Challenges for numerical relativity and gravitational-wave source modeling
Emanuele Berti, Johns Hopkins University ICERM Workshop “Advances and challenges in computational relativity” September 14 2020 Focus of this talk: what improvements in numerical relativity do we need to test GR with binary black hole mergers?
“The binary black hole problem has been solved 15 years ago”
In general relativity, for comparable-mass nonspinning nonprecessing noneccentric binaries
(and I won’t talk about neutron stars…) What keeps me up at night: Systematic errors in GR: not good enough for LISA/CE/ET spectroscopy tests sky localization Identifying “best” beyond-GR theories: specific theories vs. parametrization Punchline: NR may guide theoretical work Group
GWIC prize 2016 GWIC prize 2017 Black hole spectroscopy: a null test 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 à 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 theà 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 à 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- also agrees with that of [19,32]). As we already said, the I 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 Quasinormal (and superradiant) modes 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: Massive scalar field: the natural dynamical scale rg close to the black hole FIG. 7 (color online). The shape of the radial Schroedinger 1 7 • Ingoing waves at the horizon, • Superradiance: black hole bomb when horizon, ÀsrÀ > 10 rg. Consequently, in many cases non- potential for the eigenvalue problem in the rotating black hole linear effects, both gravitational, and due to axion self- outgoing waves at infinitybackground. Superradiant0 044026-9 Schwarzschild and Kerr quasinormal mode spectrum • One mode fixes mass and spin – and the whole spectrum! • N modes: N tests of GR dynamics…if they can be measured • Needs SNR>50 or so for a comparable mass, nonspinning binary merger [Berti-Cardoso-Will, gr-qc/0512160; EB+, gr-qc/0707.1202] 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 T´ecnico, 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 Universit´es, UPMC Universit´eParis 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. 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], main conclusionsBridging canthe be mass understood gap: gravitational relying onwave the noiseastronomy in the 2030s 16 and the detection placed some constraints on the merger power10 spectral° densities (PSDs) Sn(f) of present and fu- rates of BH binaries in the Universe [5–8]. ture detectors,17 as shown and briefly reviewed in Fig. 1, 100M 10° f = 170.2 Hz and simple back-of-the-envelope estimates. M LISA Pathfinder was successfully launched in Decem- 18 ) (Hz 21 2 and dimensionless10° spin, M and j = J/M , can be com- idea is that the dominant ` = m = 2 resonant frequency f ( putedn using22 the fittingLISA formulasAdLIGO in [26] andCE1 [27], respect- and damping time can be used to determine the rem- S 10° 2 ively (see also [28,N2A5 29]). The ringdownA+ SNRCE2⇢ widecan be es- nant’s mass M and dimensionless spin j = J/M (we p 23 timated10 by° followingN2A2 [14]. IncludingA++ redshiftCE2 factors narrow and adopt geometrical units G = c = 1 throughout this Let- N2A1 Vrt ET-B 24 ter.) In GR, all subdominant mode frequencies (e.g. the substituting10° the EuclideanO1 distanceVoyager r by theET-D luminosity distance D25L as appropriate,O2 Eq. (3.16) of [14] implies modes with ` = m = 3 and ` = m = 4 [11]) are then 10° that ⇢ is well approximated4 3 by 2 1 0 1 2 3 uniquely determined by M and j. The detection of sub- 10° 10° 10° 10° 10 10 10 10 dominant modes requires high SNR, but each mode will 1/2 f (Hz) 8 M 3✏ provide one (or more) tests of the Kerr nature of the rem- ⇢ = eq z rd , (1) D 5 S (f ) nant [12]. As first pointed out by Detweiler in 1980, grav- LFlmn n lmn [EB+, 1605.09286] itational waves allow us to do BH spectroscopy: “After 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 tion of these resonant frequencies might finally provide F ⌘ factor Qlmn(j) are given in Eqs. (E1) and (E2) of [14]. direct evidence of BHs with the same certainty as, say, The geometrical factor eq = 1 for Michelson interfero- the 21 cm line identifies interstellar hydrogen” [13]. meters with orthogonal arms, while eq = p3/2 for an Such high SNRs are known to be achievable with an eLISA-like detector (where the angle between the arms eLISA-like detector [14]. The surprisingly high SNR of is 60 ). This expression involves the non sky-averaged 33 Earth vs. space-based: ringdown detections and black hole spectroscopy 106 103 M3 M3 Q3nod 4L Q3nod 4L Q3d 4L Q3d 4L 105 M10 M10 PopIII 4L PopIII 4L M1 M1 Q3nod 6L Q3nod 6L 104 > 8 > GLRT Q3d 6L Q3d 6L PopIII 6L PopIII 6L > 8 > GLRT 103 102 102 101 events/year events/year 0 10 101 1 10 2 10 10 3 100 O1 O2 A+ Vrt ETB CE1 A++ CE2n N1A1 N1A2 N1A5 N2A1 N2A2 N2A5 ETDX CE2w Voyager AdLIGO [EB+, 1605.09286] FigureFigure 2. 2. Rates Rates of of binary binary BH BH mergers mergers that that yield yield detectable detectable ringdown ringdown signals signals (filled (filled symbols) symbols) and and allow allow for for spectroscopical spectroscopical teststests (hollow (hollow symbols). symbols). Left Left panel: panel: rates rates per per year year for for Earth-based Earth-based detectors detectors of of increasing increasing sensitivity. sensitivity. Right Right panel: panel: rates rates per per yearyear for for 6-link 6-link (solid) (solid) and and 4-link 4-link (dashed) (dashed) eLISA eLISA configurations configurations with with varying varying armlength armlength and and acceleration acceleration noise. noise. ofof BH-BH BH-BH merger merger rates, rates, and and therefore therefore model model M3 M3 should should M1M1 (M10, (M10, M3) M3) predict predict 3 3.0(2.0(2.5,.5, 0 0.57).57) events events per per year year bebe regarded regarded as as pessimistic pessimistic [8]. [8]. In In all all of of these these models models we we withwith detectable detectable ringdown ringdown in in O1; O1; 7.0 7.0 (5.8, (5.8, 1.1) 1.1) in in O2; O2; and and setset the the BH BH spins spins to to zero, zero, an an assumption assumption consistent consistent with with 4040 (35, (35, 5.2) 5.2) in in AdLIGO. AdLIGO. Model Model Q3d Q3d (Q3nod, (Q3nod, PopIII) PopIII) estimatesestimates from from GW150914 GW150914 [4]. [4]. Even Even in in the the unrealistic unrealistic predictspredicts 38 38 (533, (533, 13) 13) events events for for a a 6-link 6-link N2A5 N2A5 eLISA eLISA scenarioscenario where where all all BHs BHs in in the the Universe Universe were were maximally maximally missionmission lasting lasting 5 5 years, years, but but in in the the plot plot we we divided divided these these spinning,spinning, rates rates would would increase increase by by a a factor factor..33 (see (see Table Table numbersnumbers by by 5 5 to to facilitate facilitate a a more more fair fair comparison comparison in in terms terms 22 of of [5]). [5]). Massive Massive binaries binaries with with ringdowns ringdowns detectable detectable by by ofof events eventsperper year year.. Earth-basedEarth-based interferometers interferometers could could also also be be produced produced by by BHBH spectroscopy. spectroscopy.SupposeSuppose that that we we know know that that a a signal signal otherother mechanisms mechanisms (see (see e.g. e.g. [34–37]), [34–37]), and and therefore therefore our our containscontains two two (or (or possibly possibly more) more) ringdown ringdown modes. modes. We We ratesrates should should be be seen seen as as lower lower bounds. bounds. expectexpect the the weaker weaker mode mode to to be be hard hard to to resolve resolve if if its its amp- amp- ToTo estimate estimate ringdown ringdown rates rates from from massive massive BH BH mergers mergers litudelitude is is low low and/or and/or if if the the detector’s detector’s noise noise is is large. large. The The detectabledetectable by by eLISA eLISA we we consider consider the the same same three three models models criticalcritical SNR SNR for for the the second second mode mode to to be be resolvable resolvable can can (PopIII,(PopIII, Q3nod Q3nod and and Q3d) Q3d) used used in in [18] [18] and and produced produced with with bebe computed computed using using the the generalized generalized likelihood likelihood ratio ratio test test thethe semi-analytical semi-analytical approach approach of of [38] [38] (with (with incremental incremental (GLRT)(GLRT) [42] [42] under under the the following following assumptions: assumptions: (i) (i) using using improvementsimprovements described described in in [39–41]). [39–41]). These These models models were were otherother criteria, criteria, we we have have already already decided decided in in favor favor of of the the chosenchosen to to span span the the major major sources sources of of uncertainty uncertainty a a↵↵ect-ect- presencepresence of of one one ringdown ringdown signal; signal; (ii) (ii) the the ringdown ringdown fre- fre- inging eLISA eLISA rates, rates, namely namely (i) (i) the the nature nature of of primordial primordial BH BH quenciesquencies and and damping damping times, times, as as well well as as the the amplitude amplitude seedsseeds (light (light seeds seeds coming coming from from the the collapse collapse of of Pop Pop III III ofof the the dominant dominant mode, mode, are are known. known. Then Then the the critical critical starsstars in in model model PopIII; PopIII; heavy heavy seeds seeds originating originating from from pro- pro- SNRSNR⇢⇢GLRTGLRT toto resolve resolve a a mode mode with with either either``==mm=3=3 togalactictogalactic disks disks in in models models Q3d Q3d and and Q3nod), Q3nod), and and (ii) (ii) the the oror``==mm== 4 4 from from the the dominant dominant mode mode with with``==mm=2=2 delaydelay between between galaxy galaxy mergers mergers and and the the merger merger of of the the BHs BHs isis well well fitted, fitted, for for nonspinning nonspinning binary binary BH BH mergers, mergers, by by atat galactic galactic centers centers (model (model Q3d Q3d includes includes this this delay; delay; model model 22, ,33 1515.4597.4597 11.65242.65242 Q3nodQ3nod does does not, not, and and therefore therefore yields yields higher higher detection detection ⇢⇢GLRT == 17 17.687.687 + + ,, (2)(2) GLRT qq 11 qq rates).rates). In In all all three three models models the the BH BH spin spin evolution evolution is is fol- fol- 8383.5778.5778 4444.1125.1125 5050.1316.1316 lowedlowed self-consistently self-consistently [38, [38, 39]. 39]. For For each each event event in in the the ⇢⇢22, ,44 == 37 37.9181.9181 + + ++ ++ .(3).(3) GLRTGLRT qq qq22 qq33 catalogcatalog we we compute compute⇢⇢fromfrom Eq. Eq. (1), (1), where where✏✏rdrdisis rescaled rescaled byby a a spin-dependent spin-dependent factor factor as as necessary. necessary. TheseThese fits fits reproduce reproduce the the numerical numerical results results in in Fig. Fig. 9 9 of of DetectionDetection rates. rates.TheThe ringdown ringdown detection detection rates rates (events (events [42][42] within within 0 0.3%.3% when whenqq [1[1.01.01 100].100]. Spectroscopical Spectroscopical 2 perper year year with with⇢⇢>>88 in in a a single single detector) detector) predicted predicted by by teststests of of the the Kerr Kerr metric metric can can2 be be performed performed whenever whenever either either 2,23, 3 2,24, 4 modelsmodels M1, M1, M3, M3, M10 M10 (for (for stellar-mass stellar-mass BH BH binaries) binaries) and and modemode is is resolvable, resolvable, i.e. i.e.