Echoes from the Abyss: A highly spinning black hole remnant for the binary neutronstar merger GW170817

Jahed Abedi∗

Department of Physics, Sharif University of Technology, P.O. Box 11155-9161, Tehran, Iran andSchool of Particles and Accelerators,

Institute for Research in Fundamental Sciences (IPM)P.O. Box 19395-5531, Tehran, Iran

Niayesh Afshordi†

Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, ON, N2L 2Y5, Canada andDepartment of Physics and Astronomy, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

The first direct observation of a binary neutron star (BNS) merger was a watershed moment inmulti-messenger astronomy. However, gravitational waves from GW170817 have only been observedprior to the BNS merger, but electromagnetic observations all follow the merger event. While post-merger gravitational wave signal in general relativity is too faint (given current detector sensitivities),here we present the first tentative detection of post-merger gravitational wave “echoes” from a highlyspinning “black hole” remnant. The echoes may be expected in different models of quantum blackholes that replace event horizons by exotic Planck-scale structure and tentative evidence for themhas been found in binary black hole merger events. The fact that the echo frequency is suppressedby logM (in Planck units) puts it squarely in the LIGO sensitivity window, allowing us to build anoptimal model-agnostic search strategy via cross-correlating the two detectors in frequency/time. Wefind a tentative detection of echoes at fecho ' 72 Hz, around 1.0 sec after the BNS merger, consistentwith a 2.6-2.7 M� “black hole” remnant with dimensionless spin 0.84− 0.87. Accounting for all the“look-elsewhere” effects, we find a significance of 4.2σ, or a false alarm probability of 1.6 × 10−5,i.e. a similar cross-correlation within the expected frequency/time window after the merger cannotbe found more than 4 times in 3 days. If confirmed, this finding will have significant consequencesfor both physics of quantum black holes and astrophysics of binary neutron star mergers.

I. INTRODUCTION

One of the most controversial questions in quantumgravity today is whether quantum effects may becomesignificant outside horizons of black holes [1–4], or thatthey only play a role well inside the horizon (e.g., [5]).While the latter is the conservative assumption, theformer might come about through (so-far speculative)processes that could resolve the information paradox[2, 3, 6, 7], or the cosmological constant problem(s) [8]. Inorder to resolve the paradox, significant deviations formclassicality need to occur by the Page time ∼ M3, butthey may already be in place as early as the scramblingtime ∼M logM (or more precisely T−1

H logSBH, for tem-perature TH and entropy SBH) [9, 10], where M is theblack hole mass in Planck units.

On the observational side, different groups have foundtentative evidence (with p-values in the 0.2-2% range)for repeating echoes in the LIGO/Virgo observations ofbinary black hole mergers [11–13]. While the origin ofthis signal remains controversial (see [13–16] for differingpoints of view), they can be interpreted as signatures ofPlanck-scale deviations near black hole horizons, whichwould validate the theoretical motivations based in theinformation paradox, or the cosmological constant prob-lem. The echoes originate from the gravitational waves

∗ jahed abedi@physics.sharif.ir† nafshordi@pitp.ca

that are trapped between the angular momentum barrierand an exotic compact objects (ECOs) [17–19] (see thenext section for details), slowly leaking out and repeatingwithin time intervals of:

∆techo ' 4GMBH

c3

(1 + 1√

1−a2)× ln

(MBH

Mplanck

)' 4.7 msec

(MBH

2.7 M�

)(1 + 1√

1−a2), (1.1)

where MBH and a are the final mass and the dimension-less spin of the black hole remnant. Also, note that thistime is similar to the scrambling time for quantum “fastscramblers” with entropy of the black hole [9, 10].

Among the detected gravitational wave events, the firstobservation of a binary neutron star (BNS) coalescenceGW170817 by the LIGO/Virgo gravitational wave detec-tors [20, 21] provides a unique opportunity to test generalrelativity, as well as the nature of the merger remnant,by searching for gravitational wave echoes. After such amerger, a compact remnant forms whose nature mainlydepends on the masses of the inspiralling objects. ForGW170817, the final mass is within ∼ 2 − 3M� whichcould form either a black hole or a neutron star (NS).If it is a neutron star, it can be too massive for stabil-ity which means it will undergo a delayed collapse intoa black hole. However, if a black hole forms after themerger of GW170817, the ringdown frequency is outsidethe sensitivity band of LIGO/Virgo detectors, and thusthey remain largely blind to this signal. Indeed, no post-merger GW signal in GW170817 has been detected [21].Nevertheless, they can also lead to detectable echoes at

arX

iv:1

803.

