The generalized post-Keplerian parametrization for compact binaries on

eccentric bound orbits is established at second post-Newtonian (2PN) order in a

class of massless scalar-tensor theories. This result is used to compute the

orbit-averaged flux of energy and angular momentum at Newtonian order, which

means relative 1PN order beyond the leading-order dipolar radiation of

scalar-tensor theories. The secular evolution of the orbital elements is then

computed at 1PN order. At leading order, the closed form “Peters & Matthews”

relation between the semi-major axis $a$ and the eccentricity $e$ is found to

be independent of any scalar-tensor parameter, and is given by $a propto

e^{4/3}/(1-e^2)$. Finally, the waveform is obtained at Newtonian order in the

form of a spherical harmonic mode decomposition, extending to eccentric orbits

the results obtained in [JCAP 08 (2022) 008].

## Future Roadmap for Readers

This article provides an examination of the conclusions derived from the study on the generalized post-Keplerian parametrization for compact binaries on eccentric bound orbits in massless scalar-tensor theories. It outlines the roadmap for readers interested in the potential challenges and opportunities on the horizon.

### 1. Establishing the Generalized Post-Keplerian Parametrization

The first step is to understand the establishment of the generalized post-Keplerian parametrization at the second post-Newtonian (2PN) order in massless scalar-tensor theories. This parametrization provides a framework for studying compact binaries on eccentric bound orbits.

**Challenge:** Understanding the theoretical foundation and mathematical formalism behind the generalized post-Keplerian parametrization may require a solid grasp of post-Newtonian physics and scalar-tensor theories.

### 2. Computing the Orbit-Averaged Flux of Energy and Angular Momentum

Once the parametrization is established, the next step is to compute the orbit-averaged flux of energy and angular momentum. This calculation takes into account the effects at Newtonian order, which goes beyond the leading-order dipolar radiation of scalar-tensor theories.

**Opportunity:** Analyzing the orbit-averaged flux can provide insights into the energy and angular momentum evolution of compact binaries on eccentric orbits, potentially leading to a better understanding of these systems.

### 3. Secular Evolution of Orbital Elements

In order to study the long-term behavior of compact binaries on eccentric orbits, it is important to compute the secular evolution of the orbital elements at 1PN order. This calculation takes into account additional effects beyond the leading order.

**Challenge:** Interpreting the results of the secular evolution of orbital elements may require familiarity with the relevant mathematical techniques and concepts in celestial mechanics.

### 4. Peters & Matthews Relation for Semi-Major Axis and Eccentricity

The article highlights the discovery of the closed-form “Peters & Matthews” relation between the semi-major axis (a) and the eccentricity (e) at leading order. It is found that this relation is independent of any scalar-tensor parameter and is given by a mathematical expression: a proportional to e^(4/3) divided by (1 – e^2).

**Opportunity:** Exploring the implications of the “Peters & Matthews” relation can provide valuable insights into the relationship between the semi-major axis and eccentricity of compact binaries on eccentric orbits.

### 5. Obtaining the Waveform in Spherical Harmonic Mode Decomposition

The waveform of compact binaries on eccentric orbits is calculated at Newtonian order using a spherical harmonic mode decomposition. This extends the previous results obtained in the reference [JCAP 08 (2022) 008] to include eccentric orbits.

**Opportunity:** Studying the waveform in spherical harmonic mode decomposition can offer a deeper understanding of the gravitational radiation emitted by compact binaries on eccentric orbits, enabling further investigations into the properties of these systems.

Overall, this article provides a comprehensive exploration of the conclusions derived from studying the generalized post-Keplerian parametrization for compact binaries on eccentric bound orbits in massless scalar-tensor theories. Readers interested in this topic can follow the outlined roadmap to delve deeper into the subject, overcoming the potential challenges while seizing the available opportunities for further research.