about $10^{-2}\,{\rm Hz}$. The interferometer consists of three satellites
that are orbiting the Earth in a triangle. Figure taken from Ref. [9].

For gravitational waves sourced by phase transitions, the relevant mission is
Laser Interferometer Space Antenna (LISA). LISA is a space-based
interferometric gravitational wave detector that works with three satellites
orbiting the Earth, see Fig. 18. These are equipped with lasers and
photodetectors such as to detect small changes in the separation of the
satellites by measuring the time delays in signals sent between them, through
the technique of time delay interferometry [84, 85, 86]. The interferometer
arms have a length of $2.5\,{\rm Gm}$ such that LISA is most sensitive at
frequencies in the range $10^{-3}$ – $10^{-2}\,{\rm Hz}$. The target sources
are binary white dwarfs, merging supermassive black holes at the centres of
distant galaxies, and early-universe physics at the TeV-scale such as first-
order phase transitions. The planned launch year is 2034. In Fig. 19 we show
the projected sensitivity of LISA and the amplitudes of expected astrophysical
sources. In addition we have added the signal from a first-order phase
transition, which we discuss further below. As can be seen, LISA is expected
to measure gravitational waves coming from phase transitions, but other
sources produce a signal in the same frequency interval as well. For details
on LISA, the expected sources of gravitational waves, and its relevance for
probing fundamental physics we recommend Refs. [9, 12, 87]. Other planned
space-based gravitational missions include DECIGO [88], Taiji [89] and TianQin
[90].

Figure 19: The expected sensitivity of LISA and possible astrophysical sources
of gravitational waves. The phase transition signal, indicated by the blue
line, is predicted for a transition with nucleation temperature
$T_{\text{n}}=500$ GeV, strength parameter $\alpha=0.3$, transition rate to
Hubble rate ratio $\beta/H_{\text{n}}=100$, and wall speed $v_{\rm w}=0.4$.
Background figure taken from [9].

###  8.2 GWs from first-order phase transitions

In this section we discuss the expected gravitational waves signal produced by
a first-order phase transition. The details and derivation are quite involved
and require numerical techniques which is beyond the scope of this course.
Instead, we explain the key quantities that can be inferred from the
gravitational wave power spectrum and their relation to the results of the
previous sections. For a review we refer to Ref. [12] that discuss the
predicted power spectra for various BSMs.

The general picture is the following. Initially, when temperatures are high,
$T\gg T_{\rm c}$, the thermal Higgs potential (free energy) has a minimum only
at $\phi=0$. We recall that minima of the free energy correspond to
equilibrium states. Therefore, the Higgs is everywhere in the symmetric phase
with thermal fluctuations around the minimum. As the temperature drops, the
thermal Higgs potential develops a second minimum and the minima are separated
by a potential barrier. At a critical temperature $T=T_{\rm c}$, both minima
are degenerate. Below the critical temperature, it is thermodynamically
preferred for the Higgs to occupy the new minimum. However, it has to cross
the potential barrier. This it can do either by quantum tunneling [91, 49] or
by thermal fluctuations [50]. Here, we will study the thermal case.

Thermal fluctuations can drive the Higgs into the symmetry-breaking phase in
limited regions of space. The most probable shape of the regions is a
spherical bubble, inside which the Higgs is already in the new phase, while
outside the bubbles the Higgs is still in the metastable phase. Large enough
bubbles expand into the fluid such that the rest of the universe enters the
symmetry-broken phase.

We have already discussed the rate at which bubbles appear, and how they
expand, in sections 6 and 7. Once the bubbles start colliding, we must rely on
a mixture of numerical simulations (see e.g. Ref. [71]) and modelling to
describe the motion of the fluid and the production of gravitational waves.

