János Takátsy, Lorenz Zwick, Kai Hendriks, Pankaj Saini, Gaia Fabj, Johan Samsing

In the first part of this work, we provide a curated overview of the theoretical framework necessary for incorporating dephasing due to environmental effects (EE) in gravitational wave (GW) templates. We focus in particular on the relationship between orbital perturbations in the time-domain and the resulting dephasing in both time and frequency domain, elucidating and resolving some inconsistencies present in the literature. We discuss how commonly studied binary environments often result in several sources of dephasing that affect the GW signal at the same time. This work synthesizes insights from two decades of literature, offering a unified conceptual narrative alongside a curated reference of key formulas, illustrative examples and methodological prescriptions. It can serve both as a reference for researchers in the field as well as a modern introduction for those who wish to enter it. In the second part, we derive novel aspects of dephasing for eccentric GW sources and lay the foundations for consistently treating the full problem. Our results highlight the unique potential of modelling and searching for EE in eccentric binary sources of GWs.