with my hands. Fixing broken instruments seemed fun. Every new data point on a plot amused me. It was more than obvious that experimental physics is where my heart was and hence I went on to do a Ph.D. in condensed matter physics. When I decided to apply for postdoctoral positions, an old friend of mine, Debaleena Nandi, told me to look up the IQIM website. That was in November 2012, and I applied for the IQIM postdoctoral fellowship. My stars were probably aligned to be here. Coincidentally, Jim Eisenstein, my adviser, was in India on a sabbatical and I happened to hear him give a talk. It left such a strong impression in my mind that I was willing to give up on a trip to Europe for an interview the next day had he offered me a position. We spoke about possible problems, but no offer was in sight and hence I did travel to Europe with my mind already at Caltech. IQIM saved me from my dilemma when they offered me the fellowship a few days later.

Now, why choose IQIM! Reason number one was Jim. And reason number two was this blog which brought in this feeling that there exists a community here; where experimentalists and theorists could share their ideas and grow together in a symbiotic manner. My first project was with an earlier postdoc, Erik Henriksen, who is now a faculty at Washington University in St. Louis. It was based on a proposal by fellow IQIM professor Jason Alicea which involved decorating a film of graphene with a certain heavy metal adatom. Jason’s prediction was that if you choose the right adatom, it could endow some of its unique properties such as strong spin-orbit coupling to the underlying graphene sheet. One can thus engineer graphene to what is called a topological insulator where only the edges of the graphene sheet conduct. Erik had taken on this task and I tagged along. Working in a very small campus with a close-knit community helps bounce your ideas around others and that’s how this experiment came into being. I found it particularly interesting that Jason and his colleagues often ask us, the experimentalists, whether some of the ideas they have are actually feasible to be performed in a lab!

The IQIM fellowship allows you to work on a variety of fields that come under the common theme of quantum information and matter. In addition to providing an independent funding and research grant, the fellowship offers the flexibility to work with any mentor and even multiple mentors, especially in the theory group. In experimental groups however, that flexibility is limited but not impossible. The fellowship gives you a lot of freedom and encourages collaborations. IQIM theory folks have a very strong and friendly group with a lot of collaborations, to the extent that it is often hard to distinguish the faculty from the postdocs and students.

Apart from a yearly retreat to a beautiful resort in Lake Arrowhead, the social life at IQIM is further enhanced through the Friday seminars where you get to hear about the work from postdocs and graduate students from IQIM, as well as other universities. IQIM’s outreach activities have been outstanding. A quick look at this blog will take you from the PhD Comics animations, to teaching kids quantum mechanics through Miinecraft, to hosting middle school students at the InnoWorks academy and a host of other activities. This note will not be complete without mentioning about our repeated efforts to attract women candidates. My husband lives in India, and I live right across the globe, all for the love of science. I am not alone in this respect as we have two more women postdocs at IQIM who have similar stories to tell. So, if you are a woman and wish to pursue a quality research program, this is the place to be, for together we can bring change.

Now that I have convinced you that IQIM is something not to be missed, kindly spread the word. And if you are looking for an awesome opportunity to work at Caltech, get your CV and research statement and apply for the fellowships before Dec 5, 2014!

# Macroscopic quantum teleportation: the story of my chair

In the summer of 2000, a miracle occurred: The National Science Foundation decided to fund a new Institute for Quantum Information at Caltech with a 5 million dollar award from their Information Technology Research program. I was to be the founding director of the IQI.

Jeff Kimble explained to me why we should propose establishing the IQI. He knew I had used my slice of our shared DARPA grant to bring Alexei Kitaev to Caltech as a visiting professor, which had been wonderful. Recalling how much we had both benefited from Kitaev’s visit, Jeff remarked emphatically that “This stuff’s not free.” He had a point. To have more fun we’d need more money. Jeff took the lead in recruiting a large team of Caltech theorists and experimentalists to join the proposal we submitted, but the NSF was primarily interested in supporting the theory of quantum computation rather than the experimental part of the proposal. That was how I wound up in charge, though I continued to rely on Jeff’s advice and support.

This was a new experience for me and I worried a lot about how directing an institute would change my life. But I had one worry above all: space. We envisioned a thriving institute brimming over with talented and enthusiastic young scientists and visitors drawn from the physics, computer science, and engineering communities. But how could we carve out a place on the Caltech campus where they could work and interact?

To my surprise and delight, Jeff and I soon discovered that someone else at Caltech shared our excitement over the potential of IQI — Richard Murray, who was then the Chair of Caltech’s Division of Engineering and Applied Science. Richard arranged for the IQI to occupy office space in Steele Laboratory and some space we could configure as we pleased in Jorgensen Laboratory. The hub of the IQI became the lounge in Jorgensen, which we used for our seminar receptions, group meetings, and innumerable informal discussions, until our move to the beautiful Annenberg Center when it opened in 2009.

I sketched a rough plan for the Jorgensen layout, including furniture for the lounge. The furniture, I was told, was “NIC”. Though I was too embarrassed to ask, I eventually inferred this meant “Not in Contract” — I would need to go furniture shopping, one of my many burgeoning responsibilities as Director.

