Lots of sources...

Source: GWplotter

LISA Pathfinder

Exceeded design expectations by factor of five!

LISA Pathfinder

But that's not all...

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 submitted a year ago 1702.00786
• Officially adopted on 20.6.2017

From the proposal:

Electroweak phase transition

• This is the process by which the Higgs 'turned on'
• In the minimal Standard Model it is gentle (crossover)
• It is possible (and theoretically attractive) in extensions that it would experience a first order phase transition

EW PT in the SM

Work in the 1990s found this phase diagram for the SM:

At $m_H = 125 \, \mathrm{GeV}$, SM is a crossover
Kajantie et al.; Gurtler et al.; Csikor et al.; ...

Dimensional reduction

• At high $T$, system looks 3D for long distance physics
  (with length scales $\Delta x \gg 1/T$)
• Decomposition of fields: $$ \phi(x,\tau) = \sum_{n=-\infty}^{\infty} \phi_n(x) e^{i \omega_n \tau}; \qquad \omega_n = 2\pi n T $$
• Then integrate out $n\neq 0$ Matsubara modes due to the scale separation $$ \begin{align} Z & = \int \mathcal{D} \phi_0 \mathcal{D} \phi_n e^{-S(\phi_0) - S(\phi_0, \phi_n)} \\ & = \int \mathcal{D} \phi_0 e^{-S(\phi_0) - S_\text{eff}(\phi_0)} \end{align}$$
• The 3D theory (with most fields integrated out) is easier to study, has fewer parameters!

Using the dimensional reduction

• Using the DR'ed 3D theory, can study nonperturbatively with lattice simulations.
• This was done very successfully in the 1990s for the Standard Model:
• [Q: Can we map any other theories to the same 3D model?]

SM is a Crossover

At $m_H = 125 \, \mathrm{GeV}$, critical temperature is $159.5 \pm 1.5 \, \mathrm{GeV}$

SM is a crossover: consequences

• No real departure from thermal equilibrium
  ⇒ no significant GWs or baryogenesis
• Many alternative mechanisms for baryogenesis exist
  • Leptogenesis (add RH neutrinos, see-saw mechanism, additional leptons produced by RH neutrino decays)
  • Cold electroweak baryogenesis (non-equlibrium physics given by supercooled initial state)
  but let us instead consider additional fields which would yield a first order phase transition.

SM extensions with 1.PT

• Higgs singlet model - add extra real singlet field $\sigma$:
  quite difficult to rule out with colliders
• Two Higgs doublet model - add second complex doublet:
  many parameters, but already quite constrained
• Triplet models - add adjoint scalar field (triplet):
  few parameters, not yet widely studied, CDM candidate?

All these have unexcluded regions of parameter space for which the phase transition is first order

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

How the bubble wall moves

• Equation of motion is (schematically)
  Liu, McLerran and Turok; Prokopec and Moore; Konstandin, Nardini and Rues; ... $$ \partial_\mu \partial^\mu \phi + V_\text{eff}'(\phi,T) + \sum_{i} \frac{d m_i^2}{d \phi} \int \frac{\mathrm{d}^3 k}{(2\pi)^3 \, 2 E_i} \delta f_i(\mathbf{k},\mathbf{x}) = 0$$
  • $V_\text{eff}'(\phi)$: gradient of finite-$T$ effective potential
  • $\delta f_i(\mathbf{k},\mathbf{x})$: 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:

This equation is the realisation of this idea:

Yet another interpretation:

i.e.:

We will return to this later!

Gravitational waves from a thermal phase transition

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: $$ h^2 \Omega_\text{GW} \approx h^2 \Omega_\phi + h^2 \Omega_\text{sw} + h^2 \Omega_\text{turb}$$

Step 1: Envelope phase

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 $(\nabla \phi)^2$
• $\beta/H_*$:
  • $\beta$, inverse phase transition duration
  • $H_*$, Hubble rate at transition
  → ansatz for $h^2 \Omega_\phi$

Envelope approximation

Step 2: Acoustic phase

Coupled field and fluid system

• Scalar $\phi$ and ideal fluid $u^\mu$:
  • Split stress-energy tensor $T^{\mu\nu}$ into field and fluid bits $$\partial_\mu T^{\mu\nu} = \partial_\mu (T^{\mu\nu}_\phi + T^{\mu\nu}_\text{fluid}) = 0$$
  • 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; $\alpha_{T_*}$ ↔ $\mathcal{L}$, $v_\text{wall}$ ↔ $\tilde \eta$

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

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 $h^2 \Omega_\text{sw}$

Step 3: Turbulence

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

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

The pipeline

Next steps...

• 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?