David J. Weir - University of Helsinki - davidjamesweir

This talk: saoghal.net/slides/desy

DESY, 19 February 2018

Source: GWplotter

Exceeded design expectations by factor of five!

But that's not all...

- 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

- 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

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.*;
...

- 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 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?]

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

Source: D'Onofrio and Rummukainen

- 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)

- 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

- Bubbles nucleate and grow
- Expand in a plasma - create shock waves
- Bubbles + shocks collide - violent process
- Sound waves left behind in plasma
- Turbulence; expansion

- 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.:

$$ \partial_\mu T^{\mu\nu}_\phi + \partial_\mu T^{\mu\nu}_\text{fluid} = 0 $$We will return to this later!

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

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

- 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

**NB:** Used to be applied to shock waves (fluid KE),

now only use for bubble wall (field gradient energy)

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

- 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?

Cutting [Masters dissertation]

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

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

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

- 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 $h^2 \Omega_\text{turb}$

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

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

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

- Choose your model

(e.g. SM, xSM, 2HDM, ...) - Dim. red. model Kajantie et al.
- Phase diagram ($\alpha_{T_*}$, $T_*$);

lattice: Kajantie et al. - Nucleation rate ($\beta$);

lattice: Moore and Rummukainen - Wall velocities ($v_\text{wall}$)

Moore and Prokopec; Kozaczuk - GW power spectrum $\Omega_\mathrm{gw}$
- Sphaleron rate

Very leaky, even for SM!

- 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?