⇢⇢>>⇢⇢GLRT min(min(⇢⇢ ,,⇢⇢ ).). GLRT⌘ GLRTGLRT GLRTGLRT PopIII,PopIII, Q3d, Q3d, Q3nod Q3nod (for (for supermassive supermassive BH BH binaries) binaries) are are TheThe``==mm== 3 3 mode mode is is usually usually easier easier⌘ to to resolve resolve than than the the shownshown in in Fig. Fig. 2 2 with with filled filled symbols. symbols. For For example, example, models models ``==mm== 4 4 mode, mode, but but the the situation situation is is reversed reversed in in the the Multi-mode detectability: mass ratio and spin dependence Strongest spin dependence: [Baibhav+, 1710.02156] How many modes? Depends on spin. Best/worst case scenarios in LISA [Baibhav+EB, 1809.03500] SNR Including overtones is crucial, even in linear perturbation theory Leaver (1986): Green’s function in Schwarzschild. Overtones: agreement well before peak Zhang+: extension to Kerr (here for an ultrarelativistic infall along the z axis) 12 20 20 l=2, j=0 l=2, j=0.98 Numerical Numerical 15 n=0 15 n=0 n=1 n=1 n=2 n=2 E) n=3 E) 10 n=3 0 10 0 )/(m )/(m l l 5 5 Re(X Re(X 0 0 -5 -5 -10 -15 -10 -5 0 5 10 15 20 25 -15 -10 -5 0 5 10 15 20 25 t-r* t-r* 50 l=3, j=0 50 [Zhang+, 1710.02156]l=3, j=0.98 40 Numerical 40 Numerical n=0 n=0 n=1 30 n=1 30 n=2 n=2 E) n=3 E) n=3 0 “Excitation20 factors” in Kerr known0 20 )/(m )/(m 10 l “Excitation10 coefficients” depend onl initial data: 0 0 Re(X difficult, unsolved problem for comparableRe(X -10 -mass mergers -10 -20 -20 -30 -30 [Leaver, PRD, 1986] -15 -10 -5 0 5 10 15 20 25 [EB+Cardoso-15 -10 -5 0, gr 5-qc/0605118] 10 15 20 25 t-r* t-r* FIG. 3. Sasaki-Nakamura wave function for an ultrarelativistic infall along the symmetry axis of a Kerr BH. Solid black lines are results from the numerical solution of the perturbation equations; the other lines are results obtained by summing different numbers of overtones. The upper panels refer to l =2,thelowerpanelstol =3.Theleftpanelscorrespondsto the Schwarzschild limit (j =0),andtherightpanelstoafast-spinningKerrBHwitha =0.49 (j =0.98). In this plot, as everywhere else in the paper, we use units 2M =1. Ψ must be understood as the Sasaki-Nakamura wave V. CONCLUSIONS AND OUTLOOK function – against numerical gravitational waveforms ob- tained in this way. As in the Schwarzschild case, the In this paper we have implemented a new method, integrand appearing in the calculation of the Kerr ex- based on the MST formalism, to compute the excita- citation factors is, in general, divergent. The divergence tion factors B for Kerr QNMs. This method is simpler can be regularized following a procedure analogous to the q and more accurate than the method used by two of us Schwarzschild case (cf. Appendix A 2). in [2], allowing us to extend the calculation to higher Figure 3 confirms our basic findings from the nonrotat- angular multipoles l and to higher overtone numbers n. ing case: the convergence of the QNM expansion is not Tables of the excitation factors Bq in the Teukolsky and necessarily monotonic, and the excitation coefficient ex- Sasaki-Nakamura formalisms will be made publicly avail- pansion works better for higher values of l. Notice that a able online [39], in the hope to stimulate further research relatively small number of overtones is sufficient to repro- in this field. duce the numerical waveform at early times even when As a test of the method, we have computed the QNM the spin of the Kerr BH is rather large (j =0.98), so excitation coefficients for the classic problem of particles that one may in principle expect that a larger number of falling radially into the BH. We have compared the exci- overtones would be necessary (see e.g. [29, 54–57]). To tation coefficient expansion against numerical results for: our knowledge, the calculation presented in this Section (i) particles falling from rest (E = 1) into a Schwarzschild is the first concrete proof that an excitation-coefficient BH, (ii) large-energy particles (E = 10) falling into expansion is applicable and useful in the Kerr case: all a Schwarzschild BH, and (iii) ultrarelativistic particles calculations available in the literature so far were specific falling into a Kerr BH along the symmetry axis. In all to the Schwarzschild case (see e.g. [32, 33]). cases we found excellent agreement, validating the useful- 3 Systematic errors on QNM frequencies/mass+spin from SXS/PP waveforms 100 N =1 Top: N =2 1 10 N =3 real part (thick) 2 imaginary part (thin) 10 / 3 10 1% determination of needs one overtone 4 10 (better if two or three) 5 10 SXS, q =1 SXS, q =3 Point Particle 0 5 10 15 20 5 10 15 20 5 10 15 20 t22 t22 t22 t22 t22 t22 0 peak 0 peak 0 peak Bottom: FIG. 1. Fractional errors !r/!r (thick lines) and !i/!i (thin lines) between the fundamental ` = m =2QNM frequencies computed from BH perturbation theory and those obtained by fitting N overtones to numerical waveforms according to method (i) (see text). Left: SXSspin (thick) waveforms, q =1; middle: SXS waveforms, q =3; right: point-particle waveforms. mass (thin) 1% determination of mass and spin needs at least two modes [Baibhav+, 1710.02156] FIG. 2. Error in the spin af (thick lines) and fractional error in the mass Mf /Mf (thin lines) estimated by fitting N QNMs with ` = m =2 according to method (ii) (see text). Left: SXS waveforms, q =1; middle: SXS waveforms, q =3; right: point-particle waveforms. same as in Fig. 1, and errors decrease as we include more that the ` = m =2component of 4 in the SXS simulations overtones. contains a spurious decaying mode corresponding to the fun- The results in Figs. 1 and 2 disprove the claim of [26] that damental ` = m =4QNMs for q =1. We confirm their large-SNR detections cannot be used to perform BH spec- finding. Furthermore, as we show in Fig. 3, a multi-mode fit troscopy, but they also show that the relative error between of unequal-mass waveforms shows the presence of a spurious quantities computed in BH perturbation theory and those ex- frequency that matches quite well the fundamental QNM with tracted from numerical simulations currently saturates at ` = m =3. 3 ⇠ 10 . This “saturation effect” is less problematic for the qua- sicircular inspiral of point particles into Schwarzschild BHs, These spurious modes seem to be present only in the SXS where relative errors can be reduced by approximately one or- simulations. We did not find them in the public catalog of der of magnitude (we get worse agreement for point particles waveforms from the Georgia Tech group [52], nor in our falling into rotating BHs, where spherical-spheroidal mode own point-particle waveforms. Understanding the origin of mixing [41, 47–49] must be taken into account). these modes is beyond the scope of this work. We speculate This observation has an important implication: further nu- that they may be gauge or wave extraction artifacts, but they merical or theoretical work is required to reduce systematic are unlikely to come from spherical-spheroidal mode mixing, errors for comparable-mass binary BH mergers in the LISA which only mixes components with the same m and different band, that may have SNRs 103 or higher [50, 51]. `’s [41, 47–49]. Whatever their origin, these spurious modes ⇠ The saturation discussed above may be related to an unde- must be understood if we want to control systematics at the sired feature of SXS waveforms. It wass already noted in [28] level required to do BH spectroscopy with LISA. 3 Systematic errors on mass and spin from fitting SXS waveforms Overtones improve quality of consistency tests for GW150914, not a “genuine” spectroscopy test: 3 0 10 1.0 IV. RESULTS N =0 1 10 N =1 A. QNM overtone fits N =2 0.8 10 2 N =3 The linear superposition of the fundamental QNM and N =4 N overtones is an excellent description of the waveform 10 3 0.6 N =5 around and before the peak strain. To demonstrate this, f M N =6 4 we begin by fixing the remnant properties to the final 10 N =7 IMR 0.4values provided by the NR simulation. With the mass Mf 5 and dimensionless spin ft fixed,= 0 ms the set of frequencies 10 0 !22n(Mf , f ) is fully specifiedN by=0 perturbation theory. 6 0.2The only remaining free parameters in Eq. (1) are the 10 N =1 complex coe cients C22n and the model start time t0. N =2 7 For N included overtones, and a given choice of t0,we 10 0.0 20 10 0 10 20 30 40 50 determine50 60 the (70N + 1)80 complex90 C22n100’s using unweighted t t [M] 0 peak linear least squares,Mf [M thus] obtaining a model waveform given[Giesler by Eq. +,(1 )1903.08284;. We construct Isi+, such 1905.00869] a model waveform for t t at many start times beginning at t = t 25M FIG. 1. Mismatches as a function of time for the eight mod-FIG. 1. Remnant0 parameters inferred with di↵erent0 numberpeak els, each including up to N QNM overtones. The mismatchof overtones,and extending using data to times startingt0 = attpeak peak+ strain 60M,where amplitude.tpeak isFIG. 2. Measured quasinormal-mode amplitudes for a model associated with each model at a given t0 corresponds to theContoursthe representpeak amplitude 90%-credible of the regions complex on thestrain. remnant For each mass startwith the fundamental mode and two overtones (N =2).The purple colormap represents the joint posterior distribution mismatch computed using Eq. (2),betweenthemodelandthe(Mf )anddimensionlessspinmagnitude(time t0, we compute the mismatch f ),between obtained our from model N M NR NR waveform for t t0, where t0 specifies the lower limit used waveform, h , and the NR waveform, h , through for the three amplitudes in the N =2model: A0, A1, A2, the Bayesian analysis22 of GW150914. The inference22 model is in Eq. (3). Each additional overtone decreases the minimumthat of Eq. (1),withdi↵erent number of overtones N:0(solid as defined in Eq. (1). The solid curves enclose 90% of the achievable mismatch, with the minimum consistently shiftingblue), 1 (solid yellow), 2 (dashed purple).hNR,hN In all cases, the probability mass. A yellow curve in the A0–A1 plane, as well as =1 h 22 22i . (2)corresponding yellow dashed lines, represents the 90%-credible to earlier times. analysis uses data starting at peak strain ( t0 = t0 tpeak =0). M hNR,hNR hN ,h N measurement of the amplitudes assuming N = 1. Similarly, Amplitudes and phases are marginalizedh 22 22 ih over.22 22 Thei black blue dashed lines give the 90%-credible measurement of A0 contourIn is the the above, 90%-credible the inner regionp product obtained between from the two full IMR complex waveform, as described in the text. The intersection of the assuming N = 0. All amplitudes are defined at t = tpeak, waveforms, say x(t) and y(t), is defined by where all fits here are carried out ( t0 =0).Valueshavebeen dotted lines marks the peak of this distribution (Mf =68.5M , i `mn phase, C`mn = A`mn e , of which we make direct rescaled by a constant to correspond to the strain measured f =0.69). The top and right panels show 1D posteriors for use in Sec. IV C|. | T by the LIGO Hanford detector. Assuming N =1,themean Mf and f respectively.x(t) The,y(t linear) = quasinormalx(t)y(t) dt mode , models (3) with N>0 provide measurements of the mass and spin of the A1 marginalized posterior lies 3.6 standard deviations h i t0 Throughout, we focus on describing the dominant spher-consistent with the full IMR waveform,Z in agreement with away from zero, i.e. A1 = 0 is disfavored at 3.6 .Assuming ical harmonic mode in the NR simulation, the ` = m =2generalwhere relativity. the bar denotes the complex conjugate, the lowerN =2,A1 = A2 = 0 is disfavored with 90% credibility. mode.2 The natural angular basis in perturbation theory limit of the integral is the start time parameter t0 in is spin-weighted spheroidal harmonics [4–6], which can Eq. (1), and the upper limit of the integral T is chosen be written as an expansion in spin-weighted sphericalnumerical relativity to translate measured values of the to be a time before the NR waveform has decayed toremnant mass (abscissa) and spin magnitude (ordinate) harmonics [6, 46–48]. Decomposing the ringdown intobinary mass ratio q and component spins (~ , ~ )into numerical noise. For the aforementioned1 NR2 simulation,obtained by analyzing data starting at t with di↵er- spherical harmonics results in mixing of the spheroidalexpectedwe set remnantT = t parameters+ 90M. [50, 51]. We use posterior peak peak ent numbers of overtones (N =0, 1, 2) in the ringdown and spherical bases between the angular functions withsamplesThis on the procedure binary results parameters in mismatches made available as a function by the of t 0template of Eq. (1). The quasinormal-mode amplitudes the same m,butdi↵erent `’s, and this mixing increasesLIGOfor and each Virgo set of collaborations overtones; these [22 are, 52 presented], marginalizing in Fig. 1.The and phases have been marginalized over. For comparison, with f [6, 49]. For the SXS:BBH:0305 waveform, theover unavailablefigure shows component-spin that N = 7 overtones angles. provides the minimum we also show the 90%-credible region inferred from the ` = m = 2 spherical harmonic remains a good approx- Wemismatch consider explicit and at the deviations earliest from of times, the Kerr as compared spectrum to the imation for the ` = m = 2 spheroidal harmonic. The full IMR signal, as explained above. If the remnant is by allowingother overtone the frequency models. and The damping waveform time corresponding of the first to amplitudes of the spheroidal and spherical ` = m =2 su ciently well described as a perturbed Kerr black hole, overtonethe N to= di 7↵er overtone from the model no-hair and t values.0 = tpeak Underis visualized this in modes di↵er by a maximum of only 0.4%, which occurs and if general relativity is correct, we expect the ringdown modifiedFig. N2, where= 1 model, the model the overtone waveform angular is compared frequency to the NR roughly 15M after the peak of h.Thisdi↵erence is signif- waveform along(GR) with the fit residual. and IMR measurements to agree. As expected, this is not icantly smaller at the peak. The mixing is small becausebecomes !1 =2⇡f1 (1 + f1),with f1 a fractional the case if we assume the ringdown is composed solely of At face value, Fig. 1 provides us with(GR) a guide for de- higher (`,m) harmonics are subdominant for this wave-deviationtermining away the from times the where Kerr frequencya linear ringdownf1 for model any withthe longest-lived mode (N = 0), in which case we obtain form, but in a more general case, these higher harmonicsgivenNMQNMf and overtones f . Similarly, is applicable. the damping However, time is relying allowed on thea biased estimate of the remnant properties. In contrast, may play a more important role. (GR) to varymismatch by letting alone⌧1 can= ⌧ be1 deceiving.