Nonlinear growth and amplification of phase-transition gravitational waves induced by cosmic expansion

Technical Summary: Nonlinear Growth and Amplification of Phase-Transition Gravitational Waves Induced by Cosmic Expansion

Problem Statement
Gravitational waves (GWs) sourced by cosmological first-order phase transitions (FOPTs) are a primary target for upcoming observatories like LISA, Taiji, and TianQin, as well as current Pulsar Timing Arrays (PTA). While sound waves generated during thermal FOPTs are known to provide the dominant contribution to the stochastic gravitational wave background (SGWB), conventional theoretical predictions often neglect a critical physical ingredient: cosmic expansion. Previous semi-analytical studies incorporating expansion generally predicted a suppressed GW signal with a shifted peak frequency, relying on assumptions such as the linear superposition of sound waves. Furthermore, these approaches often fail to account for the time-dependent nature of the phase transition strength ($\alpha$) caused by the dilution of radiation energy density relative to constant vacuum energy during expansion. This oversight is particularly significant for slow phase transitions, where the interplay between expansion and nucleation dynamics may be non-trivial.

Methodology
To address these limitations, the authors perform the first systematic three-dimensional hydrodynamical simulations of cosmological FOPTs within a Friedmann–Lemaître–Robertson–Walker (FLRW) background. The study employs the following methodological framework:

• Hydrodynamical Framework: The fluid dynamics are modeled using the bag equation of state within a Higgsless framework. The energy-momentum tensor is solved in an expanding spacetime, explicitly incorporating the time-dependent scale factor $a(t)$.
• Time-Dependent Strength: The methodology factorizes thermodynamic quantities into a homogeneous background component and a relative component. This reveals that the phase transition strength parameter, $\alpha$, is inherently time-dependent, growing as the fourth power of the scale factor ($\alpha(t) \propto a(t)^4$) due to the dilution of the radiation bath.
• Simulation Setup: The authors utilize a moving boundary with a constant wall velocity ($v_w$) to simulate expanding bubbles and employ the Kurganov-Tadmor scheme to resolve shock fronts. The simulations are conducted on a $512^3$ grid with a box size $L = 40v_w/\beta$ to ensure a constant number of bubbles.
• GW Calculation: The GW energy density is computed by solving the linearized Einstein equations for metric perturbations sourced by the anisotropic stress of the fluid. The authors track the evolution of the GW energy fraction $\tilde{\Omega}^*_{GW}(t)$ and the spectral density $\Omega^*_{GW}(k)$.

Key Results
The simulations yield several counter-intuitive and significant findings regarding the impact of cosmic expansion on GW production:

1. Nonlinear Amplification: Contrary to previous predictions of signal suppression, cosmic expansion induces a highly nonlinear, nearly exponential growth in the GW energy fraction during the early stages of GW production. This leads to a substantial amplification of the final GW spectra, ranging from an order of magnitude of $O(10)$ to $O(100)$ compared to non-expanding scenarios.
2. Mechanism of Growth: This amplification is driven by the effective increase in phase transition strength $\alpha(t)$ throughout the nucleation process. As the universe expands, the latent heat conversion efficiency into fluid kinetic energy increases, enhancing the root-mean-square four-velocity ($U_f$) of the fluid. Since GW signal strength scales with $U_f^4$, this results in a significantly stronger signal.
3. Dependence on Transition Strength: The amplification effect is most pronounced for initially weak transitions ($\alpha_0 = 0.005$) compared to those of intermediate strength ($\alpha_0 = 0.05$). For weak transitions, the sound-wave lifetime is long enough to allow the exponential growth phase to dominate, whereas for intermediate transitions, the reduced mean bubble separation and shorter sound-wave lifetimes partially mitigate the enhancement.
4. Spectral Shifts: While the spectral slopes at low and high frequencies remain largely unchanged, the peak frequency shifts to higher values. This shift is attributed to the reduced mean bubble separation caused by the effectively enhanced nucleation rate in an expanding volume.
5. Role of Wall Velocity: For initially weak transitions, the amplification is stronger for lower wall velocities (deflagration regime). However, for intermediate transitions, the relationship is reversed due to the suppression of the sound-wave lifetime at lower velocities in the expanding scenario.

Significance and Claims
The paper asserts that accurately modeling slow phase transitions requires the inclusion of cosmic expansion, as neglecting it leads to a severe underestimation of the GW signal. The authors highlight that the nonlinear growth mechanism identified here cannot be captured by semi-analytical frameworks that rely on linear superposition or static strength parameters.

The study concludes that the "O(100)" amplification is a robust feature arising from the effective growth of $\alpha(t)$, which modifies the hydrodynamics of the phase transition. While the authors acknowledge that a time-dependent wall velocity (which was treated as constant here) could introduce further corrections, they argue that such effects are unlikely to eliminate the substantial amplification observed. Consequently, these results underscore the necessity of full 3D hydrodynamical simulations in expanding backgrounds for reliable theoretical predictions of GWs from the early Universe, particularly for interpreting potential signals from PTA and space-based interferometers.