For over 20 years, physicists have had reason to feel envious of certain fictional fish: specifically, the fish inhabiting the fantastic space of M.C. Escher’s Circle Limit III woodcut, which shrink to points as they approach the circular boundary of their ocean world. If only our universe had the same warped shape, theorists lament, they might have a much easier time understanding it.
Escher’s fish lucked out because their world comes with a cheat sheet — its edge. On the boundary of an Escher-esque ocean, anything complicated happening inside the sea casts a kind of shadow, which can be described in relatively simple terms. In particular, theories addressing the quantum nature of gravity can be reformulated on the edge in well-understood ways. The technique gives researchers a back door for studying otherwise impossibly complicated questions. Physicists have spent decades exploring this tantalizing link.
Inconveniently, the real universe looks more like the Escher world turned inside out. This “de Sitter” space has a positive curvature; it expands continuously everywhere. With no obvious boundary on which to study the straightforward shadow theories, theoretical physicists have been unable to transfer their breakthroughs from the Escher world.
“The closer we get to the real world, the fewer tools we have and the less we understand the rules of the game,” said Daniel Baumann, a cosmologist at the University of Amsterdam.
But some Escher advances may finally be starting to bleed through. The universe’s first moments have always been a mysterious era when the quantum nature of gravity would have been on full display. Now multiple groups are converging on a novel way to indirectly evaluate descriptions of that flash of creation. The key is a new notion of a cherished law of reality known as unitarity, the expectation that all probabilities must add up to 100%. By determining what fingerprints a unitary birth of the universe should have left behind, researchers are developing powerful tools to check which theories clear this lowest of bars in our shifty and expanding space-time.
Unitarity in de Sitter space “was not understood at all,” said Massimo Taronna, a theoretical physicist at the National Institute for Nuclear Physics in Italy. “There is a huge jump that has happened in the last couple of years.”
The unfathomable ocean that theorists aim to plumb is a brief but dramatic stretch of space and time that many cosmologists believe set the stage for all we see today. During this hypothetical era, known as inflation, the infant universe would have ballooned at a truly incomprehensible rate, inflated by an unknown entity akin to dark energy.
Cosmologists are dying to know exactly how inflation might have happened and what exotic fields might have driven it, but this era of cosmic history remains hidden. Astronomers can see only the output of inflation — the arrangement of matter hundreds of thousands of years after the Big Bang, as revealed by the cosmos’s earliest light. Their challenge is that countless inflationary theories match the final observable state. Cosmologists are like film buffs struggling to narrow down the possible plots of Thelma and Louise from its final frame: the Thunderbird hanging frozen in midair.
Yet the task may not be impossible. Just as currents in the Escher-like ocean can be deciphered from their shadows on its boundary, perhaps theorists can read the inflationary story from its final cosmic scene. In recent years, Baumann and other physicists have sought to do just that with a strategy called bootstrapping.
Cosmic bootstrappers strive to winnow the crowded field of inflationary theories with little more than logic. The general idea is to disqualify theories that fly in the face of common sense — as translated into stringent mathematical requirements. In this way, they “hoist themselves up by their bootstraps,” using math to evaluate theories that can’t be distinguished using current astronomical observations.
One such commonsense property is unitarity, an elevated name for the obvious fact that the sum of the odds of all possible events must be 1. Put simply, flipping a coin must produce a heads or a tails. Bootstrappers can tell at a glance whether a theory in the Escher-like “anti-de Sitter” space is unitary by looking at its shadow on the boundary, but inflationary theories have long resisted such simple treatment, because the expanding universe has no obvious edge.
Physicists can check a theory for unitarity by laboriously calculating its predictions from moment to moment and verifying that the odds always add up to 1, the equivalent of watching a whole movie with an eye for plot holes. What they really want is a way to glance at the end of an inflationary theory — the film’s final frame — and instantly know whether unitarity has been violated during any previous scene.
But the concept of unitarity is linked closely to the passage of time, and they’ve struggled to understand what shape the fingerprints of unitarity would take in this final frame, which is a static, timeless snapshot. “For many years the confusion was, ‘How the hell can I get information about time evolution … in an object where time doesn’t exist at all?’” said Enrico Pajer, a theoretical cosmologist at the University of Cambridge.
Last year, Pajer helped bring the confusion to an end. He and his colleagues found a way to figure out if a particular theory of inflation is unitary by looking only at the universe it produces.
In the Escher world, checking shadow theories for unitarity can be done on a cocktail napkin. These boundary theories are, in practice, quantum theories of the sort we might use to understand particle collisions. To check one for unitarity, physicists describe two particles pre-crash with a mathematical object called a matrix, and post-crash with another matrix. For a unitarity collision, the product of the two matrices is 1.
Where do physicists get these matrices? They start with the pre-crash matrix. When space holds still, a movie of a particle collision looks the same played forward or backward, so researchers can apply a simple operation to the initial matrix to find the final matrix. Multiply those two together, check the product, and they’re done.
But expanding space ruins everything. Cosmologists can work out the post-inflation matrix. Unlike particle collisions, however, an inflating cosmos looks quite different in reverse, so until recently it was unclear how to determine the pre-inflation matrix.
“For cosmology we would have to exchange the end of inflation with the beginning of inflation,” Pajer said, “which is crazy.”
Last year, Pajer, along with his colleagues Harry Goodhew and Sadra Jazayeri, figured out how to calculate the initial matrix. The Cambridge group rewrote the final matrix to accommodate complex numbers as well as real numbers. They also defined a transformation involving swapping positive energies for negative energies — analogous to what physicists might do in the particle collision context.
