Echoes from the Abyss: The Holiday Edition!

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by  Jahed Abedi

Echoes from the Abyss: The Holiday Edition!
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, Iran
2School of Particles and Accelerators, Institute for Research in
Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran
3Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, ON, N2L 2Y5, Canada
4Department of Physics, McGill University, 3600 rue University, Montreal, QC, H3A 2T8, Canada
5Department of Physics and Astronomy, University of Waterloo, Waterloo, ON, N2L 3G1, Canada
In a recent paper [1], we reported the results of the first search for echoes from Planck-scale
modifications of general relativity near black hole event horizons using the public data release by
the Advanced LIGO gravitational wave observatory. While we found tentative evidence (at (cid:39) 3σ
level) for the presence of these echoes, our statistical methodology was challenged by Ashton, et
al. [2], just in time for the holidays! In this short note, we briefly address these criticisms, arguing
thattheyeitherdonotaffectourconclusionorchangeitssignificanceby(cid:46)0.3σ. Therealtestwill
7
bewhetherourfindingcanbereproducedbyindependentgroupsusingindependentmethodologies
1
(and ultimately more data).
0
2

n Recently we reported tentative evidence of Planck- Fig. (1), which is similar to Fig. 4 in [1], but over a
a
scale structure at black hole event horizons [1] using larger range. Indeed, without any further analysis, our
J
the public data release for the three Advanced LIGO main conclusions are manifest in this figure:
2
black hole merger events GW150914, LVT151012 and For both GW150914 (the most significant reported
1
GW151226. Accounting for the “look elsewhere” effect LIGO event) and combined data from all three events,

] due to uncertainty in the echo template, we find tenta- there are ubiquitous peaks within 0.54% of x=1, which
c
tive evidence for Planck-scale structure near black hole is shown by the vertical grey bar (The width of the
q
- horizonsat2.9σ significancelevel(correspondingtofalse grey bar is the distance of the peaks from x = 1).
r
g detection probability of 1 in 270). For GW150914, the significance is 2σ (or a p-value of
[ The key property of the signal that we searched for is 5%),meaningthatcomparableSNRpeaks(fromrandom

