Echoes from the Abyss: Evidence for Planck-scale structure at black hole horizons

Jahed Abedi,1, 2, 3, ∗ Hannah Dykaar,4, 5 and Niayesh Afshordi3, 5, †

1Department of Physics, Sharif University of Technology, P.O. Box 11155-9161, Tehran, Iran2School of Particles and Accelerators, Institute for Research in

Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran3Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, ON, N2L 2Y5, Canada4Department of Physics, McGill University, 3600 rue University, Montreal, QC, H3A 2T8, Canada5Department of Physics and Astronomy, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

In classical General Relativity (GR), an observer falling into an astrophysical black hole is notexpected to experience anything dramatic as she crosses the event horizon. However, tentativeresolutions to problems in quantum gravity, such as the cosmological constant problem, or theblack hole information paradox, invoke significant departures from classicality in the vicinity of thehorizon. It was recently pointed out that such near-horizon structures can lead to late-time echoesin the black hole merger gravitational wave signals that are otherwise indistinguishable from GR.We search for observational signatures of these echoes in the gravitational wave data released byadvanced Laser Interferometer Gravitational-Wave Observatory (LIGO), following the three blackhole merger events GW150914, GW151226, and LVT151012. In particular, we look for repeatingdamped echoes with time-delays of 8M logM (+spin corrections, in Planck units), corresponding toPlanck-scale departures from GR near their respective horizons. Accounting for the “look elsewhere”effect due to uncertainty in the echo template, we find tentative evidence for Planck-scale structurenear black hole horizons at 2.9σ significance level (corresponding to false detection probability of 1 in270). Future data releases from LIGO collaboration, along with more physical echo templates, willdefinitively confirm (or rule out) this finding, providing possible empirical evidence for alternativesto classical black holes, such as in firewall or fuzzball paradigms.

There is mounting, albeit controversial, theoretical ev-idence that quantum black holes might be significantlydifferent from their classical counterparts, even in theregime where semi-classical gravity is expected to bevalid. Such strong modifications may exist, not only dueto non-perturbative quantum gravitational effects [1–4],but also at the level of semi-classical approximation [5, 6].In particular, modern versions of Hawking’s black holeinformation paradox have led to exotic alternatives toclassical black hole horizons such as fuzzball [2, 3] andfirewall paradigms [1, 7]. These should form by Pagetime ∼M3, but may emerge as early as the “scramblingtime” ∼M logM [8, 9], where M is the black hole massin Planck units.

On more phenomenological grounds, it has been pro-posed that a wholesome solution to the (old and new)cosmological constant problems replaces the black holehorizons by a Planck-scale quantum barrier, which couldnaturally explain the observed scale of dark energy [10].Furthermore, accretion into these “black holes” offers apossible origin for observed ultra high energy IceCubeneutrinos [11].

In this letter, we search for possible signatures ofquantum gravitational alternatives to black hole hori-zons in the gravitational wave data releases of black holemergers observed by the advanced Laser InterferometerGravitational-Wave Observatory (LIGO) [12]. As a sim-ple toy model, we replace the event horizon by a mirror

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

(with Dirichlet boundary conditions) at ∼ Planck properlength outside the horizon. This picture is motivated bythe realization that a thermal membrane on the stretchedhorizon, satisfying Israel junction conditions with Z2

symmetry, happens to have a thermal entropy equal tothe Bekenstein-Hawking area law [13]. Therefore, anyhorizonless microscopic model of the black hole which ac-counts for its entropy, should act as a mirror, at least forlinear long wavelength perturbations. The mirror is notperfect for particles with ω � TH (= Hawking temper-ature), as they can excite the microstates of the system,and thus be absorbed by the membrane [14], but shouldbe reflective at ω . TH as these microstates cannot beexcited. Incidentally, this is the frequency regime forgravitational waves in the ringdown phase of black holemergers. In contrast, electromagnetic emissions from ac-cretion into black holes are at much higher frequencies,where the membrane is expected to be highly absorbing,consistent with astrophysical observations [15, 16] (butalso see [11, 17]).

In spite of its simplicity, this picture is remarkably ro-bust: As first noticed in [18, 19], introduction of structurenear event horizon leads to late, repeating, echoes of theringdown phase of the black hole merger, due to wavestrapped between the near-horizon structure and the an-gular momentum barrier (Fig. 1). This is relatively in-sensitive to the nature of the structure, or indeed howone defines the Planck length, lp, as the time for reflec-tion from the stretched horizon is only logarithmicallydependent on its distance from the event horizon, i.e.∆techo = 8M log(M/lp) (+ spin corrections; see below).As a result, e.g., an order of magnitude change in this dis-

arX

iv:1

612.

