Does non-stationary noise in LIGO and Virgo affect the estimation of H0?

Simone Mozzon,a Gregory Ashton, Laura K. Nuttall, and Andrew R. WilliamsonUniversity of Portsmouth, Portsmouth, PO1 3FX, UK

(Dated: October 25, 2021)

Gravitational-wave observations of binary neutron star mergers and their electromagnetic counter-parts provide an independent measurement of the Hubble constant, H0, through the standard-sirensapproach. Current methods of determining H0, such as measurements from the early universe andthe local distance ladder, are in tension with one another. If gravitational waves are to break thistension a thorough understanding of systematic uncertainty for gravitational-wave observations isrequired. To accurately estimate the properties of gravitational-wave signals measured by LIGO andVirgo, we need to understand the characteristics of the detectors noise. Non-gaussian transients inthe detector data and rapid changes in the instrument, known as non-stationary noise, can bothadd a systematic uncertainty to inferred results. We investigate how non-stationary noise affects theestimation of the luminosity distance of the source, and therefore of H0. Using a population of 200simulated binary neutron star signals, we show that non-stationary noise can bias the estimation ofthe luminosity distance by up to 2.4%. However, only ∼15% of binary neutron star signals wouldbe affected around their merger time with non-stationary noise at a similar level to that seen in thefirst half of LIGO-Virgo’s third observing run. Comparing the expected bias to other systematicuncertainties, we argue that non-stationary noise in the current generation of detectors will notbe a limiting factor in resolving the tension on H0 using standard sirens. Although, evaluatingnon-stationarity in gravitational-wave data will be crucial to obtain accurate estimates of H0.

I. INTRODUCTION

Observations of binary neutron star (BNS) mergersand their electromagnetic (EM) counterparts can probethe expansion history of the universe [1, 2]. Gravitationalwaves from compact binary coalescences (CBC) are stan-dard sirens, which means that identical mergers alwayshave the same luminosity. Therefore, we can directly es-timate the luminosity distance of their source. EM coun-terparts, such as kilonovae or gamma-ray bursts, allowastronomers to identify the host galaxies [3, 4] from whichwe can measure the cosmological redshift. The relation-ship between the luminosity distance of the source andits redshift depends on cosmological parameters, and, forlate times, is dominated by the Hubble constant (H0).

Currently, measurements of the Cosmic MicrowaveBackground [5] are in tension with observations basedon the local distance ladder [6]; this tension has risen tothe 4.4σ level [6–8]. Being completely independent and aself-calibrated measurement of H0, standard sirens couldhave a crucial role in solving this tension. The first esti-mation ofH0 from standard sirens [9] gave results broadlyconsistent with other measurements found to date [10].Multiple multimessenger detections (i.e. gravitational-waves and EM detections) are required to improve theaccuracy; several studies predict a 1% H0 measurementaccuracy is achievable with O(100) detections [11–14].This would break the tension on H0.

An accurate estimation of H0 will depend crucially onthe understanding of the systematic uncertainties in bothEM and gravitational-wave observations. One source ofsystematic error related to the observation of the EM

a simone.mozzon@port.ac.uk

counterparts regards the peculiar velocity field of the hostgalaxy [15–17]. The uncertainty on the peculiar velocityis dominant only for extremely close events and is neg-ligible for most of the expected future detections. Anadditional bias can arise from mis-modelling the kilo-nova signal on the inclination [18]. Improved modelsof the kilonova emission could reduce this uncertainty,however, a measure of the inclination from the EM coun-terpart should not be crucial to speed up the resolutionof the Hubble tension [19]. Currently, the known dom-inant systematic uncertainty in the standard sirens ap-proach is due to the gravitational-wave data. The mainsource is the detectors calibration, with an uncertainty inthe amplitude of the calibrated strain below 2% in bothLIGO [20, 21] and Virgo [22, 23] detectors. This uncer-tainty should decrease in future observing runs, but, evenat the current level, does not limit the resolution of theH0 tension.

An unaccounted source of systematic uncertainty couldarise from mis-modelling the noise in the gravitational-wave detectors in the estimation of the luminosity dis-tance. Indeed, the most widely used inference codes forgravitational waves – (LALInference [24], Bilby [25, 26],PyCBC Inference [27] and RIFT [28]) – assume thatthe detector noise is both stationary and Gaussian [29].Gaussianity refers to the distribution of the noise andmeans that the noise can be completely characterised bya mean vector and a covariance matrix. Stationary noisemeans that the statistical properties of the noise do notvary in time.

In reality, due to broadband sources of noise of in-strumental or environmental origin, data from ground-based detectors, such as LIGO and Virgo, are both non-Gaussian and non stationary [30–33]. Non-Gaussianitiesare generally noise transients (called glitches [34, 35])

arX

iv:2

110.

