Extreme-mass ratio inspirals in Schwarzschild - de Sitter spacetime I: Weak-field orbits

The inspiral of a compact object into a black hole is a key source of low - frequency gravitational waves for future space-based detectors like LISA. While models of this process have advanced considerably, they typically focus on asymptotically flat spacetimes. In this paper, we investigate how departures from asymptotic flatness, whether driven by cosmic expansion or large-scale galactic environments, alter adiabatic orbital evolution. Using the Schwarzschild - de Sitter (SdS) metric, we parametrize these deviations via an `SdS parameter' ($\lambda$) and analyze its impact on bound orbits. We calculate how $\lambda$ shifts the separatrix between bound and plunging states and modifies the relationship between a binary's energy, angular momentum, and orbital geometry. By applying a modified quadrupole formula in the weak-field limit, we investigate the effect of $\lambda$ on circularization, plunge times, and orbital trajectories. We show that a positive SdS parameter accelerates eccentricity decay and shortens inspiral timescales. We also generate adiabatic waveforms from inspirals evolving under radiation reaction, exhibiting an increase in amplitude and a cumulative phase advance induced by $\lambda$. While these effects are negligible if $\lambda$ is strictly cosmological, they become observationally relevant if the parameter serves as a proxy for astrophysical environments that induce $\sim r^2$ potential corrections. Our results suggest that such environmental coupling could meaningfully bias event rate estimates and waveform templates for space-based gravitational-wave astronomy.