(1 + ⌧1).Fixing The n = 7f1 overtone= the ringdown and IMR measurements begin to agree with ⌧1 =decays 0 recovers away the very regular quickly,N = yet 1 analysis. Fig. 1 shows We may that then retainingthe addition of one overtone (N = 1). This is expected computethis the overtone relative still likelihood produces of small the no-hair mismatches hypothesis at timesfrom previous work suggesting that, given the network by meansbeyond of the when Savage-Dickey this mode should density no ratio longer [53 be]. numericallysignal-to-noise ratio of GW150914 ( 14 in the post-peak Resultsresolvable.. Fig. 1 Thisshows is duethe 90%-credible to overfitting regions to numerical for the noiseregion, for frequencies >154.7 Hz),⇠ we should be able to after the higher overtones in each model have su ciently decayed. We find that the turnover subsequent to the first 2 We have verified the presence and early dominance of overtones mismatch minimum in Fig. 1 is a good approximation for in other resolvable (`,m)’s in the NR waveform. when each overtone has a negligible amplitude. Mass and spin measurement with multiple modes Median errors on mass and spin combining multiple modes Sky localization and distance determination Amplitudes: Amplitude ratio: Phase difference: Relative antenna and polarization power: 8 Sky localization: the eightfold way and higher23 harmonics in the (✓, ) plane. These belong to two classes of solutions, as illustrated in Fig. 7: Assume inclination is known. Type I: the contours form a set of 8 closed rings, and Main observables: • there can be anywhere from 0 to 4 solutions at a given (top panel of Fig. 7). Type II: the contours form two ring-like structures en- • closing the north and south pole, and there are two Ignoring errors, these give solutions at any given (middle panel of Fig. 7). contours of constant These two classes of solution arise because the equations have a different number of solutions in different regions of the (QA,Q ) parameter space: ring-like solutions of Type II arise Degenerate positions when witout orbital modulation 2 1 (QA + 1) s s 1/4 of sources belonging to Type II if we assume that they functions in Fig. 5. at 7 min, magnifying the e↵ect of these features. Com- With the Frozen response, the reflected mode remains paring the two rows of the figure, we see that the higher are isotropically distributed. Notice also that the rings are exactly degenerate as anticipated in Sec. VB. All other harmonics have an e↵ect here, in making the elimination modes see their ln fall rapidly at around 2 mHz due to of secondary modes happen slightly earlier in frequency symmetric under parity (u cos ✓ u, 2⇡ ). the onset of frequency-dependencyL in the response. Al- (or time). This is expected as they start to contribute ⌘ ! ! though the frequency dependency vanishes in the limit significantly to the SNR (see Fig.4) and reach higher in [Baibhav+, 2001.10011] In practice, the rings will have finite “widths” which are f fL 0.12 Hz, note that the e↵ect is crucial even for frequency than the h22 harmonic according to (59). f/f⌧ ⇡0.02. Comparing to Fig. 5, we see that around The Full response case is essentially a superposition mainly determined by the uncertainty in ◆. L ⇡ 2 mHz the breaking of the long-wavelength approxima- of the previous two response approximations, with a [Marsat+, 2003.00357] tion still appears mild; however, the SNR starts to accu- successive onset of the time-dependency and frequency- The discussion above focused on modes with ` = m, but mulate a lot from SNR/16 40 at 2.5 h to SNR/4 160 dependency to break degeneracies. We note that over- it is also applicable to ` = m modes, with the (2, 1) mode ⇠ ⇠ being the most relevant observationally.6 The main difference FIG. 7. Top and central panels: localization contours found using relative detector amplitudes QA and phases Q for the dominant is that s = cos ◆. The (2, 1) mode also yields two families 21 (2, 2) mode. Here we consider a source at (u =cos◆, , )= of solutions, with the “phase diagram” being determined by (0.5, 60 , 60 ) and three selected values of the inclination: 45 and Eq. 29. The three Type II regions shown in Fig. 7 – which 60 (top panel) and 75 (central panel). For smaller inclinations correspond to ◆ = 60 , 75 , 80 for the (2, 2) mode – would (◆ =45 and ◆ =60 ) we get Type I contours, according to the correspond to ◆ = 36.9 , 61 , 70.3 for the (2, 1) mode. In definition in the main text. For larger inclinations (◆ =75 ) we get other words, Type II solutions are more likely for the (2, 1) Type II contours. The bottom panel shows a phase diagram of the mode: about one third of the sky gives Type II solutions for different classes of solutions in the (QA,Q ) plane for three fixed the (2, 1) mode, compared to about one fourth of the sky for values of the inclination. the (2, 2) mode. dominant modes through a mixing of spherical and spheroidal B. Localization contours using the amplitude of the (2, 1) harmonics with the same m and lower ` [1, 2, 17, 45–47]. mode The (2, 1) mode is an exception, because (i) it is not affected by mode mixing, and (ii) it can be excited to relatively large In the previous section we inferred localization contours amplitudes, especially for spinning BH binaries [15–19, 48]. 22 In this section we will focus on the localization informa- using the amplitude ratio QA = QA and phase difference 22 tion contained in the (2, 1) mode. Let us assume that the Q = Q of the dominant mode with ` = m =2, assuming that the inclination has been measured as described in Sec. III. inclination angle ◆ and the mass ratio q are known. Then a possible strategy would be to think about the two sky sen- Unfortunately we cannot extract any more information from i i the remaining modes with ` = m, because the sky sensitivity sitivities (⌦22, ⌦21) (or more precisely, the corresponding ⌦i ` 2 measurable detector amplitudes i `m ) as functions of ⌦`` sin(◆) ⌦22 for all of these modes: cf. Eq. (17). `m dL / i A / i i More information on the pattern functions F+, is encoded the corresponding antenna pattern functions (F+,F ) in each in modes with ` = m. The excitation of these modes⇥ is gen- channel [cf. Eq. (7)], and to solve these equations to⇥ determine 6 i i erally harder to quantify through numerical relativity simula- (F+,F ) in each channel. A problem with this strategy is that tions, where subdominant modes are usually contaminated by we can⇥ never obtain the antenna pattern functions themselves, Beyond GR: specific theories Testing General Relativity 14 A guiding principle to modified GR: Lovelock’s theorem In four spacetime dimensions the only divergence-free (WEP) symmetric rank-2 HigherHigher dimensions dimensions WEPWEP violations violations tensor constructed solely from the metric and its derivatives up to 2nd order, and preserving diffeomorphism invariance, LovelockLovelock theorem is the Einstein tensor plus L. theorem ExtraExtra fields fields Diff-invar.Diff-invar. violations violations Generic modifications introduce additional fields (simplest: scalars) DynamicalDynamical fields fields NondynamicalNondynamical fields fields MassiveMassive gravity gravity Lorentz-violationsLorentz-violations Minimal requirements: (SEP(SEP violations) violations) • Action principle Palatini f(R) dRGT theory Einstein-Aether • Eddington-Born-Infeld Massive bimetric Horava-Lifshitz Well-posed gravity n-DBI • Testable predictions • Black holes, neutron stars Scalars Vectors Tensors • Cosmologically viable Scalar-tensor, Metric f(R) Einstein-Aether TeVeS [Sotiriou+, 0707.2748] Horndeski, galileons Horava-Lifshitz Bimetric gravity Quadratic gravity, n-DBI [EB+, 1501.07274] Figure 2.1. This diagram illustrates how Lovelock’s theorem serves as a guide to classify modified theories of gravity. Each yellow box represents a class of modified theories of gravity that arises from violating one of the assumptions underlying the theorem. A theory can, in general, belong to multiple classes. See Table 1 for a more precise classification. 2. Extensions of general relativity: motivation and overview 2.1. A compass to navigate the modified-gravity atlas There are countless inequivalent ways to modify GR, many of them leading to theories that can be designed to agree with current observations. Cosmological observations and fundamental physics considerations suggest that GR must be modified at very low and/or very high energies. Experimental searches for beyond-GR physics are a particularly active and well motivated area of research, so it is natural to look for a guiding principle: if we were to find experimental hints of modifications of GR, which of the assumptions underlying Einstein’s theory should be abandoned? Such a guiding principle can be found by examining the building blocks of Einstein’s theory. Lovelock’s theorem [187, 188] is particularly useful in this context. In simple terms, the theorem states that GR emerges as the unique theory of gravity under specific assumptions. More precisely, it can be articulated as follows: In four spacetime dimensions the only divergence-free symmetric rank-2 tensor constructed solely from the metric gµ⌫ and its derivatives up to second differential order, and preserving diffeomorphism invariance, is the Einstein tensor plus a cosmological term. Dynamical no-hair results in scalar-tensor theories Orbital period derivative: For black hole binaries, and dipole vanishes identically Quadrupole: Result extended to higher PN orders, BH-NS, and is exact in the large mass ratio limit [Will & Zaglauer 1989; Alsing+, 1112.4903; Mirshekari & Will, 1301.4680; Yunes+, 1112.3351; Bernard 1802.10201, 1812.04169, 1906.10735] Ways around: matter (but EOS degeneracy), cosmological BCs (but small corrections), or curvature itself sourcing the scalar field: dCS, EsGB [Yagi+ 1510.02152] The EFT viewpoint Expand all operators in the action in terms of some length scale (must be macroscopic to be relevant for GW tests). Theories: sum over curvature invariants with scalar-dependent coefficients and more specifically, at order EsGB dCS (dilaton+axion) Einsteinian cubic gravity (+parity-breaking) - causality constraints [Camanho+ 1407.5597] Next order, no new DOFs [Endlich-Gorbenko-Huang-Senatore, 1704.01590] [Cano-Ruipérez, 1901.01315; Cano-Fransen-Hertog, 2005.03671. See also work by Hui, Penco…] Why Einstein-scalar-Gauss-Bonnet gravity? A loophole in no-hair theorems Horndeski Lagrangian: most general scalar-tensor theory with second-order EOMs Set: Shift symmetry: invariance under , i.e. EsGB is a special case of Horndeski and of quadratic gravity [Kobayashi+, 1105.5723; Sotiriou+Zhou, 1312.3622; Maselli+, 1508.03044] Testing General Relativity 18 action 1 S = d4xp g? [R? 2g?