### Gravitational waves

David Weir [they/he] - University of Helsinki - davidjamesweir

This talk: saoghal.net/slides/yeti2022

Durham, 12 July 2022

### Quick quiz

How much do you know about gravitational waves?

• I've never heard of them before [i.e. nothing]
• I know what they are and how they are made
• I've seen them in my GR course
• I've been working on them for a while now [i.e. lots]

You can answer (and ask questions) here: presemo.helsinki.fi/weir

### Assumed knowledge and strategy

• Not too much general relativity
• Focus on ideas relevant to BSM phenomenology
• Mostly qualitative: you can ask me or dive into the references for details

### Learning outcomes

After this lecture you will be able to:

• Describe some of the current and future ways of probing fundamental and particle physics with GWs
• Explain qualitatively how to compute the gravitational waves produced by primordial physics
• Recognise the features and processes involved in an thermal first-order early universe phase transition

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

Credit: Stephan Paul, arXiv:1205.2451

Credit: Stephan Paul, arXiv:1205.2451

Credit: Stephan Paul, arXiv:1205.2451

### What is a gravitational wave?

Stretches and squeezes a ring of matter

$\Leftrightarrow$
Sources: [CC-BY-SA] Nico 0692 on Wikimedia Commons; ESA / C. Carreau

### Q: How are they made?

A: By moving mass and energy around quickly.
[cf. electromagnetic waves, made by moving electrons]

### Q: How are they measured?

A: They change the proper length $L$ between test masses, so producing a strain $\Delta L/L$.

[in fact gravitational waves obey a form of Hooke's law]

### Detecting gravitational waves

LIGO, Virgo, LISA, etc.: compare distances to test masses in two directions with lasers

$\frac{(L_x - L_y)}{L_x + L_y} \simeq h_+(t)$

Source: (CC-BY-NC-ND) Prachatai

### GW170817 neutron star merger

Test of cosmological modified gravity:

• Gamma ray burst $\approx 1.7 \, \mathrm{s}$ after merger
• Speed of gravitational waves $|c^2_T - 1| \lesssim 10^{-15}$
• arXiv:1710.06394
• Subsequent observing runs have updated constraints
• arXiv:2112.06861

### Scales and frequencies

By considering how GWs get redshifted on the way to us, and assuming they get produced at cosmological scales:

arXiv:2008.09136

[What time do you work on? presemo.helsinki.fi/weir]

### LISA is coming!

• Three laser arms, 2.5 M km separation
• ESA-NASA mission, launch 2030s
• Mission exited 'phase A' in December 2021

arXiv:1702.00786

Source: [PD] NASA via Wikimedia Commons

### LISA: Stochastic background?

[qualitative curve, sketched on]

### How LISA will work, briefly

• LISA's arms move + there is additional frequency noise
• Solution is time-delay interferometry (TDI) gr-qc/0409034
• Measure changes in path length between spacecraft
• Cancel laser noise and arm length changes
• Construct e.g. 'Michelson' (ish) variables X, Y, Z

### So how could BSM physics produce a stochastic background?

Today's focus — first-order phase transitions — are a complementary probe of new physics that might be

• Out of sight of particle physics experiments, or
• At higher energy scales than colliders can reach

### SM 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
➥ subsequent processes make gravitational waves

### Out of equilibrium physics

1. Bubbles nucleate and grow
2. Expand in a plasma - create reaction fronts
3. Bubbles + fronts collide
4. Sound waves left behind in plasma
5. Shocks [$\rightarrow$ turbulence] $\rightarrow$ damping

### How the wall moves

• In EWPT: equation of motion is (schematically)

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

• $V_\text{eff}'(\phi)$: 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

### Velocity profile development

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

[What are you? presemo.helsinki.fi/weir]

### 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 to compute GWs?

1. Simulate your non-equilibrium primordial physics (preheating, first order phase transition, etc.)
2. Evolve Lorenz-gauge wave equation in position space $$\nabla^2 h_{ij} (\mathbf{x},t) - \frac{\partial}{\partial t^2} h_{ij}(\mathbf{x},t) = 8 \pi G T_{ij}^\text{source}(\mathbf{x},t)$$ during simulation, with appropriate $T^\text{source}_{ij}$.

### How to compute GWs?

1. Project to TT gauge only when measurement needed: $$h^{\text{TT}}_{ij}(\mathbf{k},t_\text{meas}) = \Lambda_{ij,lm}(\hat{\mathbf{k}}) h^{lm}(\mathbf{k},t)$$
2. Measure energy density in gravitational waves $$\rho_\text{GW}(t_\text{meas}) = \frac{1}{32 \pi G} \left< \dot{h}_{ij}^\text{TT} \dot{h}_{ij}^\text{TT} \right>$$
3. Redshift frequencies and energies to today.

### How are GWs produced at a first order phase transition?

• Not all phase transitions have $v_\mathrm{w} < c$ ...
• 'Vacuum' transitions with no couplings/friction
• 'Run away' transitions (but see arXiv:1703.08215)
• ... but if they do:
• Plasma motion lasts a Hubble time $1/H_*$
• Sound waves turn into turbulence on a time scale
$$\tau_\text{sh} = \frac{R_*}{\overline{U}} = \frac{\text{Bubble radius (i.e. length scale)}}{\text{Typical fluid velocity}}$$

### Using simulation results

Those simulations yield GW spectra like (sound waves):

[NB: curves scaled by $t$: collapse = constant emission]

### What matters is the SNR

[Ignoring astrophysical foregrounds here — sneaky!]
$\text{SNR} = \sqrt{\mathcal{T} \int_{f_\text{min}}^{f_\text{max}} \mathrm{d} f \left[ \frac{h^2 \Omega_\text{GW}(f)}{h^2 \Omega_\text{Sens}(f)}\right]^2}$

### A "pipeline"

Particle physics model
$\Downarrow \mathcal{L}$
Phase transition parameters from phenomenology
$\Downarrow \alpha, \beta, T_N, \ldots$
Real time cosmological simulations
$\Downarrow \Omega_\text{gw}(f)$
Cosmological GW background

### What I want you to remember

• Gravitational waves are an important probe of primordial and fundamental physics
• Phase transitions in extensions of the Standard Model are one source of such GWs

### More questions you can ask me

• How do bubbles nucleate?
• What happens when the plasma becomes turbulent?
• How do you simulate and analyse 'real' LISA data?