Testing the Kerr Paradigm with Gravitational-Wave Observations

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Testing the Kerr paradigm with gravitational-wave observations

Emanuele Berti, Mississippi/IST Lisbon/Caltech Black Hole Initiative Conference, May 8 2017 Direct detection of gravitational waves Selected for a Viewpoint in Physics week ending PRL 116, 061102 (2016) PHYSICAL REVIEW LETTERS 12 FEBRUARY 2016

Observation of Gravitational Waves from a Binary Black Hole Merger B. P. Abbott et al.* (LIGO Scientific Collaboration and Virgo Collaboration) (Received 21 January 2016; published 11 February 2016) On September 14, 2015 at 09:50:45 UTC the two detectors of the Laser Interferometer Gravitational-Wave Observatory simultaneously observed a transient gravitational-wave signal. The signal sweeps upwards in frequency from 35 to 250 Hz with a peak gravitational-wave strain of 1.0 × 10−21. It matches the waveform predicted by general relativity for theVIEWPOINT inspiral and merger of a pair of black holes and the ringdown of the resulting single black hole. The signal was observed with a matched-filter signal-to-noise ratio of 24 and a false alarm rate estimated to be lessThe than 1 event First per 203 Sounds 000 years, equivalent of toMerging a significance greater Black 160 0.03 than 5.1σ. The source lies at a luminosityHoles distance of 410−þ180 Mpc corresponding to a redshift z 0.09−þ0.04 . 5 4 ¼ In the source frame, the initial black holeGravitational masses are waves36−þ emitted4 M and by the29−þ merger4 M , of and two the black final holes black have hole been mass detected, is setting the 4 0.5 2 course for a new era of observational⊙ astrophysics.⊙ 62−þ4 M , with 3.0−þ0.5 M c radiated in gravitational waves. All uncertainties define 90% credible intervals. These observations⊙ demonstrate⊙ the existence of binary stellar-mass black hole systems. This is the first direct by Emanuele Berti , detection of gravitational waves and the first observation⇤ † of a binary black hole merger. or decades, scientists have hoped they could “lis- ten in” on violent astrophysical events by detecting DOI: 10.1103/PhysRevLett.116.061102 their emission of gravitational waves. The waves, which can be described as oscillating distortions in Fthe geometry of spacetime, were first predicted to exist by Einstein in 1916, but they have never been observed di- rectly. Now, in an extraordinary paper, scientists report that I. INTRODUCTION they have detected the waves at the Laser Interferometer Gravitational-wave ObservatoryThe (LIGO) discovery [1]. From an ofanaly- the binary pulsar system PSR B1913 16 sis of the signal, researchersby from Hulse LIGO in andthe US, Taylor and their [20] and subsequent observationsþ of In 1916, the year after the final formulationcollaborators of the field from the Virgo interferometer in Italy, infer that the gravitational waves wereits produced energy by theloss inspiral by and Taylor and Weisberg [21] demonstrated equations of general relativity, Albert Einsteinmerger predicted of two black holes (Fig. 1), each with a mass that is more than 25 times greaterthe than that existence of our Sun. Their of find- gravitational waves. This discovery, the existence of gravitational waves. Heing found provides thethat first observational evidence that black hole the linearized weak-field equations had wavebinary solutions: systems can form andalong merge in with the Universe. emerging astrophysical understanding [22], Gravitational waves are producedled to by themoving recognition masses, and Figure that 1: Numerical direct simulations observations of the gravitational waves of emitted the transverse waves of spatial strain that travel at thelike electromagnetic speed of waves, they travel at the speed of light. by the inspiral and merger of two black holes. The colored As they travel, the waves squashamplitude and stretch spacetime and phase in the ofcontours grav arounditational each black hole waves represent would the amplitude enable of the gravitational radiation; the blue lines represent the orbits of the light, generated by time variations of the massplane quadrupole perpendicular to their direction of propagation (see studies of additional relativisticblack holes and the systems green arrows represent and provide their spins. (C. new moment of the source [1,2].Einsteinunderstoodthatinset, Video 1). Detecting them, however, is exceptionally Henze/NASA Ames Research Center) hard because they inducetests very small of distortions: general even the relativity, especially in the dynamic gravitational-wave amplitudes would bestrongest remarkably gravitational waves from astrophysical events are only expected to produce relative length variations of order 21 strong-field regime. small; moreover, until the Chapel Hill conference10 . in phase, yielding no signal. A gravitational wave propagat- Experiments to detecting perpendicular gravitational to the detector waves plane disrupts began this with perfect 1957 there was significant debate about the“Advanced” physical LIGO, as the recently upgraded version of destructive interference. During its first half-cycle, the wave the experiment is called, consistsWeber of two and detectors, his resonant one in will masslengthen onedetectors arm and shorten in the the other; 1960s during[23] its sec-, reality of gravitational waves [3]. Hanford, Washington, and one in Livingston, Louisiana. ond half-cycle, these changes are reversed (see Video 1). Each detector is a Michelsonfollowed interferometer, by consisting an international of These length networkvariations alter of the cryogenicphase difference betweenreso- Also in 1916, Schwarzschild published a solutiontwo 4-km-long for the optical cavities, or “arms,” that are arranged the laser beams, allowing optical power—a signal—to reach field equations [4] that was later understood toin andescribe L shape. The a interferometernant detectors is designed so that,[24] in. Interferometricthe photodetector. With two detectors such interferometers, were LIGO first can the absence of gravitationalsuggested waves, laser beams in traveling the early in rule 1960s out spurious[25] signalsand (from, the say, 1970s a local seismic[26] wave).A black hole [5,6], and in 1963 Kerr generalizedthe the two solution arms arrive at a photodetector exactly 180 out of that appear in one detector but not in the other. study of the noise and performanceLIGO’s sensitivity is of exceptional: such detectors it can detect length[27] dif-, to rotating black holes [7]. Starting in the 1970s⇤Department theoretical of Physics and Astronomy, The University of Missis- ferences between the arms that are smaller than the size sippi, University, Mississippi 38677,and USA further conceptsof an to atomic improve nucleus. The them biggest challenge[28],ledto for LIGO is work led to the understanding of black hole quasinormalCENTRA, Departamento de Física, Instituto Superior Técnico, † proposals for long-baselinedetector noise, broadband primarily from laser seismic interferome- waves, thermal mo- modes [8 10], and in the 1990s higher-orderUniversidade post- de Lisboa, Avenida Rovisco Pais 1, 1049 Lisboa, Por- tion, and photon shot noise. These disturbances can easily – tugal mask the small signal expected from gravitational waves. Newtonian calculations [11] preceded extensive analytical ters with the potential for significantly increased sensitivity [29 32]. By the early 2000s, a set of initial detectors – c 2016 American Physical Society 11 February 2016 Physics 9, 17 studies of relativistic two-body dynamics [12,13]physics.aps.org. These advances, together with numerical relativity breakthroughs was completed, including TAMA 300 in Japan, GEO 600 in Germany, the Laser Interferometer Gravitational-Wave in the past decade [14–16], have enabled modeling of binary black hole mergers and accurate predictions of Observatory (LIGO) in the United States, and Virgo in their gravitational waveforms. While numerous black hole Italy. Combinations of these detectors made joint obser- candidates have now been identified through electromag- vations from 2002 through 2011, setting upper limits on a variety of gravitational-wave sources while evolving into netic observations [17–19], black hole mergers have not previously been observed. a global network. In 2015, Advanced LIGO became the first of a significantly more sensitive network of advanced detectors to begin observations [33–36]. *Full author list given at the end of the article. A century after the fundamental predictions of Einstein and Schwarzschild, we report the first direct detection of Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distri- gravitational waves and the first direct observation of a bution of this work must maintain attribution to the author(s) and binary black hole system merging to form a single black the published article’s title, journal citation, and DOI. hole. Our observations provide unique access to the

