J. Kissel

While reviewing the latest, best, offline-reconstructed, h(t) frames (aka C01 frames) for O3A, we noticed that the PCAL calibration line at 17.1 Hz reports a 7-10% discrepancy between the estimated displacement from PCAL and our reconstruction of h(t) at the beginning of many observation ready segments. See C01 summary pages; specifically H1-C01_HOFT_PCAL_RATIO_MAGNITUDE.png.

I think it is safe to assume that this is the low frequency sensing function response evolving as a function of time causing impact on the overall response function. Especially since we know that the O3 H1 IFO has has a long, sorted history with a confusing low frequency response as a result of avoiding ITM point absorbers, and we believe that that response is a mix between detuning of the signal recycling cavity and some parasitic length-to-angle-to-length control loop for which we have not yet accounted.

The first attachment shows the collection of O3B sensing functions, including the latest from 2019-12-04, demonstrating this issue for you.

One can immediately see from these 5 measurements, taken sparsely and sporadically over the past ~1.5 months, that there is evolution and a clustering of behavior, and it's clearly *not* "just" the response of a Fabry-Perot Michelson with a de-tuned signal recycling cavity like we've seen in O1/O2. But you knew this already. What newly discussed in this aLOG: all of these measurements are taken well in to the respective nominal low noise segment, when the IFO has thermalized.

Because the O3 IFO has never confirmed to the "simple" detuned response we developed in O1 and O2, and we don't have any physical model for what actually *is* happening -- the mix of several effects -- we've never been able to decompose the calibration lines into what we'd like -- parameters of a model of what the sensing function is doing as a function of time, such that we can correct for it. Thus, without that physical model of what's happening, we are left with treating the time-varying response as some sort of frequency dependent unknown systematic error.

To-date, we've "quantified" this unknown systematic error by running a Gaussian process regression on the *collection* of residuals between "sensing function sweep measurements of a thermalized IFO" and "the model" (i.e. the right two plots of the first attachment in this aLOG), and quite artificially craft a 68% confidence interval, uncertainty, envelope on the fit to cover all of the residuals. It's a quite arbitrary and qualitative solution determined entirely by eye, without any quantitative rigor. As such, we found that the artificial 68% CI envelope both (i) covered all of O3A's measurement residuals *and* (ii) did not significantly impact the overall response function uncertainty. 

As a reminder, the study that informed the limits of that envelop, can be found by flashing between slides 3 to 5 of the follow-up study of G1901479. That study showed "the systematic error of any one measurement show be no more than 8 (,16) % or 5 (,10) deg at 20 (,15) Hz (assuming the error is decreasing as a function of frequency as has been seen in the past) or else it will significantly impact the overall response function." So -- how do we know that the systematic error in the sensing function does not get larger than 8% at 20 Hz while the interferometer is thermalizing? The 17.1 Hz calibration line discrepancy between h(t) and PCAL seems to imply that it is indeed large and significant.

Without a physical model of what's going on -- what can we do? Well -- we can at least *phenomenologically* decompose what's happening in terms of parameters of the O1/O2 "simple" detuned model, since we know that this model -- at times -- can at least roughly approximate the shape of the more complicated, mixed, detuned SRC + L2A2L response -- e.g. like what we did at the beginning of O3 before we knew something more complicated was going on; see e.g. 2019-04-03_H1_sensingFunction.pdf.

Indeed, the GDS calibration pipeline has been doing exactly this throughout O3, using techniques similar to O2 (described in T1700106, section 2.6), to derive "optical spring" parameters. The only difference being that because the "detuned spring" has been both a "pro" (\xi^2 == f_s^2, positive, |f_s| is real in the "left-half plane", high-Q response like in August of O3) and an "anti," (\xi^2 == f_s^2 < 0, negative, |f_s| is imaginary in the "right half plane", like in O1/O2), the pipeline has been spitting out f_s^2 instead.

