Beyond the basics

Exploring gravitational waves
from strong phase transitions

David J. Weir [they/he] - University of Helsinki
davidjamesweir@mementomori.social

This talk: saoghal.net/slides/tallinn2026

3rd Nordic Cosmology Meeting, 24 September 2026

What happened in the early universe? before recombination? in dark sectors?

Credit: Stephan Paul, arXiv:1205.2451

Credit: Stephan Paul, arXiv:1205.2451

Credit: Stephan Paul, arXiv:1205.2451

Credit: Stephan Paul, arXiv:1205.2451

First order phase transition

• High temperatures: potential has one global minimum
• Below $T_\mathrm{c}$, second minimum becomes global minimum
• Nucleation temp. $T_\mathrm{n}$: ~ 1 bubble per causal volume
• 'Percolation' temp. $T_\mathrm{p}$: broken phase clusters join up

Summary: what happens?

1. Nucleation (temperatures $T_\mathrm{n}$→$T_\mathrm{p}$, duration $\beta^{-1}$)
2. Bubble walls expand in a plasma (at velocity $v_\mathrm{w}$)
   
3. Reaction fronts form around walls (with strength $\alpha$)

Then what?

4. Bubbles (size $R_*$) + fronts collide GWs
5. Sound waves left behind, plasma 4-velocity $U$ GWs
6. System becomes 'turbulent'; KE starts to decay GWs
7. GW production saturates; Hubble damping

What is a strong phase transition?

• Plasma velocities become relativistic
  ⇒ flows quickly become nonlinear
• Nonlinearities include:
  • shocks ➠ acoustic turbulence $t_{\mathrm{sh}} \sim R_* / U_{\parallel}$
  • vortices ➠ [vortical] turbulence $t_{\mathrm{ed}} \sim R_* / U_{\perp}$
  (in reality hard to separate into two types)
• Transport of energy to small scales ⇒ KE decay
• Typically strong means $\alpha$ 'large'
  (maybe better to focus on what the flows look like)

Strong simulation example

Created by José Correia for arXiv:2505.17824.

GWs from phase transitions

• Sound shell model: velocity field is superposition of self-similar sound shells
  • Convolve velocity field to get GW power spectrum
  • Works for weak transitions, linear flows, small bubble sizes
• Simulations of scalar field $\phi$ and fluid $u^\mu$,
  e.g. our SCOTTS code, 'Higgsless' code
  • Can directly source metric perturbations... $$\ddot{h}_{ij} - \nabla^2 h_{ij} = 16 \pi G T_{ij}^\text{TT}; \qquad T_{ij}^\text{(f)} = w u_i u_j $$
  • ...or convolve velocity field $u_i$ as above

Strong phase transition comparisons

Instantaneous GW production rate: arXiv:2505.17824

Detonation, $v_\mathrm{w} = 0.92$, $\alpha = 0.67$

Deflagration, $v_\mathrm{w} = 0.44$, $\alpha = 0.5$

• Sound shell model overestimates, gets wrong shape
• Convolving fluid velocity power spectrum is better

Fluid velocity power $\mathcal{P}_U$

Separated into $\mathcal{P}_{U_\parallel}$ and $\mathcal{P}_{U_\perp}$ (enthalpy-weighted $u^\mu$)

Detonation, $v_\mathrm{w} = 0.92$, $\alpha = 0.67$
Vortical $\ll$ compressional

Deflagration, $v_\mathrm{w} = 0.44$, $\alpha = 0.5$
Vortical $\sim$ compressional for some $k R_*$

Kinetic energy decay

$U_\parallel = U_\parallel^\text{max}\left[1 + \Delta t/t_*\right]^{-\zeta/2}$; the prediction is $\zeta \approx 10/7$ from 2407.05826

Detonation, $v_\mathrm{w} = 0.92$, $\alpha = 0.67$

Deflagration, $v_\mathrm{w} = 0.44$, $\alpha = 0.5$

• Higgsless: $\zeta \approx 10/7$ for deflagrations too 2409.03651
• Could metastable phase heating be responsible?

GW saturation

Asymptotic $\Omega_\text{gw}(t) = \Omega_\text{gw}^\infty \left[ 1- \left(\Delta t/t_*\right)^{-d}\right]$

Detonation
$v_\mathrm{w} = 0.92$, $\alpha = 0.67$

Deflagration
$v_\mathrm{w} = 0.44$, $\alpha = 0.5$

Can thus determine a dimensionless final 'efficiency' factor for GW production $\tilde{\Omega}_\text{gw}^\infty \approx 0.017$.

$d \approx 0.61$
$d \approx 0.79$

Next: "deep field" ➢ "wide field"

Scan in $v_\mathrm{w}$, $\alpha$, $R_*$, ... – here is a sample:

Including hybrids!

A plug

We are organising a COST action titled 'HIPPO'
(HIggs Pairs and POtential). Topics will include:

• Complementarity between collider physics and gravitational wave cosmology
• Improving the accuracy of predictions of stochastic cosmological backgrounds from BSM physics
• Providing public tools to analyse data
• Sharing expertise collider physics ⇋ GWs/cosmology (e.g. machine learning)

Interested in signing the proposal? Please email me (david.weir@helsinki.fi) by 12.00 CEST on Wed 30.9.

Conclusions

• We have extended our earlier simulation-based studies to strong phase transitions
• See evidence for turbulent motion in both compressional and longitudinal modes
• For strong phase transitions with short duration compared to Hubble damping, can extrapolate $\Omega_\text{gw}^\infty$

Questions you might like to ask me

• What phenomena might explain the discrepancies for deflagrations?
• Is there any evidence of non-Gaussianity?