µ⌫ (@ ')(@ ') V (')] + S [ ,A2(')g? ] , (2.4) 16⇡ µ ⌫ M µ⌫ Z ? ? ? where g and R are the determinant and Ricci scalar of gµ⌫ ,respectively,and V (')=A4(')U( (')). The price paid for the minimal coupling of the scalar field in the gravitational sector is the non-minimal coupling in the matter sector of the action: particle masses and fundamental constants depend on the scalar field. Testing GeneralWe Relativity remark that the actions (2.3) and (2.4) are just different representations18 of the same theory: the outcome of an experiment will not depend on the chosen action representation, as long as one takes into account that the units of physical quantities 1do scale with powers of the conformal factor A [189, 217]. It is then legitimate, when S = d4xp g? [R? 2g?µ⌫ (@ ')(@ ') V (')] + S [ ,A2(')g? ] , (2.4) 16modeling⇡ a physical process, toµ choose⌫ the conformalM frame in whichµ⌫ calculations are simpler:Z for instance, in vacuum the Einstein-frame action (2.4) formally reduces to ? ? ? where g theand GRR actionare the minimally determinant coupled and with Ricci a scalar scalar field. of gµ⌫ It,respectively,and may then be necessary to 4 V (')=Achange(')U( the (')) conformal. The price frame paid when for the extracting minimal physically coupling ofmeaningful the scalar statements field in (since the gravitationalthe scalar sector field is is the minimally non-minimal coupled coupling to matter in the matter in the sector Jordan of frame, the action: test particles particle massesfollow geodesicsand fundamental of the Jordan-frame constants dependmetric, on not the of scalar the Einstein-frame field. metric). We remarkThe that relation the actions between (2.3) Jordan-frame and (2.4) and are Einstein-frame just different representations quantites is simply = 2 2 of the sameA theory:('), 3+2 the!( outcome)=↵(') of an, where experiment↵(') willd(ln notA(')) depend/d' [2]. on Note the chosen that the theory ⌘ representation,is fixed as once long the as functionone takes! into( ) –or,equivalently, account that the units↵(') of–isfixed,andtheformofthe physical quantities do scale withscalar powers potential of the is chosen. conformal Moreover, factor A most[189, phenomenological 217]. It is then legitimate, studies neglect when the scalar Testing General Relativity 18 modelingpotential. a physical This process, approximation to choose the corresponds conformalScalar to frame neglecting-tensor in whichtheory the calculations cosmologicaland spontaneous are term,scalarization the simpler: for instance, in vacuum the Einstein-frame action (2.4) formally reduces to mass of the scalar field and§ any possible scalar self-interaction. In an asymptotically the GR action minimally coupledaction withAction a scalar(in the field.Einstein It frame): may then be necessary to flat spacetime the scalar field tends to a constant 0 at spatial infinity, corresponding change the conformal frame when extracting1 physically4 ? meaningful? ?µ⌫ statements (since 2 ? to a minimum of the potential.S = Taylor expandingd xp g [RU( 2)garound(@µ' )(@yields⌫ ') V a( cosmological')] + SM [ ,A (')g ] , (2.4) 16⇡ 0 µ⌫ the scalarconstant field is and minimally a mass coupled term for to the matter scalarZ in field the to Jordan the lowest frame, orders test [32,210]. particles follow geodesics of the Jordan-frame metric,g? notR of? the Einstein-frame metric). g? Scalar-tensor theory with§whereGravity a vanishing-andmatter coupling: scalarare the potential determinant is characterized and Ricci scalar by a of singleµ⌫ ,respectively,and The relation between Jordan-frameV (')= andA4(' Einstein-frame)U( (')). The price quantites paid for is the simply minimal = coupling of the scalar field in 2 function ↵('). The2 expansion of this function around the asymptotic value '0 can be A ('), 3+2!( )=↵(') , wherethe↵ gravitational(') d(ln A sector('))/d is the' [2]. non-minimal Note that coupling the theory in the matter sector of the action: written in the form ⌘ is fixed once the function !( ) –or,equivalently,particle masses and↵(' fundamental) –isfixed,andtheformofthe constants depend on the scalar field. Testing General↵(')= Relativity↵0 + 0(' '0)+... (2.5) 19 scalar potential is chosen. Moreover, mostWe phenomenological remark that the actions studies (2.3) neglect and (2.4) the scalar are just different representations potential.As This mentioned approximation above, corresponds theof the choice same to↵ theory:(' neglecting)= the↵0 outcome the=constant cosmological of an (i.e., experiment term,!( )= thewillconstant) not depend on the chosen In the§representation, EinsteinField equations: frame, as thelong field as one equations takes into are account that the units of physical quantities mass of thecorresponds scalar field to and Brans-Dicke any possible theory. scalar A more self-interaction. general formulation, In an asymptotically proposed by Damour do scale with powers of the conformal factor A [189, 217]. It is then legitimate, when flat spacetimeand Esposito-Farèse, the scalar field tends is parametrized to a? constant by ↵00 atand spatial 1 0?[111,112]. infinity, correspondingAnother1 ? simple variant? modelingGµ⌫ =2 a physical@µ'@⌫ process,' gµ to⌫ @ choose'@ ' the conformalgµ⌫ V (')+8 frame⇡Tµ in⌫ , which calculations(2.7a) are to a minimumis massive of the Brans-Dicke potential. Taylor theory, expanding in whichU(↵ ()'around) is2 constant, 0 yields but a cosmological2 the potential is non- simpler: for1 instance,✓ in vacuum2 the Einstein-frame◆ action (2.4) formally reduces to constantvanishing and a mass and term has for the the form scalartheU( GR )= field action2 U to00( theminimally 0)( lowest ? 0 coupled1) ordersdV,sothatthescalarfieldhasamass with [32,210]. a scalar field. It may then be necessary to 2 ⇤g? ' = 4⇡↵(')T + , (2.7b) m U 00( 0).Notethatsincethescalarfield' 4ind' the action (2.4) is dimensionless, Scalar-tensors ⇠ theory with a vanishingchange the scalar conformal potential frame is when characterized extracting byphysically a single meaningful statements (since ↵the(') function ↵(') and thethe constants scalar field↵0, is minimally0 are dimensionless coupled to as matter well.' in the Jordan frame, test particles function . The expansion of this? µ⌫ function around1/2 the asymptotic2 ? ? value 0 can be The field equationswhere T offollow scalar-tensor= geodesics2( g) oftheory theSMJordan-frame( in,A thegµ Jordan⌫ )/ gmetric,µ⌫ frameis the not Einstein-frameare of the (see Einstein-frame e.g. [218], stress-energy metric). written in the form ? ? µ⌫ ? [Damour+Esposito-Farese, PRL 70, 2220 (1993); PRD 54, 1474 (1996)] tensor of matterThe fields relation and betweenT = g Jordan-frameTµ⌫ (see e.g. and [34]). Einstein-frame Eq. (2.7b) shows quantites that ↵ is(' simply) = [219]) ↵(')=↵02 + 0(' '0)+... 2 (2.5)1 couples theA scalar('), 3+2 fields !( to )= matter↵(') [220], , where as does↵('(3) +d 2(ln!( A))( '))in/d' the[2]. Jordan Note frame:that the theory ⌘ As mentioned above,8⇡ the!cf.( choice Eq.) (2.6b)].is↵ fixed(')= once1 ↵ the0 function=constant!( )1 (i.e.,–or,equivalently,!( )=constant)↵(') –isfixed,andtheformoftheU( ) G = T + @ @ g @ @ + ( g ) g , correspondsµ⌫ to Brans-Dicke µ⌫ theory.2 Astrophysicalµscalar A more⌫ potential observations general2 µ⌫ is chosen. formulation, set Moreover, bounds r proposedµ onr most⌫ the phenomenological parameter byµ⌫ Damour⇤g space 2 studiesof scalar-tensorµ⌫ neglect the scalar and Esposito-Farèse, is parametrizedtheories.✓ potential. In by the↵0 caseand This of approximation0 Brans-Dicke[111,112].◆ theory, Another corresponds the simple best to neglecting observational variant the bound cosmological (↵0 < term, the 3 (2.6a) 3.5 10 mass) comes of the from scalar the field Cassini and measurement any possible scalar of the self-interaction. Shapiro time delay. In an In asymptotically the is massive Brans-Dicke theory,⇥ in which ↵(') is constant, but the potential is non- 1 more generalflat1 spacetime case@T withd the !0 2 scalar=0,currentconstraintson field tendsdU to a constant(↵ 00, at0) spatialhave been infinity, obtained corresponding vanishing⇤ andg = has the form U(8 ⇡)=T 216U 00⇡ ( 0)( 0)@6 ,sothatthescalarfieldhasamass @ + 2U( ) , (2.6b) 2 3+2!( )by observations to a minimum of@ NS-NS ofd the and potential. NS-WD binaryTaylord expanding systems [33],U( and) around will be 0 discussedyields a cosmological in m U 00( ).Notethatsincethescalarfield' in the action (2.4) is dimensionless, s ⇠ 0 Section✓ 6constant (cf. Figure and 6.3). a mass Observations term for the of scalar compact field◆ tobinary the lowest systems orders also [32,210]. constrain the function ↵(')µand⌫ the constants1/2 ↵0, 0 are dimensionless as well. where T = 2(massiveg) Brans-Dicke SMScalar-tensor( ,gµ theory,⌫ )/ g theoryµ⌫ leadingis the with to Jordan-frame exclusiona vanishing regions scalar stress-energy in potential the (↵0,m is characterizedtensors) plane [32]. by a single The field equations of scalar-tensor µ⌫ theory↵(') in the Jordan frame are (see e.g. [218], ' of matter fields, and TAn= function interestingg Tµ⌫ . feature. The expansion of scalar-tensor of this function gravity around is the the prediction asymptotic of certain value 0 can be [219]) characteristicwritten physical in the phenomena form which do not occur at all in GR. Even though we know from observations that ↵0 ↵1('and)= that↵0 + GR 0(' deviations'0)+... are generally small, (2.5) 8⇡ !( ) 1 1 ⌧ U( ) Gµ⌫ = Tµ⌫ + @µ @these⌫ phenomenaAsgµ mentioned⌫ @ @ may lead above,+ to( observable theµ ⌫ choice g consequences.µ↵⌫(')=g ) ↵ = Thereconstantgµ⌫ , are (i.e., at least!( three)=constant) 2 2 r r ⇤ 0 2 ✓ possible smokingcorresponds guns to◆ of Brans-Dicke scalar-tensor theory. gravity. A more The general first is formulation, the emission proposed of dipolar by Damour gravitational radiation from compact binary systems [218,221], which will be discussed and Esposito-Farèse, is parametrized by ↵0 and 0(2.6a)[111,112]. Another simple variant in Section 5.1. Dipolar gravitational radiation is “pre-Newtonian,” i.e. it occurs at 1 @Tis massived! Brans-Dicke dU theory, in which ↵(') is constant, but the potential is non- g = 8⇡T 16lower⇡ PN order than@ @ quadrupole + radiation,2U( ) and1 , it does not exist(2.6b)2 in GR. The second is ⇤ vanishing and has the form U( )= 2 U 00( 0)( 0) ,sothatthescalarfieldhasamass 3+2!( ) @ 2 d d ✓ the existencem ofU non-perturbative00( ).Notethatsincethescalarfield NS solutions◆ in which' thein scalar the action field (2.4) amplitude is dimensionless, is s ⇠ 0 µ⌫ 1/2 finite eventhe for function↵0 1↵.( This') andspontaneous the constants scalarization↵ , arephenomenon dimensionless [111, as 112] well. will be where T = 2( g) SM ( ,gµ⌫ )/ gµ⌧⌫ is the Jordan-frame stress-energy0 0 tensor discussedµ⌫ in detailThe field in Section equations 4.2. of Here scalar-tensor we only remark theory that in the spontaneous Jordan frame scalarization are (see e.g. [218], of matter fields, and T = g Tµ⌫ . would significantly[219]) affect the mass and radius of a NS, and therefore the orbital motion of a compact binary system, even far from coalescence. The third example is 8⇡ !( ) 1 1 U( ) also non-perturbative,G = T and+ it involves@ @ massive g fields.@ @ The + coupling( of massiveg scalar ) g , µ⌫ µ⌫ 2 µ ⌫ 2 µ⌫ rµr⌫ µ⌫ ⇤g 2 µ⌫ fields to matter in orbit around✓ rotating BHs leads to a◆ surprising effect: because of superradiance, matter can hover into “floating orbits” for which the net gravitational (2.6a) energy loss at infinity1 is entirely provided@ byT thed! BH’s rotationaldU energy [222]. = 8⇡T 16⇡ @ @ + 2U( ) , (2.6b) The⇤ phenomenologyg 3+2!( ) of scalar-tensor @ theory d in vacuum spacetimes,d such as BH spacetimes, is less interesting.