Tests of general relativity through the direct detection of gravitational waves

Chris Van Den Broeck

Virtual Conference of the Polish Society on Relativity 2020 24-25 September 2020 Coalescing binary neutron stars and black holes: inspiral-merger-ringdown

• 13 published binary black hole detections so far: GW150914, GW151012, GW151226, GW170104, GW170608, GW170729, GW170809, GW170814, GW170818, GW170823, GW190412, GW190814, GW190521 • Binary neutron star GW170817; possible binary neutron star GW190425 Access to strongly curved, dynamical spacetime Curvature of spacetime spacetime of Curvature

Characteristic timescale Yunes et al., PRD 94, 084002 (2016) Fundamental physics with gravitational waves (from coalescing binaries)

1. The strong-field dynamics of spacetime

2. The propagation of gravitational waves

3. The nature of compact objects 1. The strong-field dynamics of spacetime

Ø Inspiral-merger-ringdown process • Post-Newtonian description of inspiral phase

• Merger-ringdown governed by additional parameters βn, ⍺n

Ø Look for possible deviations in these parameters:

LIGO + Virgo, PRL 118, 221101 (2017)

Ø Rich physics: Dynamical self-interaction of spacetime, spin-orbit and spin-spin interactions, … 1. The strong-field dynamics of spacetime

Ø Inspiral-merger-ringdown process • Post-Newtonian description of inspiral phase

• Merger-ringdown governed by additional parameters βn, ⍺n

Ø Combine information from multiple sources:

LIGO + Virgo, PRD 100, 104036 (2019) 1. The strong-field dynamics of spacetime Ø Consistency between inspiral and merger-ringdown? • Masses and spins during inspiral can be used to predict mass and spin of the final object • Compare prediction from inspiral with what follows from merger-ringdown

