Usually, when physicists want to model the merger of two black holes, they use supercomputers. Now, however, physicists in the US have found a possible shortcut. In a study published in Physical Review Letters, they show that, of the final states permitted by energy and angular momentum, the remnant of a non-spinning black hole merger ends up in a state quite close to one that maximizes the system’s entropy – a surprising and puzzling result that reinforces the deep connection between thermodynamics and black holes.

“Our group has long believed that black hole thermodynamics could provide a new perspective on binary black hole mergers,” says team member Monica Rincon-Ramirez, a postdoctoral researcher at Pennsylvania State University. “While black hole thermodynamics is well established for equilibrium systems, much less is understood in the highly dynamical regime of mergers.”

Merging black holes

Black hole mergers occur when two black holes that orbit each other spiral inward with increasing speed, losing both energy and angular momentum by emitting gravitational waves until they eventually converge in a violent clash. After the moment of merger, the remnant oscillates vigorously, “ringing down” into a quiescent, rotating object known as a Kerr black hole that can be described in terms of just two quantities: mass and spin.

Meanwhile, the gravitational waves – ripples in the fabric of spacetime – from this energetic collision propagate through space virtually uninterrupted. These gravitational waves encode information about their source, and when we detect them on Earth (millions of light-years away from their origin), we can use them to predict the remnant’s size and spin. However, doing this requires solving complex equations from Einstein’s general theory of relativity using computationally expensive numerical simulations.

Connecting black holes and thermodynamics

Thermodynamics is the branch of physics concerned with the relationships between energy, heat, work and temperature. For ordinary matter, thermodynamics can describe how energy flows from one system to another in simple terms, even for complex systems with numerous particles and degrees of freedom. For example, rather than meticulously tracking each individual particle, concepts such as entropy – a measure of a system’s disorder – can be used to describe the final state of the entire system.

Physicists have long known that black holes obey laws that closely parallel the laws of thermodynamics, with the event horizon and surface gravity of black holes being analogous to entropy and temperature, respectively. Building on this established connection, Rincon-Ramirez and colleagues at the University of Mississippi, the University of California, Berkeley, and Northwestern University as well as Penn State investigated whether black hole thermodynamics could go beyond describing the properties of isolated, stationary black holes and provide, in addition, a principle for determining the final state of a merger.

The maximum-entropy conjecture

After accounting for the energy and angular momentum carried away by gravitational waves, the researchers tracked the evolving mass and angular momentum of a binary, mapped these values to those of hypothetical Kerr black holes, and calculated the corresponding entropy. They found that the Kerr entropy reaches a maximum at a mass and spin strikingly close to the final mass and spin predicted by numerical relativity simulations. Indeed, the two methods agreed with each other to within a few percent. “The Kerr entropy turned out to be remarkably meaningful for the binary system as a whole,” Rincon-Ramirez says.

Interpreting and presenting these results proved to be a challenge, however. “What we found was completely unexpected,” says Rincon-Ramirez. After deliberating among themselves as to why the final black hole’s parameters should lie so close to those of the maximum-entropy black hole, they ultimately chose to extend the conversation to the scientific community. By presenting their puzzling results and a broad conjecture, which they term the “maximum entropy conjecture for black hole mergers”, they hope to sharpen their findings through rigorous testing and eventually place them on a firm mathematical foundation within general relativity.

In the meantime, the researchers plan to extend their conjecture to more general binary black hole systems. By studying systems such as highly spinning and eccentric binaries, they hope to gain further understanding of the role of thermodynamic quantities in general relativity. “Ultimately, we hope this work will contribute to a more complete thermodynamic framework for highly dynamical spacetimes,” Rincon-Ramirez says.

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Physicists have developed a model that predicts how gravitational-wave emission changes when a binary black hole system moves through dark matter. Gravitational waves are ripples in spacetime generated by compact objects spiralling together in a cosmic dance that ends in a violent collision. In the presence of dark matter, these waves may carry an imprint of the surrounding environment. Despite making up over 80% of all matter, dark matter does not interact with the electromagnetic force, and its existence has been surmised only by its gravitational interactions, leaving its precise nature a mystery. Gravitational waves may provide a new way to probe the characteristics of dark matter.

One proposed dark-matter candidate is the light scalar boson, an extremely small particle orders of magnitude lighter than an electron, that can exhibit coherent wave-like behaviour. In the presence of spinning black holes, a process called superradiance can transfer rotational energy into the surrounding field of ultralight boson particles, amplifying the dark matter into a dense cloud around the black hole. These clouds can modify the dynamics of black hole binaries, encoding observable signatures into the gravitational-wave signals that they emit.

To understand what these signatures may look like, an international team of researchers developed a model for gravitational-wave emission from a binary black hole system moving through dark matter. Developing their model required meticulous numerical relativity simulations, but ensuring their results were accurate and reliable proved to be a challenge.

“The hardest part was making sure we truly understood what the simulations were telling us. The danger in numerics is fooling yourself,” explains Josu Aurrekoetxea of Massachusetts Institute of Technology. “You run a simulation, you see an effect, and you have to ask – is that dark matter, or is that my code?”

Working with Katy Clough of Queen Mary University of London and Pedro Ferreira of the University of Oxford, Aurrekoetxea spent several years developing these simulations, before joining forces with Soumen Roy of Université Catholique de Louvain and Rodrigo Vicente of the University of Amsterdam, whose expertise in analytical modelling and data analysis helped the team construct and validate the novel waveform model.

Most work in the field of gravitational waves and dark matter focuses on observations with future space-based detectors such as the Laser Interferometer Space Antenna. “Our motivation was to ask what could already be learned about specific dark-matter models using the LVK [LIGO–Virgo–KAGRA] gravitational-wave data available today,” says Vicente. “We were surprised to find that some of the more extreme models can already begin to be tested with signals observed by current experiments.”

Searching for these dark-matter imprints, they applied their model to 28 publicly available signals observed by the LVK collaboration, a global gravitational-wave detector network that has observed signals from hundreds of mergers. For each event, the team compared the signal to their predictive dark matter-modulated waveform and the standard waveform for black holes merging in a vacuum.

After rigorous testing, the researchers found that 27 of the 28 events were consistent with gravitational-wave signals expected from binary black holes evolving in empty space. One event, GW190728, showed a slight preference for the dark-matter model, indicating that its signal contains properties consistent with modulations from the presence of dark matter. However, the statistical evidence is not strong enough to claim a confident detection of a dark-matter imprint.

Although only one event demonstrated a preference for the predictive dark-matter model and no definitive claim of a dark-matter signature could be made, the researchers have demonstrated a novel method to probe and characterize dark matter through signatures encoded in gravitational-wave signals.

“Our findings from GW190728 suggest that, if this interpretation is correct and we are seeing a genuine environmental signature, similar effects may appear in future LVK events,” says Roy. Therefore, the next step for the researchers is to extend their analysis to LVK’s fourth observing run. Roy also says that if they can estimate the number and types of events that should carry dark-matter imprints, they can combine multiple signals to test their hypothesis more robustly.

“It has become clear that gravitational waves have tremendous potential to help us understand how matter behaves in the extreme gravitational fields surrounding black holes. This is particularly relevant for dark matter, which may interact only through gravity,” says Vicente. And as more gravitational-wave signals are observed, they provide a promising channel to uncover the elusive nature of dark matter.

The research is described in Physical Review Letters.

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