✓ When the matter action SM can be neglected,◆ the µ⌫ 1/2 Einstein-framewhere formulationT = 2( ofg) the theorySM ( ,g isµ equivalent⌫ )/ gµ⌫ is to the GR Jordan-frame minimally coupledstress-energy to tensor µ⌫ ascalarfield.BHsinBergmann-Wagonertheoriessatisfythesameof matter fields, and T = g Tµ⌫ . no-hair theorem as in GR, and thus the stationary BH solutions in the two theories coincide [51, 54]. Moreover, dynamical (vacuum) BH spacetimes satisfy a similar generalized no-hair theorem: the dynamics of a BH binary system in Bergmann-Wagoner theory with vanishing potential are the same as in GR, up to at least 2.5 PN order for generic mass ratios [223] and at any PN order in the extreme mass-ratio limit [224] (see Section 5.1.1). These no-hair theorems will be discussed in Section 3.2. Tensor-multi-scalarTensor-multi-scalar theories theories 3 3 Tensor-multi-scalar theories 3 Tensor-multi-scalar1. IntroductionTensor-multi-scalarTensor-multi-scalar theoriesTensor-multi-scalar1. Introduction theories theories theories 3 33 3 1. Introduction ⇤g⇤ ' = 4⇡↵(')T ⇤ g ' = 4⇡↵(')T ⇤ (1.1) (1.1) 1. Introduction1.1. Introduction Introduction ⇤ ⇤ Tensor-multi-scalar1. theories Introduction 3 ⇤g⇤ ' = 4⇡↵(')T ⇤ (1.1) g⇤ ' ↵=(')=4Scalarization⇡↵ (g'g'')⇤T'=⇤= 4⇡↵threshold4⇡↵('(')T)T⇤ ⇤: a back-of-the(1.1)-envelope argument(1.1)(1.1) ⇤ ⇤⇤0⇤ ⇤g⇤ '↵=(')=4⇡↵ 0('')T ⇤ (1.2) (1.1)(1.2) 1. Introduction ↵(')=3 m0' 3 m (1.2) 4 ↵(')= 0' ↵4↵('(')=)= 0 '0' (1.2) (1.2)(1.2) T ⇤ = A (⇤✏⇤g⇤ '3=p⇤) T4⇡↵⇤ =('A)T(✏⇤⇤ for3↵p(⇤'))=r '0 m/R' =R0.2=⇡R/2 ⇡ /R2R4 ⇡/⇡2R/ 2 ⇡/42 (1.7)(1.6)(1.8)(1.7) (1.7)(1.7) (1.7)(1.8) c ⇠cos(⇠R) ⇠ ⇠ 0 ⇠⇠ ⇠⇠ [Leonardo: Introduction[Leonardo: extended] IntroductionModifications extended] of generalModifications relativity[Damour+Esposito of general (GR) relativity-Farese, PRL (GR) 70, 2220 (1993)] generically lead to thegenerically introduction lead to of the additional introduction 4 degrees4 of4 additionalof4 freedom degrees [?]. The of simplest freedom(1.8) (1.8) [?]. 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These strongly These effects constrained e areffects strongly are strongly by constrained constrained by by of compactaffect the starsbinary masses can pulsar admit and observations radii non-perturbative of neutron [?], but stars. solutions could These still with eff leaveects large are a signature amplitudesstrongly constrainedin the of the late inspiral by of binary pulsarbinary observationsbinary pulsar pulsar[? observations], but observations could [ still?], but leave [?], could but a signature could still leave still in a leave the signature late a signature inspiral in the of in late the inspiral late inspiral of of scalarbinary fields.compact pulsar This observations phenomenom, binaries, the [? so-called], called but couldspontaneous dynamical still leave scalarization scalarization? a? signature,cansignificantly [?, in?], the which late could inspiral be observedof compact binaries,compact thecompact so-called binaries, binaries, dynamical the so-called the so-calledscalarization dynamical dynamical [ scalarization, ], which scalarization could [?,?], be [which? observed,?], which could could be observed be observed byaffect advancedcompact the massesby gravitational-wave binaries, advanced and radii the gravitational-wave of so-called neutron detectors. dynamical stars. detectors.These scalarization effects are [? strongly,?], which constrained could be observed by by advancedby advanced gravitational-wave gravitational-wave detectors. detectors. binaryby pulsar advanced observations gravitational-wave [?], but could detectors. still leave a signature in the late inspiral of compact binaries, the so-called dynamical scalarization [?,?], which could be observed by advanced gravitational-wave detectors. No-hair conditions in EsGB Matter: zero in vacuum GB contribution Kerr is a solution with constant scalar field if: Dilatonic theories and shift-symmetric theories do not have a GR limit! No-hair theorem: in addition, [Silva+, 1711.02080] A proof, and a heuristic argument Integrate by parts, divergence theorem: The RHS vanishes for stationary, asymptotically flat spacetimes; if both terms on the LHS vanish separately, i.e. In alternative, linearize the scalar field equation: is an effective mass for the perturbation – tachyonic instability? [Silva+, 1711.02080] EsGB: black hole scalarization and other solutions Einstein-dilaton-Gauss-Bonnet: [Mignemi-Stewart 93, Kanti+ 96, Kerr not a solution, minimum BH mass Pani-Cardoso 09, Yunes-Stein 11…] Shift-symmetric Gauss-Bonnet: [Sotiriou-Zhou 14, Barausse-Yagi 15, Kerr not a solution! Benkel+ 16…] Minimal scalarization model: [Silva+ 1711.02080] Nonminimal scalarization model: [Doneva+ 1711.01187] Polynomial, inverse polynomial, logarithmic… [Antoniou+ 1711.03390/07431; Brihaye-Ducobu 1812.07438…] Radial instability of the minimal scalarization model Are the solutions stable under radial perturbations? Dashed/dotted: stable (no unstable modes, or positive potential); solid: unstable : Schwarzschild is stable; threshold is coupling-independent Intermediate mass: nodeless scalarized BHs with exponential coupling are stable No BHs are stable below a certain mass (as in EdGB) (scalar charge Q) charge (scalar [Blazquez-Salcedo+, 1805.05755] What about spin? • For each spin J, scalarized BHs exist between two critical mass values (lowest bound: zero in static limit) • Scalarized black holes are entropically favored • Spin reduces difference between scalarized/unscalarized solutions: differences nearly unmeasurable for (LIGO range!) • Coupling dependence? [Cunha+, 1904.09997] Spin-induced scalarization • Take a second look at • Gauss-Bonnet invariant changes sign for ! Spin-induced instability • Even for quadratic coupling, spin-induced scalarized black holes are entropically favored over Kerr • Small differences in charge, mass etc between quadratic and exponential coupling • Quasinormal modes? May be easier via time evolutions • Dynamical stability? [Dima+, 2006.03095; Herdeiro+, 2009.03904; EB+, 2009.03905] Black hole thermodynamics, skeletonization and the two-body Lagrangian • Analytical solutions for generic coupling functions (up to fourth order in the EsGB coupling constant) satisfy the first law of black hole thermodynamics • Skeletonization à la Eardley: slow inspiral means that black holes evolve with constant Wald entropy and a nontrivial asymptotic value of the scalar field • Padé resummation seems to correctly predict poles in coupling of black holes to scalar fields • EsGB-induced corrections to the (conservative) two-body Lagrangian: formally 1PN contribution, but for small coupling, effectively a 3PN term “Miraculously”, no regularization is needed to solve the EOMs at 1PN: “simple” Fock integral • d+1 formulation: nonlinear canonical momenta, multivalued ADM Hamiltonian, strong-field breakdown [Julié+EB 1909.05258, 2004.00003; Witek+ 2004.00009; Ripley+Pretorius] Binaries in Einstein-scalar-Gauss-Bonnet 14 • Post-Newtonian calculations in the weak coupling limit: Interestingly, while the scalar signal for l 2 is qualita- [Yagi+, 1110.5990] tively similar to the gravitational waveform and displays Higher-order in coupling, generic EsGB: the classical chirp, the dipole is qualitatively di↵erent. As shown in Fig. 7, the frequency of the dipole mode [Julié+, 1909.05258] grows as expected during the merger, but the amplitude remains almost constant. This is a strong-field behavior • Dynamical scalarization: that is not captured by the PN approximation. A po- tential explanation of this behaviour is that the scalar [Khalil+, 1906.08161] configuration ceases to be dominantly dipolar before the merger, i.e. the dynamical evolution of the scalar is more • Numerical simulations (weak coupling limit): complex and it involves additional oscillations and recon- Scalar waveforms figuration. Our simulations, and in particular the time evolution of the scalar distribution, do seem to be con- Scalar-led QNMs + gravitational-led QNMs sistent with this explanation, though limitations in reso- [Witek+, 1810.05177] lution do not allow us to make a conclusive statement. In the post-merger phase the background approaches • a stationary spinning BH, so we expect to observe the Related work in dynamical Chern-Simons (weak coupling limit): same multiple ringing discussed in the previous section Scalar and gravitational waveforms FIG. 6. (Color online) Scalar waveforms, rescaled by the for an isolated BH. This is confirmed in the insets of [Okounkova+ 1705.07924, 1906.08789] extraction radius Rex =100M, sourced by an equal-mass, Figs. 6 and 7 and by the postmerger ringdown frequen- nonspinning BH binary whose waveform 4,22 is displayed in the bottom panel for comparison. tˆ =0M indicates the cies extracted from the scalar waveform using the two- merger time. We show the l = m =2(toppanel)andl = mode fit (60) and presented in Table IV. Note that, in m = 4 (mid panel) modes of the scalar field. During the contrast to the single BH case, the background is now a inspiral phase we also display the PN waveform (black dashed perturbed BH plus gravitational radiation, both of which lines, see Appendix B 2). In the right panels we zoom in on modify the source term of the scalar field. In particular, the merger-ringdown phase and observe a modulation due to gravitational radiation seems to cause an enhancement of the presence of both scalar-led and gravitational-led modes. the gravitational-led quadrupole modes, which dominate over the scalar-led one in some configurations (see, e.g., the l = m = 2 case for q =1, 1/2 in Table IV). Finally, the scalar field monopole for the same values dx =1.8M and dx =1.0M. We estimate the numer- c h of the mass ratio is shown in Fig. 8. The pre-merger ical error to be about (i) / 1.5% in the 4,22 4,22 . amplitude is larger for smaller values of the mass ratio q, gravitational waveforms; (ii) / 1.5% in the 22 22 . while the final amplitude is approximately independent scalar waveforms; and (iii) 0.5% in both the gravita- . of q. This behaviour can be understood noting that the tional and scalar phases. The corresponding convergence post-merger amplitude is, as a first approximation, plot is shown in Fig. 5. ↵ /(2Mr) (see App. B 1 a), where (by construction) the' Waveforms: We present the background gravita- GB total mass M is the same in all cases, i.e.,independent tional waveform and the (✏) scalar waveform for bi- of the mass ratio. Instead, when the two BHs are well naries with mass ratio q =O 1 and q =1/2, 1/4 in Figs. 6 separated, the scalar field amplitude is and 7, respectively. All presented waveforms are shifted in time such that t/Mˆ =(t tmerger Rex)/M =0 ↵ ↵ ↵ 1 GB + GB = GB , (63) indicates the maximum in the dominant gravitational '2m1r 2m2r 2Mr ⌘ mode as measure for the time of merger. The wave- forms exhibit the typical morphology: a sinusoid with for su ciently large radii encompassing the entire binary. increasing frequency that is driven by the orbital mo- Therefore, the ratio between the pre-merger and the post- tion of the BHs, the highly nonlinear merger followed by merger amplitude is expected to be determined by the the exponentially damped ringdown. During the inspi- (inverse of the) symmetric mass ratio ⌘. In particular we ral we compare the numerical results to the analytical have 1/⌘ = 4, 4.5 and 6.25 for mass ratios q = 1, 1/2 expressions obtained at leading PN order in [24] (see Ap- and 1/4. These values are in agreement with Fig. 8. pendix B 2). We remark that these expressions depend Energy and momentum fluxes: Next, we inves- on the time-dependent orbital frequency ⌦(t), which, at tigate the energy radiated in gravitational and scalar this PN order, can not be obtained with good approxi- waves. We compute their energy fluxes using (52)– (53) mation. Therefore, we extract the orbital frequency from with (56), i.e. accounting for both the canonical scalar’s the numerical data (measuring, at each half-cycle, the and Gauss–Bonnet contributions to the energy flux. We wavelength of the gravitational waveform). Within this furthermore estimate the total radiated energy by inte- approach, which is similar to that used in [43], the com- grating Eqs. (52) and (53) in time, measuring it at dif- parison between PN and numerical results concerns the ferent extraction radii and performing the extrapolation amplitudes of the scalar waveforms, while their phases GW GW E /M =E /M + B/Rex , (64) agree by definition. 