AutoTherm: Automated Thermal Field Theory rates for cosmology

In the earliest moments of the universe, just after the Big Bang, the cosmos was not the cold, empty expanse we see today. Instead, it was a seething, superheated soup of particles, a dense thermal bath where matter and energy collided with such frequency that they were locked in a state of perfect equilibrium. As this primordial fireball expanded and cooled, particles began to drift apart, and some of them fell out of sync with the rest, becoming the dark matter, the neutrinos, or the gravitational waves that we still detect today. To understand how these particles were born and how they shaped the history of the universe, physicists must calculate the rates at which they interact with the hot plasma. It is a question of timing: how fast does a particle interact with its neighbors, and how long does it take to reach a state of balance or to escape it entirely?

For decades, calculating these interaction rates has been a laborious task, often requiring teams of researchers to spend months or years manually working through complex equations. The difficulty arises when particles interact by exchanging massless force carriers, a process that mathematically leads to infinities unless the collective behavior of the hot medium is taken into account. This collective behavior, where the plasma acts like a single, fluid-like entity rather than a collection of individual particles, is essential for getting the right answer. Without it, the calculations break down, leaving cosmologists without a clear picture of how the universe evolved.

A new software tool called AutoTherm, developed by researchers at SUBATECH in Nantes, France, has changed this landscape by automating these difficult calculations. The program is designed to take a description of a particle physics model—essentially a list of the particles involved and how they talk to one another—and automatically compute the rate at which a new, out-of-equilibrium particle would be produced in a thermal bath. The researchers built this tool to handle the specific mathematical headaches that arise when particles exchange massless messengers, a scenario that is common in theories about dark matter, neutrinos, and the early universe. By automating the process, AutoTherm allows scientists to test a wide variety of theories quickly and reliably, something that was previously too slow and prone to human error to do systematically.

The core innovation of AutoTherm lies in its ability to handle a specific type of mathematical divergence that occurs when particles interact at very small angles. In a hot plasma, when a particle exchanges a massless force carrier with another, the calculation can blow up to infinity if one ignores the fact that the plasma itself modifies the properties of that force carrier. The plasma gives the massless particle an effective mass, a phenomenon known as a thermal mass, which tames the infinity and makes the calculation finite. While physicists have known how to do this manually for specific cases, doing it for a new, complex model usually meant starting from scratch and deriving the thermal masses and interaction rates by hand. AutoTherm does this automatically. It reads the model file, identifies the particles that need these thermal corrections, calculates their effective masses within the hot environment, and then performs the necessary resummation—a sophisticated mathematical reorganization of the equations—to produce a finite, physical result.

The researchers tested their tool against a wide range of existing results in the literature, covering particles like right-handed neutrinos, axions, gravitons, and dark photons. In every case, the software successfully reproduced the known results, confirming that the automation was working correctly. More importantly, the tool found and corrected a small but significant sign error in a previous calculation regarding the production of a specific type of dark matter candidate. This kind of correction is exactly why automation is valuable; it removes the fatigue and oversight that can creep into long, manual derivations. The software also provided a way to estimate the uncertainty in these calculations. By offering three slightly different, but mathematically equivalent, ways to handle the thermal corrections, the program allows researchers to see how much the final answer might vary due to unknown higher-order effects. This spread gives a realistic measure of the theory's uncertainty, which is crucial for making precise predictions about the early universe.

The impact of this work extends beyond just getting the numbers right faster. The tool is built to be modular, meaning that different parts of the calculation can be used independently. A researcher could use the software just to calculate the thermal mass of a particle in a specific model, or just to perform the final numerical integration, without needing to run the entire pipeline. This flexibility makes it a versatile instrument for the community. The developers have already applied it to complex scenarios involving supersymmetry, a theoretical framework that proposes a partner particle for every known particle. In these high-energy environments, the number of interactions is vast, and the manual calculation of production rates for gravitinos—the supersymmetric partners of gravitons—was a notoriously difficult and error-prone endeavor. AutoTherm handled this complexity with ease, reproducing known results and clarifying discrepancies in previous studies that stemmed from inconsistent mathematical treatments.

The software is now available to the scientific community, packaged with examples and documentation that allow users to plug in their own models and get results immediately. It works by taking a standard file format used by particle physicists to define their theories and converting it into a detailed map of all possible interactions. It then uses advanced algorithms to sift through these interactions, identifying which ones are sensitive to the collective effects of the plasma and applying the necessary corrections. For the first time, the process of determining how a new particle would have been produced in the hot early universe is no longer a bespoke, manual art but a streamlined, automated procedure. This shift allows cosmologists to focus less on the tedious algebra of calculation and more on the physics of what these particles tell us about the origins of our universe.

The success of AutoTherm suggests that the future of thermal field theory calculations will be increasingly automated. The researchers plan to expand the tool in the future to handle even more complex scenarios, such as processes where particles are produced in pairs or where the interactions are not just at the simplest level but include higher-order corrections. For now, the tool stands as a robust solution for the most common and challenging cases, providing a reliable way to explore the thermal history of the cosmos. By turning a months-long manual struggle into a matter of minutes, it opens the door to testing a much wider array of theories about dark matter and the early universe, ensuring that our understanding of the cosmos is built on calculations that are both precise and reproducible.