1173

1v1

[as

tro-

ph.C

O]

22

Oct

202

1

2

that last on the order of a second. Non-stationary noise,instead, can vary the detector sensitivity on the order oftens of seconds affecting especially long duration signals.Noise transients are more obvious within gravitational-wave data compared to non-stationary noise. As such,analyses can be performed with noise transients eithersubtracted or the frequency range of analyses restrictedto limit the impact of the noise transient (e.g. [36–38]).It is not possible to employ these workaround techniqueswith non-stationary noise as the noise tends to be moresubtle, harder to model and the noise usually impacts alarge frequency range.

A recent study on simulated data [39] showed that theuncertainty on the posteriors of parameters recovered innon-stationary noise were mis-estimated, in particular forBNS signals. Therefore, non-stationary noise could af-fect the estimation of the luminosity distance, and, con-sequently, the number of detections necessary to reacha few percent measurement of H0. More importantly,non-stationary noise could bias the estimation of H0.

In this paper we investigate how non-stationary noiseaffects the estimation of the luminosity distance of BNSsignals from gravitational-wave data, assuming the de-tection of an electromagnetic counterpart. We use pub-licly available LIGO and Virgo data from the first half ofthe third observing run (O3a) [32, 40]. While previousstudies have investigated how to obtain more accurateparameter estimation in non-stationarity data [41, 42],fully accounting for non-stationarity in parameter esti-mation is still computationally prohibitive. Therefore,we aim to determine if this effort is necessary to solvethe Hubble tension.

The rest of the paper is organised as follows. SectionII describes the Bayesian approach which is widely usedto estimate the parameters of gravitational-wave signalsand how non-stationarity breaks its basic assumptions.In section III we introduce our investigation on the effectof non-stationary noise in the estimation of the luminos-ity distance. We also present our results discussing thepossible consequences on the estimation of H0 throughgravitational-wave data. In section IV we summarise ourmain results and conclude.

II. PARAMETER ESTIMATION INNON-STATIONARY NOISE

The output of a gravitational-wave interferometer is atime series d(t) such that:

d(t) =

{n(t) + h(t), if a signal is present,

n(t), otherwise,(1)

where n(t) is the detector noise and h(t) is agravitational-wave signal. Gravitational-wave transientsearches identify signals by matched-filtering the datawith a number of templates which sample the wave-form parameter space [43, 44]. Once the merger time

is identified, the posterior probability densities for thesource parameters are extracted using a Bayesian ap-proach [45, 46]. This approach requires the computationof the likelihood function, which represents the proba-bility of observing data d assuming that the signal hasparameters θ. If the noise is Gaussian with zero mean,the single detector likelihood takes the form [36, 47]

L = p(d|h(θ)) =1

det(2πCn)e−

12 r†(f)C−1

n r(f) (2)

where the residual r(f) = d(f) − h(f, θ) is assumedto have the same distribution as the noise. Cn =〈n∗(f)n(f ′)〉 is the noise covariance matrix, where the an-gle brackets denote averaging over different realisationsof the noise. We use the variables t and f to indicatewhether a quantity is in the time or frequency domain.If the data are stationary the frequencies are completelyuncorrelated. In this case, 〈n∗(f)n(f ′)〉 is diagonal and isfully described by the noise power spectral density (PSD)Sn(f):

〈n∗(f)n(f ′)〉 =T

2Sn(|f |)δ(f − f ′) , (3)

where T is the duration of the analysed data. Com-bining Equations (2) and (3) we obtain the likelihoodfunction typically used for gravitational-wave parameterestimation [48]. This model is accurate for short seg-ments of data from ground based interferometers. How-ever, the assumption of stationary breaks down for pe-riods of 64 seconds [49], a smaller window than what istypically needed to analyse binary neutron star mergers.Non-stationarity appears in Equation (2) as off-diagonalterms in the covariance matrix [50]. Ref. [39] showedthat ignoring these terms would affect the width of theposterior, with more evident effects when the signal islonger in duration. Nevertheless, accounting for non-stationarity would increase the computational cost fromO(N) to O(N2), where N2 is the number of elementsin the covariance matrix. This would be prohibitive inparticular for longer signals.

Even assuming a diagonal covariance matrix as a fairapproximation, non-stationary noise would bias the es-timation of the noise spectrum. As a workaround, acommon approach is to compute the PSD “off-source”,using data close to, but not containing, the detected sig-nal. However, this approach has an intrinsic uncertaintywhich could introduce new biases in the parameter esti-mation [51]. Moreover it produces poor estimates of thenoise for long signals [52]. A better approach is to esti-mate the noise spectrum “on-source” using parametrizedmodels [49, 53] and to marginalise over the noise estima-tion uncertainty [54].