1 Well posedness • Papallo-Reall: EsGB not strongly hyperbolic in the generalized harmonic gauge [Papallo-Reall, 1704.04730] • Ripley-Pretorius: evidence for hair in spherical collapse in the weak-coupling limit, but well-posedness issues in spherical collapse in shift-symmetric EdGB “there are open sets of initial data for which the character of the system of equations changes from hyperbolic to elliptic type in a compact region of the spacetime […] it is conceivable that a well-posed formulation of EdGB gravity (at least within spherical symmetry) may be possible if the equations are appropriately treated as mixed-type” [Ripley-Pretorius, 1902.01468] [Ripley-Pretorius, 1903.07543] • Does Horndeski generally lead to caustics? Do global solutions exist? [Babichev 1602.00735] [Bernard-Lehner-Luna, 1904.12866] Time evolution of the full Einstein-scalar-Gauss-Bonnet theory? • Built action generalizing Gibbons-Hawking-York and Myers to ESGB theories • ADM Lagrangian and Hamiltonian and d+1 decomposition generalized to EsGB • Canonical momenta of ESGB theories are nonlinear in the extrinsic curvature: “accelerations” are functions of metric, scalar field and their first derivatives • Implications: 1) the ADM Hamiltonian is generically multivalued, and the associated Hamiltonian evolution is not predictable 2) the d+1 equations of motion are quasilinear and may break down in strongly curved, highly dynamical regimes • Ripley-Pretorius: evidence for hair in spherical collapse in the weak-coupling limit, but well-posedness issues in spherical collapse in shift-symmetric EdGB [Julié+EB, 2004.00003; see also Witek+, 2004.00009] Specific Theories Summary • EsGB: subclass of Horndeski theory that evades no-hair theorems • Scalarized solution exist, are radially stable (as long as backreaction is included), can differ sensibly from GR • Stable, nonspinning scalarized solutions are well motivated in EFT • Scalarized solution become close to GR for spins of interest to LIGO remnant - more interesting phenomenology for spin-induced scalarization • BHBs produce dipolar radiation [Yagi+ 1510.02152; Julié+, 1909.05258] • Binaries have been simulated in the weak-coupling limit [Witek+ 1810.05177] • Full merger? Open issues with well posedness in the strong-coupling limit [Papallo-Reall, Ripley-Pretorius, Bernard+, Julié+EB…] Beyond GR: parametrization Extreme gravity tests with gravitational waves… Page 13 of 45 46 data. Indeed, a similar approach was successfully pursued when carrying out tests with Solar System observations, which led to the development of the parameterized PN framework of Will and Nordtvedt [152–155]. The first attempt at such a generic test consisted of verifying the PN structure of the waveform phase [156]. The idea was to decompose the Fourier-domain waveform model into a frequency-dependent amplitude and a frequency-dependent phase, and to then rewrite the phase as2 n 7 = 5 n Ψ ( f ) αnv( f )− + , (18) = n 0 != where αn are PN coefficients, which in GR are known functions of the parameters of the binary (to be more precise , the individual masses m1 and m2 for non-spinning black hole binaries), and v( f ) (πmf)1/3 is the orbital velocity, with m the binary’s total mass. The proposal was then= to treat all of these coefficients as independent and find the best-fit values by comparing the above template waveform with the data. One can then draw error regions of each coefficient in the m1–m2 plane assuming GR is correct to check for consistency, namely to check if there is a region where all of error regions overlap. Later the authors only considered three out of eight coefficients, so that correlations among parameters could be reduced and one could carry out a stronger test by shrinking the error regions [157,158]. This procedure resembles binary pulsar tests in the parameterized post-Keplerian formalism [7,159]. Although feasible in principle, the above test has a few limitations. First, it has the strong bias of assuming Nature follows the same exact functional structure of the PN approximation in GR, i.e. that the Fourier phase can be expressed as a series in 5 integer powers of velocity, with the leading-order term starting at v− .Indeed,many examples of modified gravity effects and modified gravity theories exist which do not 7 admit this structure; examples of this include dipole emission ( v− ), variability of 13 ∝ 7.3 the fundamental constants ( v− ), parity violation in eccentric binaries ( v− ), ∝ 9.3 ∝ and massive gravitons in eccentric binaries ( v− ), to name a few. Second, the framework does not allow for tests of modified gravity∝ theories that lead predominantly to amplitude modifications, without affecting the phase evolution much; examples of this include gravitational birefringence [160–162]. Third, the framework assumes that polynomials in velocity are a good basis to expand the Fourier phase during the entire inspiral, including right up to plunge and merger. Today, we know that this is not the case, with the series requiring arctangent corrections [163,164]. An extension and generalization of this method that resolves all of the above prob- lems is theInspiral parameterized: GR solution post-Einsteinianknown, parametrized (ppE) approachpost [165-Einstein]. In this framework, one extends the GR waveform model via b a iΨGR( f ) iβppE v( f ) h( f ) AGR( f ) 1 αppE v( f ) e + , (19) ˜ = ˜ + " # 2 The terms α5 and α6 contain contributions that depend on ln v,whichtheauthorstreatasconstantin [156]. In their follow-up papers [157,158], they modified Eq. (18)byaddingfurthertermsoftheform k αn,l ln v. $ 123 [Yunes-Pretorius+, 0909.3328] 46 Page 16 of 45 E. Berti et al. Mapping to theories – can we do the same for ringdown? Table 2 Mapping of ppE parameters to those in each theory for a black hole binary Theory βppE b Scalar–tensor 5 φ2η2/5 m sST m sST 2 7 1792 ˙ 1 1 2 2 [36,179, − − − 180] ! " 2 m2sGB m2sGB EdGB, D2GB 5 ζ 1 2 − 2 1 7 7168 GB # m4η18/5 $ [23] − − 1549225 ζCS 231808 2 16068 2 dCS [181] 1 η χ 1 η χ 2δm χs χa 1 11812864 η14/5 − 61969 s + − 61969 a − − %# $ #2 $ & 3 c14 1 2c14c 3c14 EA [182] 128 1 2 Æ + 2 Æ Æ 1 5 − − w + (c c c c ) w + 2w (2 c14) − − ' ( 2 ++ −− − + 1 0 − ) * ! " 3 1 3βKG Khronometric 128 (1 βKG) KG KG 1 5 − − w 2w (1 βKG) − − [182] ' ( 2 0 − ) * 25 dm 3 26η 34η2 Extra − + 13 851968 dt η2/5(1 2η) − dimension # $ − [183] Varying G 25 G 13 65536 ˙ [151] − M − 2 αMDR D 1 αMDR Mod. disp. rel. π − αMDR − 3(α 1) (1 α ) 2 αMDR M 1 αMDR MDR MDR λ − (1 z) − − [184] − A + In scalar–tensor theories, black holes acquires a scalar charge for a cosmologically evolving scalar field [179,180]. Such a scalar charge is proportional to sST 1 (1 χ2 )1/2 /2. sGB is related to the black hole A ≡ [ + − A ] A scalar charge µ in D2GB in Eq. (37)asµGB 2(α /m2 )sGB. The dimensionless coupling constant in A = GB A A 2 4 Æ,KG quadratic-curvature theories is defined by ζGB,CS 16πα /m . Propagation speeds w in Lorentz- = GB,CS i violating theories are summarized in Table 1. dm/dt dm1/dt dm2/dt can be calculated from Eq. (17). 1/(α 2) = + λ h A ,whereh is the Planck constant. The distance DαMDR is defined in Eq. (22) A ≡ − the results of [191–193]tonon-spinningbinarysystemsonquasi-circularorbitsin scalar–tensor gravity at 2PN relative order.3 Such waveforms introduce PN corrections to the mapping presented in Table 2.JuliéandDeruelle[195,196]usethesehigherorder PN results to begin to extend the effective-one-body (EOB) formalism of Buonanno and Damour [197]toscalar–tensorgravity.Suchresummedwaveformmodelscannot be analytically mapped to the ppE waveforms directly. All the mappings in Table 2 (except for the last one) originate from non-GR effects created at the level of generation of GWs, while such waves in general acquire modi- fications also at the level of their propagation. The dispersion relation of the graviton 3 By imposing the stringent constraints set by current astrophysical observations (cf. Table II of [194]), they find that dipolar radiation is subdominant to quadrupolar radiation for most prospective GW sources: in the absence of spontaneous scalarization, the dipole term can dominate only at frequencies f ! 100 µHz in binary neutron star or neutron-star/stellar-mass-black-hole systems, and at frequencies f ! 5 µHz in neutron-star/intermediate-mass-black-hole systems. Therefore, ground- and space-based GW detectors would only observe binary systems whose inspiral is driven by the next-to-leading order flux. 123 Scalar, electromagnetic and gravitational perturbations in GR Gravitational perturbations of a Schwarzschild BH: Regge-Wheeler/Zerilli equations Isospectrality: the odd/even potentials have the same quasinormal mode spectrum [Chandrasekhar-Detweiler 1975] Scalar, electromagnetic and (odd) gravitational perturbations: [e.g. EB+, 0905.2975] Generic (but decoupled) corrections to GR potentials Modifications to the gravity sector and/or beyond Standard Model physics: expect • small modifications to the functional form of the potentials – parametrize! • coupling between the wave equations (more later) Maximum of is , so corrections are small if: [Cardoso+, 1901.01265] VITOR CARDOSO et al. Correction coefficients andPHYS. their REV. asymptotic D 99, 104077 (2019) behavior QNM frequency correction coefficients by direct integration [Pani, 1305.6759] Asymptotics: Damped oscillatory behavior for large j PARAMETRIZED BLACK HOLE QUASINORMAL RINGDOWN: … PHYS. REV. D 99, 104077 (2019) FIG. 1. Real and imaginary parts of the components e− defined in (10) for j 1; …; 20 and odd-parity gravitationalFitting perturbations. the numerics by since the amplitude of fδVj is proportional to 1=j for large j ¼ j, we can expect that ej → 0 for j → ∞. Following [32],a III. RESULTS The linearized superpartner relations now yield perturbative expansion of the QNM frequency yieldsThe the most important results of our analysis are the s dδW numerical values of the complex factors ejÆ and dj, which 2 δV− − δV ; 14 following correction: dr ¼ þ ð Þ were obtained via high-precision direct integration of the confirms this. relevant equations. Figure 1 and Table I show representa- W0 s ∞ tive results for the coefficients e and d in (10) and (11) 4 δW δV δV−; 15 2 2 jÆ j f ¼ þ þ ð Þ ejαjÆ ∝ dr frHδVjΦ0; for odd19 gravitational and scalar modes, respectively. à ð Þ Z−∞ Results for all fields and the lowest multipole numbers with δW the induced change in the superpotential W with l s , s 1 ; …; 10 are available online [30]. Our respect to its GR value W0 [Eq. (13)]. [Pani, 1305.6759] ¼ j j j þ j s − where 0 is the unperturbed mode itself. calculated ejÆ (and dj) have a five-digit accuracy for j 30. Let us admit that the theory predicts a set of αj . The Φ ≲ relations above give− us a unique possibility for αþ, namely, Let us assume that fδVj has support only in FIG. 2. Values of the frequency components ej (red dots) for j A. Isospectrality r ln j r σ r r ln j r σ, with a constant the odd tensor modes with l 2, compared against the− numeri- − − H − H ≤ ≤− H H αþ α ; αþ α ; 16 à þ A remarkable resultcal fit in GR of concerns Eq. (22) the(black isospectrality dashed¼ of curve). 0 ¼ 0 1 ¼ 1 ð Þ σ O 1 (the particular choice of σ does not affectpotentials our(2) and (3). These potentials are sometimes ¼ ð Þ 6 α− α− results). Then the integral can be estimated as follows:referred to as “superpartners,” since they can both be − 0 − 1 α2þ α2 ð Þ ; …: 17 obtained from a “superpotential” W [4,31]: ¼ þ λ λ 2 ð Þ such that a0 mapping exists between the α andð þ theÞ −r ln j r σ jÆ H þ H 2 dW λ2 λ 2 2 These relations showa that isospectrality is very fragile, and e α dr fδV fV W2 f 0 ; 12 2 j j jÆ ∝ jΦ0 coefficients0 − ofð theþ2 Þ theory aj, i.e.,that inα generalÆ r it willj . be To broken. study Ã Æ ¼ ∓ dr 36r ð Þ j H r Z−rH ln j−rHσ H ¼ H the convergence of the QNM expansionIV. ASYMPTOTIC(10), we BEHAVIOR can AND −r ln j r σ 3rH rH − r λ λ 2 H H αjÆ 2 2 W ð Þ − ð þ Þ : 13 CONVERGENCE OF SERIES þ − r rH ln j = 2rH −i2ω0r 0 2 dr e ð Ãþ Þ ð Þe à ¼computer 3rH λr the ratio6rH ð Þ ð þ Þ “ ” ¼ −rH ln j−rHσ à ej Having defined the basis vectors ej, it is crucial to Z When the potentials are perturbed from their GR values investigate their behavior, and the overall convergence of α as in Eq. (6), isospectrality will in general be broken, unless αnÆthe1e QNMn 1 frequency expansion (10). For large j, the jÆ 2ir ω j H 0 × constant: the coefficients20 α are related in some wayϒ to α−. lim þ þ : 23 jþ ¼j n→∞ αcontributione to the potential fromð Þ the jth term can be ¼ j ð Þ describednÆ n by a Gaussian profile depending on the tortoise TABLE I. Real and imaginary part of a few of the first coordinate 2 r , i.e., − ⋆ frequency componentsThe for series odd-parity converges gravitational when (ej ) andϒ < 1. Replacing the explicit In conclusion we find 0 scalar field (d ) perturbations with l 2. The full set of j j rH αjÆ 2 2 ¼ 2 − r rH ln j = 2rH frequencies up to jform100 is of provided the expansion online [30]. coefficientsfrαHjÆδV,j andfαjÆ using≃ thee factð Ãþ Þ ð Þ; 18 ¼ ¼ r ej ð Þ 2iω0rH 2irHωR−2rHωI 2irHωR ln j — j j e jrthate− the frequency basisr d0 vectors ej behave as in Eq. (22) ej ∝ ; 21 H j H j showing that the location of the peak is proportional to 1 2rHωI or equivalently Eq. (21)—we find j ¼ j ¼ j þ 0 ð 0.