0031-9007=16=116(6)=061102(16) 061102-1 Published by the American Physical Society 8 Population synthesis rates: detection not unexpected… TABLE 3 Detection rates for second-generation detectors in the low-end metallicity scenario AdV [ρ ≥ 8] KAGRA [ρ ≥ 8] aLIGO [ρ ≥ 8] 3-det network [ρ ≥ 10(12)] fcut =20Hz fcut =10Hz fcut =20Hz fcut =20Hz Model Insp PhC (EOB) Insp PhC (EOB) Insp PhC(EOB) PhC (spin) Insp PhC yr−1 yr−1 yr−1 yr−1 yr−1 yr−1 yr−1 yr−1 yr−1 NS-NS Standard 0.3 0.3 0.7 0.6 1.1 1.0 - 2.3 (1.3) 2.2 (1.3) Optimistic CE 0.8 0.7 1.8 1.7 2.9 2.7 - 6.0 (3.5) 5.6 (3.3) Delayed SN 0.4 0.4 1.0 0.9 1.5 1.4 - 3.2 (1.8) 2.9 (1.7) High BH Kicks 0.3 0.3 0.7 0.6 1.0 1.0 - 2.1 (1.3) 2.0 (1.2) BH-NS Standard 0.3 0.2 0.7 0.5 1.1 0.8 1.2 2.3 (1.3) 1.8 (1.0) Optimistic CE 1.4 1.2 3.6 2.8 5.5 4.4 5.7 12 (6.7) 9.4 (5.4) Delayed SN 0.2 0.1 0.5 0.4 0.8 0.6 0.9 1.7 (0.9) 1.3 (0.7) High BH Kicks 0.04 0.03 0.09 0.07 0.1 0.1 0.3 0.6 (0.2) 0.5 (0.2) BH-BH Standard 56 66 (61) 106 153 (140) 183 246 (235) 610 369 (226) 514 (292) Optimistic CE 287 324 (297) 629 828 (745) 1124 1421 (1339) 3560 2384 (1336) 3087 (1633) Delayed SN 53 64 (59) 97 152 (139) 171 241 (231) 596 345 (213) 501 (291) High Kick 0.9 1.5 (1.4) 1.4 3.8 (3.6) 3.2 5.9 (5.8) 19 6.6 (4.0) 13 (7.2) a Same as Table 2, but for the low-end metallicity scenario. form models in both cases. Additionally,[Dominik+, 1202.4901; 1308.1546; 1405.7016] Pannarale et al. ing distribution of detectable DCO parameters) depend (2013) found that in the nonspinning case, the SNR dif- on our detection criteria. We ignore a variety of com- ference between the mergers of disrupted BH-NS systems plications of the detection pipelines, such as the diffi- and the undisruptedShown: rates per year for Advanced LIGO at design sensitivity systems modeled with PhC is less culty of searching for precessing sources, noise artifacts than 1%. (non-stationary, non-Gaussian “glitches” in the instru- Including the merger portion of the signal is impor- ments) which can make searches for shorter, high-mass tant for BH-BH systems.Rescaled to actual O1 duration (46 For illustration, let us focus signals-48 days): about 30 events less sensitive, and the limited uptime of detec- on the StandardO1 sensitivity: roughly factor 8 below design, 4 events (LIGO: three candidates!) Model: if we use PhC waveforms rather tors. Instead, we have assumed several simplistic detec- than the restricted PN approximation, we find a ∼ 25% tion thresholds on single-detector or network SNR that increase in the detection rates of BH-BH systems, from are constant across all masses and mass ratios. 117 (183) to 148 (246) in the high-end (low-end) metal- Moreover, achieving good detector sensitivity at low licity scenario. frequencies may prove particularly difficult. We have The rates predicted by EOB and PhC models agree only included bandwidth above specified low-frequency 2 quite well . This can be understood by looking again cutoffs(fcut = 20 Hz in most cases) for detection-rate at Figure 2, which shows that different approximations calculations. However, the specific choice of low fre- of the strong-field merger waveform agree rather well (at quency cutoff has minimal impact on our results. For least in the equal-mass limit) on the SNR ρ and hence on example, using a lower cutoff fcut =10Hzratherthan the predicted event rates, which scale with the cube of fcut = 20 Hz in the single-detector, high-end metallic- the SNR. Waveform differences produce systematic rate ity aLIGO rate calculation would increase the Standard uncertainties significantly less than a factor of 2, much Model BH-BH rates from 117 to 128 in the inspiral case, smaller than astrophysical differences between our pre- and from 148 to 161 in the IMR case. The effect is even ferred models. smaller for BH-NS and NS-NS rates. Our detailed calculation shows that typically PhC The impact of spins on the predicted detection rates models overestimate the rates by about 10% when com- can be important. We only consider BH spins, since NSs pared to EOB models. This agreement is nontrivial, be- in compact binaries are not expected to be rapidly spin- cause the two families of models are very different in ning (e.g., Mandel & O’Shaughnessy 2010) and the dy- spirit and construction: the PhC family is a frequency- namical impact of NS spin will be small. In Tables 2 and domain model that can be easily implemented in rate 3 we use the PhC model to estimate the possible impact calculations, while the time-domain EOB model is more of BH spin on BH-NS and BH-BH detection rates by as- accurate in its domain of validity and more computation- suming that all BHs are nearly maximally spinning (i.e., ally demanding. It is important to note that in order to with dimensionless spin parameter χ1 = χ2 =0.998) and use the two families of models in rate calculations we aligned with the orbital angular momentum. Aligned BH must compute waveforms and SNRs in regions of the spins cause an orbital hang-up effect that increases the parameter space where the models were not tuned to nu- overall power radiated in the merger, produces a rapidly merical relativity simulations. In particular, both models spinning merger remnant, and therefore increases the become less accurate for small mass-ratio binaries. range to which high-mass binaries can be detected. Besides systematic errors in waveform modeling, the We find that spin effects may increase BH-BH detec- detection rates reported in this work (and the result- tion rates by as much as a factor of 3. These increased rates are a direct result of the increased horizon distance 2 We also carried out calculations using PhB models, which over- to spinning binaries. For example, a (30+30) M⊙ binary estimate rates by about 10% with respect to PhC models. We de- can be observed to roughly 1.3 times farther and be decided not to report these results in the Tables, because the PhB tected ≃ (1.3)3 ≃ 2 more often with near-maximal spins model is less accurate than PhC, although it is easier to implement than with zero spin. Additionally, spin dynamics can and less computationally expensive.

New physics: gravity sector or matter sector?