a series of damping echoes within the time intervals of: noise) can be found within ∆x 0.0054/0.05 =0.11, as
1 (cid:39)
can be seen with other peaks at x 0.91 and 1.16. For
5v 0.2925±0.00916 I =GW150914 thecombinedevents,thesignificanc(cid:39)eis2.9σ(orap-value
8 ∆techo,I(sec)= 0.1013 0.01152 I =GW151226 of1/270),i.e. comparablepeakscanonlybefoundwithin
4 0.1778±0.02789 I =LVT151012 ∆x=0.0054 270=1.46,whichisalsodemonstratedin
3 ± ×
Fig. (1)asnohigherpeakcanbeseenwithinaninterval
0 (1)
of ∆x=1.2.
.
1 This prediction follows from combining the linear per- Indeed, the fact that the highest SNR peak in Fig.
0
turbation theory with the Planck-scale hypothesis, using (1) is within x 1=0.054 of the theoretical prediction,
17 the reported constraints on the final redshifted masses while no higher−peaks exists within a range of ∆x (cid:38) 1
: andspinsoftheremnantblackholes[3]. Assuch,there- is a clear indication that this is unlikely to be mere co-
v
portederrorsaredominatedbytheLIGOdetector noise. incidence. Even for GW150914 alone, which has a lower
i
X There is further theoretical uncertainty on the time- significance, after applying the time-delay due to finite
r delay from merger until the first echo, given that the speed of light, both Hanford and Livingston detectors
a
metric perturbations are non-linear close to the merger see simultaneous SNR peaks near x = 1 (Fig. 2), which
event: sounds very unlikely due to random chance.
t t With this introduction, let us now address the specific
echo merger
x − =1 (1%). (2) criticisms raised by Ashton et al. [2]:
≡ ∆t ±O
echo
Ourprimarymethodwasthentomaximizethesignal- 1. Ashton et al. point out that we find a slightly
to-noise ratio (SNR) for the echo template and see higherSNR forechoesinLVT151012,compared
best
whether there is a significant peak within the predicted to GW150914, even though the SNR for the main
range given by Eq. (2). The significance is then quanti- event is lower by a factor of 2.4. Is this surprising?
fied by how often a higher peak can be found elsewhere,
In fact, this is expected as constraints on final
within a similar interval. This is best demonstrated in
massandspinofLVT151012aresignificantlyworse
than GW150914. As a result, the relative error
on ∆t is 5 times higher for LVT151012, com-
echo
∗ jahed [email protected] pared to GW150914. This leads to larger values
† [email protected] of SNR across the board, as we are searching
best
2
SNRT otal SNRGW150914
50
3σ 99.7%
2.9σ− 99.6%
−
2.5σ 98.8%
45 −
2σ 95.4%
−
40 1.5σ 86.6%
−
2 1σ 68.3%
R −
N35
S
t
fi
t
s
Be30
25
20
2σ 95.4%
−
1.5σ 86.6%
15 −
1σ 68.3%
−
10
0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6
techo−tmerger
∆techo
FIG. 1: Same as Fig. 4 in [1], but over an extended range of x= techo−tmerger. The SNR peaks at the predicted value
∆techo
of x=1 have 2.0σ and 2.9σ significance, for GW150914 and combined events respectively.
a larger region of parameter space. This, however,
12
does not necessarily lead to increased significance,
Hanford
as the same would be true for all values of x.
10 Livingston
If there was no real echo signal in LVT151012 and
8 GW151226,addingthemtoGW150914wouldonly
2
R
N dilute the significance of the peak near x=1. The
S 6 fact that the opposite happens suggests that, in
spite of larger variations in SNR due to higher un-
4
certaintyin∆t ,thereisstillsignificantenhance-
echo
ment in SNR near x=1 .
2
Weshouldalsocautionaboutcomparingthesignif-
0 icanceoftheechoeswiththatofthemergerevents,
0.95 0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04 1.05
techo−tmerger astheyhaveverydifferentfrequencystructures(see
∆techo
Fig. 3) leading to different SNR ratios, especially
FIG. 2: SNR2 near the expected time of merger echoes given the non-trivial frequency dependence of the
(Eq. 1) for GW150914 in Hanford (red) and Livingston LIGO detector noise.
(green) detectors. Interestingly, their SNR ratio
Finally,weshouldwarnaboutover-interpretingour
2.74/3.37=0.81 is comparable to the SNR ratio for the
quoted significances. Even though we gain compa-
main event 13.3/18.6=0.72. Note that, unlike Fig. (1),
rable evidence for echoes by including LVT151012
here we have fixed the echo parameters to their best fit andGW151226,i.e. 22+22 2.92,itdoesn’tmean
values for combined detectors. (cid:39)
that they have the same significance: A 2σ peak
could be a 1-σ fluctuation of a 1-σ or a 3-σ under-
lying signal.
3
GW150914 GW151226 LVT151012
50
|A | 0.091 0.33 0.34
best,I
SNR 4.13 3.83 4.52
best,I 40
TABLE I: The best fit SNR’s and amplitudes of
individual events, for our joint echo template fit to the 30
three events (see [1] for details).
20
LIGO template Echoes Hanford ASD Livingston ASD 10
H1/L1 ASD and template around event GW150914
10-21
0
0.96 0.98 1.00 1.02 1.04
10-22 techo−tmerger
∆techo
10-23 FIG. 4: Resulting prior distribution on x= techo−tmerger,
∆techo
assuming a random phase for the echo template.
/rtHz)10-24 102 frequency (Hz) 103
n
(strai10-21 H1/L1 ASD and template around event GW151226 rameters. However, that has not been our goal, as
D
S
A the parameters only quantify a toy model for the
oise10-22 echoes. The goal was rather to find whether the
n
n
Strai10-23 best-fit toy model, within the parameter range, is
√f, consistent with random noise. As we discussed in
× the introduction, we find that has a probability of
h(f)10-24 102 103 < 1%.
Template10-21 H1/L1 ASD and temfprelaquteen acryo (uHnz)d event LVT151012 Arasmweteerasrgoufeaitnoy[1m],ordaetlhteor tthheainr epxutsrheimnges,thine opuar-
opinion, it will be much more fruitful to find more
10-22
physical echo templates.
10-23 3. Perhaps the most serious objection of Ashton, et
al. concernsourestimationofsignificance,orfalse-
10-24 102 103 detection probability (p-value). As we outlined in
frequency (Hz)