0026

6v1

[gr

-qc]

1 D

ec 2

016

2

event

1st echo

2nd echotim

e

r*

mem

bran

e / fi

rew

all

tmerger

techo

Δtecho

angu

lar m

omen

tum

bar

rier

FIG. 1: Spacetime depiction of gravitational waveechoes from a membrane/firewall on the stretched

horizon, following a black hole merger event.

tance only affects the time of the echoes at 2− 3% level.While ∆techo is determined by linear physics, the timebetween the main merger event and the first echo couldbe further affected by non-linear physics during merger,i.e. techo − tmerger = ∆techo + O(M) (see Fig. 1), orequivalently:

techo − tmerger

∆techo= 1±O(1%), (1)

where ∆techo is predicted from the final (redshifted) massand spin measurements for each event.

Quite surprisingly, we find statistical evidence for thesedelayed echoes in LIGO events: GW150914, GW151226,and LVT151012 at a combined significance of 2.9σ. Weshall first describe our theoretical framework for theechoes, and then our statistical methodology and results.

Echo time-delays: At the linear order, per-turbed black holes are described by quasi-normal modes(QNM’s) which satisfy the boundary conditions of purelyoutgoing waves at infinity and purely ingoing waves at thehorizon. The transition (from ingoing to outgoing) takesplace continuously at the peak of the angular momentumpotential barrier of the black hole.

In our case, the ingoing modes of the ringdown re-flect back from the membrane (e.g., fuzzball or firewall)near horizon and passes back through the potential bar-rier. Part of the wave goes to infinity with a time delay.We call this the 1st echo (see Fig. 1). This time delaycorresponds to twice the tortoise coordinate distance be-tween the peak of the angular momentum barrier (rmax)and the membrane (which diverges logarithmically if themembrane approaches the horizon) . The remaining partof the 1st echo returns back towards the membrane andthe process repeats itself. Assuming Dirichlet boundaryconditions at the membrane (discussed above), the re-

flected waves must be phase inverted, i.e. even echoeshave opposite phase with respect to the odd ones.

For a Kerr black hole with dimensionless spin param-eter a, this implies:

∆techo = 2× r∗|rmax

r++∆r = 2×∫ rmax

r++∆r

r2 + a2M2

r2 − 2Mr + a2M2dr

= 2rmax − 2r+ − 2∆r + 2r2+ + a2M2

r+ − r−ln(

rmax − r+

∆r)

−2r2− + a2M2

r+ − r−ln(

rmax − r−r+ − r− + ∆r

),

(2)

where r± = M(1 ±√

1− a2), and ∆r is the coordinatedistance of the membrane and the (would-be) horizon.

The peak of the angular momentum barrier, rmax, isgiven by the roots of a sixth-order polynomial [20]:

2r4max(rmax − 3)2

+4r2max[(1− µ2)r2

max − 2rmax − 3(1− µ2)]a2

+(1− µ2)[(2− µ2)r2max + 2(2 + µ2)rmax

+(2− µ2)]a4 = 0, (3)

where µ = m/(l + 12 ) and rmax = rmax/M . For the

dominant QNM, rmax < 3M and (l,m) = (2, 2) resultingin µ = 0.8.

We further posit that the location of the membraneshould be near a Planck proper length from the horizon.This assumption is required to explain the observed den-sity of cosmological dark energy within the gravitationalaether proposal [10], but is also expected from genericquantum gravity scalings, such as the brick wall model[21], or trans-Planckian effects [22, 23]. This implies:∫ r++∆r

r+

√grrdr|θ=0 ∼ lp ' 1.62× 10−33 cm, (4)

which fixes the location of the membrane:

∆r|θ=0 =

√1− a2l2p

4M(1 +√

1− a2)(5)

With this set-up, we note that ∆techo '8M log(M/lp)

[1 +O(a2)

]is comparable to the scram-

bling time: the time over which the black hole state isexpected to thermalize [8, 9, 24, 25].

Using the measurements of the final black hole (red-shifted) mass and spin by the LIGO collaboration, wecan constrain ∆techo for each merger event. Assuminggaussian errors, we find (see the appendix):

∆techo,I(sec) =

0.2925± 0.00916 I = GW1509140.1013± 0.01152 I = GW1512260.1778± 0.02789 I = LVT151012

(6)

Data and the Echo template: In this analysis, weuse three datasets for each event. One is the theoretical

3

best-fit waveform for the BH merger event, provided bythe LIGO and Virgo collaborations [26–28] . The othertwo are the observed strain datastream from Hanford andLivingston detectors. We call these MI(t), hH,I(t) andhL,I(t), respectively. We used the strain data at 4096 Hzand for 32 sec duration. The waveform model consists ofthree phases: inspiral, merger, and ringdown.