By this time, Ann Harvey was in place as IQI administrator, a huge relief. But furniture was something I thought I knew about, because I had designed and furnished a common area for the particle theory group a couple of years earlier. As we had done on that previous occasion, my wife Roberta and I went to Krause’s Sofa Factory to order a custom-made couch, love seat, and lounge chair, in a grayish green leather which we thought would blend well with the carpeting.

Directing an institute is not as simple as it sounds, though. Before the furniture was delivered, Krause’s declared bankruptcy! We had paid in full, but I had some anxious moments wondering whether there would be a place to sit down in the IQI lounge. In the end, after some delay, our furniture was delivered in time for the grand opening of the new space in September 2001. A happy ending, but not really the end of the story.

Before the move to Annenberg in 2009, I ordered furniture to fill our (much smaller) studio space, which became the new IQI common area. The Jorgensen furniture was retired, and everything was new! It was nice … But every once in a while I felt a twinge of sadness. I missed my old leather chair, from which I had pontificated at eight years worth of group meetings. That chair and I had been through a lot together, and I couldn’t help but feel that my chair’s career had been cut short before its time.

I don’t recall mentioning these feelings to anyone, but someone must have sensed by regrets. Because one day not long after the move another miracle occurred … my chair was baaack! Sitting in it again felt … good. For five years now I’ve been pontificating from my old chair in our new studio, just like I used to. No one told me how my chair had been returned to me, and I knew better than to ask.

My chair today. Like me, a bit worn but still far from retirement.

Eventually the truth comes out. At my 60th birthday celebration last year, Stephanie Wehner and Darrick Chang admitted to being the perpetrators, and revealed the whole amazing story in their article on “Macroscopic Quantum Teleportation” in a special issue of Nature Relocations. Their breakthrough article was enhanced by Stephanie’s extraordinary artwork, which you really have to see to believe. So if your curiosity is piqued, please follow this link to find out more.

Why, you may wonder, am I reminiscing today about the story of my chair? Well, is an excuse really necessary? But if you must know, it may be because, after two renewals and 14 years of operation, I submitted the IQI Final Report to the NSF this week. Don’t worry — the Report is not really Final, because the IQI has become part of an even grander vision, the IQIM (which has given birth to this blog among other good things). Like my chair, the IQI is not quite what it was, yet it lives on.

The nostalgic feelings aroused by filing the Final Report led me to reread the wonderful volume my colleagues put together for my birthday celebration, which recounts not only the unforgettable exploits of Stephanie and Darrick, but many other stories and testimonials that deeply touched me.

Browsing through that book today, one thing that struck me is the ways we sometimes have impact on others without even being aware of it. For example, Aram Harrow, Debbie Leung, Joe Renes and Stephanie all remember lectures I gave when they were undergraduate students (before I knew them), which might have influenced their later research careers. Knowing this will make it a little harder to say no the next time I’m invited to give a talk. Yaoyun Shi has vivid memories of the time I wore my gorilla mask to the IQI seminar on Halloween, which inspired him to dress up as “a butcher threatening to cut off the ears of my students with a bloody machete if they were not listening,” thus boosting his teaching evaluations. And Alexios Polychronakos, upon hearing that I had left particle theory to pursue quantum computing, felt it “was a bit like watching your father move to Las Vegas and marry a young dancer after you leave for college,” while at the same time he appreciated “that such reinventions are within the spectrum of possibilities for physicists who still have a pulse.”

I’m proud of what the IQI(M) has accomplished, but we’re just getting started. After 14 years, I still have a pulse, and my chair has plenty of wear left. Together we look forward to many more years of pontification.

# John Preskill and the dawn of the entanglement frontier

Editor’s Note: John Preskill’s recent election to the National Academy of Sciences generated a lot of enthusiasm among his colleagues and students. In an earlier post today, famed Stanford theoretical physicist, Leonard Susskind, paid tribute to John’s early contributions to physics ranging from magnetic monopoles to the quantum mechanics of black holes. In this post, Daniel Gottesman, a faculty member at the Perimeter Institute, takes us back to the formative years of the Institute for Quantum Information at Caltech, the precursor to IQIM and a world-renowned incubator for quantum information and quantum computation research. Though John shies away from the spotlight, we, at IQIM, believe that the integrity of his character and his role as a mentor and catalyst for science are worthy of attention and set a good example for current and future generations of theoretical physicists.

Preskill’s legacy may well be the incredible number of preeminent research scientists in quantum physics he has mentored throughout his extraordinary career.

When someone wins a big award, it has become traditional on this blog for John Preskill to write something about them. The system breaks down, though, when John is the one winning the award. Therefore I’ve been brought in as a pinch hitter (or should it be pinch lionizer?).

The award in this case is that John has been elected to the National Academy of Sciences, along with Charlie Kane and a number of other people that don’t work on quantum information. Lenny Susskind has already written about John’s work on other topics; I will focus on quantum information.