Ø Here too, combine information from multiple sources

LIGO + Virgo, PRD 100, 104036 (2019) 1. The strongR.- ABBOTTfieldet al.dynamics of spacetime PHYS. REV. D 102, 043015 (2020) energy to be detectable in the data; no further modeling Ø Full structure of a gravitational wave signal: input is needed, although we do not require phase coher- h+ ih = h`m(t) 2Y`m(✓, ) ence along each track. ⇥ The resulting Y α for GW190412 is shown in the h+ ih = h`m(t) `2Y2`m`(✓m, `) ð Þ ⇥ X X  h+ ih = bottom panelh`m of(t) Fig.28Y.` Itm has(✓, a global) peak at α 1, ` 2 ` m ` h+ ih ⇥= correspondingh` tom the(t) dominant2Y` (2,2)m(✓ multipole,, ) and¼ a   h ih = ⇥h (t`) 2 Y` m(✓`, ) X X + `m 2prominent`m  local peak at α 1.5, corresponding to the ⇥ `X2 X` m ` ¼ • ` 2 ` m ` X mX3 multipoles. We also calculate Y α from different Dominant harmonic: ` = m =2 f22(t)=2X forb X  segments¼ surrounding, but not includingð Þ GW190412, to • Frequency as function of time of higher-order` = mmodes:=2 f`m(t)=(capturem the/2) detectorf22(t noise) characteristics;↵ f22(t) in this case we call| the| quantity N α . The⇥ ensemble average μ α h+ ih = h`m(t) 2Y`m(✓, ) ð Þ ð Þ¼ ⇥ • hSearch+ ih for= sub`=-dominantm =2`h`=m(tfm )modes`m2=2(Yt)=(`m( ✓with, mf)` m/frequency2)(tf)=(22(t) m ↵ f / 22 2) ( t ) f 22, mwhere(2t/m)N α1 ↵ and f 0is22 standard. 3left(t)m free deviation2/mm21/mσ α01.of1 N α0.are3 alsom2/m1 0.1 ⇥ | | h+ ih =plottedh ð Þi for⇠ referenceh and`m to( highlightt) ⇠2ðYÞ the`m relative(✓ð, Þ) strength ` 2 ` m ` ` 2 ` m ` h+ |ih | = ⇥ h`m(t) 2Y`m(✓, )⇠ ⇠ X X  X X  ⇥ of` the2 GW` signalm present` in the on-source segment. Ø Sub-dominant modes more prominent when strongly unequal` 2component` XInsteadm ` ofX estimatingmasses the significance of the m 3 X multipolesX  from comparing Y α to its background¼ at α 1.5, we perform a more powerfulð Þ statistical analysis ` = m =2 f`m(`t)=(= m =2m R./2) ABBOTT•ff`m22((tGW190412:t)=(et) al. ↵mf22/(2)t)f 22 m ( t 2 ) /m ↵ 1 f 22 ( 0t .) 3 m; GW190814:m2/m2/m1 1 0.3PHYS.0. 1 m REV.2/m D1102, 0430150.1 (2020) | | ⇠ ⇠ in¼ which we test the hypotheses that the data contain either | | ⇠ ` = ⇠m =2 f`m(t)=(m /2) f22(t) ↵f22(t) m2/m1 0.3 m2/m1 0.1 only noise (H0), or noise and a dominant-multipole ` = energym =2 to be detectablef`m in(t the)=( data; nom further/2) modelingf| 22|(t)maximum↵f22 likelihood(t) signalm2/m (H ),1 or noise0 and.3⇠ a maximumm2/m1 0.⇠1 input is needed, although we do not require| | phase coher- 1 ⇠ ⇠ ence along each track. likelihood signal that includes m 2 and m 3 multipoles Ø(HIn). both By maximizing cases the likelihood¼ of observing¼ Y α The resulting Y α for GW190412 is shown in the 2 ð Þ bottom panel of Fig.ð Þ 8. It has a global peak at α 1, given each hypothesis over a free amplitude parameter for ¼ strong evidence corresponding to the dominant (2,2) multipole, and a each multipole, we obtain likelihood ratios for H1 and H2, prominent local peak at α 1.5, corresponding to the andfor their modes difference at is in turn incorporated into a detection m 3 multipoles. We also calculate¼ Y α from different statistic β [see Eq. (7) in [153] ]. segments¼ surrounding, but not includingð Þ GW190412, to ↵From=1 the on-source.5 m data=3 segment taken from the LIGO capture the detector noise characteristics; in this case we Livingston detector, we found the detection statistic β call the quantity N α . The ensemble average μ↵α =1.i.e.5 m =3 −4 ¼ ð Þ ð Þ¼ 6.1 with a p-value of 6.4 × 10 for EOBNR HM model; N α and standard deviation σ α of N α are also −4 plottedh ð Þi for reference and to highlightð Þ the relativeð Þ strength and β 6.2 with a p-value < 6.4 × 10 for Phenom HM model,¼ which strongly supports the presence of m 3 of the GW signal present in the on-source segment. ¼ Instead of estimating the significance of the m 3 modes in the signal. The full distribution of β from off- multipoles from comparing Y α to its background¼ at source data segments from the LIGO Livingston detector α 1.5, we perform a more powerfulð Þ statistical analysis surrounding the trigger time of GW190412 is shown in the in¼ which we test the hypotheses that the data contain either inset of Fig. 8. only noise (H0), or noise and a dominant-multipoleLIGO + Virgo, PRD 102, 043015 (2020) maximum likelihood signal (H1), or noise and a maximumLIGO + Virgo, ApJ Lett.D. 896 Signal, L44 reconstructions (2020) FIG.likelihood 8. Top panel: signal Time-frequencythat includes m spectrogram2 and m 3 ofmultipoles data con- (H ). By maximizing the likelihood¼ of observing¼ Y α As an additional test of consistency, and an instructive taining GW190412,2 observed in the LIGO Livingston detector.ð Þ The horizontalgiven each axis hypothesis is time (in over seconds) a free relative amplitude to the parameter trigger time for visual representation of the observed GW signal, we (1239082262.17).each multipole, The we amplitude obtain likelihood scale of ratios the detector for H1 outputand H2 is, compare the results of two signal reconstruction methods. normalizedand their by differencethe PSD of is the in noise.turn incorporated To illustrate into the method,a detection the One is derived from the parameter-estimation analysis predictedstatistic trackβ for[see the Eq.m (7)3 inmultipoles[153] ]. is highlighted as a dashed presented in Sec. III, the second uses the model-agnostic ¼ line, aboveFrom the the track on-source from the datam segment2 multipoles taken that from are the visible LIGO in wavelet-based burst analysis BayesWave [154] which was the spectrogram.Livingston detector, Bottom panel: we found¼ The the variation detection of Y statisticα , i.e.,β the also used to generate PSDs. A detailed discussion of such −4 ð Þ ¼ energy6.1 inwith the pixelsa p-value of the of top6.4 × panel,10 alongfor EOBNR the track HM defined model; by signal comparisons for previous BBH observations can be and β 6.2 with a p-value < 6.4 × 10−4 for Phenom HM fα t αf22 t , where f22 t is computed from the Phenom HM found in [155]. ð Þ¼model,¼ð whichÞ stronglyð Þ supports the presence of m 3 analysis. Two consecutive peaks at α 1.0 and α 1.5 (thin For GW190412, both signal reconstruction methods modes in the signal. The full distribution¼ of β¼from¼ off- dashed line) indicate the energy of the m 2 and m 3 multi- agree reasonably well as illustrated in Fig. 9. To quantify poles,source respectively. data segments Inset: The from distribution the LIGO of¼ the Livingston detection¼ detector statistic β in noise,surrounding used to the quantify trigger timep-values of GW190412 for the hypothesis is shown that in the the the agreement for each signal model from the Phenom and data containsinset of Fig.m 82. and m 3 multipoles (red dashed line). EOB families, we compute the noise-weighted inner ¼ ¼ product [128,151] between 200 parameter-estimation sam- Δf 1=5 Hz alongD. the Signal frequency reconstructions axis. By doing so, we ples and the BayesWave median waveform. The BayesWave FIG. 8. Top panel: Time-frequency spectrogram of data con-can¼ decouple the energy in individual multipoles of the waveform is constructed by computing the median values at taining GW190412, observed in the LIGO Livingston detector. As an additional test of consistency, and an instructive signal.visual Once representationf22 t is defined, of the this observed is a computationally GW signal, we every time step across samples. Similar comparison strat- The horizontal axis is time (in seconds) relative to the trigger timeefficient way to analyzeð Þ which multipoles have sufficient egies have been used in [3,5,34,156]. (1239082262.17). The amplitude scale of the detector output is compare the results of two signal reconstruction methods. normalized by the PSD of the noise. To illustrate the method, the One is derived from the parameter-estimation analysis predicted track for the m 3 multipoles is highlighted as a dashed presented in Sec. III, the second uses the model-agnostic ¼ line, above the track from the m 2 multipoles that are visible in wavelet-based burst analysis BayesWave [154] which was the spectrogram. Bottom panel:¼ The variation of Y α , i.e., the also used to generate PSDs. A detailed discussion of such043015-10 ð Þ energy in the pixels of the top panel, along the track defined by signal comparisons for previous BBH observations can be fα t αf22 t , where f22 t is computed from the Phenom HM found in [155]. analysis.ð Þ¼ Twoð Þ consecutiveð peaksÞ at α 1.0 and α 1.5 (thin ¼ ¼ For GW190412, both signal reconstruction methods dashed line) indicate the energy of the m 2 and m 3 multi- agree reasonably well as illustrated in Fig. 9. To quantify poles, respectively. Inset: The distribution of¼ the detection¼ statistic β in noise, used to quantify p-values for the hypothesis that the the agreement for each signal model from the Phenom and data contains m 2 and m 3 multipoles (red dashed line). EOB families, we compute the noise-weighted inner ¼ ¼ product [128,151] between 200 parameter-estimation sam- Δf 1=5 Hz along the frequency axis. By doing so, we ples and the BayesWave median waveform. The BayesWave can¼ decouple the energy in individual multipoles of the waveform is constructed by computing the median values at signal. Once f22 t is defined, this is a computationally every time step across samples. Similar comparison strat- efficient way to analyzeð Þ which multipoles have sufficient egies have been used in [3,5,34,156].