Following the numerical results of [18, 19], we constructa phenomenological gravitational wave template for theechoes using five free parameters:

1. ∆techo is the time-interval in between successiveechoes, which we vary within the 1−σ range, fixedby the uncertainties in (redshifted) mass and angu-lar momentum of the final black hole (Eq. 6).

2. techo is the time of arrival of the first echo, whichcan be affected by non-linear dynamics near merger(Eq. 1).

3. t0 quantifies which part of the GR merger templateis truncated to produce the subsequent echo tem-plates1. To do this, we introduce a smooth cut-offfunction with a free parameter t0,

ΘI(t, t0) ≡ 1

2

{1 + tanh

[1

2ωI(t)(t− tmerger − t0)

]},

(7)where ωI(t) is frequency of model as a functionof time [30] and tmerger is the time at which theGR template peaks. As the intermediate region(merger) is before tmerger, we assume t0 is negative,and vary it within the range t0 ∈ (−0.1, 0)∆techo.Using this definition, we can define the truncatedtemplate:

MT,I(t, t0) ≡ ΘI(t, t0)MI(t). (8)

4. γ is damping factor of successive echoes, whichshould be between 0 and 1. In our analysis, wevary this free parameter within the range (0.1, 0.9)and look for the best fit.

5. A is the over-all amplitude of the echo template(with respect to the main event) which we fit for,assuming a flat prior.

The truncated model with echoes and all the free pa-rameters is then given by:

MTE,I(t) ≡

A

∞∑n=0

(−1)n+1γnMT,I(t+ tmerger − techo − n∆techo, t0)

(9)

1 Note that the wavelength of gravitational waves in the inspiralphase is much longer than the size of the black holes, which leadsto an echo signal suppressed at 4PN order [29].

0.0 0.2 0.4 0.6 0.8 1.0

6

4

2

0

2

4

6

1e 22

Tem

plate

withechoes

t − tmerger (s)

techo − tmerger

∆techo

FIG. 2: LIGO original template for GW150914, alongwith our best fit template for the echoes.

102 10310-24

10-23

10-22

10-21

10-20

Tem

plate

h(f)×

√f,

Strain

noiseASD

(strain/rtHz)

H1/L1 ASD and template around event GW150914

frequency (Hz)

LIGO templateEchoesLIGO template with echoes

Hanford ASDLivingston ASD

FIG. 3: Amplitude Spectral Densities (ASD’s) of ourbest fit echo template (Eq. 9) and the main event, for

GW150914. Since we have a quasi-periodic model, thereare resonances in the spectrum. The ASDs are the

square root of the power spectral densities, which areaverages of the square of the fast Fourier transforms(FFTs) of the data. The noise spectra from Hanford

and Livingston detectors are also shown.

where the term (−1)n+1 is due to the phase inversion ofthe truncated model in each reflection. Fig. (2) showsour best fit for this template for GW150914 within theparameter space described above, along with the mainmerger event. In the frequency domain we expect to seeresonances of these echoes (Fig. 3).

Results: Our strategy is to search for the best fit forthe echo template (9), by maximizing its signal-to-noiseratio, SNR, within the conjectured parameter space de-scribed above at fixed techo. We then identify the highestpeak within the range techo − tmerger = (1± 0.05) ∆techo

(Eq. 1). We quantify the significance of this peak byhow often a higher SNR peak is achieved within an in-terval of duration |techo,peak− tmerger−∆techo,peak| (graybox in Fig. 4), at a random point (away from the mainevent) in the data stream. It is worth noting that due

4

0.95 0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04 1.05

10

15

20

30

35

40

45

50BestfitSNR

2 SNRGW150914SNRT otal

techo−tmerger

∆techo

1σ − 68.3%

1.5σ− 86.6%

2σ− 95.4%

1σ− 68.3%

1.5σ− 86.6%

2σ− 95.4%

2.5σ − 98.8%

2.9σ − 99.6%3σ − 99.7%

FIG. 4: Best fit (or maximum) SNR2 near the expectedtime of merger echoes (Eq’s. 1 and 6), for the combined(top) and GW150914 (bottom) events. The significance

of the peaks is quantified by the p-value of theirSNRmax within the gray rectangle.

to different angles and locations of each detector, a com-plex model is analysed. Therefore in calculation of SNR’swe subtracted the phase of the main event from complextemplate and obtained two real templates correspondingto each detector (Hanford/Livingston). Then we set theoriginal gravitational wave peak at t = 0 by removingthe offset from SNRs (see [26–28]).

