Our Universe - within its first second - evolved in a very high-energy stage, typically involving new physics beyond the Standard Model (SM). Due to the weakness of gravitational interactions, gravitational waves (GW) from primordial sources convey unaltered information about the Universe from their production time until today. With the present and future generations of GW observatories, it is possible to map the Universe’s expansion history and study particle physics at energy scales unreachable by other means of experiments. This seminar will focus on particle physics beyond the SM that can change the cosmic-expansion history beyond the standard radiation-dominated Universe. Specifically, we will discuss two examples: the intermediate matter era from heavy-unstable particles and the novel intermediate “kination” from the rotating “axion.” These scenarios have smoking-gun GW signatures detectable by next-generation observatories, providing a unique window into the early Universe and particle physics beyond the SM.