the introduction, it is already clear from Fig. (1)
FIG. 3: Best fit templates for LIGO main events and that the p-value for our SNR peak near x = 1
echoes (using the joint best fit described in [1]), in should be (cid:46) 0.1 and (cid:46) 0.01, for GW150914 and
Fourier space (similar to Fig. 3 in [1]). The amplitude combined events, respectively.
spectral distribution (ASD) for each detector is shown The main criticism of Ashton, et al. stems from us
for comparison. quantifying our p-values by considering how often
random intervals of size ∆x=0.0054 have an SNR
bigger than the peaks we observe at x = 1.0054,
For completeness, the individual amplitudes of the
while we should have actually allowed for different
best joint fit are listed in Table (I). We note that,
choices of ∆x. This would depend on the prior for
even though best fit SNR’s are comparable for the
∆x: the larger the prior, the the higher would be
three events, the errors on the amplitude: ∆A
the p-value.
= A /SNR is much smaller for GW150914,
best best
However, we already have a decent idea about this
given that A is the smallest. Therefore, as ex-
best
prior from Eq. (2) which suggests ∆x = (0.01),
pected, GW150914 which is the most significant of O
not far from what we used. We can get a more
the 3 LIGO events, would also dominate the com-
concrete handle on this prior by assuming that the
bined constraint on the echo amplitude.
echo template acquires a random phase (with re-
2. Ashton, et al. worry that railing up of the best- specttothemainevent)duetononlinearpropaga-
fitSNRvaluesneartheboundaryoftheparameter tioneffects. Figure(4)showstheresultingprioron
range, particularly the damping factor γ =0.9, ∆x,whichwefindbyreplacingthedatainourSNR
best
mightposeaproblemforouranalysis(anissuethat computation(forGW150914)bytheechotemplate
we discussed at length in [1]) . This indeed would with a random phase, and finding the position of
be the case if the goal was to measure these pa- the peak. This results in a near top-hat prior with
4
0.040 as the evidence for echoes remains marginal and
nearlyallLIGOdata(awayfromthemergerevent)
0.035
is dominated by noise. Secondly, p-value quan-
0.030 tifies the probability of null hypothesis, i.e. how
often you see the echoes, assuming that there are
0.025
none. As such, to find p-value one should assume
e
u
al0.020 thatLIGOdata,awayfromthemainevent,ispure
v
p- noise and use that to quantify detection probabil-
0.015
ity,whichwaswhatwedid. Therefore,wefindthis
0.010 criticism ill-founded.
0.005 Ashton et al. further advocate using larger
stretches of LIGO data (which is publicly avail-
0.000
0.000 0.005 0.010 0.015 0.020 able) to define p-value more precisely. While this
σ
echo is in principle correct, LIGO noise is known to sig-
nificantly vary and be very non-gaussian over long
FIG. 5: An alternative false detection probability
time-scales (see Fig’s 14-15 in [4]), which makes
(p-value) as a function of uncertainty in t defined in
echo
the interpretation of p-value ambiguous. The 9-38
Eq. (3).
∆t interval used is quite adequate to quan-
echo,I
×
tify the p-value for our signal, as otherwise we
would see a sharp cut-off in our SNR cumulative
0.01 < ∆x < 0.01 (an interval of 0.02 rather
− distribution (Fig. 5 in [2]).
than 0.0054), which slightly increases the p-value
to 0.011 (or significance of 2.54σ).
Toconclude,whiletheauthorsof[2]haveraisedimpor-
Yet another way to quantify the significance would tant questions about our tentative evidence for Planck-
be to define a “loudness” function which averages scale structure near black hole horizons [1], we believe
themaximumlikelihoodfortheechoeswithagaus- they do not affect our conclusions significantly. We have
sian prior x=1 σecho, i.e. : provided a careful assessment of these issues, along with
±
various quantitative and qualitative arguments for why
(cid:104) (cid:105)
L(x,σecho) (cid:90) exp(cid:20)SNR2total(x(cid:48))(cid:21) exp(cid:112)−(2xσ−e2xch(cid:48))o2 dx(cid:48). t>he2.f6aσlsseigdneitfieccatinocne)p.rTobhaebseiliwtyillrebmeaailnsosfluesrsthtehradnis1c%uss(eodr
≡ 2 × 2πσ2
echo and expanded in an upcoming revision of our original
(3)
arXivsubmission. Ofcourse,therealtestwillbewhether
We again use the LIGO data stream within the
this evidence can be reproduced in independent analyses
range 9-38 ∆t after the merger event, to
echo,I
× by other groups, and/or using other merger events.
quantify how often L(x,σ ) exceeds L(1,σ ),
echo echo Acknowledgments—WethankVitorCardosoandLuis
foragivenσ . ThisplottedinFig. (5),andpro-
echo
Lehner for helpful comments and discussions. We also
vides an alternative p-value (or probability of false
thank the authors of [2] for their very constructive feed-
detection). Thisisalsominimizedatσ 0.5%,
echo
(cid:39) back, as well as providing us with fruitful holiday activi-
with p-value of 0.01 (or significance of 2.6σ).
ties!
4. Ashton, et al. claim that our reported significance
of2.9σbasedonp-valueof3.7 10 3correspondsto
−
×
2.7σ with a one sided significance convention. We
disagree: wealreadyconsideronesidedsignificance
[1] J. Abedi, H. Dykaar, and N. Afshordi, (2016),
inourp-valueestimation,sincewejustconsiderthe
arXiv:1612.00266v1 [gr-qc].
absolute value of the SNRs.
[2] G.Ashton,O.Birnholtz,M.Cabero,C.Capano,T.Dent,
B.Krishnan,G.D.Meadors,A.B.Nielsen,A.Nitz, and
5. Ashton et al. are concerned that the range 9-38
J. Westerweck, (2016), arXiv:1612.05625 [gr-qc].
∆techo,I after the merger event, which we use to [3] B. P. Abbott et al. (Virgo, LIGO Scientific), (2016),
×
quantify false detection probability, might be con- arXiv:1606.04856 [gr-qc].
taminated by the echoes and somehow affect our [4] D.V.Martynovetal.(LIGOScientific),Phys.Rev.D93,
significance estimation. Firstly, this is unlikely, 112004 (2016), arXiv:1604.00439 [astro-ph.IM].