Gravitational waves as a probe of early universe phase transitions

David J. Weir [they/he] - Helsinki - davidjamesweir

This talk:

EuCAPT colloquium, January 2022

While you were waiting

You were watching a movie of vorticity $\nabla \times \mathbf{v}$ in a simulation of 2D acoustic turbulence by Jani Dahl



  1. Context: early universe, LISA, pipeline
  2. Pause for questions
  3. Microphysics: beyond the Standard Model
  4. Pause
  5. Macrophysics: out of equilibrium
  6. Wrapping up: key points and questions to ask

What happened in the early universe? when the universe was optically opaque? in dark sectors?

LISA is coming!

  • Three laser arms, 2.5 M km separation
  • ESA-NASA mission, launch 2030s
  • Mission exited 'phase A' in December 2021


LISA: "Astrophysics" signals

LISA: Stochastic background?

[qualitative curve, sketched on]

This talk is about:

How a background might be made by a phase transition

This talk is not about:

How to infer the background's existence from LISA data

[but that is a cool topic all the same]

arXiv:2106.05984; arXiv:2107.06275; etc.

Questions so far

You could ask...

Why focus on LISA (for potentially detecting a cosmological stochastic background)?

Image by ed_needs_a_bicycle on Flickr [CC-BY-NC-SA]

How does a stochastic cosmological background come about, then?

  • What happens during first order phase transitions?
  • What are the consequences for gravitational waves?

One approach

Particle physics model
$\Downarrow \mathcal{L}_{4\mathrm{d}}$
Dimensional reduction

$\Downarrow \mathcal{L}_{3\mathrm{d}}$
Lattice Monte Carlo simulations

$\Downarrow \alpha, \beta, T_N, v_\mathrm{w}, \ldots$
Real time cosmological simulations

$\Downarrow \Omega_\text{gw}(f)$
Cosmological GW background

Model-independent parameters bridge the gap


  • $\alpha$, the phase transition strength
  • $\beta$, the inverse phase transition duration
  • $T_N$, the temperature at which bubbles nucleate
  • $v_\mathrm{w}$, the speed at which bubbles expand

A "pipeline"


Particle physics model
$\Downarrow \mathcal{L}_{4\mathrm{d}}$
Dimensional reduction
$\Downarrow \mathcal{L}_{3\mathrm{d}}$
Lattice Monte Carlo simulations

$\Downarrow \alpha, \beta, T_N, \ldots$
Real time cosmological simulations
$\Downarrow \Omega_\text{gw}(f)$
Cosmological GW background

My focus: extensions of the Standard Model

$$ \mathcal{L}_{4\mathrm{d}} = \mathcal{L}_\text{SM}[\text{SM fields}] \color{red}{+ \mathcal{L}_\text{BSM}[\text{SM fields},\ldots ?]} $$

SM electroweak phase transition

  • Process by which the Higgs 'switched on'
  • In the Standard Model it is a crossover
  • Possible in extensions that it would be first order
    ➥ subsequent processes make gravitational waves

SM electroweak phase diagram

arXiv:hep-ph/9605288 ; arXiv:hep-lat/9704013; arXiv:hep-ph/9809291

How? Dimensional reduction

  • At high $T$, system looks 3D at distances $\Delta x \gg 1/T$
  • Match Green's functions at each step to desired order
  • Handles the infrared problem, light fields can be studied on lattice arXiv:hep-ph/9508379

Using the dimensional reduction

  • Simulate DR'ed 3D theory on lattice arXiv:hep-let/9510020
  • With DR, integrate out heavy new physics and recycle

How to get strong transitions?

Theories that look SM-like in the IR ⇒ not observable!


When new physics is heavy

Benchmark: ● 4d PT vs ● 3d PT vs ● NP (lattice)


DR: $\Sigma$SM (triplet) example

Perturbation theory doesn't see the phase transition!


Key points so far

  • Dimensional reduction + lattice simulations a well-proven method for studying BSM theories
  • Higher dimensional operators or light new physics needed for observable gravitational waves
  • Should benchmark perturbation theory with DR + lattice, particularly for strong transitions

Questions so far

You could ask...

But how do you calculate the wall velocity?


Particle physics model
$\Downarrow \mathcal{L}_{4\mathrm{d}}$
Dimensional reduction ✅
$\Downarrow \mathcal{L}_{3\mathrm{d}}$
Lattice Monte Carlo simulations ✅
$\Downarrow \alpha, \beta, T_N, \ldots$
Real time cosmological simulations
$\Downarrow \Omega_\text{gw}(f)$
Cosmological GW background

Out of equilibrium physics

  1. Bubbles nucleate and grow
  2. Expand in a plasma - create reaction fronts
  3. Bubbles + fronts collide
  4. Sound waves left behind in plasma
  5. Shocks [$\rightarrow$ turbulence] $\rightarrow$ damping

Explore $\Omega_\text{gw}(f)$ with

Model ⟶ ($\alpha$, $\beta$, $T_N$, $v_\mathrm{w}$ ) ⟶ this plot


Explore $\Omega_\text{gw}(f)$ with

Assumes GW emission stops when nonlinearities form.



  • Nonlinearities during the transition:
    • Generation of vorticity
    • Droplets
  • Nonlinearities after the transition:
    • Shocks
    • Turbulence (and acoustic turbulence)
  • Let's take a look at droplets and acoustic turbulence

Strong deflagrations ⇒ droplets

[$\alpha_{T_*} = 0.34$, $v_\mathrm{w} = 0.24$ (deflag.)], velocity $\mathbf{v}$

Droplets form ➤ walls slow down

At large $\alpha_{T_*}$ reheated droplets form in front of the walls

Sound waves ➤ acoustic turbulence

  • Thermal phase transitions produce sound waves
  • Over time, sound waves steepen into shocks
  • Overlapping field of shocks = 'acoustic turbulence'
  • Distinct from, but related to Kolmogorov turbulence


2d acoustic turbulence

Acoustic turbulence: GWs

Spectral shape $S$ as function of $k$ and integral scale $L_0$:

Different from sound waves and Kolmogorov turbulence!
⇒ all must be taken into consideration.


  • Students:
    Jani Dahl, Anna Kormu, Lauri Niemi, Satumaaria Sukuvaara, Essi Vilhonen
  • Postdocs:
    Daniel Cutting, Oliver Gould
  • Collaborators:
    Jonathan Kozaczuk, Mark Hindmarsh, Stephan Huber, Hiren Patel, Michael Ramsey-Musolf, Kari Rummukainen, Tuomas Tenkanen

What I want you to remember

  • Dimensional reduction is a valuable field theory tool
    $\Rightarrow$ lattice Monte Carlo simulations of phase transitions
  • Nonlinearities matter when studying phase transitions
    $\Rightarrow$ large-scale real-time cosmological simulations

More questions you can ask me

  • How accurate are bubble nucleation calculations?
  • What are the consequences of droplet formation?
  • What about other types of turbulence?