From equations of state to gravitational wave spectra of first-order phase transitions in the early universe

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

Tampere Cosmology Meeting
5.9.2025

In collaboration with David Weir, Mark Hindmarsh, Deanna Hooper & Lorenzo Giombi

Contents of the talk

Phase transitions in the early universe
- Introduction
- Types of simulations
- Self-similar fluid shells
- Equations of state
Theoretical tools developed by our group
- Sound Shell Model
- Correction factors
Simulation software and results
- PTtools
- Results
- PTPlot
Summary

First-order: potential barrier
⇒ phase boundary ⇒ bubble nucleation
SM Higgs: crossover, BSM Higgs: first-order
⇒ Experimental testing of BSM theories

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
- This is what we are working on

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)

https://ptplot.org

GW spectrum is an experimentally testable result of BSM theories
The equation of state has a significant effect on the GW spectrum
Sound Shell Model enables quick computation of the GW spectrum
Utilities:
- PTtools: https://pttools.readthedocs.io
- PTPlot: https://www.ptplot.org