Adiabatic waveforms for extreme mass-ratio inspirals via multivoice decomposition in time and frequency

Before we get to our new paper, our previous letter was published today in Physical Review Letters. In the letter we demonstrated a method to rapidly compute fully relativistic EMRI waveforms. As part of that work we created the Fast EMRI Waveforms (FEW). Since we put out the letter the framework has been extended to have a generic waveform interface. With this new interface you can now access generic Kerr kludge waveforms computed using the Augmented Analytic Kludge (AAK). Via the generic interface you can also calculate our fast, fully relativistic Schwarzschild waveform in the detector frame. The plan is that the FEW framework will be used in future LISA Data Challenges (LDCs), and as such the older EMRI Kludge Suite is now discontinued.

 

Ok, on to our new paper out today. In this work we look at computing adiabatic waveforms for EMRIs in both the time and frequency domain. There are a few nice result in the paper but the main two are (i) direct calculation of EMRI waveforms in the frequency domain and (ii) the calculation of waveforms for generic inspirals into Kerr black hole. Let’s look at each of these.

 

Frequency domain waveforms: Often we think of EMRI waveforms as a sum of many “voices”, where each voice corresponds to a particular multipolar and frequency mode of the radiation. Each voice has a simple chirping behaviour that is similar to the well known waveforms for quasi-circular binaries. If you like you can think of an EMRI waveform as being equivalent to thousands of simultaneous chirping binaries (with a precise phase and amplitude relation between them). There is one key difference though between the quasi-circular case and the voices of the EMRI: the latter can both chirp (\(\dot{f} >0\)) and backwards chirp (\(\dot{f} < 0\)). The figure below shows the amplitude (top panel) and frequency (bottom panel) of a voice for an eccentric, equatorial Kerr inspiral with \(a=0.9, p=12, e=0.7\). In the bottom panel you can clearly see that \(\dot{f} = 0\) near the end of the inspiral

Screenshot 2021-02-05 at 16.57.08

Ok, so what? Well, as EMRIs evolve slowly we can appeal to the stationary phase approximation (SPA) to compute the Fourier transform of the waveform. Unfortunately the standard SPA breaks down when \(f=0\) and so in this work we extended the SPA to handle this case. When then apply our ‘extended SPA’ to real EMRI data and compute the frequency-domain waveform. As a check on our calculation we compare the results with the Discrete Fourier Transform (DFT) of the equivalent time-domain waveform.

Screenshot 2021-02-05 at 17.00.42

Generic inspirals into a Kerr black hole:  The second main result this paper is the calculation of waveforms (in both the time and frequency domain) for generic (eccentric and inclined) inspirals into a Kerr black hole. This involves computing the fluxes and mode amplitudes across a region of the 3D parameter space. From the fluxes we can compute the (phase space) inspiral and we can stitch the amplitudes together along this inspiral to compute the waveform in the time-domain, or frequency domain using the extended SPA discussed above. There are some additional phase corrections required in this calculation that have been discussed elsewhere but here we implement them for the first time. We verify our ‘stitched together’ waveforms against waveforms computed directly using a 2+1 time-domain Teukolsky code. The result is beautiful generic Kerr EMRI waveforms (as well as spherical and eccentric, equatorial orbits).

Screenshot 2021-02-05 at 17.08.55

What Next?  The present work can calculate frequency domain and generic Kerr waveform (driven by adiabatic fluxes). Our next goal is to incorporate this work in to the Fast EMRI Waveforms (FEW) framework. There are some new challenges here: the parameter space is now 3D (rather than 2D as in Schwarzschild) and we need methods to rapidly compute or interpolation the spheroidal harmonics. Implementing extended SPA in an efficient way will also require some thought. These are tasks for another day.