These lecture notes are based on a course given by Mark Hindmarsh at the
24th Saalburg Summer School 2018 and written up by Marvin Lüben, Johannes Lumma and
Martin Pauly.
The aim is to provide the necessary basics to understand first-order phase transitions
in the early universe, to outline how they leave imprints in gravitational waves,
and advertise how those gravitational waves could be detected in the future.
A first-order phase transition at the electroweak scale is a prediction
of many theories beyond the Standard Model, and is also motivated
as an ingredient of some
theories attempting to provide an explanation for the matter-antimatter asymmetry in our Universe.
Starting from bosonic and fermionic statistics, we derive Boltzmann's equation
and generalise to a fluid of particles with field dependent mass.
We introduce the thermal effective potential for the field in its lowest order approximation,
discuss the transition to the Higgs phase in the Standard Model and beyond,
and compute the probability for the field to cross a potential barrier.
After these preliminaries, we provide a hydrodynamical description
of first-order phase transitions as it is appropriate for describing the early Universe.
We thereby discuss the key quantities characterising a phase transition, and
how they are imprinted in the gravitational wave power spectrum that might be detectable
by the space-based gravitational wave detector LISA in the 2030s.
Implications of the detection of primordial gravitational waves for the Standard Model