But had they found the right transformation?
Pajer then set out to verify that these two matrices really do capture unitarity. Using a more generic theory of inflation, Pajer and Scott Melville, also at Cambridge, played the birth of the universe forward frame by frame, looking for illegal unitarity violations in the traditional way. In the end, they showed that this painstaking process gave the same result as the matrix method.
The new method allows them to skip the moment-by-moment calculation. For a general theory involving particles of any mass and any spin communing via any force, they could see if it is unitary by checking the final outcome. They had discovered how to reveal the plot without watching the movie.
The new matrix test, known as the cosmological optical theorem, soon proved its power. Pajer and Melville found that a lot of possible theories violated unitarity. In fact, the researchers ended up with so few valid possibilities that they wondered if they could make some predictions. Even without a specific theory of inflation in hand, could they tell astronomers what to search for?
Cosmic Triangle Test
One revealing imprint of inflation is the way galaxies are distributed across the sky. The simplest pattern is the two-point correlation function, which, roughly speaking, gives the odds of finding two galaxies separated by particular distances. In other words, it tells you where the universe’s matter is.
Our universe’s matter is spread out in a special way, observations have found, with dense spots stuffed full of galaxies that come in a variety of sizes. The theory of inflation arose in part to explain this peculiar finding.
The universe started out quite smooth overall, the thinking goes, but quantum wiggles imprinted space with tiny dollops of extra matter. As space expanded, these dense spots stretched out even as the tiny ripples continued to arise. When inflation stopped, the young cosmos was left with dense spots ranging from small to large, which would go on to become galaxies and galaxy clusters.
All theories of inflation nail this two-point correlation function. To distinguish between competing theories, researchers need to measure subtler, higher-point correlations — relationships between the angles formed by a trio of galaxies, for instance.
Typically, cosmologists propose a theory of inflation involving certain exotic particles, and then play it forward to calculate the three-point correlation functions it would leave in the sky, giving astronomers a target to search for. In this way, researchers tackle theories one by one. “There are many, many, many possible things you could look for. Infinitely many, in fact,” said Daan Meerburg, a cosmologist at the University of Groningen.
Pajer has turned that process around. Inflation is thought to have left ripples in the fabric of space in the form of gravitational waves. Pajer and his collaborators started with all possible three-point functions describing these gravitational waves and checked them with the matrix test, eliminating any functions that failed unitarity.
In the case of a certain type of gravitational wave, the group found that unitary three-point functions are few and far between. In fact, only three pass the test, the researchers announced in a preprint posted in September. The result “is very remarkable,” said Meerburg, who was not involved. If astronomers ever detect primordial gravitational waves — and efforts are ongoing — these will be the first signs of inflation to look for.
The cosmological optical theorem guarantees that the probabilities of all possible events add up to 1, just as a coin is certain to have two sides. But there is another way of thinking about unitarity: The odds of each event must be positive. No coin can have a negative chance of landing on tails.
Victor Gorbenko, a theoretical physicist at Stanford University, Lorenzo Di Pietro of the University of Trieste in Italy, and Shota Komatsu of CERN in Switzerland recently approached unitarity in de Sitter space from this perspective. What would the sky look like, they wondered, in bizarro universes that broke this law of positivity?
Taking inspiration from the Escher world, they were intrigued by the fact that anti-de Sitter space and de Sitter space share one fundamental feature: Viewed properly, each can look the same at all scales. Zoom in near the boundary of Escher’s Circle Limit III woodcut, and the shrimpy fish have identical proportions to the whoppers in the middle. Similarly, quantum ripples in the inflating universe generated dense spots large and small. This common property, “conformal symmetry,” recently allowed Taronna, who has been working with Charlotte Sleight, a theoretical physicist at Durham University in the U.K., to port a popular mathematical technique for breaking apart boundary theories between the two worlds.
Gorbenko’s group further developed the tool, which let them take the end of inflation in any universe — the hodgepodge of density ripples — and break it into a sum of wavelike patterns. For unitary universes, they found, each wave would have a positive coefficient. Any theories predicting negative waves would be no good. They described their test in a preprint in August. Simultaneously, an independent group led by João Penedones of the Swiss Federal Institute of Technology Lausanne arrived at the same result.
The positivity test is more exact than the cosmological optical theorem, but less ready for real data. Both positivity groups made simplifications, including stripping out gravity and assuming flawless de Sitter structure, that will need to be modified to fit our messy, gravitating universe. But Gorbenko calls these steps “concrete and doable.”
Cause for Hope
Now that bootstrappers are closing in on the notion of what unitarity looks like for the outcome of a de Sitter expansion, they can move on to other classic bootstrapping rules, such as the expectation that causes should come before effects. It’s not currently clear how to see the traces of causality in a timeless snapshot, but the same was once true of unitarity.
“That’s the most exciting thing that we still don’t fully understand,” said Taronna. “We don’t know what is not causal in de Sitter.”
As bootstrappers learn the ropes of de Sitter space, they hope to zero in on a few correlation functions that next-generation telescopes might actually spot — and the few theories of inflation, or even gravity, that could have produced them. If they can pull it off, our swollen universe might someday look as transparent as the world of Escher’s fish.
“After many years of working in de Sitter,” Taronna said, “we are finally starting to understand what the rules of a mathematically consistent theory of quantum gravity are.”
Update: November 12, 2021
This article has been edited to clarify how the positivity test for unitarity was developed. Massimo Taronna and Charlotte Sleight first imported a tool for breaking apart boundary theories from Anti de Sitter space to de Sitter Space. Then the groups of Gorbenko and Penedones extended it to arrive at the positivity test.