Status of hydrodynamical simulations of early universe phase transitions

David J. Weir [they/he]
University of Helsinki

This talk: saoghal.net/slides/coswg2023

10th LISA Cosmology Working Group Workshop

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

arXiv:2112.12013

Scales and frequencies

By considering how GWs get redshifted on the way to us, and assuming they get produced at cosmological scales:

[order-of-magnitude calculation!]
arXiv:2008.09136

Could BSM physics produce a stochastic background?

First-order phase transitions are a complementary probe of new physics that might be

• Out of sight of particle physics experiments, or
• At higher energy scales than colliders can reach

For example: electroweak PT

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

[what BSM physics might there be?]
Particle physics model
$\Downarrow \mathcal{L}_{4\mathrm{d}}$
Magical field theory stuff OG's talk!
(perturbative or lattice)
$\Downarrow$
Phase transition parameters
$\Downarrow \alpha, \beta, T_N, v_\mathrm{w}, \ldots$
Real time cosmological simulations
$\Downarrow \Omega_\text{gw}(f)$
Cosmological GW background
[what would we see as a result?]

[what BSM physics might there be?]
Particle physics model
$\Downarrow \mathcal{L}_{4\mathrm{d}}$
Magical field theory stuff OG's talk!
(perturbative or lattice)
$\Downarrow$
Phase transition parameters
$\Downarrow \alpha, \beta, T_N, v_\mathrm{w}, \ldots$
Real time cosmological simulations
$\Downarrow \Omega_\text{gw}(f)$
Cosmological GW background
[what would we see as a result?]

[what BSM physics might there be?]
Particle physics model
$\Downarrow \mathcal{L}_{4\mathrm{d}}$
Magical field theory stuff OG's talk!
(perturbative or lattice)
$\Downarrow$
Phase transition parameters
$\Downarrow \alpha, \beta, T_N, v_\mathrm{w}, \ldots$
Real time cosmological simulations
$\Downarrow \Omega_\text{gw}(f)$
Cosmological GW background
[what would we see as a result?]

Dynamics of phase transitions (probably) rely on relatively few parameters

Including:

• $\alpha$, 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

Phase transition = out of equilibrium

1. Bubbles nucleate (temperature $T_\mathrm{N}$, on timescale $\beta^{-1}$)
2. Bubble walls expand in a plasma (at velocity $v_\mathrm{w}$)
3. Reaction fronts form around walls (with strength $\alpha$)
4. Bubbles + fronts collide GWs
5. Sound waves left behind in plasma GWs
6. Shocks [$\rightarrow$ turbulence] $\rightarrow$ damping GWs

How are GWs produced at a first order phase transition?

• Not all phase transitions have $v_\mathrm{w} \ll c$ ...
  • 'Vacuum' transitions with no couplings/friction
  • 'Run away' transitions arXiv:1703.08215
• ... but if they do:
  • Shear stress sourced most efficiently in first $1/H_*$
  • Fluid motion becomes nonlinear on a time scale
    $$\tau_\text{sh} = \frac{R_*}{\overline{U}} = \frac{\text{Bubble radius (i.e. typical length scale)}}{\text{Typical fluid velocity}}$$

Nonlinearities?

• Nonlinearities during the transition:
  • Generation of vorticity
  • Formation of droplets
• Nonlinearities after the transition:
  • Shocks
  • Turbulence (Kolmogorov and acoustic)

Let's take a look at droplets and turbulence

Strong deflagrations ⇒ droplets

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

A closer look in spherical geometry

arXiv:2204.03396

Droplets form ➤ walls slow down

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

arXiv:2204.03396

Droplets may suppress GWs

arXiv:1906.00480

Droplets and phenomenology

• Long-lived regions with moderate wall velocities
• Could this help with baryogenesis? For strong transitions, walls often move too quickly.
• Proposals for dark strongly interacting matter 'nuggets' where the phases coexist e.g. arXiv:1810.04360 arXiv:1912.02830
• Other applications?
• Further 3D simulations will be needed

arXiv:2112.12013, arXiv:2205.02588

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.

Kolmogorov turbulence: GWs

Validated theoretical modelling of GWs from Kolmogorov turbulence with large-scale simulations

arXiv:2205.02588

Thanks

• Students:
  Jani Dahl, Ethan Edwards, Jenni Häkkinen, Anna Kormu, Tiina Minkkinen, Satumaaria Sukuvaara, Essi Vilhonen
• Postdocs:
  Deanna C. Hooper, Lauri Niemi
• Collaborators:
  including Pierre Auclair, Chiara Caprini, Daniel Cutting, Oliver Gould, Mark Hindmarsh, Kari Rummukainen, Dani Steer, Tuomas Tenkanen