John Wheeler: “Spacetime tells mass how to move, mass tells spacetime how to curve” 8⇡G G = T µ⌫ c4 µ⌫

Can we use gravitational waves to look for modified gravity (LHS) or physics beyond the Standard Model (RHS)? Strong gravity and beyond Standard Model physics

Modification Can we test…

• Modified GR (Ultraviolet/Infrared) • Kerr dynamics? Most theories: same BHs as GR No-hair tests with ringdown Dynamics can be different Modified gravity: Horndeski, EdGB…

Gµ⌫ • Quantum corrections at the horizon? Firewalls Echoes

• Beyond Standard Model physics • Ultralight dark matter candidates? Axions: superradiant instabilities Tµ⌫ • Exotic compact objects & beyond SM? Rule out exotica (boson stars, Proca stars, gravastars…) EXPLORING THE STRING AXIVERSE WITH PRECISION ... PHYSICAL REVIEW D 83, 044026 (2011) after switching to the tortoise coordinate (14) and introduc- To relate the tunneling exponent with the rate of super- ing É r2 a2 1=2R the radial Eq. (12) takes the form of radiance instability let us consider again the energy flow the Schro¼ð¨dingerþ equationÞ Eq. (6). Integrating it over the horizon we obtain 2 d É dE 2 VÉ 0 (20) ! mw ! Y R r ; (23) dr2 À ¼ dt ¼ ð þ À Þ horizon j ð Þ ð þÞj Ã Z with the potential where E is the energy in the axion cloud. The energy is maximum in the Keplerian region, so that in the limit 2 2 2 4rgram! a m where we only keep track of the dependence on the ex- V ! À I ¼À þ r2 a2 2 ponent eÀ we can write ð þ Þ 2 2 2 2I 2 Á 2 l l 1 k a E R rc e R r ; ð þ Þþ /j ð Þj ’ j ð þÞj þ r2 a2 a þ r2 a2 and, consequently, to rewrite (23) as 2þ 2 þ 2 3r 4rgr a 3Ár À2 2þ2 2 2 3 : (21) dE 2I þ r a À r a const mw ! eÀ E: (24) ð þ Þ ð þ Þ dt ¼ Áð þ À Þ We include the !2 term in the definition of the poten- ðÀ Þ In other words, the WKB approximation for the super- tial, because even if we were to separate it, there would be a radiance rate gives1 residual dependence on !. We present the qualitative shape 2I of the potential V for a typical choice of parameters in À mw ! eÀ ; (25) Fig. 7. One can clearly see the potential well where the ¼ ð þ À Þ where the normalization prefactor is determined mainly by bound Keplerian orbits are localized and a barrier separat- the spread of the wave function in the classically allowed ing this region from the near-horizon region where super- region. We will limit ourself by calculating the exponential radiant amplification takes place. part À. We leave the technical details for the Appendix, and Consequently, the axion wave function at the horizon present only the final result here. Namely, the final answer r r (corresponding to r ) is suppressed relative for the tunneling integral in the extremal Kerr geometry to¼ theþ wave function in theÃ vicinity¼À1 of the Keplerian orbit takes the form by a tunneling exponent, I R r R rc eÀ ; I 2 2 1 ; (26) j ð þÞj’j ð Þj ¼ À ð À Þ where the tunneling integral I is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi which translates in the following superradiant rate, 2 2 r r2 r2 pV r a Ãð Þ 7 1 2 2 p2 7 1 3:7 I dr pV dr ð þ Þ ; (22) ÀWKB 10À rgÀ eÀ ð À Þ 10À rgÀ eÀ ; (27) ¼ r r1 Ã ¼ r1 ffiffiffiffi Á Z Ãð Þ ffiffiffiffi Z where we took the large limitffiffi in (26) and chose the with r1;2 being the boundaries of the classically forbidden prefactor to match the low results of Sec. II B (this value region. We will only follow the leading exponential depen- I also agrees with that of [19,32]). As we already said, the dence on eÀ and do not aim at calculating the normaliza- exponent in (27) is larger than that in [19] by a factor of two. tion prefactor in front of the exponent. As explained in the Appendix, the rate (27) provides an Black hole dynamics: wave scattering upper envelope for superradiance rates at different l in the Ergo-region Barrier Potential Well region large limit. We have presented (27) by a dotted line in Exponential Fig. 5; it agrees reasonably well with the previous =l 1 growth region “Mirror” results. at r~1/µ

III. DYNAMICS OF SUPERRADIANCE Potential Let us turn now to discussing the dynamical consequen- ces of the superradiant instability. One important property of the rates calculated in Sec. II is that the time scale for the

Black Hole Horizon r* [Arvanitaki+Dubovsky, 1004.3558] development of the instability is quite slow compared to Quasinormal modes: Light massive fields: the natural dynamical scale rg close to the black hole FIG. 7 (color online). The shape of the radial Schroedinger 1 7 horizon, ÀsrÀ > 10 rg. Consequently, in many cases non- q potentialIngoing waves at the horizon, for the eigenvalue problemq Superradiance in the rotating: black hole bomb black hole outgoing waves at infinity when 0 < w < mW linear effects, both gravitational, and due to axion self- background. Superradiant modes are localized in a potentialH well interactions, become important in the regime where the q regionDiscrete spectrum of damped created by the mass ‘‘mirror’’q Hydrogen from the spatial-like, unstable inﬁnity on theexponentials (“ right, and byringdown the centrifugal”) barrierbound states: radiation at from the ergo-region andw ~ µ horizon[EB++, 0905.2975] on the left. [Detweiler, Zouros+Eardley…] 1Note, that at this stage we still agree with [19].

044026-9 Kerr ringdown: black hole spectroscopy 15 Critical SNR for black hole spectroscopy q In GR, black holes oscillate [EB+, gr-qc/0707.1202] 4 in a set of discrete 10 complex-frequency modes l’=m’=3 l’=m’=4 (quasinormal modes) ρboth

3 determined only by 10 mass M and spin a crit ρGLRT

q One mode: (M,a) SNR 2 10 q Any other mode frequency: ρcrit

No-hair theorem test 1 10 1 1.5 2 2.5 3 3.5 4 q q Feasibility depends on SNR: for GW150914-like binaries (q~1), need SNR ~ 80 FIG. 9: Minimum SNR required to resolve two modes, as functionofthebinary’smassratioq.Ifρ > ρGLRT we can tell theGW150914: presence of aringdown second modeSNR in the~ 7 waveform, if ρ > ρcrit we can resolve either the frequency or the damping time, and if ρ > ρboth we can resolve both. Mode “1” is assumed to be the fundamental mode with l = m =2;mode“2”iseitherthe fundamental mode with l = m =3(solidlines)orthefundamentalmodewithl = m =4(dashedlines).

V. CONCLUSIONS

In this paper we analyze the detectability of ringdown waves by Earth-based interferometers. Confirming and extending previous analyses, we show that Advanced LIGO and EGO could detect intermediate-mass black holes of 3 mass up to 10 M⊙ out to a luminosity distance of a few Gpc. Using recent∼ results for the multipolar energy distributionfromnumericalrelativitysimulationsofnon-spinning binary black hole mergers [10] to estimate the relative amplitude of the dominant multipolar components, we point out that the single-mode templates presently used for ringdown searches in the LIGO data stream could produce asignificanteventloss(> 10% in a large interval of black hole masses). A similar event loss should affect also next-generation Earth-based detectors, as well as the planned space-based interferometer LISA. Single-mode templates are useful for detection of low-mass systems, but they produce large errors in the estimated values of the parameters (and especially of the quality factor). We estimate that, unfortunately, more than 106 templates would be needed for a single-stage multi-mode search. For this reason we recommend a “two stage” search∼ to save on computational costs: a single-mode template couldbeusedtodetectthesignal,andamulti-modetemplate (or even better, Prony methods [32]) could be used to estimateparametersonceadetectionhasbeenmade. In Appendix B we introduce a criterion to decide for the presence of more than one mode in a ringdown signal. By updating estimates of the critical signal-to-noise ratiorequiredtoresolvethefrequenciesofdifferent QNMs using results from numerical relativity, we show that second-generation Earth-based detectors and LISA both have the potential to perform tests of the Kerr nature of astrophysical black holes. In the future we plan to use numerical waveforms (possibly including spin effects) to refine our estimates. We also plan to carry out Monte Carlo simulations to study the information that can be extracted on the source position and orientation using a network of Earth-based detectors. The possibility to constrain the black hole spin’s direction from the multipolar distribution of the merger-ringdown radiation should be particularly interesting (eg. for coincident electromagnetic observations of jets that could be emitted along the black hole spin’s axis).