On the research side of quantum information, John is probably best known for his work on fault-tolerant quantum computation, particularly topological fault tolerance. John jumped into the field of quantum computation in 1994 in the wake of Shor’s algorithm, and brought me and some of his other grad students with him. It was obvious from the start that error correction was an important theoretical challenge (emphasized, for instance, by Unruh), so that was one of the things we looked at. We couldn’t figure out how to do it, but some other people did. John and I embarked on a long drawn-out project to get good bounds on the threshold error rate. If you can build a quantum computer with an error rate below the threshold value, you can do arbitrarily large quantum computations. If not, then errors will eventually overwhelm you. Early versions of my project with John suggested that the threshold should be about $10^{-4}$, and the number began floating around (somewhat embarrassingly) as the definitive word on the threshold value. Our attempts to bound the higher-order terms in the computation became rather grotesque, and the project proceeded very slowly until a new approach and the recruitment of Panos Aliferis finally let us finish a paper with a rigorous proof of a slightly lower threshold value.

Meanwhile, John had also been working on topological quantum computation. John has already written about his excitement when Kitaev visited Caltech and talked about the toric code. The two of them, plus Eric Dennis and Andrew Landahl, studied the application of this code for fault tolerance. If you look at the citations of this paper over time, it looks rather … exponential. For a while, topological things were too exotic for most quantum computer people, but over time, the virtues of surface codes have become obvious (apparently high threshold, convenient for two-dimensional architectures). It’s become one of the hot topics in recent years and there are no signs of flagging interest in the community.

John has also made some important contributions to security proofs for quantum key distribution, known to the cognoscenti just by its initials. QKD allows two people (almost invariably named Alice and Bob) to establish a secret key by sending qubits over an insecure channel. If the eavesdropper Eve tries to live up to her name, her measurements of the qubits being transmitted will cause errors revealing her presence. If Alice and Bob don’t detect the presence of Eve, they conclude that she is not listening in (or at any rate hasn’t learned much about the secret key) and therefore they can be confident of security when they later use the secret key to encrypt a secret message. With Peter Shor, John gave a security proof of the best-known QKD protocol, known as the “Shor-Preskill” proof. Sometimes we scientists lack originality in naming. It was not the first proof of security, but earlier ones were rather complicated. The Shor-Preskill proof was conceptually much clearer and made a beautiful connection between the properties of quantum error-correcting codes and QKD. The techniques introduced in their paper got adopted into much later work on quantum cryptography.

Collaborating with John is always an interesting experience. Sometimes we’ll discuss some idea or some topic and it will be clear that John does not understand the idea clearly or knows little about the topic. Then, a few days later we discuss the same subject again and John is an expert, or at least he knows a lot more than me. I guess this ability to master
topics quickly is why he was always able to answer Steve Flammia’s random questions after lunch. And then when it comes time to write the paper … John will do it. It’s not just that he will volunteer to write the first draft — he keeps control of the whole paper and generally won’t let you edit the source, although of course he will incorporate your comments. I think this habit started because of incompatibilities between the TeX editor he was using and any other program, but he maintains it (I believe) to make sure that the paper meets his high standards of presentation quality.

This also explains why John has been so successful as an expositor. His
lecture notes for the quantum computation class at Caltech are well-known. Despite being incomplete and not available on Amazon, they are probably almost as widely read as the standard textbook by Nielsen and Chuang.

Before IQIM, there was IQI, and before that was QUIC.

He apparently is also good at writing grants. Under his leadership and Jeff Kimble’s, Caltech has become one of the top places for quantum computation. In my last year of graduate school, John and Jeff, along with Steve Koonin, secured the QUIC grant, and all of a sudden Caltech had money for quantum computation. I got a research assistantship and could write my thesis without having to worry about TAing. Postdocs started to come — first Chris Fuchs, then a long stream of illustrious others. The QUIC grant grew into IQI, and that eventually sprouted an M and drew in even more people. When I was a student, John’s group was located in Lauritsen with the particle theory group. We had maybe three grad student offices (and not all the students were working on quantum information), plus John’s office. As the Caltech quantum effort grew, IQI acquired territory in another building, then another, and then moved into a good chunk of the new Annenberg building. Without John’s efforts, the quantum computing program at Caltech would certainly be much smaller and maybe completely lacking a theory side. It’s also unlikely this blog would exist.

The National Academy has now elected John a member, probably more for his research than his twitter account (@preskill), though I suppose you never know. Anyway, congratulations, John!

-D. Gottesman

# Of magnetic monopoles and fast-scrambling black holes

Editor’s Note: On April 29th, 2014, the National Academy of Sciences announced the new electees to the prestigious organization. This was an especially happy occasion for everyone here at IQIM, since the new members included our very own John Preskill, Richard P. Feynman Professor of Theoretical Physics and regular blogger on this site. A request was sent to Leonard Susskind, a close friend and collaborator of John’s, to take a trip down memory lane and give the rest of us a glimpse of some of John’s early contributions to Physics. John, congratulations from all of us here at IQIM.

John Preskill was elected to the National Academy of Sciences, an event long overdue. Perhaps it took longer than it should have because there is no way to pigeon-hole him; he is a theoretical physicist, and that’s all there is to it.

