Exploring the parameter space of first-order phase transitions with PTtools

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

LISA Cosmology Working Group workshop
4.6.2026

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

Online slides

• First-order: potential barrier
  ⇒ phase boundary ⇒ bubble nucleation
• SM Higgs: crossover, BSM Higgs: first-order
  ⇒ Observational constraints of BSM theories

• Wall velocity $v_\text{wall}$
• Transition strength $\alpha_n$
• Speed of sound $c_s(T,\phi)$

• 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

• 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)

• Calibrating with 4D hydrodynamic lattice simulations
  • Gowling & Hindmarsh (2021, arXiv:2106.05984)
  • Suppression factor $\Sigma(v_\text{w},\alpha_n)$
  $$\mathcal{P}_\text{gw} = \Sigma(v_\text{w},\alpha_n) \mathcal{P}_\text{gw}^\text{ssm}$$
• Thermal suppression of bubble nuclation
  • Ajmi & Hindmarsh (2022, arXiv:2205.04097)
  • Deflagrations heat the plasma outside the bubble wall
    → suppression of bubble nucleation
  • Bubble spacing enlargement factor $\Lambda$
  $$R_* = \Lambda(\beta) R_{*,0}$$
• Finite lifetime of the acoustic source
  • Giombi et al. (2024, arXiv:2409.01426)
  • Acoustic source duration $\frac{\Delta \eta_\text{v}}{\eta_*}$ → source lifetime factor $\Upsilon$
  $$\tilde{P}_\text{gw}(z) \equiv \Upsilon_\ell \tilde{P}_\text{gw}^\text{ssm}(z) \quad \Upsilon_\ell \equiv \frac{1}{\ell(\nu)} \left(1 - \left(1 + \frac{\Delta \eta_\text{v}}{\eta_*} \right)^{-\ell(\nu)} \right)$$

• Giombi et al. (2024, arXiv:2409.01426), (2026, arXiv:2504.08037)
• GW spectral density $$\tilde{P}_\text{gw}(z) = \frac{\tau_*}{\pi^2 z^3} \int_0^\infty \int_{|x-z|}^{x+z} d\tilde{x} \rho(z,x,\tilde{x}) \tilde{P}_v(x) \tilde{P}_v(\tilde{x}) \Delta(z,x,\tilde{x},\tau,\tau_*,\tau_\text{end})$$
• Sound Shell Model (2019): $\Delta_\text{high} \propto \delta(z - c_s(x + \tilde{x})) \ \rightarrow \ k^9, \ k^{-3}$
• Low frequencies: $\Delta_\text{low} \ \rightarrow \ k^3$
• Intermediate frequencies: $\Delta_\text{int} \ \rightarrow \ k^1$

• PTtools
  • Simulation framework (Python library), doi:10.5281/zenodo.15268219
  • Based on the Sound Shell Model by Hindmarsh et al.
  • Included in LISA ProtoLab
• PTPlot
  • Online plotting utility (Django web app), arXiv:1910.13125
  • Supports several models for the GW spectrum: broken power law (BPL), double-broken power law (DBPL), Sound Shell Model (SSM) (in development)
• Input
  • Equation of state: $p(T,\phi) = \frac{\pi^2}{90} g_p(T,\phi) T^4 - V(T,\phi)$
  • Wall speed $v_\text{wall}$ (computed e.g. with WallGo)
  • Transition strength parameter $\alpha_n \equiv \frac{4D\theta(T_n)}{3w_n} = \frac{4}{3} \frac{\theta_+(T_n) - \theta_-(T_n)}{w_n}$
  • Nucleation rate parameter $\tilde{\beta} = \frac{\beta}{H_*}$
• Output:
  • GW power spectrum today $\Omega_{\text{gw},0}(f)$
  • Signal-to-noise ratio (SNR) plots
• Use case: estimate the likelihood of various Standard Model extensions with LISA data

                            
                                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")
                            
                        

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