Phase transitions in the early universe
  These lecture notes are based on a course given by Mark Hindmarsh at the 24th
Saalburg Summer School 2018 and written up by Marvin L\"uben, 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.
