Simple Symmetry
In March 2014, scientists with the BICEP2 telescope announced that they had detected swirls in the sky imprinted by pairs of gravitons during cosmic inflation. The swirl pattern was quickly determined to come from galactic dust rather than events from the dawn of time, but in the course of the debacle many physicists, including Arkani-Hamed and Maldacena, started thinking anew about inflation.
Combining their expertise, the two physicists realized that they could treat cosmic inflation like an ultrapowerful particle collider. The energy of the inflaton field would have fueled the copious production of pairs of particles, whose interactions and decay would have yielded higher-point correlations similar to the cascades of particles that fly out of collisions at Europe’s Large Hadron Collider.
Ordinarily, this reframing wouldn’t help; particle interactions can proceed in innumerable ways, and the standard method for predicting the likeliest outcomes—essentially, taking a weighted sum of as many possible chains of events as you can write down—is a slog. But particle physicists had recently found shortcuts using the bootstrap. By leveraging symmetries, logical principles and consistency conditions, they could often determine the final answer without ever working through the complicated particle dynamics. The results hinted that the usual picture of particle physics, in which particles move and interact in space and time, might not be the deepest description of what is happening. A major clue came in 2013, when Arkani-Hamed and his student Jaroslav Trnka discovered that the outcomes of certain particle collisions follow very simply from the volume of a geometric shape called the amplituhedron.
With these discoveries in mind, Arkani-Hamed and Maldacena suspected that they could arrive at a simpler understanding of the dynamics during cosmic inflation. They used the fact that, according to inflationary cosmology, the exponentially expanding universe had almost exactly the geometry of “de Sitter space,” a sphere-like space that has 10 symmetries, or ways it can be transformed and still stay the same. Some of these symmetries are familiar and still hold today, like the fact that you can move or turn in any direction and the laws of physics stay the same. De Sitter space also respects dilatation symmetry: When you zoom in or out, all physical quantities stay the same or at most become rescaled by a constant number. Lastly, de Sitter space is symmetric under “special conformal transformations”: When you invert all spatial coordinates, then shift the coordinates by a translation, then invert them again, nothing changes.
The duo found that these 10 symmetries of an inflating universe tightly constrain the cosmological correlations that inflation can produce. Whereas in the usual approach, you would start with a description of inflatons and other particles that might have existed; specify how they might move, interact and morph into one another; and try to work out the spatial pattern that might have frozen into the universe as a result, Arkani-Hamed and Maldacena translated the 10 symmetries of de Sitter space into a concise differential equation dictating the final answer. In a 2015 paper, they solved the equation in the “squeezed limit” of very narrow triangles and quadrilaterals, but they couldn’t solve it in full.
Daniel Baumann and Hayden Lee, then a professor and graduate student, respectively, at Cambridge University, and Pimentel in Amsterdam soon saw how to extend Arkani-Hamed and Maldacena’s solution to three- and four-point correlation functions for a range of possible primordial fields and associated particles. Arkani-Hamed struck up a collaboration with the young physicists, and the four of them bootstrapped their way further through the math.
They found that a particular four-point correlation function is key, because once they had solved the differential equation dictating this function, they could bootstrap all the others. “They basically showed that symmetries, with just a few extra requirements, are strong enough to tell you the full answer,” said Xingang Chen, a cosmologist at Harvard University whose own calculations about higher-point correlations helped inspire Arkani-Hamed and Maldacena’s 2015 work.
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