John has long been one of my heroes in theoretical physics. There is something very special about his work. It has exceptional clarity, it has vision, it has integrity—you can count on it. And sometimes it has another property: it can surprise. The first time I heard his name come up, sometime around 1979, I was not only surprised; I was dismayed. A student whose name I had never heard of, had uncovered a serious clash between two things, both of which I deeply wanted to believe in. One was the Big-Bang theory and the other was the discovery of grand unified particle theories. Unification led to the extraordinary prediction that Dirac’s magnetic monopoles must exist, at least in principle. The Big-Bang theory said they must exist in fact. The extreme conditions at the beginning of the universe were exactly what was needed to create loads of monopoles; so many that they would flood the universe with too much mass. John, the unknown graduate student, did a masterful analysis. It left no doubt that something had to give. Cosmology gave. About a year later, inflationary cosmology was discovered by Guth who was in part motivated by Preskill’s monopole puzzle.

John’s subsequent career as a particle physicist was marked by a number of important insights which often had that surprising quality. The cosmology of the invisible axion was one. Others had to do with very subtle and counterintuitive features of quantum field theory, like the existence of “Alice strings”. In the very distant past, Roger Penrose and I had a peculiar conversation about possible generalizations of the Aharonov-Bohm effect. We speculated on all sorts of things that might happen when something is transported around a string. I think it was Roger who got excited about the possibilities that might result if a topological defect could change gender. Alice strings were not quite that exotic, only electric charge flips, but nevertheless it was very surprising.

John of course had a long standing interest in the quantum mechanics of black holes: I will quote a passage from a visionary 1992 review paper, “Do Black Holes Destroy Information?

“I conclude that the information loss paradox may well presage a revolution in fundamental physics.”

At that time no one knew the answer to the paradox, although a few of us, including John, thought the answer was that information could not be lost. But almost no one saw the future as clearly as John did. Our paths crossed in 1993 in a very exciting discussion about black holes and information. We were both thinking about the same thing, now called black hole complementarity. We were concerned about quantum cloning if information is carried by Hawking radiation. We thought we knew the answer: it takes too long to retrieve the information to then be able to jump into the black hole and discover the clone. This is probably true, but at that time we had no idea how close a call this might be.

It took until 2007 to properly formulate the problem. Patrick Hayden and John Preskill utterly surprised me, and probably everyone else who had been thinking about black holes, with their now-famous paper “Black Holes as Mirrors.” In a sense, this paper started a revolution in applying the powerful methods of quantum information theory to black holes.

We live in the age of entanglement. From quantum computing to condensed matter theory, to quantum gravity, entanglement is the new watchword. Preskill was in the vanguard of this revolution, but he was also the teacher who made the new concepts available to physicists like myself. We can now speak about entanglement, error correction, fault tolerance, tensor networks and more. The Preskill lectures were the indispensable source of knowledge and insight for us.

Congratulations John. And congratulations NAS.

-L. S.

# Tsar Nikita and His Scientists

Once upon a time, a Russian tsar named Nikita had forty daughters:

Every one from top to toe
Was a captivating creature,
Perfect—but for one lost feature.

So wrote Alexander Pushkin, the 19th-century Shakespeare who revolutionized Russian literature. In a rhyme, Pushkin imagined forty princesses born without “that bit” “[b]etween their legs.” A courier scours the countryside for a witch who can help. By summoning the devil in the woods, she conjures what the princesses lack into a casket. The tsar parcels out the casket’s contents, and everyone rejoices.

“[N]onsense,” Pushkin calls the tale in its penultimate line. A “joke.”

The joke has, nearly two centuries later, become reality. Researchers have grown vaginas in a lab and implanted them into teenage girls. Thanks to a genetic defect, the girls suffered from Mayer-Rokitansky-Küster-Hauser (MRKH) syndrome: Their vaginas and uteruses had failed to grow to maturity or at all. A team at Wake Forest and in Mexico City took samples of the girls’ cells, grew more cells, and combined their harvest with vagina-shaped scaffolds. Early in the 2000s, surgeons implanted the artificial organs into the girls. The patients, the researchers reported in the journal The Lancet last week, function normally.

I don’t usually write about reproductive machinery. But the implants’ resonance with “Tsar Nikita” floored me. Scientists have implanted much of Pushkin’s plot into labs. The sexually deficient girls, the craftsperson, the replacement organs—all appear in “Tsar Nikita” as in The Lancet. In poetry as in science fiction, we read the future.

Though threads of Pushkin’s plot survive, society’s view of the specialist has progressed. “Deep [in] the dark woods” lives Pushkin’s witch. Upon summoning the devil, she locks her cure in a casket. Today’s vagina-implanters star in headlines. The Wall Street Journal highlighted the implants in its front section. Unless the patients’ health degrades, the researchers will likely list last week’s paper high on their CVs and websites.