043015-10

1 1

1 1 1

1

1 1

1 2. The propagation of gravitational waves

Ø Dispersion of gravitational waves? E.g. as a result of non-zero graviton mass: • Dispersion relation:

2 2 2 2 4 E = p c + mgc g = h/(mgc) h+ ih = h`m(t) 2Y`m(✓, ) • Group⇥ velocity:g = h/(mgc) ` 2 ` m ` X X 2 4 2 vg/c =1 mgc /2E 1 2 h h (t) 1 i'0 i(2⇡fts(f) (ts(f))) 2 42 2 ix + A(ts(f)) e e vg/c =1 mgc /E dx e ⇥ ` = m =22 f`m(t)=(m /2) f22(t) ↵f¨22(t(st()f))m2/m1 0.3 m2/m1 0.1 • Modification to gravitational2✓ ◆ waveZ1 phase: | | = ⇡Dc/[g(1 + z) f] ⇠ ⇠ 2 1 = ⇡Dc/[g(11 + z) f] g = h/(mgc) int 1 1 L = QLQL = QL L 2`!` 2`! E Ø Bound on`=1 graviton> 1.6 10mass:13 km`=1 Xg ⇥ X 23 2 dxµm=(g dx40.,dx7 110,dx2,dxeV3/c)  ⇥ 23 2 m 7.7 10 eV/c g  ⇥ 2/3 4 G 5/3 ⇡f (s)(tret) h (tret)= Mc s c s a(t )r c2 c emis ✓ ◆ ✓ ◆

x =0 LIGO + Virgo, PRD 100, 104036 (2019)

dnˆ F A(nˆ) F A(nˆ) 4⇡ 1 2 A Z X t

1 1

1 1

1 I I M˜ = 1 2 cos(2! t)+const 11 2 rot I I M˜ = 1 2 cos(◆)sin(2! t)+const 12 2 rot I I M˜ = 1 2 cos2(◆)cos(2! t)+const 22 2 rot

m2 µ x1(t)= R eˆ(t)= R eˆ(t) m1 + m2 m1 m1 µ x2(t)= 2. TheR eˆ (propagationt)= R eˆ(t) of gravitational waves m1 + m2 m2 E2 = p2c2 + Ap↵c↵ Ø MoreE2 = generalp2c2 + formsAp↵c ↵of dispersion: E2 = p2c2 + Ap↵c↵ E2 = p2c2 + Ap↵c↵ E2 = p2c2 + Ap↵c↵ ↵ =0 corresponds to violation of local Lorentz invariance 6 § ↵ =2.5 multi-fractal spacetime 1 2 1 i'0 i(2⇡ft§s(f↵) =3(ts(f)))doubly 2special relativity ix A(ts(f)) e e ↵=4 dx e 2 § higher¨(t-sdimensional(f)) theories ✓ ◆ Z1

1 1 int 1 1 L = QLQL = QL L 2`!` 2`! E X`=1 X`=1 dxµ =(dx0,dx1,dx2,dx3)

2/3 4 G 5/3 ⇡f (s)(tret) h (tret)= Mc s c s a(t )r c2 c emis ✓ ◆ ✓ ◆ x =0 LIGO + Virgo, PRD 100, 104036 (2019) dnˆ F A(nˆ) F A(nˆ) 4⇡ 1 2 A Z X t

1

1 1

1 1 2. The propagation of gravitational waves

Ø Does the speed of gravity equal the speed of light? Ø The binary neutron star coalescence GW170817 came with gamma ray burst, 1.74 seconds afterwards

Ø With a conservative lower bound on the distance to the source: -15 -16 -3 x 10 < Δv/vEM < +7 x 10

Ø Excluded certain alternative theories of gravity designed to explain dark matter or dark energy in a dynamical way

LIGO + Virgo + Fermi-GBM + INTEGRAL, ApJ. 848, L13 (2017) .

Polarization from 3-detector2. The propagation observation of gravitational of GW170814 waves

Ø Metric theories of gravity allow up to 6 polarizations Ø Distinct antenna patterns:six polarizations distinct antenna patterns !

Isi & Weinstein, PRD 96, 042001 (2017) In GR: GW are transverse, traceless Ø In the case of GW170817, sky positiononly tensor waspolarizations known from EM counterpart § Pure tensor / pure vector = 1021 / 1 § Pure tensor / pure scalar = 1023 / 1 pure tensor / pure scalar =LIGO 1000 + Virgo, / PRL 1 123, 011102 (2019) pure tensor / pure vector = 200 / 1

Isi & Weinstein (2017) Need multiple detectors: thanks to Virgo! Abbott et al.,PRL119,141101(2017)

12 of 20 3. What is the nature of compact objects?

Ø Black holes, or still more exotic objects?

• Boson stars

• Dark matter stars

• Gravastars

• Wormholes

• Firewalls, fuzzballs

• The unknown 3. What is the nature of compact objects?