A. Non-stationary noise in LIGO and Virgo data

To obtain an accurate measurement of H0, the stan-dard sirens method requires us to combine several multi-messenger detections of binary neutron stars. We saw

3

FIG. 1. PSD variation (vs) distribution in LIGO Livingstonbased on data for O2 (blue), O3 (orange) and, estimated, O4(green). The black curve shows the expected distribution inGaussian stationary noise based on O3 data. The vertical reddashed line shows the limit over which we consider data to benon-stationary.

that non-stationary noise can affect the parameter es-timation for longer signals. Now we want to estimatehow many signals could be detected, on average, in non-stationary data.

We identify non-stationary noise using the approachdescribed in Ref. [55]. This method relies on modellingthe relation between the noise spectrum computed overa short stretch of data (typically 8 seconds) and a longersegment of time (512 seconds) with a frequency indepen-dent factor, vs, such that Sn(short) = vsSn(long). Thetime series vs(t), also called PSD variation, has beenproved effective to track non-stationarity in LIGO andVirgo data during the third observing run (O3) [33]. ThePSD variation at each time depends only on the ampli-tude of the non-stationarity and is completely indepen-dent of the shape of the noise.

In Gaussian and stationary noise, the PSD variationstatistic is well modelled by a Gaussian distribution withmean 1 and variance dependent on the bandwidth of thedetector, as shown by the black curve in Figure 1 basedon O3 data. As shown in Figure 1, non-stationarity ap-pears as a tail of high vs values. For simplicity, here weconsider vs > 1.2 as an indicator of non-stationarity inthe data. With this approximation we analyse LIGO andVirgo data. We found that the fraction of non-stationarydata in the LIGO detectors almost doubles between thesecond observing run (O2) and O3a. During O3a ∼2%of LIGO Hanford and LIGO Livingston data and ∼1% ofVirgo data are non-stationary. Randomly placing 10,000signals of 128 seconds in duration in the data, we foundthat 15% of the signals would lie in non-stationary noisein at least one detector for 10 or more seconds aroundthe merger time. Therefore, an average of more than 1in 7 BNS detections could have been affected by non-stationary noise during O3a.

Assuming the fraction of non-stationary data to be lin-early dependent with the sensitivity of the detectors wepredict the levels of non-stationarity in LIGO Livingstonfor the next two observing runs [56]. We found that 3.5%and 8% of data will be non-stationary respectively forO4 and O5. Figure 1 shows the measured distributionof the PSD variation of LIGO Livingston for O2, O3aand the estimation for O4. For O4, we assume an av-erage BNS range of 180 Mpc. Similar values can bepredicted for LIGO Hanford, showing that consideringnon-stationarity will be increasingly important in the fu-ture.

III. ESTIMATING THE EFFECT OFNON-STATIONARY NOISE

In order to investigate the effect of non-stationary noisein the estimation of the luminosity distance we add apopulation of 400 simulated binary neutron stars to O3aLIGO and Virgo data. We target 28 separate periodsof non stationary data in LIGO Livingston, with vary-ing duration between 25 and 200 seconds. In the se-lected segments, we require the data in LIGO Hanfordand Virgo to be stationary. In fact, coincident periodsof non-stationarity are rare for ground-based detectors,representing less then 2% of the total non-stationaritytime. In future observing runs we expect more periodsof coincident non-stationarity due to the larger fractionof non-stationary data; although it is unlikely this willrepresent the dominant scenario.

From the simulated signals we randomly select 200signals with signal-to-noise ratio (SNR) greater than 12in the LIGO Livingston observatory and we perform aBayesian analysis to estimate the luminosity distance.We choose this detection threshold in order to analysesignals which are confidently detected by search pipelineseven in non-stationary noise. Finally, we compare the re-sults of an equivalent analysis made over stationary noiseclose in time (but not overlapping) with the targetednon-stationarity segments. The detectors sensitivity togravitational-wave signals varies considerably during O3adue to adjustment in the configuration of the interferom-eters. Considering adjacent times reduces the possibilityof any variation which could affect our investigation.

We targeted moderate non-stationary noise with aPSD variation value between 1.2-3, which constitutes80% of all non-stationarity. Higher values generally in-dicate extreme non-stationarity or very short bursts ofexcess power [33, 57, 58] that are likely to be identifiedand removed before performing the parameter estima-tion analysis. Figure 2 shows two time-frequency spec-trograms of LIGO Livingston data for one targeted timeand its adjacent closest period of stationary noise. Non-stationarity appears as power excess of unknown origindistributed around 50 Hz.

4

FIG. 2. Time-frequency spectrogram of stationary (left) and non-stationary data (right) measured by LIGO Livingston duringthe LIGO and Virgo third observing run.