1045

4v1

[gr

-qc]

28

Mar

201

8

2

event

time

r*

tmerger

techo

Δtecho

Echoes

FIG. 1: Spacetime depiction of gravitational waveechoes from a membrane on the stretched horizon,

following a collapse from binary neutron star mergerevent.

lower freqeuncies in case an ECO is formed [18].In nature, a BNS merger can develop in four possible

ways: [21]:

1. A black hole forms immediately after the merger.

2. A hypermassive NS is formed, then within . 1 secit collapses into a black hole.

3. A supramassive NS is formed that collapses to ablack hole on timescales of 10 - 104 sec.

4. A stable neutron star is formed [20].

A neutron star with mass greater than the maximummass of a uniformly rotating star is hypermassive, as it isprevented from collapse through support from differentialrotation and thermal gradients, caused by rapid coolingthrough neutrino emission. Eventually, magnetic brakingof the differential rotation causes such merger remnantsto collapse within . 1 sec after formation [22, 23].

For GW170817, a broad range of NS equations of statelead to a post-merger mass which remains within the hy-permassive NS regime [20]. Accordingly, in this paper wepresent a search for short (.1 sec) duration GW echosignal potentially emitted from post-merger remnants inscenarios (1) or (2). We find tentative detection of echoessourced by a “black hole” (or ECO) remnant of the neu-tron star collapse which is consistent with scenario (2) atfecho ' 72 Hz. Specifically, we find signatures of quan-tum gravitational alternatives to black hole horizons (asremnant of the BNS merger) in the gravitational wavedata with statistical significance of 4.2σ1, or a false de-

1 In this paper, we use 1-tailed gaussian probability to assign a sig-

0.65 0.70 0.75 0.80 0.85 0.90 0.95Spin

2.4

2.6

2.8

3.0

3.2

3.4

Fina

lred

shift

edBH

Mas

s/M

Echoes at 72 HzEchoes at 73 Hz

LIGO low-spin prior 90% region

LIGO high-spin prior90% region

FIG. 2: Constraints on mass and spin of the “blackhole” remnant of GW170817 from the gravitational

waves emitted during inspiral (shaded regions; assuming2-5% energy loss during merger) and post-merger

echoes (lines). The low (high) BNS spin priorsconstrain the final dimensionless spin to be within

0.84-0.87 (0.70-0.87).

tection probability of 1.56× 10−5. This would constrainthe spin of the “black hole” remnant to be within 0.84-0.87 (0.70-0.87) for the low (high) spin prior total BNSmasses, reported by the LIGO/Virgo collaboration [20](see Fig. 2), consistent with the expected range fromnumerical simulations (e.g., [24]).

We shall first describe the expected properties ofechoes, then our optimal search methodology, and finallyresults and discussions.

II. PROPERTIES OF ECHOES

Late time perturbations around Kerr black holes are of-ten described in terms of quasi-normal modes (QNM’s).In classical GR, it is assumed that these modes are purelyoutgoing at infinity and purely ingoing at the horizon.The continuous transition from ingoing to outgoing hap-pens at the peak of the black hole angular momentumpotential barrier. Correspondingly, as first noticed in[17, 18], if a membrane near the horizon (exotic com-pact object, or ECO) exists, it can partially reflect backthe ingoing modes of the ringdown. In the “geometricoptics” approximation, the reflected wavepacket reaches

nificance to a p-value, e.g., 1−p-value= 84% and 98% correspondto 1σ and 2σ respectively.

3

the angular momentum barrier after ∆techo = 8M logM(+ spin corrections; see below) which is equal to twicethe tortoise coordinate distance between the peak of theangular momentum barrier (rmax) and the membrane.The wavepacket would be partially transmitted throughthe barrier, appearing as delayed echo, and partially re-flected, leading to subsequent echoes that repeat with aperiod of ∆techo (see Fig. 1).

These echoes have two natural frequencies: The reso-nance (or harmonic) frequencies of the “echo chamber”fn = n/∆techo (also known as w-modes), where n is aninteger number, as well as the “black hole” ringdown(or QNM) frequencies that set the initial conditions forthe echoes. The overlap of these frequencies will definethe echo waveform. The higher harmonics (n � 1) thatare primarily excited by the merger event decay quickly,while the lower harmonics are much more long lived (evenmarginally unstable [19, 25, 26]) as they are trapped bythe angular momentum barrier.

Since the final mass of BNS merger (2-3 M�) is muchsmaller than that of the binary black hole mergers [11],the lowest harmonic 1/∆techo (' 80 Hz), is shifted tothe regime of LIGO sensitivity. This enables us to devisea model-agnostic strategy (such as in [12]) to search forlowest harmonics, independent of the initial conditions.

