Testing Strong-Field Gravity with Space-Based Detectors

Testing strong-field gravity with space-based detectors

Emanuele Berti, University of Mississippi/Caltech LISA Symposium X, Gainesville, May 21 2014 Outline 1) Weak-field tests of gravity • Laboratory and Solar System tests • Radiation from binary pulsars

2) Strong-field tests of gravity: why do we need (e/u)LISA? • Earth-based detectors: AdLIGO/Virgo, KAGRA, LIGO India, ET… • Pulsar timing: IPTA, SKA • Space-based detectors

3) Strong-field tests of gravity: “external” vs. “internal” tests q External tests: extensions of general relativity q Internal tests: the no-hair theorem q Ringdown tests: Consistency with inspiral / area theorem Nonlinearities Importance of localization/distance determination q Propagation: graviton mass q Can we beat binary pulsar constraints? One example: superradiance and floating orbits The Confrontation between General Relativity and Experiment 9

the wire and g were not quite parallel because of the centripetal acceleration on the apparatus due to the Earth’s rotation; the apparatus was rotated about the direction14 of the wire. In the Dicke Cli↵ord M. Will and Braginsky experiments, g was that of the Sun, and the rotation of the Earth provided the modulation of the torque at a period of 24 hr (TEGP 2.4 (a) [281]). Beginning in the late 1980s, numerous experiments were carried out primarily to search for a “fifth force” (see Section 2.3.1), but their null results also constituted tests of WEP. In the “free-fall GalileoThe first experiment” successful, performed high-precision redshift measurement was the series of Pound–Rebka–Snider 57 at the University of Colorado, the relative free-fall acceleration ofexperiments two bodies made of 1960 of uranium – 1965 that and measured the frequency shift of gamma-ray photons from Fe as copper was measured using a laser interferometric technique. Thethey “Eöt-Wash” ascended or experiments descended car- the Je↵erson Physical Laboratory tower at Harvard University. The ried out at the University of Washington used a sophisticated torsionhigh accuracy balance tray achieved to compare – one the percent – was obtained by making use of the Mössbauer e↵ect to accelerations of various materials toward local topography on Earth,produce movable a narrow laboratory resonance masses, line whose shift could be accurately determined. Other experiments 13 the Sun and the galaxy [249, 19], and have reached levels of 3since10 1960[2]. measured The resulting the shift upper of spectral lines in the Sun’s gravitational field and the change in ⇥ limits on ⌘ are summarized in Figure 1 (TEGP 14.1 [281]; for arate bibliography of atomic of clocks experiments transported up to aloft on aircraft, rockets and satellites. Figure 3 summarizes the 1991, see [107]). importantThe redshiftfoundaons experiments that have been performed of general since 1960 (TEGP 2.4 (c) [281]). relavity TESTS OF THE TESTS OF WEAK EQUIVALENCE PRINCIPLE LOCAL POSITION INVARIANCE40 Cli↵ord M. Will

-8 Pound Rebka Millisecond Pulsar [Will, gr-qc/0510072] 10 Eötvös 10 -1 Renner -9 Free-fall Null 10 Redshift -2 Fifth-force 10 Pound searches Snider 10-10 Boulder η Saturn Princeton Eöt-Wash 10 -3 10-11 α H maser Moscow Eöt-Wash 10-12 LLR 10 -4 THE PARAMETER (1+γ)/2 12 Cli↵ord M. Will Null Redshift -13 10 Solar spectra 1.10 a -a 2 Radio η= 1 10 -5 Clocks in rockets DEFLECTION (a +a )/2 10-14 1 2 spacecraft & planes OF LIGHT Optical 1900 1920 1940 1960 1970 1980 1990 2000 1960 1970 1980 1990 2000 1.05 2X10-4 VLBI YEAR OF EXPERIMENT YEAR OF EXPERIMENT TESTS OF 2 Figure 1: Selected tests of theLOCAL weak equivalence LORENTZ principle, INVARIANCE showing bounds on ⌘, which measures Δν/ν = (1+α)ΔU/c 1.00 fractional di↵erence in acceleration of di↵erent materials or bodies. The free-fall and Eöt-Wash experiments were originally performed to search for a fifth force (green region, representing many experiments). The blue band shows evolving bounds on ⌘ for gravitatingFigure 3: Selected bodies fromWeak Equivalence Principle+ tests lunar of local laser position invariance via gravitational redshift experiments, showing ranging (LLR). -2 bounds on ↵, which measures degree of deviation of redshift from the formula ⌫/⌫ = U/c2.In 10 JPL Michelson-Morley null redshift experiments, the bound is on the di↵erence in ↵ between di↵erent kinds of clocks. Hipparcos Joos )/2 0.95

TPA Local Lorentz Invariance+ γ 10-6 Centrifuge After almost 50 years of inconclusive or contradictory measurements, the gravitational redshift Living Reviews in Relativity of solarCavities spectral lines was ﬁnally measured reliably. During the early years of GR, the failure http://www.livingreviews.org/lrr-2006-3Local Posion Invariance= (1+ to measure this e↵ect in solar lines was siezed upon by some as reason to doubt the theory. -10 10 Brillet-Hall Unfortunately, theEinstein’s Equivalence Principle measurement is not simple. Solar spectral lines are subject to the “limb e↵ect”, PSR 1937+21 δ a variation of spectral line wavelengths between the center of the solar disk and its edge or1.05 “limb”; this e↵ect is actually a Doppler shift caused by complex convective and turbulent motions in the SHAPIRO 10-14 Hughes Drever photosphere and lower chromosphere, and is expected to be minimized by observing at the solar TIME Voyager NIST Implies gravity is a metric theory: DELAY 10-18 Harvard 1.00 Living Reviews ingravity is Relativity spaceme curvature U. Washington 10-22 http://www.livingreviews.org/lrr-2006-3 Viking Cassini (1X10-5) δ = 1/c2 - 1 Best test of spaceme curvature: 0.95 10-26

1900 1920 1940 1960 1970 1980 1990 2000 2010 Cassini bound 1920 1940 1960 1970 1980 1990 2000 YEAR OF EXPERIMENT YEAR OF EXPERIMENT

Figure 2: Selected tests of local Lorentz invariance showing the bounds on the parameter , which measures the degree of violation of Lorentz invariance in electromagnetism. The Michelson–Morley, Joos, Brillet–Hall and cavity experiments test the isotropy of the round-trip speed of light. The centrifuge, two-photon absorption (TPA) and JPL experiments test the isotropy of light speed Figure 5: Measurements of the coecient (1 + )/2 from light deﬂection and time delay measure- using one-way propagation. The most precise experiments test isotropy of atomic energy levels. ments. Its GR value is unity. The arrows at the top denote anomalously large values from early 1 The limits assume a speed of Earth of 370 km s relative to the mean rest frame of the universe. eclipse expeditions. The Shapiro time-delay measurements using the Cassini spacecraft yielded an 3 agreement with GR to 10 percent, and VLBI light deﬂection measurements have reached 0.02 percent. Hipparcos denotes the optical astrometry satellite, which reached 0.1 percent.

Living Reviews in Relativity http://www.livingreviews.org/lrr-2006-3

Living Reviews in Relativity http://www.livingreviews.org/lrr-2006-3 Indirect detecon of gravitaonal waves: the binary pulsar 1993 Nobel Prize to Hulse and Taylor: discovery of the binary pulsar 1913+16 -3 Strongest test of GR: PSR J0348+0432, P=2.46hr, v/c=2x10 [Antoniadis+, 1304.6875]

[Weisberg+Taylor] Probes and Tests of Strong-Field Gravity with Observations in the Electromagnetic Spectrum 13

Strong ﬁeld: gravitaonal ﬁeld vs. curvature; probing vs. tesng -10 10 -13 Neutron Stars 10 Black Holes -16 10 -19 ) 10 -2

10-22

(cm Grav Prob B 2 -25 c AGN 3 10 Eclipse -28 Hulse-Taylor 10

=GM/r -31 Over the Horizon ξ 10 -34 Moon 10 Mercury -37 10

10-40 10-15 10-12 10-9 10-6 10-3 100 2 ε=GM/rc [Psals, Living Reviews in Relavity]

Figure 2: Tests of general relativity placed on an appropriate parameter space. The long-dashed line represents the event horizon of Schwarzschild black holes.

Living Reviews in Relativity http://www.livingreviews.org/lrr-2008-9 Gravitaonal-wave tests Energy ﬂux: E.g. scalar-tensor theories predict dipole radiaon because (ineral mass)≠(gravitaonal mass)

Polarizaon: Up to six polarizaon states

Propagaon:

If e.g. mgraviton≠0, gravitaonal waves would travel slower than EM waves

Massive black hole inspirals -26 can yield bounds mgraviton<∼10 eV (104-106 beer than Solar System)

Direct detecon: Strong-ﬁeld dynamics [e.g. Gair+,1212.5575] Gravitational-wave astronomy: a timeline!

