GWs from first-order phase transitions

David J. Weir, University of Helsinki

NORDITA, 5 July 2017

arXiv:1705.01783 and references therein

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 arXiv:1702.00786
  • Officially adopted on 20.6.2017

From the LISA 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: $$ h^2 \Omega_\text{GW} \approx h^2 \Omega_\phi + h^2 \Omega_\text{sw} + h^2 \Omega_\text{turb}$$
  • Equation of motion is (schematically)
    Liu, McLerran and Turok; Prokopec and Moore $$ \partial_\mu \partial^\mu \phi + V_\text{eff}'(\phi,T) + \sum_{i} \frac{d m_i^2}{d \phi} \int \frac{\mathrm{d}^3 k}{(2\pi)^3 2 E_i} \delta f_i(\mathbf{k},\mathbf{x}) = 0$$
    • $V_\text{eff}'(\phi)$: gradient of finite-$T$ effective potential
    • $\delta f_i(k,x)$: deviation from equilibrium phase space density of $i$th species
    • $m_i$: effective mass of $i$th species:
      • Leptons: $m^2 = y^2 \phi^2/2$
      • Gauge bosons: $m^2 = g_w^2 \phi^2/4$
      • Also Higgs and pseudo-Goldstone modes

Put another way:

$$ \overbrace{\partial_\mu T^{\mu\nu}}^\text{Force on $\phi$} - \overbrace{\int \frac{d^3 k}{(2\pi)^3} f(\mathbf{k}) F^\nu }^\text{Force on particles}= 0 $$

This equation is the realisation of this idea:

Yet another interpretation:

$$ \overbrace{\partial_\mu T^{\mu\nu}}^\text{Field part} - \overbrace{\int \frac{d^3 k}{(2\pi)^3} f(\mathbf{k}) F^\nu }^\text{Fluid part}= 0 $$


$$ \partial_\mu T^{\mu\nu}_\phi + \partial_\mu T^{\mu\nu}_\text{fluid} = 0 $$

We will return to this later!

Envelope approximation

Envelope approximation

Kosowsky, Turner and Watkins; Kamionkowski, Kosowsky and Turner
  • 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 near-vacuum/runaway transitions]

Envelope approximation

Coupled field and fluid system

Ignatius, Kajantie, Kurki-Suonio and Laine
  • Scalar $\phi$ and ideal fluid $u^\mu$:
    • Split stress-energy tensor $T^{\mu\nu}$ into field and fluid bits $$\partial_\mu T^{\mu\nu} = \partial_\mu (T^{\mu\nu}_\phi + T^{\mu\nu}_\text{fluid}) = 0$$
    • Parameter $\eta$ sets the scale of friction due to plasma $$\partial_\mu T^{\mu\nu}_\phi = \tilde \eta \frac{\phi^2}{T} u^\mu \partial_\mu \phi \partial^\nu \phi \quad \partial_\mu T^{\mu\nu}_\text{fluid} = - \tilde \eta \frac{\phi^2}{T} u^\mu \partial_\mu \phi \partial^\nu \phi $$
    • $V(\phi,T)$ 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 $\tilde \eta$

Cutting [Masters dissertation]

Simulation slice example

Velocity power spectra and power laws

Fast deflagration

  • 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

  • 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 - $h^2 \Omega_\text{gw}$ arXiv: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 arXiv: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?