by Blake Moore
Humankind has been obsessed with circles for a long time. It comes as no surprise then that the modeling of gravitational waves had focused until recently on those emitted by black holes or neutron stars in circular orbits around each other. But in the case of gravitational wave modeling, there is good reason for this obsession. Gravitational waves remove energy and angular momentum from a binary, forcing the eccentricity to decay and the orbit to circularize rapidly. Since the 1960s, the expectation has then been that the gravitational waves that ground-based detectors would observe would correspond to circular binaries.
But as with most things in physics, Nature adores the complex if one looks closely enough. Several astrophysical studies have recently shown that binaries may form with moderate eccentricities at orbital separations at which they would be emitting gravitational waves that ground-based detectors could observe very soon. These binaries would form near a supermassive black hole or in globular clusters, where three- or many-body interactions may source eccentricity. And if they are detected, they could shed light on their true population and formation scenario.
Unfortunately, the detailed and accurate characterization of a signal requires a similarly detailed and accurate model, which for eccentric binaries seems simply hopeless. In the circular case, the equations of motion can be solved in a weak-field and small velocity expansion. But in the eccentric case, the orbital equations are transcendental, a fact known to Kepler himself in the 1600s, and the inclusion of relativistic effects do not make these equations any simpler.
As physicists, of course, we know very well what to do: turn the nearly impossible problem into one that we have solved before. An ellipse looks an awful lot like a circle, so why not attempt to represent eccentric orbits as a sum of a sequence of circular orbits? Or better yet, one can try to represent an eccentric gravitational wave as a sum of circular ones. And what’s more, one can do this in the time domain or in the frequency domain, with the latter option being particularly appealing for data analysis applications.
Clearly then, one route to incorporate eccentricity in gravitational wave models requires a deep understanding of the circular case. Circular binaries emit gravitational waves at twice their orbital frequency. That is, if a circular binary were not inspiraling due to the emission of gravitational waves, the spectrum would be a delta function centered at twice the orbital frequency. Gravitational waves, however, chirp as the orbital separation shrinks due to gravitational wave energy loss, causing the delta function to broaden.
Let us use this insight to now consider the eccentric case. If eccentric gravitational waves can be represented as a sum of circular ones, its spectrum should be simply a sum of delta functions (if we momentarily ignore radiation-reaction), each centered at multiples of the mean orbital frequency of the eccentric orbit. Thus, as the orbital frequency chirps due to gravitational wave energy loss, each of these delta functions widens and interferes with one another, producing a messy looking spectrum.
The nature of an eccentric signal is thus better understood through a time-frequency representation, which we have provided in the included figure. Each of the separate tracks in time-frequency represents a term that looks like what one finds in the circular case. For each of these harmonics, there are then straightforward techniques to analytically compute the Fourier transform in the stationary phase approximation. The full eccentric signal is then the sum of all of the Fourier transformed harmonics.
The analogy presented above works quite well, but there are of course other complications that must be accounted for. Of utmost importance is the eccentric corrections to the energy and angular momentum flux due to gravitational radiation, which effectively controls the chirping rate. In our work, we have been able to solve for the phase evolution exactly to leading post-Newtonian order, but we have also been able to approximate this evolution in a computationally efficient manner. This analytic prescription for the Fourier phase evolution, together with the Fourier amplitudes described earlier then provide an analytic and computationally efficient model for gravitational waves emitted in eccentric inspirals.
While analytic and fast are appealing, the true value of the model must be gauged by analyzing its accuracy. We monitored this accuracy by computing the match between our model and a fully numerical one obtained by numerically solving the post-Newtonian equations of motion. The match is like a normalized dot product between two models, so a value of unity indicates perfect agreement, while a value of 0 indicates perfect disagreement. Our model leads to matches that are near 0.99 for initial eccentricities as high as 0.9. We are currently in the process of incorporating higher post-Newtonian order corrections to the model, which when completed will allow for a highly accurate, analytic and computationally efficient scheme to model eccentric waveforms.
About the author:
Blake Moore is a graduate student at Montana State University working with Professor Nico Yunes.
Read the full article in Classical and Quantum Gravity:
Towards a Fourier domain waveform for non-spinning binaries with arbitrary eccentricity
Blake Moore et al 2018 Class. Quantum Grav. 35 235006
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