Interesting Sensing Function Results: Confirmation of Evolving Detuned SRC Optical Spring Seen By Time-Dependent Correction Factors, Impacting Response Function Systematic Error

Using the data collected above, I'm able to 
    - Confirm that the ~40 minute time constant for the regular evolution / thermalization of the signal recycling cavity's detuned optic spring frequency seen in the first attachment (an example from the 2019-09-25 summary page) and recently stacked over several re-acquisitions (see LHO aLOG 52139, because it's present at the start of every recent observation stretch) is a real effect during the interferometer's settling after powering up to 37 W. (Note: in the acquisition process, the time between "the input laser power has reached 37 W" and "we're ready for observation" is typically around 5 minutes, much less time than the thermalization takes.)

    - Though the estimate of the spring frequency squared (\xi^2 == f_s^2) from the calibration lines appears to suggest an evolution from "pro-spring" (\xi^2 == f_s^2 > 0, positive, like in August of O3) to just-barely-anti-spring (\xi^2 == f_s^2 < 0, negative, like in O1/O2) -- if any spring at all -- this contradicts the sweep data, which suggests the evolution instead goes from what we know to be an anti-spring response to just-barely-a-pro-spring.

    - At the beginning of the lock stretch, this does indeed result in significant magnitude and phase systematic error in the over all response function, contrary to popular belief that tracking this carefully ''doesn't matter, because the spring frequency is well below the DARM loop UGF.'' 

    - We also can confirm that during this evolution, that systematic error is NOT covered by our recent estimate of the time-independent, 68% confidence interval of the uncertainty budget (from LHO aLOG 52132)
   
    - Finally, once thermalized, however, the measured systematic error is remarkably consistent between lock stretches, and our propagated estimate of that error (via the guassian progress regression) is consistent with measurement.

Attachments to support this:
(1) H1-LOCKED_8DAD23_TIMESERIES-1253404818-86400_TDCFs_f_s_squared_During_Powerup.png: Again, this is an example report from the 2019-09-25 summary page for time-dependent correction factors (a.k.a. TDCFs; see full page here). However, it perfectly demonstrates one unperturbed acquisition (first decay of f_s^2), and the decay in the middle of which I was performing all of the 2019-09-25 measurements.

(2) 2019-09-25_H1_PrevsPostThermalize_sensingFunction_referenceModel_vs_allMeasurements.pdf: This is the standard analyzed plot for sensing functions, comparing against the new 2019-09-09 model -- which has no spring response in the model -- and it shows 
    (a) all of the sensing function measurements that went in to informing the 2019-09-09 model (all taken well in to a long observation stretch), and
    (b) the "Pre" and "Post" thermalized data from 2019-09-25. 
One can clearly see the anti-spring response in the 2019-09-25 Pre-thermalized data -- and note that this was taken at ~20:14 UTC, ~45 minutes after the start of the relevant observation ready segment 19:24 UTC. 

(3) 2019-09-25_H1_PreThermalize_sensingFunction.pdf Looking to gather estimates of what the spring frequency is during this already-45-minutes-thermalized anti-spring "pre-thermalized" measurement, in order to compare it against the calibration-line-reported-estimate of f_s^2, I used our standard MCMC infrastructure, and tested fits with a range from 5 to 5000, and from 20 to 5000 Hz (the latter of which we've used for thermalized data to inform the 2019-09-09 model). The results for the 5 Hz and up fit, which I believe are better are
    f_cc = 416.4 +/- 0.75 Hz
    f_s  = 2.137 +/- 0.03 Hz #anti-spring.
(Note: the optical gain / \kappa_C estimate in both cases agrees with the reference model to within 0.5%, and the Q_s estimate has always been poor / meaningless because of the remaining unknown response we think may be due to parasitic L2A2L coupling.)
This MCMC fit f_s = 2.137 Hz would correspond to an f_s^2 of 4.6 Hz^2, which is indeed roughly consistent with the typical time-series from the calibration line estimate, given that the data was taken ~2/3rd of the way down the exponential decay (and considering the time-constant analysis in LHO aLOG 52139).

(4) https://alog.ligo-wa.caltech.edu/aLOG/uploads/52145_20190926092701_2019-09-25_H1_PostThermalize_sensingFunction.pdf Now MCMC fitting the "Post Thermalization" data, taken around 21:33 UTC well after thermalization, I argue the fit using only-above-20 Hz is better, and it reveals 
     f_cc = 411.1 +/- 0.75 Hz
     f_s = 0.105 +/- 0.06 Hz #pro-spring
which is very much consistent with what's used in the 2019-09-09 model (411.3 Hz for f_cc, and 0.0 Hz for f_s, i.e. no detuning) as expected.

(5) 2019-09-26_H1_deltaL_over_pcal.pdf This is our money plot, comparing two ratios:
    (a) R_sample / R_MAP = "the uncertainty budget" = the reconstructed response function ratio, where the numerator is the 68% confidence interval of a distribution of response functions created by sampling the correlated posterior distributions of uncertainty in both the original model parameters and the estimated residual unknown systematic error (and its uncertainty), R_sample, and the denominator is the response function using the static parameters of the 2019-09-09 model. 
    (b) Delta L_PCAL / Delta L_EXT = The one-time measurements of a driven transfer function between PCAL and our front-end produced estimate of DELTAL_EXTERNAL, corrected for every known flaw of the front-end produced h(t), i.e. the ratio of R_GDS / R_CALCS. (For more details on the correction and why we need it, see T1900169).

For (b), I show 5 measurements -- 
    (i) a previous, 2019-09-16 broad-band injection when the IFO was well-thermalized. Consistent with uncertainty budget. Good!
    (ii) the broadband injection taken as soon as we went out of observing to start the 2019-09-25 calibration suite (~3 minutes of data at starting 19:51 UTC, ~25 minutes after achieving 37 W).
    (iii) the 2019-09-25 \Delta L / PCAL swept sine data taken at the exact same time as the "pre-thermalized" sensing function DARM_IN1/PCAL sweep described in (2) above that reports a spring frequency of 2.1 Hz (only!)
    (iv) the 2019-09-25 \Delta L / PCAL swept sine data taken at the exact same time as the "post-thermalized" sensing function DARM_IN1/PCAL sweep described in (2) that reports no spring.
    (v) a final broadband injection at the close of the measurement period.

(ii) shows a deviation of the systematic error on the level of 1% (at 60 Hz) to 2% (at worst at 25 Hz) in magnitude, and 1 deg in phase at 40 Hz.
(iii) shows that we're mostly "recovered" to the thermalized level of systematic error / uncertainty by 45 minutes after power up.
(i), (iv), (v) are all quite consistent.

See conclusions above. Stay tuned for further action.
    
Scripts to produce the attached plots:
    (1) is from the summary pages, as linked.
/ligo/svncommon/CalSVN/aligocalibration/trunk/Runs/O3/H1/Scripts/FullIFOSensingTFs/
    (2) process_sensingmeas_collection_20190925_PrevsPostThermalize.py
    (3) process_sensingmeas_20190925.py
    (4) process_sensingmeas_20190925_PostThermalize.py
/ligo/svncommon/CalSVN/aligocalibration/trunk/Runs/O3/H1/Scripts/CALCS_FE/
    (5) process_broadband_pcal2darmtf_collection_20190925.py