Acknowledgements

We are grateful to Alessandra Buonanno, Kostas Kokkotas, Clifford Will and Nicolas Yunes for discussions. This work was partially funded by Funda¸cão para a Ciência eTecnologia(FCT)-Portugalthroughprojects PTDC/FIS/64175/2006 and POCI/FP/81915/2007, by the National Science Foundation under grant numbers PHY 03-53180 and PHY 06-52448, and by NASA under grant number NNG06GI60 to Washington University. LIGO-T15TBI–v1 dictates that we begin planning now for detectors that may begin operation 20 years from now. A typical detector cycle includes: Simulation of ideas and concepts; Experimental tests; Conceptual designLIGOLooking and prototyping UPGRADEforward: phases;advanced ProposalTIMELINEdetectors and engineering; construction and Installation; Commissioning and observing phases.

Explorer R&D + Design Explorer – New Facility

Si, Cryo, 2um R&D Voyager – Current Facility

Coating, Suspension R&D A+

Sqz R&D A+ Color Code: Simulation Installation

Advanced Experiment Commissioning

Design Data

2015 2020 2025 2030

Figure 2: Timeline for A+, LIGO Voyager and LIGO Cosmic Explorer. The timeline shown is for a single LIGO detector. Upgrading/replacing each detector in the network would need to be staged to optimize science outcomes of the prevailing global array. (NB: 2 µm is used to indicate any wavelength between 1.5 µmand2.2 µm.)

We envisage potentially three detector epochs post Advanced LIGO baseline over the next 25 years with working titles A+, LIGO Voyager and LIGO Cosmic Explorer, see Figure 2. The funds required to implement the upgrades are classiﬁed as: modest, less than $10M to $20M; medium, $50M to $100M; major, greater than $150M. This strategy will be modiﬁed according to signals observed, technology readiness and funds available.

2.1 A+

A+ would essentially be a modest cost upgrade to aLIGO, implemented in stages. It would have a binary neutron star inspiral range approximately 1.7 times aLIGO (around 340 Mpc), (see Figure 3). For A+ to begin operation around 2017-18, the ﬁrst 2 phases of the detector cycle (simulation and experimental testing) need to have already been completed for stage 1, and should be well underway for stage 2. frequency dependent squeezed light, implemented in stage 1, • better mirror coatings and possibly slightly bigger laser beam sizes in the optical cav- • ities to reduce coating thermal noise, implemented in a second stage.

Miller et al [3] have shown that squeezing of the light’s quantum noise and coating thermal noise reduction must be combined to achieve maximum beneﬁt. The goal is to minimize page 7 Looking forward: LISA Pathfinder launch (Dec 3 2015) Pathfinder Launch:Launch 2/3: 2034? December 2016 Spectroscopy of Kerr black holes with Earth- and space-based interferometers

Emanuele Berti1,2, Alberto Sesana3, Enrico Barausse4,5, Vitor Cardoso2,6, Krzysztof Belczynski7 1 Department of Physics and Astronomy, The University of Mississippi, University, MS 38677, USA 2 CENTRA, Departamento de F´ısica, Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal 3 School of Physics and Astronomy, The University of Birmingham, Edgbaston, Birmingham B15 2TT, UK 4 Sorbonne Universités, UPMC UniversitéParis 6, UMR 7095, Institut d’Astrophysique de Paris, 98 bis Bd Arago, 75014 Paris, France 5 CNRS, UMR 7095, Institut d’Astrophysique de Paris, 98 bis Bd Arago, 75014 Paris, France 6 Perimeter Institute for Theoretical Physics, 31 Caroline Street North Waterloo, Ontario N2L 2Y5, Canada and 7 Astronomical Observatory, Warsaw University, Al. Ujazdowskie 4, 00-478 Warsaw, Poland

We estimate the potential of present and future interferometric gravitational-wave detectors to test the Kerr nature of black holes through “gravitational spectroscopy,” i.e. the measurement of multiple quasinormal mode frequencies from the remnant of a black hole merger. Using population synthesis models of the formation and evolution of stellar-mass black hole binaries, we ﬁnd that Voyager-class interferometers will be necessary to perform these tests. Gravitational spectroscopy in the local Universe may become routine with the Einstein Telescope, but a 40-km facility like Cosmic Explorer is necessary to go beyond z 3. In contrast, eLISA-like detectors should carry out a few – or even hundreds – of these tests every⇠ year, depending on uncertainties in massive black hole formation models. Many space-based spectroscopical measurements will occur at high redshift, testing the strong gravity dynamics of Kerr black holes in domains where cosmological corrections to general relativity (if they occur in nature) must be signiﬁcant.

Introduction. The first binary black hole (BH) mer- GW150914 raised the question whether current detect- ger signal detected by the LIGO Scientific Collaboration, ors at design sensitivity should routinely observe ring- GW150914 [1], had a surprisingly high combined signal- down signals loud enough to perform gravitational spec- to-noise ratio (SNR) of 24 in the Hanford and Livingston troscopy. Leaving aside conceptual issues about ruling detectors. The quasinormal mode signal (“ringdown”) out exotic alternatives [15–17], here we use our current from the merger remnant is consistent with the predic- best understanding of the astrophysics of stellar-mass tions of general relativity (GR) for a Kerr BH, but it was and supermassive BHs to compute the rates of events observed with a relatively low SNR ⇢ 7 [2]. The large that would allow us to carry out spectroscopical tests. 2 masses of the binary components [3] have⇠ interesting im- Below we provide the details of our analysis, but the plications for the astrophysics of binary BH formation [4], 16 mainEarth vs. conclusionsspace can- bebased understoodinterferometers relying on the noise 10 and the detection placed some constraints on the merger power spectral densities (PSDs) Sn(f) of present and fu- 10 17 rates of BH binaries in the Universe [5–8]. ture detectors, as shown and briefly reviewed in Fig. 1, 18 10 and simple back-of-the-envelope estimates. )

LISA Pathﬁnder was successfully launched in Decem-2 19 /

1 10 Ringdown SNR. Consider the merger of two BHs with ber 2015, paving the way for a space-based detector such 20 as eLISA [9, 10], which will observe mergers of massive10 source-frame masses (m1,m2), spins (j1, j2), total mass