Geometric echo time delay for Kerr black hole withdimensionless spin parameter a is given by [11]:

∆techo = 2rmax − 2r+ − 2∆r

+2r2+ + a2M2

r+ − r−ln

(rmax − r+

∆r

)−2

r2− + a2M2

r+ − r−ln

(rmax − r−

r+ − r− + ∆r

), (2.1)

where r± = M(1 ±√

1− a2), ∆r is the coordinate dis-tance of the membrane and the (would-be) horizon, andrmax is the position of the peak of the angular momentumbarrier [11].

Placing the membrane at ∼ a Planck proper distanceoutside the horiozn:∫ r++∆r

r+

√grrdr|θ=0 ∼ lp ' 1.62× 10−33 cm, (2.2)

would be equivalent to setting ∆techo ∼ scrambling timefor the black hole [9, 10, 27, 28]. This choice can alsobe motivated by generic quantum gravity considerationssuch as the brick wall model [29], trans-Planckian effects[30, 31], 2-2 holes [32], or solving the cosmological con-stant problem [8], which all fix the location of the mem-

brane at ∆r|θ=0 ∼√

1− a2l2p/[4M(1 +√

1− a2)] [11].To find the prior for final BH mass, we use the 90%

credible region for the total (redshifted or detector frame)mass from Table I of [20] (for the more realistic low-spinprior for BNS), but assume that 2 − 5% of the energyis lost during the merger [33]. For the dimensionless BHspin a = 0.72−0.89 (e.g., [24]), as well as an order of mag-nitude change in Planck length, we can constrain ∆techo

to be within:

0.0109 < ∆techo(sec) < 0.0159. (2.3)

This range in echo time delay corresponds to echo fre-quency (fecho = ∆t−1

echo) range:

63 ≤ fecho(Hz) ≤ 92, (2.4)

which we shall use for our echo search and backgroundestimates, for the rest of the paper.

III. MODEL-AGNOSTIC SEARCH FORECHOES

The simplicity of a method indicates its degree of cer-tainty and minimizes any ambiguity. Motivated by themodel-agnostic echo search proposed in [12], here we de-vise an optimal (minimum variance) method to extractperiodic gravitational wave bursts in the cross-correlationof the two detectors:

h(t) ∝∑n

δD(t− n∆techo − t0), (3.1)

or

hf ∝∑n

δD(f − nfecho), (3.2)

in frequency space, δD is the Dirac delta function. Thisis clearly a simplification, but can be justified as:

1. The structure of individual echoes cannot be re-solved, as they are set by the QNM frequency thatis beyond the LIGO sensitivity window for the BNSmasses.

2. Echoes decay slowly in the low frequency and/orsuperradiance regime [19].

Therefore, given that the noise is uncorrelated betweenthe two detectors while the signal should be the same, wesearch for periodic peaks of equal amplitude in the cross-power spectrum of the two detectors at integer multiplesof fecho.

In this analysis, we use the observed strain datas-tream of GW170817 [34] occured at GPS time tmerger =1187008882.43 == August 17 2017, 12:41:04.43 UTCfrom the two detectors Hanford and Livingston. Wecall these hH(t) and hL(t), respectively. We used thestrain data at 16 kHz and for 2048 sec duration. Firstwe calculate the offset between detectors. By correlat-ing the two datasets (before the merger) we obtain thatfor GW170817, Hanford event is δt ' 2.62 msec delayedwith respect to Livingston (see Appendix A for details).This is consistent with what has already been reported[35]. Similarly, we can confirm the relative polarizationof −1 between the two detectors.

Based on general properties of echo signal outlinedabove, we build our model-agnostic search as follows:

1. We vary the fundamental frequency of echoesfecho = ∆t−1

echo within the 90% credible regionrange, as outlined in Sec. II:

63 ≤ fecho(Hz) ≤ 92.

4

0 25 50 75 125 150 175 200100Frequency [Hz]

5

0

5

10

15

20

25

30

35

40t

t mer

ger [

sec]

−1 × 1039

.6 43

5.94

5.37

4.69

3.85

2.64

−1

FIG. 3: Time-frequency representations of X(t, f) before and after the merger for the BNS mergergravitational-wave event GW170817, observed by the LIGO, and Virgo detectors. The possible peak of echoes foundin the time-frequency and amplitude-frequency plots are marked with a green squares. As can be seen by the color

scale, the peak at fpeak = 72 (±0.5) Hz and t− tmerger ' 1.0 sec, with an amplitude of X(tpeak, fpeak) = −6.48×1039,is the highest peak in this diagram, from before and after the BNS merger (see Figs. 4 and 6 below for more

details). A secondary tentative peak at the same frequency but t− tmerger ' 32.9 sec is also highlighted.