Much as Dr. Atlántida Raya-Rivera, the paper’s lead author, differs from Pushkin’s witch, the visage of Pushkin’s magic wears the nose and eyebrows of science. When tsars or millenials need medical help, they seek knowledge-keepers: specialists, a fringe of society. Before summoning the devil, the witch “[l]ocked her door . . . Three days passed.” I hide away to calculate and study (though days alone might render me more like the protagonist in another Russian story, Chekhov’s “The Bet”). Just as the witch “stocked up coal,” some students stockpile Red Bull before hitting the library. Some habits, like the archetype of the wise woman, refuse to die.

From a Russian rhyme, the bones of “Tsar Nikita” have evolved into cutting-edge science. Pushkin and the implants highlight how attitudes toward knowledge have changed, offering a lens onto science in culture and onto science culture. No wonder readers call Pushkin “timeless.”

But what would he have rhymed with “Mayer-Rokitansky-Küster-Hauser”?

“Tsar Nikita” has many nuances—messages about censorship, for example—that I didn’t discuss. To the intrigued, I recommend The Queen of Spades: And selected works, translated by Anthony Briggs and published by Pushkin Press.

# IQIM Presents …”my father”

Debaleena Nandi at Caltech

Following the IQIM teaser, which was made with the intent of creating a wider perspective of the scientist, to highlight the normalcy behind the perception of brilliance and to celebrate the common human struggles to achieve greatness, we decided to do individual vignettes of some of the characters you saw in the video.

We start with Debaleena Nandi, a grad student in Prof Jim Eisenstein’s lab, whose journey from Jadavpur University in West Bengal, India to the graduate school and research facility at the Indian institute of Science, Bangalore, to Caltech has seen many obstacles. We focus on the essentials of an environment needed to manifest the quest for “the truth” as Debaleena says. We start with her days as a child when her double-shift working father sat by her through the days and nights that she pursued her homework.

She highlights what she feels is the only way to growth; working on what is lacking, to develop that missing tool in your skill set, that asset that others might have by birth but you need to inspire by hard work.

Debaleena’s motto: to realize and face your shortcomings is the only way to achievement.

As we build Debaleena up, we also build up the identity of Caltech through its breathtaking architecture that oscillates from Spanish to Goth to modern. Both Debaleena and Caltech are revealed slowly, bit by bit.

This series is about dissecting high achievers, seeing the day to day steps, the bit by bit that adds up to the more often than not, overwhelming, impressive presence of Caltech’s science. We attempt to break it down in smaller vignettes that help us appreciate the amount of discipline, intent and passion that goes into making cutting edge researchers.

Presenting the emotional alongside the rational is something this series aspires to achieve. It honors and celebrates human limitations surrounding limitless boundaries, discoveries and possibilities.

Stay tuned for more vignettes in the IQIM Presents “My _______” Series.

But for now, here is the video. Watch, like and share!

(C) Parveen Shah Production 2014

# Inflation on the back of an envelope

Last Monday was an exciting day!

After following the BICEP2 announcement via Twitter, I had to board a transcontinental flight, so I had 5 uninterrupted hours to think about what it all meant. Without Internet access or references, and having not thought seriously about inflation for decades, I wanted to reconstruct a few scraps of knowledge needed to interpret the implications of r ~ 0.2.

I did what any physicist would have done … I derived the basic equations without worrying about niceties such as factors of 3 or $2 \pi$. None of what I derived was at all original —  the theory has been known for 30 years — but I’ve decided to turn my in-flight notes into a blog post. Experts may cringe at the crude approximations and overlooked conceptual nuances, not to mention the missing references. But some mathematically literate readers who are curious about the implications of the BICEP2 findings may find these notes helpful. I should emphasize that I am not an expert on this stuff (anymore), and if there are serious errors I hope better informed readers will point them out.

By tradition, careless estimates like these are called “back-of-the-envelope” calculations. There have been times when I have made notes on the back of an envelope, or a napkin or place mat. But in this case I had the presence of mind to bring a notepad with me.

Notes from a plane ride

According to inflation theory, a nearly homogeneous scalar field called the inflaton (denoted by $\phi$)  filled the very early universe. The value of $\phi$ varied with time, as determined by a potential function $V(\phi)$. The inflaton rolled slowly for a while, while the dark energy stored in $V(\phi)$ caused the universe to expand exponentially. This rapid cosmic inflation lasted long enough that previously existing inhomogeneities in our currently visible universe were nearly smoothed out. What inhomogeneities remained arose from quantum fluctuations in the inflaton and the spacetime geometry occurring during the inflationary period.

Gradually, the rolling inflaton picked up speed. When its kinetic energy became comparable to its potential energy, inflation ended, and the universe “reheated” — the energy previously stored in the potential $V(\phi)$ was converted to hot radiation, instigating a “hot big bang”. As the universe continued to expand, the radiation cooled. Eventually, the energy density in the universe came to be dominated by cold matter, and the relic fluctuations of the inflaton became perturbations in the matter density. Regions that were more dense than average grew even more dense due to their gravitational pull, eventually collapsing into the galaxies and clusters of galaxies that fill the universe today. Relic fluctuations in the geometry became gravitational waves, which BICEP2 seems to have detected.