We recall that near equilibrium, the system can be described as an
ultrarelativistic fluid coupled to a Higgs field, according to equations
(5.26) and (5.24). For concreteness, we state the equations of motion as used
in Ref. [71] ready for numerical simulations. As before, the fluid four-
velocity is given by $u^{\mu}=\gamma(1,\vec{v})^{\mu}$. The Higgs field
satisfies the relativistic equation

$-\ddot{\phi}+\nabla^{2}\phi-V^{\prime}=\eta\gamma(\dot{\phi}+v^{i}\partial_{i}\phi)\,,$
(8.15)

with $\eta$ the Higgs-fluid coupling and $V^{\prime}=\partial_{\phi}V$, c.f.
Eq. (7.2). Further the relativistic fluid equations read

$\displaystyle\dot{E}+\partial_{i}(Ev^{i})+p(\dot{\gamma}+\partial_{i}(\gamma
v^{i}))-V^{\prime}\gamma(\dot{\phi}+v^{i}\partial_{i}\phi)$ $\displaystyle=$
$\displaystyle\eta^{2}\gamma^{2}(\dot{\phi}+v^{i}\partial_{i}\phi)^{2}$
$\displaystyle\dot{Z}_{i}+\partial_{j}(Z_{i}v^{j})+\partial_{i}p+V^{\prime}\partial_{i}\phi$
$\displaystyle=$
$\displaystyle-\eta\gamma(\dot{\phi}+v^{j}\partial_{j}\phi)\partial_{i}\phi\,,$

which follow from Eq. (5.24) together with Eq. (5.25). Here, $E=\gamma e$ is
the fluid energy density and $Z_{i}=\gamma(e+p)u_{i}$ the fluid momentum
density with $u_{i}=\gamma v^{i}$. The uncertainty in the precise form of
$\eta$ turns out not to matter much: different choices affect only the precise
profile of the domain wall as it moves through the fluid, and the
hydrodynamics ensures that the fluid flows around expanding bubbles are the
same for a given wall speed $v_{\rm w}$ and phase transition strength
$\alpha_{\rm n}$.

Finally, the metric perturbation is described by Eq. (8.3). In practice, one
evolves an unconstrained symmetric tensor $u_{ij}$ with the tensor source

$\Pi_{ij}=(e+p)\gamma^{2}v_{i}v_{j}+\partial_{i}\phi\partial_{j}\phi,$ (8.16)

and projects out the transverse traceless part $h_{ij}(t,\vec{k})$ from the
Fourier transform $u_{kl}(t,\vec{k})$ with the appropriate projector. In the
bubbles of a thermal transition, the scalar field part of the source is
confined to a microscopically thin wall, while the fluid source is distributed
over a significant volume. The ratio of the energies can be estimated as
$\ell_{\rm w}/R$, where $\ell_{\rm w}$ is the wall thickness, and $R$ is the
bubble radius. The wall thickness $\ell_{\rm w}$ is very roughly the inverse
temperature, and given that the total energy density is $e\sim T^{4}$, the
ratio can be estimated as

$\frac{\ell_{\rm w}}{R}\sim\frac{\sqrt{G}T}{HR}\,,$

where we have used $H\sim\sqrt{G}T^{2}$ as a consequence of Friedmann’s
equation. At electroweak temperatures $T\sim 100$ GeV, we have $\sqrt{G}T\sim
10^{-17}$, while bubbles typically grow to a few orders of magnitude of the
Hubble length. Hence, we find that $\ell_{\rm w}/R\ll 1$ and the energy-
momentum of the scalar field is generally negligible.

An exception is if the bubble wall is very weakly coupled to the fluid and
continues to accelerate rather than reaching a terminal velocity, or there is
a lot of supercooling and the fluid energy density becomes negligible compared
with the potential energy in the scalar field (see [92] for a recent study).
In this case, one can study the scalar field only [55, 93].

Based on simulations and modelling of thermal transitions, it is possible to
identify three stages of gravitational wave production.

  1. 1.

Initially the bubbles of the stable phase collide and merge. This stage is of
short duration and subdominant compared to the subsequent stages of
gravitational wave production, unless the bubbles grow as large as the Hubble
length. We will denote the contribution following from the collision of
bubbles $\Omega_{\rm bc}$.

  2. 2.

After the bubbles have collided and merged, the shells of fluid kinetic energy
continue to expand into the plasma as sound waves. These different waves
overlap and source gravitational waves. The power spectrum sourced by this
‘acoustic’ stage is denoted $\Omega_{\rm sw}$.

  3. 3.

