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

12.3.2025

Phase transitions in the early universe

Potential
  • First-order: potential barrier
    ⇒ phase boundary ⇒ bubble nucleation
  • SM Higgs: crossover, BSM Higgs: first-order
    ⇒ Experimental testing of BSM theories
Fluid movement ⇒ sound waves ⇒ gravitational waves

Self-similar hydrodynamics

Relativistic combustion
  • Self-similar: time-independent solution
  • Black circle: phase boundary, aka. bubble wall
  • Colour: velocity of moving plasma
  • $c_s$: speed of sound
  • $v_\text{w}$: wall speed
  • $c_\text{J}$: Chapman-Jouguet speed

PTtools: GW spectra from equations of state

  • Python-based simulation framework
  • Based on the Sound Shell Model by Hindmarsh et al.
    • Computationally efficient compared to lattice simulations
  • Simulation of a single expanding self-similar spherically symmetric bubble
    ⇒ GW power spectrum
  • Key parameters from particle physics: $v_\text{wall}, \alpha_n, r_*, T_n,$
    equation of state: $p(T,\phi)$

General equation of state

  • General equation of state: $p(T,\phi) = \frac{\pi^2}{90} g_p(T) T^4 - V(T,\phi)$
    • Sound speed $c_s \equiv \frac{dp}{de} \neq \frac{1}{\sqrt{3}}$

PTtools

  • Python-based simulation framework
  • Based on the Sound Shell Model
    • Single bubble
    PTtools
    PTtools repository

From fluid profiles to GW spectra

Bubbles

🠮

Spectra
Fluid shell velocity profiles
  • Boundary conditions
  • ODE integration
GW spectra
  • Sine transform
  • Conversion to observable $f$ etc.
  • Experimentally testable by LISA

How to use PTtools

						
							from pttools.analysis import plot_spectra_omgw0, plot_bubbles_v
							from pttools.bubble import Bubble
							from pttools.models import ConstCSModel
							from pttools.omgw0 import Spectrum

							alpha_n = 0.2
							model = ConstCSModel(css2=1/3, csb2=1/4, alpha_n_min=alpha_n)
							bubbles = [
								Bubble(model, v_wall=0.3, alpha_n=alpha_n),
								Bubble(model, v_wall=0.5, alpha_n=alpha_n),
								Bubble(model, v_wall=0.9, alpha_n=alpha_n)
							]
							spectra = [Spectrum(bubble, r_star=0.1, Tn=200) for bubble in bubbles]

							plot_bubbles_v(bubbles, path="bubbles.svg", v_max=0.5)
							plot_spectra_omgw0(spectra, path="spectra.svg")
PTtools
PTtools repository

Thank you!

Fluid velocity spectra

Velocity spectra

Gravitational wave power spectra

GW power spectra