High-order asymptotics for the Spin-Weighted Spheroidal Equation at large real frequency



We recently put out a paper to compute the high-order, large-frequency expansions for the eigenvalue and the eigenfunction of the spin-weighted spheroidal equation. The spin-weighted spheroidal equation turns up in a number of places in physics, and I most often encounter it using the Teukolsky formalism to model gravitational wave emission. A great strength of our paper is the combination of analytic and numerical results. The topic of high-frequency expansion of spin-weighted harmonics has been addressed before. Evaluating the results of that work relied on the results of an earlier work which had an error in it. Careful comparison with numerical calculations brought this to light and our recent paper corrects and extends the literature.


Our work provides formula to compute a high-frequency expansion of the eigenvalue of the spin-weighted spheroidal equation. Code to compute this series expansion has also been made publicly available in the SpinWeightedSpheroidalHarmonics Mathematica package of the Black Hole Perturbation Toolkit. Details are provided in the paper. We also provide an example notebook which shows how to use the new feature and gives code to compute the coefficients of the expansion of the harmonic.


This work is in collaboration with Marc Casals and Adrian Ottewill.



LISA consortium: waveform working group

Shortly after the LISA consortium reboot it was clear that in addition to the Cosmology, Fundamental Physics and Astrophysics Working Groups under the LISA Science Group there was as a need to have a Waveform Working Group. I happy to say this group has now been created with Deirdre Shoemaker, Helvi Witek, Maarten van de Meent and myself as the co-chairs.

If you are a full or associate member of the consortium and interested in waveform modeling for LISA then join the new Waveform Working Group!

PhD studentship

I have funding for a PhD student to work on modeling gravitational wave emission from extreme mass-ratio inspirals with me at University College Dublin, Ireland. For more information, and to contact me if you are interested, see the advert. If you have any questions about the project please do not hesitate to get in touch.


Gravity and Light

The gravitational wave news just keeps coming. This time it is really big news: the first gravitational wave event with an electromagnetic counterpart. Roughly 130 million years ago in the galaxy NGC 4993 two neutron stars ended their billions of years long dance with a violent collision. In their final throes they disturbed space and time just enough for the outgoing gravitational waves to be detectable after their 130 million light year journey to Earth. Shortly after the gravitational waves were detected a gamma-ray burst was detected by the Fermi space telescope and so began the era of multi-messenger astronomy. A great many follow up observations were made and a lot of exciting science was done by thousands of scientists around the world.

With all the information released today it will take a lot of time to digest it all. Better get reading!

Screenshot 2017-09-27 18.20.52

Three-detector observation of gravitational waves from a binary black hole coalescence

Congratulations to the LIGO-Virgo collaboration on the 4th detection of gravitational waves! The source of this detection is again merging black holes. This one is special as it is the first time the Virgo collaboration has joined the detection. Having a third detector results in substantially better sky localization (down from ~1000 deg^2 to ~100 deg^2) and the ability to test the polarization states of the incoming waves.

Information on the detection and the detection announcement paper can be found at http://www.ligo.org/detections/GW170814.php. Mark Hannam also has a nice blog post about the event.

Screenshot 2017-09-14 19.43.37

Happy 2nd birthday GW150914!

It’s hard to believe that it has already been two years since the first detection of gravitational waves*. So much has happened since then: two more confirmed detections of merging black holes, the success of the LISA Pathfinder mission, and the funding by the European Space Agency of the LISA mission. With rumors swirling again about a possibly detection of merging neutron stars, it remains a very exciting time to be gravitational wave scientist.

There have been a few things released today to celebrate GW150914’s birthday. I enjoyed this introduction to gravitational waves on TEDEd:

* After 1.5 years, the detection paper has over 1700 citations already!


Royal Society – Science Foundation Ireland University Research Fellowship

I have been very fortunate to have been awarded a Royal Society – Science Foundation Ireland University Research Fellowship. The Fellowship will see me continuing to work at University College Dublin for 5 years after my Marie Curie Fellowship finishes at the end of September.

During the Fellowship I will continue my work on black hole perturbation theory. As well as computing new perturbative results a particular focus of the Fellowship will be incorporating those results into practical waveform template generation schemes ready for when LISA launches.

Further information about the Fellowship can be found at the Royal Society webpage along with the full list of awardees.


Kerr orbit visualizer

In March I wrote a code to compute the frequencies and constants of motion associated with generic, bound, timelike Kerr geodesics. Back then I said I wanted to write an online tool for visualizing the associated orbits. A few weeks back I did just that and I finally have time to share it and write a little about it now. Rather than trying to embed it in the WordPress layout the Kerr timelike orbit visualizer can be found here.

The tool is basic but it will plot most orbits. Note it wont tell you if you enter parameters that do not correspond to Kerr a bound geodesic, you’ll just get a blank plot. Also, special cases like \(a=0, e=0\) and \(\theta_\text{inc}=0\) are not implemented (but you can set a very close to zero value to get it to work).

You can plot the orbit in Boyer-Lindquist coordinates and also in co-rotating coordinates [as defined in Eq. (3.2) of arXiv:0904.3810] where the orbit usually looks much simpler. A nice little feature, which works in most modern browsers, is you can `play’ the orbital frequencies*. I like really like this feature as it allows you to hear the character of each orbit. If you find a set of parameters near a resonance you can hear the beating between the fundamental frequencies.

This visualizer is pretty basic but I’ve had discussions with Leo Stein and Scott Hughes about improving it. In particular, it would be nice to make it more user friendly, provide more information as the orbits are plotted, give a nice set of example orbits, plot the black hole, allow animation of the orbit, etc. It might also be possible to speed up the calculating using fast Fourier transform methods.

The current code uses plotly.js to visualize the orbit. The orbit is computed by numerically integrating the geodesic equations with a fixed-step RK4 integrator found in JSXgraph. Much of the inspiration to create this online visualizer came from Leo Stein’s visualizer for bound, spherical null geodesics in Kerr. Documenting the equations the code solves, as Leo does so nicely, is something that also needs to be done.

* the frequencies do not correspond to any astrophysical extreme mass-ratio binary as such systems have frequencies in the milli-Hertz regime which cannot be heard by humans. Instead I just increase all the frequencies by a constant multiple to make them audible.