Calibration uncertainty for Advanced LIGO’s first and second observing runs

Calibration of the Advanced LIGO detectors is the quantification of the detectors’ response to gravitational waves. Gravitational waves incident on the detectors cause phase shifts in the interferometer laser light which are read out as intensity fluctuations at the detector output. Understanding this detector response to gravitational waves is crucial to producing accurate and precise gravitational wave strain data. Estimates of binary black hole and neutron star parameters and tests of general relativity require well-calibrated data, as miscalibrations will lead to biased results. We describe the method of producing calibration uncertainty estimates for both LIGO detectors in the first and second observing runs.

Figure 1

Simplified Advanced LIGO interferometer layout. Laser light enters the interferometer at the lower left through a power-recycling mirror, and is split by a $50/50$ beamsplitter. Two 4 km long Fabry-Pérot resonating arm cavities are formed from four highly reflective test masses. Laser light builds up in the cavities, reaching about 100 kW of laser power in O1 and O2. A signal-recycling mirror between the beamsplitter and GW readout photodetector modifies the detector response to increase the detector bandwidth. Inset: one of the quadruple pendulum suspension systems which holds each of the four test masses is shown.

Figure 2

DARM control loop and calibration procedure. The DARM control loop is shown in the grey box on the left. The sensing plant $C$ produces the detector output $d_{\mathrm{err}}$ in linear response to residual differential arm motion $\mathrm{\Delta }{L}_{\mathrm{res}}$. The digital filters $D$ are known filters conditioning the detector output $d_{\mathrm{err}}$ into a control signal $d_{\mathrm{ctrl}}$. The actuation plant $A$ takes the control signal $d_{\mathrm{ctrl}}$ and actuates on the optics by $\mathrm{\Delta }{L}_{\mathrm{ctrl}}$ to maintain cavity resonance. The pink box on the right shows the calibration procedure, consisting of an inverse sensing model $1/C^{(\mathrm{model})}$ and actuation model $A^{(\mathrm{model})}$. The output of the calibration pipeline is GW strain data $h(t)$.

Figure 3

Sensing measurements $C^{(\mathrm{meas})}(f)$, sensing model $C^{(\text{model})}(f,t,{\overrightarrow{\lambda }}_C)$, and their residuals $\delta C(f,t)/C^{(\text{model})}(f,t,{\overrightarrow{\lambda }}_C)$. The H1 sensing reference measurement from January 4th, 2017 is shown in the four panels on the left in red. The L1 sensing reference measurement from November 26th, 2016 is in the four right panels in blue. The first and third columns are the Bode plots of $C$, while the second and fourth columns are the residuals. The model parameters ${\overrightarrow{\lambda }}_C$ were found via an MCMC. Physically, the magnitude Bode plots represent how many milliamps of current are generated at our transimpedence photodetector per picometer of differential arm motion from 5 to 5000 Hz. The drop in sensitivity at low frequencies shows the effect of detuning at both detectors. The 180 degree phase difference between H1 and L1 is a sign convention difference between the detectors.

Figure 4

Actuation stage measurements $[H_i{A}_i(f){]}^{(\mathrm{meas})}$ and models $[H_i{A}_i(f,{\overrightarrow{\lambda }}_i){]}^{(\text{model})}$. Each index $i$ is one of the actuation stages $U$, $P$, or $T$. The H1 actuation reference measurements from January 4th, 2017 are shown in the two left plots in red. The L1 actuation reference measurements from November 26th, 2016 are in the two right panels in blue. The model parameters ${\overrightarrow{\lambda }}_A$ for $A_i(f,{\overrightarrow{\lambda }}_i)$ have been found via the MCMC method. The actuation strength magnitude is in units of meters per $d_{\mathrm{ctrl}}$ count. Notches seen in the magnitude plot are purposefully placed to avoid ringing up suspension violin modes at specific frequencies. Each stage’s phase is sensible for frequencies at which each actuation stage dominates, but then rolls rapidly as it loses authority at high frequencies. For this reason, the UIM and PUM stage phase plots are cut off at 300 Hz and 400 Hz respectively.

Figure 5

Time dependent parameters in L1 around GW170104. The time series of the optical gain scale factor ${\kappa }_C(t)$ and coupled cavity pole $f_{CC}(t)$ in L1 for the two hours surrounding GW170104. The reference values are plotted as a red line, while the data is blue dots with uncertainty bars. The values at the time of GW170104 are plotted in orange. Changes in ${\kappa }_C(t)$ are updated in the reference sensing model $C^{(\text{model})}(f,t)$. Changes in $f_{CC}(t)$ are not reflected in the reference sensing model. This represents a systematic error in the calibration pipeline strain data $h(t)$.

