Simulating a strongly first-order thermal phase transition

David Weir [he/him/his] - davidjamesweir

University of Helsinki

This talk: saoghal.net/slides/nordita2019

Nordita, 18 September 2019

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 ...

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

Thermal phase transitions

  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

  • $T_*$, 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

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

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

This equation is the realisation of this idea:

Field-fluid system

Using a flow ansatz for the wall-plasma system:

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

i.e.:

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

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

astro-ph/9309059

What sources GWs?

  • Bubbles nucleate + expand, reaction fronts 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: $$ h^2 \Omega_\text{GW} + h^2 \Omega_\text{sw} + h^2 \Omega_\text{turb}$$
  • Bubbles nucleate + expand, reaction fronts 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: $$ h^2 \Omega_\text{GW} + h^2 \Omega_\text{sw} + h^2 \Omega_\text{turb}$$

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.

Weak transitions

$\alpha_{T_*} \ll 1$

arXiv:1704.05871 ∙ arXiv:1512.06239 ∙ arXiv:1504.03291 ∙ arXiv:1304.2433

Velocity profile development: detonation vs deflagration

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

Simulation slice example

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

Velocity power spectra

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

GW power spectra

[curves scaled by $t$]

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

Strong transitions

$\alpha_{T_*} \sim 1$

arXiv:1906.00480

[with Daniel Cutting, Mark Hindmarsh]

Fluid profile around the wall

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

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

[Movies: Cosmic Defects Channel]

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

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

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

[Movies: Cosmic Defects Channel]

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

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

Aside: GW 'reach' of SM EFT with acoustic results

arXiv:1903.11604