5

X(t

peak,f)

[strain]−

2

Frequency [Hz]

Fewer than 4 similar peaks in 3 days data1 : 1

2 : 1

2 : 3

1 : 2

1 : 4

2 : 9

2500

Frequency series of X(t, f)at t = tpeak = 1.0s after the mergerat t = trandom after the merger

2σ

3σ

4.2σ

50 150 200100

1

0

−1

−2

−3

−4

−5

−6

−7

1039

FIG. 4: Amplitude-frequency representations of first peak (red) at 1.0 sec after the merger for the BNS mergergravitational-wave event GW170817, observed by x-correlating the LIGO detectors. Here the minimum of the firstpeak at 1.0 sec after the merger is −6.48× 1039. We also plot the same amplitude-frequency for a random time indata (yellow). Other significant peaks associated with resonance frequencies are 2

9 : 14 : 1

2 : 23 : 1 : 2 of 72 Hz. Solid

area between 63 Hz and 92 Hz was our prior range to search for Planckian echoes.

We shall then refer to fpeak as the best fit value offecho, which maximizes (−1×) cross-correlation ofthe two detectors.

2. We search for the echo signal within the range 0 <t− tmerger ≤ 1 sec. This is the prior range alreadyused by Abbott et al. 2017 [21] in search of post-merger gravitational waves from remnant of BNSmerger GW170817.

3. We obtain the amplitude spectral density (ASD)with the same method introduced in [36] which wasused for whitening the data. As we are interestedto combine different frequencies to obtain the echosignal, we Wiener filter (rather than whiten) thedata by dividing by noise variance ASD2 (ratherthan ASD). This ensures the optimal combinationof different frequencies, which receives minimum

variance from noise:

H(t, f) = Spectrogram[IFFT

(FFT(hH(t−δt))

ASD2L

)],

L(t, f) = Spectrogram[IFFT

(FFT(hL(t))

ASD2L

)].

(3.3)

Here the same setup as LIGO code [36] to ob-tain the spectrogrm with mlab.specgram() func-tion in Python with NFFT=fs = 16384, andmode=’complex’ is used. Same as the methodused in [36], the number of points of overlap be-tween blocks is NOVL = NFFT × 15/16. Withweighting the frequencies and after a right normal-ization the resulting time series becomes in unitsof strain−1. For consistency checks, instead ofmlab.specgram() we also used one-dimensional dis-

6

crete Fourier transform for real input np.fft.rfft()taking 1 sec data segments, with steps of 1/16 ofa second, and after right normalization we repro-duced same results. As another consistency check,same results is reproduced using 4096 Hz data.Here the ASDs are obtained for all the 2048 secrange of data.

4. We then cross-correlate the resulting spectrogramsand sum over all the resonance frequencies of nf .More specifically, we define

X(t, f) =

10∑n=1

< [H(t, nf)× L∗(t, nf)] , (3.4)

which is the Wiener-filtered cross-power spectrumof the two detectors for the echo signal 3.2. Thechoice of n ≤ 10 does not affect our results, as theLIGO noise (or ASD2) blows up at high frequencies,and thus they do not contribute to X(t, f). Due tothe opposite polarizations of the LIGO detectorsmentioned above, the real gravitational wave sig-nals show up as peaks in −X(t, f).

5. For obtaining the p-value we used 2048 sec dataof GW170817 at 16384 Hz after noise subtrac-tion. We picked 1500 sec = 25 min of this data(∼ (−1600,−100) sec before time of merger). Thenwith shifting detectors we produced ∼14 days of“data” out of the original 25 minutes. This datais “time-shifted”, i.e. a non-physical time lag is in-troduced between the detectors analyzed so as toremove correlated gravitational-wave signals. Thisdata is also “off-source”, i.e. it is outside of the 100sec window of event. Assuming that the detectornoise properties remain reasonably steady through-out the observation time, this analysis produces in-dependent realizations of noise (or background) forX(t, f), which can be used to estimate the false-alarm probability for a given possible detection.

The strength of this method is in the simplicity of thecomputation, which only involves a straightforward x-correlation in frequency space, without any arbitrary orad-hoc cuts or parameters.

