Generating gravitational wave spectra from equations of state for phase transitions in the early universe

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

Physics Days
6.3.2026

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

Phase transitions in the early universe

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

Fluid shell velocity profiles

GW spectra

PTtools
- Simulation framework (Python library), doi:10.5281/zenodo.15268219
- Based on the Sound Shell Model by Hindmarsh et al.
- Developed by Mark Hindmarsh, Chloe Hopling (f.k.a. Gowling), Mika Mäki & Lorenzo Giombi
PTPlot
- Online plotting utility (Django web app), arXiv:1704.05871
- Developed by David Weir, Deanna Hooper, Jenni Häkkinen & Mika Mäki
- Supports several models for the GW spectrum
  - Broken power law
  - Double broken power law (in development)
  - Sound Shell Model (in development)
Input
- Equation of state: $p(T,\phi) = \frac{\pi^2}{90} g_p(T,\phi) T^4 - V(T,\phi)$
- 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 $\Omega_{gw,0}(f)$
- Signal-to-noise ratio (SNR) plots
Use case: estimate the likelihood of various Standard Model extensions with LISA data

https://www.ptplot.org

Broken power law (BPL)

Double broken power law (DBPL)

Sound Shell Model (SSM)

Example particle physics model: Singlet scalar
- SM extended with a general real singlet scalar field $S$
  $$ \Delta V = b_1 S + \frac{1}{2} b_2 S^2 + \frac{1}{2} a_1 S \left| H \right|^2 + \frac{1}{2} a_2 S^2 \left| H \right|^2 + \frac{1}{3} b_3 S^3 + \frac{1}{4} b_4 S^4 $$
- SNR curves plotted with $v_\text{wall} = 0.95, T_* = 50.0 \ \text{GeV}, g_* = 107.75$
- Example points sampled from the parameter space of the model
Significant change in the shape of the SNR curves with improved simulations

GW spectrum is an experimentally testable result of many BSM theories
The equation of state has a significant effect on the sound speed and therefore on the GW spectrum
Sound Shell Model improves the power spectrum and SNR accuracy compared to broken power laws, while maintaining computational efficiency
Utilities:
- PTtools: https://pttools.readthedocs.io
- PTPlot: https://www.ptplot.org

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