Þ24725 0.092643i 0.15782 0.054078i − ln j (in terms of the tortoise coordinate r ). Moreover, 1 0.15985 þ 0.018208i 0.11307 þ 0.015119i à 2 0.096632 þ 0.0024155i 0.076570 þ 0.00016782i − β 2 where ω0 ωR iωI. Equation (21) shows that the3 basis0.058491 0.0037179i a0n.0511211 nþ 0.0032973sin γ lni n The1 Gaussianζ fit (18) representsan a1 good approximation for the − − 2 ¼ þ 1 2r ω 4 0.036679 0ϒ.00043870limi 0.034527þ 0.0024724½ i ð peakþ ofÞþ the potential ’slim modificationsþ fr: HδVj. Although its pre- frequency e decays as 1=j H I , and oscillates as − n β− 1 n j þ 10 0.0036853 0.0065244¼ →i ∞ 0.a0050350n n þ 0.0037363rH sini γ lncisionn is limitedζ for¼ large→ values∞ ar n, far from the maximum of the 3 þ þ ½ function,þ the fit is accurate enough ⋆ for the purposes of this section. sin 2rHωR ln j . For tensor perturbations with l 2, ð Þe1.5i ln j ¼ The previous equation implies that, in our approach, the we have ej ≃ 0.64 . This result is also confirmed by a j convergence of the QNM frequencies depends on the numerical study of e for large j. We fitted our large-j 104077-4 j behavior of the coefficients of the (expanded) potential numerical results to the following functional form: for the specific theory of gravity. κ As a nontrivial example to test the formalism, consider ej ∼ sin γ ln j ζ ; 22 the case jβ ð þ Þ ð Þ ρk ∞ ρ kj with κ; β; γ; ζ numerical coefficients to be determined. We r2 δV 10−2 0 10−2 1 j 1 0 ; 24 ð Þ H k k − þ find β ≃ 0.66 and γ ≃ 1.5, in very good agreement with the ¼ ρ0 r ¼ ð Þ r ð Þ þ j 1 analytical estimate of Eq. (21). Figure 2 compares the fit X¼ j 1 −2 kj with the actual data for ej. The agreement is extremely where ρ0 is a constant, and αjÆ −1 þ 10 ρ0=rH . good, and it is strong evidence in support of the asymptotic For this potential the convergence¼ð criterionÞ reducesð toÞ behavior (21). n 1 Having assessed the asymptotic properties of the basis ρ0þ ϒ lim n ρ0 ; 25 ej, we can now study in detail the convergence of the ¼ n→∞ ρ ¼ j j ð Þ 0 frequencies ωQNM. We assume that the effective potential of ∞ the specific theory under consideration is C , such that we and therefore the perturbative formalism is valid if ρ0 < 1. ∞ aj can expand it for large distances as δV j 0 rj , where ¼ ¼ the a ’s are constant coefficients. Moreover, following our V. EXAMPLES j P formalism developed in Sec. II, we expand the corrections In this section we will provide some specific examples of with respect to the Schwarzschild term according to Eq. (6), theories of gravity whose gravitational perturbations can be described in terms of a master equation with the same 3Note that Eq. (19) holds regardless of the overtone number functional form of Eq. (1). In order to determine the domain and of the multipole number of the mode, if j is sufficiently large. of validity and the accuracy of our semianalytical approach, 104077-5 Generic isospectrality breaking Isospectrality follows from the existence of a “superpotential” such that: Perturb to find conditions for isospectrality to hold: Preserving isospectrality needs fine tuning! [Chandrasekhar-Detweiler 1975] Example 1: EFT EFT corrections quartic in the curvature lead to a modified Regge-Wheeler equation: Trivially read off the correction coefficient: Plug into to find in agreement with numerical integrations. [Cardoso+, 1808.08962] Example 2: Reissner-Nordström Odd gravitational perturbations of Reissner-Nordström satisfy A simple change of variables brings the wave equation in our “canonical” form, with PARAMETRIZED BLACK HOLE QUASINORMAL RINGDOWN: … PHYS. REV. D 99, 104077 (2019) for small charge. TABLE II. Relative percentage errors on the real and imaginary TABLE III. Relative percentage errors in the real and imaginary Read off coefficients to find: parts of the QNMs for RN BHs, as a function of the charge-to- parts of the QNM frequencies for scalar perturbations around a mass ratio Q=M. slowly spinning black hole, as a function of the BH angular momentum a=M. Q=M ΔR ΔI a=M 0.00 0% 0% ΔR ΔI 0.05 0.11% 0.042% 0 0% 0% 0.10 0.43% 0.17% 10−4 0.0050% 0.83% 0.20 1.7% 0.66% 10−3 0.049% 5.1% 0.30 3.8% 1.5% 10−2 0.49% 34% 0.40 6.8% 2.6% 0.50 11% 4.2% 0 am 0 0 0 0 ωQNM ω0 2 − 2amω0 d0 d1 d2 : 40 ¼ þ rH ð þ þ Þ ð Þ ΔI Im ωQNM − ωGR =Im ωfull − ωGR − 1 ; 36 ¼ j ð Þ ð Þ j ð Þ Comparing these results with numerical data for l where ω is computed by numerically solving the exact full m 2 [4,30], we find the relative percentage errors¼ master equation without any approximation, and ωQNM is ¼ ΔR; ΔI listed in Table III. Our formula is a good given by Eq. (34). For Q=M 0.2, for example, the QNMs ðapproximationÞ for very small a=M 10−4. For a=M frequencies derived using Eq.¼ (34) agree to within 0.05% ∼ ∼ 10−2 the agreement gets worse, especially in the imaginary with those in Ref. [33]. The errors obtained for different −2 values of Q=M are listed in Table II. component. We find that a=M ≪ 10 is a necessary condition for our approximation to be valid. The relative uncertainties ΔR;I grow with Q=M up to ∼10% (4%) for the real (imaginary) component of the mode’s frequency when Q=M 0.5. This behavior is VI. STATISTICAL ERRORS consistent with the assumptions¼ made to obtain Eq. (34), In this section we analyze the detectability of the in which we neglected terms that are order O α 2 , the ðð jÆÞ Þ modifications of the QNM spectrum by space and terres- latter corresponding to ignoring O Q4 corrections to the ð Þ trial GW interferometers. We follow the approach described RN perturbations. Therefore, for large values of Q, a better in Ref. [5], and we refer the reader to this paper and accuracy would require to compute ωQNM by including references therein for further details. Since we are inter- − second-order terms in both αjþ and αj . ested only in an order-of-magnitude estimate of the associated observational errors, we assume that the mass C. Scalars around a slowly spinning black hole of the BH is known (and therefore that the fundamental GR frequencies are known). For a more sophisticated analysis, The massless Klein-Gordon equation □Ψ 0 around a ¼ we refer the reader to Refs. [7,35]. slowly rotating Kerr BH can be written as The gravitational waveform measured by the interfer- ometers is a linear superposition of two polarization states d d 2 4amMω of the form h h F h F , where F and F denotes f f Φ ω − fV0 − Φ 0; 37 × × × dr dr þ r3 ¼ ð Þ the standard pattern¼ þ functionsþ þ (which dependþ on the source orientation with respect to the detector and on a “polari- at O a , where f 1–2M=r, a=M is the BH angular zation angle”). In the frequency domain, the two GW ð Þ ¼ −iωt momentum, and we assumed Ψ e Ylm θ; ϕ Φ r =r. components are simply given by We can rewrite this equation in the¼ followingð form:Þ ð Þ 2 Aþ d d am 2amω ˜ lmn iϕþ −iϕþ h f e lmn Slmnb f e lmn Sl⋆mnb− f ; f f Φ ω − 2 − f V0 − 2 þð Þ¼ p2 ½ þð Þþ ð Þ dr dr þ rH rH × 2 iAlmn iϕ× iϕ× 2amω r 2amω r ˜ lmn − lmn H H h× f − ffiffiffi e Slmnb f e Sl⋆mnb− f ; − − Φ 0; 38 ð Þ¼ p ½ þð Þþ ð Þ r2 r r2 r ¼ ð Þ 2 H H ffiffiffi 2 ;× where we used rH 2M O a . Thus where the amplitude coefficients Alþmn and the phase ¼ þ ð Þ ;× coefficients ϕlþmn are real, Slmn represent the (complex) β0 β0 β0 −2amω0; 39 spin-weighted spheroidal harmonics of spin weight 2, 0 ¼ 1 ¼ 2 ¼ 0 ð Þ which depend on the polar and azimuthal angles, and and we find b f are the Breit-Wigner functions: Æð Þ 104077-7 Example 3: Klein-Gordon in slowly rotating Kerr At linear order in the spin parameter: i.e. CorrectionPARAMETRIZED coefficients BLACK to the HOLE scalar QUASINORMAL wave equation: RINGDOWN: … PHYS. REV. D 99, 104077 (2019) TABLE II. Relative percentage errors on the real and imaginary TABLE III. Relative percentage errors in the real and imaginary parts of the QNMs for RN BHs, as a function of the charge-to- parts of the QNM frequencies for scalar perturbations around a mass ratio Q=M. slowly spinning black hole, as a function of the BH angular momentum a=M. Q=M ΔR ΔI a=M 0.00 0% 0% ΔR ΔI 0.05 0.11% 0.042% 0 0% 0% 0.10 0.43% 0.17% 10−4 0.0050% 0.83% 0.20 1.7% 0.66% 10−3 0.049% 5.1% 0.30 3.8% 1.5% 10−2 0.49% 34% 0.40 6.8% 2.6% 0.50 11% 4.2% 0 am 0 0 0 0 ωQNM ω0 2 − 2amω0 d0 d1 d2 : 40 ¼ þ rH ð þ þ Þ ð Þ ΔI Im ωQNM − ωGR =Im ωfull − ωGR − 1 ; 36 ¼ j ð Þ ð Þ j ð Þ Comparing these results with numerical data for l where ω is computed by numerically solving the exact full m 2 [4,30], we find the relative percentage errors¼ master equation without any approximation, and ωQNM is ¼ ΔR; ΔI listed in Table III. Our formula is a good given by Eq. (34). For Q=M 0.2, for example, the QNMs ðapproximationÞ for very small a=M 10−4. For a=M frequencies derived using Eq.¼ (34) agree to within 0.05% ∼ ∼ 10−2 the agreement gets worse, especially in the imaginary with those in Ref. [33]. The errors obtained for different −2 values of Q=M are listed in Table II. component. We find that a=M ≪ 10 is a necessary condition for our approximation to be valid. The relative uncertainties ΔR;I grow with Q=M up to ∼10% (4%) for the real (imaginary) component of the mode’s frequency when Q=M 0.5. This behavior is VI. STATISTICAL ERRORS consistent with the assumptions¼ made to obtain Eq. (34), In this section we analyze the detectability of the in which we neglected terms that are order O α 2 , the ðð jÆÞ Þ modifications of the QNM spectrum by space and terres- latter corresponding to ignoring O Q4 corrections to the ð Þ trial GW interferometers. We follow the approach described RN perturbations. Therefore, for large values of Q, a better in Ref. [5], and we refer the reader to this paper and accuracy would require to compute ωQNM by including references therein for further details. Since we are inter- − second-order terms in both αjþ and αj . ested only in an order-of-magnitude estimate of the associated observational errors, we assume that the mass C. Scalars around a slowly spinning black hole of the BH is known (and therefore that the fundamental GR frequencies are known). For a more sophisticated analysis, The massless Klein-Gordon equation □Ψ 0 around a ¼ we refer the reader to Refs. [7,35]. slowly rotating Kerr BH can be written as The gravitational waveform measured by the interfer- ometers is a linear superposition of two polarization states d d 2 4amMω of the form h h F h F , where F and F denotes f f Φ ω − fV0 − Φ 0; 37 × × × dr dr þ r3 ¼ ð Þ the standard pattern¼ þ functionsþ þ (which dependþ on the source orientation with respect to the detector and on a “polari- at O a , where f 1–2M=r, a=M is the BH angular zation angle”). In the frequency domain, the two GW ð Þ ¼ −iωt momentum, and we assumed Ψ e Ylm θ; ϕ Φ r =r. components are simply given by We can rewrite this equation in the¼ followingð form:Þ ð Þ 2 Aþ d d am 2amω ˜ lmn iϕþ −iϕþ h f e lmn Slmnb f e lmn Sl⋆mnb− f ; f f Φ ω − 2 − f V0 − 2 þð Þ¼ p2 ½ þð Þþ ð Þ dr dr þ rH rH × 2 iAlmn iϕ× iϕ× 2amω r 2amω r ˜ lmn − lmn H H h× f − ffiffiffi e Slmnb f e Sl⋆mnb− f ; − − Φ 0; 38 ð Þ¼ p ½ þð Þþ ð Þ r2 r r2 r ¼ ð Þ 2 H H ffiffiffi 2 ;× where we used rH 2M O a . Thus where the amplitude coefficients Alþmn and the phase ¼ þ ð Þ ;× coefficients ϕlþmn are real, Slmn represent the (complex) β0 β0 β0 −2amω0; 39 spin-weighted spheroidal harmonics of spin weight 2, 0 ¼ 1 ¼ 2 ¼ 0 ð Þ which depend on the polar and azimuthal angles, and and we find b f are the Breit-Wigner functions: Æð Þ 104077-7 Coupled perturbations We really want to solve the coupled system where each matrix element is perturbed: If the background spectra are nondegenerate, coupling will induce quadratic corrections. Allow to depend on . We need • quadratic corrections in , besides the linear diagonal terms • coupling-induced corrections (Einstein summation) [McManus+, 1906.05155] Correction coefficients The degenerate case Degenerate spectra (e.g. even/odd gravitational perturbations) need special care. Why? Diagonalize: Corrections are linear in a Use degenerate perturbation theory: Example 1: scalar/odd gravitational in dynamical Chern-Simons 7 0.75 Tensor-led 1.4 Scalar-led Spectra are nondegenerate 0.70 1.3 The perturbed potentials read: 0.65 1.2 0.60 1.1 0.55 1.0 0.50 -0.18 -0.20 -0.20 -0.22 -0.22 0.25 -0.24 -0.24 0 0 -0.26 -0.1 -0.26 -0.25 -0.2 Corrected frequencies: -0.28 -0.5 -0.28 0 0.05 0.1 0 0.05 0.1 -0.30 -0.30 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.08 0.10 FIG. 3. Fundamental axial ` = 2 QNM frequencies of Schwarzschild