Timeline: Earth-based detectors 2014: BICEP2 – inflationary GWs?

2015: AdLIGO LISA Pathfinder

2018: AdLIGO design sensitivity FIRST DETECTIONS?

<~2022: First PTA detections?

<2030 AdVirgo, KAGRA, LIGO India

<2030? Einstein Telescope?

2034: ESA L3 launch – eLISA

<2044: DECIGO? BBO?

2015: AdLIGO LISA Pathfinder

2018: AdLIGO design sensitivity FIRST DETECTIONS?

<~2022: First PTA detections?

<2030 AdVirgo, KAGRA, LIGO India

<2030? Einstein Telescope?

2034: ESA L3 launch – eLISA

<2044: DECIGO? BBO?

2015: AdLIGO LISA Pathfinder

2018: AdLIGO design sensitivity FIRST DETECTIONS?

<~2022: First PTA detections?

<2030 AdVirgo, KAGRA, LIGO India

<2030? Einstein Telescope?

2034: ESA L3 launch – eLISA

<2044: DECIGO? BBO?

Decadal 2010: The panel recommends that the LISA mission be given the highest priority for a new start in the next decade, given the extensive technology development that has already been completed, the expected short me unl the LISA Pathfinder mission (LPF) launch, and the need to maintain momentum in the U.S. community and guarantee a smooth transion to a joint NASA-ESA mission. The panel recommends that NASA funding of LISA begin immediately, with connuaon beyond LPF conngent on the success of that mission. Strong gravity tests: “external” vs.“internal”! “External” tests of general relavity – against what? Telescope [13] and a space-based, LISA-like mission [14, 15, 16] in the long term) is precisely their potential to test GR in strong-field, high-velocity regimes inaccessible to Solar System and binary pulsar experiments. The strength of these tests will depend on two key elements: (i) the signal-to-noise ratio (SNR) of individual observations [17], that also a↵ects accuracy in binary parameter estimation, and (ii) the number N of observations that can be used to constrain GR. The reason is that, given a theory whose deviations from GR can be parametrized by one or more universal parameters (e.g. coupling constants), the bounds on these parameters will scale roughly with pN (as a matter of fact the bounds could improve faster than pN if some events are particularly loud: see e.g. [18, 19] for detailed analyses addressing specific modifications to GR in the Advanced LIGO/eLISA context, respectively). Second-generation interferometers such as Advanced LIGO are expected to detect a large number of compact binary coalescence events [20, 21]. Unfortunately from the point of view of testing GR, most binary mergers detected by Advanced LIGO/Virgo are expected to have low signal-to-noise ratios (a possible exception is the observation of intermediate-mass BH mergers [22], that would be a great discovery in and by itself). Third generation detectors such as the Einstein Telescope will perform significantly better in terms of parameter estimation and tests of alternative theories [23, 24]. As I will argue below, a LISA-like mission will be the ideal instrument to put GR to the test, constraining the cosmological growth of supermassive BHs and testing gravity in unprecedented ways [15, 16]. Before turning to a discussion of tests of GR that will be possible in the near future, I will present a short overview of GR alternatives that are considered as plausible contenders at the time of writing and single out one of them (namely, massive scalar-tensor theories) as particularly interesting.

2. Testing general relativity: against what? Several extensions of GR have been proposed recently (see e.g. [2] for an excellent review). Among these models, the ones whose observational consequences have been better explored in various contexts (including cosmology, Solar System experiments, the structure of compact stars and gravitational radiation from binary systems) can be summarized via the following Lagrangian density:

= f0( )R (1) L | | a 2 ( )@a⇤@ V ( )+ mat ,A ( )gab “External| | ” tests of general | |relavityL – against what| | ? 2 abcd +f ( ) + f ( )R ⇤R § Acon principle 1 | | RGB 2 | | abcd ⇥ ⇤ § Well-posed § Testable predicons § Cosmologically viable = f ()R (2) L 0 !()@ @a M()+ ,A2()g a Lmat ab 2 ab abcd +f1()(R 4RabR + RabcdR⇥ ) ⇤ abcd +f2()Rabcd⇤R + Lorentz-violang vector fields… Alternave theories usually: where Rabcd is the Riemann Introduce more fields (scalars, vectors) or higher-curvature terms tensor corresponding to the metric gab, Rab is the Ricci tensor, Need strong-field tests! Challenge pillars of general relavity: R is the Ricci scalar and denotes matter fields. The functions fi( )(i =0,... ,2), V ( ) § and A( ) are in principle arbitrary, Equivalence principle but they are not all independent| | (in the sense that,| for| | | § Lorentz invariance (Einstein-aether, TeVeS…) example, some of them can§ be set equal to one via field redefinitions without loss of generality). Parity conservaon… [Gair+,1212.5575; Clion+, 1106.2476] This Lagrangian allows for all models in which gravity is coupled to a single (generically charged) scalar field in all possible ways, including all linearly independent quadratic curvature corrections to GR. The requirement that the field equations should be second-order means that “Internal” tests: the black-hole paradigm Compact massive objects in galacc centers coevolve with galaxies

Evidence based on correlaons:

MBH – σ, MBH – Lbulge

Are these Kerr black holes? Good reasons for theorecal bias

Alternaves are ugly/indisnguishable • dense star clusters: unstable • fermion stars: ruled out

• exoc maer: violate energy condions Figure 3 Correlations of dynamically measured black hole masses and bulk properties of host galaxies. (a) Black hole mass, MBH, versus stellar velocity | [McConnell+, 1112.1078] dispersion, , for 65 galaxies with direct dynamical measurements of MBH. For galaxies with spatially resolved stellar kinematics, is the luminosity-weighted average within one effective radius (Supplementary Information). (b) Black hole mass versus V -band bulge luminosity, LV (L ,V , solar value), for 36 early- • M 9 naked singularies: unstable, violate causality type galaxies with direct dynamical measurements of BH. Our sample of 65 galaxies consists of 32 measurements from a 2009 compilation , 16 galaxies with masses updated since 2009, 15 new galaxies with MBH measurements, and the two galaxies reported here. A complete list of the galaxies is given in Supplementary Table 4. BCGs (defined here as the most luminous galaxy in a cluster) are plotted in green, other elliptical and S0 galaxies are plotted in red, • boson stars/gravastars: formaon? and late-type spiral galaxies are plotted in blue. The black hole masses are measured using dynamics of masers (triangles), stars (stars), or gas (circles). Error bars, 68% confidence intervals. For most of the maser galaxies, the error bars in MBH are smaller than the plotted symbol. The solid black line in (a) 1 shows the best-fitting power law for the entire sample: log10(MBH/M )=8.29 + 5.12 log10[/(200 km s )]. When early-type and late-type galaxies 1 • log (M /M )=8.38 + 4.53 log [/(200 km s )] are fit separately, the resulting power laws are BH for elliptical and S0 galaxies (dashed red line), and black holes in alternave theories: 10 10 1 log10(MBH/M )=7.97 + 4.58 log10[/(200 km s )] for spiral galaxies (dotted blue line). The solid black line in (b) shows the best-fitting power law: 11 log10(MBH/M )=9.16 + 1.16 log10(LV /10 L ). We do not label Messier 87 as a BCG, as is commonly done, as NGC 4472 in the Virgo cluster is ü in scalar-tensor theories (and many other theories), 0.2 mag brighter. soluons are the same [Thorne-Dykla, Hawking, Soriou-Faraoni…] ü when soluons are not the same (EDGB, Chern-Simons) deviaons are (most likely) astrophysically unmeasurable

4 The (massive) black-hole paradigm: posive evidence? Tests based on electromagnec observaons: [Narayan’s Bad Honnef talk, 2014] ü Luminous, rapidly variable ü Radio VLBI: Sgr A* size ∼ few (2GM/c2) – does not imply this is a black hole [Shen+ 2005, Doeleman+ 2008, Fish+ 2010] ü Arguments against the presence of a surface if Sgr A* is powered by accreon: 10 Radio ﬂux implies that brightness temperature TB > 10 K Radiang gas must have temperature T ≥ TB If it were a blackbody, the luminosity L = 4πR2σT4 ~ 1062 erg/s – enormous! If Sgr A* had a surface, IR emission at least ∼1036 erg/s – not seen Therefore accreon cannot come from a surface – evidence for event horizon

Proposed tests based on electromagnec observaons: ü Measurement of quadrupole moment/higher moments of Sag A*: [Will, Merri…]

Problem: need to understand stellar dynamics

If we had one good argument, we would not need so many!

