Hello, I’m wondering if anyone could help me understand how the waveforms from GWOSC are generated. In the tutorials (here), we learned how to generate a waveform with certain parameters. In the Simulating Extreme Spacetimes (here) software, you are allowed to do the same but there seems to be more of a dependence on overtones. Could someone explain how GWOSC creates its waveforms and if anyone is familiar with SXS, could they explain what they believe the primary differences are between it and GWOSC in terms of generating waveforms? Thanks.

The procedure for generating waveforms through PyCBC given in the GWOSC tutorial allows you to generate waveforms using a large number of semianalytical waveform models coded up in the LALSuite library. The SEOBNRv4_opt waveform model used in the tutorial is what is known as an effective-one-body model that combines together resummed information from post-Newtonian and black hole perturbation theory calculations along with phenomenological terms that are tuned to reproduce the results of full numerical relativity (NR) simulations to generate binary black hole waveforms to good accuracy. The waveforms are generated in the time domain by solving a set of coupled ODEs, as described in the paper linked there. This model produces the dominant l = 2, m = +/- 2 spherical harmonic modes of the waveform for quasicircular systems with spins aligned with the orbital angular momentum (so no spin precession), but there is also the SEOBNRv4HM model (described in this paper), which includes the leading higher modes and a few additional ones, as well as SEOBNRv4PHM (described in this paper), which includes both spin precession and higher modes, and is commonly used in LVK data analysis, e.g., of all the black hole binaries detected in GWTC-3. I think that these higher modes are what you are referring to by overtones in your question. See Sec. III A in the GW190412 paper for an introduction to the higher modes; in the gravitational wave literature, *overtones* usually refers to more strongly damped quasinormal modes in a given spherical harmonic in the ringdown, as discussed in, e.g., this paper.

However, there are many other waveform models available through LALSuite, some of which are quite out of date and mostly kept for historical purposes/backwards compatibility, but others of which are up to date and of comparable accuracy to the SEOBNRv4 family and also used in LVK data analysis. Unfortunately, there is no publicly available overview of all the waveform models in LALSuite of which I am aware, but I will mention a few of the other models here (all for quasicircular binary black holes), which construct the waveforms in rather different ways than the SEOBNRv4 family does:

IMRPhenomXPHM (described in this paper) generates binary black hole waveforms in the frequency domain without solving ODEs, making it quite fast and thus frequently used for GW data analysis (e.g., of all the black hole binaries in GWTC-3), which is mostly carried out in the frequency domain. It is also constructed by tuning phenomenological additions to post-Newtonian and black hole perturbation theory results to reproduce the results of NR simulations, but here directly describing the frequency-domain waveform, rather than the time-domain dynamics and wave generation, as in the effective-one-body construction. It also includes a slightly different selection of higher modes than SEOBNRv4[P]HM does. (There is also a frequency-domain reduced order model for SEOBNRv4HM, SEOBNRv4HM_ROM, described in this paper, which allows it to be generated much more quickly, since one is interpolating directly in the frequency domain, not solving ODEs. There is also a recent reduced order model for SEOBNRv4PHM described in this paper, but it is not in LALSuite, likely because most work is devoted to the updated SEOBNRv5 family of models that is still in development.)

NRSur7dq4 (described in this paper) is an NR surrogate model, which directly interpolates the results of NR simulations of precessing binary black holes in the time domain, so it is quite accurate, and includes all the spherical harmonic modes of the waveform through l = 4, while the other models mentioned just include a selection of these. However, it is restricted by the parameter space coverage and length of the NR simulations (since long NR simulations are quite expensive), and thus can only be applied to analyze high-mass binary black hole signals, where the signal in the sensitive band of the detectors is quite short (e.g., GW190521), and also cannot describe binaries with a mass ratio larger than 6 to 1 (already extrapolating from its training region of binaries with a mass ratio up to 4 to 1).

There are also waveform models for binary neutron stars and black hole-neutron star systems, which I can mention if you are particularly interested in these, as well as the NR_hdf5 model, which is not a waveform model, but rather an interface to numerical relativity simulations (from the SXS collaboration and other groups) that have been converted to the appropriate format.

All the LALSuite waveform models except NR_hdf5 can generate a waveform for arbitrary masses and spins (up to restrictions on the parameter space from the waveform model, as discussed above, e.g., for NRSur7dq4). The SXS simulations, however, cover a discrete set of mass ratios and spins (though the binary black hole waveforms can be scaled to any total mass) and are all restricted in length by the length of the simulation. However, they include more spherical harmonic modes than do any of the waveform models (all the ones up to l = 8, though the very low-amplitude ones are not reliable, as discussed in the tutorial). They are also the most accurate predictions, since they come directly from solving the Einstein equations (which involves solving nonlinear PDEs and thus requires serious computation), so their accuracy is limited only by the resolution of the simulation as well as any inaccuracies in the initial data and waveform extraction (see, e.g., this paper for a description of how the simulations are carried out).

From a practical standpoint, the waveforms you generate using the method from the GWOSC tutorial are always for a source at a given distance and inclination angle, while the SXS waveforms start from the spherical harmonic modes, which you can then evaluate for a binary at a given orientation using the evaluate method, but which still does not scale by the distance.

Thank you for such a detailed response. It will take me some time to go through these links. This was extremely helpful.