Into the heart of a dynamical neutron star

(Press-News.org) Neutron stars harbor some of the most extreme environments in the universe: their densities soar to several times those of atomic nuclei, and they possess some of the strongest gravitational fields of any known objects, surpassed only by black holes. First observed in the 1960s, much of the internal composition of neutron stars is still unknown. Scientists are beginning to look to gravitational waves emitted by binary neutron-star inspirals—pairs of mutually orbiting neutron stars—as possible sources of information about their interiors.

Physicists at the University of Illinois Urbana-Champaign, together with colleagues at the University of California, Santa Barbara, Montana State University, and the Tata Institute of Fundamental Research in India have made a major theoretical breakthrough in understanding how inspiraling binary neutron stars respond to tidal forces, a key step in elucidating neutron stars’ makeup. The team has proven that the time-dependent tidal responses of such stars can be described in terms of their oscillatory behavior, or modes, extending an analogous result from Newtonian gravity to the relativistic setting.

This research was published as an Editors’ Suggestion in the journal Physical Review Letters on February 18, 2026, and paves the way to probing the internal structure of neutron stars and some of nature’s most extreme types of matter using gravitational waves.

Neutron stars: a natural laboratory to study extreme matter

As their name suggests, neutron stars are partly made of neutrons, which can form when protons and electrons are squeezed to pressures so high that they essentially “fuse” together. But neutrons aren’t the whole story. Leading theories suggest that heavy elements, free electrons, and free protons are significant components too. Some even suspect that quantum superfluid and superconducting phases arise deeper down. These conjectures, however, are difficult to verify, and much of the interior composition—especially inside the core—is still a giant question mark.

But neutron stars aren’t just interesting in their own right. Scientists believe they can tell us about extreme physics in general. Theorists surmise that neutron stars represent one instance of a more general kind of matter known as a quark-gluon plasma, a highly dense, hot state of matter composed of quarks, the elementary building blocks of protons and neutrons. Such matter exists in only the most extreme environments, such as the early universe in the first few microseconds following the Big Bang. 

The only way to study quark-gluon plasma on Earth is by smashing high-energy particles together in particle colliders, which probe such plasmas at extraordinarily high temperatures. At lower temperatures, though, no lab-based methods exist.

Illinois Physics Professor Nicolás Yunes said, “It’s very hard to study the physics of matter at such high densities and, relatively speaking, low temperatures. But the universe provides a natural lab to study this kind of matter through neutron stars.”

Obviously, because neutron stars can’t be studied on Earth, physicists must infer their properties from astrophysical observations, which have traditionally been limited to electromagnetic observations. With the advent of gravitational-wave astronomy, however, physicists have realized a powerful alternative that may enable them to peer into the very heart of a neutron star.

Whispers in gravitational waves

Sometimes neutron stars form binary systems, where two stars move about a common center of mass. Caught in each other’s orbit, they begin to spiral in toward each other, losing energy to gravitational waves—vibrations in spacetime that propagate outward at the speed of light. As they spiral in, each star tugs on its partner through gravity, producing tidal forces like the Moon does on Earth, before finally merging in a violent collision.

Depiction of a pair of neutron stars during an inspiral. Each star exerts tidal forces on its neighbor, which deforms and excites frequency patterns within, leaving imprints on the gravitational waves emitted. Researchers can analyze these gravitational waves to ‘hear’ what’s going on inside of the stars. Image generated by Abhishek Hegade and Nicolás Yunes using OpenAI ChatGPT Pro.

Former Illinois Physics graduate student and current Princeton University postdoctoral scholar Abhishek Hegade shared, “As they get closer, tidal forces from one star begin to deform the other and vice versa. The amount of deformation depends on what’s inside of the stars.”

These deformations excite oscillatory patterns, called modes, within the stars, just like a hammer excites ringtones when it strikes a bell. These modes leave imprints on the emitted gravitational waves, which can be picked up by sensitive detectors on Earth. By “listening in” to these imprints, scientists may be able to infer what’s going on inside.

Yunes explained, “If we can understand the mode frequencies of oscillation and their decay times, we might be able to determine the composition of neutron stars in a regime not accessible on Earth.”

Getting the tidal response right

To decipher the mode imprints, scientists must first understand how neutron stars respond to tidal forces, a difficult task because the forces—and thus the tidal response—are dynamical, changing rapidly as functions of time, especially during the late stages of inspiral.

For the dynamical tidal responses of non-relativistic Newtonian bodies, the solutions to Newton’s gravitational equations are the modes, which behave like dampened springs, or as physicists put it, damped harmonic oscillators. Moreover, the object’s tidal response can be expressed entirely in terms of these modes—nothing more—forming what’s called a “complete” set.

Yunes stressed that expressing tidal responses in this way is crucial, pointing out, “Without a complete set of modes, it’s entirely possible that you could miss part of the tidal response when you model it, as there could possibly be other pieces you’re omitting from the response’s mathematical description needed to capture all the physics.”