Anomalous effects during inspiral

Ringdown of newly formed object

Gravitational wave echoes Anomalous tidal effects during inspiral

• Tidal field of one body causes quadrupole deformationQ in =the other:(EOS; m ) ij Eij Qij = (EOS; m) ij Q = E(EOS; m) where (EOS; ij m ) depends0 > on0 ij < 0 ⌘ E (EOS; m) 0 > 0 Q 0 < 0 E ⌘ ������ • Boson stars, (EOS;dark matterm) stars:0 > 0 < 0 △ ������� □ ⌘ □ (EOS; m)• Gravastars0 > 0: < 0 × ������� □ 1000 □ ⌘ □ □ ��������� □ 500 □ □ □ • Enters inspiral phase at 5PN order, through □ □ □ □ △ △ 5 5 |[%] △ (m)/m (R/m)

Λ 100 △ / / △ △

Λ △ 50 △

σ △

| 2 5 △ △ △ △ • O(10 - 10 ) for neutron stars × (EOS; m) 0 > 0 < 0 × × ⌘ × • Can also be measurable for black hole 10 × × × × 5 × × × × × mimickers, e.g. boson stars 10 20 30 40 50

M [M⊙] Cardoso et al., PRD 95, 084014 (2017) Johnson-McDaniel et al., arXiv:1804.0826 FIG. 1. Relative percentage errors on the average tidal deformability ⇤ for BS-BS binaries observed by AdLIGO (left panel), ET (middle panel), and LISA (right panel), as a function of the BS mass and for di↵erent BS models considered in this work (for each model, we considered the most compact configuration in the stable branch; see main text for details). For terrestrial interferometers we assume a prototype binary at d = 100Mpc, while for LISA the source is located at d =500Mpc.The horizontal dashed line identifies the upper bound ⇤/⇤ = 1. Roughly speaking, a measurement of the TLNs for systems which lie below the threshold line would be incompatible with zero and, therefore, the corresponding BSs can be distinguished from E BHs. Here ⇤ is given by Eq. (72), the two inspiralling objects have the same mass, and ⇤/⇤ kE /k2 . ⇠ 2

TABLE I. Tidal Love numbers (TLNs) of some exotic compact objects (ECOs) and BHs in Einstein-Maxwell theory and modified theories of gravity; details are given in the main text. As a comparison, we provide the order of magnitude of the TLNs for static NSs with compactness C 0.2(theprecisenumberdependsontheneutron-starequationofstate;seeTableIIIformoreprecisefits).ForBSs, the table provides the⇡ lowest value of the corresponding TLNs among di↵erent models (cf. Sec. III A) and values of the compactness. In the polar case, the lowest TLNs correspond to solitonic BSs with compactness C 0.18 or C 0.20 (when the radius is defined as that containing 99% or 90% of the total mass, respectively). In the axial case, the lowest⇡ TLNs correspond⇡ to a massive BS with C 0.16 or C 0.2(againforthetwodefinitionsoftheradius,respectively)andinthelimitoflargequarticcoupling.ForotherECOs,weprovide⇡ ⇡ expressions for very compact configurations where the surface r0 sits at r0 2M and is parametrized by ⇠ := r0/(2M) 1; the full results are available online [65]. In the Chern-Simons case, the axial l =3TLNisa⇠↵ected by some ambiguity and is denoted by a question mark [see Sec. IV C for more details]. Note that the TLNs for Einstein-Maxwell and Chern-Simons gravity were obtained under the assumption of vanishing electromagnetic and scalar tides. Tidal Love numbers E E B B k2 k3 k2 k3 NSs 210 1300 11 70 Boson star 41.4402.8 13.6 211.8 Wormhole 4 8 16 16 ECOs 5(8+3 log ⇠) 105(7+2 log ⇠) 5(31+12 log ⇠) 7(209+60 log ⇠) 8 8 32 32 Perfect mirror 5(7+3 log ⇠) 35(10+3 log ⇠) 5(25+12 log ⇠) 7(197+60 log ⇠) 16 16 32 32 Gravastar 5(23 6log2+9log⇠) 35(31 6log2+9log⇠) 5(43 12 log 2+18 log ⇠) 7(307 60 log 2+90 log ⇠) 1 Einstein-Maxwell 00 0 0 Scalar-tensor 00 0 0 BHs 2 2 ↵CS ↵CS Chern-Simons 001.1 M 4 11.1 M 4 ? 1

TLNs as [6, 8]

1 l(l 1) 4⇡ M kE l , l ⌘2 M 2l+1 2l +1 r El0 (3) B 3 l(l 1) 4⇡ Sl kl 2l+1 , ⌘2 (l + 1)M 2l +1 l0 selection rules that allow to define a wider class of “rotational” r B TLNs [22, 23, 75, 76]. In this paper, we neglect spin e↵ects to 1 leading order. where M is the mass of the object, whereas l0 (respec- 1 E 1

