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<latexit sha1_base64="jYfVTIpELl6Vbm+ImvxJeEbWqUs=">AAAB/XicbVDLSsNAFJ3UV42v+Ni5CRbBVUmqoIsuim66s4J9QBPCZDpph85kwsxEqKH4K25cKOLW/3Dn3zhps9DWA5d7OOde5s4JE0qkcpxvo7Syura+Ud40t7Z3dves/YOO5KlAuI045aIXQokpiXFbEUVxLxEYspDibji+yf3uAxaS8PheTRLsMziMSUQQVFoKrCOn7nGGh7DOvNu8B03TDKyKU3VmsJeJW5AKKNAKrC9vwFHKcKwQhVL2XSdRfgaFIojiqemlEicQjeEQ9zWNIcPSz2bXT+1TrQzsiAtdsbJn6u+NDDIpJyzUkwyqkVz0cvE/r5+q6MrPSJykCsdo/lCUUltxO4/CHhCBkaITTSASRN9qoxEUECkdWB6Cu/jlZdKpVd3zau3uotK4LuIog2NwAs6ACy5BAzRBC7QBAo/gGbyCN+PJeDHejY/5aMkodg7BHxifP3uJk/E=</latexit> modes< ! <m are localized⌦H [Press in-Teukolsky a potential1972] well • Spectrum of damped modes (“ringdown”) • Hydrogen-like, unstable bound states interactions, become important in the regime where the region created by the mass ‘‘mirror’’ from the spatial infinity on [EB+, 0905.2975] [Detweiler 1980, Zouros+Eardley, Dolan…] the right, and by the centrifugal barrier from the ergo-region and horizon on the left. 1Note, that at this stage we still agree with [19]. 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 <latexit sha1_base64="dimHfGHVtE3izdRc0XFTwzmYZTQ=">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</latexit> Ringdown10° SNR.