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

This talk: saoghal.net/slides/yeti2022

Durham, 12 July 2022

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

- 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

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

Credit: Stephan Paul, arXiv:1205.2451

Credit: Stephan Paul, arXiv:1205.2451

Credit: Stephan Paul, arXiv:1205.2451

gravitational waves help?

Stretches and squeezes a ring of matter

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

A: By moving mass and energy around quickly.

[cf. electromagnetic waves, made by moving electrons]

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]

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

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

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]

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

Source: [PD] NASA via Wikimedia Commons

Source: arXiv:1702.00786

[qualitative curve, sketched on]

- 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

Source: arXiv:1908.00546

Today's focus — **first-order phase transitions** — are a

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

- 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

arXiv:hep-ph/9605288 ; arXiv:hep-lat/9704013; arXiv:hep-ph/9809291

- Bubbles nucleate and grow
- Expand in a plasma - create reaction fronts
- Bubbles + fronts collide
**Sound waves**left behind in plasma- Shocks [$\rightarrow$ turbulence] $\rightarrow$ damping

- 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

This equation is the realisation of this idea:

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/9309059Deflagration $v_\mathrm{w} < c_\mathrm{s}$

Detonation $v_\mathrm{w} > c_\mathrm{s}$

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

- $T_*$, 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

- Simulate your non-equilibrium primordial physics (preheating, first order phase transition, etc.)
- 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 $T^\text{source}_{ij}$.

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

- 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

$$\tau_\text{sh} = \frac{R_*}{\overline{U}} = \frac{\text{Bubble radius (i.e. length scale)}}{\text{Typical fluid velocity}}$$

Those simulations yield GW spectra like (sound waves):

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

[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} $

**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**

- 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

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