1 On the empirical verification of the black hole no-hair conjecture from gravitational-wave observations

1,2 2 3 4 2 Gregorio Carullo ,⇤ Laura van der Schaaf , Lionel London , Peter T. H. Pang , Ka Wa Tsang , Otto A. Hannuksela4, Jeroen Meidam2, Michalis Agathos5, Anuradha Samajdar2, Archisman Ghosh2, Tjonnie G. F. Li4, Walter Del Pozzo1,6, and Chris Van Den Broeck2,7 1 Dipartimento di Fisica “Enrico Fermi”, Universit`adi Pisa, Pisa I-56127, Italy 2 Nikhef – National Institute for Subatomic Physics, Science Park, 1098 XG Amsterdam, The Netherlands 3 School of Physics and Astronomy, Cardi↵ University, The Parade, Cardi↵ CF24 3AA, UK 4 Department of Physics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong 5 DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom 6 INFN sezione di Pisa, Pisa I-56127, Italy and 7 Van Swinderen Institute for Particle Physics and Gravity, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands (Dated: May 12, 2018) We show that second-generation gravitational-wave detectors at their design sensitivity will allow us to directly probe the ringdown phase of binary black hole coalescences. This opens the possibility to test the so-called black hole no-hair conjecture in a statistically rigorous way. Using state-of-the- art numerical relativity-tuned waveform models and dedicated methods to e↵ectively isolate the quasi-stationary perturbative regime where a ringdown description is valid, we demonstrate the capability of measuring the physical parameters of the remnant black hole, and subsequently deter- mining parameterized deviations from the ringdown of Kerr black holes. By combining information from (5) binary black hole mergers with realistic signal-to-noiseRingdown ratios achievableof with newly the current formed black hole generationO of detectors, the validity of the no-hair conjecture can be verified with an accuracy of 1.5% at 90% confidence. ⇠ • Ringdown regime: Kerr metric + linear perturbations Introduction – The detection of gravitational waves form• Ringdown signal is a superposition of quasi-normal modes (GWs) by the LIGO and Virgo Collaborations [1, 2] has t/⌧ h(t)= e nlm cos(! t + ) . (1) opened up a variety of avenues for the observational ex- Anlm nlm nlm!nlm ⌧nlm ploration of the dynamics of gravity and of the nature nlmX !nlm ⌧nlm of black holes. GW150914 [3] and subsequent detections • Characteristic frequencies ! nlm and⌧nlm damping! nlmtimes ⌧nlm For black holes in GR, all frequencies !nlm and damp- [4–9] have enabled unique tests of general relativity (GR) ing times ⌧nlm are completely!nlm determined= !nlm( byM thef ,a blackf ) ⌧nlm = ⌧nlm(Mf ,af ) !nlm = !nlm(Mf ,af ) ⌧nlm = ⌧nlm(Mf ,af ) [5, 6, 8, 10]. Among the several detections, GW150914 hole’s mass and spin.1 This can be viewed as a mani- still holds a special place, not only because it was the• Nofestation-hair ofconjecture: the black hole stationary, no-hair conjecture, electrically which es- neutral black hole first and the loudest binary black hole event detected, but completelysentially states characterized that in GR, a stationary by mass axisymmetric M f M, spinfaf af also because it was the kind of textbook signal that al- black hole is determined uniquely by its mass, intrinsic • ! ⌧ lowed measurements of the frequency and damping time angularLinearized momentum, Einstein andnlm electric equationsnlm charge around (with the Kerr latter background enforce specific of what has been interpreted as the least damped quasi-!nlmexpecteddependences:⌧nlm to be zero for astrophysical objects) [19–26]; see normal mode (QNM) of the presumed remnant black hole [27] for!nlm a review.= !nlm( ThisMf ,a connectionf ) ⌧nlm is= key⌧nlm to(M severalf ,af ) tests (BH) resulting from a binary black hole merger [10]. This that have been proposed in the literature [28–36]. So far sparked considerable interest in the community,!nlm = !nlm since(M itf ,af )the possibility⌧nlm = ⌧nlm to(M verifyf ,af ) (or refute) experimentally the opened up the prospect of more in-depth empirical stud- no-hair conjecture has beenMf exploredaf mostly in the con- ies of quasi-stationary Kerr black holes [11, 12] in the near • MtextLooka of third-generation for deviations ground-based from the expressions [37, 38] or space- for frequencies, damping times: future, as the sensitivity of the Advanced LIGO and Ad- fbasedf [39] gravitational-wave detectors. In this Letter,we ! (M ,a ) (1 + !ˆ ) ! (M ,a ) vanced Virgo detectors is progressively improved [1, 13]. show thatlmn thef existingf ! advancedlmn interferometriclmn f f detector Consistency with the prediction of GR hinted that the ⌧ (M ,a ) (1 + ⌧ˆ ) ⌧ (M ,a ) network,lmn whenf operatingf ! at designlmn sensitivity,lmn f willf be ca- end result of GW150914 was indeed a Kerr black hole pable of testing the no-hair conjecture with an accuracy [14], but inability to detect more than one QNM did not of a few percent with the observation of the ringdown Carulllo et al., PRD 98, 104020 (2018) yet allow tests of some key GR predictions for these ob- signal already for (5) GW events. Brito et al., PRD 98, 084038 (2018) jects. As first predicted by Vishveshwara [15] and fur- O ther investigated by Press [16], and Chandrasekhar and Detweiler [17], in the regime where linearized general rel- ativity is valid, the strain of the emitted gravitational- 1 BH perturbation theory alone cannot predict the amplitudes wave signal, at large distances from the BH and neglect- nlm and relative phases nlm;inthecaseofblackholesre- sultingA from a binary merger, these are set by the properties of ing subdominant power-law tail contributions, takes the the parent binary black hole system; see e.g. [18].

1

1 1 1 1 1

1  =16M =16RingdownM of newly formed black hole

Ø Assume!ˆ lmnAdvanced LIGO/Virgo at design sensitivity !ˆlmn Ø 6 sources similar to GW150914 § !ˆ220 measurable⌧ˆ220 to O(2%) !ˆ220 § ⌧ˆ220 measurable to O(10%) h(t)=F+(↵, , )h+(t)+F (↵, , )h (t) 0.15 ⇥ ⇥ h(t)=F+(↵, , )h+(t)+F (↵, , )h (t) 0.10⇥ ⇥ Mf ,af , ⌘, ↵, , ◆, ,r,tc, 'c { 0.05 } Mf ,af , ⌘, ↵, , ◆, ,r,tc, 'c

{ 220 }

(↵, )(ˆ 0.00◆, )(t , ' ) ! c c 0.05 (↵, )(◆, )(N tc, 'c) 1 2 () 0.10 D ()+detCI () B ⌘ N I N I=1 1 X0.q15 2 () DI ()+det1 C2 I (3 ) 4 5 6 probability density B ⌘ N N I=1 events X q

Simulation Carulllo et al., PRD 98, 104020 (2018)

1

1 3

0.15 servable imprint in the GW signal at late times. From black hole 3 Eq. (6), a typical time delay reads 0.10 0.15 servable imprint in the GW signal at late times. From black hole Eq. (6), a typical time delay reads`/LP 0.05 outgoing at infinity t 54(n/4) M30 1 0.01 log ms , (7) ingoing at horizon0.10 ⇠ M  ✓ 30 ◆ 0.00 `/LP 0.15 0.05 outgoing at infinity t 54(n/4) M30 1 0.01 log ms , (7) ingoing at horizon wormhole where M30 :=⇠M/(30M ). M30  ✓ ◆ 2 0.00 0.10 The picture of GW signal scattered o↵ the potential 0.15 wormhole ) M where M30 := M/(30M ). * barrier is also supported by two further features shown 2 0.05 outgoing at infinity0.10 The picture of GW signal scattered o↵ the potential V(r trapped outgoing at infinity in Fig. 2, namely the modulation and the distortion of ) M

