Gravitational waves
from the early universe

David J. Weir - University of Helsinki - davidjamesweir

Particle Physics Day, Jyväskylä, 23.11.2018

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

Start of the GW astrophysics era

(CC-BY) arXiv:1710.05833

Cosmological sources

Early universe processes that could produce observable GWs:

  • Inflation (and how it ended)
    - CMB experiments? see Elina's talk
  • Cosmic strings and other defects
    - see Asier's talk
  • First-order phase transitions
    - this talk!

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 for LISA

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 ...

Let's focus on these first-order phase transitions...

Electroweak phase transition

  • 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

Thermal phase transition: what, when?

  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

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

This equation is the realisation of this idea:

Field-fluid system

Using a flow ansatz for the wall-plasma system:

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


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

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
    ☛ 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

What the makes the GWs at a first-order 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: $$ h^2 \Omega_\text{GW} h^2 \Omega_\text{sw} + h^2 \Omega_\text{turb}$$

Velocity profile development: detonation vs deflagration

Simulation slice example

Velocity power spectra

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

GW power spectra and power laws

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

NB: curves scaled by $t$

Shocks and turbulence?

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

From (beta!)

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

Final conclusion

  • Now have good understanding of thermal history of first-order thermal phase transitions.
  • Can make good estimates of the GW power spectrum.
  • Turbulence still a challenge.
  • Acoustic waves can enhance the source considerably.
  • LISA provides a model-independent probe of first-order phase transitions around $100~\mathrm{GeV}$.