Scientists the world over have hoped that a complete set of modes for binary neutron stars in Einstein’s theory of general relativity exists too. But inspiraling neutron stars are highly relativistic: they’re extremely dense and can approach speeds near 40 percent the speed of light before they merge, strongly distorting spacetime around them. This complex picture and the sheer complexity of the Einstein equations have thwarted physicists’ attempts to determine whether neutron-star modes form a complete set of harmonic oscillators.

First, because there are two stars in a binary system, it's difficult to separate out the effects of one on the other, a situation where the solutions of the stars’ governing equations no longer satisfy the right mathematical constraints, or boundary conditions, required for complete modes to emerge.

“Furthermore,” lead author Hegade added, “a star’s own gravity changes the equations inside and outside of itself. This doesn’t happen in Newtonian gravity, where everything happens in a vacuum. To interpret the star’s tidal response in terms of its modes, you need to know the tidal field both outside and inside of the star too.

“In addition, the loss of energy to gravitational radiation isn’t accounted for by Newtonian theory either. If your system is losing energy, then its modes cannot be complete, so you cannot decompose any perturbation in terms of the modes.”

Finding the modes

To address these hurdles, Yunes’s team broke down the problem into simpler pieces, focusing on one star and viewing its partner as a tidal source. If they could apply the boundary conditions in just the right way, they might be able to find a complete set of modes. Starting with a set of linearized Einstein-Euler equations, which describe how matter generates gravitational fields and evolves in spacetime, they divided the interior and exterior of the star into distinct regions (see diagram): a strong-gravity zone and a weak-gravity zone.

Hegade elaborated, “Physically, it’s a very intuitive way to conceptualize the system. Inside of the star as well as near its surface, gravity is strong. But far away, gravity is weak.

“This process is called a matched-asymptotic expansion, where you zoom in at different scales and then find approximate solutions. Finally, you stitch the solutions together to get something uniform across all scales.”

Decomposing the system in this way and carefully stitching together the strong- and weak-zone solutions enabled the researchers to impose the appropriate boundary conditions piece by piece. Crucially, the incorporation of the weak-gravity zone successfully eliminated radiation in the team’s analysis.

“Our near-zone decomposition ensured that we accounted for the tidal field,” Hegade remarked. “By restricting to the near zone, we took care of radiation by subtracting it out and treating it as a small correction. This allowed us to obtain a complete set of modes.”

The researchers also devised a method for finding the tidal field within the star. By manipulating the Einstein-Euler equations in a suitable way, they discovered they could view the interior tidal field as a driver of oscillations. Specifically, they found that, as long as the tidal field varies without any sudden jumps or sharp corners, the equations spit out harmonic-oscillator modes—just as in Newtonian theory.

From modeling to real data

With the neutron-star’s complete set of harmonic-oscillator modes now in hand, the researchers accomplished exactly what they had set out to do.

Hegade summarized, “We showed two major things. First, we were able to subtract off radiation, finding that a neutron star’s modes do indeed form a complete set. Second, we found that if you consistently solve a certain set of equations using a tidal field that’s sufficiently ‘smooth,’ it’s a solution to the interior of the star, and you can do all the same things in general relativity as in Newtonian gravity.”

The researchers are now eager to see what their new framework might unearth.

Yunes said, “One hope is that we’ll be able to get some information about the neutron-star equation of state at densities found in the inner core of a neutron star. Is there really a quark core, as some have recently claimed? Are there phase transitions occurring inside that we don’t know about yet?”

But answering these questions may have to wait.

Yunes noted, “The signal-to-noise ratios obtained by the LIGO collaboration in their most recent data from 2017 aren’t large enough for us to see the features we’ve captured in our model. In addition, current detectors aren’t that sensitive to sufficiently high frequencies, where most of the information about the neutron-star oscillation modes sit.”

Many are hoping that newer generations of detectors, which are expected to come online in the next few years, together with lucky discoveries of nearby merger events, will ramp up the signal-to-noise ratios and sensitivities required to see more details in the data.

Until then, physicists have lots of time to gear up for the anticipated detectors. Yunes’s team already has some proposed directions: their current framework holds for non-rotating stars only, so they hope to extend it to include rotation as well, as most neutron stars rotate fast. They also plan to repeat their analysis for nonlinear tidal forces and include non-gravitational fields, such as magnetic fields. In terms of their new generalized model, though, they’ve overcome the most challenging obstacle.

Hegade said, “The nice thing about our new framework is that we’ve figured out the hard part—gravity. Now it’s just a matter of applying our models to more realistic configurations.”

 

This research was funded by the Simons Foundation under Grant No. 896696; by the National Science Foundation under Grant Nos. PHY-2207650, 2012086, 2309360, and 2308415; by NASA under Grant No. 80NSSC22K0806; by the Montana NASA EPSCoR Research Infrastructure Development program under Grant No. 80NSSC22M0042; by the Alfred P. Sloan Foundation under Grant No. FG-2023-20470; and by the Binational Science Foundation under Grant No. 2022136. Any opinions, findings, conclusions or recommendations expressed in this material are those of the researchers and do not necessarily reflect the views of the funding agencies.

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