Both the density perturbations and the gravitational waves have been detected via their influence on the inhomogeneities in the cosmic microwave background. The 2.726 K photons left over from the big bang have a nearly uniform temperature as we scan across the sky, but there are small deviations from perfect uniformity that have been precisely measured. We won’t worry about the details of how the size of the perturbations is inferred from the data. Our goal is to achieve a crude understanding of how the density perturbations and gravitational waves are related, which is what the BICEP2 results are telling us about. We also won’t worry about the details of the shape of the potential function $V(\phi)$, though it’s very interesting that we might learn a lot about that from the data.

Exponential expansion

Einstein’s field equations tell us how the rate at which the universe expands during inflation is related to energy density stored in the scalar field potential. If a(t) is the “scale factor” which describes how lengths grow with time, then roughly

$\left(\frac{\dot a}{a}\right)^2 \sim \frac{V}{m_P^2}$.

Here $\dot a$ means the time derivative of the scale factor, and $m_P = 1/\sqrt{8 \pi G} \approx 2.4 \times 10^{18}$ GeV is the Planck scale associated with quantum gravity. (G is Newton’s gravitational constant.) I’ve left our a factor of 3 on purpose, and I used the symbol ~ rather than = to emphasize that we are just trying to get a feel for the order of magnitude of things. I’m using units in which Planck’s constant $\hbar$ and the speed of light c are set to one, so mass, energy, and inverse length (or inverse time) all have the same dimensions. 1 GeV means one billion electron volts, about the mass of a proton.

(To persuade yourself that this is at least roughly the right equation, you should note that a similar equation applies to an expanding spherical ball of radius a(t) with uniform mass density V. But in the case of the ball, the mass density would decrease as the ball expands. The universe is different — it can expand without diluting its mass density, so the rate of expansion $\dot a / a$ does not slow down as the expansion proceeds.)

During inflation, the scalar field $\phi$ and therefore the potential energy $V(\phi)$ were changing slowly; it’s a good approximation to assume $V$ is constant. Then the solution is

$a(t) \sim a(0) e^{Ht},$

where $H$, the Hubble constant during inflation, is

$H \sim \frac{\sqrt{V}}{m_P}.$

To explain the smoothness of the observed universe, we require at least 50 “e-foldings” of inflation before the universe reheated — that is, inflation should have lasted for a time at least $50 H^{-1}$.

Slow rolling

During inflation the inflaton $\phi$ rolls slowly, so slowly that friction dominates inertia — this friction results from the cosmic expansion. The speed of rolling $\dot \phi$ is determined by

$H \dot \phi \sim -V'(\phi).$

Here $V'(\phi)$ is the slope of the potential, so the right-hand side is the force exerted by the potential, which matches the frictional force on the left-hand side. The coefficient of $\dot \phi$ has to be $H$ on dimensional grounds. (Here I have blown another factor of 3, but let’s not worry about that.)

Density perturbations

The trickiest thing we need to understand is how inflation produced the density perturbations which later seeded the formation of galaxies. There are several steps to the argument.

Quantum fluctuations of the inflaton

As the universe inflates, the inflaton field is subject to quantum fluctuations, where the size of the fluctuation depends on its wavelength. Due to inflation, the wavelength increases rapidly, like $e^{Ht}$, and once the wavelength gets large compared to $H^{-1}$, there isn’t enough time for the fluctuation to wiggle — it gets “frozen in.” Much later, long after the reheating of the universe, the oscillation period of the wave becomes comparable to the age of the universe, and then it can wiggle again. (We say that the fluctuations “cross the horizon” at that stage.) Observations of the anisotropy of the microwave background have determined how big the fluctuations are at the time of horizon crossing. What does inflation theory say about that?

Well, first of all, how big are the fluctuations when they leave the horizon during inflation? Then the wavelength is $H^{-1}$ and the universe is expanding at the rate $H$, so $H$ is the only thing the magnitude of the fluctuations could depend on. Since the field $\phi$ has the same dimensions as $H$, we conclude that fluctuations have magnitude

$\delta \phi \sim H.$

From inflaton fluctuations to density perturbations

Reheating occurs abruptly when the inflaton field reaches a particular value. Because of the quantum fluctuations, some horizon volumes have larger than average values of $\phi$ and some have smaller than average values; hence different regions reheat at slightly different times. The energy density in regions that reheat earlier starts to be reduced by expansion (“red shifted”) earlier, so these regions have a smaller than average energy density. Likewise, regions that reheat later start to red shift later, and wind up having larger than average density.