IV. RESULTS

We now present the outcome of the search for gravi-tational wave echo signal from the remnant of the BNSmerger GW170817 using data from LIGO/Virgo collabo-ration, within . 1 sec after the merger. The signal couldbe a consequence of a hypermassive NS collapse, whichhas collapsed to form a “black hole” (or ECO) [21]. Wefind a significant negative peak in the spectrogram inFigs. 3 and 4, at t − tmerger ' 1.0 sec after the merger(see Fig. 6), at fecho ' 72 Hz. As it is shown in Fig.4, it is also accompanied by secondary resonances, as all

−X (t

5 × 1039

peak, fpeak)

3 × 1039 4 × 1039 6 × 1039 7 × 1039

Search result Search background

Number

of e

ven

ts

2σ 3σ 4σ 4.5σ

10 5

10 4

10 3

10 2

10 1

100

Echoes

:4.2σ

FIG. 5: Average number of noise peaks higher than aparticular -X(t,f) within a frequency-intervals of 63-92

Hz and time-intervals of 1 sec for LIGO noise nearGW170817 event. The red square shows the observed−X(tpeak, fpeak) peak at 1.0 sec after the merger. The

horizontal bar shows the correspondence betweenX(t, f) values and their significance. This histogram

obtained from producing ∼2 weeks data out ofoff-source 2048 sec available data [34].

the periodic signals (Eq. 3.2) with a rational ratio of pe-riods, have a non-vanishing overlap (also see [12] for asimilar effect). Accounting for the “look elsewhere” ef-fect, we find tentative detection of Planck-scale structurenear black hole horizons at false detection probability of1.56 × 10−5 (corresponding to 4.2σ significance level inFig. 5). This means fewer than 4 similar peaks can befound in three days of “data”, as discussed in the Sec.III.

As we hinted in the Introduction above, the frequencyof this peak would constrain the dimensionless spin ofthe “black hole” or ECO remnant to be within 0.84-0.87(0.70-0.87) for the low (high) spin prior total BNS masses[20] (see Fig. 2). For the more realistic low-spin prior onBNS spins, the tight constraint of a = 0.84− 0.87 is ex-actly in line with numerical simulations of BNS mergers(e.g., [24]), as opposed to a ' 0.67 for binary BH mergers(e.g., [37]).

Along with this significant peak, there exists a lowerpeak with nearly identical resonance pattern at 73 Hzand t − tmerger = 32.9 sec after the merger (see Figs. 3,7 and 8). While this secondary peak could be due todetector noise, it might also originate from post-mergerinfall into the “black hole” or ECO, which would causea slight change in spin and mass.

The observed peak at t − tmerger ' 1.0 sec is consis-tent with Scenario 2 in Section I, which is a hypermas-sive NS remnant [21] which collapses into a BH within∼ a second. Interestingly, this merger might lead to theformation of a torus around this black hole, providingthe sufficient power for a short γ-ray burst (GRB) [24]from its rotational energy, fuelling a jet that has been de-

7

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00t tmerger [sec]

0

1

2

3

4

5

6

1×X(

t,f p

eak)

[stra

in]

21e39

Time series of X(t, fpeak)

2σ

3σ

Fewer than 4 similar peaks in 3 days data4.2σ

FIG. 6: Amplitude-time representations of first (andmost significant) echo peak at 1.0 sec after the mergerand frequency of 72 Hz. The shaded region is 0-1 sec

prior range after the merger, first adopted in [21], whichwe use to estimate p-value . The maximum of the peak

is 6.48× 1039.

tected (170817A) by the Fermi Gamma-ray Burst Moni-tor [38, 39] 1.7 sec after the merger.

V. DISCUSSIONS

In this section, we shall go over some sanity checks:

1. The most immediate question on reader’s mindmight be whether our tentative detection of echoesat 72 Hz, around 1.0 second after the merger, mightbe due to some type of leakage from BNS inspi-ral signal. Now, this signal is computed fromx-correlation of detectors within 0.5 sec < t −tmerger < 1.5 sec, and thus is not expected tobe affected by the pre-merger inspiral signal, orthe LIGO-Livingston glitch at 1.1 second beforemerger. Furthermore, the most significant peak inX(t, f) in Fig. (3) is indeed our echo peak at 72Hz and 1 sec, while the pre-merger signal is less

significant (at any particular frequency), and doesnot appear to have any specific feature at 72 Hz.Furthermore, recurrence of the same frequency at33 seconds after the merger makes the possibilityof leakage from before merger less likely.