Figure 6

Posterior distribution on the H1 sensing parameters ${\overrightarrow{\lambda }}_C$. Each column represents one of the five sensing parameters: optical gain $H_C$, coupled cavity pole $f_{CC}$, time delay correction $\delta {\tau }_C$, optical spring $f_S$, and optical spring inverse quality factor $Q_S^{-1}$. Each point represents a sample in five dimensional parameter space. The diagonal plots represent the variance on each parameter, while the off-diagonal plots show the covariance of each parameter with another. The dashed vertical lines on the diagonal plots represent the median and $1\sigma$ values for each parameter.

Figure 7

Gaussian process regression of L1’s sensing systematic error $\delta C^{GP}(f)$. The dark blue points are all the sensing measurement residuals, $\delta C/C^{(\text{model})}(f,t,{\overrightarrow{\lambda }}_C)$, taken over the entire observation run. This includes the residuals from the L1 reference measurement in the far right plots of Fig. 3. The light blue line is the mean function representing systematic error. The light orange envelope is the $1\sigma$ uncertainty on the systematic error.

Figure 8

Total calibration error and uncertainty budget at the time of GW170104. The uncertainty in the calibrated response function for the H1 detector is on the left, and for L1 is on the right. The $y$ axis is relative response error $\delta R/R^{(\text{model})}$ and uncertainty ${\sigma }_R/R^{(\text{model})}$, with magnitude on top and phase on the bottom. The solid line is the median relative response, interpreted as the frequency dependent systematic error on the model response $R^{(\text{model})}$. The dashed lines represent the $1\sigma$ uncertainty on this error. Stacking ten thousand drawn response function samples produces the numerical uncertainty budget shown here. The extreme $1\sigma$ uncertainties are presented in Table 3.

Figure 9

Total calibration uncertainty percentiles for observing run two. The percentiles are created for all of O2 data from November 30, 2016 to August 25th, 2017. H1’s uncertainty is on the left, and L1’s is on the right. The $y$ axis is relative response $\delta R/R^{(\text{model})}$ magnitude (top) or phase (bottom), stacked for all times in the observing run. The dashed white line is the median relative response, while the colors represent the $1\sigma$ calibration uncertainty for 68%, 95%, and 99% of the run’s time. The largest changes in the calibration at H1 were due to clipping of the photon calibrator laser misreporting the strength of our response. The largest calibration changes at L1 were due to fluctuations in the coupled cavity pole, which changes in time but is not yet corrected for in our calibrated data.

Figure 10

Total calibration uncertainty budgets for GW150914, LVT151012, and GW151226. Uncertainties for GW150914 are on top, LVT151012 in the middle, and GW151226 on the bottom. The uncertainty in the calibrated response function for the H1 detector is on the left, and for L1 is on the right. The $y$ axis is relative response error $\delta R/R^{(\text{model})}$ and uncertainty ${\sigma }_R/R^{(\text{model})}$, with magnitude on top and phase on the bottom. The uncertainties for these O1 events have been calculated using the refined O2 methods described in this paper. All the budgets are quite similar with slight differences coming from the uncorrected time dependent cavity pole. The jagged lines around 300–500 Hz come from the actuation function notches, like those seen in Fig. 4. These budgets report a smaller uncertainty in the region around 100 Hz than reported in past calibration uncertainty publications, with significantly smaller systematic error fluctuations [16]. The refined calibration uncertainty budget methods give a more sensible uncertainty budget for O1 events, with uncertainty expanding at extreme frequencies and reduced in the lowest noise regions.

Figure 11

Total calibration uncertainty percentiles for all of observing run one. The uncertainty in the calibrated response function for the H1 detector is on the left, and for L1 is on the right. The $y$ axis is relative response error $\delta R/R^{(\text{model})}$ and uncertainty ${\sigma }_R/R^{(\text{model})}$, with magnitude on top and phase on the bottom. The dashed white line is the median relative response, while the colors represent the $1\sigma$ calibration uncertainty for 68%, 95%, and 99% of the run’s time. Both interferometer’s uncertainty and systematic error are fairly consistent over the course of the run, with the larger fluctuations coming from changes in the coupled cavity pole, which is not corrected for in O1 calibrated data.