Primordial gravitational waves from first-order phase transitions

David J. Weir
University of Helsinki

This talk: saoghal.net/slides/cau2023

2023 Chung-Ang University Beyond the Standard Model Workshop

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

Credit: Stephan Paul, arXiv:1205.2451

Credit: Stephan Paul, arXiv:1205.2451

Credit: Stephan Paul, arXiv:1205.2451

How could
gravitational waves help?

LISA is coming!

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

arXiv:1702.00786

LISA: "Astrophysics" signals

LISA: Stochastic background?

[qualitative curve, sketched on]

Scales and frequencies

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

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

[what BSM physics might there be?]
Particle physics model
$\Downarrow \mathcal{L}_{4\mathrm{d}}$
Dimensional reduction
$\Downarrow \mathcal{L}_{3\mathrm{d}}$
Phase transition parameters
from lattice simulations
$\Downarrow \alpha, \beta, T_N, v_\mathrm{w}, \ldots$
Real time cosmological simulations
$\Downarrow \Omega_\text{gw}(f)$
Cosmological GW background
[what would we see as a result?]

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

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

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

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

When new physics is heavy

• Comparison at benchmark point in minimal SM
• Compare: ● 4d PT vs ● 3d PT vs ● NP (= lattice)

How to get strong transitions?

• Theories that look SM-like in the IR ⇒ not observable!
• But what happens with additional light fields?

Lattice Monte Carlo benchmarks

Need for accuracy: $\Sigma$SM (triplet) example arXiv:2005.11332

Perturbation theory out by 10% or more!

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

Particle physics model ✅
$\Downarrow \mathcal{L}_{4\mathrm{d}}$
Dimensional reduction ✅
$\Downarrow \mathcal{L}_{3\mathrm{d}}$
Phase transition parameters from lattice simulations ✅
$\Downarrow \alpha, \beta, T_N, v_\mathrm{w}, \ldots$
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

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 $v_\mathrm{w} < c$ ...
  • 'Vacuum' transitions with no couplings/friction
  • 'Run away' transitions arXiv:1703.08215
• ... but if they do:
  • Plasma motion lasts a Hubble time $1/H_*$
  • Fluid motion becomes nonlinear on a time scale
    $$\tau_\text{sh} = \frac{R_*}{\overline{U}} = \frac{\text{Bubble radius (i.e. length scale)}}{\text{Typical fluid velocity}}$$

Using simulation results

Those simulations yield GW spectra like (sound waves):

[NB: curves scaled by $t$: collapse = constant emission]

What matters is the SNR

$\text{SNR} = \sqrt{\mathcal{T} \int_{f_\text{min}}^{f_\text{max}} \mathrm{d} f \left[ \frac{h^2 \Omega_\text{GW}(f)}{h^2 \Omega_\text{Sens}(f)}\right]^2} $

Still need to handle astrophysical foregrounds properly!

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

arXiv:2204.03396

Droplets may suppress GWs

Suppression compared to sound waves (redder = worse)

arXiv:1906.00480

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, 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.

Thanks

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