[Paper Review] Energy losses by gravitational radiation in inspiralling compact binaries to five halves post-Newtonian order

Luc Blanchet|arXiv (Cornell University)|Mar 26, 1996

Pulsars and Gravitational Waves Research1 references71 citations

TL;DR

This paper computes the energy loss rate due to gravitational radiation in inspiralling compact binaries to 2.5 post-Newtonian (2.5PN) order using a post-Minkowski wave generation formalism. It finds that the 2.5PN contribution to the energy loss is entirely due to tail effects in the wave zone, and derives the corresponding evolution laws for orbital frequency and phase, which are crucial for accurate gravitational-wave data analysis in LIGO and VIRGO detectors.

ABSTRACT

This paper derives the total power or energy loss rate generated in the form of gravitational waves by an inspiralling compact binary system to the five halves post-Newtonian (2.5PN) approximation of general relativity. Extending a recently developed gravitational-wave generation formalism valid for arbitrary (slowly-moving) systems, we compute the mass multipole moments of the system and the relevant tails present in the wave zone to 2.5PN order. In the case of two point-masses moving on a quasi-circular orbit, we find that the 2.5PN contribution in the energy loss rate is entirely due to tails. Relying on an energy balance argument we derive the laws of variation of the instantaneous frequency and phase of the binary. The 2.5PN order in the accumulated phase is significantly large, being grossly of the same order of magnitude as the previous 2PN order, but opposite in sign. However finite mass effects at 2.5PN order are small. The results of this paper should be useful when analyzing the data from inspiralling compact binaries in future gravitational-wave detectors like VIRGO and LIGO.

Motivation & Objective

• To compute the gravitational wave energy loss rate in inspiralling compact binaries to 2.5PN order in general relativity.
• To extend the post-Minkowski wave generation formalism to 2.5PN accuracy for arbitrary slowly-moving sources.
• To determine the contribution of nonlinear tail effects to the energy loss at 2.5PN order in the case of two point masses on quasi-circular orbits.
• To derive the evolution laws for the instantaneous orbital frequency and phase using an energy balance argument.
• To provide a theoretical foundation for constructing high-precision templates in gravitational-wave data analysis for LIGO and VIRGO.

Proposed method

• Uses a post-Minkowski wave generation formalism that combines post-Minkowskian expansions in the wave zone with post-Newtonian solutions in the near zone.
• Applies a matching procedure between the exterior (post-Minkowskian) and interior (post-Newtonian) solutions in the near-zone boundary.
• Computes mass multipole moments and tail contributions to the wave zone field up to 2.5PN order.
• Relies on the formalism developed in previous works (Blanchet & Damour, 1986; Blanchet et al., 1992) to include nonlinear effects such as tails.
• Derives the energy loss rate via the 2.5PN-accurate flux of gravitational waves at future null infinity.
• Uses an energy balance argument to relate the 2.5PN energy loss to the rate of change of the binary’s binding energy, assuming consistency with 5PN-order equations of motion.

Experimental results

Research questions

• RQ1 What is the 2.5PN-accurate expression for the total energy loss rate due to gravitational radiation in a compact binary system?
• RQ2 How do nonlinear tail effects contribute to the energy loss at 2.5PN order in the post-Newtonian expansion?
• RQ3 What is the role of the 2.5PN-order radiation reaction in the evolution of the orbital frequency and phase of the binary?
• RQ4 How does the 2.5PN energy loss compare in magnitude and sign to the 2PN-order contribution?
• RQ5 To what extent can the energy balance equation be trusted at 2.5PN order, and what assumptions are required?

Key findings

• The 2.5PN contribution to the gravitational wave energy loss rate in a two-point-mass binary system is entirely due to tail effects in the wave zone.
• The 2.5PN correction to the accumulated orbital phase is of the same order of magnitude as the 2PN term but opposite in sign, indicating a significant relativistic correction.
• Finite mass effects at 2.5PN order are found to be small, consistent with the test-body limit where the result reduces to known perturbative results.
• The energy balance equation at 2.5PN order is validated by showing consistency between the 2.5PN binding energy loss and the Newtonian quadrupole formula for the flux, up to O(6) post-Newtonian order.
• The derived 2.5PN energy loss rate provides a critical input for constructing high-precision gravitational-wave templates used in data analysis for LIGO and VIRGO.
• The formalism confirms that the 2.5PN radiation reaction forces are consistent with the 5PN-order equations of motion, assuming energy balance holds at this order.