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

21.3.2025

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

Self-similar hydrodynamics

• 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 repository

From fluid profiles to GW spectra

🠮

Fluid shell velocity profiles

• Boundary conditions
• ODE integration

GW spectra

• Sine transform
• Conversion to observable $f$ etc.
• Experimentally testable by LISA

Detection of primordial gravitational waves

• LISA: Laser Interferometer Space Antenna
• Will be launched in the 2030s
• Three satellites connected by 2.5 million kilometer long laser beams
• Measures gravitational waves at milli-Hertz frequencies
• "Listening to the sound of a cosmic kettle"
• Direct testing and comparison of particle physics theories beyond the Standard Model

Thank you!

Fluid velocity spectra

Gravitational wave power spectra