From dimensional reduction to droplets

Understanding strong first-order phase transitions

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

This talk: saoghal.net/slides/d2d

Gravitational Wave Probes of Physics Beyond the Standard Model, July 2021

Temperature slice from a simulation by Daniel Cutting

Hot, red areas are shrinking droplets

• How do droplets form?
• What are the consequences for gravitational waves?

Particle physics model
$\Downarrow \mathcal{L}_{4\mathrm{d}}$
Dimensional reduction
$\Downarrow \mathcal{L}_{3\mathrm{d}}$
Phase transition parameters from lattice simulations
$\Downarrow \alpha, \beta, T_N, \ldots$
Real time cosmological simulations
$\Downarrow \Omega_\text{gw}(f)$
Cosmological GW background

Key parameters bridge the gap

Including:

• $\alpha$, the phase transition strength
• $\beta$, the inverse phase transition duration
• $T_N$, the nucleation temperature

A "pipeline"

Particle physics model
$\Downarrow \mathcal{L}_{4\mathrm{d}}$
Dimensional reduction
$\Downarrow \mathcal{L}_{3\mathrm{d}}$
Phase transition parameters from lattice simulations

My focus: extensions of the Standard Model

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
  ➥ colliding bubbles then make gravitational waves

SM electroweak phase diagram

How? Dimensional reduction

• At high $T$, system looks 3D at distances $\Delta x \gg 1/T$

• Each step involves matching Green's functions in the effective and full theories to the 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

When new physics is heavy

Benchmark: ● 4d PT vs ● 3d PT vs ● lattice

How to get strong transitions?

Need light physics or dim-6 operators

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 a strong phase transition
• Should benchmark perturbation theory with DR + lattice, particularly for strong transitions

Particle physics model ✅
$\Downarrow \mathcal{L}_{4\mathrm{d}}$
Dimensional reduction ✅
$\Downarrow \mathcal{L}_{3\mathrm{d}}$
Phase transition parameters from lattice 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 $\Omega_\text{col}(f)$
4. Sound waves left behind in plasma $\Omega_\text{sw}(f)$
5. Shocks [$\rightarrow$ turbulence] $\rightarrow$ damping $\Omega_\text{turb}(f)$

Simulating weak transitions: $\alpha \ll 1$

• Focusing on GWs from sound waves... arXiv:1704.05871

• ...$\Omega_\text{sw}(f)$ fairly close to broken power law arXiv:1512.06239
• ...and the linear sound shell model arXiv:1608.04735

Explore $\Omega_\text{sw}(f)$ with PTPlot.org

Model ⟶ ($T_*$, $\alpha_{T_*}$, $v_\mathrm{w}$, $\beta$) ⟶ this plot

[Here: $Z_2$-symmetric xSM points from arXiv:1910.13125]

But what about strong transitions?

• Nonlinearities during the transition:
  • Generation of vorticity
  • Droplets
• Nonlinearities after the transition:
  • Shocks
  • turbulence

Let's take a look at droplets

Strong simulation velocity slice

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

Walls slow, droplets form

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

Isolated spherical droplets

In the spherical case, we can get a self-similar droplet. We see the same wall velocity slowdown:

Droplets may suppress GWs

Suppression compared to sound waves (redder = worse)

arXiv:1906.00480

Thanks

• Students:
  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, a valuable field theory tool
  $\Rightarrow$ test perturbative studies of phase transitions
• Strong phase transitions: hot droplets slow completion
  $\Rightarrow$ also suppress GW production

Questions you can ask me

• How accurate are bubble nucleation calculations?
• What about the onset of shocks and turbulence?
• What other physics could explain the GW suppression seen in strong transitions?