  Ongoing and future gravitational wave collaborations explore different
frequency ranges of the gravitational wave spectrum, probing different stages
of the early universe and Beyond Standard Model physics. Due to the very high
energies involved, accelerators cannot probe these earlier stages. Therefore,
until some years ago, knowledge about new physics was limited and relied on
bounds from CMB observations and theoretical assumptions about higher energy
scales. While models could be constrained by CMB data, they were left
unconstrained at shorter wavelength scales. Nonetheless, each one of these
models has a gravitational wave density spectrum at these shorter wavelength
scales that can ultimately be compared to data from ground-based, space-born,
and pulsar timing array searches. These lecture notes review the formalism of
gravitational waves in General Relativity and introduce stochastic
gravitational waves, with a focus on primordial sources and commenting on
detection efforts. These lecture notes were inspired by the course
"Gravitational Waves from the Early Universe" given at the 27th W.E. Heraeus
"Saalburg" Summer School 2021 by Valerie Domcke.
