Merger-Ringdown Consistency: A New Test of Strong Gravity using Deep Learning
MMerger-Ringdown Consistency: A New Test of Strong Gravity using Deep Learning
Swetha Bhagwat ∗ and Costantino Pacilio † Dipartimento di Fisica, “Sapienza” Universit`a di Roma & Sezione INFN Roma1, Roma 00185, Italy (Dated: February 18, 2021)The gravitational waves emitted during the coalescence of binary black holes are an excellentprobe to test the behaviour of strong gravity at different length scales. In this paper, we proposea new test called the merger-ringdown consistency test that focuses on probing the imprints of thedynamics in strong-gravity around the black-holes during the plunge-merger and ringdown phase.Furthermore, we present a scheme that allows us to efficiently combine information across multipleringdown observations to perform a statistical null test of GR using the detected BH population. Wepresent a proof-of-concept study of this test using simulated binary black hole ringdowns embeddedin the next-generation ground-based detector noise. We demonstrate the feasibility of our test usinga deep learning framework, setting a precedence for performing precision tests of gravity with neuralnetworks.
INTRODUCTION
The detection of gravitational waves (GWs) emittedduring the binary black holes (BH) mergers present uswith an unparalleled opportunity to test the behaviourof strong gravity around BHs. GW are emitted as theBHs slowly spiralling in towards the common center ofmass (a.k.a. the inspiral phase). This is followed bya rapid plunge and merger (a.k.a. the plunge-mergerphase) where the two BHs coalesce forming a remnantBH which then rings down and settles to a final state(a.k.a. the ringdown phase). We focus on the latter twophases. While both plunge-merger and ringdown con-tain imprints of the dynamics in the strong field regimeat few times the horizon-length scale, the plunge-mergeris dictated by non-linear dynamics and the ringdown isprescribed by linear perturbation theory.Ringdown corresponds to the evolution of linear per-turbations on the space-time metric of the remnant BHand the perturbation conditions are setup during theplunge-merger phase. Given an underlying theory ofgravity, both the symmetries during the merger and theproperties of the remnant BH are dictated by the ini-tial symmetries of the system and are not independentquantities. We propose a novel test that checks if theperturbation conditions set by the merger phase are con-sistent with the properties of the remnant BH formedafter the ringdown phase across a population of binaryBH ringdown observations. If GR were to be modified ina way that it affects a binary BH evolution, the plunge-merger phase and the ringdown phase could be alteredinconsistently and we design a test to probe this. Theproposed test checks for the consistency between plunge-merger and ringdown phase by simultaneously using thefrequency content and amplitudes of excitation in theringdown signals. Furthermore, it stacks the informationfrom multiple GW observations efficiently to provide astatistical ‘null’ test. Henceforth, we call it the merger-ringdown consistency test .The test is particularly suited to the third-generation (3g) detectors such as the Einstein Telescope (ET) [1],the Cosmic Explorer [2] and LISA [3] where the ring-downs are expected to be loud and the number of de-tections can be ∼ − / year [4, 5]. Analyzing alarge number of events demands for a rapid and compu-tationally efficient inference algorithms. We demonstratethe feasibility of our test entirely using a deep learningframework to speed up the parameter inference by ordersof magnitude [6–8]. We train a neural network (NN) ar-chitecture called a conditional variational autoencoder(CVAE) [9–11] to infer posterior distributions of the pa-rameter set { M, χ f , q } from a set of simulated ringdownwaveforms. Following the deep learning application toGW science — e.g., detection [12–16] and parameter es-timation (PE) [6, 7, 17–20], our work also sets a prece-dence for precision tests of GR using deep learning algo-rithms. MERGER-RINGDOWN CONSISTENCY TESTTheory