When we compare different regions of comparable size, we can find the typical (root-mean-square) fluctuations $\delta t$ in the reheating time, knowing the fluctuations in $\phi$ and the rolling speed $\dot \phi$:

$\delta t \sim \frac{\delta \phi}{\dot \phi} \sim \frac{H}{\dot\phi}.$

Small fractional fluctuations in the scale factor $a$ right after reheating produce comparable small fractional fluctuations in the energy density $\rho$. The expansion rate right after reheating roughly matches the expansion rate $H$ right before reheating, and so we find that the characteristic size of the density perturbations is

$\delta_S\equiv\left(\frac{\delta \rho}{\rho}\right)_{hor} \sim \frac{\delta a}{a} \sim \frac{\dot a}{a} \delta t\sim \frac{H^2}{\dot \phi}.$

The subscript hor serves to remind us that this is the size of density perturbations as they cross the horizon, before they get a chance to grow due to gravitational instabilities. We have found our first important conclusion: The density perturbations have a size determined by the Hubble constant $H$ and the rolling speed $\dot \phi$ of the inflaton, up to a factor of order one which we have not tried to keep track of. Insofar as the Hubble constant and rolling speed change slowly during inflation, these density perturbations have a strength which is nearly independent of the length scale of the perturbation. From here on we will denote this dimensionless scale of the fluctuations by $\delta_S$, where the subscript $S$ stands for “scalar”.

Perturbations in terms of the potential

Putting together $\dot \phi \sim -V' / H$ and $H^2 \sim V/{m_P}^2$ with our expression for $\delta_S$, we find

$\delta_S^2 \sim \frac{H^4}{\dot\phi^2}\sim \frac{H^6}{V'^2} \sim \frac{1}{{m_P}^6}\frac{V^3}{V'^2}.$

The observed density perturbations are telling us something interesting about the scalar field potential during inflation.

Gravitational waves and the meaning of r

The gravitational field as well as the inflaton field is subject to quantum fluctuations during inflation. We call these tensor fluctuations to distinguish them from the scalar fluctuations in the energy density. The tensor fluctuations have an effect on the microwave anisotropy which can be distinguished in principle from the scalar fluctuations. We’ll just take that for granted here, without worrying about the details of how it’s done.

While a scalar field fluctuation with wavelength $\lambda$ and strength $\delta \phi$ carries energy density $\sim \delta\phi^2 / \lambda^2$, a fluctuation of the dimensionless gravitation field $h$ with wavelength $\lambda$ and strength $\delta h$ carries energy density $\sim m_P^2 \delta h^2 / \lambda^2$. Applying the same dimensional analysis we used to estimate $\delta \phi$ at horizon crossing to the rescaled field $h/m_P$, we estimate the strength $\delta_T$ of the tensor fluctuations as

$\delta_T^2 \sim \frac{H^2}{m_P^2}\sim \frac{V}{m_P^4}.$

From observations of the CMB anisotropy we know that $\delta_S\sim 10^{-5}$, and now BICEP2 claims that the ratio

$r = \frac{\delta_T^2}{\delta_S^2}$

is about $r\sim 0.2$ at an angular scale on the sky of about one degree. The conclusion (being a little more careful about the O(1) factors this time) is

$V^{1/4} \sim 2 \times 10^{16}~GeV \left(\frac{r}{0.2}\right)^{1/4}.$

This is our second important conclusion: The energy density during inflation defines a mass scale, which turns our to be $2 \times 10^{16}~GeV$ for the observed value of $r$. This is a very interesting finding because this mass scale is not so far below the Planck scale, where quantum gravity kicks in, and is in fact pretty close to theoretical estimates of the unification scale in supersymmetric grand unified theories. If this mass scale were a factor of 2 smaller, then $r$ would be smaller by a factor of 16, and hence much harder to detect.

Rolling, rolling, rolling, …

Using $\delta_S^2 \sim H^4/\dot\phi^2$, we can express $r$ as

$r = \frac{\delta_T^2}{\delta_S^2}\sim \frac{\dot\phi^2}{m_P^2 H^2}.$

It is convenient to measure time in units of the number $N = H t$ of e-foldings of inflation, in terms of which we find

$\frac{1}{m_P^2} \left(\frac{d\phi}{dN}\right)^2\sim r;$

Now, we know that for inflation to explain the smoothness of the universe we need $N$ larger than 50, and if we assume that the inflaton rolls at a roughly constant rate during $N$ e-foldings, we conclude that, while rolling, the change in the inflaton field is

$\frac{\Delta \phi}{m_P} \sim N \sqrt{r}.$

This is our third important conclusion — the inflaton field had to roll a long, long, way during inflation — it changed by much more than the Planck scale! Putting in the O(1) factors we have left out reduces the required amount of rolling by about a factor of 3, but we still conclude that the rolling was super-Planckian if $r\sim 0.2$. That’s curious, because when the scalar field strength is super-Planckian, we expect the kind of effective field theory we have been implicitly using to be a poor approximation because quantum gravity corrections are large. One possible way out is that the inflaton might have rolled round and round in a circle instead of in a straight line, so the field strength stayed sub-Planckian even though the distance traveled was super-Planckian.

Spectral tilt

As the inflaton rolls, the potential energy, and hence also the Hubble constant $H$, change during inflation. That means that both the scalar and tensor fluctuations have a strength which is not quite independent of length scale. We can parametrize the scale dependence in terms of how the fluctuations change per e-folding of inflation, which is equivalent to the change per logarithmic length scale and is called the “spectral tilt.”

