Gravitational waves from a first order
electroweak phase transition

David J. Weir, University of Helsinki

UMass Amherst, 8 April 2017

https://tinyurl.com/acfi-gws

Lots of sources...

Source: http://rhcole.com/apps/GWplotter/

LISA Pathfinder

Exceeded design expectations by factor of five!

What's next: LISA

• LISA: three arms (six laser links), 2.5 M km separation
• Launch as ESA’s third large-scale mission (L3) in (or before) 2034
• Proposal officially submitted earlier this year 1702.00786

From the proposal:

First order thermal phase transition:

1. Bubbles nucleate and grow
2. Expand in a plasma - create shock waves
3. Bubbles + shocks collide - violent process
4. Sound waves left behind in plasma
5. Turbulence; expansion

Thermal phase transitions

• Standard Model is a crossover
  Kajantie et al; Karsch et al.; ...
• First order possible in extensions (xSM, 2HDM, ...)
  Andersen et al., Kozaczuk et al., Carena et al.,
  Bödeker et al., Damgaard et al., Ramsey-Musolf et al.,
  Cline and Kainulainen, ...
• Baryogenesis?
• GW PS ⇔ model information?

What the metric sees at a thermal phase transition

• Bubbles nucleate and expand, shocks form, then:
  1. $h^2 \Omega_\phi$: Bubbles + shocks collide - 'envelope phase'
  2. $h^2 \Omega_\text{sw}$: Sound waves set up - 'acoustic phase'
  3. $h^2 \Omega_\text{turb}$: [MHD] turbulence - 'turbulent phase'


• Sources add together to give observed GW power: $$ h^2 \Omega_\text{GW} \approx h^2 \Omega_\phi + h^2 \Omega_\text{sw} + h^2 \Omega_\text{turb}$$

Envelope approximation

Envelope approximation

• Thin, hollow bubbles, no fluid
• Stress-energy tensor $\propto R^3$ on wall
• Solid angle: overlapping bubbles → GWs
• Simple power spectrum:
  • One length scale (average radius $R_*$)
  • Two power laws ($\omega^3$, $ \sim \omega^{-1}$)
  • Amplitude
  ⇒ 4 numbers define spectral form

NB: Used to be applied to shock waves (fluid KE),
now only use for bubble wall (field gradient energy)

Envelope approximation

4-5 numbers parametrise the transition:

• $\alpha_{T_*}$, vacuum energy fraction
• $v_\mathrm{w}$, bubble wall speed
• $\kappa_\phi$, conversion 'efficiency' into gradient energy $(\nabla \phi)^2$
• Transition rate:
  • $H_*$, Hubble rate at transition
  • $\beta$, bubble nucleation rate
  → ansatz for $h^2 \Omega_\phi$

[only matters for vacuum/runaway transitions]

Envelope approximation

Coupled field and fluid system

• Scalar $\phi$ and ideal fluid $u^\mu$:
  • Split stress-energy tensor $T^{\mu\nu}$ into field and fluid bits $$\partial_\mu T^{\mu\nu} = \partial_\mu (T^{\mu\nu}_\text{field} + T^{\mu\nu}_\text{fluid}) = 0$$
  • Parameter $\eta$ sets the scale of friction due to plasma $$\partial_\mu T^{\mu\nu}_\text{field} = \tilde \eta \frac{\phi^2}{T} u^\mu \partial_\mu \phi \partial^\nu \phi \quad \partial_\mu T^{\mu\nu}_\text{fluid} = - \tilde \eta \frac{\phi^2}{T} u^\mu \partial_\mu \phi \partial^\nu \phi $$
  • $V(\phi,T)$ is a 'toy' potential tuned to give latent heat $\mathcal{L}$
  • $\beta$ ↔ number of bubbles; $\alpha_{T_*}$ ↔ $\mathcal{L}$, $v_\text{wall}$ ↔ $\tilde \eta$

Begin in spherical coordinates: what sort of solutions does this system have?

Velocity profile development: small $\tilde \eta$ ⇒ detonation (supersonic wall)

Velocity profile development: large $\tilde \eta$ ⇒ deflagration (subsonic wall)

$v_\mathrm{w}$ as a function of $\tilde \eta$

Simulation slice example

Velocity power spectra and power laws

Fast deflagration

Detonation

• Weak transition: $\alpha_{T_*} =0.01$
• Power law behaviour above peak is between $k^{-2}$ and $k^{-1}$
• “Ringing” due to simultaneous nucleation, unimportant

GW power spectra and power laws

Fast deflagration

Detonation

• Causal $k^3$ at low $k$, approximate $k^{-3}$ or $k^{-4}$ at high $k$
• Curves scaled by $t$: source until turbulence/expansion

→ power law ansatz for $h^2 \Omega_\text{sw}$

Transverse versus longitudinal modes – turbulence?

• Short simulation; weak transition (small $\alpha$): linear; most power in longitudinal modes ⇒ acoustic waves, turbulent
• Turbulence requires longer timescales $R_*/\overline{U}_\mathrm{f}$
• Plenty of theoretical results, use those instead
  Kahniashvili et al.; Caprini, Durrer and Servant; Pen and Turok; ...

→ power law ansatz for $h^2 \Omega_\text{turb}$

Putting it all together - $h^2 \Omega_\text{gw}$ 1512.06239

• Three sources, $\approx$ $h^2\Omega_\phi$, $h^2\Omega_\text{sw}$, $h^2 \Omega_\text{turb}$
• Know their dependence on $T_*$, $\alpha_T$, $v_\mathrm{w}$, $\beta$
  Espinosa, Konstandin, No, Servant
• Know these for any given model, predict the signal...

(example, $T_* = 100 \mathrm{GeV}$, $\alpha_{T_*} = 0.5$, $v_\mathrm{w} =0.95$, $\beta/H_* = 10$)

Putting it all together - physical models to GW power spectra

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

... which tells you if it is detectable by LISA (see 1512.06239)

Detectability from acoustic waves alone

• In many cases, sound waves dominant
• Parametrise by RMS fluid velocity $\overline{U}_\mathrm{f}$ and bubble radius $R_*$ (quite easily obtained Espinosa, Konstandin, No and Servant)

Sensitivity plot:

The pipeline

Questions, requests or demands...

• Turbulence
  • MHD or no MHD?
  • Timescales $H_* R_*/\overline{U}_\mathrm{f} \sim 1$, sound waves and turbulence?
  • More simulations needed?
  
  
• Interaction with baryogenesis
  • Competing wall velocity dependence of BG and GWs?
  • Sphaleron rates in extended models?
  
  
• The best possible determinations for xSM, 2HDM, $\Sigma$SM, ...
  • What is the phase diagram?
  • Nonperturbative nucleation rates?

Dynamic range issues

• Most realtime lattice simulations in the early universe have a single [nontrivial] length scale
• Here, many length scales important
• Recently completed simulations with $4200^3$ lattices, $\delta x = 2/T_\mathrm{c}$ → approx 1M CPU hours each (17.6M total)

Implementation: special relativistic hydrodynamics

Different things live in different places...

With this discretisation, evolution is second-order accurate!