2. Fig. 9 shows the simple cross-power spectrum ofwhitened, time-shifted data from the two detectors,around the time of observed echo peaks, 1.0 secand 32.9 sec after the merger. Given that, withinthe assumption of gaussian detector noise, the noiseshould be uncorrelated and uniform on this plot,we can assess the significance of various harmonics.The vertical bands show the integer multiples ofour tentative detection fecho(Hz) = 72.0± 0.5. Wesee that different harmonics have different levels ofexcitement at different times. Interestingly, the 2nd(1st) harmonic has the most significant amplitudefor 1st (2nd) echo peak. Moreover, for the 1st (moresignificant) echo peak at 1.0 sec, we switch fromeven harmonics 2 and 4, to odd ones, i.e. 7 and(possibly) 9. There is also a curious peak at 48Hz = 2

3fecho (for both echo peaks), which is notcaptured by the linear echo model (3.2) and mayindicate resonance with nonlinear interactions, e.g.,h3 at the source.

3. Another worry is the leakage from LIGO calibrationlines at 34 Hz-37 Hz, which are nearly 1/2 of ourobserved fecho = 72 Hz2. While this is a possibility,it is not clear why this would only happen at 1second after BNS merger, and not two weeks of pre-merger “data” we use for our background search. Infact, our most significant peaks at 144 Hz and 288Hz in the whitened cross-power spectrum (see Fig.9) do not appear to correspond to any particularfeature in the LIGO noise curve. Moreover, themost significant x-correlation peaks in Fig. 9 areall negative, as expected for real gravitational wavesignal from GW170817, while there is no physicalprocess that would correlate the detector noise fromHanford and Livingston detectors.

4. Phase shifts upon reflections off the angular mo-mentum barrier, or the ECO/membrane may shiftthe natural frequencies of the echo3. At frequen-cies much lower than the QNM frequencies (∼ 104

Hz; associated with 1/thickness of the angular mo-mentum barrier, or the Hawking temperature asso-ciated with the “firewall” or “fuzzball”), we expectthe phase shifts, and thus the shift in natural fre-quencies to approach a constant, i.e.

fn =

(n− φ1 + φ2

2π

)fecho ' nfecho + const., (5.1)

2 We thank Ofek Birnholtz for bringing this point to our attention3 We thank Jing Ren for bringing this point to our attention.

8

X(t,f

peak)

[strain]−

2

Fewer than 4 similar peaksin 3 days data

Time series of X(t, fpeak)at fpeak = 72 ± 0.5 Hz,and fpeak = 73 ± 0.5 Hz

2σ

3σ

4.2σ

1

0

−1

−2

−3

−4

−5

−6

−7

10392

−100 − 75 − 50 − 25 0 25 50 75 100

t− tmergr [sec]

FIG. 7: Amplitude-time representations of peaks before and after the merger t− tmerger =(-100s,100s) for the BNSmerger gravitational-wave event GW170817, observed by the LIGO, and Virgo detectors. As highlighted, the firstand second highest negative peaks appear 1.0 sec and 32.9 sec after the merger respectively. This plot is for fixedfrequencies of 72 (±0.5) Hz and 73 (±0.5) Hz. First solid area is 1 sec prior range after the merger determined in

[21]. As can be seen minimum of first peak is −6.48× 1039 and minimum of second peak is −4.54× 1039.

9

X(t

peak,f)

[strain]−

2

Frequency [Hz]

1 : 1

2 : 1

2 : 3

1 : 2

1 : 4

2 : 9

Frequency series of X(tpeak, f)at tpeak = 1.0s after the mergerat tpeak = 32.9s after the merger

Fewer than 4 similar peaksin 3 days data

2σ

3σ

4.2σ

50 100 150 200

1

0

−1

−2

−3

−4

−5

−6

1039

−7

FIG. 8: Amplitude-frequency representations of data after the merger for the peaks at 1.0 sec (red) and 32.9 sec(green) after the merger for the BNS merger gravitational-wave event GW170817, observed by the LIGO, and Virgodetectors. Here the resonance frequencies for the first peak at 1.0 sec and second peak at 32.9 sec after the merger

are shown in this plot. For the second peak there is identical pattern with a factor 1.0139 in these frequencies. Herethe maximum of first peak is −6.48× 1039 and maximum of second peak is −4.54× 1039. Solid area between 63 Hz

and 92 Hz is the region we expect to see Planckian echoes.

where φ1 and φ2 are phase shifts at the angular mo-mentum barrier and the ECO/membrane, respec-tively. However, note that this does not change thespacing of the fn’s. The spacing of our two mostsignificant frequency peaks in Fig. 9 (1 second afterthe merger) is ∆f ' 288 Hz − 144 Hz = 144 Hz.Now, considering the frequencies within our priorrange fecho = 63 − 92 Hz (Eq. 2.4), the only fre-quency that can be multiplied by an integer to give∆f , is fecho = 72 Hz, consistent with our prior find-ing. This also suggests that the sum of the phaseshifts φ1 +φ2 must be an integer multiple of 2π (atleast, within a couple of percent).