) (Hz 21 M = m + m , mass ratio q m /m 1 and sym- 10 tot 1 2 1 2

BHs throughout the Universe with very large SNRs andf

( ⌘2

n 22 metricN2A5 mass ratioO1 ⌘ = m1Voyagerm2/Mtot. The remnant mass

test the Kerr nature of the merger remnants. The basicS 10 and dimensionlessN2A2 O2 spin, MCE1and j = J/M2, can be com- 23 idea is that the dominant ` = m = 2 resonant frequency10 N2A1 AdLIGO CE2 wide putedN1A5 using theA+ fitting formulasCE2 narrow in [26] and [27], respect- and damping time can be used to determine the rem- 24 2 10 ivelyN1A2 (see alsoA++ [28, 29]). TheET-B ringdown SNR ⇢ can be es- nant’s mass M and dimensionless spin j = J/M (we 25 N1A1 Vrt ET-D 10 timated by following [14]. Including redshift factors and adopt geometrical units G = c = 1 throughout this Let- 4 3 2 1 0 1 2 3 substituting10 10 the10 Euclidean 10 distance10 r10by the10 luminosity10 ter.) In GR, all subdominant mode frequencies (e.g. the f (Hz) distance DL as appropriate, Eq. (3.16) of [14] implies modes with ` = m = 3 and ` = m = 4 [11]) are then 2 uniquely determined by M and j. The detectionFigure 1. of Noise sub- PSDs forthat various⇢ space-basedis wellf approximated= and170.2 (10 advanced Earth-basedM bysun)/M detectorHz designs. “NiAk” refers to non sky-averaged eLISA PSDs with pessimistic (N1) and optimistic (N2) acceleration noise and armlength L = k Gm (cf. [18]). In the high- dominant modes requires high SNR, but eachfrequency mode regime, will we show noise PSDs for (top to bottom): the first AdLIGO observing1/2 run (O1); the expected sensitivity for 8 M 3✏ provide one (or more) tests of the Kerr naturethe of second the observing rem- run (O2) and the Advanced LIGOeq design sensitivityz (AdLIGO)rd [19]; the pessimistic and optimistic ranges of AdLIGO designs with squeezing (A+, A++)⇢ = [20] ; Vrt and Voyager [21]; Cosmic Explorer, (CE1), basically(1) A+ in a 40-km DL lmn 5 Sn(flmn) nant [12]. As first pointed out by Detweiler infacility 1980, [22]; grav- CE2 wide and CE2 narrow, i.e. 40-kmF detectors with Voyager-type technology[EB+, but1605.09286] di↵erent signal extraction itational waves allow us to do BH spectroscopy:tuning [23]; “After and two possible Einstein Telescope designs, namely ET-B [24] and ET-D in the “xylophone” configuration [25]. where Mz = M(1 + z). Fits of the mass-independent di- the advent of gravitational wave astronomy, the observa- mensionless frequency lmn(j) 2⇡Mzflmn and quality F ⌘ 1/2 21 1/2 tion of these resonant frequencies might finallynoise PSD provideSn(f), and we have used the approximation of the N2A5 eLISA detector is at SN2A5 10 Hz . factor Qlmn(j) are given in Eqs. (E1) and (E2)2 of3 [14].4 ⇠ direct evidence of BHs with the same certainty4Qlmn as,1. say, The ringdown eciency for nonspinning This noise level is 10 (10 , 10 ) times larger than binaries is well approximatedThe geometrical by the matched-filtering factor eq =the 1 best for sensitivity Michelson⇠ of AdLIGO interfero- (Voyager, Einstein Tele- the 21 cm line identifies interstellar hydrogen” [13]. 2 4 estimate of Eq. (4.17)meters in [11]: ✏rd with=0.44 orthogonal⌘ .Whenus- arms,scope), while respectively.eq = Howeverp3/2 eLISA for an BHs are 10 times ing the best-fit parameters inferred for GW150914 [3], more massive, yielding SNRs that are larger⇠ by a factor Such high SNRs are known to be achievableEq. (1) with yields an a ringdowneLISA-like SNR ⇢ 7.7 detector in O1 (in agree- (where the106 angle. Astrophysical between rate the calculations arms are very di↵er- eLISA-like detector [14]. The surprisinglyment high with SNR [2]) ofand ⇢ is16. 602 in). AdLIGO.' This expression involvesent⇠ in the the twonon frequency sky-averaged regimes, but these qualitative ' Due to the orbital hang-up e↵ect, spinning binaries arguments explain why only Einstein Telescope-class de- with aligned (antialigned) spins radiate more (less) than tectors will achieve SNRs nearly comparable to eLISA. their nonspinning counterparts. The dominant spin- Astrophysical models. We estimate ringdown de- induced correction to the radiated energy is proportional tection rates for Earth-based interferometers (detection to the sum of the components of the binary spins along rates for the full inspiral-merger-ringdown signal are the orbital angular momentum [26, 30, 31]. We es- higher) using three population synthesis models com- timate this correction by rescaling the radiated energy puted with the Startrack code: models M1, M3 and by the factor Erad(m1,m2, j1, j2)/Erad(m1,m2, 0, 0), M10. Models M1 and M3 are the “standard” and “pess- where the total energy radiated in the merger Erad is imistic” models described in [8]. The “standard model” computed using Eq. (18) of [26]. We find that spin- M1 and model M10 predict very similar rates for Ad- dependent corrections change ⇢ by at most 50%. LIGO at design sensitivity. In both of these models, It is now easy to understand why Einstein Telescope- compact objects receive natal kicks that decrease with class detectors are needed to match the SNR of eLISA- the compact object mass, with the most massive BHs like detectors and to perform BH spectroscopy. The receiving no natal kicks. This decreases the probability quantity (j) is a number of order unity [12, 14]. of massive BHs being ejected from the binary, increasing Flmn The physical frequency is flmn 1/Mz: for example, merger rates. Model M1 allows for BH masses as high an equal-mass merger of nonspinning⇠ BHs produces a as 100 M . On the contrary, model M10 includes the remnant with j 0.6864 and fundamental ringdown fre- e↵ect⇠ of pair-instability mass loss, which sets an upper ' 2 quency f220 170.2(10 M /Mz) Hz. So Earth-based limit of 50M on the mass of stellar origin BHs [32]. detectors are' most sensitive to the ringdown of BHs with In model⇠ M3, all compact objects (including BHs) ex- 2 Mz 10 M , while space-based detectors are most sens- perience high natal kicks drawn from a Maxwellian with ⇠ 6 1 itive to the ringdown of BHs with Mz 10 M .Thecru- = 265km s based on the natal kick distribution cial point is that, according to Eq. (1),⇠⇢ M3/2 at fixed measured for single pulsars in our Galaxy [33]. The as- redshift and noise PSD. As shown in Fig.⇠ 1, the “bucket” sumption of large natal kicks leads to a severe reduction Astrophysical population models

Stellar mass black hole binaries: Massive black hole binaries: Startrack (Belczynski+) Barausse+ qM1 qPopIII Standard Light seeds

M<100Msun qM10 qQ3nod Pair-instability mass loss Heavy seeds

M<50Msun No delays qModel M3 qQ3d Pessimistic Heavy seeds High kicks, lower limit on rates Final parsec problem 3 Earth vs. space-based: detection rates 106 M3 M3 105 M10 M10 M1 M1 104 > 8 > GLRT

