What happened in the early universe? when the universe was optically opaque? in dark sectors?

After LIGO comes LISA

• Three laser arms, 2.5 M km separation
• ESA-NASA mission, launch by 2034
• Mission adopted 2017 arXiv:1702.00786

LISA: "Astrophysics" signals

LISA: Early Universe Cosmology

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 ...

We're getting ahead of ourselves...

Electroweak phase transition

• Process by which the Higgs 'switched on'
• In the Standard Model it is a crossover
• Possible in extensions that it would be first order
  ➥ colliding bubbles then make gravitational waves
• Can also be probed with experimental particle physics

Source: arXiv:1206.2942

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.; ...

How? Dimensional reduction

• At high $T$, system looks 3D at distances $\Delta x \gg 1/T$
• Decomposition of fields into Matsubara modes: $$ \phi(\tau, \mathbf{x}) = T \sum_{n=-\infty}^{\infty} \tilde{\phi}_n(\mathbf{x}) e^{i \omega_n \tau}; \quad \omega_n = \begin{cases} 2n \pi T & \text{bosons} \\ (2n +1) \pi & \text{fermions} \end{cases} $$
• Integrate out $\omega_n\neq 0$ due to scale separation
  (heavy modes $\gtrsim \pi T$) $$ Z = \int \mathcal{D} \phi_0 \mathcal{D} \phi_n e^{-S(\phi_0) - S(\phi_0, \phi_n)} \to \int \mathcal{D} \phi_0 e^{-S(\phi_0) - S_\text{eff}(\phi_0)} $$
• Also handles the infrared problem, leaving light fields to be studied nonperturbatively.

Dimensional reduction: sketch

Each step involves matching Green's functions in the effective and full theories to the desired order.

Using the dimensional reduction

• Simulate DR'ed 3D theory on lattice Kajantie et al.
• Applied to SM very successfully in the 1990s:
• NB: if 'new physics' heavy, can be integrated out in DR

DR: xSM example

Out of equilibrium physics

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

How the wall moves

• In EWPT: equation of motion is (schematically)
  PRD 46 2668; hep-ph/9503296; arXiv:1407.3132; ... $$ \partial^2 \phi + V_\text{eff}'(\phi,T) + \sum_{i} \frac{d m_i^2}{d \phi} \int \frac{\mathrm{d}^3 k}{(2\pi)^3 \, 2 E_i} \delta f_i(\mathbf{k},\mathbf{x}) = 0$$
  • $V_\text{eff}'(\phi)$: 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:

Field-fluid system

Using a flow ansatz for the wall-plasma system:

i.e.:

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

Key parameters for GW production

• $T_*$, temperature
  • $T_* \sim 100 \, \mathrm{GeV} \longrightarrow \mathrm{mHz}$ today
• $\alpha_{T_*}$, vacuum energy fraction
  • $\alpha_{T_*} \ll 1$: 'weak'
  • $\alpha_{T_*} \gtrsim 1$: 'strong'
• $v_\mathrm{w}$, bubble wall speed
• $\beta/H_*$, 'duration'
  • $\beta$: inverse phase transition duration
  • $H_*$: Hubble rate at transition

What sources GWs?

Why focus on sound waves?

• "Envelope phase" is short-lived, one-off
  • Relevant only in exotic models where $\alpha_{T_*} \gg 1$ or vacuum transitions...
  • ... but then dynamics aren't so simple arXiv:1802.05712
• "Turbulent phase" may never occur if Hubble damping happens on a shorter timescale
  • Turbulence: bubble radius/average fluid velocity
  • Hubble damping: Hubble time

➤ Focus on the hydrodynamics of the transition
and immediate aftermath.

Velocity profile development: detonation vs deflagration

Simulation slice example

[$\alpha_{T_*} = 0.01$, $v_\mathrm{w} = 0.68$ (detonation)]

Velocity power spectra

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

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

GW power spectra

[curves scaled by $t$]

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

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

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

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

(example, $T_* = 200~\mathrm{GeV}$, $\alpha_{T_*} = 0.1$, $v_\mathrm{w} =0.9$, $\beta/H_* = 50$) $\mathrm{SNR} = 27$ ☺️
[From ptplot.org / arXiv:1910.13125]

PTPlot.org

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

[Here: $Z_2$-symmetric xSM points from arXiv:1910.13125]

GW reach of SM EFT

Fluid profile around the wall

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

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

NB: Deflagration front ends at a shock ($\approx c_\mathrm{s}$)

[Movies: Cosmic Defects Channel]

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

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

NB: Deflagration front ends at a shock ($\approx c_\mathrm{s}$)

[Movies: Cosmic Defects Channel]

Strong simulation slice 1

[$\alpha_{T_*} = 0.5$, $v_\mathrm{w} = 0.92$ (detonation)], velocity $\mathbf{v}$

Fluid profile around the wall

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

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

Strong simulation slice 2

[$\alpha_{T_*} = 0.5$, $v_\mathrm{w} = 0.44$ (deflag.)], velocity $\mathbf{v}$

Strong deflagrations: walls slow

At large $\alpha_{T_*}$ reheated droplets form in front of the walls

Strong simulation slice 3

[$\alpha_{T_*} = 0.34$, $v_\mathrm{w} = 0.24$ (deflag.)], vorticity $\nabla \times \mathbf{v}$

Strong deflagrations: vortical modes

GW production can be suppressed

Suppression relative to LISA CosWG ansatz

🚰 A pipeline

1. Choose your model (e.g. SM, xSM, 2HDM, ...)
2. Dim. red. model
   Kajantie et al.
3. Phase diagram ($\alpha_{T_*}$, $T_*$);
   lattice: Kajantie et al.
4. Nucleation rate ($\beta$) + Sphaleron rate;
   lattice: Moore and Rummukainen
5. Wall velocities ($v_\text{wall}$)
   Moore and Prokopec; Kozaczuk
6. GW power spectrum $\Omega_\mathrm{gw}$

🔧 Not watertight, even for SM!

Conclusions

• Weak transitions: good estimates of power spectrum
  ☛ ptplot.org
• Strong transitions still an emerging topic:
  ☛ acoustic gravitational wave production suppressed
• Causes: vortical mode production and slower walls
• Worst for large $\alpha_{T_*}$, small $v_\mathrm{w}$
• Turbulence still a challenge, work ongoing
• In any case: LISA provides a model-independent probe of first-order phase transitions around $100~\mathrm{GeV}$
  ☛ ... but new physics must be dynamical