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 ...
We're getting ahead of ourselves...
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
➥ colliding bubbles then make gravitational waves
Can also be probed with experimental particle physics
Work in the 1990s found this phase diagram for the SM:
At $m_H = 125 \, \mathrm{GeV}$, SM is a crossover Kajantie et
al.; Gurtler et al.;
Csikor et al.;
...
How? Dimensional reduction
At high $T$, system looks 3D at distances $\Delta x \gg 1/T$
Decomposition of fields into Matsubara modes:
$$ \phi(\tau, \mathbf{x}) = T \sum_{n=-\infty}^{\infty}
\tilde{\phi}_n(\mathbf{x}) e^{i \omega_n \tau}; \quad \omega_n =
\begin{cases}
2n \pi T & \text{bosons} \\
(2n +1) \pi & \text{fermions}
\end{cases}
$$
Integrate out $\omega_n\neq 0$
due to scale separation (heavy modes
$\gtrsim \pi T$)
$$
Z = \int \mathcal{D} \phi_0 \mathcal{D} \phi_n
e^{-S(\phi_0) - S(\phi_0, \phi_n)} \to \int \mathcal{D} \phi_0 e^{-S(\phi_0) -
S_\text{eff}(\phi_0)} $$
Also handles the infrared problem, leaving light
fields to be studied nonperturbatively.
Dimensional reduction: sketch
Each step involves matching Green's functions in the
effective and full theories to the desired order.
Using the dimensional reduction
Simulate DR'ed 3D theory
on lattice Kajantie et al.
Applied to SM very successfully in the 1990s:
NB: if 'new physics' heavy, can be integrated out
in DR
DR: xSM example
[$Z_2$-symmetric; plot of trilinear higgs-singlet
coupling $\lambda_{221}$ vs. singlet mass eigenstate $m_2$] Source: arXiv:1903.11604
[with Oliver Gould, Jonathan Kozaczuk, Lauri Niemi, Michael
Ramsey-Musolf and Tuomas Tenkanen]
Phase diagram ($\alpha_{T_*}$, $T_*$);
lattice: Kajantie et
al.
Nucleation rate ($\beta$) + Sphaleron
rate;
lattice: Moore and
Rummukainen
Wall velocities ($v_\text{wall}$) Moore and
Prokopec; Kozaczuk
GW power spectrum
$\Omega_\mathrm{gw}$
🔧 Not watertight, even for SM!
Conclusions
Weak transitions: good estimates of
power spectrum
☛ ptplot.org
Strong transitions still an emerging topic:
☛ acoustic gravitational wave production suppressed
Causes: vortical mode production and slower walls
Worst for large $\alpha_{T_*}$, small
$v_\mathrm{w}$
Turbulence still a challenge, work ongoing
In any case: LISA provides a model-independent probe of
first-order phase transitions around
$100~\mathrm{GeV}$
☛ ... but new physics must be dynamical