So -- in the hopes that I can reconcile the f_s^2 computed during all observation ready segments for O3B, against the thermalized sweeps -- and convince ourselves that this time-series may serve as a "good enough" approximation to what's happening at low frequency, allow us to approximate the sensing function systematic error while the IFO is thermalizing -- I attach a time series of the entire ~1 month of O3B data of 
   - the optical gain relative to the 2019-09-09 reference value ("\kappa_C"), 
   - the DARM coupled cavity pole ("f_cc"), and 
   - the "detuned optical spring pole frequency squared" (f_s^2), 
compared against the times at which we took the sensing function sweeps (which are *not* in observation ready segments, so the GDS computation stops). The colored vertical lines in the time series match the time and color of the transfer function colors shown in the first attachment. The first page of this second attachment is the entire data set, so you can see very long term trends. The following pages are zooms in on each week, such that you can better see what's going on at the start of each segment, and assess for yourself whether the sweep data roughly agrees with the f_s^2 time series. For the ~1 month trend, the minor tick marks are each day, and in the 1 week zooms, the tick marks are every 2 hours.

Observations from this time series that give me confidence that at least the trends in time dependence are real:
    (1) The sign convention appears to be discrepant between my impression of the response of a positive f_s^2 vs. a negative f_s^2 as shown by the sweeps, and what the GDS pipeline reports. A high Q feature, like what the 2019-11-04 and 2019-11-11 data show, is what I would call a "pro" spring, with a positive f_s^2, but the gds computed number says that f_s^2 was negative around the time of those sweeps. 
    (2) Later in this portion of the run, GDS reports that f_s^2 is "around zero" during the time of the 2019-11-20, 11-27, and 12-04 sweeps, which is roughly consistent with the flat response seen in those sweeps.  
    (3) This has already been concluded before in LHO aLOGs 52145 and 52139, but it's worth re-emphasizing here -- while reducing from wherever the "f_s^2" starts to 1/e of that takes 45 minutes, it takes about 2 hours for the IFO to completely thermalize.
    (4) During this thermalization time, one can also see the DARM cavity pole starting out "high" at ~420 Hz, and decaying to ~412-415 Hz, also over the same ~2 hour period. This is not unsurprising that these features are correlated.

Conclusions:
Thus -- if you, like me, based on the above, believe that this GDS time series of "f_s^2," regardless of its flaws and that it's an approximation to a more complex mixing of effects, is reporting at least something close to the right response at low frequency, then I conclude that:
    (a) the IFO response is swinging from a very *anti* spring-like response (repo rrted f_s^2 at "+50 Hz" == really -50 Hz >> |f_s| = 7 Hz) to a mildy pro-spring-like response once thermalized for the first few weeks of O3B. 
    (b) something happens after maintenance day on 2019-11-19, because we come back from it with the next observation ready segment and "f_s^2" settles to a near-zero value for the remainder of the run thus far -- which again is confirmed by sweeps subsequent to this date.
    (c) Something is happening to *even the thermalized settling point** in terms of the (phenomenological, reported) "detuned" response in between the sweeps -- so the sweeps definitely do not cover the entire representative response of the IFO.
    (d) The relative optical gain, \kappa_C pops back up to near unity on 2019-12-04 07:00 UTC, which is likely when our problems with the PSL's rotation stage started, and we're no longer delivering the full 37+ W to the IFO, but more like 36+ W, which is similar to the power level at the time of the 2019-09-09 model (remember, we regained a little bit of power during the O3 October break, because we touched up the PSL alignment).
    (e) Thus far, the "detuning" has gotten as bad as either a "anti" spring of (reported f_s^2 at "+70 Hz" == really -80 Hz >> |f_s| = ) 8.4 Hz during the start of the lock stretch on 2019-11-29 17:00 UTC, or a "pro" spring of (reported as f_s^2 "-20 Hz" == really +20 Hz >> |f_s| = ) 4.5 Hz during some mid-lock excursions on 2019-11-02 17:30 UTC and 2019-11-03.

There's *lots* more to dive in to as a result of this study, but I wanted to at least get these plots up, so you get periodic updates instead of a super blast. 

As a result of these plots, we're going to make an attempt to measure the sensing function "live" by installing many calibration lines tomorrow during maintenance, such that upon arrival at the power-up of the IFO, and subsequent hit of nominal low noise, we can watch the sensing function evolve as a function of time, through the thermalization.