A. Simulations

We simulate a population of non-spinning binary neu-tron stars with detector-frame chirp massMdet [59] uni-formly distributed between 1.7 and 1.9 M� and a massratio between 0.75− 1. We distribute mergers uniformlyin Euclidean volume, extracting the signals from a priorin luminosity distance π(dL) ∝ d2L [60, 61] between 20and 200 Mpc. This approximation is appropriate todescribe the observed population of BNS in the lumi-nosity range considered [26]. To reduce the computa-tional cost we neglect tidal effects; tidal parameters donot contribute to the waveform amplitude and are notcorrelated with the luminosity distance. The choice ofnon-spinning injection is justified by the expected lownumber of events with high spins (e.g. [62]). Moreover,none of the BNS signals detected so far by the LIGOand Virgo Collaborations were consistent with high spins(e.g. [63]). We generate the signals using the waveformmodel IMRPhenomPv2 [64, 65] with a low frequency cut offof 20 Hz. This model has a low computational cost andprovides a good approximation for a BNS system if thetidal effects are neglected. We fix the sky position of thesignals to the optimal location for our targeted detector,i.e. on the zenith for LIGO Livingston. With this choiceeach signal has the highest SNR in the detector whichpresents non-stationary noise.

We injected the signals in individual stretches of dataand we separate the merger times between simulationsby 4 seconds. This is to avoid correlation between theestimations [66–68]. We calculate the PSD using theoff-source approach as described in Ref [69], using theWelch method for 1024 seconds of data. Despite beingsub-optimal compared to the “on-source” method, thisapproach is much faster for longer signals and is there-fore preferable for population studies. We then use Bilby

with the Dynesty sampler [70] to estimate the parametersof the injections. For each signal, we analyse 128 secondsof data using IMRPhenomPv2 model in its reduced orderquadrature approximation to reduce the computationalcost [71, 72]. We use priors consistent with the gener-ated population of signals. Assuming the EM counter-part would allow us to uniquely identify the host galaxy,we fix the sky location to the injected value. While thisis an optimistic scenario, this assumption drastically im-proves the accuracy in the estimation of the luminositydistance, helping to isolate and highlight the effect ofnon-stationarity. We assume the EM counterpart doesnot provide any information on the binary inclinationangle.

B. Bias in luminosity distance

We compare the luminosity distance posteriors forevents injected in stationary and non-stationary noisecomputing the normalised cumulative fraction of trueluminosity distances which lie within a measured con-fidence interval [73]. This approach is commonly usedto identify biases in the inference on gravitational-wavesources [26, 27, 74–77]. Figure 3 shows the results for 200injections in stationary and non-stationary noise. This isgenerally referred to as a P-P plot. If the inference isunbiased the fraction of events in a particular confidenceinterval is drawn from a uniform distribution. Hence,the expected cumulative distribution would lie on thediagonal of the plot with some scatter due to Poisson er-ror. The shaded regions delimits the expected 1, 2 and3-sigma error given the number of events. For signalsinjected in non-stationary noise the cumulative distribu-tion is systematically below the diagonal, exceeding the3-sigma error for confidence intervals around the 95%level.

5

FIG. 3. Results of 200 injections recovered in stationaryand non-stationary noise. The grey regions cover the cumula-tive 1,2 and 3 σ confidence intervals accounting for samplingerrors. The blue lines represent the cumulative fraction ofreal luminosity distances found within this confidence inter-val (C.I.). Luminosity distance p-values for stationary andnon-stationary noise are displayed in parentheses in the plotlegend. The luminosity distance p-value for stationary noise,is 0.256, consistent with the p-value being drawn from a uni-form distribution.

SNR cut 12 13 14

stationary 0.256 0.357 0.141non-stationary 0.043 0.042 0.010

TABLE I. Luminosity distance p-values for stationary andnon-stationary noise as a function of the SNR cut imposed.

We test the consistency between the measured curvesand the diagonal line using the Kolmogorov-Smirnov(KS) statistic. For unbiased parameter estimations thetwo-tailed p-value is uniformly distributed between 0 and1. Therefore, a p-value < 0.05 will occur once in 20times. For the curves in Figure 3 the resulting p-valuesare 0.256 and 0.043 for stationary and non-stationarynoise respectively. A smaller p-value indicates that themeasured curve is unlikely to be randomly extracted fromthe assumed distribution if the sampler is unbiased. Inparticular, there is just a 4% chance to obtain a more ex-treme curve than the one measured from events in non-stationary noise. As shown in Table I, we obtained p-values of the same order of magnitude when increasingthe cut in SNR, showing that the P-P plot distortion isconsistent for louder signals.

This inconsistency can be interpreted as a system-atic bias in the estimated luminosity distance in non-stationary noise. Power excess in the data can increasethe matched-filter SNR of the detection, decreasing theestimated luminosity distance. This hypothesis is sup-ported by our analysis in non-stationary noise, with the

mean luminosity distance underestimated for 62% of thesignals (compared to the 53% for signals in stationarynoise). However, a lower p-value can also arise fromover-constraining the posterior. If the posterior is over-constrained we would see a larger fraction of events inlower confidence interval and smaller fraction for higherintervals. This distortion with respect to the predictedcurve would lower the p-value.