These are not tests of GR: Kerr metric is a soluon in most alternave theories We need to test the dynamics of the theory – gravitaonal waves! EXPLORING THE STRING AXIVERSE WITH PRECISION ... PHYSICAL REVIEW D 83, 044026 (2011) after switching to the tortoise coordinate (14) and introduc- To relate the tunneling exponent with the rate of super- ing É r2 a2 1=2R the radial Eq. (12) takes the form of radiance instability let us consider again the energy flow the Schro¼ð¨dingerþ equationÞ Eq. (6). Integrating it over the horizon we obtain 2 d É dE 2 VÉ 0 (20) ! mw ! Y R r ; (23) dr2 À ¼ dt ¼ ð þ À Þ horizon j ð Þ ð þÞj Ã Z with the potential where E is the energy in the axion cloud. The energy is maximum in the Keplerian region, so that in the limit 2 2 2 4rgram! a m where we only keep track of the dependence on the ex- V ! À I ¼À þ r2 a2 2 ponent eÀ we can write ð þ Þ 2 2 2 2I 2 Á 2 l l 1 k a E R rc e R r ; ð þ Þþ /j ð Þj ’ j ð þÞj þ r2 a2 a þ r2 a2 and, consequently, to rewrite (23) as 2þ 2 þ 2 3r 4rgr a 3Ár À2 2þ2 2 2 3 : (21) dE 2I þ r a À r a const mw ! eÀ E: (24) ð þ Þ ð þ Þ dt ¼ Áð þ À Þ We include the !2 term in the definition of the poten- ðÀ Þ In other words, the WKB approximation for the super- tial, because even if we were to separate it, there would be a radiance rate gives1 residual dependence on !. We present the qualitative shape 2I of the potential V for a typical choice of parameters in À mw ! eÀ ; (25) Fig. 7. One can clearly see the potential well where the ¼ ð þ À Þ where the normalization prefactor is determined mainly by bound Keplerian orbits are localized and a barrier separat- the spread of the wave function in the classically allowed ing this region from the near-horizon region where super- region. We will limit ourself by calculating the exponential radiant amplification takes place. part À. We leave the technical details for the Appendix, and Consequently, the axion wave function at the horizon present only the final result here. Namely, the final answer r r (corresponding to r ) is suppressed relative for the tunneling integral in the extremal Kerr geometry to¼ theþ wave function in theÃ vicinity¼À1 of the Keplerian orbit takes the form by a tunneling exponent, I R r R rc eÀ ; I 2 2 1 ; (26) j ð þÞj’j ð Þj ¼ À ð À Þ where the tunneling integral I is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi which translates in the following superradiant rate, 2 2 r r2 r2 pV r a Ãð Þ 7 1 2 2 p2 7 1 3:7 I dr pV dr ð þ Þ ; (22) ÀWKB 10À rgÀ eÀ ð À Þ 10À rgÀ eÀ ; (27) ¼ r r1 Ã ¼ r1 ffiffiffiffi Á Z Ãð Þ ffiffiffiffi Z where we took the large limitffiffi in (26) and chose the with r1;2 being the boundaries of the classically forbidden prefactor to match the low results of Sec. II B (this value region. We will only follow the leading exponential depen- I also agrees with that of [19,32]). As we already said, the dence on eÀ and do not aim at calculating the normaliza- exponent in (27) is larger than that in [19] by a factor of two. tion prefactor in front of the exponent. As explained in the Appendix, the rate (27) provides an Dynamics: wave scaering in rotang black holes 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 consequences 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 horizon, À 1 > 107r . Consequently, in many cases non- q Ingoing waves at the horizon, q Superradiance: black hole bomb srÀ g potential for the eigenvalue problem in the rotating black hole linear effects, both gravitational, and due to axion self- outgoing waves at infinity when 0 < ω < mΩ background. Superradiant modes are localized in a potentialH well interactions, become important in the regime where the q regionDiscrete spectrum of damped created by the mass ‘‘mirror’’q Hydrogen-like, from the spatial infinity on theexponenals (“ right, and byringdown the centrifugal”) barrierunstable bound states from the ergo-region and horizon[EB++, 0905.2975] on the left. [Detweiler, Zouros+Eardley…] 1Note, that at this stage we still agree with [19].

044026-9 Are massive scalar ﬁelds viable? Bounds from: ü Shapiro me delay: ω >40,000 [Perivolaropoulos, 0911.3401] BD 3 ü Lunar Laser Ranging ü Binary pulsars: ω >25,000 [Freire++, 1205.1450] 5 BD 10 0 10 Cassini 4 J1141-6545 10 J1012+5307 LLR -1 3 10 +3/2) 10 ξ BD ω 2 -2 10 10

1 10 -3 10 0 10 Upper bound on Cassini J1141-6545 -4 Lower bound on ( -1 J1012+5307 10 10 LLR

-2 -5 10 -21 -20 -19 -18 -17 -16 -15 10 -21 -20 -19 -18 -17 -16 -15 10 10 10 10 10 10 10 10 10 10 10 10 10 10 ms(eV) [Alsing, EB, Will & Zaglauer, 1112.4903] ms(eV)

FIG. 1. Left: Right: Massive black hole mergers: black hole spectroscopy [Visualizaon: NASA Goddard] q In GR, black holes oscillate in a set of complex-frequency modes determined only by mass and spin q One mode: (M,a) Any other mode frequency: No-hair theorem test Relave mode amplitudes: pre-merger parameters [Kamaretsos+,Gossan+] q Feasibility depends on SNR: -2 6 f = 1.2 x 10 (10 Msun)/M Hz Need SNR>30 [EB+, 2005/07] 6 τ = 55 M/(10 Msun) s 1) Noise S(fQNM) 1/2 2) Signal h∼E , E=εrdM 2 2 εrd∼0.01(4η) for comparable-mass mergers, η=m1m2/(m1+m2) (e/u)LISA vs. AdLIGO: strongest tests from space -2 6 f = 1.2 x 10 (10 Msun)/M Hz 6 τ = 55 M/(10 Msun) s

[Schutz] SNR=h/S: S comparable, h∼ηM1/2 eLISA: Astrophysics and cosmology in the millihertz regime

SNR for eLISA/NGO

20 50 20 18 10 10 16 20 14 50 50 20 10 z 12 100 10 10 20 50 50 100 Redshift 8 20 10 200 6 10 100 20 4 100 50 50 300 200 300 2 20 10 200 1000 100 500 2500 2500 2 3 4 5 6 7 8 9 10 log(M/M ) Figure 16: Constant-contour levels of the sky and polarisation angle-averaged signal-to-noise ratio (SNR) for equal mass non-spinning binaries as a function of their total mass M and cosmological redshift z. The total mass M is measured in the rest frame of the source. The SNR is computed using PhenomC waveforms (Santamaría et al., 2010), which are inclusive of the three phases of black hole coalescence (in jargon: inspiral, merger, and ring-down, as described in the text).

33 of 81 Ringdown: no-hair tests, merger dynamics and black hole formaon q LISA parameter esmaon studies: merger-tree models Light or heavy seeds? Accreon mode? [Arun+, 0811.1011] [Sesana+, 1011.5893] q based on these models, >10 binaries can be used for no-hair tests q Accurate spin observaons will constrain SMBH formaon [Barausse’s talk] q Ringdown could provide consistency tests with inspiral [Kamaretsos+, Gossan+] q Possible evidence for nonlinearies in general relavity [London+, 1404.3197] [EB, 1302.5702] Constraint maps eLISA measures redshied masses and spins Angular moon and three-arm detector:

ü Measure (luminosity) distance DL(z,cosmology) ü Locate source

Assuming cosmology is known, ﬁnd z(DL) and remove degeneracy Problem: weak lensing errors larger than eLISA measurement errors Possible soluon: redshi from EM counterparts

* Each SMBH ringdown measurement is a unique (local) probe of strong-ﬁeld GR on cosmological scales * Can tell ΛCDM from modiﬁed gravity models * Lower bounds on z are good enough

Need a three-arm LISA-like instrument [Arun+, 0811.1011]: ü Source localizaon within 10 deg2 or even 1 deg2 ü Distance determinaon within ∼10% [see Sathya’s talk, Colpi’s talk; Sesana 1209.4671] RAPID COMMUNICATIONS

PHYSICAL REVIEW D 84, 101501(R) (2011) Graviton mass bounds from space-based gravitational-wave observations of massive black hole populations

