Generalized Perturbed Kepler Problem: Gravitational Wave Imprints from Eccentric Compact Binaries

TL;DR

This work introduces the Unified Perturbed Keplerian (UPK) framework to study binaries evolving under a perturbed Keplerian potential, capturing both conservative and dissipative effects on eccentric orbital dynamics and gravitational-wave emission. The method parameterizes deviations via radial potential corrections and derives the perturbed orbit, GW and EM fluxes, and GW phasing, providing explicit dephasing expressions dependent on perturbation coefficients that reveal distinct frequency trends for different perturbation orders. The analysis shows how small deviations from Keplerian dynamics can imprint measurable signatures on long-lived GW signals, with eccentricity generally suppressing perturbative effects except for certain higher-order terms, highlighting the potential to test GR and environmental physics with next-generation detectors. By offering a physically interpretable alternative to ppE/PPN formalisms, UPK enables source-specific constraints for eccentric binaries and paves the way for Bayesian analyses on real events, including the roles of environment and spin in future work.

Abstract

Observations of astrophysical binaries may reveal departures from pure Keplerian orbits due to environmental influences, modifications to the underlying gravitational dynamics, or signatures of new physics. In this work, we develop a unified framework to systematically study such perturbations in the ambit of the perturbed Kepler problem and explore their impact on eccentric orbital dynamics and gravitational wave emission. Unlike traditional parametrized frameworks such as post-Newtonian and post-Einsteinian expansions, our approach offers a more source-specific modeling strategy, making it more natural to trace the physical origins of eccentric binary model parameters. Starting from a general perturbed potential, we derive the modified orbit and compute the associated gravitational fluxes and phase evolution, assessing their observational relevance for both current and future detectors. This framework thus offers a general and physically transparent toolkit for probing such subtle deviations from standard dynamics in gravitational wave data.

Figures (7)

• Figure 1: Variation of the ratio $|n_k|/\nu_k$ with frequency for $k=\{1,\cdots,6\}$ for some arbitrarily chosen values of $n_k$ (only one at a time) for a GW150914-like system with masses $36 M_\odot$ and $29 M_\odot$. The solid cyan line represents a reference line where the ratio is $10^{-5}$, which is well within the domain of our approach.
• Figure 2: Orbits ($x=r(\phi) \cos\phi,\, y=r(\phi)\sin\phi$) with $r(\phi)$ given by Eq. \ref{['neworbit']} for $k=\{3,5\}$ (only one at a time) and $e=0.2$. Twenty cycles are shown and orbit sizes are normalized to unity ($\hat{a}=1$).
• Figure 3: Variation of GW energy flux differences from the corresponding Keplerian value as a function of GW frequency for a GW150914-like system. We have chosen $\{k=2, \cdots,6\}$ (only one at a time) represented by the same colored lines as indicated in the upper-left subplot. The corresponding values of $\bar{n}_k$ are fixed such that $\bar{n}_k/\nu_k=10^{-5}$ at $f_{ISCO} = 67.26\, \text{Hz}$.
• Figure 4: Instantaneous dephasing for a GW150914-like system for different $k$ values with $\bar{n}_k/\nu_k=10^{-5}$ at $f_{ISCO}$. The vertical axis shown is minus dephasing. We have chosen $e_0=0.1$ at $f_0=4$ Hz. For $k=2,3,5,6$, dephasing is negative, whereas for $k=4$ it is positive. For $k=2,3, 4$ it decreases with frequency, whereas for $k=5, 6$ it increases with increase in frequency. There is a switch over at $k=4$.
• Figure 5: Cumulative dephasing for a GW150914-like system for different $k$ values as a function of different $\bar{n}_k$'s (only one at a time) and with $e_0=0.1$ (solid) and $e_0=0.4$ (dashed) at $f_0=4$ Hz. These $\bar{n}_k$-ranges are chosen in a window around the corresponding values of $\bar{n}_k$ for which $\bar{n}_k/\nu_k=10^{-5}$. For the cases shown, increasing eccentricity results in a decrease in the net cumulative dephasing. However, the case with $k=6$ shows a different nature, where eccentricity contribution enhances cumulative dephasing. Since variation for the chosen eccentricities is small, we depict the difference curve $\Delta_{e=0.4-0.1}$ as a gray dotted line (scale given on the RHS $y$-axis), which exhibits a positive value ($\sim 10^{-4}$) and a positive slope.