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A new study published in Physical examination letters explored the possibility that a highly supercooled first-order phase transition in the early universe could explain the gravitational wave signals observed by Pulsar Timing Arrays (PTAs).

Gravitational waves, first proposed by Albert Einstein in his general theory of relativity, are ripples in the fabric of spacetime caused by turbulent processes such as black hole mergers.

They were first discovered by LIGO in 2016, confirming Einstein’s predictions almost a century later. The most common sources of black holes are merging black holes, spinning neutron stars, and supernovae.

Recently, NANOGrav, or the North American Nanohertz Gravitational-Wave Observatory, discovered the presence of the Stochastic Gravitational Wave Background (SGWB) from pulsar time arrays (PTAs).

SGWBs are different because they are isotropic, meaning they spread out equally in all directions, indicating that their source is distributed evenly throughout the universe.

This discovery prompted scientists in PRL study to investigate the origin of these waves, which could be from first-order phase transitions (FOPTs) in the early universe.

Phys.org spoke with the study’s co-authors, Prof. Yongcheng Wu, Prof. Chih-Ting Lu, Prof. Peter Athron, and Prof. Lei W of Nanjing Normal University to learn more about their work.

“Our study of the early universe is limited to the period after the formation of the CMB [cosmic microwave background]. Although we have some indirect hints about what happened before the CMB, gravitational waves are currently the only method to study the very early universe,” Yongcheng said.

Prof Ley added: “Over the past few years, supercooled FOPT has been widely considered as a possible source of SGWB.”

“A new signal seen by PTA may be evidence of this – a very exciting possibility,” said Prof Atron.

Prof. Chih-Ting said he wanted to understand the relationship between the Higgs field and the Higgs boson and its connection to the electroweak symmetry breaking mechanism. “Relating gravitational wave signals of different frequencies to cosmic phase transitions opened another window for me to explore this,” he said.

First order phase transitions

FOPTs are phase transitions where the system switches between different phases abruptly or discontinuously. One such example that we see in our daily life is the freezing of water.

“Water can remain in a liquid state even if the temperature is below freezing. Then, with some embarrassment [change], suddenly turns to ice. The key signature is that the system remains in phase for a long time below the transition temperature,” Prof. Yongcheng explained.

The electroweak force is a unified description of two of the four fundamental forces of nature: the electromagnetic force and the weak nuclear force.

“We know that in our universe one drastic change – the breaking of the electroweak symmetry that predicts all weak nuclear interactions – generates the masses of all the fundamental particles we have observed today,” said Prof Atron.

This led to the separation of the electroweak force into electromagnetic and weak forces via the Higgs field (which gives all particles their mass). The process by which this occurs is the first-order strong electroweak phase transition.

A supercooled FOPT is one in which the temperature drop during the phase transition is sudden. The researchers wanted to know if such a FOPT could be the source of the SGWB observed by the NANOGrav collaboration.

A potential mechanism for generating SGWB

The idea behind the theory is that the early universe was in a high-temperature state known as a false vacuum state, meaning that its energy was not the lowest possible energy.

As the universe expands and cools, the potential energy decreases. Below a critical temperature, the false vacuum state becomes unstable.

At this temperature, quantum fluctuations (random motions) can initiate the formation of true vacuum states, which are the lowest energy states. This happens through the process of nucleation (formation) of bubbles.

Bubbles represent regions where FOPT from false vacuum to true vacuum has occurred.

Once nucleated, these bubbles of true vacuum grow and expand. They can collide and merge, eventually penetrating space. Percolation refers to the formation of a connected network of regions of true vacuum.

The phase transition is complete when a sufficient fraction of the universe is in a state of true vacuum. This completion usually requires the bubbles to penetrate a significant portion of the universe.

During this process, the collisions and dynamics of the expanding bubbles generate SGWB, which the NANOGrav collaboration has observed.

Modification of the Higgs potential

The researchers’ work began with the construction of a theoretical model to study supercooled FOPTs and the possibility of generating SGWBs.

Prof Ley explained: “In the case of supercooled FOPTs, models can predict the conditions under which such transitions can occur, including the temperature at which the phase transition occurs and the characteristics of the transition process.”

The researchers began by modifying the Higgs potential, which explains how the Higgs field interacts with itself and other fundamental particles.

They added a cubic term to facilitate the dynamics of the supercooled FOPT in the early universe.

Here they define four key parameters to explore the challenges of fitting the nano Hz (nHz) signal (discovered by the NANOGrav collaboration) to this cubic potential:

1. The seepage temperature is the temperature at which bubbles from the true vacuum state nucleate and grow sufficiently to form a connected network throughout the universe.
2. The termination temperature is the temperature at which the phase transition is completely complete, with the entire universe transitioning to a state of true vacuum.
3. Benchmark 1 represents a scenario with a significant degree of supercooling while meeting the percolation and completion criteria.
4. Benchmark 2 represents a scenario where stronger supercooling is achieved with a nominal seepage temperature of around 100 MeV, but does not meet realistic seepage criteria and does not complete the transition.

Both temperature measures are essential for understanding the dynamics and timing of the phase transition. They ensure that the transition is thorough and complete, which is necessary to generate a gravitational wave signal.

The comparison points, on the other hand, bring to light the challenges of supercooled FOPT for SGWB generation.

Limitations of the model

The researchers identified two major challenges that rule out the supercooled FOPT model as an explanation for the nHz signal detected by the NANOGrav collaboration.

The first challenge is the percolation and completion of the supercooled FOPT. When the temperature of the universe falls below a critical value, the phase transition will not occur.

This is because the energy required for the formation and growth of bubbles from the new phase (true vacuum) is low.

“Only a few bubbles form and do not grow fast enough to fill the universe,” explained Prof. Atron.

Therefore, the completion of the phase transition, where the entire universe moves to the new phase, becomes less likely.

The second challenge is overheating. Even if one considers a scenario where completion is somehow achieved, the energy released during the phase transition releases heat into the universe. This process raises the temperature of the universe, a process known as reheating.

“This makes it difficult to maintain the conditions necessary for SGWB production,” added Prof Ley.

Gravitational waves produced in this scenario will not have the same frequency as those observed by PTA, typically in the nHz range.

Conclusion and future work

A supercooled FOPT as an explanation for SGWB can help to avoid limitations on standard model modifications and link the nHz signal to new higher-scale physics, such as those involved in the electroweak phase transition or beyond.

However, as the researchers have shown, challenges suggest that supercooled FOPT may not be the source of the observed SGWB.

The researchers have plans to investigate other FOPTs that could explain the observed signal.

“If the unknown dark sector is capable of generating chiral phase transitions similar to those in quantum chromodynamics, thereby additionally producing nHz gravitational wave signals, it can naturally explain such low-frequency gravitational wave signals,” explained Prof. Chih-Ting.

Prof. Yongcheng added: “The supercooled phase transition can trigger the formation of primordial black holes, which may be part of the dark matter component of our universe. The turbulent process of supercooled FOPT and the much higher energy released during the procedure, can also provide an environment for particle production, which is much more important if we are considering the production of dark matter.”

Prof Ley also mentioned the study of wider cosmological implications such as supermassive black hole binaries.

The researchers also plan to release the software and calculations they developed in this work.

“We plan to release public software with a complete calculation from the particle physics model to the gravitational wave spectra that is fully state-of-the-art and as precise as can be achieved today, so that other teams can easily apply the same level of rigor as we have,” Prof. Atron concluded.