Ringdowns is modelled as a linear superposition ofdamped sinusoids with characteristic BH frequencies( f lm ) and damping times ( τ lm ) known as the quasi-normal-mode (QNM) spectra. It is generally decomposedin spin-2 weighted spheroidal harmonic basis Y lm ( ι ),where ( ι ∈ [0 , π )) is the inclination angle. Ringdowns See [21] for a recent review. a r X i v : . [ g r- q c ] F e b take the following analytical form h + ( t ) = MD L (cid:88) l,m> Y lm + ( ι ) A lm e − t/τ lm cos(2 πf lm t − φ lm ) , (1a) h × ( t ) = MD L (cid:88) l,m> Y lm × ( ι ) A lm e − t/τ lm sin(2 πf lm t − φ lm ) . (1b)Here { + , ×} are the GW polarizations and D L is theluminosity distance of the system. The QNMs are in-dexed by the angular multipole numbers ( l, m ) and theyare determined by the final mass and final spin { M, χ f } ,i.e., f lm = f lm ( M, χ f ) and τ lm = τ lm ( M, χ f ). A lm and φ lm are the amplitudes and the phases of excitationsof QNMs that are dictated by (and contain informationabout) the perturbation conditions set during the plunge-merger. For a non-spinning binaries, the initial system iscompletely characterized by the total-mass M tot and thebinary mass ratio q . While M tot sets the overall ampli-tude scale, q sets the symmetry during the plunge-mergerphase and determines the relative excitations of QNMs,i.e., A lm /A = A Rlm ( q ) and φ − φ lm = δφ lm ( q ). Thus,the ringdown waveform can be parameterized by a setof three parameters { M, χ f , q } and Eq. (1) can be re-written as, h + ( t ) = h + ( t ; M, χ f , q ) , h × ( t ) = h × ( t ; M, χ f , q ) . (2)Using the ringdown phase of the GW event one can infer { M, χ f , q } by treating them as independent quantities ina Bayesian PE setup.Next, in GR, a given set of { M tot , q } can be determinis-tically mapped to { M, χ f } for a non-spinning binary BHsystem. The three ringdown parameters { M, χ f , q } arenot truly independent and χ f can be mapped to q usingthe fitting formula presented in [24] (see also [23, 25, 26]) χ f = 2 √ η − . η + 4 . η + O ( η ) (3)where η = q/ (1 + q ) .The test checks if the independent measurements of { M, χ f , q } from the ringdowns are consistent with the re-lation between χ f and q as predicted by GR. Specifically,we check if the χ f directly measured from the ringdown For simplicity, we approximate the spin-2 weighted spheroidalharmonics as spin-2 weighted spherical harmonics basis — a rea-sonable assumption for the range of spins used in this study [22]. The remnant BH can be expressed in terms of the initial binaryBH parameters by fitting the numerical relativity simulations.The relationship can be expressed explicitly in approximate an-alytical forms [23–29], or implicitly using machine learning algo-rithms [30–32]. agrees with the χ f calculated by plugging the measuredvalue of q in Eq. (3).Although we use only ringdowns for this test, we callit a merger-ringdown consistency test because it checks ifthe perturbation conditions setup during merger — thatis dictated by strong non-linear gravity — is compatiblewith the ringdown of a remnant BH predicted by a GRevolution. Prescription for the Merger-ringdown consistency
Let a population of non-spinning binary BHs ring-downs be detected by a GW observatory. Note that thequantities directly measured in PE have a superscript‘meas’ and those inferred using Eq. (3) have ‘infer’.1. Parameterize the ringdown as in Eq. (2) and es-timate { M meas , χ meas f , q meas } for each event. Weused the median of the marginalized posterior dis-tribution as the ‘measured’ value.2. For each event, compute the χ infer f from the medianvalue of q meas in step 1 using the relation in Eq. (3).3. Make a scatter plot with { χ infer , χ meas } using allringdown observations. In GR, one expects thatall the data should lie along the χ infer = χ meas linein a 2-D scatter plot, with the noise in the dataleading to a spread around this line. To performthe merger-ringdown consistency test, we express χ meas f = a + b χ infer F (4)and fit for the parameters { a, b } . If the best-fitparameters for Eq. (4) are compatible with { a =0 , b = 1 } the observations are consistent with GR,providing a statistical null test. Details of Implementation