To keep things simple, let’s suppose that the rate of rolling is constant during inflation, at least over the length scales for which we have data. Using $\delta_S^2 \sim H^4/\dot\phi^2$, and assuming $\dot\phi$ is constant, we estimate the scalar spectral tilt as

$-\frac{1}{\delta_S^2}\frac{d\delta_S^2}{d N} \sim - \frac{4 \dot H}{H^2}.$

Using $\delta_T^2 \sim H^2/m_P^2$, we conclude that the tensor spectral tilt is half as big.

From $H^2 \sim V/m_P^2$, we find

$\dot H \sim \frac{1}{2} \dot \phi \frac{V'}{V} H,$

and using $\dot \phi \sim -V'/H$ we find

$-\frac{1}{\delta_S^2}\frac{d\delta_S^2}{d N} \sim \frac{V'^2}{H^2V}\sim m_P^2\left(\frac{V'}{V}\right)^2\sim \left(\frac{V}{m_P^4}\right)\left(\frac{m_P^6 V'^2}{V^3}\right)\sim \delta_T^2 \delta_S^{-2}\sim r.$

Putting in the numbers more carefully we find a scalar spectral tilt of $r/4$ and a tensor spectral tilt of $r/8$.

This is our last important conclusion: A relatively large value of $r$ means a significant spectral tilt. In fact, even before the BICEP2 results, the CMB anisotropy data already supported a scalar spectral tilt of about .04, which suggested something like $r \sim .16$. The BICEP2 detection of the tensor fluctuations (if correct) has confirmed that suspicion.

Summing up

If you have stuck with me this far, and you haven’t seen this stuff before, I hope you’re impressed. Of course, everything I’ve described can be done much more carefully. I’ve tried to convey, though, that the emerging story seems to hold together pretty well. Compared to last week, we have stronger evidence now that inflation occurred, that the mass scale of inflation is high, and that the scalar and tensor fluctuations produced during inflation have been detected. One prediction is that the tensor fluctuations, like the scalar ones, should have a notable spectral tilt, though a lot more data will be needed to pin that down.

I apologize to the experts again, for the sloppiness of these arguments. I hope that I have at least faithfully conveyed some of the spirit of inflation theory in a way that seems somewhat accessible to the uninitiated. And I’m sorry there are no references, but I wasn’t sure which ones to include (and I was too lazy to track them down).

It should also be clear that much can be done to sharpen the confrontation between theory and experiment. A whole lot of fun lies ahead.

Okay, here’s a good reference, a useful review article by Baumann. (I found out about it on Twitter!)

From Baumann’s lectures I learned a convenient notation. The rolling of the inflaton can be characterized by two “potential slow-roll parameters” defined by

$\epsilon = \frac{m_p^2}{2}\left(\frac{V'}{V}\right)^2,\quad \eta = m_p^2\left(\frac{V''}{V}\right).$

Both parameters are small during slow rolling, but the relationship between them depends on the shape of the potential. My crude approximation ($\epsilon = \eta$) would hold for a quadratic potential.

We can express the spectral tilt (as I defined it) in terms of these parameters, finding $2\epsilon$ for the tensor tilt, and $6 \epsilon - 2\eta$ for the scalar tilt. To derive these formulas it suffices to know that $\delta_S^2$ is proportional to $V^3/V'^2$, and that $\delta_T^2$ is proportional to $H^2$; we also use

$3H\dot \phi = -V', \quad 3H^2 = V/m_P^2,$

keeping factors of 3 that I left out before. (As a homework exercise, check these formulas for the tensor and scalar tilt.)

It is also easy to see that $r$ is proportional to $\epsilon$; it turns out that $r = 16 \epsilon$. To get that factor of 16 we need more detailed information about the relative size of the tensor and scalar fluctuations than I explained in the post; I can’t think of a handwaving way to derive it.

We see, though, that the conclusion that the tensor tilt is $r/8$ does not depend on the details of the potential, while the relation between the scalar tilt and $r$ does depend on the details. Nevertheless, it seems fair to claim (as I did) that, already before we knew the BICEP2 results, the measured nonzero scalar spectral tilt indicated a reasonably large value of $r$.

Once again, we’re lucky. On the one hand, it’s good to have a robust prediction (for the tensor tilt). On the other hand, it’s good to have a handle (the scalar tilt) for distinguishing among different inflationary models.

One last point is worth mentioning. We have set Planck’s constant $\hbar$ equal to one so far, but it is easy to put the powers of $\hbar$ back in using dimensional analysis (we’ll continue to assume the speed of light c is one). Since Newton’s constant $G$ has the dimensions of length/energy, and the potential $V$ has the dimensions of energy/volume, while $\hbar$ has the dimensions of energy times length, we see that

$\delta_T^2 \sim \hbar G^2V.$

Thus the production of gravitational waves during inflation is a quantum effect, which would disappear in the limit $\hbar \to 0$. Likewise, the scalar fluctuation strength $\delta_S^2$ is also $O(\hbar)$, and hence also a quantum effect.

Therefore the detection of primordial gravitational waves by BICEP2, if correct, confirms that gravity is quantized just like the other fundamental forces. That shouldn’t be a surprise, but it’s nice to know.