Higgs physics and cosmology:
Gravitational waves from the EWPT


David J. Weir - University of Helsinki - davidjamesweir

cHarged 2018±

What happened when the universe was optically opaque?

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 ...

Electroweak phase transition

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 bubble wall moves

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

Phys. Rev. D 46, 2668

This equation is the realisation of this idea:

Yet another interpretation:

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


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

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

Detonations vs deflagrations

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

Velocity profile development: detonation vs deflagration

Simulation slice example

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

CosWG report arXiv:1512.06239

  • Results for a variety of models, at "benchmark points"
  • Key result: parametric plots with contours at $\mathrm{SNR}_\text{thr}$

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

Current EWPT work in LISA CosWG

  • In preparation: update to first report on PTs (arXiv:1512.06239)
    • "Final" sensitivity curve
    • Updated model 'showcase'
    • New theoretical work (including no runaways)
  • PTPlot web tool for computing SNR
    • Modular, containerised
    • Code will be open, can be run locally
  • Coming soon 😅


Recent results 1: parameter space

  • "Non-perturbative" results for triplet model arXiv:1802.10500
  • Dimensional reduction, mapping to existing theory
  • Light green region - first order phase transition
  • Dark green + gray regions - new simulations required

Recent results 2: spectral shape

  • Each simulation: ~1M CPU hours arXiv:1704.05871
  • Validate spectral shape used in WG reports

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 ($\beta$);
    lattice: Moore and Rummukainen
  5. Wall velocities ($v_\text{wall}$)
    Moore and Prokopec; Kozaczuk
  6. GW power spectrum $\Omega_\mathrm{gw}$
  7. Sphaleron rate

Very leaky, even for SM!

What I am thinking about

  • Turbulence
    • MHD or no MHD?
    • Timescales $H_* R_*/\overline{U}_\mathrm{f} \sim 1$, sound waves and turbulence?
    • More simulations needed?

  • Complementarity of GW signal and BG
    • 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?

Final conclusion

  • Now have good understanding of thermal history of first-order electroweak phase transitions
  • 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 $100~\mathrm{GeV}$