To investigate these two scenarios and quantify thebias, we repeat our analysis adding 200 signals in sim-ulated Gaussian stationary noise. For consistency withthe analysis in real data we require SNR>12 in LIGOLivingston, which leaves us to estimate the luminositydistance posteriors for the remaining 154 signals. Foreach signal we then artificially bias the estimated lumi-nosity distance posterior dL by shifting the distributionby a constant value ∆dL, such that:

dL,biased = dL −∆dL × dL,inj (4)

where dL,inj is the injected luminosity distance. Finallywe compute the p-value of the biased distribution. Toinvestigate the relation between the p-value and the biaswe repeat this procedure for increasing values of ∆dL, re-estimating the p-value for each iteration. We perform asimilar test to understand the effect of over-constrainingthe posterior. In this case we systematically reduce thesamples standard deviation σ(dL) by a factor σb for eachsignal. For example, a σb = 0.1 refers to a 10% reduc-tion of the standard deviation of the luminosity distancedistribution for each signal.

The top plots of Figure 4 show the evolution of theluminosity distance p-value as a function of the bias in-troduced in the posteriors for the two cases. The relationbetween the p-value and the bias is monotonic, with lowerp-values indicating greater biases. Indeed, greater distor-tions of the posterior distribution makes the assumptionthat the sampler is unbiased more unlikely. From theseplots we can infer the bias associated with the p-valueobserved for events in non-stationary noise, which is in-dicated in Figure 4 with a red dashed line. The mea-sured p-value is consistent with a 2.4% systematic under-estimation of the measured luminosity distance or a 18%reduction in the dispersion of the posteriors distribution.In the bottom plots of Figure 4 we show how these twobiases distort the P-P plots. Reducing the samples vari-ance (left plot) induces an “S-shape” in the cumulativefraction of injections found in each confidence interval,increasing it for lower confidence intervals and decreas-ing it for higher levels. Instead, reducing the mean ofthe posteriors (right plot) systematically decreases thecumulative fraction of real luminosity distances in eachconfidence interval. Qualitatively comparing these plotswith Figure 3, we can conclude that the measured domi-nant effect of non-stationary noise is a systematic under-estimation of the luminosity distance.

6

FIG. 4. Results for 154 BNS injections in simulated Gaussian noise. The top plots show the variation of the luminositydistance p-value as a function of the bias introduced in the luminosity distance posterior samples. The red dashed line showsthe p-value measured for events in non-stationary noise. In the bottom plots we show how the bias distorts the cumulativefraction of injected luminosity distances found within the confidence interval in two particular cases: for σb = 0.2 (Left) and∆dL = 0.024 (Right). These values correspond to the biases associated with the non-stationary p-values in the upper plots

C. Considerations on the estimation of H0

In principle, a 2.4% systematic under-estimation ofthe luminosity distance of the source combined with theother expected systematic uncertainties could dramati-cally affect the accuracy of the estimation of H0 usingstandard sirens. Despite this inaccuracy, standard sirenswould still help to break the H0 tension. Let us con-sider the worst case, in which the systematics due tonon-stationarity and calibration simply add up. Con-sidering a 1% calibration error, this would correspondto a 3.4% systematic under-estimation of the luminos-ity distance, i.e. we would infer 3.4% higher valuesof H0. Assuming the early universe estimation of H0

(67.4 ± 0.5 km s−1 Mpc−1 [5]) to be correct, we would

obtain H0 = 69.7 ± 0.7 km s−1 Mpc−1, in which we as-sumed our estimation to be Gaussian distributed with a1% error. In this case, despite the effect of non-stationarynoise the standard sirens estimation would still favour theearly universe measurement over the local distance lad-der estimation of H0 (74.03 ± 1.42 km s−1 Mpc−1 [6]),but none of the two hypothesis would be confidently ex-cluded.

However, our results represent an upper limit in theshift of the luminosity distance due to non-stationarynoise, with the measured p-value likely to be a result ofvarious effects. Moreover, to reach the precision of a fewpercent required to solve the current tension on the esti-mation of H0 it may be necessary to combine at least ∼50standard sirens [13]. Of them, just a fraction will be af-

7

fected by non-stationary noise, hence the error on the es-timation will be reduced. On the other hand, the numberof standard sirens required to attain the necessary pre-cision may be reduced by additional EM constraints one.g., source orientation, as with GW170817 [78, 79], andthis could further compound any present bias. Therefore,assessing the level of non-stationarity for individual BNSdetections will be important in confidently presenting es-timates of H0 free from significant bias.

