HydroGrav: Precise hydrodynamics and gravitational waves for cosmological phase transitions

Technical Summary of "HydroGrav: Precise hydrodynamics and gravitational waves for cosmological phase transitions"

Problem Statement
The detection of a stochastic gravitational wave background (SGWB) from cosmological first-order phase transitions is a primary target for future space-based observatories like LISA. The dominant contribution to this signal arises from sound waves generated by expanding bubbles of the broken phase. However, predicting the resulting gravitational wave spectrum requires accurate modeling of the hydrodynamic evolution of the plasma. Current literature predominantly relies on simplified equations of state (EOS)—specifically the "bag" model and the $\mu\nu$ (improved bag) model—which assume a constant speed of sound ($c_s^2 = 1/3$) or fixed sound speeds in the vacuum phases. These approximations ignore the temperature-dependent thermodynamic properties derived directly from the underlying particle physics effective potential. The authors argue that using simplified EOS for hydrodynamics while employing fitting formulas (derived from those same simplifications) to predict gravitational waves introduces a conceptual inconsistency. Furthermore, the validity of these approximations in determining the hydrodynamic mode (deflagration, detonation, or hybrid) and the spectral shape has not been rigorously tested against fully model-dependent calculations.

Methodology
The paper introduces HydroGrav, a C++ code designed to compute self-similar fluid profiles and gravitational wave spectra using the exact equation of state derived directly from a finite-temperature effective potential. The framework operates as follows:

1. Effective Potential and Thermodynamics: The code calculates thermodynamic quantities (pressure $p$, energy density $e$, enthalpy $w$, entropy $s$) directly from the effective potential $V_{\text{eff}}(\phi, T)$, rather than assuming a radiation-dominated form. This allows for a fully model-dependent calculation of the speed of sound $c_s^2(T) = dp/de$.
2. Relativistic Hydrodynamics: HydroGrav solves the relativistic hydrodynamic equations of motion for a perfect fluid in a self-similar coordinate system ($\xi = r/t$). It constructs fluid profiles ($v(\xi)$, $w(\xi)$, $T(\xi)$) for three hydrodynamic modes: deflagration, detonation, and hybrid.
   • For simplified EOS (bag and $\mu\nu$), the code uses algebraic matching conditions to determine shock fronts and wall velocities.
   • For the exact EOS, the code employs numerical root-finding algorithms (bounded 2D Newton's method) to satisfy matching conditions at the bubble wall and shock fronts, as the algebraic forms of thermodynamic quantities are unknown.
3. Gravitational Wave Calculation: Instead of using fitting formulas calibrated to limited simulations, HydroGrav implements the Sound Shell Model (SSM). This semi-analytic approach constructs the gravitational wave spectrum directly from the velocity and energy-density perturbation profiles generated by the hydrodynamic solution. This preserves the connection between the microscopic thermodynamics and the macroscopic signal, capturing spectral features (such as double-peak structures) lost in broken power-law fits.
4. Validation and Implementation: The code is validated against unit tests for numerical solvers (Golden section minimization, Newton's method, RK4 integration). It is demonstrated using a $Z_2$-symmetric real singlet extension of the Standard Model (xSM), where the effective potential is constructed using 3D effective field theory (3dEFT) via the DRalgo package.

Key Contributions

• HydroGrav Framework: A new, open-source C++ tool that bridges particle physics models and gravitational wave predictions using exact thermodynamics, supporting both simplified and exact EOS treatments.
• Exact EOS Implementation: A robust numerical method for solving hydrodynamic profiles using the exact equation of state, avoiding the assumptions of constant sound speed inherent in bag and $\mu\nu$ models.
• Sound Shell Model Integration: A consistent pipeline that uses the same fluid profiles for both hydrodynamic evolution and gravitational wave source calculation, eliminating the mismatch between sophisticated hydrodynamics and simplified spectral fitting.
• Systematic Comparison: A comprehensive scan of the xSM parameter space comparing fluid profiles and gravitational wave spectra across bag, $\mu\nu$, and exact EOS models.

Results
The authors performed a parameter scan of the xSM model (1,037 points) and analyzed specific benchmark points for deflagration, hybrid, and detonation modes:

• Hydrodynamic Modes: The $\mu\nu$ model generally reproduces the hydrodynamic mode (deflagration vs. detonation) predicted by the exact EOS more accurately than the bag model. However, discrepancies exist; the bag model incorrectly predicted the hydrodynamic mode for approximately 5% of sampled points, often misidentifying detonations as hybrids or deflagrations. The $\mu\nu$ model showed exceptional consistency, failing only in $\lesssim 0.02\%$ of cases.
• Fluid Profiles: The largest deviations from the radiation-dominated assumption occur in the broken phase, where the temperature dependence of thermodynamic quantities causes the sound speed to deviate from $c_s^2 = 1/3$. This is most evident in temperature profiles and the position of shock fronts.
• Gravitational Wave Spectra:
  • Spectral Shape: The Sound Shell Model reveals a double-peak structure encoding the mean bubble separation and sound shell thickness, which single broken power-law fitting formulas fail to capture.
  • Amplitude and Shape Differences: While the $\mu\nu$ model generally agrees better with the exact EOS than the bag model, significant differences in peak amplitude and spectral shape (measured by Wasserstein distance) persist, particularly for hybrid solutions and in regions where the sound speed deviates significantly from the radiation limit.
  • Peak Amplitude: For the xSM model, the $\mu\nu$ approximation captures the peak amplitude within 5% of the exact result for roughly 76% of deflagration points, 50% of hybrid points, and 47% of detonation points.
• Signal-to-Noise Ratio (SNR): Using the exact EOS, the authors estimated the SNR for LISA across the xSM parameter space. The predicted signals for this specific model remain below the nominal detection threshold ($SNR \approx 1$), though the authors note this is a demonstration of the pipeline rather than a definitive exclusion of the model, given theoretical uncertainties.

Significance and Claims
The paper claims that HydroGrav provides a necessary step toward "precision gravitational wave cosmology" by establishing a fully model-dependent pipeline. The authors emphasize that:

1. Consistency: Using the exact EOS for hydrodynamics while retaining simplified fitting formulas creates a conceptual inconsistency; the Sound Shell Model resolves this by using the same fluid profiles for both dynamics and signal generation.
2. Validity of Approximations: While the $\mu\nu$ model is a significant improvement over the bag model, simplified EOS are not universally reliable. They can lead to qualitatively different descriptions of plasma dynamics (e.g., incorrect hydrodynamic modes) and quantitatively different gravitational wave predictions, particularly in regions of parameter space where the sound speed deviates strongly from the radiation-dominated limit.
3. Future Necessity: As observatories like LISA approach the sensitivity required to probe electroweak-scale phase transitions, fully model-dependent treatments of hydrodynamics will become increasingly important to establish robust connections between observations and underlying particle physics.

The authors modestly conclude that their work demonstrates the capability and necessity of such precise treatments, rather than claiming that the specific xSM model considered is definitively detectable or that simplified models are always invalid. The framework is presented as a tool for quantifying the regime of validity of existing approximations.