Nonlinear growth of gravitational waves from cosmological phase transitions under the influence of the Universe's expansion ✛
Context
Why it matters: Gravitational waves from first-order cosmological phase transitions are one of the most promising windows into physics beyond the Standard Model. Such transitions might have occurred in the early Universe, when the cooling cosmic plasma spontaneously rearranged, giving rise to bubbles of a new phase that collided and produced shock sound waves. Until now, all serious calculations neglected the expansion of space, considering it a minor effect. However, for slow transitions lasting a fraction of the Hubble time, this approximation is not justified—and our work shows for the first time the dramatic consequences of including the background cosmological dynamics.
Methods
How it was done: We conducted a series of three-dimensional hydrodynamic simulations within the 'Higgsless' approach, which replaces the microscopic dynamics of the Higgs field with bubble walls moving at constant speed. Unlike all predecessors, Einstein's field equations were directly solved in the Friedmann–Lemaître–Robertson–Walker (FLRW) metric, taking into account cosmic expansion governed by Hubble's law and Friedmann's equations (the link between Hubble and expansion). The secret to success lies in simultaneously accounting for two effects: the decrease in average radiation density, causing the transition strength parameter α(t) to grow as $a^4$, and the increase in bubble number due to the accelerating expansion of physical volume. The simulations reproduced a Poisson nucleation process in an expanding cosmology; the gravitational waves radiated by sound waves were computed from the transverse-traceless part of the stress-energy tensor.
Results
What we got: The main finding is a violation of the linear growth of gravitational wave energy after percolation. In a static Universe, GW energy increases proportionally to the lifetime of sound waves. Under expansion, however, a nonlinear feedback kicks in: growing α(t) amplifies the kinetic energy of the plasma, which forces the generation of metric perturbations—and in the first few β^−1 units, the growth becomes nearly exponential. As a result, the amplitude of the $Ω_{GW}$ spectrum increases by tens of times (for initially intermediate-strength transitions, α0 = 0.05) and even hundreds of times (for weak transitions, α0 = 0.005). For example, with a wall velocity $v_w$ = 0.4 and β/$H_n$ = 20, the amplification reaches O(50). The peak frequency shifts upward due to the smaller average distance between bubbles. The spectral slopes at low and high frequencies remain virtually unchanged.
Implications
Significance: The results overturn previous ideas about the role of cosmic expansion in generating gravitational waves. Semi-analytical sound shell models predicted signal suppression; we instead see a giant amplification. This means that slow transitions in the early Universe, previously considered unpromising for observation, may in fact dominate the stochastic gravitational-wave background. The interpretation of existing radio astronomical observations from pulsar timing arrays and future observations with laser interferometers will need to reassess the connection between transition parameters and the signal.
Future development
Future development: The work opens several directions. First, the variable bubble wall velocity must be taken into account—in reality, it depends on the transition strength and could further enhance the effect. Second, the sound wave lifetime is crucial: we used a simple estimate based on the eddy turnover time, whereas in an expanding medium turbulence decays differently. Third, transitions with strong supercooling, where bubble collisions dominate over sound, may have their own surprises. Finally, incorporating fully microphysical simulations (with the Higgs field dynamics) will eliminate artifacts of the 'Higgsless' approximation.
Impact
Impact on: The results directly affect the interpretation of data from pulsar timing arrays (NANOGrav, EPTA, PPTA, CPTA) and the search strategy for gravitational waves by future space-based interferometers (LISA, Taiji, TianQin), as well as ground-based LIGO in the high-frequency band. They are also important for indirectly constraining models beyond the Standard Model, including various extensions of the Higgs sector and grand unified theories.
Next steps
Next steps: The immediate tasks are to conduct simulations over a wider parameter range (especially for strong and ultra-relativistic transitions) and to study in detail the decay time of acoustic waves by including physical viscosity and turbulence. It is also necessary to develop efficient spectrum emulators suitable for multidimensional data analysis from radio telescopes and gravitational-wave observatories.