Other methods to estimate H0 which rely on shortersignals like binary black holes will also be important toimprove the accuracy on H0 [80–82]. These approachesrequire shorter periods of data, therefore the effect ofnon-stationary noise will be less important.

IV. SUMMARY AND CONCLUSIONS

In this paper we have investigated whether the pres-ence of non-stationarity in LIGO and Virgo data intro-duces a new source of systematic error in the estimationof the Hubble constant through the standard sirens ap-proach. The problem is particularly important for longerduration signals such as BNS, for which longer periods ofdata are required, making the parameter estimation morevulnerable to fluctuations in the detector noise. Indeed,we found that during O3a non-stationarity accounts for2% of the overall LIGO data. By placing simulated BNSsignals of 128 seconds in length throughout O3a data,we find that 1 in 7 BNS signals merger times could havefallen in non-stationary noise. More importantly, thisfraction is predicted to increase, with non-stationaritythat will reach an estimated 3.5% and 8% of the overalldata for O4 and O5 respectively.

We explore the issue of non-stationarity and how itaffects the estimation of the luminosity distance of thesource, by adding simulated BNS signals in stationaryand non-stationary data from O3a. We compare the lu-minosity distance posteriors obtained in the two casescalculating the cumulative fraction of true luminosity dis-tances which lie within a measured confidence interval.We employ the Kolmogorov-Smirnov test to estimate theconsistency of the results with the theoretical predic-tions. We found a lower p-value (0.043) for events innon-stationary noise, showing that the null hypothesis ofan unbiased estimation is unlikely. In order to under-stand the magnitude of the misestimation, we artificiallybias the posteriors of BNS signals estimated in simulatedGaussian and stationary noise. We found that the mea-sured p-value for non-stationary noise is consistent with asystematic under-estimation of the measured luminositydistance by up to 2.4%.

The estimated bias in the luminosity distance is anupper limit and does not automatically translate to anexpected systematic error in the estimation of H0. First,just a fraction of the BNS-like gravitational-wave detec-

tions will be measured in non-stationary noise. It is esti-mated that O(100) joint gravitational-wave and EM de-tections are needed in order to infer H0 to an accuracy of1%. Therefore the combination of these O(100) signals,of which ∼ 15% may be affected by non-stationarity, isunlikely to have a large effect on the accuracy of H0.Moreover, binary black holes detections are expected togive an important contribution to improve the accuracyof the H0. The duration of these signals is on the orderof seconds, making the effect of non-stationary noise lessimportant. Therefore, we do not expect non-stationarynoise to be a crucial factor in the accuracy of H0 us-ing data from second generation (2G) detectors. How-ever, until gravitational-wave inference methods fully ac-count for non-stationary noise, assessing the level of non-stationarity in the data, in particular for louder signals,will be crucial to exclude biases in the H0 estimation.

The next generation (3G) of gravitational-wave detec-tors, such as the Einstein Telescope [83, 84] and Cos-mic Explorer [85, 86], with their increased sensitivityat lower frequencies, will detect much longer durationgravitational-wave signals than current 2G detectors.Non-stationarity will still be an issue in these detectors,although we can not estimate, just yet, whether it willbe at similar levels or worse than what we see in the 2Gdetectors. Either way, non-stationarity will have to beconsidered in the interpretation of long duration signalsin 3G detectors to ensure this form of noise does notimpact key scientific conclusions.

ACKNOWLEDGMENTS

We are grateful to the LIGO/Virgo Cosmology groupfor insightful comments on this work, as well as SylviaBiscoveanu and Ian Harry for useful discussions on thispaper.

SM was supported by a STFC studentship. GA andLKN thank the UKRI Future Leaders Fellowship for sup-port through the grant MR/T01881X/1. ARW thanksthe STFC for support through the grant ST/S000550/1.

This research has made use of data, software and/orweb tools obtained from the Gravitational Wave OpenScience Center (https://www.gw-openscience.org), a ser-vice of LIGO Laboratory, the LIGO Scientific Collabo-ration and the Virgo Collaboration. LIGO is funded bythe U.S. National Science Foundation. Virgo is fundedby the French Centre National de Recherche Scientifique(CNRS), the Italian Istituto Nazionale della Fisica Nucle-are (INFN) and the Dutch Nikhef, with contributions byPolish and Hungarian institutes. The authors are grate-ful for computational resources provided by the LIGOLaboratory and supported by National Science Founda-tion Grants PHY-0757058 and PHY-0823459.

This work carries LIGO Document number P2100316.

