The construction and use of dephasing prescriptions for environmental effects in gravitational wave astronomy

TL;DR

This work develops a cohesive framework to incorporate environmental effects (EE) as dephasing in gravitational-wave templates, clarifying how time-domain perturbations translate into Fourier-domain phase shifts through the stationary phase approximation. It distinguishes between dissipation-driven and conservative EE mechanisms and provides explicit prescriptions linking energy fluxes, radial potentials, and Doppler shifts to changes in the GW chirp and phase, including for eccentric sources where EE imprint additional, often indirect, dephasing via eccentricity evolution. The paper demonstrates that eccentricity can substantially enhance EE detectability, even for modest eccentricities at 10 Hz, by amplifying higher GW harmonics in the δSNR, and it offers illustrative EMRI and stellar-mass binary examples. These results have important implications for waveform modeling, parameter estimation biases, and the prospects of probing astrophysical environments with current and future GW detectors. The framework is intended as both a reference and a practical entry point for researchers entering the EE literature, and it lays the groundwork for systematic analyses of EE in eccentric binaries.

Abstract

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. Importantly, we demonstrate that the detectability of EEs can be significantly enhanced in the presence of eccentricity, even for $e_\mathrm{10Hz}\lesssim0.2$, substantially increasing the prospects for detection in ground based detectors. Our results highlight the unique potential of modeling and searching for EE in eccentric binary sources of GWs.

Figures (8)

• Figure 1: Depiction of the various aspects analysed in this work. Dephasing in GWs can be produced by both observer dependent redshift-like effects (top-panel) as well as intrinsic perturbations to the binary source secular evolution (bottom panel). These manifest as observables in Fourier domain waveforms in two different ways. They are best thought of as a time delay $\Delta t$ in the former case, and as a binary phase shift $\delta \phi$ at a given time $t$ in the latter case. Dephasing in eccentric binaries is mostly unexplored, and is a significantly richer problem than the corresponding circular case.
• Figure 2: Illustration of the stationary phase approximation. The main contribution to the Fourier domain amplitude for a given frequency $f$ is given by the part of the signal where $\mathrm{d}\phi/\mathrm{d}t=2\pi f$. In the close vicinity of this point the signal can be approximated by a sinusoidal of that frequency (red). The sinusoidal's phase has to be matched to the GW phase $\phi(t_{f})$ at the time $t(f)$ at which the GW frequency is $f$ (blue dot). Thus, the Fourier representation contains information on both $\phi(t_{f})$ and $t(f)$.
• Figure 3: Illustration of the different types of dephasing addressed in section \ref{['ssec:deph_circ_tF']}. The coloured dots on the phase circle correspond to the same dots on the GW signals and represent the different phases. The perturbed waveform $h_{\rm tot}$ (yellow) has a different phase from the vacuum waveform $h_{\rm vac}$ at a given reference time, yielding a dephasing $\delta \phi(t)$. It may also have a different phase at a given reference frequency, yielding a dephasing $\delta \varphi (f)$. The two signals reach a given reference frequency with a time difference $\delta t(f)$. The Fourier dephasing is then related to these quantities by $\delta \psi = 2\pi f \delta t(f) - \delta\varphi(f) \approx -\delta \phi(t)$.
• Figure 4: Illustration of the epicyclic expansion of $h(t)$ for a single period of an orbit with $e=0.5$. The time domain waveform $h(t)$ has been rescaled so that its amplitude is comparable to those of the various harmonics. The transparency and amplitude of the harmonics both reflect their relative importance.
• Figure 5: Illustration of an eccentric waveform in Fourier space (computed with the Newtonian model described in 2009yunes) and the harmonics it is composed of. Note how the waveform appears noisy, while in reality it is composed of a sum of smooth harmonics, the phase of which adds incoherently. As the binary evolves in frequency, its eccentricity decreases and the GW waveform is better described by the $\ell=2$ mode. In this example, the minimum eccentricity of the binary is $e_0 = 0.04$ and the maximum is $e_0 = 0.29$. The amplitude and frequency can be rescaled arbitrarily by changing binary parameters.