Emanuele Berti,1,2,* Jonathan Gair,3,† and Alberto Sesana4,‡ 1Department of Physics and Astronomy, The University of Mississippi, University, Mississippi 38677, USA 2California Institute of Technology, Pasadena, California 91109, USA 3Institute of Astronomy, University of Cambridge, Cambridge, CB3 0HA, UK 4Max-Planck Institut fu¨r Gravitationasphysik (Albert-Einstein-Institut), Am Mu¨hlenberg 1, D-14476, Potsdam, Germany RAPID COMMUNICATIONS (Received 18 July 2011; published 4 November 2011) PHYSICAL REVIEW D 84, 101501(R) (2011) We study the bounds that space-based gravitational-wave detectors could realistically place on the Graviton mass bounds from space-based gravitational-wave observations graviton Compton wavelength !g h= mgc by observing multiple inspiralling black hole binaries. ¼ ð Þ 15 of massive black hole populations Observations of individual inspirals will yield mean bounds !g 3 10 km, but the combined bound Â 16 from observing 50 events in a two-year mission is about ten times better: !g 3 10 km (mg Emanuele Berti,1,2,* Jonathan Gair,3,† and Alberto Sesana4,‡ ’ Â ’ 4 10 26 eV). The bound improves faster than the square root of the number of observed events, because 1Department of Physics and Astronomy, The University of Mississippi, University, Mississippi 38677, USA Â À 2California Institute of Technology, Pasadena, California 91109, USA typically a few sources provide constraints as much as three times better than the mean. This result is only 3Institute of Astronomy, University of Cambridge, Cambridge, CB3 0HA, UK mildly dependent on details of black hole formation and detector characteristics. The bound achievable in 4Max-Planck Institut fu¨r Gravitationasphysik (Albert-Einstein-Institut), Am Mu¨hlenberg 1, D-14476, Potsdam, Germany (Received 18 July 2011; published 4 November 2011) practice should be an order of magnitude better than this figure, because our calculations ignore the merger/ringdown portion of the waveform. We study the bounds that space-based gravitational-wave detectors could realistically place on the graviton Compton wavelength ! h= m c by observing multiple inspiralling black hole binaries. g ¼ ð g Þ DOI: 10.1103/PhysRevD.84.101501 PACS numbers: 04.30.Tv, 04.50. h, 04.70.Bw, 04.80.Cc Observations of individual inspirals will yield mean bounds ! 3 1015 km, but the combined bound g Â À from observing 50 events in a two-year mission is about ten times better: ! 3 1016 km (m g ’ Â g ’ 4 10 26 eV). The bound improves faster than the square root of the number of observed events, because Â À typically a few sources provide constraints as much as three times better than the mean. This result is only mildly dependent on details of black hole formation and detector characteristics. The bound achievable in The formulation of gravitational theories with non- Our analysis will show that, in hierarchical models of practice should be an order of magnitude better than this figure, because our calculations ignore the zero mass for the graviton that are consistent with cosmo- massive black hole (BH) formation [6,7], the bound on !g merger/ringdown portion of the waveform. logical observations is an important open problem. from GW observations of a population of merger events is DOI: 10.1103/PhysRevD.84.101501 PACS numbers: 04.30.Tv, 04.50. h, 04.70.Bw, 04.80.Cc Attempts to construct such theories led to well-known about an order of magnitude better than the mean bound À conceptual difficulties, such as the so-called van Dam- from GW observations of individual mergers. Veltman-Zakharov discontinuity [1–3], due to the fact that To put our results in context, in Table I we present a The formulation of gravitational theories with non- Our analysis will show that, in hierarchicalthe models helicity-0 of component of the graviton does not de- nonexhaustive summary of current and proposed bounds zero mass for the graviton that are consistent with cosmo- massive black hole (BH) formation [6,7], the boundcouple on from!g matter when the putative mass of the graviton logical observations is an important open problem. from GW observations of a population of merger events is on !g that do not rely solely on GW observations. Attempts to construct such theories led to well-known mg 0. To circumvent pathologies related to the van These bounds can be roughly divided into three classes. about an order of magnitude better than the mean! bound conceptual difficulties, such as the so-called van Dam- from GW observations of individual mergers. Dam-Veltman-Zakharov discontinuity, various versions (i) Static bounds. If the graviton has nonzero mass, Veltman-Zakharov discontinuity [1–3], due to the fact that To put our results in context, in Table I weof present Lorentz-violating a massive graviton theories have been it is reasonable to expect that the Newtonian gravitational the helicity-0 component of the graviton does not de- nonexhaustive summary of current and proposed bounds proposed in recent years [4]. Massive graviton signatures potential will be modified to the Yukawa form in the non- couple from matter when the putative mass of the graviton on !g that do not rely solely on GW observations. mg 0. To circumvent pathologies related to the van These bounds can be roughly divided into threein the classes. CMB and possible constraints on mg from cosmo- radiative near zone of any body of mass M: V r ! Dam-Veltman-Zakharov discontinuity, various versions ð Þ¼ (i) Static bounds. If the graviton has nonzerological mass, observations are an active area of research (see, GM=r exp r=!g . Talmadge et. al. [9] investigated de- of Lorentz-violating massive graviton theories have been it is reasonable to expect that the Newtonian gravitationale.g., [5]). ð Þ ðÀ Þ proposed in recent years [4]. Massive graviton signatures viations from Kepler’s third law for the inner planets of the potential will be modified to the Yukawa form inIn the this non- paper we are interested in hypothetical massive in the CMB and possible constraints on mg from cosmo- radiative near zone of any body of mass M: V r Solar System. By translating the uncertainties in these ð Þ¼ logical observations are an active area of research (see, GM=r exp r=! . Talmadge et. al. [9] investigatedgraviton de- theories as ‘‘straw men’’ for alternative theories of ð Þ ðÀ gÞ e.g., [5]). viations from Kepler’s third law for the inner planetsgravity of the in which the propagation speed of gravity differs In this paper we are interested in hypothetical massive Solar System. By translating the uncertaintiesfrom in thatthese of electromagnetic waves, leading to a modified TABLE I. Graviton mass bounds. For proposed methods we graviton theories as ‘‘straw men’’ for alternative theories of dispersion relation. Therefore we will adopt a phenomeno- quote the best achievable bounds. In the notation of the main gravity in which the propagation speed of gravity differs text, a dagger (†) denotes static bounds; a number sign (#) TABLE I. Graviton mass bounds. For proposed methods we from that of electromagnetic waves, leading to a modified logical point of view and ask the following question: if the dynamical bounds; an asterisk (*) bounds that could be achieved dispersion relation. Therefore we will adopt a phenomeno- quote the best achievable bounds. In the notation of the main text, a dagger (†) denotes static bounds; a numbergraviton sign (#) mass were nonzero, what upper bounds on mg by comparing GW and electromagnetic observations. logical point of view and ask the following question: if the dynamical bounds; an asterisk (*) bounds that couldcould be achieved we set by gravitational-wave (GW) observations of graviton mass were nonzero, what upper bounds on mg by comparingBlack GW and electromagnetichole mergers observations. and graviton mass bounds Current bounds ! km m eV Reference could we set by gravitational-wave (GW) observations of inspiralling compact binaries with future space-based de- g½ g½ Current bounds !g km mg eV tectors?Reference Using ! h= m c , upper limits on the graviton 10 20 inspiralling compact binaries with future space-based de- ½ ½ g g Binary pulsars# 1:6 10 7:6 10À [8] tectors? Using ! h= m c , upper limits on the graviton 10 20 ¼ ð Þ Â 12 Â 22 g g Binary pulsars# 1:6 10 7:6 10À mass[8m] g (in eV) can be expressed as lower limits on its Solar system† 2:8 10 4:4 10À [9,10] ¼ ð Þ Â 12 Â 22 mass mg (in eV) can be expressed as lower limits on its Solar system† 2:8 10 4:4 10À [9,10] Â 19 Â 29 1 Â 19 Â 29 1Compton wavelength !g (in km), since Clusters† 6:2 10 h0 2:0 10À h0À [11] Compton wavelength !g (in km), since Clusters† 6:2 10 h0 2:0 10À h0À [11] Â 22 Â 32 Weak lensing† 1:8Â 1022 6:Â9 10 32 [12] Weak lensing† 1:8 10 6:9 10À [12] 9 À 9 Â Â !g km mg eV 1:24 10À : (1) Â Â !g km mg eV 1:24 10À : (1) ½ Â ½ ¼ Â Proposed bounds ! km m eV Reference ½ Â ½ ¼ Â Proposed bounds ! km m eV Reference g½ g½ g½ g½ 13 23 Pulsar timing# 4:1 10 3:0 10À [13] Pulsar timing# 4:1 1013 3:0 10 23 [13] * Â 14 RAPIDÂ COMMUNICATIONS24 RAPID COMMUNICATIONS À [email protected] White dwarfs* 1:4 10 8:8 10À [14] † * Â 14 Â 24 [email protected] 15Â 16 24Â 25 [email protected] White dwarfs* 1:4 10 8:8 10À [14] EM counterparts* 10 –10 10À –10À † [15] ‡[email protected] BERTI, JONATHAN GAIR, AND ALBERTO SESANA PHYSICAL REVIEW D 84, 101501(R) (2011) [email protected] MASS BOUNDS FROM SPACE-BASED ... PHYSICALEM REVIEW counterparts* D 84, 101501(R)10 (2011)15Â–1016 10 24Â–10 25 [15] -14 ‡ À À 10 One Michelson Two Michelsons [email protected] Michelson Two Michelsons would suggest that N identical sources should provide a Classic LISA -15 400 LISA bound pN times better than the bound from a single 10 New LISA C2 5000 LISA SE SE New LISA C5 New LISA C2 New LISA C2 New LISA C5 source. Our combined bound is typically about 3 times 4000 New LISA C5 300 ffiffiffiffi

