What happened during the electroweak phase transition? when the universe was optically opaque? in dark sectors?

What's next: LISA mission

• Three laser arms, 2.5 M km separation
• ESA-NASA mission, launch by 2034
• Proposal submitted last year arXiv:1702.00786
• Officially adopted on 20.6.2017

LISA Pathfinder

Exceeded design expectations by factor of five!

Possible signals

Key science for LISA

Science Investigation 7.2: Measure, or set upper limits on, the spectral shape of the cosmological stochastic GW background.

Operational Requirement 7.2: Probe a broken power-law stochastic background from the early Universe as predicted, for example, by first order phase transitions ...

First order thermal phase transition

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

Key parameters for GW production

4 numbers parametrise the transition:

• $T_*$, temperature ($\approx T_\mathrm{n} \lesssim T_\mathrm{c}$)
• $\alpha_{T_*}$, vacuum energy fraction
• $v_\mathrm{w}$, bubble wall speed
• $\beta/H_*$:
  • $\beta$, inverse phase transition duration
  • $H_*$, Hubble rate at transition

How the wall moves

• In EWPT: 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:

Force interpretation

This equation is the realisation of this idea:

Fluid interpretation

Yet another interpretation:

i.e.:

Can simulate as effective model of field $\phi$ + fluid $u^\mu$.

What the makes the GWs 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?

Hope: envelope approximation

• Finite $v_\mathrm{w}$
• 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

Hope: Envelope approximation

Source: arXiv:1604.08429

When $v_\mathrm{w}$ finite, nice $k^{-1}$ power law - agrees with lattice

Reality: Vacuum transitions (large $v_\mathrm{w}$)

Source: arXiv:1802.05712

When $v_\mathrm{w}$ ultrarelativistic, power law steeper

Reality: Envelope approximation

• For most transitions: most energy in plasma
• Ultra-relativistic: disagrees with envelope approximation

Claim: Envelope approximation can be safely ignored

Step 1: Envelope phase?

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 $\alpha_{T_*}$
  • $\beta/H_*$ ↔ number of bubbles, $v_\text{wall}$ ↔ $\tilde \eta$

What sort of solutions does this system have?

Reaction front

• At a reaction front: chemical transformation. Chemically and physically different on each side.
• Different from a shock front, where the energy density and entropy change.
• We have a reaction front as $\langle \phi \rangle = 0$ before and $\langle \phi \rangle \neq 0$ after.

Detonations vs deflagrations

• If $\phi$ wall moves supersonically and the fluid $u^\mu$ enters the wall at rest, we have a detonation
  ☛ Generally good for GWs
• If $\phi$ wall moves subsonically and the fluid $u^\mu$ enters the wall at its maximum velocity, it's a deflagration
  ☛ Generally bad for GWs

Velocity profile development: detonation vs deflagration

Simulation slice example

Velocity power spectra

$v_\mathrm{w} < c_\mathrm{s}$

$v_\mathrm{w} > c_\mathrm{s}$

Computing $\Omega_\text{GW}$

• Simply evolve numerically: $$ \square \, h_{ij}(x,t) = 16 \pi G \, T_{ij}^\text{source}(x,t). $$
• When $T_{ij}^\text{source} = (\epsilon + p) u_i u_j$ ➸ convolution of fluid velocity
  arXiv:0909.0622
• To measure the energy in gravitational waves, project out transverse-traceless part: $$ t_{\mu\nu}^\text{GW}= \frac{1}{32 \pi G}\langle \partial_\mu h^\text{TT}_{ij} \partial_\nu h^\text{TT}_{ij} \rangle; \quad \rho_\text{GW} = \frac{1}{32 \pi G} \langle \dot{h}^\text{TT}_{ij} \dot{h}^\text{TT}_{ij} \rangle. $$
• We can then redshift this to present day to get $\Omega_\text{GW} h^2$.

GW power spectra and power laws

$v_\mathrm{w} < c_\mathrm{s}$

$v_\mathrm{w} > c_\mathrm{s}$

NB: curves scaled by $t$

Step 1: Envelope phase?

Step 2: Acoustic phase 👍

Step 3: Turbulence 🤔

Shocks and turbulence?

• Require longer timescales (fluid turnover time) $R_*/\overline{U}_\mathrm{f}$,
  thus: may not develop at all
• Plenty of theoretical results, but little agreement
  arXiv:0705.1733; arXiv:0909.0622; arXiv:1510.02985; ...

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

• For any given theory, can get $T_*$, $\alpha_{T_*}$, $\beta/H_*$, $v_\mathrm{w}$ arXiv:1004.4187
• It's then easy to predict the signal...

(example, $T_* = 94.7~\mathrm{GeV}$, $\alpha_{T_*} = 0.066$, $v_\mathrm{w} =0.95$, $\beta/H_* = 105.9$) $\mathrm{SNR} = 95$ ☺️

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)

PTPlot.org

Update this plot?

Final conclusion

• Now have good understanding of thermal history of first-order thermal phase transitions ([near-]vacuum still developing).
• Can make good estimates of the GW power spectrum.
• Turbulence still a challenge.
• Recently appreciated contributions, like acoustic waves, enhance the source considerably.
• LISA provides a model-independent probe of first-order phase transitions around $100~\mathrm{GeV}$.