* barrier is also supported by two further features shown 0.05 outgoing at infinity the echo signal. In general, modulation is due to the 0.00 V(r outgoing at infinity in Fig. 2, namely the modulation and the distortion of 0.15 trapped slow leaking of the echo modes, which contain less en- 0.00 centrifugal barrier star-like ECO the echo signal. In general, modulation is due to the 0.15 ergy thanslow the leaking initial of the one. echo In modes, the wormhole which contain case, less this en- 0.10 centrifugal barrier star-like ECO e↵ect isergy stronger than due the toinitial the one.fact that In the modes wormhole can also case, leak this 0.10 0.05 outgoing at infinity to the “othere↵ect is universe” stronger due through to the fact tunneling that modes at the can second also leak regular at the center 0.05 trapped outgoing at infinitypeak ofto the the potential. “other universe” While the through amplitude tunneling of the at the echoes second regular at the center 0.00 trapped is model-dependent,peak of the potential. for a given While model the amplitude it depends of the only echoes -50 -400.00 -30 -20 -10 0 10 20 30 40 50 is model-dependent, for a given model it depends only -50 -40 -30r /M -20 -10 0 10 20 30 40mildly 50 on `. Distortion is also due to the potential bar- * r /M mildly on `. Distortion is also due to the potential bar- * rier, which acts as a low-pass filter and reflects only the rier, which acts as a low-pass filter and reflects only the low-frequency, quasibound echo modes. This implies that FIG. 1. QualitativeFIG. 1. features Qualitative of the features e↵ective of the potential e↵ective potential felt by felt by low-frequency, quasibound echo modes. This implies that perturbationsperturbations of a Schwarzschild of a Schwarzschild BH compared BH compared to the case to the caseeach echoeach is echo a low-frequency is a low-frequency filtered filtered version version of the of the previ- previ- of wormholesof [12] wormholes and of star-like [12] and of ECOs star-like with ECOs a regular with a cen- regular cen-ous oneous and one the and original the original shape shape of the of mode the mode gets gets quickly quickly ter [22]. The preciseter [22]. location The precise of the location center of of the the center star of is the model- star is model-washedwashed out after out a after few a echoes few echoes1. 1. dependent and4dependent was chosen and for was visual chosen clarity. for visual The clarity. maximum The maximum and minimumand of the minimum potential of the corresponds potential corresponds approximately approximately to to 6 0.4 the location of the unstable and stable PS, and the correspon- B. Waves generated by infalling or scattered 4 wormhole the locationwormhole of the unstable and stable PS, and the correspon- B. Waves generated by infalling or scattered 0.2 dence is exact in the eikonal limit of large angular number l. 2 dence is exact in the eikonal limit of large angular number l. particles 0 0.0 In the wormhole case, modes can be trapped between the particles -2 -0.2 In the wormhole case, modes can be trapped between the -4 -0.4 PSs in the two “universes”. In the star-like case, modes are PSs in the two “universes”. In the star-like case, modes are The features above are observed in a simple scattering 4 empty shell 0.5 empty shell trapped between the PS and the centrifugal barrier near the 2 The features above are observed in a simple scattering (t) (t) Gravitationaltrapped between thewave PS and echoes the centrifugal barrier near the process, but are also evident in the GW signal produced 10 10 0 0.0 center of the star [28–30]. In3 all cases the potential is of fi- Ψ Ψ process, but are also evident in the GW signal produced -2 -0.5 center of thenite star height, [28–30]. and In the all modes cases leak the away, potential with is higher-frequency of fi- by head-on collisions or close encounters, in the test- 0.15 servable imprint in the GW signal at late times. From -4 black hole nite height,Eq. (6), andmodes a typical the time leaking modes delay reads on leak shorter away, timescales. with higher-frequency by head-onparticle collisions limit. The or latterclose diencounters,↵er from the in radial the test- plunge gravastar 0.10 gravastar 4 0.5 • Exotic objects with corrections near 2 `/LP particlestudied limit. in The Ref. latter [12] in di that↵er their from pericenter the radialr plunge> 3M, 0.05 modesoutgoing at infinity leaking on shorter timescales. min 0.0 t 54(n/4) M30 1 0.01 log ms , (7) 0 ingoing at horizon ⇠ horizon: M inner30 potential barrier for 0.00  ✓ ◆ i.e. the particle does not cross the radius of the PS -2 0.15-0.5 studied in Ref. [12] in that their pericenter rmin > 3M, wormhole where M30 := M/(30M ). -4 2 The picture of GWradial signal scattered motion o↵ the potential (in fact, scattered particles in the Schwarzschild geom- -100 0 100 200 300 400 0.10 100 200 300 400 i.e. the particle does not cross the radius of the PS ) M t/M * t/M barrierwhere is also supported• rmin byis two the further location features shown of the minimum of the potential 0.05 outgoing at infinity etry can never get inside the r =4M surface). In V(r outgoing at infinity in Fig. 2, namely theAftermodulation formation/ringdown:and the distortion of continuing trapped (in fact, scattered particles in the Schwarzschild geom- the echo signal. In general, modulation is due to the 0.00 shown in Fig. 1. If we consider a microscopic correction order to compute the GW signal, we use the Regge- FIG. 2. Left: A dipolar (l =1,m = 0) scalar wavepacket scattered0.15 o↵ a Schwarzschild BH and o↵wheredi↵erent ECOsrminslow withis leaking the of thelocation echobursts modes, of whichof the radiation contain minimum less en- called of theechoes potential 6 centrifugal barrier star-like ECO etry can never get inside the r =4M surface). In ` =10 M (r0 =2.000001M). The right panel shows the late-time behavior of the waveform. The result for a wormhole,ergy than aat the the initial horizon one. In the scale wormhole (` case, thisM), then the main contribution Wheeler-Zerilli decomposition reviewed in Appendix B 0.10 e↵ect is stronger• due to the fact that modes can also leak gravastar, and a simple empty shell of matter are qualitatively similar and display a series of “echoes”shown which are modulated in Fig.to 1. the If timeIf we microscopic consider delay comes a⌧ horizon microscopic near the modification radius correction of the star order and to compute the GW signal, we use the Regge- in amplitude and distorted in frequency. For this compactness, the0.05 delay time in Eq. (6) reads t 110outgoingM at infinityfor wormholes,to the “other universe” through tunneling at the second (cf. Ref. [31] for details). regular at the center ⇡ t 82M for gravastars, and t 55M for empty shells, respectively. trapped at the horizonpeak of the scale potential. ( While` the amplitudeM),then then of the echoestime the mainbetween contribution ⇡ ⇡ 0.00 is model-dependent,therefore, for a given model it depends only Wheeler-Zerilli decomposition reviewed in Appendix B -50 -40 -30 -20 -10 0 10 20 30 40 50 ⌧ We have studied the GW emitted during collisions or r /M mildly on `. Distortion is also due to the potential bar- * to the time delay comessuccessive near echoes the radius of the star and rier, which acts as a low-pass filter and reflects only the (cf. Ref.scatters [31] for between details). point particles and ECOs; again the FIG. 1. Qualitative features of the e↵ective potentialtherefore, felt by low-frequency, quasibound echo modes. This implies that ` perturbations of a Schwarzschild BH compared to the case each echo is a low-frequency filteredt versionnM of thelog previ- , ` M, (6)We havegeneral studied qualitative the GW features emitted are duringthe same collisions as those or dis- Infall of wormholes [12] and of star-likeScattering ECOs with a regular cen- ous one and the original shape of the mode gets quickly 1.0 1 ⇠ M ⌧ cussed in Section II A and independent of the nature ter [22].0.2 The precise location of the center of the star is model- washed out after a few echoes . ✓ ◆ scatters between point particles and ECOs; again the dependent and was chosen for visual clarity. The maximum ` and minimum of the potential corresponds approximately to where n set by nature of object: generalof qualitative the ECO. To features be specific, are the we show same in as Fig. those 3 the dis- Zer- the location of the unstable and stable PS, and the correspon- B.where Wavest generatednnMis by alog infalling factor or of scattered order, ` unityM, that takes into(6) account dence is exact in the eikonal limit of large angular number l. illi wavefunction for a point particle plunging into (left 0.5 0.0 ⇠ particles• n = 8 Mfor wormholes⌧ cussed in Section II A and independent of the nature 0.7 In the wormhole case, modes can be trapped between the (t)] the structure of✓ the◆ objects. For wormholes, n =8to 6 (t) PSs22 in the two “universes”.0.1 In the star-like case, modes are panel) or scattering o↵ a wormhole with ` = 10 M,with 22 Ψ The features above are• observed in a simple scattering of the ECO. To be specific, we show in Fig. 3 the Zer- Ψ trapped between the PS and the centrifugal barrier near the account forn the = 6 fact for thatthin-shell the signal gravastars is reflected by the