The last stage is the so-called turbulent phase, where the intrinsic non-
linearity in the fluid equations becomes important. Through the non-
linearities, the previous phases might produce vorticity and turbulence, and
the sound waves eventually develop shocks. The spectrum of this phase is
labelled by $\Omega_{\rm tu}$.

These different sources are relevant on different length scales and at
different time scales, and the total power spectrum can be approximately
written as the sum

$\Omega_{\rm gw}=\Omega_{\rm bc}+\Omega_{\rm sw}+\Omega_{\rm tu}\,.$ (8.17)

In general, $\Omega_{\rm sw}$ is thought to yield the dominant contribution,
unless the transition is very strong, or the bubbles are as large as the
cosmological horizon.

The key quantities which determine these power spectra are the following:

  * •

$T_{\text{n}}$, the cosmological phase transition nucleation temperature;

  * •

$\alpha_{\rm n}$, the phase transition strength parameter at the nucleation
temperature, which is related to the scalar potential energy released during
the phase transition and given in Eq. (7.40);

  * •

$v_{\text{w}}$, the bubble wall speed.

  * •

$\beta$, the transition rate parameter (7.12), which can be thought of as the
inverse phase transition duration, and in combination with $v_{\text{w}}$
determines the mean bubble centre separation $R_{*}$ through Eq. (7.21).

The belief is that only these four parameters determine the power spectrum,
and therefore that the gravitational wave power spectrum is potentially a
precise probe of these quantities, which are in principle computable from the
Lagrangian of a specific theory.777Recent work has demonstrated that a fifth
parameter, the sound speed, should be added to the list [69, 94]. Holographic
calculations indicate that at phase transitions in strongly coupled theories
the sound speed can be quite different from $1/\sqrt{3}$ [95] Hence, there is
great excitement about the potential of LISA to probe physics beyond the
Standard Model 888While the four parameters listed above allow one to fit the
gravitational wave spectra obtained from numerical simulations, it has been
found that two of these parameters, the strength of the phase transition
$\alpha$, and the inverse phase transition duration $\beta$, are not entirely
independent from each other [72, 73]: Stronger phase transitions,
characterized by larger values of $\alpha$, take more time to complete and
hence also predict a smaller value of $\beta$. Conversely, weaker phase
transitions, predicting smaller values of $\alpha$, proceed rather quickly. In
simple words, the underlying reason for this relationship is rooted in the
fact that the strength of the phase transition is related to the potential
difference between the metastable and the stable ground state, $\Delta V$. As
the temperature drops, the difference between the metastable and the stable
ground state becomes larger. Strong phase transitions are a result of a
sufficient amount of supercooling, i.e., the phase transition proceeds well
below the critical temperature, where the two minima are degenerate, and hence
take more time to complete..

One can estimate that the gravitational wave density parameter after a phase
transition is [71, 96]

$\Omega_{\rm
gw}\sim(H_{\text{n}}\tau_{\text{v}})(H_{\text{n}}\tau_{\text{ac}})K^{2},$
(8.18)

where $\tau_{\text{v}}$ is the lifetime of the shear stresses,
$\tau_{\text{ac}}$ is the autocorrelation time of the fluid flow, and $K$ is
the kinetic energy fraction, defined in Eq. (7.36). The timescales depend on
the properties of the fluid flow through its characteristic length scale
$L_{\rm f}$ and a characteristic flow speed $V_{\rm f}\simeq\sqrt{K}$, which
both ultimately depend on the parameters $R_{*}$, $\alpha_{\rm n}$ and $v_{\rm
w}$. The Hubble time $H_{\text{n}}^{-1}$ is a maximum time scale, as the
expansion of the universe reduces the factor $H_{\text{n}}^{2}$ in (8.18) and
therefore the effective power of the source. Initially, one can estimate the
characteristic length scale as $L_{\rm f}\sim R_{*}$. The kinetic energy
fraction $K(v_{\rm w},\alpha_{\rm n})$ of an individual bubble is a good
estimate of the global kinetic energy fraction [71, 96], unless the transition
is strong and proceeds by deflagrations [97]. In that case, there is
significant suppression of the fluid kinetic energy, which is associated with
the slowing of the bubble walls when they encounter a hot region around
another bubble.

