From equations of state to gravitational wave spectra

Mika Mäki, Doctoral Researcher,
Computational Field Theory group, University of Helsinki

Advancing gravitational wave predictions from cosmological first-order phase transitions
CERN, 28.8.2025

Different types of phase transition simulations

Lattice simulations
- Capture multiple types of effects: bubble expansion, collision and turbulence
- Computationally expensive
Analytical approximations
- Broken power law, double broken power law
- Useful but rather crude approximations
Semi-analytical approximation: Sound Shell Model
- Captures dependence on phase transition parameters
- Reproduces the results of lattice simulations for intermediate-strength transitions

Source: Hindmarsh et al. (2021, arXiv:2008.09136), hybrids discovered by Kurki-Suonio & Laine (1995, arXiv:hep-ph/9501216)

Friction → constant $v_\text{wall}$ → self-similar solution

Three types of solutions, determined by

Explanation of the figure

Equation of state for an ultrarelativistic plasma with multiple degrees of freedom $$p(T,\phi) = \frac{\pi^2}{90} g_p(T,\phi) T^4 - V(T,\phi)$$
The rest can be deduced with thermodynamics
- Entropy density $s = \frac{dp}{dT}$, enthalpy density $w = Ts$,
  energy density $e = w - p$, sound speed $c_s \equiv \sqrt{\frac{dp}{de}}$
Bag model: equation of state with constant degrees of freedom $$p_\pm = a_\pm T^4 - V_\pm \quad \Rightarrow \quad c_s^2 \equiv \frac{dp}{de} = \frac{1}{3}$$
Constant sound speed model
(aka. $\mu, \nu$ model or template model, arXiv:2004.06995, arXiv:2010.09744) $$p_\pm = a_\pm \left( \frac{T}{T_0} \right)^{\mu_\pm - 4} T^4 - V_\pm {\color{gray} \approx a_\pm T^{\mu_\pm} - V_\pm } \quad \Rightarrow \quad c_{s\pm}^2 = \frac{1}{\mu_\pm - 1}$$
From an arbitrary particle physics model: $g_p(T,\phi), \ V(T,\phi)$
- This can also be done with e.g. WallGo (arXiv:2411.04970)

Hindmarsh (2018, arXiv:1608.04735), Hindmarsh & Hijazi (2019, arXiv: 1909.10040)
When the transition strength $\alpha_n \ll 1$ and $v_\text{wall} < 1$
- Sound waves are the majority contribution
  → collisions and turbulence can be neglected → no interaction between the bubbles

Giombi et al. (2024, arXiv:2409.01426): Changing the sound speed → source lifetime factor $\Lambda$
$$\begin{align} \tilde{P}_\text{gw,corr.}(k) &\equiv \Lambda \tilde{P}_\text{gw}(k) & \Lambda &\equiv \frac{1}{1 + 2\nu} \left(1 - \left(1 + \frac{\Delta \eta}{\eta_*} \right) \right)^{-1-2\nu} \\ \nu_\text{gdh2024} &\equiv \frac{1-3\omega}{1 + 3\omega} & \omega(T,\phi) &\equiv \frac{p(T,\phi)}{e(T,\phi)} \quad \textcolor{gray}{\omega_\text{bag} = \frac{1}{3} \rightarrow \nu_\text{bag} = 0} \end{align}$$
Gowling & Hindmarsh (2021, arXiv:2106.05984): Suppression factor
- From comparison with 3D hydrodynamic simulations
$$\Omega_\text{gw} = \Sigma(v_\text{w},\alpha_n) \Omega_\text{gw}^\text{ssm}(z)$$
Giombi et al. (2024, arXiv:2409.01426)
- Low-frequency tail of the spectrum

Suppression factor
Low-k

Low-frequency tail

Python-based simulation framework, doi:10.5281/zenodo.15268219
Based on the Sound Shell Model by Hindmarsh et al.
- Computationally efficient compared to lattice simulations
Developed by Mark Hindmarsh, Chloe Hopling (f.k.a. Gowling), Mika Mäki & Lorenzo Giombi
Input
- Equation of state: $p(T,\phi)$
- Wall speed $v_\text{wall}$
- Transition strength parameter $\alpha_n \equiv \frac{4D\theta(T_n)}{3w_n} = \frac{4}{3} \frac{\theta_+(T_n) - \theta_-(T_n)}{w_n}$
- Hubble-scaled mean bubble spacing $r_* \equiv H_n R_*$
Output: GW power spectrum today $\Omega_{gw,0}(f)$
Use case: estimate the likelihood of various Standard Model extensions with LISA data

https://pttools.readthedocs.io

Fluid shell velocity profiles

GW spectra

Online plotting utility, arXiv:1704.05871
- Based on the Python web framework Django
- Developed by David Weir, Deanna Hooper, Jenni Häkkinen & Mika Mäki
Supports several models for the GW spectrum
- Broken power law
- Double broken power law (upcoming)
- Sound Shell Model (upcoming)