We can calculate how much detector calibration uncertainty affects the estimation of astrophysical parameters using the following method:
Let be set of astrophysical parameters (like component masses, distance etc), be set of detector parameters (like detector pole, gain or simply transfer function vaue for each frequency bin). If true GW waveform is given by , and the detector transfer function is given by , then the detected gravitational waveform becomes:
One can calculate a derivative of waveform with respect to the different parameters and calculate Fisher matrix as (see correction in 40m/17017):
where the bracket denotes iner product defined as:
where is strain noise PSD of the detector.
With the gamma matrix in hand, the error propagation from detector parameter fractional errors to astrophysical paramter fractional errors is given by (eq 26 in Evan et al 2019 Class. Quantum Grav. 36 205006):
where and .
Using the above mentioned formalism, I looked into two ways of calculating error propagation from detector calibration error to astrophysical paramter estimations:
Using detector response function model:
If we assume detector response function as a simple DC gain (4.2 W/nm) and one pole (500 Hz) transfer function, we can plot conversion of pole frequency error into astrophysical parameter errors. I took two cases:
- Binary Neutron Star merger with star masses of 1.3 and 1.35 solar masses at 100 Mpc distance with a of 500. (Attachment 1)
- Binary black hole merger with black masses of 35 and 30 at 400 MPc distance with spin along z direction of 0.5 and 0.8. (I do not fully understand the meaning of these spin components but a pycbc waveform generation model still lets me calculate the effect of detector errors) (Attachment 2)
The plots are plotted in both loglog and linear plots to show the order of magnitude effect and how the error propsagation slope is different for different parameters. 'm still not sure which way is the best to convey the information. The way to read this plot is for a given error say 4% in pole frequency determination, what is the expected error in component masses, merger distance etc. I
Note that the overall gain of detector response is not sensitive to astrophysical error estimation.
Using detector transfer function as frequency bin wise multi-parameter function
Alternatively, we can choose to not fit any model to the detector transfer function and simply use the errors in magnitude and phase at each frequency point as an independent parameter in the above formalism. This then lets us see what is the error propagation slope for each frequency point. The hope is to identify which parts of the calibration function are more important to calibrate with low uncertainty to have the least effect on astrophysical parameter estimation. Attachment 3 and 4 show these plots for BNS and BBH cases mentioned above. The top panel is the error propagation slope at each frequency due to error in magnitude of the detector transfer function at that frequency and the bottom panel is the error propagation slope at each frequency due to error in phase of the detector transfer function.
The calibration error in magnitude and phase as a function of frequency would be multiplied by the curves and summed together, to get total uncertainty in each parameter estimation.
This is my first attempt at this problem, so I expect to have made some mistakes. Please let me know if you can point out any. Like, do the order of magnitude and shape of error propagation makes sense? Also, comments/suggestions on the inference of these plots would be helpful.
Finally, I haven't yet tried seeing how these curves change for different true values of the merger event parameters. I'm not yet sure what is the best way to extract some general information for a variety of merger parameters.
Future goals are to utilize this information in informing system identification method i.e. multicolor calibration scheme parameters like calibration line frequencies and strength.
Code location |