Gravitational waves from the sound of a first-order phase transition

David J. Weir - University of Helsinki - davidjamesweir

This talk: saoghal.net/slides/desy

DESY, 19 February 2018

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

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

Source: Morrissey and Ramsey-Musolf

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 $159.5 \pm 1.5 \, \mathrm{GeV}$

Source: D'Onofrio and Rummukainen

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
few parameters, not yet widely studied, CDM candidate?

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

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:

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

This equation is the realisation of this idea:

Yet another interpretation:

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

i.e.:

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

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

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

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; $\alpha_{T_*}$ ↔ $\mathcal{L}$, $v_\text{wall}$ ↔ $\tilde \eta$

What sort of solutions does this system have?

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

Cutting [Masters dissertation]

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

Step 3: Turbulence

Source: Wikimedia Commons/Gary Settles (CC-BY-SA)

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)

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?