For simplicity, we restrict our study to non-spinning binary BHs. We compute the QNM spectra { f lm ( M, χ f ) , τ lm ( M, χ f ) } using the data in [33]. Fur-ther, we focus our attention on the dominant mode( l, m ) = (2 ,
2) and the two most excited subdominantangular modes for the case of non-spinning systems —( l, m ) = { (2 , , (3 , } [34, 35]. We concentrate solelyon the dominant overtone, i.e., n overtone = 0 [36, 37].We use these simplifying assumptions for this proof-of-concept study. However, note that including more angu-lar modes and overtones is a tangible extension to ourwork.Key to our study is the expression of the QNM excita-tion amplitudes and phases, as functions of q . Followingthe prescription in [35], we express A Rlm and δφ lm as A Rlm ( q ) = a + a q + a q + a q , (5a) δφ ( q ) = b + b b + q , (5b)where we use the convention in which q >
1. An updatedlist of coefficients { a i , b i } is provided in the SupplementalMaterial. Further, for the dominant mode’s amplitude,we use A = 0 . η [38]. Lastly, we assume a uniformsupport in φ ∈ [0 , π ] and generate the waveforms ex-pressed in Eq. (1).The parameters { M, χ f , q } can be reconstructed fromthe waveform through variational inference. We follow[6–8] and train a CVAE, a NN architecture well suitedto posterior sampling. Details of the implementation areprovided in the Supplemental Material. RESULTSNetwork training
To train the network, we simulate a data-set of 10 ringdowns by sampling the waveform parameters uni-formly in the ranges indicated in Tab. I. Signal-to-noiseratio (SNR) is used to set the waveform scale w.r.t. thenoise. The SNR is computed as in [39, 40]. When per-forming PE, we only estimate posteriors for { M, χ f , q } .For simplicity, we only consider the + polarization and Parameter Symbol RangeFinal BH mass M [25 , M (cid:12) Final BH spin χ f [0 , . q [1 , φ [0 , π ] radSignal-to-noise ratio SNR [40 , fix the inclination angle to ι = π/
3. The ringdowns areembedded in simulated ET-like noise segments [41, 42].To test the network, we generate a new data-set of10 simulated ringdown waveforms, whose parametersare sampled again from the ranges indicated in Tab. I.For each input waveform, the CVAE samples 10 distinctpoints to build the posterior. The total time to analyzeall the samples is approximately 40 s on a single GPU,corresponding to 40 ms per waveform. For illustration,Fig. 1 shows the contour plot obtained from the PE ofone signal from the test dataset.An account of the network performances is detailed inthe Supplemental Material. M = 64.89 +1.361.28 . . . . f = 0.80 +0.030.03 . . . . . . . . . .
70 0 .
75 0 .
80 0 .
85 1 . . . . . q = 1.56 +0.170.16 t (s) strain
1e 22
SNR=60noiseless
FIG. 1. Contour plot for the PE of the ringdown signalshown in the lower panel, with (
M, χ f , q ) = (65 , . , .
5) andSNR= 60. Blue lines indicate the true values. Dashed linesmark the 95% confidence interval. The plot titles indicate 1 σ uncertainties. Proof-of-concept Merger-Ringdown test
We simulate a data-set consisting of 10 ringdownswith parameter ranges in Tab. I except for χ f . χ f isnow inferred from q by imposing the relation (3).First, the posteriors produced by our CVAE for a singleevent can be used to check if the event follows the spinrelation (3). In Fig. 2, we show the posterior samples for χ meas f and χ infer f — where χ infer f is determined by q meas asper Eq. (3), for a randomly picked event from our data-set. Further, a quantitative confidence in this null-testof GR can be assigned using this plot. For this event, wesee that χ meas f = χ infer f (blue dashed line) lies within the68% contour of the posteriors, asserting that this event’splunge-merger-ringdown phase is consistent with a GRevolution.Further, we use the scheme outlined in the previoussection to efficiently combine the information across mul-tiple ringdown observations for a more stringent testof GR. In GR and in the absence of noise, we expect χ meas f = χ infer f . However, our inferences are probabilis- measf = 0.67 +0.040.04 .
