Tails from Eccentric Encounters

by Nicholas Loutrel.


Nicholas Loutrel is a Graduate Student in the eXtreme Gravity Institute at Montana State University.

A new method of computation aims to fill in the gaps in our knowledge of gravitational waves from eccentric binaries.

The modeling of gravitational waves (GWs) suitable for detection with ground-based detectors has been mostly focused on binary systems composed of compact objects, such as neutron stars (NSs) and black holes (BHs). Binaries that form with wide orbital separations are expected to have very small orbital eccentricity, typically less than 0.1, by the time their GW emission enters the detection band of these instruments. However, in dense stellar environments, unbound encounters between multiple compact objects can result in the formation of binaries with high orbital eccentricity (close to, but still less than unity) and whose GW emission is in band for ground-based detectors. Such systems are expected to be a small subset of the total number of detectable signals; however, they can provide critical information about the population of compact objects in globular clusters and galactic nuclei, and about the gravitational interaction in a regime that can normally only be probed in the very last stages of quasi-circular inspirals.

The GW emission from highly eccentric systems is peaked strongly around pericenter passage, where the orbital velocity is highest. Thus, the observed signal will resemble a set of bursts in time and frequency, instead of the continuous chirp expected from quasi-circular binaries. In addition, the system spends very little time in pericenter passage, resulting in a low signal-to-noise ratio for any one burst. This presents a unique issue since very few accurate models exist for systems with such high eccentricity, making detection rather difficult. One solution is to consider a power stacking approach to detection, whereby the power in the individual bursts is stacked to enhance the signal-to-noise ratio of the total signal. This method relies on understanding the effect of radiation reaction, specifically how much orbital energy and angular momentum is lost due to GW emission. However, precise expressions for these quantities have been difficult to obtain for highly eccentric systems, especially when considering the back-reaction due to the scattering of GWs off of the background spacetime of the binary, the so-called GW tails.


Nicolás Yunes is an Associate Professor in the Physics Department at Montana State University and the co-founder of the eXtreme Gravity Institute.

The effects of GW tails on the loss of orbital energy and angular momentum had been calculated previously in terms of infinite sums of Bessel functions that depend on the orbital eccentricity. Our work has focused on resumming these expressions through techniques from asymptotic analysis. After replacing the Bessel functions with their uniform asymptotic expansion, we converted the infinite sums into integrals and analytically expanded the resulting expression in the high eccentricity limit. The resulting asymptotic series can be compared to numerical results obtained from the infinite sums of Bessel functions to find so called superasymptotic series, i.e. an asymptotic expansion that achieves the minimum relative error when compared to numerical results. These superasymptotic expressions are the first, closed-form, analytic expressions for the tail energy and angular momentum fluxes; they are accurate to 10-4 near the circular limit and 10-9 or better for highly eccentric systems. We have further improved the accuracy of these expressions by matching them to previously-obtained, small eccentricity expansions of the tail fluxes, resulting in the construction of hyperasymptotic series. These final series are accurate to approximately 10-8 or better at all orbital eccentricities, making them the most accurate analytic representations to date for the tail fluxes.

Despite this incredible accuracy, much work remains. While comparing our results to numerical calculations has shown promise, it does not provide sufficient information to determine whether these analytic expressions are sufficiently accurate to be used in template-based or burst searches for GWs. A rigorous data analysis study would need to be performed to understand how these expressions affect our ability to detect GW signals from eccentric binaries, as well as how any errors might bias the recovered parameters of the system generating the signal. This work has hopefully taken us one step closer to computing accurate analytic waveforms for generically eccentric systems within the post-Newtonian formalism.

Read the full article in Classical and Quantum Gravity:
Hereditary effects in eccentric compact binary inspirals to third post-Newtonian order
Loutrel and Yunes 2017 Class. Quantum Grav. 34 044003

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