For sound waves the characteristic speed is the sound speed
$c_{\text{s}}\simeq 1/\sqrt{3}$, and the autocorrelation time is therefore
$\tau_{\text{ac}}\sim R_{*}/c_{\text{s}}$. The decay of a fluid flow is a non-
linear process. From the non-linear terms in the fluid equations (8.2) one can
form the non-linearity timescale $\tau_{\text{nl}}\sim L_{\rm f}/V_{\rm f}\sim
R_{*}/\sqrt{K}$. The approximate lifetime of the gravitational wave source is
then $\tau_{\text{v}}=\min(\tau_{\text{nl}},H_{\text{n}}^{-1})$.

A more careful calculation of gravitational wave production in an expanding
universe [98] shows that a better approximation for the effective source
lifetime is

$\tau_{\text{v}}=H_{\text{n}}^{-1}\left(1-\frac{1}{\sqrt{1+2\tau_{\text{nl}}R_{*}}}\right)\,,$
(8.19)

which assumes that the RMS fluid velocity stays constant until
$\tau_{\text{nl}}$, when it immediately vanishes. Hence we can estimate

$\Omega_{\rm
gw}\sim\left(1-\frac{K^{1/4}}{\sqrt{K^{1/2}+2H_{\text{n}}R_{*}}}\right)(H_{\text{n}}R_{*})K^{2},$
(8.20)

which goes as $(H_{\text{n}}R_{*})^{2}K^{3/2}$ for fluid flow lifetimes much
less than the Hubble time. We recall from Eq. (7.43) that the kinetic energy
fraction is

$K\simeq\frac{\kappa\alpha_{\rm n}}{1+\alpha_{\rm n}}\,,$ (8.21)

where $\kappa(\alpha_{\rm n},v_{\text{w}})$ is the efficiency with which Higgs
potential energy is turned into kinetic energy, discussed around Eq. (7.44).

A more detailed survey of the expected lifetimes and correlation times of the
various sources can be found in Ref. [12]. It is important to note that
numerical simulations and modelling show that the dimensionless constant of
proportionality in Eq. (8.20) is O($10^{-2}$) and not of order unity [71].

The shape of the gravitational wave spectrum is best understood for acoustic
production, which has been explored by numerical simulations [71] and are now
described by a precise theoretical framework, the sound shell model [66]. This
is only the most recent development in a long history of modelling of
gravitational wave production at a thermal phase transition [68, 99, 100, 101,
102].

As sketched in Fig. 20 (left) the power rises as $k^{9}$ to a domed peak at
$kR_{*}\sim 10$, and then decreases as $k^{-3}$. If the wall speed is close to
the speed of sound, $v_{\text{w}}\sim c_{\rm s}$, then the domed peak becomes
a slow $k^{1}$ increase towards a peak wavenumber determined by the sound
shell thickness $\Delta R_{*}$. At very low wavenumbers, where the signal is
generically small, different power laws may emerge [103, 104]. For wall speeds
away from the sound speed, the shape of the domed peak can be approximated by
a simpler broken power law rising as $k^{3}$ and falling as $k^{-4}$ [71].

Figure 20: Left panel: shape of the gravitational wave spectrum from sound
waves, showing the power laws and the characteristic wavenumber scales, which
are determined by the mean bubble centre separation $R_{*}$ and the sound
shell thickness $\Delta R_{*}$. Right panel: survey of predictions of the
power spectrum from modelling of turbulent gravitational wave production, from
[105] (green), [102] (“top-hat” autocorrelation, black), and [106] (blue).

The turbulent stage, however, is far less well understood, as simulations of
turbulent flows are very challenging [107, 97], and consequently the modelling
[105, 102, 106] relies on untested assumptions. The main assumptions that go
into the modelling are how the flow autocorrelation time depends on
wavenumber, and also how the global quantities such as kinetic energy and
correlation length evolve with time. The differences between these assumptions
explain the different predictions for the power laws at low and high
wavenumber, sketched in Fig. 20 (right). Only dedicated numerical simulations
can resolve these issue. The simulations of turbulent vortical flow [107] seem
to disagree with all the predictions, having a slow rise of around $k^{1}$ to
the peak, and a $k^{-8/3}$ decay at high wavenumber.