56 0 .
64 0 .
72 0 . . . . . .
68 0 .
52 0 .
56 0 .
60 0 .
64 0 . inferf = 0.63 +0.020.03 FIG. 2. Contour plot for the posteriors of χ meas f and χ infer f for a signal with ( M, χ f , q ) = (52 . , . , .
53) and SNR= 60.The signal is extracted from the second test data-set, wherethe relation (3) is enforced. The blue dashed line represents χ meas f = χ infer f line. inferred spin m e a s u r e d s p i n m e a s u r e d m a ss r a t i o FIG. 3. Scatter plot of χ meas f vs χ infer f . The color bar indicatesthe value q meas for each observation. The black dotted linemarks the χ meas f = χ infer f . tic and with noise, leading to measurement uncertain-ties which translated as a scatter around the diagonalline. This is illustrated in Fig. 3. In Fig. 3, we con-firm that our data-set lies around the χ meas f = χ infer f line. An ordinary least squared (OLS) fit of Eq. (4) gives a ∈ [ − . , . b ∈ [0 . , . χ meas f = χ infer f line. Also, as expected, lower values of q give higher val-ues of χ f . Fig. 3 thus observationally validates (3).Next, we study the convergence of a and b , and therebythe efficiency of this test with the number of ringdownobservations. In Fig. 4 we present the best-fit values for a and b with their 2 σ confidence levels as a function ofthe number of observations.In Fig. 4, we find that the mean of a and b are ∼ a observations ba FIG. 4. Evolution of the best-fit values (continuous lines)and 2 σ contours (shaded regions) for b and a in Eq. (4), ver-sus the number of observations. In GR, the noiseless best-fitcorresponds to b = 1 and a = 0. and b , i.e., { σ b , σ a } decrease with increasing number ofobservations as a power-law. For instance, with 20 ob-servations we can constrain { σ b , σ a } = { . , . } andwith 100 observations we have { σ b , σ a } = { . , . } .Concretely, σ a and σ b scale with the number of observa-tions as σ a ( n ) = 0 . √ n , σ b ( n ) = 0 . √ n , (6)which is consistent with our expectations given our noise.Thus, while the merger-ringdown consistency test is pow-erful when combining a large number of ringdowns, it isalso feasible to perform it with just a few observations( ∼ CONCLUSION AND OUTLOOK
We demonstrated a proof-of-concept study for anovel test of GR called the merger-ringdown consistencytest that checks for statistical consistency between theplunge-merger-ringdown phase across a set of ringdowndetections using a deep learning framework. The testaims at increasing the sensitivity to the merger-ringdownby simultaneously incorporating information on boththe amplitude-phase excitations and the QNM frequencyspectra in the ringdown. It uses the fact that the QNMexcitation amplitudes and the spectrum in the ringdownare not independent quantities in GR and are dictatedby the initial binary’s symmetries. Furthermore, the testprovides an efficient way of stacking ringdowns.This work illustrates that Bayesian deep learningmethods can be applied to infer the posteriors of ring-down parameters to conduct precision tests of GR. Un-like the traditional Monte-Carlo sampling, neural net-works perform the PE in fractions of a second, providinga crucial edge when dealing with the large number of ob-servations that are expected with future GW detectors.Note that our test is akin in spirit (and complemen-tary) to the more traditional inspiral-merger-ringdown (IMR) test [43–46]. However, the two tests aim at ad-dressing slightly different questions: the standard IMRtest checks for the overall consistency of the evolutionof a binary BH system by independently computing theremnant mass and spin using the pre-merger and thepost-merger signals; our test focuses solely on the merger-ringdown phase and checks for the statistical consistencybetween the QNM excitation amplitudes and the QNMspectrum in a population of ringdown detections. Fur-thermore, for possible future detections of heavy massbinary systems (that is set precedence by the discoveryof [47]), our test can be used to check the consistencyof initial BH parameters with the ringdown alone if theinspiral is not well measured.Here we have used non-spinning