) -16 1550-7998=201110=84(10)=101501(6) 101501-1 Ó 2011 American Physical Society better than the bound from the best event, but the median

-1/2 3000 200 bound is typically an order of magnitude worse than -17 (Hz 10 2000 the best, and hence 30 times worse than the combined 1/2 1550-7998100 =2011=84(10)=101501(6) 101501-1 2011 American Physical Society 1000 Ó (f)] -18 bound. A typical realization has 50 events, so our analy- 10 NSA 0 0 n sis shows that we can beat the pN extrapolation from the -19 400 [S 5000 LE LE 10 median bound by a considerableffiffiffiffi margin. If 5=6 links -20 4000 300 (two Michelsons) are available instead of 4 links (one 10 3000 Michelson), the bound typically improves by a factor p2. 200 -21 10 2000 -5 -4 -3 -2 -1 0 ffiffiffi 10 10 10 10 10 10 100 1000 f (Hz) III. CONCLUSIONS AND OUTLOOK 0 0 15 16 15 16 16 17 16 17 10 10 10 10 10 10 10 10 We assessed the capability of future space-based inter- FIG. 1 (color online). Non-sky-averaged noise power spectral λ λ λ λ ferometers, such as ‘‘Classic LISA’’ and the proposed density SNSA f for New LISA C2 (black, solid line), New g g g g ESA-led ‘‘New LISA,’’ to constrain the mass of the gravi- LISA C5 (blue,ð Þ dash-dotted line) and Classic LISA (red, dashed line). FIG. 2 (color online). Distribution of bounds on the graviton FIG. 3 (color online). Distribution of combined bounds over ton by combining observations of a population of massive Compton wavelength for individual observations. We consider 1000 realizations of the MBH population. Line styles are as in BH binaries. We found that: (1) by using a population of 1000 realizations of the massive BH population and three differ- Fig. 2. merging BH binaries we can obtain a bound on !g that is ent detector designs: New LISA C2 (black, thick line), New 10 times better than the mean bound on individual ob- Here we consider the ‘‘Classic LISA’’ design along with LISA C5 (blue, medium line) and Classic LISA (red, dashed servations; (2) quite independently of the detector’s design two different designs for the proposed ESA-led space- For each model we consider 1000 realizations of the line). and of details of the massive BH formation models, the based detector, that we will call by the working name Universe. Each of these realizations typically produces combined bound from inspiral observations will be ! of ‘‘New LISA.’’ ‘‘Classic LISA’’ consists of three space- 30–50 events observable with signal-to-noise ratio larger g ’ The non-sky-averaged noise power spectral densities for 3 1016 km. This figure is likely to underestimate the craft forming an equilateral triangle with laser power than 8. The distribution of bounds on !g resulting from Â P 2W, telescope diameter d 0:4mand armlength all three configurations are shown in Fig. 1; they are related individual observations is shown in Fig. 2. bound achievable in practice by about an order of magni- ¼ 9 ¼ to the sky-averaged power spectral density by SNSA f tude, as we have ignored the merger and ringdown portion L 5 10 m, trailing 20 behind the Earth at an incli- ð Þ¼ The top half of Table II shows that the mean bound ¼ Â 3 SSA f (see [19] for a discussion of sky averaging). over individual observations is 3 1015 km for New of the waveform [24], but further work is required to nation of 60 with respect to the ecliptic. The authors are 20 ð Þ members of a Science Performance Task Force that is These curves include galactic confusion noise, estimated LISA C2, and only slightly better forÂ the other designs. confirm this expectation. considering several different LISA-like configurations using methods similar to [34] (which in turn was based on This conclusion is quite robust, in the sense that numbers In conclusion, space-based observations of a population of merging BHs should set bounds in the range ! with different characteristics and sensitivities. The con- [35]). In our study we consider the noise power spectral vary only mildly for different seeding mechanisms and g 2 2 1016; 1018 km on the graviton Compton wavelength, figurations that we call New LISA C2 (C5, respectively) density to be infinite below a cutoff frequency fcutoff different detector characteristics. ½ Â 5 ¼ depending on details of the detector and on the specific consist of three spacecraft forming an equilateral triangle 10À Hz. As shown in [19], bounds on !g drop signifi- In most alternative theories, deviations from general with armlength L 109 m (L 2 109 m), laser power cantly at masses * 2 106M if the noise cannot be relativity can be parametrized by one or more global waveform model used to set the bounds. This is compa- ¼ ¼ Â 4 Â rable to the (static and model-dependent) bounds from P 2Wand telescope diameter d 0:4m(d 0:28 m). trusted below 10À Hz. This assumption has a mild effect parameters (such as ! ) which are the same for every ¼ ¼ ¼ g cosmological-scale observations quoted in Table I but it ‘‘New LISA’’ should be deployed 10 behind the Earth, on our results, because most binaries in our models have system. It is natural to expect that one can obtain better is very different in nature, because gravitational radia- gradually drifting to 25 behind the Earth in five years. mass lower than this. constraints on these parameters, as well as other universal $ tion tests the dynamical regime of Einstein’s general constants, by combining multiple observations (see, e.g., relativity. TABLE II. Top: mean (in parentheses: median) bound on !g for different BH formation [25,28,29,36–39]). Assuming that estimates for individual models, using one or two detectors, in units of 1015 km. Bottom: mean (in parentheses: median) sources are independent and Gaussian posteriors for each ACKNOWLEDGMENTS of the combined bound on !g over 1000 realizations of the massive BH population, in units source, consistent with the Fisher matrix approximation, 16 2 of 10 km. the width ' of the combined posterior on 1=!g is given by E. B. is supported by NSF Grant No. PHY-0900735 and 2 2 2 Mean (median) of individual events (1015 km) 'À i'iÀ , where 'i is the width of the posterior for by NSF CAREER Grant No. PHY-1055103. J. G.’s work is the ith¼ source. The bound on ! can thus be obtained by P g supported by the Royal Society. The authors wish to thank Detector SE, 1 Mich. LE, 1 Mich. SE, 2 Mich. LE, 2 Mich. adding the individual bounds in quadrature. The results are Martin Elvis and the ‘‘New LISA’’ Science Performance Classic LISA 4.26(2.60) 6.83(5.77) 4.87(2.72) 9.13(7.72) shown in Fig. 3 and in the bottom half of Table II. The Task Force for discussions, Marta Volonteri for sharing her New LISA C2 3.03(2.44) 3.62(3.27) 3.60(2.80) 4.76(4.29) combined bound obtained from the whole BH population is BH formation models, and the Aspen Centre for Physics New LISA C5 3.41(2.53) 4.63(4.13) 4.02(2.84) 6.15(5.48) about an order of magnitude better than the average bound (where this work was started) for providing a very stimu- Mean (median) of combined bound (1016 km) obtained from typical observations. A rough estimate lating environment. Detector SE, 1 Mich. LE, 1 Mich. SE, 2 Mich. LE, 2 Mich. Classic LISA 4.93(4.87) 5.67(5.59) 6.51(6.45) 7.50(7.37) New LISA C2 2.29(2.25) 2.73(2.71) 3.09(3.04) 3.66(3.64) New LISA C5 3.10(3.07) 3.64(3.62) 4.16(4.12) 4.85(4.82)

101501-4 101501-5 RAPID COMMUNICATIONS

EMANUELE BERTI, JONATHAN GAIR, AND ALBERTO SESANA PHYSICAL REVIEW D 84, 101501(R) (2011) -14 10 One Michelson Two Michelsons Classic LISA -15 10 New LISA C2 5000 LISA SE New LISA C5 New LISA C2 4000 New LISA C5