Other unresolved questions are how the vortical component of the flow, which
leads to turbulence, is generated in the first place, how important it is
relative to the the compressional part, and how efficient they are at
generating gravitational waves. On the first question, recent simulations [97]
show that vorticity is generated by bubble collisions in strong phase
transitions. For example, in the simulation shown in Fig. 21), the vorticity
is associated with regions where bubbles collide, and about 20% of the fluid
kinetic energy is in vortical flow. The figure also shows the above-mentioned
heating of the symmetric phase, here confined to shrinking asymmetrical
droplets.

Figure 21: Slice through a numerical simulation of the field-fluid system,
showing temperature (left) and vorticity (right) following a strong phase
transition, with $\alpha_{\rm n}=0.5$ and $v_{\text{w}}=0.44$ [97] . The
temperature is highest inside shrinking droplets of the symmetric phase. The
vorticity (shown in units of the critical temperature $T_{\text{c}}$) is
highest around the sites of bubble collisions. (Figure courtesy D. Cutting.)

There is clearly much work to be done before we are able to accurately compute
a gravitational wave power spectrum for a phase transition of any strength. At
the moment, this means regions where simulations have been carried out [71]
and models [66] have reasonable agreement in the range $0.4\lesssim v_{\rm
w}\lesssim 0.9$ and $\alpha_{\rm n}\lesssim 0.1$.

For ultra-strong transitions ($\alpha_{\rm n}\gg 1$), the energy density of
the universe becomes dominated by the potential energy of the scalar field,
and the expansion of the universe starts to accelerate. This is the original
idea behind cosmic inflation [4], and it was quickly realised that the
nucleation rate parameter cannot be much smaller than the Hubble rate, as
otherwise the majority of the Universe would still be inflating [65, 108].
Close to this boundary, the bubble separation $R_{*}$ would be of order the
Hubble length, and so the gravitational signal would be maximal, motivating
recent study [109, 92, 110]. In particular, bubble dynamics in the near-vacuum
state is rather different from the thermal energy dominated case we consider
in these lectures [92]. Recent numerical simulations involving just the scalar
field are helping to develop a picture of what happens when the fluid plays no
role at all [93, 55, 111]. One old idea which can be tested at the same time
is that vacuum bubble collisions could have generated primordial black holes
[108, 112, 113, 111].

The gravitational wave power spectrum at much larger scales than $R_{*}$ is
also not fully understood. There are predicted to be some characteristic power
laws in the gravitational wave power spectrum at long wavelengths connected
with expanding shells of shear stress [103, 104, 114, 115], which in the
strong supercooling case would not be dominated by the sound shell model
signal. The fluid flow length scale is also expected to grow due to non-
linearities in the fluid equations, which may also contribute to the large-
scale power [116].

###  8.3 Comparison with GW observations

Finally for this section, we see how calculations of the gravitational wave
signal match up to the potential for observations. First, we study how the
gravitational wave frequency and intensity change between their generation and
their observation.

The frequency scale of the spectrum when it is produced can be taken to be
$f_{*}=R_{*}^{-1}$, which is also an estimate of the peak frequency. The
frequency today $f_{*,0}$ is determined by the redshifting of the
gravitational waves since the time of the phase transition. Using the
Friedmann equation, the conservation of entropy, and a photon temperature
today of $T_{\gamma,0}=2.725$ K, one can show that

$f_{*,0}\simeq
2.62\left(\frac{1}{H_{\text{n}}R_{*}}\right)\left(\frac{T_{\text{n}}}{10^{2}\,\text{GeV}}\right)\left(\frac{g_{\text{eff}}}{100}\right)^{\frac{1}{6}}\;\mu\text{Hz}\,.$
(8.22)

Hence LISA will be most sensitive to electroweak-scale ($100$ – $1000$ GeV)
transitions with bubble separations between $10^{-2}$ and $10^{-3}$ of the
Hubble length.