centerRe[ of the star [28–30]. In all cases the potential is of fi- process, but are also evident in the GW signal produced initial Lorentz boost E =1.5. The coordinate system we -0.2 where n is a factor of order unity that takes into account nite height, and the modes leak away, with higher-frequency by head-on collisions or close encounters, in the test- 0.0 0.6 two maxima• n in = Fig.4 for 1, empty whereas shell for our thin-shell gravas-illi wavefunction for a point particle plunging into (left modes leaking on shorter0.0 timescales. the structureparticle limit. of The the latter objects. di↵er from the radial For plunge wormholes, n =8to use is such that the particles are moving along6 the equa- studiedtar in Ref. model [12] in that and their pericenter the empty-shellrmin > 3M, model it is easy to checkpanel) or scattering o↵ a wormhole with ` = 10 M,with i.e. the particle• doesFor not crossGW150914 the radius of the (M PS = 65 Msun), tor, and it di↵ers - by a ⇡/2 rotation - from the coordinate -0.4 account(in for fact, the scattered fact particles that in the Schwarzschild the signal geom- is reflected by the 0.5 -0.1 that n = 6 and n = 4, respectively. The results showninitial in Lorentz boost E =1.5. The coordinate system we -0.5 0 200 400 600 where rmin is the location of the200 minimum 400 of the 600 potential 800 etry can never get insidetaking the r ℓ=4=Mℓ6surface).Planck, Inand n = 4: axis used in Ref. [12]. As such, the l = 2 Zerilli-Moncrief shown in Fig. 1. If we consider a microscopictwo correction maxima in Fig. 1, whereas for our thin-shell gravas- -200 0 200 400 600 800 -200 0 200 400 600 800order toFig. compute 2 the for GW` signal,= 10 we useM the Regge-are perfectly consistent with this at the horizon scale (` M), then the main contribution use is suchwavefunction, that the particles for example, are moving has contributions along the from equa- az- t/M ⌧ t/M Wheeler-Zerilli decomposition�t = 117 reviewed ms in Appendix B to the time delay comes near the radius oftar the star model and (cf. and Ref.picture, [31] the for details). empty-shell with the wormhole model it case is easy displaying to check longer echo imuthal numbers m =0, 2. Note also that it is easy to therefore, We have studied the GW emitted during collisions or tor, and it di↵ers - by a ⇡/2 rotation - from the coordinate FIG. 3. Left panel: The waveform for the radial infall of a particle with specific energy E =1.5 into a wormhole withdelays than the other cases with the same compactness. ± 6 that n =scatters 6 and betweenn = point 4, particles respectively. and ECOs; again the The results shown in ` =10 M, compared to the BH case. The BH ringdown, causedCardoso by oscillations et al., ofPRL the 116 outer,` 171101 PS as the (2016) particle crosses through, axis used in Ref. [12]. As such, the l = 2 Zerilli-Moncrief t nM log , ` M, (6) general qualitative6 features are the same as those dis- are also present in the wormhole waveform. A part of this pulse travels inwards⇠ and is absorbedM by the⌧ eventFig. horizon 2 for (forcussed BHs)` = inOur Section 10 results II AM andare independent show perfectly that of the the nature dependenceconsistentTsang et al., PRD 98 with on, 024023` is this (2018) indeed log- Cardoso et al., PRD 94✓, 084031◆ (2016) wavefunction, for example, has contributions from az- or then bounces o↵ the inner (centrifugal or PS) barrier for ECOs, giving rise to echoes of the initial pulse. This is a low-passof the ECO.arithmically To be specific, we for show all in Fig. the 3 the ECOs Zer- we studied. cavity which cleans the pulse of high-frequency components. Atwhere late times,n is a factoronly a of lower order frequency, unity that takeslong-livedpicture, into account signal is present, withilli wavefunction the for wormhole a point particle plunging case into displaying (left longer echo the structure of the objects. For wormholes, n =8to 6 imuthal numbers m =0, 2. Note also that it is easy to well described by the QNMs of the ECO. Right panel: the same for a scattering trajectory, with pericenter rmin =4.3panel)M,o↵ or scattering o↵ a wormhole with ` = 10 M,with 1 6 account for the fact that the signal is reflected by the As argued in Ref. [12], the logarithmic dependence dis- a wormhole with ` =10 M. The main pulse is generated now through the bremsstrahlung radiationdelays emitted as the than particleinitial Lorentz the boost otherE =1. cases5. The coordinate with system the we same compactness. Incidentally, we note± that all these features (namely time delay, approaches the pericenter. The remaining main features are astwo before. maxima We in Fig. show 1, whereas only the for real our part thin-shell of the gravas- waveform,use isthe such that the particles are moving along the equa- tar model and the empty-shell model it is easy to check played in Eq. (6) implies that even Planckian corrections echoes, modulation, and high-frequency filtering) are precisely imaginary part displays the same qualitative behavior. Our resultstor, and show it di↵ers - that by a ⇡/2 rotation the dependence - from the coordinate on ` is indeed log- that n = 6 and n = 4, respectively. The results shown in 33 what one would expect by the scattering of sound waves in a 6 axis used(` in Ref. [12].LP As such,=2 the l = 210 Zerilli-Moncrief cm) appear relatively soon after Fig. 2 for ` = 10 M are perfectly consistentarithmically with this wavefunction, for⇡ for all example, the has ECOs contributions⇥ we from studied. az- finite-size cavity. picture, with the wormhole case displaying longer echo imuthalthe numbers mainm =0, burst2. Note also of that radiation, it is easy to so they might leave an ob- express these results in a rotated frame [32, 33], and we delayswith than our theprevious other cases study with [12] the2. same compactness. ± 1 checked that the waveforms agree up to numerical errors Our results show that the dependence on ` is indeedAs log- argued in Ref. [12], the logarithmic dependence dis- Incidentally, we note that all these features (namely time delay, arithmically for all the ECOs we studied. As argued in Ref. [12], the logarithmic dependenceplayed dis- in1 Eq.Incidentally, (6) we note implies that all these features that (namely even time delay, Planckian corrections echoes, modulation, and high-frequency filtering) are precisely played in Eq. (6) implies that even Planckian corrections echoes, modulation, and high-frequency33 filtering) are precisely 33 what one would expect by the scattering of sound waves in a (`2 LP =2 10 cm) appear relatively(` soon afterLP what=2 one would expect10 by thecm) scattering ofappear sound waves in relatively a soon after ⇡Note however⇥ the following typo in the original publication: the the main burst of radiation, so they mightthe leave⇡ an main ob- finite-size burst cavity.⇥ of radiation, so they might leave an ob- finite-size cavity. Gravitational wave echoes Ø Detection claims and counter-claims:

§ Abedi et al. 2016: tentative evidence (2.5�) for Planck scale structure near horizon from GW150914, GW151012, GW151226 Abedi et al., PRD 96, 082004 (2017) § Abedi et al. 2018: claimed significance 4.2� for GW170817 Abedi et al., arXiv:1803.10454 (2018)

§ Westerweck et al. 2018 • More extensive background calculation for the BBHs, reduced significance • Re-evaluation of prior boundaries and density distributions of waveform templates Westerweck et al., PRD 97, 124037 (2018)

§ Other non-detection statements Lo et al., PRD 99, 084052 Nielsen et al., PRD 99, 104012 (2019) Uchikata et al., PRD 100, 062006 (2019)

§ Most of the above relied on template waveforms § Theoretical predictions still in early stages, and there may be exotic objects that have yet to be envisaged! Conklin et al., PRD 98, 044021 (2018) § Need for morphology-independent search methods Tsang et al., PRD 98, 024023 (2018) Tsang et al., PRD 101, 064012 (2019) Summary

Ø The first direct detection of gravitational waves has enabled unprecedented tests of general relativity: § First access to genuinely strong-field dynamics of vacuum spacetime § Propagation of gravitational waves over large distances § Polarizations of gravitational waves

Ø How sure can we be that the massive compact objects we see are the standard black holes from GR? § Look for anomalous tidal effects during inspiral § Probing ringdown will enable indirect tests of no-hair conjecture § Searching for echoes

Ø Ultra-high precision tests with next-generation observatories: LISA, Einstein Telescope, Cosmic Explorer § Higher accuracy § Larger number of sources § Propagation of gravitational waves over cosmological distances