SXS Gravitational Wave Data: Initial Conditions Explained

[BillKet]

Hello! I need to do some analysis for a project with the SXS gravitational wave data: https://data.black-holes.org/waveforms/catalog.html but I am a bit confused about the initial conditions of their simulations. I read the paper they published about the data (it can be found at that website) but i am still not sure I understand. I attached several weaveforms I extracted from their database (the y-axis has $h_+$). So it seems like the starting time (in their units) is the same for all the waveforms. But I am not sure how that is defined. More specifically, for the purpose of my project, I would like to know what defines the initial frequency of each gravitational wave in their simulations. Can someone help me with this? Thank you!

 

[phyzguy]

It sounds complicated. What I read in their paper is the following:

"We iteratively construct BBH initial data, tuning the initial data to achieve a BBH
with the desired properties. Our iterative scheme uses two nested loops. The inner loop
solves the XCTS equations, adjusting our choices for the free data (conformal metric,
trace of extrinsic curvature, and the time derivatives of each) and boundary conditions,
until the resulting BBHs have the desired mass ratio and spins [132, 133]. The outer
loop briefly (typically for a few orbits) evolves the initial data resulting from the inner
loop, and adjusts the initial coordinate velocities to yield a BBH with small orbital
eccentricity [46,47,134], typically e0 ∼ 10−4 as defined in Eq. (17). For some simulations
in the SXS catalog, we intentionally omit the eccentricity-reduction loop, to obtain initial
data for BBHs with non-negligible orbital eccentricity."

They also say, "Here Ncyc is approximated by doubling the
number of orbits during inspiral up to merger (when a common horizon forms), as
determined by the coordinate trajectories of the black holes."

So it sounds to me like they are doing the following:
(1) They have some definition of the start of the inspiral, probably determined by the rate of change of the orbital period.
(2) They then count the number of cycles (call it N) from the start of inspiral until coalescence.
(3) They then double this to give Ncyc = 2N. So they start N cycles before the start of inspiral, as defined in step 1.

If this is the case, then the only question is how they determine the start of insprial in step 1. You could try looking at your extracted waveforms Ncyc/2 cycles from the beginning and see if you can determine what they are doing. Or in the "Help and Documentation" tab, there is a link to send them an Email, so you could ask them.

 

[Ibix]

@BillKet - it would be helpful to link to the paper you refer to, rather than say "it can be found at the website", since there are multiple arXiv postings linked in various places. I think you refer to this one.

(3) They then double this to give Ncyc = 2N. So they start N cycles before the start of inspiral, as defined in step 1.

I disagree with this - I think Ncyc is the number of wave cycles, which is twice the number of orbits. This is because every time the black holes complete half orbits, you're back to something that looks like the start position, so the gravitational wave must have completed a cycle. I think this is approximate because unequal mass holes don't look exactly the same after a half orbit (they've swapped position) and I would expect there to be some qualitative difference between even and odd numbered peaks.

But yes, it looks to me like they define an initial orbit period (presumably more or less the same in all cases) and tune their simulation to give this.