The intensity of the gravitational wave signal is also affected by the
expansion of the Universe. While the energy density of the Universe is
dominated by radiation, the density parameter of gravitational waves
$\Omega_{\rm gw}$ remains constant, as the energy densities of both components
redshift the same way. However, once radiation gives way to non-relativistic
matter as the dominant component, the density parameter of gravitational waves
$\Omega_{\rm gw}$ decreases, until today it is reduced by a factor

$F_{{\rm gw},0}=(3.57\pm 0.05)\times
10^{-5}\left(\frac{100}{g_{\text{eff}}}\right)^{\frac{1}{3}},$ (8.23)

where the uncertainty comes mainly from the measurement of the Hubble rate
today, taken here to be the Planck 2015 best-fit value $H_{0}=67.8\pm
0.9\,\textrm{km}\,\textrm{s}^{-1}\,\textrm{Mpc}^{-1}$ [117].

We can now present a simple model for the gravitational wave power spectrum
from first-order phase transitions, which captures the parametric
understanding of the power outlined in this section, gives a simple functional
form based on numerical simulations [71, 12], and includes an explicit
attenuation factor due to the decay of the flow [98]. The contribution to the
fractional density in gravitational waves in a logarithmic frequency interval
$\differential\ln f$ from a phase transition with mean bubble centre
separation to Hubble length ratio $H_{\text{n}}R_{*}$ generating a kinetic
energy fraction $K$ is

$\frac{\differential\Omega_{{\rm gw},0}}{\differential\ln f}=2.061F_{{\rm
gw},0}\tilde{\Omega}_{\rm
gw}\frac{(H_{*}R_{*})^{2}}{\sqrt{K}+H_{*}R_{*}}K^{2}C(f/f_{\text{p},0})\,,$
(8.24)

where $\tilde{\Omega}_{\rm gw}=0.012$ is a numerically determined constant,
representing the approximate efficiency with which kinetic energy is turned
into gravitational waves, $C(s)$ is the function

$C(s)=s^{3}\left(\frac{7}{4+3s^{2}}\right)^{7/2}\,,$ (8.25)

describing the gravitational wave power around the peak, and
$f_{\text{p},0}=10f_{*,0}$ is an estimate of the peak frequency, again from
the numerical simulations. The number $2.061$ is an approximation to
$3/\int_{0}^{\infty}d\ln(s)C(s)$. In this very simple model the power at small
$s$ is over-estimated, and at large $s$ under-estimated, compared to the sound
shell model, and has significant deviations around the peak for wall speeds
near the speed of sound. Its domain of reliability is $\alpha_{\rm n}\lesssim
0.1$, $0.4\lesssim v_{\rm w}\lesssim 0.5$ and $v_{\rm CJ}(\alpha_{\rm
n})\lesssim 0.9$, where $v_{\rm CJ}$ is the minimum speed of a detonation,
given in Eq. (7.34). This represents the parameter range where gravitational
wave power spectra have been extracted from numerical simulations. The model
does not get the shape right for deflagrations with $v_{\rm w}$ near or above
the speed of sound (see Fig. 20).

We are now in a position to estimate the expected density fraction in
gravitational waves from a phase transition, or equivalently the expected
characteristic strain. As the power spectrum has a definite peak, the estimate
represents both the total power and the peak power. Taking $T_{\text{n}}=500$
GeV, $\alpha_{\rm n}=0.3$, $\beta/H_{\text{n}}=10^{2}$, and $v_{\rm w}=0.4$,
we find from Eqs. (8.22), (8.20), (8.21), and (7.44) that

$\Omega_{{\rm gw},0}\sim 10^{-12}\,,$ (8.26)

and that the peak frequency estimate is $f_{*,0}\simeq 10^{-3}$ Hz. At this
frequency, the characteristic strain of a signal with a gravitational wave
density parameter of order $10^{-12}$ is

$h_{c}\sim 10^{-21}\,,$ (8.27)

where we have used Eq. (8.10). Estimates of the power spectrum, in the
simplified broken power law approximation in Eq. (8.24), can be generated by
the PTPlot tool [118, 12].

In Fig. 19 we have plotted the signal from a phase transition predicted by the
more refined sound shell model [119, 66], with the parameters given above.
While the strength of the transition is in the range where the amplitude and
shape of the signal are merely indicative, the amplitude agrees with our
order-of-magnitude estimate above, and such a transition would be detected
very clearly by LISA, despite the competing astrophysical signals. The most
significant of these will probably be the foreground from white dwarf binaries
in our galaxy, of which there are thought to be tens of millions. This
foreground will always be present, unlike the transient signals from merging
supermassive black holes in distant galaxies. However, the anisotropy of the
galactic white dwarf foreground should make it vary in strength throughout the
year as LISA orbits the sun and the relative orientation of the plane of the
satellite constellation and the galaxy changes. This will make it
distinguishable from the isotropic background expected from phase transitions
(see [83] for a review of gravitational wave signals and detection methods).