BHs in dCS gravity as function of ¯ for the tensor-led (left) and scalar-led (right) modes. Solid lines refer to Eqs. (33) and (34); bullets were computed through a direct integration method. The inset shows the relative di erence between the two calculations for the real (solid black lines) and imaginary (dashed red lines) part of the modes. One might have hoped that the corrections to the QNM gravitational perturbations is the same as in GR, whereas axial spectra from the coupling of[Cardoso the two degenerate-Gualtieri, fields would 0907.5008;gravitational Molina+, perturbations 1004.4007] and scalar perturbations lead to a allow for an expansion analogous to Eq. (4). Indeed, in the coupled system of the form (1)[42, 71] with the following absence of coupling, the argument of the square root in Eq. (28) potentials, in the notation of Eqs. (3), (6) and (8): becomes a square, and we do recover a linear sum over single- field expectation values. However, in general, the presence of V11 = V , (30) couplings makes this relation nonlinear. Therefore we provide k s 1 12 ` + 2 ! rH 5 the values of the quantities V(+) V+( ) for k, s = 0,...10 and V12 = V21 = ⇡ ( ) , (31) 2 2 ` ` = 2 ...10 online [60]. In Table III we also list a small rH p rH s 2 ! r ( ) ⇣ ⌘ k k 8 sample of values of V(+) V+( ) for ` = 2. In order to find 1 144⇡` ` + 1 rH V22 = Vs=0 + ( ) . (32) linear corrections to the QNM frequencies, these quantities r2 r4 r k H H ⇣ ⌘ must be plugged into Eqs. (28) and (29). Note that V++( ) and k V( ) are just the coe cients d k for uncoupled (axial or polar) The parameter appearing in the dCS action has dimensions gravitational perturbations. ( ) L 4 and it sets the strength of the coupling, playing a role [ ] similar to the Brans-Dicke parameter !BD. It is useful to intro- 1 2 2 duce a small dimensionless coupling parameter ¯ / rH VI. EXAMPLES such that the equations decouple in the GR limit ¯ ⌘ 0. ! We first study how the parameter ¯ modifies the tensor-led For illustration, we now apply the formalism to compute mode. Using Eq. (4) and reading o the relevant coe cients QNM spectra for some classes of modified theories of grav- from the potentials (30)–(32) we find ity that are known to lead to coupled perturbation equations. 2 Specifically, we consider two models where the coupling is ` + 2 ! between scalar and tensor modes (dCS gravity [44] and Horn- ! = ! + e1221 12 ¯ ⇡ ( ) . (33) 0 55 ` 2 ! deski gravity [34]) and a model where the coupling is between ( ) s ( ) ! axial and polar gravitational perturbations (the EFT inspired model [46] not considered in detail in Paper I), so that the Proceeding similarly for the scalar-led mode, we find background QNM spectra are degenerate. 2 2 2 ! = !0 + 2d 8 144⇡` ` + 1 ¯ + e 88 144⇡` ` + 1 ¯ ( ) ( ) ( ) ( ) 2 A. Dynamical Chern-Simons gravity ` + 2 ! ⇥ ⇤ + e1221 12 ¯ ⇡ ( ) . (34) 55 ` 2 ! ( ) s ( ) ! In dCS gravity, an e ective low-energy theory with an ad- ditional scalar degree of freedom [41], nonspinning BHs are These expressions illustrate the importance of specifying the described by the Schwarzschild metric. The polar sector of form of the coupling: since the o -diagonal terms V12 and V21 Example 2: scalar-led perturbations in Horndeski The scalar-led perturbation is related to background coupling functions in the Horndeski Lagrangian: Corrected frequencies read (can set ): [Tattersall+, 1711.01992] Example 3: odd/even gravitational coupling in EFT (degenerate) 9 0.84 0.76 QNM frequency corrections are linear in the perturbations. The quartic-in-curvature EFT leads to a degenerate ] Our results significantly simplify the task of computing H 0.75 0.82 r perturbed eigenvalue problem: QNM frequencies in any modified theory of gravity, or any [ 0.74 Re theory allowing for additional fields. The general recipe for ] 0.8 perturbed solution H 0.73 r this calculation can be summarized as follows: 0.0 0.005 0.01 0.015 0.02 [ 0.78 Re 1) Derive the master equations for the perturbation vari- 0.76 ables in the given theory; 0.74 where off-diagonal perturbations are given in 2) Eliminate first-order (friction-like) terms in the field [Cardoso+, 1808.08962] derivatives through field redefinitions, as described in 0.2 Appendix A; 0.19 k ↵( ) Direct integration vs. degenerate parametrization: 3) Identify the relevant coe cients ij in the perturbed 0.2 0.18 ] potentials Vij [Eq. (3)] appearing in the general coupled H ] good agreement, but quadratic corrections r 0.18 H r [ system of Eq. (1); if these coe cients are frequency- 0.17 could be useful Im 0.16 s [ - dependent, compute their frequency derivative ↵0(pq) . 0.14 Im 0.16 - 0.0 0.02 0.04 4a) If any two unperturbed spectra are nondegenerate, com- 0.15 pute corrections to the QNM frequencies by simple mul- direct integration tiplications and additions using Eq. (4) and the tabulated 0.14 values of dij and eijpq, which are available online [60]. 0.0 0.02 0.04 0.06 0.08 0.1 k ks ( ) ( ) 4b) If any two unperturbed spectra are degenerate, com- pute corrections to the QNM frequencies using Eq. (28) FIG. 5. Perturbations of the fundamental QNM frequencies with ` = k k and the tabulated values of V( ) and V( ) defined in 2 for axial and polar gravitational perturbations, coupled according ±± ±⌥ to the EFT model of [46]. The bullet points correspond to frequency Eqs. (D7) and (D8), which are also available online [60]. values obtained through a direct integration method. The insets show a zoom for small ✏ where the behaviour of ! deviate from the linear In Sec. VI we illustrate this procedure for three classes of trend, and therefore from the linear approximation given by Eq. (28). modified theories of gravity leading to coupled perturbation equations: two models coupling the scalar and tensor modes (dCS gravity [44] and Horndeski gravity [34]) and an EFT Fitting the two branches of QNM frequencies to a seventh- model coupling the axial and polar gravitational perturba- order polynomial in ✏ gives a linear coe cient 0.1479 tions [46], where the background QNM spectra are degenerate. ( 0.2729i rH for the solid black branch, and 0.1480 + While our expansion is theory-agnostic, we make assump- ) ( 0.2719i rH for the dashed red branch. The linear corrections tions about the e ect of the modified gravity theory: the back- are nonzero,) as they should, because the uncoupled axial and ground should be perturbatively close to the Schwarzschild polar spectra are degenerate. metric, and the corrections to the “ordinary” potentials in the GR master equations should be amenable to a power-series expansion in inverse powers of the radial variable. This results by construction in small corrections to the GR VII. CONCLUSIONS AND A COMPUTATIONAL RECIPE QNM spectra (4). In general, as discussed in Paper I, new nonperturbative frequencies (e.g., quasibound states emerging We have extended the formalism of Paper I to compute QNM from zero frequency for massive scalars) may appear in the frequencies of coupled fields as long as the perturbations they spectrum, and these are not captured by our formalism. induce are small deviations from the perturbation equations The assumption that the background is only perturbatively for the Schwarzschild geometry in GR. Our main result is a di erent from the Schwarzschild solution is slightly less re- convenient, ready-to-use recipe to compute QNM frequencies strictive than one might think. For example, in Paper I we at quadratic order in the perturbations. We crucially allow showed that slowly rotating Kerr BHs can be accommodated for the possibility of coupling between the master equations. within the formalism. Recent work on higher-derivative cor- First-order (friction-like) terms in the field derivatives can be rections to the Kerr geometry [75] and on QNM frequencies accommodated through field redefinitions (cf. Appendix A). of rotating solutions for small coupling [53, 54] may allow We have found the expansion of the QNM frequencies for us to make progress on the calculation of QNMs in modified uncoupled wave equations to quadratic order. Perhaps our gravity for rotating BH remnants, such as those observed by most interesting findings concern the QNM spectra of coupled the LIGO/Virgo collaboration [76] (see [77, 78] for a related fields. When the coupling occurs between fields with nonde- attempt at parametrizing deviations from the Kerr QNM spec- generate spectra at zero order in the perturbations, linear-order trum). corrections to the QNM frequencies vanish. However, when Acknowledgments. We thank Macarena Lagos, Oliver Tat- the coupling occurs between fields with degenerate spectra, the tersall and Aaron Zimmerman for useful discussions. E.B. Parametrized merger/ringdown: a summary Modifications to the gravity sector and/or beyond Standard Model physics: • small modifications to the potentials • coupling between the (matrix-valued) wave equations We parametrized modifications by power laws, then computed perturbed QNMs for: • linear corrections to diagonal terms [Cardoso+, 1901.01265] • quadratic corrections + coupling [McManus+, 1906.05155] The formalism is very general! Examples: • EFT, Reissner-Nordström, Klein-Gordon in Kerr for slow rotation • scalar/odd gravitational dCS, scalar-led Horndeski, odd/even gravitational EFT Needed generalizations: • higher-order corrections (in particular, in degenerate coupled case) • coupled gravitational modes with rotation – LIGO/Virgo remnants have spins 0.7 or so! Parametrized spectroscopy: adding rotation Rotating BH QNMs in modified gravity: the EFT viewpoint QNM calculations: limited sample for specific theories (EdGB/EsGB, dCS) and nonrotating BHs [Blazquez-Salcedo+ 1609.01286 (EdGB), 2006.06006 (EsGB); Molina+ 1004.4007 (dCS)] Cano’s work: systematic small-rotation expansion + scalar QNMs Theories: sum over curvature invariants with scalar-dependent coefficients and more specifically, at order EsGB dCS (dilaton+axion) Einsteinian cubic gravity (+parity-breaking) - causality constraints [Camanho+ 1407.5597] Next order, no new DOFs [Endlich-Gorbenko-Huang-Senatore, 1704.01590] [Cano-Ruipérez, 1901.01315; Cano-Fransen-Hertog, 2005.03671. See also work by Hui, Penco…] No calculation of rotating BH QNMs in modified gravity: the EFT viewpoint Background solutions: algorithm to compute small-coupling corrections, up to order 14 in rotation Scalar QNM calculations: “quasi-separable” For zero coupling, can be separated in terms of spin-weighted spheroidal harmonics End of the story: second-order radial ODEs can be cast as wave equations via redefinitions of the radial variable/radial WF, and solved either numerically or via WKB Note: not all potentials vanish at the horizon [Cano-Ruipérez, 1901.01315; Cano-Fransen-Hertog, 2005.03671] Parametrized spectroscopy: how many observations do we need? Use a small-spin expansion and add parametric deviations to frequency and damping time Assume you detect N sources, and q QNM frequencies for each source modes/source Order in the spin expansion: need at least 4 or 5 in GR sources Expansion coefficients in GR Small, universal non-GR corrections How many parameters? If for all sources , reabsorb How many observables? Need only [Maselli+, 1711.01992] No calculation of rotating BH QNMs in modified gravity: the EFT viewpoint Complication: the coupling is often dimensionful Use Bayesian inference (MCMC), , (one mode), simple source distributions Einstein Telescope: constrain first three frequency coeffs and only the first damping coeffs Width at 90% confidence gets better as we get more observations: [Maselli+, 1711.01992] Take-home messages “Null” spectroscopic tests of GR: High-SNR (LISA/CE/ET) GW astronomy will need better control over systematic errors [Ferguson+ 2006.04272] Study excitation factor with “true” merger initial data. Nonlinearities? “Real” tests of GR with black hole inspiral/merger/ringdown: Need full nonlinear simulations in beyond-GR theories • Identify interesting theories: EsGB is one such • Study 3+1 decomposition and well-posedness (EFT - or better, full theory) Parametrized tests of GR with black hole ringdown: Nonrotating case quite well understood – but irrelevant to most “real” mergers Rotation: • Can use perturbation theory and slow-rotation approximation, but hard • Numerical, single black hole time-evolutions could get the QNM spectrum in specific theories first (similar work has been done for rotating stars)