We do the analysis once for GW150914 (LIGO’s mostsignificant detection), and repeat it for the 3 recordedevents combined, by maximizing:

SNR2total ≡

∑I

SNR2I . (10)

In doing so, we assume the same γ and t0/∆techo for allthree events, while keeping ∆techo and A’s independent.The results are shown in Fig’s (4-5) and Tables I-II.

Fig. (4) shows that there is indeed a significantpeak with SNRmax = 4.21(6.96) for echoes following theGW150914 (combined) merger event(s), within 0.54% ofthe predicted echo time delay. To find the significance of

Search result Search background

Number

of e

ven

ts

100

10−3

10−1

10−2

SNR25525 4530 35 40 50

2σ 2.5σ 3σ 3.5σ

Echoes: 2

.9σ

Search result Search background

Number

of e

vents

100

10−1

10−2

10−3

SNR26 10 14 18 22

Echoes: 2

σ

2σ 2.5σ1.5σ 3σ

FIG. 5: Average number of noise peaks higher than aparticular SNR-value within a time-interval

0.54%×∆techo for combined (top) and GW150914(bottom) events. The red dots show the observed SNRpeak at techo = 1.0054∆techo (Fig. 4). The horizontal

bar shows the correspondence between SNR values andtheir significance.

finding this peak so close to the predicted value, we di-vide up the data steam (within the range 9-38 ×∆techo,I

after the merger) into intervals of 0.54%×∆techo,I , andcompute the average number of points in the intervalthat exceed SNR (Fig. 5). This yields an estimate of thesignificance of SNR peaks observed near the predictedecho times, at 2.0σ (2.9σ) for the GW150914 (combined)merger event(s).Conclusions and Discussion: In this letter, we

have searched advanced LIGO’s public data release ofthe first observed gravitational wave signals from blackhole merger events for signatures of Planck-scale struc-ture near their event horizons. By building a phenomeno-logical template for successive echoes from such exoticstructures expected in e.g., firewall or fuzzball paradigms,after marginalizing over its parameters, we report a firsttentative evidence for these echoes at 2.9σ significance

5

Range GW150914 Combined

(techo − tmerger)/∆techo (0.95,1.05) 1.0054 1.0054

γ (0.1,0.9) 0.89 0.9

t0/∆techo (-0.1,0) -0.084 -0.1

Amplitude 0.0992 0.124

SNRmax 4.21 6.96

p-value 4.6 × 10−2 3.7 × 10−3

significance 2.0σ 2.9σ

TABLE I: Best fit values for echo parameters of thehighest SNR peak near the predicted ∆techo, and their

significance.

GW150914 GW151226 LVT151012

∆techo,pred(sec) 0.2925 0.1013 0.1778

± 0.00916 ± 0.01152 ± 0.02789

∆techo,best(sec) 0.30068 0.09758 0.19043

SNRbest,I 4.13 3.83 4.52

TABLE II: Theoretical expectations for ∆techo’s of eachmerger event (Eq. 6), compared to their best combinedfit within the 1σ credible region, and the contribution of

each event to the joint SNR for the echoes (Eq. 10).

(or p-value of 1/270) in LIGO data proceeding its re-ported merger events. Future data releases from LIGOcollaboration at much higher sensitivity, will be able todefinitively confirm or rule out this finding.

One may wonder how including GW151226 andLVT151012 may improve the significance of the echo sig-nal, even though their ringdown phase was not detectablein LIGO data. We should note that while GW150914 andthe two other events have similar numerical contributionsto the significance, the nature of their contributions arequite different: GW150914 appears to pin down the echotemplate parameters, while the others help improve evi-dence for this template. Furthermore, the repeating na-ture of the echoes gives them a low frequency structure(Fig. 3) which may be detectable, even if the ringdownitself falls below the detector noise at high frequencies forGW151226 and LVT151012 (Fig. 1 in [12]).