8

[1] B. F. Schutz, Nature (London) 323, 310 (1986).[2] D. E. Holz and S. A. Hughes, The Astrophysical Journal

629, 15–22 (2005).[3] N. Dalal et al., Physical Review D 74 (2006),

10.1103/physrevd.74.063006.[4] S. Nissanke et al., The Astrophysical Journal 725, 496

(2010).[5] Aghanim, N. et al. (Planck Collaboration), A&A 641,

A6 (2020).[6] A. G. Riess et al., The Astrophysical Journal 876, 85

(2019).[7] D. Valentino et al., Class. Quant. Grav. 38, 153001

(2021), arXiv:2103.01183 [astro-ph.CO].[8] W. L. Freedman, The Astrophysical Journal 919, 16

(2021).[9] B. P. Abbott et al. (The LIGO Scientific and Virgo Col-

laborations), The Astrophysical Journal 848, L13 (2017).[10] W. L. Freedman, Nature Astronomy 1, 0121 (2017).[11] S. Nissanke et al., arXiv e-prints , arXiv:1307.2638

(2013), arXiv:1307.2638 [astro-ph.CO].[12] H.-Y. Chen et al., Nature 562, 545–547 (2018).[13] S. M. Feeney et al., Physical Review Letters 122 (2019),

10.1103/physrevlett.122.061105.[14] D. J. Mortlock et al., Phys. Rev. D 100, 103523 (2019),

arXiv:1811.11723 [astro-ph.CO].[15] C. Howlett and T. M. Davis, Monthly Notices of the

Royal Astronomical Society 492, 3803 (2020).[16] C. Nicolaou et al., Monthly Notices of the Royal Astro-

nomical Society 495, 90 (2020).[17] S. Mukherjee et al., Astronomy & Astrophysics 646, A65

(2021).[18] H.-Y. Chen, Physical Review Letters 125 (2020),

10.1103/physrevlett.125.201301.[19] S. Mastrogiovanni, R. Duque, E. Chassande-Mottin,

F. Daigne, and R. Mochkovitch, Astronomy & Astro-physics 652, A1 (2021).

[20] L. Sun et al., Classical and Quantum Gravity 37, 225008(2020).

[21] The LIGO Scientific Collaboration et al., Class. Quant.Grav. 32, 074001 (2015), arXiv:1411.4547 [gr-qc].

[22] F. Acernese et al. (VIRGO), Class. Quant. Grav. 32,024001 (2015), arXiv:1408.3978 [gr-qc].

[23] D. Estevez et al., Classical and Quantum Gravity 38,075007 (2021).

[24] J. Veitch et al., Phys. Rev. D 91, 042003 (2015).[25] G. Ashton, , et al., The Astrophysical Journal Supple-

ment Series 241, 27 (2019).[26] I. M. Romero-Shaw et al., Monthly Notices

of the Royal Astronomical Society 499, 3295(2020), https://academic.oup.com/mnras/article-pdf/499/3/3295/34052625/staa2850.pdf.

[27] C. M. Biwer et al., Publications of the Astronomical So-ciety of the Pacific 131, 024503 (2019).

[28] J. Lange et al., arXiv e-prints (2018), arXiv:1805.10457[gr-qc].

[29] C. Rover, Phys. Rev. D 84, 122004 (2011).[30] B. P. Abbott et al. (The LIGO Scientific and Virgo Col-

laborations), Classical and Quantum Gravity 33, 134001(2016).

[31] B. P. Abbott et al. (The LIGO Scientific and Virgo Col-laborations), Classical and Quantum Gravity 37, 055002

(2020).[32] R. Abbott et al. (The LIGO Scientific and Virgo Collab-

orations), SoftwareX 13, 100658 (2021).[33] D. Davis et al. (The LIGO Scientific and Virgo Collab-

orations), Classical and Quantum Gravity 38, 135014(2021).

[34] L. K. Nuttall et al., Classical and Quantum Gravity 32,245005 (2015).

[35] M. Zevin et al., Classical and Quantum Gravity 34,064003 (2017).

[36] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev.Lett. 119, 161101 (2017), arXiv:1710.05832 [gr-qc].

[37] C. Pankow et al., Physical Review D 98 (2018),10.1103/physrevd.98.084016.

[38] R. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. X11, 021053 (2021), arXiv:2010.14527 [gr-qc].

[39] O. Edy et al., Physical Review D 103 (2021),10.1103/physrevd.103.124061.

[40] M. Vallisneri et al., Journal of Physics: Conference Series610, 012021 (2015).

[41] N. J. Cornish, Phys. Rev. D 102, 124038 (2020).[42] N. J. Cornish et al., Phys. Rev. D 103, 044006 (2021).[43] C. Cutler et al., Physical Review Letters 70, 2984–2987

(1993).[44] B. Allen et al., Physical Review D 85 (2012),

10.1103/physrevd.85.122006.[45] L. S. Finn, Phys. Rev. D 46, 5236 (1992).[46] J. D. Romano and N. J. Cornish, Living Reviews in Rel-

ativity 20 (2017), 10.1007/s41114-017-0004-1.[47] J. Veitch and A. Vecchio, Physical Review D 81 (2010),

10.1103/physrevd.81.062003.[48] J. Veitch et al., Physical Review D 91 (2015),

10.1103/physrevd.91.042003.[49] T. B. Littenberg and N. J. Cornish, Phys. Rev. D 91,

084034 (2015).[50] C. Talbot et al., “Inference with finite time series: Ob-

serving the gravitational universe through windows,”(2021), arXiv:2106.13785 [astro-ph.IM].