) -16 10

-1/2 3000 -17 (Hz 10 2000 1/2 1000 (f)] -18 10

NSA 0 n -19 [S 10 5000 LE

-20 4000 10 3000 -21 10 -5 -4 -3 -2 -1 0 2000 10 10 10 10 10 10 1000 f (Hz) 0 15 16 15 16 10 10 10 10 FIG. 1 (color online). Non-sky-averaged noise power spectral λ λ density SNSA f for New LISA C2 (black, solid line), New g g LISA C5 (blue,ð Þ dash-dotted line) and Classic LISA (red, dashed line). FIG. 2 (color online). Distribution of bounds on the graviton Compton wavelength for individual observations. We consider 1000 realizations of the massive BH population and three different detector designs: New LISA C2 (black, thick line), New Here we consider the ‘‘Classic LISA’’ design along with LISA C5 (blue, medium line) and Classic LISA (red, dashed two different designs for the proposed ESA-led space- line). based detector, that we will call by the working name of ‘‘New LISA.’’ ‘‘Classic LISA’’ consists of three spacecraft forming an equilateral triangle with laser power The non-sky-averaged noise power spectral densities for P 2W, telescope diameter d 0:4mand armlength all three configurations are shown in Fig. 1; they are related ¼ 9 ¼ to the sky-averaged power spectral density by SNSA f L 5 10 m, trailing 20 behind the Earth at an incli- ð Þ¼ ¼ Â 3 SSA f (see [19] for a discussion of sky averaging). nation of 60 with respect to the ecliptic. The authors are 20 ð Þ members of a Science Performance Task Force that is These curves include galactic confusion noise, estimated considering several different LISA-like configurations using methods similar to [34] (which in turn was based on with different characteristics and sensitivities. The con- [35]). In our study we consider the noise power spectral figurations that we call New LISA C2 (C5, respectively) density to be infinite below a cutoff frequency fcutoff 5 ¼ consist of three spacecraft forming an equilateral triangle 10À Hz. As shown in [19], bounds on !g drop signifi- with armlength L 109 m (L 2 109 m), laser power cantly at masses * 2 106M if the noise cannot be P 2Wand telescope¼ diameter¼d Â0:4m(d 0:28 m). trusted below 10 4 Hz.Â This assumption has a mild effect ¼ ¼ ¼ À ‘‘New LISA’’ should be deployed 10 behind the Earth, on our results, because most binaries in our models have gradually drifting to 25 behind the Earth in five years. mass lower than this. $ Black hole mergers and graviton mass bounds

TABLE II. Top: mean (in parentheses: median) bound on !g for different BH formation models, using one or two detectors, in units of 1015 km. Bottom: mean (in parentheses: median) of the combined bound on !g over 1000 realizations of the massive BH population, in units of 1016 km.

Mean (median) of individual events (1015 km) Detector SE, 1 Mich. LE, 1 Mich. SE, 2 Mich. LE, 2 Mich. Classic LISA 4.26(2.60) 6.83(5.77) 4.87(2.72) 9.13(7.72) New LISA C2 3.03(2.44) 3.62(3.27) 3.60(2.80) 4.76(4.29) New LISA C5 3.41(2.53) 4.63(4.13) 4.02(2.84) 6.15(5.48) Mean (median) of combined bound (1016 km) Detector SE, 1 Mich. LE, 1 Mich. SE, 2 Mich. LE, 2 Mich. Classic LISA 4.93(4.87) 5.67(5.59) 6.51(6.45) 7.50(7.37) New LISA C2 2.29(2.25) 2.73(2.71) 3.09(3.04) 3.66(3.64) New LISA C5 3.10(3.07) 3.64(3.62) 4.16(4.12) 4.85(4.82)

101501-4 [Will, gr-qc/9709011; many iteraons… EB+,1107.3528] Extreme mass-rao inspirals By the no-hair theorem of GR, a black-hole spaceme has (mass and current) mulpoles determined only by mass and spin:

l l+1 Ml+iSl=(ia) M eLISA-like detectors may observe

∼tens of 1-10Msun BHs (or NSs) 6 spiralling into ∼10 Msun BHs.

∼104-105 cycles, periapsis/orbital plane precession. Payoff: ü map Kerr spaceme from gravitaonal wave signal ü measure masses of stellar-mass BHs and low-mass SMBHs ü probe nature of central object (boson star/gravastar very different) ü test GR (NS inspirals emit dipole radiaon in scalar-tensor theories) ü probe astrophysical perturbaons (e.g. by a second SMBH) ü ergodicity of orbits, resonances ü smoking gun of alternave theories? [Amaro-Seoane+, astro-ph/0703495] Introduction Standard scalar-tensor theories Modified Newtonian Dynamics Conclusions Expected interferometer constraints on scalar-tensor gravity LISA tests vs. binary pulsar tests of scalar-tensor gravity matter 0| ϕ [Esposito-Farése] LLR 100

LIGO/VIRGO LIGO/VIRGO NS-BH NS-NS B1534+12 10 SEP

B1913+16

LISA NS-BH LLR 10 PSR-BH J0737–3039 J1141–6545 Cassini J1738+0333 10 ALLOWED THEORIES

matter 10 ϕ 0 ϕ −6 −4 −2 0 2 4 6 Binary pulsars and strong-ﬁeld scalar-tensor gravity Heraeus Seminar Gilles Esposito-Farese, IAP, CNRS, France • week ending PRL 107, 241101 (2011) PHYSICAL REVIEW LETTERS 9 DECEMBER 2011

Floating and Sinking: The Imprint of Massive Scalars around Rotating Black Holes

Vitor Cardoso,1,2 Sayan Chakrabarti,1 Paolo Pani,1 Emanuele Berti,2,3 and Leonardo Gualtieri4 1CENTRA, Departamento de Fı´sica, Instituto Superior Tećnico, Universidade Tećnica de Lisboa-UTL, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal 2Department of Physics and Astronomy, The University of Mississippi, University, Mississippi 38677, USA 3California Institute of Technology, Pasadena, California 91109, USA 4Dipartimento di Fisica, Universita` di Roma ‘‘Sapienza’’ & Sezione, INFN Roma1, Piazzale Aldo Moro 5, 00185, Roma, Italy (Received 18 August 2011; published 7 December 2011) We study the coupling of massive scalar fields to matter in orbit around rotating black holes.2 It is generally expected that orbiting bodies will lose energy in gravitational waves, slowly inspiraling into the black hole. Instead, we show that the coupling2 of the field2 to matter leads to a surprising effect: because of ness we focus on source terms of the form superradiance,where matter candr/dr hover into∗ = ‘‘floating∆/(r orbits’’+ a for), which the net gravitational energy loss at infinity is entirely provided by the black hole’s rotational energy. Orbiting bodies remain floating until they extract α ∗ dτ¯ (4) sufficient angular momentum from the black hole, or until perturbations or nonlinear effects disrupt the = mpδ (x X(¯τ)) , (3) Tlmω = t Slm(π/2)δ(r r0) mpδ(mΩp ω) , (9) T ! g¯(0) − orbit. For slowly rotating and nonrotating−U black holes floating− orbits are unlikely− to exist, but resonances at − orbital frequencies corresponding to quasibound states of the scalar field can speed up the inspiral, so that " the orbiting bodyand sinks. the These effective effects could potential be a smoking for gun wave of deviations propagation from generalV relativity.is corresponding to the trace of the stress-energy tensor of a given (e.g.) in [19]. Let us consider two independent point particle with mass m ,where¯g(0) is the backgroundDOI: 10.1103/PhysRevLett.107.241101r+ ∞ PACS numbers: 04.70. s, 04.25.Nx, 04.30. w, 04.80.Cc p solutions Xlmω and Xlmω to the homogeneousÀ equationÀ (Kerr) metric. In scalar-tensor theories, for example, satisfying the following boundary conditions: I. Introduction.—Massive scalars are ubiquitous in phys- hover in a ‘‘floating orbit’’ [9,10]. Here we show that α = 8π/(2 + ωBD)(s 1/2), where ωBD is the Brans- ∞,r 1− ics. For example, light scalars spanning several+ ordersik∞ of,H r∗floating orbits, for which the net gravitational energy loss Dicke" (BD) parameter and s is an object-dependent Xlmω e as r ,r+ , magnitude in mass are predicted in string-theory∼ scenarios at infinity→∞ is entirely provided by the BH’s rotational “sensitivity factor” [4, 19]. [1–3]. Massive scalars are observationally viable in scalar- energy, can exist for a wide range of scalar-field masses. where k = ω mΩ and k∞ = ω2 µ2.LetW Weak-field gravitational radiation circularizestensor generalizations the or- of Einstein’sH general relativity− H [4] Orbiting bodies will− floats until they extract sufficient an- bit (see below for a proof in the presentand context), can be regarded so we as anbe effective their propagating Wronskian. degree The of fluxesgular momentum of scalar" from energy the BH at or the until disruptive (perhaps consider equatorial circular orbits aroundfreedom a Kerr in f R BH,theorieshorizon [5,6]. In this and Letter at weinfinity consider are nonlinear) effects stop the process. When the BH rotates generic massiveð Þ scalar fields coupled to matter in orbit slowly the condition for superradiance at these resonances but most results apply to more general orbits. Using s r+,∞ 2 around a rotating black hole (BH). E˙ = mΩpkH,∞ Zis not met,, but we show that resonances(10) at small orbital r+,∞ | lmω | the “adiabatic approximation” we assumeAwell-known that the ra- phenomenon in BH physics is the Penrose∞,r+ frequencies∗ (corresponding to large positive scalar fluxes diation reaction timescale is much longerprocess than (for the particles) or- and the associatedr+,∞ superradiantX am-lmω (goingr0) S intolm(π the/2) horizon) still exist, and that they cause the Z α mp/M . (11) plification (for waves) [7,8]. Considerlm aω Kerr BH of mass tobject to2 inspiral2 faster. bital timescale, and compute the total energy flux E˙ T ≡− WU r + a IIA. Setup.—0 The process we consider is quite general. It for geodesic orbits. For prograde orbits, energy,M, angular angular momentum J aM, and horizon radius r " p 2 2 IIC.¼ Analytic solutionþ ¼ occurs at low in all frequencies. theories of gravity with Kerr BHs as back- momentum and frequency of a particle atMr = Mr reada , so that the angular velocity of the horizon The þ 0 À Massive bosonic fieldsground and solutionssuperradiant and a scalar field instabilies of mass s coupled to H a=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Mr (here andscalar below we flux set atG infinityc 1). A can be computed in the low- wave¼ with frequencyþ !µs, scalar ra- √M mentally by the Hulse-Taylordiation binaryStrongest dominates pulsar. instability This over follows: µ gravitationalsM∼1 radiation: compare Ωp = 3/2 . from energy balance:(6) if the orbital[Dolan energy, 0705.2880] of the particle is ˙ g a√M + r Eq. (12) with the standard quadrupole formula E∞ = 0 Ep, and the total (gravitational plus scalar)−5 energy2 flux2 is E_ E_ g E_ s, then 32/5(r0/M ) mp/M . This result10 is oblivious to the T For µs=1eV, M=Msun : µsM∼10 The four-velocity of the particle on a timelike¼ geodesicþ presence of the rotating BH. In fact, for ω >µs,the _ _ g _ s reads r2m U α =((r2 + a2)Q/∆ + a(L aE ), 0, 0,L Ep E ENeed0 light : scalars (or (1)primordial black holes!) 0 p 0 p p p þ fluxesþ ¼ at the horizon are negligible.FIG. 1 (color online). However, Pictorial for description fre- of floating orbits. 2 − 2 _ g _2s − aEp + aQ/∆), where ∆ = r 2Mr + Usuallya , Q E=(rE0 >+0, andquencies therefore the close orbit to shrinksµs, awith resonanceAn orbiting occurs body excites at [24]: superradiant scalar modes close to the 2 − þ Negave scalar flux at the horizon close to superradiant resonances at a )Ep aLp. time. However, it is possible that, due to superradiance, BH horizon. Since the scalar field is massive, the flux at infinity − E_ g E_ s 0. In this case E_ 0, and the particle can consists2 solely of gravitational radiation. IIB. Wave emission. Because of the coupling to p µsM þ ¼ ¼ω2 = µ2 µ2 ,n=0, 1, ... [(13)Detweiler 1980] matter, the orbiting object emits both gravitational and res s − s )l +1+n* scalar radiation. Gravitational radiation0031-9007 can be=11 com-=107(24)=241101(5) 241101-1 Ó 2011 American Physical Society ˙ s puted using Teukolsky’s formalism [22]. The relevant From Eq. (10) we see that Er+ < 0 in the superradiant equations and their solution are presented by Detweiler regime (kH < 0). Close to resonance we get (cf. also [24]) [23]. Here we focus on scalar wave emission. Defining ∞ − 2 X rl+1e µsMr/(l+1+n) . (14) lmω ∼ imφ−iωt Xlm(ω,r) ϕ(t, r, Ωp)= dωe Slm(θ) , (7) ! √r2 + a2 We have verified this result numerically, finding very &l,m good agreement with the analytical prediction. For the fundamental mode n =0,atresonance,wefind: we get the non-homogeneous equation for the scalar field