We should note that the ad hoc nature of our echo tem-plate construction is not entirely satisfactory and couldlead to some ambiguity in interpreting the statistical sig-nificance of our finding. In particular, the fact that thecombined SNR is maximized on the edge of our param-eter range (see Table I) points to a need for a betterphysical prior on parameters, or simply a more physicalecho template. This does not change the statistical sig-nificance of our SNR peaks, but suggests higher peaksmay lie beyond this range. However, extending the anal-ysis beyond this range (in particular γ > 0.9) requiresanalyzing a much larger portion of LIGO data which wedefer to future work.

Future numerical simulations of merging black holeswith a membrane can sharpen the echo template, possi-bly increasing the detection significance. We thus predictthat a synergy of improvements in observational sensi-tivity and theoretical modelling can provide conclusiveevidence for quantum gravitational alternatives to blackhole horizons.

Acknowledgments— We thank Vitor Cardoso, WillEast, Sabine Hossenfelder, Matt Johnson, Luis Lehner,Rafael Sorkin, Nico Yunes, and Aaron Zimmerman forhelpful comments and discussions. We also thank all theparticipants in our weekly group meetings for their pa-tience during our discussions. JA would like to thankPerimeter Institute for Theoretical Physics (PI) for thegreat hospitality during the course of this work, as wellas Hessamaddin Arfaei for all his supports. He alsothanks Mansour Karami, Seyed Faroogh Moosavian andYasaman Yazdi for their kind hospitality during his visit.This work has been partially supported by Ministryof Science, Research and Technology of Iran, Institutefor Research in Fundamental Sciences (IPM), Univer-sity of Waterloo, and Perimeter Institute for TheoreticalPhysics (PI). Research at PI is supported by the Govern-ment of Canada through the Department of Innovation,Science and Economic Development Canada and by theProvince of Ontario through the Ministry of Research,Innovation and Science.

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The statistics of gaussian credible region

We approximate the uncertainty in the final redshiftedmassM and angular momentum a of the LIGO black holemerger events by a gaussian probability distribution:

P (∆a,∆M)

=

√detM

2πexp(−1

2α∆a2 − 1

2β∆M2 − γ∆a∆M) (11)

where we assume 〈M〉 and 〈a〉 are the best fit reportedvalues, while their inverse covariance matrix is given by,

Mij =

[α γ

γ β

]. (12)

Here

∆a = a− 〈a〉 , ∆M = M − 〈M〉. (13)

We can then obtain the probability distribution of

∆techo,

P (∆t)

=

∫δD(∆t(a,M)−∆techo)P (∆a,∆M)dadM

=

∫dM

|∂∆t∂a |

P (∆a,∆M) (14)

This leads to,

P (∆t) '√αβ − γ2√

2π(αµ2 + β + 2γµ) |∂∆t/∂a|a,M

× exp(−1

2

αβ − γ2

αµ2 + β + 2γµ

(∆t−∆t)2

(|∂∆t/∂a|a,M )2) (15)

where µ = − ∂∆t/∂M |a,M

∂∆t/∂a|a,M.

Using contour of 50% credible regions reported in [12],we can obtain the angles of the eigenvectors of the co-variance matrix. This gives a relation between α, β, andγ:

tan(2θI) =2γI

αI − βI=

0.013848 I = GW150914

0.0072280 I = GW151226

−0.0038272 I = LV T151012

(16)

For the mean of the distribution, using the detector frame(or redshifted) masses we obtain (see [12] Table I),(

MI/M�, aI , ∆tpred(aI , MI))

=

67.8 0.68 0.29559s I = GW150914

22.6 0.74 0.10246s I = GW151226

42 0.66 0.17962s I = LV T151012

,

(17)

while 〈∆a2〉, and 〈∆M2〉 are 68% credible region (see [12]Table I),(〈∆M2

I 〉/M2�, 〈∆a2

I〉)

=

(αI

αIβI − γ2I

,βI

αIβI − γ2I

)

=

4.6866 0.0012058 I = GW150914

6.7632 0.0016633 I = GW151226

36.091 0.0034056 I = LV T151012

(18)

These can be combined with θI ’s (16) to give:(αI , βI/M

2�, γI/M�

)=

1019.1 0.26221 7.0546 I = GW150914

634.92 0.15615 2.2940 I = GW151226

305.49 0.028826 −0.58452 I = LV T151012

(19)

With these values we can obtain the gaussian posteriorfor ∆techo’s,

7

P (∆tI) =

77.187s−1√π

exp(−5957.8s−2(∆t− 0.2925s)2) I = GW15091461.372s−1√π

exp(−3766.5s−2(∆t− 0.1013s)2) I = GW15122625.357s−1√π

exp(−643.00s−2(∆t− 0.1778s)2) I = LV T151012

(20)