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Entry  Mon Jul 18 04:42:54 2022, Anchal, Update, Calibration, Error propagation to astrophysical parameters from detector calibration uncertainty BNSparamsErrorwrtfdError-merged.pdfBBHparamsErrorwrtfdError-merged.pdfBNSparamsEPSwrtCalError.pdfBBHparamsEPSwrtCalError.pdf
    Reply  Mon Jul 18 15:17:51 2022, Hang, Update, Calibration, Error propagation to astrophysical parameters from detector calibration uncertainty 
       Reply  Tue Jul 19 07:34:46 2022, Anchal, Update, Calibration, Error propagation to astrophysical parameters from detector calibration uncertainty BNSparamsErrorwrtfdError.pdfBBHparamsErrorwrtfdError.pdfBNSparamsEPSwrtCalError.pdfBBHparamsEPSwrtCalError.pdf
          Reply  Sun Jul 24 08:56:01 2022, Hang, Update, Calibration, Error propagation to astrophysical parameters from detector calibration uncertainty 
Message ID: 17017     Entry time: Tue Jul 19 07:34:46 2022     In reply to: 17011     Reply to this: 17029
Author: Anchal 
Type: Update 
Category: Calibration 
Subject: Error propagation to astrophysical parameters from detector calibration uncertainty 

Addressing the comments as numbered:

  1. Yeah, that's correct, that equation normally \Delta \Theta = -\mathbf{H}^{-1} \mathbf{M} \Delta \Lambda but it is different if I define \Gamma bit differently that I did in the code, correct my definition of \Gamma to :
    \Gamma_{ij} = \mu_i \mu_j \left( \frac{\partial g}{\partial \mu_i} | \frac{\partial g}{\partial \mu_j} \right )
    then the relation between fractional errors of detector parameter and astrophysical parameters is:
    \frac{\Delta \Theta}{\Theta} = - \mathbf{H}^{-1} \mathbf{M} \frac{\Delta \Lambda}{\Lambda}
    I prefer this as the relation between fractional errors is a dimensionless way to see it.
  2. Thanks for pointing this out. I didn't see these parameters used anywhere in the examples (in fact there is no t_c in documentation even though it works). Using these did not affect the shape of error propagation slope function vs frequency but reduced the slope for chirped Mass M_c by a couple of order of magnitudes.
    1. I used the get_t_merger(f_gw, M1, M2) function from Hang's work to calculate t_c by assuming f_{gw} must be the lowest frequency that comes within the detection band during inspiral. This function is:
      t_c = \frac{5}{256 \pi^{8/3}} \left(\frac{c^3}{G M_c}\right)^{5/3} f_{gw}^{-8/3}
      For my calculations, I've taken f_{gw} as 20 Hz.
    2. I used the get_f_gw_2(f_gw_1, M1, M2, t) function from Hang's work to calculate the evolution of the frequency of the IMR defined as:
      f_{gw}(t) = \left( f_{gw0}^{-8/3} - \frac{768}{15} \pi^{8/3} \left(\frac{G M_c}{c^3}\right)^{5/3} t \right)^{-3/8}
      where f_{gw0} is the frequency at t=0. I integrated this frequency evolution for t_c time to get the coalescence phase phi_c as:
      \phi_c = \int^{t_c}_0 2 \pi f_{gw}(t) dt
  3. In Fig 1, which representation makes more sense, loglog of linear axis plot? Regarding the affect of uncertainties on Tidal amplitude below 500 Hz, I agree that I was also expecting more contribution from higher frequencies. I did find one bug in my code that I corrected but it did not affect this point. Maybe the SNR of chosen BNS parameters (which is ~28) is too low for tidal information to come reliably anyways and the curve is just an inverse of the strain noise PSD, that is all the information is dumped below statistical noise. Maybe someone else can also take a look at get_fisher2() function that I wrote to do this calculation.
  4. Now, I have made BBH parameters such that the spin of the two black holes would be assumed the same along z. You were right, the gamma matrix was degenerate before. To your second point, I think the curve also shows that above ~200 Hz, there is not much contribution to the uncertainty of any parameter, and it rolls-off very steeply. I've reduced the yspan of the plot to see the details of the curve in the relevant region.
Quote:

1. In the error propogation equation, it should be \Delta \Theta = -H^{-1} M \Delta \Lambda, instead of the fractional error. 

2. For the astro parameters, in general you would need t_c for the time of coalescence and \phi_c for the phase. See, e.g., https://ui.adsabs.harvard.edu/abs/1994PhRvD..49.2658C/abstract.

3. Fig. 1 looks very nice to me, yet I don't understand Fig. 3... Why would phase or amplitude uncertainties at 30 Hz affect the tidal deformability? The tide should be visible only > 500 Hz.

4. For BBH, we don't measure individual spin well but only their mass-weighted sum, \chi_eff = (m_1*a_1 + m_2*a_2)/(m_1 + m_2). If you treat S1z and S2z as free parameters, your matrix is likely degenerate. Might want to double-check. Also, for a BBH, you don't need to extend the signal much higher than \omega ~ 0.4/M_tot ~ 10^4 Hz * (Ms/M_tot). So if the total mass is ~ 100 Ms, then the highest frequency should be ~ 100 Hz. Above this number there is no signal.

 

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