103

102

101 events/year

100

1 10

2 10

3 10

O1 O2 A+ Vrt ETB CE1 A++ CE2n ETDX CE2w Voyager AdLIGO

Figure 2. Rates of binary BH mergers that yield detectable ringdown signals (filled symbols) and allow for spectroscopical tests (hollow symbols). Left panel: rates per year for Earth-based detectors of increasing sensitivity. Right panel: rates per year for 6-link (solid) and 4-link (dashed) eLISA configurations with varying armlength and acceleration noise. of BH-BH merger rates, and therefore model M3 should M1 (M10, M3) predict 3.0(2.5, 0.57) events per year be regarded as pessimistic [8]. In all of these models we with detectable ringdown in O1; 7.0 (5.8, 1.1) in O2; and set the BH spins to zero, an assumption consistent with 40 (35, 5.2) in AdLIGO. Model Q3d (Q3nod, PopIII) estimates from GW150914 [4]. Even in the unrealistic predicts 38 (533, 13) events for a 6-link N2A5 eLISA scenario where all BHs in the Universe were maximally mission lasting 5 years, but in the plot we divided these spinning, rates would increase by a factor . 3 (see Table numbers by 5 to facilitate a more fair comparison in terms 2 of [5]). Massive binaries with ringdowns detectable by of events per year. Earth-based interferometers could also be produced by BH spectroscopy. Suppose that we know that a signal other mechanisms (see e.g. [34–37]), and therefore our contains two (or possibly more) ringdown modes. We rates should be seen as lower bounds. expect the weaker mode to be hard to resolve if its amp- To estimate ringdown rates from massive BH mergers litude is low and/or if the detector’s noise is large. The detectable by eLISA we consider the same three models critical SNR for the second mode to be resolvable can (PopIII, Q3nod and Q3d) used in [18] and produced with be computed using the generalized likelihood ratio test the semi-analytical approach of [38] (with incremental (GLRT) [42] under the following assumptions: (i) using improvements described in [39–41]). These models were other criteria, we have already decided in favor of the chosen to span the major sources of uncertainty a↵ect- presence of one ringdown signal; (ii) the ringdown fre- ing eLISA rates, namely (i) the nature of primordial BH quencies and damping times, as well as the amplitude seeds (light seeds coming from the collapse of Pop III of the dominant mode, are known. Then the critical stars in model PopIII; heavy seeds originating from pro- SNR ⇢GLRT to resolve a mode with either ` = m =3 togalactic disks in models Q3d and Q3nod), and (ii) the or ` = m = 4 from the dominant mode with ` = m =2 delay between galaxy mergers and the merger of the BHs is well fitted, for nonspinning binary BH mergers, by at galactic centers (model Q3d includes this delay; model 2, 3 15.4597 1.65242 Q3nod does not, and therefore yields higher detection ⇢ = 17.687 + , (2) GLRT q 1 q rates). In all three models the BH spin evolution is fol- 83.5778 44.1125 50.1316 lowed self-consistently [38, 39]. For each event in the ⇢2, 4 = 37.9181 + + + .(3) GLRT q q2 q3 catalog we compute ⇢ from Eq. (1), where ✏rd is rescaled by a spin-dependent factor as necessary. These fits reproduce the numerical results in Fig. 9 of Detection rates. The ringdown detection rates (events [42] within 0.3% when q [1.01 100]. Spectroscopical per year with ⇢ > 8 in a single detector) predicted by tests of the Kerr metric can2 be performed whenever either 2, 3 2, 4 models M1, M3, M10 (for stellar-mass BH binaries) and mode is resolvable, i.e. ⇢ > ⇢GLRT min(⇢GLRT, ⇢GLRT). PopIII, Q3d, Q3nod (for supermassive BH binaries) are The ` = m = 3 mode is usually easier⌘ to resolve than the shown in Fig. 2 with filled symbols. For example, models ` = m = 4 mode, but the situation is reversed in the 3 Earth vs. space-based: detection rates 103 Q3nod 4L Q3nod 4L Q3d 4L Q3d 4L PopIII 4L PopIII 4L Q3nod 6L Q3nod 6L Q3d 6L Q3d 6L PopIII 6L PopIII 6L > 8 > GLRT 102 events/year

101

100

N1A1 N1A2 N1A5 N2A1 N2A2 N2A5

Earth vs. space-based: redshift distribution 104

103

102

dN/dz 101

100 > 8 1 10 O1 A++ ET-B 103 O2 Vrt CE1 AdLIGO Voyager CE2w A+ ET-D CE2n 102

dN/dz 101

100

> GLRT 10 1 0 5 10 15 20 z

Figure 3. Left: redshift distribution of events with ⇢ > 8(top)and⇢ > ⇢GLRT (bottom) for model M1 and Earth-based 2 detectors. In the bottom-left panel, the estimated AdLIGO rate ( 2.6 10 events/year) is too low to display. Right: same for models Q3nod, Q3d and PopIII. Di↵erent eLISA design choices⇡ have⇥ an almost irrelevant impact on the distributions. comparable-mass limit q 1, where the amplitude of would be able to do spectroscopy at z 10. ! ⇡ odd-m modes is suppressed [11, 43]. Extreme mass-ratio Conclusions. Using our best understanding of the calculations [44] and a preliminary analysis of numerical formation of field binaries, we predict that AdLIGO at waveforms show that the ratio of mode amplitudes is, to design sensitivity should observe several ringdown events a good accuracy, spin-independent, therefore this SNR per year. However routine spectroscopical tests of the threshold is adequate for our present purpose. dynamics of Kerr BHs will require the construction and The rates of events with ⇢ > ⇢GLRT are shown in operation of detectors such as the Einstein Telescope [45– Fig. 2 by curves with hollow symbols. The key obser- 47], and 40-km detectors [22, 23] will be necessary to vation here is that, although ringdown detections should reach cosmological distances. Many of the mergers for be routine already in AdLIGO, high-SNR events are ex- which eLISA can do BH spectroscopy will be located at ceedingly rare: reaching the threshold of 1 event/year z 1. These systems will test GR in qualitatively dif- requires Voyager-class detectors, while sensitivities⇠ com- ferent regimes than any low-z observation by AdLIGO: parable to Einstein Telescope are needed to carry out BH spectroscopy with eLISA will test whether gravity such tests routinely. This is not the case for space-based behaves locally like GR even at the very early epochs of interferometers: typical ringdown detections have such our Universe, possibly placing constraints on proposed high SNR that 50% or more of them can be used to extensions of Einstein’s theory [48]. ⇡ do BH spectroscopy. The total number of eLISA detec- Given the time lines for the construction and operation tions and spectroscopic tests depends on the underlying of these detectors, it is likely that the first instances of BH formation model, but it is remarkably independent of BH spectroscopy will come from a space-based detector. detector design (although the N1A1 design would sens- This conclusion is based on the simple GLRT criterion ibly reduce rates in the most optimistic models). introduced in [42], and it is possible that better data Perhaps the most striking di↵erence between Earth- analysis techniques (such as the Bayesian methods ad- and space-based detectors is that a very large fraction vocated in [46, 47]) could improve our prospects for grav- of the “spectroscopically significant” events will occur at itational spectroscopy with Earth-based interferometers. cosmological redshift in eLISA, but not in Einstein tele- We hope that our work will stimulate the development scope. This is shown very clearly in Fig. 3, where we of these techniques and their use on actual data. plot redshift histograms of detected events (top panel) As shown in Fig. 2, di↵erences in rates between models and of events that allow for spectroscopy (bottom panel). M1 and M10 become large enough to be detectable in eLISA can do spectroscopy out to z 5 (10, or even 20!) A+. We estimate 34 (29) ringdown events per year for for PopIII (Q3d, Q3nod) models, while⇡ even the Einstein M1 (M10) in A+, and 89 (66) events per year in A++. Telescope is limited to z . 3. Only 40-km detectors with Rate di↵erences are even larger when we consider the cosmological reach, such as Cosmic Explorer [22, 23], complete signal. Therefore, while the implementation 4

Earth vs. space-based: redshift distribution

Tobs = 5 years 102 > 8

101 dN/dz

100

1 10 PopIII, N2A5 Q3d, N2A5 Q3nod, N2A5 102 PopIII, N2A2 Q3d, N2A2 Q3nod, N2A2 PopIII, N2A1 Q3d, N2A1 Q3nod, N2A1 > GLRT 101 dN/dz

100

10 1 0 5 10 15 20 z

increases SNR r33 of the subdominant mode

[Yang+, 1701.05808] GRAVITATIONAL-WAVEPlanck scale structures SIGNATURESnear the horizon OF EXOTICand …echoes PHYSICAL REVIEW D 94, 084031 (2016) 0.15 black hole On the other hand, when the Schwarzschild horizon is 0.10 replaced by a surface (as, e.g., in the gravastar case) or by a throat (as in the wormhole case), the potential also develops 0.05 outgoing at infinity ingoing at horizon a minimum (i.e., an innermost stable PS) which can trap 0.00 low-frequency modes [12,15,28 30] (cf. Fig. 1). This inner 0.15 – wormhole PS can also be thought of as being caused by the centrifugal 2 0.10 barrier, and it may become nonlinearly unstable [12]. These ) M * modes make their way to the waveforms in Fig. 2 in the 0.05 outgoing at infinity V(r trapped outgoing at infinity form of “echoes” of the initial PS modes after they leak 0.00 0.15 through the potential barrier: the radiation pulse generated centrifugal barrier star-like ECO at the potential barrier peak (the PS modes) is then trapped 0.10 in a semipermeable cavity bounded between the two PSs.