5. Let us next estimate the gravitational wave energyemitted in echoes. We start by noticing that thenegative peaks at t − tmerger ' −0.9 and −5.4 secprior to the merger in Fig. 7, correspond to theinspiral signal at 144 and 72 Hz, respectively (see

also Fig. 3). We now can use the energy emittedduring the inspiral:

∆EGW = −M2

∆(πfGM)2/3, (5.2)

in terms of chirp mass M ' 1.188 M� [20], toestimate that the fraction of inspiral energy radi-ated within 72-73 Hz and 144-145 Hz, compared toall energy emitted in LIGO band (∼ 0.025 M�c2)is 0.5%. We would gain an extra factor of ∼ 3,by including the 4th harmonic at 288 Hz, which isthe only other significant harmonic present in dataup to 500 Hz (given that GW energy scales at f2×amplitude2). Since the amplitudes of inspiral peaksin Fig. 7 are a factor of ∼ 2 smaller than the echoes(within 1 Hz frequency windows), we can estimatethat the fraction of radiated post-merger echo toradiated BNS inspiral energy (within LIGO sensi-

10

tivity band) is:

Eecho(. 500 Hz)

Einspiral(. 500 Hz)∼ 3%, (5.3)

for the first and most prominent echo peak . 1sec after the merger. Any estimate beyond thisfrequency range will be highly model-dependent.

6. In order to further test the consistency of the sig-nal, we shifted detectors within the range of (-10ms, 10 ms) and minimized X(t− tmerger, f) withinthe range 0.5 sec < t − tmerger ≤ 1.5 sec. Fig.10 shows this minimization and confirms that thenegative peak at fpeak=72 Hz is indeed the min-imum. Similarly, Figs. 11 (a)-(b) confirm thatthe peaks at −0.5 < tpeak(sec) − 1.0 < 0.5, and−0.5 < tpeak(sec) − 32.9 < 0.5 show a minimumwhen detector times are shifted by 2.62 ms, whichis exactly the same time delay seen for the mainevent [20]. Moreover, the plots confirm that rel-ative polarization of the two detectors which areopposite to each other for the main event, as wellas the peaks at 1.0 sec and 32.9 sec after the merger.

7. During the course of this work, an updated versionof [12] appeared on arXiv, claiming evidence for anecho frequency of f ′echo ' (0.00719 sec)−1 = 139 Hzfor GW1708174, with a p-value of 1/300. Given theproximity of this value to the second harmonic ofour echo frequency 2× fecho = 144 Hz, it is plausi-ble that the two different methods are indeed seeingthe same echo signal. However, the method ap-plied in [12] is sub-optimal, as they whiten (ratherthan Wiener filter) the data (see Sec. III above),and thus could underestimate the significance ofthe correlation peak at fecho ' f ′echo/2. Indeed,our p-value of 1.6 × 10−5 is lower by more than 2orders of magnitude. Moreover, their f ′echo = 139Hz is outside our expected prior range (2.4) for theremnant of the the GW170817 BNS merger, andcannot account for our odd harmonic at f ' 504Hz either.

VI. CONCLUSIONS

In the past two years, theoretical and observationalstudies of black hole echoes have emerged as a surpris-ing and yet powerful probe of the quantum nature ofblack hole horizons. So far, three independent groupshave found tentative evidence for echoes in LIGO/Virgogravitational wave observations of BH (or BNS) mergersat 2-3σ level (i.e. p-values of 0.2-2%, including the lookelsewhere effects) [11–13]. In this paper, we presented an

4 Based on our correspondence with the authors of [12], it appearsthat there is a typo in the value of ∆techo in the text of the arXivv.2, while the figures are correct.

optimal and model-agnostic method to search for echosignals well below the ringdown frequency of the blackhole quasinormal modes. As such, this method is well-suited to find echoes in the LIGO/Virgo observations ofthe BNS merger GW170817. This yielded the first ten-tative detection of the echoes at 4.2σ significance, corre-sponding to a false detection probability of 1.6 × 10−5.The peak signal for echoes occurs 1.0 second after themerger at fundamental frequency of 72 Hz, with anothertentative peak at 33 seconds. If confirmed, this couldalso be the first evidence for formation of a hypermassiveneutron star that collapses to form a 2.6-2.7 M� “blackhole” with dimensionless spin of 0.84 − 0.87 within ∼ 1second of the BNS merger. We further estimate that thegravitational wave energy emitted in post-merger echoesis ∼ 3% of the energy emitted in the BNS inspiral, withinthe LIGO sensitivity band.