progenitor BHs wherethe QNM excitation amplitudes and phases are fullyparametrizable by its mass ratio q and the χ f − q re-lation is approximated by the simple analytical expres-sion in Eq. (3). However, our method can be extendedto encompass spinning progenitor BHs where the QNMexcitations depend on both q and χ , . The dependenceof the remnant spin χ f on the binary BH parametersshould then be replaced by implicit non-analytical rela-tions such as those in [30, 31]. Our OLS fit strategy doesnot rely on analytical relations and new parameters canbe estimated by increasing the output dimensions of theCVAE.This study used stellar-mass BH ringdowns targetingthe ET-like data. Similar results can be expected to holdfor CE. However, LISA will detect ringdowns from su-permassive BHs [4, 40], with loud SNRs. We plan toextend our analysis to include LISA-like data in the fu-ture. Finally, while this work demonstrates the feasibilityof a null test of GR, we are working on an extension toquantify possible deviations from GR — such as thoseparametrized in [48]. SUPPLEMENTAL MATERIALExcitation amplitudes and phases
We update the fits presented in [35] by additionallyrequiring A lm /A → q → t peak + 12 M . Note that the goodness of thefits do not change significantly between the version hereand in [35]. Details on the CVAE implementation
We use a deep learning framework to estimate the pos-terior distributions of x = { M, χ f , q } from the ringdown (3 ,
3) (2 , a a -0.555401 -1.1035 a a b b b l, m ) =(3 ,
3) and (2 , strain h = y ( x ). We follow [6–8] and use a CVAE ar-chitecture. The CVAE acts as an inverse nonlinear mapfrom y to the posteriors of x CVAE : y → p ( x | y ) . (7)It has the structure of a variational autoencoder [9, 10]: itis made of two serial neural network units (the ‘encoder’and the ‘decoder’) separated by a stochastic latent layer.The first neural network (encoder) maps the input y intothe latent layer. The second neural network (decoder)maps the latent representation of the input into the out-put probability distribution p ( x | y ).The CVAE is trained by introducing a third neuralnetwork unit, called the auxiliary encoder or the ‘guide’,at training time [49]. The training consists in optimisinga loss function. For the CVAE, the loss naturally splitsinto [7, 49]:1. the Kullback-Leibler (KL) divergence L KL , mea-suring the similarity between the outputs of theencoder and the guide; it quantifies the ability ofthe encoder to produce a meaningful mapping ofthe input into the latent space;2. the reconstruction loss L recon , measuring the prob-ability that the true values x true falls within thedecoder distribution.The total loss to be optimised is L tot = L recon + β L KL . (8)When β = 1, L tot coincides with the standard ELBOloss [9, 10]. The additional parameter β gives flexibilityin implementing effective training strategies.After the training, the guide is dropped out. At pro-duction time, only the encoder and the decoder are usedto sample the posteriors p ( x | y ). Fig. 5 contains a flow-diagram of the training and production steps. Details ofthe neural network architecture and our code is providedin a dedicated git repository [50].The variables which determine the CVAE training arelisted in Tab. III. We train the CVAE in batches of 512waveforms for 500 epochs, i.e., 500 forward-backwardpasses of the entire training set. The loss is minimized x true y { ~µ , ~σ } { ~µ , ~σ } z { ~µ , ~σ } Guide Encoder L KL Decoder L recon sampling y { ~µ , ~σ } z { ~µ , ~σ } EncoderDecoder x pred Repeat samplingsamplingTraining Production