2 2 l+iP Γ[l +1]Γ[l +1 2iP ] d2 ∆ W i r+ + a (r+ r−) − , ∼ # − Γ[2l +1]Γ[1 2iP ] 2 + V Xlmω(r)= 2 2 3/2 Tlmω , (8) − 'dr∗ ( (r + a ) where P = 2Mr+kH /(r+ r−). Finally we can estimate the peak− flux close to the− resonant frequencies. At large distances and for l = m =1,n=0wefind 1 Measurements of the Shapiro time delay require ωBD > 40, 000 2 r0 2 ∼ O 3α m M for µs =0[20],butcouplingsoforderωBD (1) are obser- ˙ s,peak M p −17 Er+ , (15) vationally allowed when µs ! 10 eV, and no bounds on ωBD ∼− 2 "2 a M 3/2 −16 16πr+ (M a ) ( ) exist when µs ! 10 eV [21]. Considering a supermassive BH 2r+ r0 5 − + − , F of mass M ∼ 10 M⊙ and a typical sensitivity s ∼ 0.2, these −3 2 bounds translate into α " 8 · 10 when µs =0,α " 0.9when with =1+4P . Quite surprisingly the scalar flux at µ M>10−2 and no bounds on α when µ M>0.1. F s s the horizon grows in magnitude with r0 and it is negative, Floating and sinking: the imprint of massive scalars around rotating black holes

Vitor Cardoso,1, 2 Sayan Chakrabarti,1 Paolo Pani,1 Emanuele Berti,2, 3 and Leonardo Gualtieri4 1CENTRA, Departamento de F´ısica, Instituto Superior Técnico, Universidade Técnica de Lisboa - UTL, Av. Rovisco Pais 1, 1049 Lisboa, Portugal. 2Department of Physics and Astronomy, The University of Mississippi, University, MS 38677, USA. 3California Institute of Technology, Pasadena, CA 91109, USA 4Dipartimento di Fisica, Universitàdi Roma “Sapienza” & Sezione, INFN Roma1, P.A. Moro 5, 00185, Roma, Italy. We study the coupling of massive scalar fields to matter in orbit around rotating black holes. It is generally expected that orbiting bodies will lose energy in gravitational waves, slowly inspiralling into the black hole. Instead, we show that the coupling of the field to matter leads to a surprising effect: because of superradiance, matter can hover into “floating orbits” for which the net gravitational energy loss at infinity is entirely provided by the black hole’s rotational energy. Orbiting bodies remain floating until they extract sufficient angular momentum from the black hole, or until perturbations or nonlinear effects disrupt the orbit. For slowly rotating and nonrotating black holes floating orbits are unlikely to exist, but resonances at orbital frequencies corresponding to quasibound states of the scalar field can speed up the inspiral, so that the orbiting body “sinks”. These effects could be a smoking gun of deviations from generalrelativity.

I. Introduction. Massive scalars are ubiquitous in responsible for many interesting effects [9–16]. Here we physics. For example, light scalars spanning several or- explore the interesting possibility that an object in or- ders of magnitude in mass are predicted in string-theory bit around a rotating BH may excite superradiant modes scenarios [1–3]. Massive scalars are observationally vi- to appreciable amplitudes. As the object orbits around able in scalar-tensor generalizations of Einstein’s general the BH it loses energy in gravitational waves, slowly spi- relativity [4] and can be regarded as an effective propa- ralling in, as shown experimentally by the Hulse-Taylor gating degree of freedom in f(R) theories [5, 6]. In this binary pulsar. This follows from energy balance: if the paper we consider generic massive scalar fields coupled orbital energy of the particle is Ep, and the total (gravi- to matter in orbit around a rotating black hole (BH). ˙ ˙ g ˙ s tationalLight scalars plus scalar): floang energy orbits flux (Press & is ET =TeukolskyE + E 1972) ,then A well-known phenomenon in BH physics is the Pen- 3 g s rose process (for particles) and the associated superradi- E˙ p + E˙ + E˙ =0. (1) -2 ant amplification (for waves)−2 [7,˙ s, peak 8]. Consider a Kerr BH 10 µsMr0/M (resonance) (αmp/M ) Er αcrit -3 of mass M, angular momentum+J = aM and horizon ra- 10 ˙ g ˙ s −1 2 2 −1 Usually-4 E + E > 0, and therefore the orbit shrinks 10 4.33400288873563dius r+ = M + √M −0.a1828, so that the 1.1 angular· 10 velocity 10 with time.-5 However it is possible that, due to superradi- −2 − −3 10 10 21.4020987080510of the horizon ΩH =−a/0.48812Mr+ (here 1 and.6 · 10 below we set ˙ g ˙ s s ˙ ance, -6E + E = 0. In this case Ep = 0, and the particle -(dE /dt) -4 /dt) 10 11 r −3 G = c = 1). A wave with frequency ω