We should note that, unlike the binary BH mergers,there are additional frequencies associated with the ac-cretion disk around BNS merger remnants. While theirnature remains mysterious, observed low frequency quasiperiodic oscillations (QPOs) in black hole X-ray binarieshave similar frequencies to the echoes found here [40]5.Could it be that we are simply detecting the QPOs in(a much more massive) post-merger accretion disk? Orrather, are all low frequency QPO’s (of BH accretiondisks) sourced by echoes of exotic compact objects? Weshall defer a more detailed analysis of this point to fu-ture work, but it suffices to say that the behavior of postBNS merger accretion disks over long time scales are verypoorly understood (but see e.g., [41] for recent progressin this direction).

In our last paper [11], we predicted that: “ ... a synergyof improvements in observational sensitivity and theoret-ical modelling can provide conclusive evidence for quan-tum gravitational alternatives to black hole horizons.”With the tentative detection reported here, along withother complementary developments, it appears that weare much closer to that possibility.

ACKNOWLEDGMENTS

We dedicate this work to the memories of StephenHawking and Joe Polchinski, two of the champions ofthe black hole information paradox, as well as muchof modern theoretical physics, as we know it. Theirpresence is sorely missed in our community.

We thank Avery Broderick, Ofek Birnholtz, Vitor Car-doso, Bob Holdom, Jing Ren, and Daniel Siegel for help-ful comments and discussions. We also thank Lam Huiand Shinji Mukohyama for encouraging us to look forechoes in GW170817 BNS merger. This work was sup-ported by the University of Waterloo, Natural Sciences

5 We thank Avery Broderick for pointing this out.

11

×10−4

5

2.5

0

−2.5

−5

−7.5

−10

−12.5

Frequency

48 72 144 216 288 360 432 504 576 648

1:1x− correlation(H,L) at tpeak = 1.0 sec after the mergerx− correlation(H,L) at tpeak = 32.9 sec after the merger

x−correlation(H

, L)

23 :1

2:1

4:1

7:1

FIG. 9: The simple cross-power spectrum of Hanford and Livingston detectors (with a 2.62 msec time delay), afterwhitening each dataset independently at t− tmerger(sec) = 1.0 (red) and 32.9 (green). The vertical bands show all

the integer multiples of fecho(Hz) = 72.0± 0.5 in frequency (which contribute to X(t, fecho) in Eq. 3.4), as well as acurious peak at 2

3fecho. Different harmonics of the “echo chamber” appear to be excited at different times

and Engineering Research Council of Canada (NSERC),Institute for Research in Fundamental Sciences (IPM),and the Perimeter Institute for Theoretical Physics. Re-search at the Perimeter Institute is supported by the

Government of Canada through Industry Canada, andby the Province of Ontario through the Ministry of Re-search and Innovation.

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[strain]−

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0.0100 0.0075 0.0050 0.0025 0.0000 0.0025 0.0050 0.0075 0.0100Detector time shift [sec]

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FIG. 11: (a) Peak of 1.0 sec after the merger. (b) Peak of 32.9 sec after the merger. We see that for both of thepeaks the minimums (red), maximums (green) are at 2.62 ms. Minimized (red) Maximized (green) X(t, f)

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Appendix A: Effective time delay between Hanfordand Livingston

In order to find the time delay, δt, between the twodetectors, we follow these steps:

1. We shift one of the whitened strain series within(-10 msec, 10 msec).

2. We obtain complex spectrogram for both Hanfordand Livingston detectors, H(t, f) and L(t, f) with16k HZ data.

3. Given that the two detectors have nearly 180 de-grees phase difference, we maximized |H(t−δt, f)−L(t, f)| for constant f within the range t =(-30 s,+0.5 s) .

4. We only keep the 100 highest values of

|H(t− δt, f)− L(t, f)|f=fmax(t),

14

and sum over them (smaller values are more likelyto be affected by noise).

5. Finally, we find the δt that maximizes this sum,which happens at the value of 2.62 ms.

Note that as spectrogram uses time ±0.5 sec to Fouriertransform around each time, we require to go up to 0.5sec, to include the merger event. However, there is no

overlap between this range used to calculate δt, and(0.5s, 1.5s) range which contributes to our primary echopeak.

Also note that, while δt = 2.62 sec is consistent withthe 3 ms time delay reported by LIGO [20], it is notexactly the same as the geometric time delay, as it alsocaptures the small phase difference (away from 180 de-grees) between the two detectors.