[51] C. Talbot and E. Thrane, Physical Review Research 2,043298 (2020), arXiv:2006.05292 [astro-ph.IM].

[52] K. Chatziioannou et al., Physical Review D 100 (2019),10.1103/physrevd.100.104004.

[53] N. J. Cornish and T. B. Littenberg, Classical and Quan-tum Gravity 32, 135012 (2015).

[54] S. Biscoveanu, C.-J. Haster, S. Vitale, and J. Davies,Physical Review D 102 (2020), 10.1103/phys-revd.102.023008.

[55] S. Mozzon et al., Classical and Quantum Gravity 37,215014 (2020).

[56] B. P. Abbott et al., Living Reviews in Relativity 23(2020), 10.1007/s41114-020-00026-9.

[57] B. P. Abbott et al. (LIGO Scientific, Virgo), Class.Quant. Grav. 33, 134001 (2016), arXiv:1602.03844 [gr-qc].

[58] L. K. Nuttall, Philosophical Transactions of the RoyalSociety of London Series A 376, 20170286 (2018),arXiv:1804.07592 [astro-ph.IM].

[59] B. S. Sathyaprakash and B. F. Schutz, Living Reviews inRelativity 12 (2009), 10.12942/lrr-2009-2.

[60] B. P. Abbott et al. (LIGO Scientific Collaboration andVirgo Collaboration), Phys. Rev. Lett. 116, 241102

9

(2016).[61] B. P. Abbott et al. (LIGO Scientific Collaboration and

Virgo Collaboration), Phys. Rev. X 9, 031040 (2019).[62] X. Zhu et al., Phys. Rev. D 98, 043002 (2018),

arXiv:1711.09226 [astro-ph.HE].[63] R. Abbott et al. (The LIGO Scientific and Virgo Collab-

orations), Phys. Rev. X 11, 021053 (2021).[64] P. Schmidt et al., Phys. Rev. D 91, 024043 (2015).[65] M. Hannam et al., Phys. Rev. Lett. 113, 151101 (2014).[66] R. Smith et al., Physical Review Letters 127 (2021),

10.1103/physrevlett.127.081102.[67] A. Samajdar et al., Physical Review D 104 (2021),

10.1103/physrevd.104.044003.[68] P. Relton and V. Raymond, Physical Review D 104

(2021), 10.1103/physrevd.104.084039.[69] B. Allen et al., Phys. Rev. D 85, 122006 (2012).[70] J. S. Speagle, MNRAS 493, 3132 (2020),

arXiv:1904.02180 [astro-ph.IM].[71] P. Canizares et al., Physical Review Letters 114 (2015),

10.1103/physrevlett.114.071104.[72] R. Smith et al., Physical Review D 94 (2016),

10.1103/physrevd.94.044031.[73] S. R. Cook et al., Journal of Computa-

tional and Graphical Statistics 15, 675 (2006),

https://doi.org/10.1198/106186006X136976.[74] J. R. Gair and C. J. Moore, Phys. Rev. D 91, 124062

(2015), arXiv:1504.02767 [gr-qc].[75] C. P. L. Berry et al., The Astrophysical Journal 804, 114

(2015).[76] T. Sidery et al., Phys. Rev. D 89, 084060 (2014).[77] C. Pankow et al., Phys. Rev. D 92, 023002 (2015).[78] K. Hotokezaka et al., Nature Astron. 3, 940 (2019),

arXiv:1806.10596 [astro-ph.CO].[79] S. Mukherjee et al., Astron. Astrophys. 646, A65 (2021),

arXiv:1909.08627 [astro-ph.CO].[80] M. Fishbach et al., The Astrophysical Journal 871, L13

(2019).[81] M. Soares-Santos et al., The Astrophysical Journal 876,

L7 (2019).[82] R. Gray et al., Physical Review D 101 (2020),

10.1103/physrevd.101.122001.[83] M. Punturo et al., Class. Quant. Grav. 27, 194002 (2010).[84] M. Maggiore et al., JCAP 03, 050 (2020),

arXiv:1912.02622 [astro-ph.CO].[85] B. P. Abbott et al. (LIGO Scientific), Class. Quant. Grav.

34, 044001 (2017), arXiv:1607.08697 [astro-ph.IM].[86] D. Reitze et al., Bull. Am. Astron. Soc. 51, 035 (2019),

arXiv:1907.04833 [astro-ph.IM].