Key open problems
Connection to unsolved problems: The generation of a stochastic background of gravitational waves is directly linked to three big questions: what is the nature of dark matter (a phase transition in the early Universe could have produced dark matter particles); how did baryon asymmetry arise (electroweak baryogenesis in extensions of the Standard Model); and is there a 'desert' between the electroweak scale and the Planck energy, or is physics rich with intermediate transitions that we can hear.
The knowledge machine's view
This section is addressed to the author of the paper and their colleagues. Neighbouring works were found by meaning-based search across our archive, not by word overlap. We do not review: we show the field around the work.
What the work does
For the first time, the expansion of the universe is taken into account in hydrodynamic simulations of a first-order phase transition. Using a Higgsless approach and the FLRW metric, the authors show that cosmic expansion leads to a nonlinear growth of gravitational wave energy after percolation, boosting the spectrum amplitude by tens to hundreds of times.
Where its strength is
The strength of the work lies in direct numerical simulations that include cosmological expansion, revealing a qualitatively new effect of nonlinear amplification of gravitational waves. This paves the way for more accurate predictions for slow transitions.
Where it could go next
- It might be interesting to see how including expansion affects other sources of gravitational waves from phase transitions, for example, bubble collisions beyond the horizon, where expansion could play an even more significant role.based on: Giant Bubbles Beyond the Horizon: Their Collisions Are Heard in Gravitational Waves arXiv:2607.24734v1
- A related work discusses a pause in expansion that leaves a gravitational imprint. One could think about how such non-monotonic expansion scenarios would change bubble dynamics and the gravitational wave spectrum.based on: The Frozen Universe: How a Pause in Expansion Left a Gravitational Imprint arXiv:2607.03537
- It would be interesting to compare with studies of oscillons, which also arise in the early universe and may contribute to gravitational waves. Possibly, expansion affects these two sources differently.based on: Ripples in the Fabric of the Early Universe arXiv:2607.29013v1
Worth trying
- Investigate how the nonlinear amplification depends on the wall velocity and the parameter β/H_n over a wider range.
- Compare predictions with the sensitivity of future detectors such as LISA for slow transitions.
- Consider scenarios with a varying equation of state of the background to see how this alters the growth of gravitational waves.
Why this may matter
Including expansion could substantially change estimates of the detectability of gravitational waves from phase transitions, especially for weak transitions previously considered invisible. This broadens the window into physics beyond the Standard Model.
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Original abstract
as written by the authors arXiv:2606.31571
We perform the first three-dimensional hydrodynamical simulations of cosmological first-order phase transitions in an expanding background. These simulations consistently incorporate the effects of the evolving phase transition strength throughout the full nucleation process of slow phase transitions. We find that, in addition to reducing mean bubble separations via an effectively enhanced nucleation rate, cosmic expansion unexpectedly induces highly nonlinear growth in the gravitational wave energy fraction, ultimately leading to a significant $$\mathcal{O}(10)$$ to $$\mathcal{O}(100)$$ amplification of the gravitational wave spectra. This amplification is more pronounced for initially weak transitions than for those of initially intermediate strength. Our results highlight the challenge and importance of accurately modelling slow phase transitions while accounting for cosmic expansion.
If we lived in a static Universe, gravitational waves from a phase transition would continue to grow linearly forever, as long as sound waves exist. But in an expanding Universe, the growth itself becomes explosive, then naturally fades due to the decreasing plasma density.
In Greg Egan's novel 'Schild's Ladder', vacuum phase transitions are explored that can reshape spacetime. Our work shows that even in 'gentle', non-catastrophic transitions, the Universe's expansion can generate unexpectedly powerful gravitational waves — a kind of echo of cosmic restructuring that may be louder than we thought.
Key numbers
- amplitude amplification of the spectrum: up to 100 times (for weak transitions)
- transition strength parameter α0: 0.005 (weak) and 0.05 (intermediate)
- ratio β/H_n: from 15 to 25 (transition duration)
- bubble wall velocity v_w: 0.4–0.8 in fractions of the speed of light
- characteristic peak frequency: ~10-6 Hz for T_n ~ 100 GeV