FIG. 5. A schematic representation of the CVAE architecture. On the left, a single training step is represented. First, the signal y is mapped by the encoder into a latent stochastic distribution, which is a multivariate diagonal Gaussian with means andstandard deviations { (cid:126)µ , (cid:126)σ } ; similarly, the couple ( y, x true ) is mapped by the guide into a second Gaussian with parameters { (cid:126)µ , (cid:126)σ } ; the two are then combined into the L KL loss. Next, a latent variable z is sampled from the guide distribution andmapped by the decoder into a third Gaussian with parameters { (cid:126)µ , (cid:126)σ } . This final distribution is eventually used to computethe L recon loss. On the right, a single step at production time is shown. Now, the latent representation is sampled from theencoder and a predicted output x pred is sampled from the decoder; this step is repeated n samples times to produce an informativeposterior distribution for x ; in text, we fix n samples = 10 . Note the final distribution of x pred is not a Gaussian, but is a complexdistribution resulting from the convolution of the two serial sampling steps.Batch size 512Epochs 500Optimizer AdamInitial lr 10 − lr decay × . β annealing 3 × [10 − , , , , , using the Adam optimizer with an initial learning rate setto 10 − . The learning rate is decreased by a factor of 2every 80 epochs. To monitor the convergence of the loss,we set aside 10% of the training data-set and we use it forvalidation. Following [6], we use β to implement a cyclicannealing schedule. Annealing improves the efficiencyof the training and allows autoencoders to express moremeaningful latent variables [51]. We increase β from 0to 1 in steps of [10 − , , , , ,
1] and these steps are re-peated 3 times. After this, β is definitively fixed to 1.Next, the training performances improve when the in-puts y are standardized to zero mean and unit variance,and when the outputs x are normalized to have support in [1 , x is then scaled back to the original normal-ization at production time.The training data-set consists of ringdown waveformssampled at 4096 Hz with a total signal duration of 31 . L recon andthe KL divergence L KL separately. Further, we show theloss evaluated on the 90% training data-set and on the10% validation data-set. Notice that the training andvalidation losses are consistent, substantiating that thenetwork is not overfitting. For a quantitative diagnostic of the network perfor-mances, we present the P-P plot in Fig. 7: the plotshows marginalized cumulative distribution CDF of the The initial oscillations which are visible in L KL are due to thecyclic annealing. epochs recon (training) recon (validation) KL (training) KL (validation) FIG. 6. Evolution of the reconstruction and KL losses acrossthe training epochs. p C D F ( p ) final mass M final spin f mass ratio q FIG. 7. P-P plot of test data-set. For each observable, theplot shows the cumulative distribution CDF of the true valuesas a function of the p % confidence interval of the posteriordistribution. true values x true as a function of p % confidence interval.A diagonal P-P plot means that x true is contained p % ofthe times within p % confidence interval of the marginal-ized posteriors for x pred . Note that all the CDFs are con-sistent with the diagonal, demonstrating that our CVAErecovers the posterior samples for { M, χ f , q } from theringdowns reliably. ACKNOWLEDGMENTS
We thank P. Pani and A. Maselli for their produc-tive comments on an early version of this manuscriptand G. Sanguinetti for his advices on neural networks.We thank N. Johnson-McDaniel for useful clarificationsabout the IMR test. CP is indebted to E. Barausse andL. Heltai for their invaluable support in getting familiarwith neural networks and gravitational wave physics. Weacknowledge financial support provided under the Eu-ropean Union’s H2020 ERC, Starting Grant agreement no. DarkGRA–757480. We also acknowledge supportunder the MIUR PRIN and FARE programmes (GW-NEXT, CUP: B84I20000100001), and from the AmaldiResearch Center funded by the MIUR program “Dipar-timento di Eccellenza” (CUP: B81I18001170001).
Software.
The
PyCBC [52, 53] library was used to gen-erate the noise realizations and
LalSuite [54] to gen-erate the ringdowns. The neural network model usedin this work was developed using the
PyTorch library[55] for
Python . The OLS test was performed withthe statsmodels package [56] for
Python . The cor-ner plots have been produced with the corner package[57] for
Python . Code availability.
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