##  9 Summary

Gravitational waves are an extraordinary new tool for astronomy and cosmology,
and can be used to directly observe processes which happened in the early
universe. In particular, first-order phase transitions in the early universe
give rise to gravitational waves with characteristic power spectra. The source
of the gravitational waves is the shear stresses in the fluid set up by the
expansion and collision of bubbles of the low-temperature symmetry-broken
phase during the phase transition. The shear stresses are initially
overlapping pressure waves, or sound, which may be strong enough to generate
turbulence when they collide.

The dynamics of the phase transition can be described in terms of only four
properties of the phase transition: the bubble nucleation temperature
$T_{\text{n}}$, the strength parameter $\alpha_{\rm n}$, the transition rate
parameter $\beta$, and the bubble wall speed $v_{\text{w}}$. These are all in
principle computable from an underlying particle physics model, and in these
lectures we have gained some insight into the methods. We have also seen how
relativistic combustion theory explains how kinetic energy is generated in the
fluid, which eventually shows up as the sound waves and turbulent motion.

The current excitement is that information about these phase transition
parameters is contained in the power spectrum of the gravitational waves, and
that the space mission LISA could detect these gravitational waves if the
phase transition took place at a time around $t=10^{-11}{\rm s}$ [12], when
the temperature was in the range 100 GeV to 1 TeV. This is the scale of
electroweak symmetry-breaking. Hence LISA is a particle physics experiment, as
well as an astrophysical observatory. A first-order electroweak phase
transition is possible only if there is physics beyond the Standard Model,
perhaps in the form of extra Higgs fields. LISA has the potential to discover
new fundamental physics.

This is a dynamic and developing field. In the last section we already
mentioned the current uncertainty about the hydrodynamics of strong phase
transitions ($\alpha_{\rm n}\gtrsim 0.1$). It is also not yet known how
accurately LISA can measure the phase transition parameters, but even a 20%
measurement of (say) the wall speed is not yet matched by the accuracy of the
calculations from underlying particle physics models. Furthermore, the
framework of homogeneous nucleation theory is taken from condensed matter,
where nucleation by thermal fluctuation has never been observed to take place.
There is a long-standing mystery about the lifetime of the metastable
superfluid 3He-A phase towards decay into 3He-B. The theory described in these
lectures predicts lifetimes of experimental samples of superfluid 3He-A of
order $10^{1000000}$ seconds, while in the laboratory, even when taking great
care over impurities, 3He-A lasts only a few minutes (see e.g. [120]). There
is much to be done to realise the goal of making LISA into a particle physics
experiment to complement the Large Hadron Collider.

## Acknowledgements

These notes are based on a course given by Mark Hindmarsh at the 24th W.E.
Heraeus Summer School “Saalburg” for Graduate Students on “Foundations and New
Methods in Theoretical Physics” in September 2018. We would like to thank the
organisers and all participants for the warm and productive atmosphere. MH
would also like to thank the organisers of the 2017 Benasque School for
Gravitational Waves for Cosmology and Astrophysics, where an earlier version
of these lectures were given.

We are grateful to Fëanor Reuben Ares, Giordano Cintia and Anupam Mazumdar for
careful reading and useful comments on the manuscript. We further thank Daniel
Cutting for Figs. 9, 14, 21 and Chloe Gowling for scripts used in the
production of Fig. 19.

##### Funding information

MH acknowledges support from the Science and Technology Facilities Council
(grant number ST/L000504/1) and the Academy of Finland (project numbers
286769, 333609). MP acknowledges support by the Heidelberg Graduate School of
Fundamental Physics and through a scholarship of the German Academic
Scholarship Foundation. JL acknowledges support by an Emmy-Noether grant of
the DFG under grant number Ei/1037-1. ML is supported by a PhD grant from the
Max Planck Society.

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