Higgs and cosmology:

Gravitational waves from a first-order
electroweak phase transition

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

This talk: saoghal.net/slides/blois2023

34th Rencontres de Blois

LISA is coming!

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

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:

Event Time (s) Temp (GeV) $\mathbf{g}_*$ Frequency (Hz)
QCD phase transition $10^{-3}$ $0.1$ $\sim 10$ $10^{-8}$
EW phase transition $10^{-11}$ $100$ $\sim 100$ $10^{-5}$ LISA!
??? $10^{-25}$ $10^9$ $\gtrsim 100$ $100$
End of inflation $\gtrsim 10^{-36}$ $\lesssim 10^{16}$ $\gtrsim 100$ $\gtrsim 10^8$

[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

[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

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

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
    ➥ 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

When new physics is heavy

LISA SNR curves
Inverse phase
transition duration
Phase transition strength
  • Comparison at benchmark point in minimal SM
  • Compare: ● 4d PT vs ● 3d PT vs ● NP (= lattice)
arXiv:1903.11604

How to get strong transitions?

GW signal may be
observable
in this corner
Inverse phase
transition duration
Phase transition strength
  • Theories that look SM-like in the IR ⇒ not observable!
  • But what happens with additional light fields?
arXiv:1903.11604

Lattice Monte Carlo benchmarks

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

Perturbation theory out by 10% or more!

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 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?]

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
      $$\tau_\text{sh} = \frac{R_*}{\overline{U}} = \frac{\text{Bubble radius (i.e. length scale)}}{\text{Typical fluid velocity}}$$

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!

Square of the
inverse ratio of this... ➘
...to this... ➚
... integrated over frequency ↓ and time.

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

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

Consequences of droplets

  • May suppress GWs (redder = worse) arXiv:1906.00480
    Worst
    case
    scenario ➘
    Wall velocity
    Peak fluid 3-velocity
  • May enhance baryogenesis(?) by slowing walls down
  • Further 3D studies needed...

Thanks

  • Students:
    Jani Dahl, Jenni Häkkinen, Anna Kormu, Tiina Minkkinen, 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

What I want you to remember

  • Early universe processes can probe BSM physics
    ... but we need precise predictions of key parameters $\Rightarrow$ lattice Monte Carlo simulations of phase transitions
  • Nonlinearities matter when studying phase transitions
    $\Rightarrow$ large-scale real-time cosmological simulations

Some questions you can ask me

  • How accurate are bubble nucleation calculations?
  • What are the consequences of droplet formation?
  • What about turbulence? How does that affect things?