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To study the origin and evolution of the universe, physicists rely on theories that describe the statistical relationships between different events or fields in spacetime, broadly referred to as cosmological correlations. Kinematic parameters are essentially the data that specify a cosmological correlation—the positions of particles, or the wavenumbers of cosmological fluctuations.

Changes in cosmological correlations influenced by variations in kinematic parameters can be described using so-called differential equations. These are a type of mathematical equation that connect a function (i.e., a relationship between an input and an output) to its rate of change. In physics, these equations are used extensively as they are well-suited for capturing the universe's highly dynamic nature.

Researchers at Princeton's Institute for Advanced Study, the Leung Center for Cosmology and Particle Astrophysics in Taipei, Caltech's Walter Burke Institute for Theoretical Â鶹ÒùÔºics, the University of Chicago, and the Scuola Normale Superiore in Pisa recently introduced a new perspective to approach equations describing how cosmological correlations are affected by smooth changes in kinematic parameters.

Their paper, in Â鶹ÒùÔºical Review Letters, introduces a framework, called "kinematic flow," that could help to derive these differential equations from basic combinatorial rules (i.e., principles to determine the number of ways in which a number of objects in a set can be arranged or selected).

"We started working on this paper around September 2020," Guilherme Leite Pimentel, one of the authors of the paper, told Â鶹ÒùÔº. "Our motivation was to understand why this mathematical structure—differential equations in kinematic space—kept appearing in cosmology.

"Was it because of symmetry? Some simpler reason? There was suspicion that we were trading time evolution for spatial separation—the idea is that looking at a snapshot of the early universe at long distances probed the earliest phases of the universe—and we wanted to make that precise."

Over the past decades, equations in kinematic space have proved to be useful for predicting what might have happened in the early days of the universe, more so than approaches that entail delineating a possible timeline and evolving it. As similar approaches were also used by particle physicists, Pimentel and his colleagues tried to apply some of the constructs they relied on to the cosmology problem that was central to their study.

"The story told in the paper came about when the number of equations started to become very large," said Pimentel. "To keep track of what's going on, we decided to draw pictures of the terms in the equations, rather than write formulas. Staring at these pictures eventually made it clear that there was a pattern. The pictures had a 'life of their own' and we could predict the equations by a simple set of rules that involved cutting and coloring the pictures."

In their theoretical analyses, the researchers first considered the kinematic space of a given correlator (i.e., tool measuring the strength of a relationship between two quantities). Subsequently, they applied a set of rules to this correlator.

The idea has a few parts: first, we consider the kinematic space of a given correlator. Then, we applied a set of rules to it: "We sliced it, drew tubes (or, alternatively, triangulated it) and then applied some seemingly esoteric recipe of growing these tubes, merging them, absorbing etc.," explained Pimentel.

"If you buy these rules, then you can predict all terms that appear in a mathematically consistent family of equations. The solutions to these equations—which depended only on spatial separations, or the kinematics of the correlators—are time integrals. In that sense, time emerges. The appearance of this structure hints that there are other ways of organizing these calculations, without invoking explicit time evolution."

This recent work by Pimentel and his colleagues introduces a simpler approach to understanding and describing complex cosmological correlations. Their proposed framework is aligned with other themes rooted in quantum gravity, such as the use of boundary observables and the emergence of spacetime, and could potentially contribute to a timeless description of cosmology.

"We introduced a new array of techniques to compute early universe cosmological correlators, borrowing from other areas of theoretical physics," said Pimentel.

"We expect many of the techniques to be widely applicable, beyond the toy models we described in the letter. The usage of techniques from other fields triggered the interest of mathematicians and particle physicists, who are looking for ways to bring their expertise and contribute. These connections hint that there is probably more knowledge that can be leveraged across these fields."

This team's recent paper could soon inspire further theoretical studies building on the framework they introduced. Meanwhile, Pimentel and his colleagues are seeking further examples in which their described concept of kinematic flow could appear and more sophisticated models could consider in additional research.

"In our next studies, we could also apply these techniques to models which are closer to what we expect the early universe to look like," said Pimentel.

"Another frontier for future research will be to push beyond the leading term—meaning, going from trees to loops—where many new theoretical and mathematical challenges appear. I've also been deeply involved in finding a new description of spinning particles in cosmology, where we are seeing glimpses of new mathematical structures that also previously appeared in particle physics."

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