### Gravitational waves from the early universe

#### saoghal.net/slides/ppd2018/

David J. Weir - University of Helsinki - davidjamesweir

Particle Physics Day, Jyväskylä, 23.11.2018

#### Start of the GW astrophysics era

Source:
(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

### LISA Pathfinder

Exceeded design expectations by factor of five!

### 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

• 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)
$$\partial_\mu \partial^\mu \phi + V_\text{eff}'(\phi,T) + \sum_{i} \frac{d m_i^2}{d \phi} \int \frac{\mathrm{d}^3 k}{(2\pi)^3 \, 2 E_i} \delta f_i(\mathbf{k},\mathbf{x}) = 0$$
• $V_\text{eff}'(\phi)$: 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:

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

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

## 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

### 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 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?

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

#### 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 ptplot.org (beta!)

### PTPlot.org

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 $100~\mathrm{GeV}$.