0.05 outgoing at infinity Indeed, the time delay between two consecutive echoes is regular at the center trapped roughly the time that light takes for a round trip between the 0.00 -50 -40 -30 -20 -10 0 10 20 30 40 50 potential barrier. In general, this delay time reads r*/M [Cardoso+, 1602.07309 and 1608.08637; Abedi/Dykaar/Afshordi, 1612.00266] 3M dr FIG. 1. Qualitative features of the effective potential felt by Δt ∼ 2 ; 5 r pFB ð Þ perturbations of a Schwarzschild BH compared to the case of Z min wormholes [12] and of starlike ECOs with a regular center [22]. ffiffiffiffiffiffiffi The precise location of the center of the star is model dependent where r is the location of the minimum of the potential and was chosen for visual clarity. The maximum and minimum of min shown in Fig. 1. If we consider a microscopic correction at the potential corresponds approximately to the location of the unstable and stable PS, and the correspondence is exact in the the horizon scale (l ≪ M), then the main contribution to eikonal limit of large angular number l. In the wormhole case, the time delay comes near the radius of the star and modes can be trapped between the PSs in the two “universes.” In therefore, the starlike case, modes are trapped between the PS and the l centrifugal barrier near the center of the star [28–30]. In all cases Δt ∼−nM log ; l ≪ M; 6 the potential is of finite height, and the modes leak away, with M ð Þ higher-frequency modes leaking on shorter timescales. where n is a factor of order unity that takes into account the structure of the objects. For wormholes, n 8 to account for the fact that the signal is reflected by the¼ two maxima in

6 0.4 4 wormhole wormhole 2 0.2 0 0.0 -2 -0.2 -4 -0.4

4 empty shell 0.5 empty shell 2 (t) (t) 10 10 0 0.0 Ψ Ψ -2 -0.5 -4

gravastar gravastar 4 0.5 2 0 0.0 -2 -0.5 -4 -100 0 100 200 300 400 100 200 300 400 t/M t/M

FIG. 2. Left: A dipolar (l 1, m 0) scalar wave packet scattered off a Schwarzschild BH and off different ECOs with l 10−6M ¼ ¼ ¼ (r0 2.000001M). The right panel shows the late-time behavior of the waveform. The result for a wormhole, a gravastar, and a simple empty¼ shell of matter are qualitatively similar and display a series of “echoes” which are modulated in amplitude and distorted in frequency. For this compactness, the delay time in Eq. (6) reads Δt ≈ 110M for wormholes, Δt ≈ 82M for gravastars, and Δt ≈ 55M for empty shells, respectively.

084031-3 Signatures of quantum corrections near the horizon?

Delay time between echoes: r t 2M log 0 1 echoes ⇠ 2M ⇣ ⌘ Beyond toy models?

Formation

Stability (ergoregion instability stronger as )r 2M 0 ! Model independent constraint?

Planck-scale corrections within reach? (timescale/amplitude)

Which SNR? Data analysis? Dark matter detection via GWs? Ultralight fields / dark matter candidates

Superradiance when w < mWH - strongest instability: l=m=1, µM~1 [Dolan, 0705.2880] 10 For µ=1eV, M=Msun : µM~10 Need light scalars (or primordial black holes!) 1.0 spin-0 0.8

0.6 Evolution of instability 1e-12eV 1e-14eV BH spin 0.4 Nonlinear effects? 1e-16eV 0.2 EM 1e-18eV 1e-20eV Perturbation theory is ok GW 1e-22eV [East-Pretorius, 1704.04791] spin-1 0.8

0.6 GW signatures 0.4 Periodic sources BH spin 0.2 Stochastic background spin-2 Regge holes 0.8

0.6

Rates for LIGO/LISA? 0.4 BH spin

0.2

0.0 100 102 104 106 108 1010

BH mass [M⊙] 2

GWs from bosonic condensates around BHs. The evolution assume that the observed GWs are entirely produced after the of the condensate occurs on two different time scales [30]. instability has saturated, leading the BH from an initial state First the condensate grows on the instability time scale ⌧ (M ,J) to a final state (M ,J ), and we compute h using inst ⌘ i i f f 1/!I until the superradiant condition is saturated; then it is the final BH parameters. It can be shown that MS scales lin- dissipated through GW emission over a time scale ⌧GW that early with Ji [34], so the GW strain h also grows with Ji. depends on the mass MS of the final condensate and on the Resolved signals and stochastic background GW emission rate. These time scales can be computed ana- -�� lytically when Mµ 1, with the result [34] �� ⌧ -�� 1 �� 5 8 9 ⌧inst 10 yr M6 µ17 , (1) ⇠ -�� 1 �� 11 14 15 ⌧GW 5 10 yr M6 µ17 , (2) ⇠ ⇥ ��-�� 6 17 where M6 = M/(10 M ), µ17 = ms/(10 eV), and -�� J/M2 is the dimensionless spin parameter. These rela- �� ⌘ tions are a reasonably good approximation even when Mµbe- ��-�� comes of order unity [34]. Since ⌧ ⌧ M, the ax- GW inst ��-�� ion condensate has enough time to grow, and the evolution of ��-� ��-� ��-� ��-� ��-� �� the BH-condensate system can be studied in a quasi-adiabatic approximation [30] using Teukolsky’s formalism [35, 36]. Vertical lines: a/M=0.9, z=0.01-3.01 (right to left), µM grows along vertical lines The field stress-energy tensor is typically small compared to [Brito+, in preparation] the BH energy density, so the boson field can be considered FIG. 1. GW strain produced by boson condensates compared to Ad- vanced LIGO and LISA PSDs (black thick curves), assuming an almost stationary and its backreaction can be neglected [30]. observation time of Tobs =4yrin both cases. Near-vertical lines Over the observation time T ⌧ ⌧ the GW obs ⌧ inst ⌧ GW are computed for BHs with initial spin i =0.9 and for fixed ms signal can be considered almost monochromatic, with fre- (as indicated). Each line corresponds to a single source at redshift quency f0 =2⇡(2!R) 4⇡µ, and in this sense it is simi- z (0.001, 3.001) (from right to left, in steps of z =0.2); recall ⇠ 2 lar to continuous sources, such as LIGO pulsars or LISA ver- that GW detectors measure frequencies f = fs/(1 + z), where fs ification binaries. The signal-to-noise ratio (SNR) for con- is the source-frame frequency. The strain increases along a nearly straight vertical line as we increase the BH mass, so that Mµ spans tinuous signals is given by (see e.g. [37]) SNR hpTobs , pSh(f0) the range (0, 0.43). Thin lines correspond to the stochastic back- ' 2 where h is the signal’s root-mean-square (rms) strain ampli- ground produced by the whole population of astrophysical BHs, assuming that the boson has mass ms. The PSD of DECIGO [40] tude, Tobs is the observation time, and Sh(f0) is the noise (dashed line) is also shown for reference. power spectral density (PSD) at f0. Coherent all-sky searches where the location the sources is typically unknown are computationally expensive. One can resort to semicoherent meth- In Fig. 1 we compare the GW strain of Eq. (3) with the ods (e.g. [38]), dividing the signal in coherent segments PSD for Advanced LIGO and LISA. Each near-vertical line N 2 with time length Tcoh. Then the typical sensitivity threshold is corresponds to a BH with i = Ji/Mi =0.9 at redshift 25 Sh(f0) z, for bosons with mass ms. The GW strain increases along hthr 1/4 , where hthr is the minimum rms strain Tcoh ' N a nearly vertical line as we vary Mµ in the range (0, 0.43). amplitude detectableq over the observation time Tcoh. In N ⇥ Thin continuous curves correspond to the stochastic back- our estimate below we consider both full coherent and semi- ground produced by the whole population of astrophysical coherent searches. BHs, assuming that the boson has mass ms. Note that the We compute h numerically within BH perturbation the- stochastic background becomes itself a source of noise when ory [36] in the entire (a/M, µM) plane (see [34] for details). 17 m 10 eV, complicating the detection of individual s ⇡ Our results are much more accurate than the analytical ap- sources. Figure 1 suggests that axion-like particles in the mass proximations made in Refs. [29, 30]. The GW amplitude, av- 19 11 range 10 eV . ms . 10 eV (with a small gap around eraged over source and detector orientations, reads f 1 Hz which might be filled by DECIGO [40]) could ⇠ in principle be detectable with LIGO and LISA, at least for GM MS p2A h = , (3) highly spinning BHs. Below we quantify this expectation. c2r M p5⇡(2⇡f M)2 ✓ ◆ 0 BH population models. An assessment of the detectability where r is the distance to the source, MS is the mass of the of GWs from superradiant instabilities requires astrophysical scalar condensate, and the dimensionless constant A (related models for BH populations. [Paolo: I think we will have to cut to the GW energy flux) is computed from BH perturbation this part even further, but let’s see after we finish the writing] theory [34]. We take into account correction factors due to the For massive BHs we adopt the same astrophysical models geometry of the detector and we sky-average the signal [39]. as in [41]. The main assumptions concern (cf. [34]): To take into account cosmological effects it is sufficient to (i) Mass and spin distribution of isolated massive BHs. multiply source masses by redshift factors (1 + z), and to re- We adopt an optimistic model where the redshift-dependent 2 place r by the luminosity distance DL. We conservatively BH number density d n/(d log10 Mda) is computed using 4 Stochastic background for different astrophysical models ��-� ��-� ��-� ��-� ��-� ��-�� -�� -�� -�� -�� -� ��-� ��-� ��-� ��

Frequency-integrated sensitivity curves [Thrane-Romano, 1310.5300] [Brito+, in preparation] FIG. 2. Left panel: stochastic background in the LIGO and LISA bands. For LIGO, the different spectra for each scalar field mass correspond to a uniform spin distribution with (from top to bottom) [0.8, 1], [0.5, 1], [0, 1] and [0, 0.5]. The black lines are the power-law integrated curves of Ref. [47] for a threshold SNR2 = 1 taken from2 Ref. [48],2 corresponding to2 LIGO’s first two observing runs, O1 and O2, and for the design sensitivity O5. By definition, SNR 1 if a power-law spectrum intersects the black line. For LISA, the three different curves correspond to different mass-spin distributions [34] (with the optimistic model on the top). Right panel: SNR for LIGO and LISA for the backgrounds shown in the left panel. We assumed Tobs =2yrfor LIGO and Tobs =4yrfor LISA.

4 3 This assumption is conservative and should not significantly sity. Since the BH mass density is ⇢BH a few 10 M /pc ⇠ ⇥ affect the final results, because subsequent annihilation sig- in the mass range 104 107M relevant for LISA, this yields 9 nals (if they occur at all) are much weaker [34]. In order to ⌦ 10 for LISA. For LIGO, let us first note that the GW, ax ⇠ ensure that the instabilitity timescale is shorter than the typi- background of GWs from BH binaries can be approximated as cal accretion and merger timescales we only consider BHs for ⌦ f f ⇢ /⇢ , where f =1 10% is the bi- gw, bin ⇠ gw bin bh c gw which ⌧ < t. Furthermore, we only consider events for nary’s mass fraction emitted in GWs [63], and f 1%[50] inst bin ⇠ which the average number of mergers Nm within the instabil- is the fraction of stellar-mass BHs in binaries. Therefore ity timescale is smaller than unity. ⌦GW, ax/⌦GW, bin fax/(fgwfbin) 100 1000. Since ⇠ ⇠ 9 The SNR for the stochastic background is [47] from the O1 results one has ⌦GW, bin 10 at f = 25 7 ⇠ 6 Hz [48], we obtain ⌦GW, ax 10 10 . 1/2 ⇠ fmax 2 [Emanuele: Do PTA bounds [64] matter?] p ⌦GW SNR = T df 2 , (7) [Paolo: The argument above assumes that the BH mass dis- f ⌦ Z min sens ! tribution peaks around the value that would maximize the in-

2 2 LIGO Sh(f) 2⇡ 3 LISA 2⇡ 3 stability, right? Or, equivalently, it requires an axion with a where ⌦sens = 2 f and ⌦sens = Sh(f) 2 f 2IJ(f) 3H0 3H0 mass that corresponds to µM (0.1) relative to the peak ⇠ O for LIGO [61] and LISA [62], respectively. In the LIGO case of the BH mass distribution. This is why the argument doesn’t we assume the same Sh for the Livingston and Hanford detec- work for PTAs, since the latter would require most BHs to 11 tors, and IJ denotes their overlap reduction function [47]. have M & 10 eV.] Figure 2 presents a summary of our results. Remarkably, Event rates for resolvable sources. [Paolo: Add LIGO.] For the SNR for the stochastic GW background from bosonic con- LIGO sources, we estimate the number of detectable events as densates can be very high. For our optimistic mass-spin distri- 13 bution, axion-like particles in the entire range 2 10 eV . N = XXXXX , (8) 12 19 ⇥16 m 10 eV (5 10 eV m 5 10 eV) would s . ⇥ . s . ⇥ yield SNR > 8 with LIGO (LISA). On the other hand, the number of events detectable by LISA The order of magnitude of the stochastic background can be as individual GW sources can be estimated as estimated by a simple back-of-the-envelope calculation. The mass fraction of an isolated BH that is emitted by the axion 1 d2n N = (M,a,z) cloud is fax 1% and fax 10% in the mass ranges rel- t d log Mda ⇠ ⇠ 0 ZSNR>8 10 evant for LIGO and LISA, respectively. [Paolo: Why is it dD t(M,a,µ)4⇡D (z)2d log Mda c dz , (9) different? It should only depend on Mµ...] Since the emitted ⇥ c 10 dz gravitational radiation is concentrated in a relatively small frequency window ln f 1 for both LISA and LIGO (because where Dc = DL/(1 + z) is the comoving distance. ⇠ the GW signal is almost monochromatic with frequency de- Figure 3 summarizes our results for the number of individ- termined by the BH and axion masses), we obtain ⌦GW, ax = ually resolvable events. In the left panel we consider our opti- 1/⇢ d⇢ /d ln f f ⇢ /⇢ , where ⇢ is the critical den- mistic models for BH mass and spin distributions and different c⇥ gw ⇠ ax BH c c New physics with gravitational wave observations ü Tests of Kerr dynamics

AdLIGO: no-hair theorem tests possible with coherent stacking [Yang+, 1701.05808] Third generation detectors, LISA: tests from single sources at larger redshift [EB+, 1605.09286] Planck structure at the horizon? [Cardoso+, 1602.07309]

ü Dark matter searches

Light fields of mass µ~10-12eV already excluded by O1? LISA could detect/rule out light bosons with µ~10-17eV [Brito+, in preparation] Exotic compact objects, primordial black holes [Cardoso+, 1608.08638; Cholis+, 1603.00464] Parameter space testable by LIGO/LISA: is it interesting from a HEP/cosmological point of view? [Giudice-McCullough-Urbano, 1605.01209]