Real-time cosmological simulations of the early universe

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

This talk: saoghal.net/slides/nordic

Nordic Lattice Meeting 22.8.2022

Aim of today's talk:

A brief introduction to the simulations needed to understand early universe phase 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

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

Focus on 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

Using 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

The electroweak phase transition

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

How to get strong transitions?

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

Lattice Monte Carlo benchmarks

$\Sigma$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

Model-independent parameters bridge the gap

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

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

How the wall moves

• In EWPT: equation of motion is (schematically)
  PRD 46 2668; hep-ph/9503296; arXiv:1407.3132; ... $$ \partial^2 \phi + V_\text{eff}'(\phi,T) + \sum_{i} \frac{d m_i^2}{d \phi} \int \frac{\mathrm{d}^3 k}{(2\pi)^3 \, 2 E_i} \delta f_i(\mathbf{k},\mathbf{x}) = 0$$
  • $V_\text{eff}'(\phi)$: gradient of finite-$T$ effective potential
  • $\delta f_i(\mathbf{k},\mathbf{x})$: deviation from equilibrium phase space density of $i$th species
  • $m_i$: effective mass of $i$th species

Force interpretation

This equation is the realisation of this idea:

Field-fluid system

Using a flow ansatz for the wall-plasma system:

i.e.:

Can simulate as effective model of field $\phi$ + fluid $u^\mu$

Relativistic hydro sim example

But fitting everything in is hard

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

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

arXiv:1910.13125

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

Assumes GW emission stops when nonlinearities form.

arXiv:1910.13125

Nonlinearities?

• 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

Droplets may suppress GWs

Suppression compared to sound waves (redder = worse)

arXiv:1906.00480

3d Kolmogorov turbulence

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

arXiv:2112.12013

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.

Thanks

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