“Long Ringdown” of Neutron Star Mergers Provides Insights into the Equation of State

Scientists at Goethe University Frankfurt have made a groundbreaking discovery that could revolutionize our understanding of neutron stars. By analyzing the “long ringdown” phase, a distinct pure-tone gravitational wave signal emitted after the collision of neutron stars, they have identified a powerful new method to explore the inner structure of these mysterious objects. Their findings, recently published in Nature Communications, promise to shed light on the composition and behavior of matter at the extreme densities and pressures found within neutron stars.

The Enigma of Neutron Stars

Neutron stars are among the most fascinating and extreme objects in the universe. With masses greater than that of the entire solar system, they are packed into an object just 12 kilometers across—about the size of a small city. These dense remnants of supernova explosions exhibit extraordinary gravitational and electromagnetic properties, making them a focal point of study in astrophysics. Despite their immense density, neutron stars are incredibly hard to study, as their interiors are composed of matter in conditions far beyond what we can recreate in laboratories on Earth.

The interior composition and structure of neutron stars have long remained a mystery because of the extreme conditions present. Their interiors are thought to consist of a strange form of matter, where nuclear forces become so intense that protons and neutrons are packed together in an unusual and highly compressed state. However, due to the lack of direct observational data, scientists have struggled to determine the exact equation of state (EoS)—a mathematical description of how matter behaves under these extreme conditions.

Neutron Star Collisions: A Unique Window into the Unknown

In recent years, the collision of two neutron stars has provided an unprecedented opportunity to study these objects in detail. One such event was the groundbreaking detection of the GW170817 merger in 2017, the first observed neutron star merger. When two neutron stars spiral toward each other and eventually collide, they emit powerful gravitational waves—ripples in spacetime that can be detected by observatories like LIGO (Laser Interferometer Gravitational-Wave Observatory) and Virgo. The gravitational waves emitted by the spiraling system contain invaluable information about the stars’ masses, spins, and the properties of matter inside their cores.

However, the most intense gravitational wave signal is not emitted during the inspiral phase but rather in the milliseconds after the merger. During this brief moment, the collision generates a rapidly rotating post-merger remnant—a massive object that can either collapse into a black hole or settle into a stable neutron star. This post-merger object emits gravitational waves that hold clues about the equation of state of neutron-star matter, offering an opportunity to understand the composition of the densest regions of a neutron star’s core.

The “Long Ringdown” Phase: A New Discovery

Prof. Luciano Rezzolla and his team at Goethe University Frankfurt have made a significant breakthrough in understanding these post-merger gravitational waves. They have discovered a previously unrecognized phase in the post-merger signal, which they have termed the “long ringdown.” This phase occurs after the merger and is characterized by a pure-tone gravitational wave signal, much like the ringing of a tuning fork after being struck.

In the aftermath of the merger, the amplitude of the gravitational wave signal gradually diminishes over time, but it does so in a highly regular and predictable pattern. As the post-merger remnant settles into a more stable configuration, the frequency of the emitted gravitational waves becomes increasingly pure, eventually stabilizing at a single frequency. This phase, lasting for a fraction of a second, carries crucial information about the post-merger object, and by analyzing its properties, scientists can gain insights into the neutron star’s interior.

Just as different materials produce different pure tones when struck, remnants formed from neutron stars with different equations of state will “ring down” at distinct frequencies. By analyzing these frequencies, scientists can potentially determine the equation of state of the matter inside the neutron star and understand how it behaves under extreme pressure and density. This discovery opens the door to answering some of the fundamental questions about the nature of matter in the most extreme environments.

A New Tool for Understanding Extreme Matter

Prof. Rezzolla emphasizes the significance of this discovery, noting that it provides a novel way to probe the extreme conditions inside neutron stars. “Just like tuning forks of different material will have different pure tones, remnants described by different equations of state will ring down at different frequencies. The detection of this signal thus has the potential to reveal what neutron stars are made of,” he says.

Using general-relativistic simulations of merging neutron stars, the researchers were able to create models of the post-merger remnants and simulate their gravitational wave signals. These models, constructed with a carefully chosen set of equations of state, allowed the team to explore how the long ringdown signal varies for different neutron-star compositions. The findings revealed that this phase of the signal is directly related to the properties of matter in the densest regions of the neutron star’s core, providing new ways to measure the equation of state at these extreme densities.

Reducing Uncertainties in the Equation of State

As Dr. Christian Ecker, the first author of the study, explains, “Thanks to advances in statistical modeling and high-precision simulations on Germany’s most powerful supercomputers, we have discovered a new phase of the long ringdown in neutron star mergers. It has the potential to provide new and stringent constraints on the state of matter in neutron stars. This finding paves the way for a better understanding of dense neutron star matter, especially as new events are observed in the future.”

Dr. Tyler Gorda, a co-author of the study, further explains the computational innovation behind their work. “By cleverly selecting a few equations of state, we were able to effectively simulate the results of a full statistical ensemble of matter models with considerably less effort. Not only does this result in less computer time and energy consumption, but it also gives us confidence that our results are robust and will be applicable to whatever equation of state actually occurs in nature.”

This approach is crucial because the equation of state of neutron star matter remains one of the most uncertain aspects of nuclear physics. Current methods of observing neutron stars—such as studying their rotation rates or observing their gravitational interactions—can only provide limited information. The long ringdown, however, offers a more direct and specific way to probe the matter at the core of these stars, potentially narrowing the range of possible equations of state.

Looking Toward the Future: A New Era of Gravitational-Wave Astronomy

While current gravitational wave detectors have not yet observed the long ringdown signal, scientists are optimistic that future detectors will make this possible. The Einstein Telescope, a next-generation gravitational wave observatory set to become operational in Europe within the next decade, is expected to have the sensitivity required to detect these signals with high precision. When this detection occurs, the long ringdown will become an invaluable tool in neutron star research.

The long ringdown phase offers an unprecedented opportunity to study the most extreme states of matter in the universe. By analyzing this phase, scientists can gain new insights into the equation of state of neutron stars and, by extension, the behavior of nuclear matter under high densities and pressures. This will deepen our understanding not only of neutron stars but also of fundamental physics and the nature of matter at its most extreme.

Conclusion

The discovery of the long ringdown phase is a game-changer in the field of astrophysics and gravitational wave astronomy. The ability to analyze the pure-tone gravitational waves emitted during the post-merger phase of neutron star collisions opens up a new window into the interiors of these enigmatic objects. As we move toward the next generation of gravitational-wave detectors, the long ringdown promises to provide crucial new data that will help scientists unravel the mysteries of neutron stars and the behavior of matter in the most extreme environments.

With this discovery, the study of neutron stars enters a new phase, one in which the equation of state and the fundamental properties of nuclear matter can be explored with unprecedented precision. As future gravitational wave events are observed, the long ringdown will undoubtedly become a key tool in answering some of the most pressing questions about the universe.

Reference: Christian Ecker et al, Constraining the equation of state in neutron-star cores via the long-ringdown signal, Nature Communications (2025). DOI: 10.1038/s41467-025-56500-x