-2 10 amplified in a scattering process, the excess energy com- -7 at infinity-9 is entirely provided10 by the BH’s rotational en- TABLE I. Orbital radius at resonance and peak scalar flux for /M) 10 -8 ing from the BH’s rotational energy. Superradiance is p -10 10 ≪ ergy, can exist for a wide-9 range of scalar-field masses. n =0,l = m =1,a =0.99M and several values of µsM 1. (m 10 10 -11 -16 -16 Orbiting10 bodies will float until-2×10 they0 2 extract×10 sufficient an- For comparison, a typical extreme mass ratio inspiral becomes r /r -12 Δ 0 res ∼ gular10 momentum from the BH or until disruptive (per- detectablearXiv:1109.6021v1 [gr-qc] 27 Sep 2011 by space-based interferometers at radii r0/M -13 6 −4 −2/3 haps10 nonlinear) effects stop the process. When the BH 50 [(10 M⊙/M )(fcut/10 Hz)] ,wherefcut is the lower -14 cutoff for the sensitivity threshold of the interferometer. A rotates10 slowly5 the condition 10 for 15 superradiance 20 at 25 these r /M resonances is not met, but0 we show that resonances at floating orbit occurs for α > αcrit.Noticethatαcrit is well small orbital frequencies (corresponding[Cardoso++ 1109.6021; to large Yunes positive++, 1112.3351] below current observational bounds [21] for any µs. FIG. 2.scalar Dominant fluxes going fluxes into of the scalar horizon) and gravitational still exist, and energy that −2 (l = mthey=1and causel the= m object=2,respectively)for to inspiral faster.µsM =10 , α = −2 due to superradiance, at sufficiently large distances (for 10 andIIA.a Setup.=0.99MThe.Theinsetisazoomaroundresonance. process we consider is quite general. 2l−3/2 It occurs in all theories of gravity with Kerr BHs as back- generic l, the peak flux would scale as E˙ s,peak r ). ! r+ ∝ 0 ground solutions and a scalar field of mass µs coupled to For very small a the peak flux at resonance is instead pos- matter (see e.g. [17, 18]). At first order in perturbation itive, and it can also be very large: for the Schwarzschild theory, the field equations for the scalar field reduce to 4 ˙ s,peak 2 2 2 geometry, 32πM Er 3α r0mp. FIG. 1.+ Pictorial∼ description of floating orbits. An orbit- In the adiabatic approximation,! µ2 ϕ = α the. mass and angular(2) ing body excites superradiant scalar modes close to the BHmomentum of the background− s spacetimeT are constant. III. Floating orbits. From the previous discussion it ! " horizon. Since the scalar field is massive, the flux at infinityHowever, the negative energy flux at the horizon re- follows that,consists for any solelyµ M of gravitational1, there radiation. exists a frequency Our main results will be to a large extent independent of s duces the BH mass and angular momentum (δM<0, ω ! µ for which the total≪ flux E˙ s + E˙ s + E˙ g + the source term on the right-hand side, but for concrete- res s ∞ r+ ∞ δJ<0). In order to estimate how long a particle can ˙ g Er+ = 0, because the negative scalar flux at the horizon stay in a floating orbit we must go beyond the adiabatic is (in modulus) large enough to compensate for the other approximation. Under ideal conditions, floating would positive contributions. This expectation is confirmed by stop only when the peak of the scalar flux at the hori- a full numerical integration of Teukolsky’s equation: see zon is too small to compensate for the gravitational flux, Fig. 2 and Table I. The width of the peak is proportional ˙ g ˙ s E > Epeak . From the balance condition (1) we find to the imaginary part of the resonant mode ω µ4l+5 | | | | I s that δEp = 0, which, using Eq. (4), can be written as [24]. For l = 1, more explicitly, ∝ 2 8 δMr0 a +3(2M r0)r0 +2a r0/M (r0 3M) (µsM) δr = " − − $ , ωI = µs (a/M 2µsr+) . (16) 0 2 # 24 − M 3a 8a√Mr0 +(6M r0)r0 − − % & As µs 0 the imaginary part becomes tiny, and an accu- where we used the relation δM = Ω δJ, valid for → p rate fine-tuning is needed to numerically resolve the res- circular orbits. Substituting the equation above into onance. For example, to resolve the peak at r 100M, ˙ s ˙ g 0 Eq. (15) and approximating δM/δt Er+ = E∞ = −3 ∼ ∼ − corresponding to µsM =10 , we tuned the location of −5 2 2 32/5(r0/M ) mp/M at resonance, we obtain, for r0 to 25 decimal places. Computing the imaginary part of − l = m = 1 and in the limits µs 0anda 1/µs, the unstable modes when µ 0 is also challenging, but → ≫ s → we were able to obtain stable results for the resonance ˙ s, peak 2 4 3 δEr+ 12α M mp M location and for the height of the peak. A fit to numer- = 2 , δt − 5π a " M $ ' r0 ( ical results for 10M ! r0 ! 100M (cf. Table I) yields E˙ s, peak r0.51, to be compared with E˙ s, peak r1/2 in which is negative: BH mass loss decreases the height of r+ ∼ 0 r+ ∼ 0 Eq. (15). Close to floating orbits the peak on a timescale ˙ s, peak 2 7/2 Er+ 5a M r0 dEp dE˙ T , (Ep Ef ) , (17) ˙ s, peak 32 'm ( M ! δEr+ /δt ∼ p " $ dt ∼− − dEp ! !Ep=Ef ! which does not depend on the coupling constant α. ! where Ef is the energy of the particle at the floating The delayed inspiral may have observational conse- orbit, and we used the balance condition (1). During in- quences. In particular, notice that in the absence of spiral, right before reaching the floating orbit, the time scalar fields the evolution of orbital frequency scales as needed for the particle to increase its binding energy from ˙ 2 5/3 E ϵ to E diverges logarithmically. Therefore, float- ΩGR/Ωp 96/5(mp/M )(MΩp) . (18) | f |− | f | ∼ ing orbits are expected to last much longer than a typical Close to a floating orbit we find instead that inspiral timescale, with a potentially striking observational signature in the gravitational-wave spectrum. Ω˙ /Ω2 32(m /M )2(MΩ )7/3 . (19) floating p ∼ p p PANI et al. PHYSICAL REVIEW D 81, 084011 (2010) bital frequency. The source term can be written as dE E E_ R GW lim GW dt ¼ T T !1 Tlm !; r ! m!K SlmÃ ; 0 Ulm 1 dEGW ð Þ¼ ð À Þ 0 2 0 lim d! (2.21) ¼ T T d! lm !1 Xlm Z S ; 0 U 1 lmÃ lm ^ þÀ 2 1 can be written in terms of Alm ! as follows: À ð Þ m2 2 SlmÃ ; 0 Ulm ; (2.16) _ R 0 ^ 2 _ R þÀ 2 2 E m!K 2 Alm m!K Elm: À ð Þ¼ 4 m!K j ð Þj Xlm ð Þ Xlm where the functions sUlm are explicitly given in (2.22) Refs. [16,24]. 0 1 The solution of Eq. (2.14) satisfying the boundary con- In order to evaluate Élm and Élm, we integrate the BPT ditions of pure outgoing radiation at radial infinity and equation by an adaptive Runge-Kutta method. Close to a matching continuously with the interior solution can be resonance the solutions must be computed very accurately, found by the Green’s functions technique. The amplitude since the Wronskian (2.18) is the difference between two of the wave at radial infinity can be shown to be [15] terms that almost cancel each other. When required, the tolerance parameter in the adaptive integration routines is decreased to achieve convergence. Since the orbital fre- 1 1 dr0 1 Alm ! 2 Élm !; r0 Tlm !; r0 ; quency is related to the orbital velocity v and to the semi- ð Þ¼ÀW ! R Á ð Þ ð Þ lmð Þ Z latus rectum (which for circular orbits is simply (2.17) p R =M) by the relations ¼ 0 1=3 1=2 where Wlm ! is the Wronskian of the two independent v M!K pÀ ; (2.23) solutions ofð theÞ homogeneous BPT equation ¼ð Þ ¼ the energy flux E_ R can also be considered as a function of v 1 or p. In the following we shall normalize E_ R to the W ! É1 @ É0 É0 @ É1 : (2.18) lmð Þ¼Á ½ lm r lm À lm r lm Newtonian quadrupole energy flux 32 m2 The two solutions É0 and É1 satisfy different boundary _ N 0 10 lm lm E 2 v : (2.24) conditions: ¼ 5 M Then the energy flux emitted in gravitational waves nor- 0 0 3 i!r LBPTÉ !; r 0; É !; r r e Ã ; malized to the Newtonian quadrupole energy flux is given lmð Þ¼ lmð !1Þ¼ 1 1 by PANI et al. PHYSICAL REVIEW D 81, 084011 (2010) LBPTÉlm !; r 0; Élm !; a Élm !; a ; ð Þ¼ ð Þ¼ ð Þ TABLE I. Values of the compactness ", angular momentum pressure in the shell are far below Planckian and the 1 E_ R number5 l, QNM frequency,M2 orbital velocity v, and GW frequencyGRAVITATIONALgeometry WAVE ... can. II. EXTREME still be MASS described... reliably by Einstein’sPHYSICAL REVIEW D 81, 084011 (2010) @rÉlm !; a @rÉlm !; a : (2.19) #GW of the circular orbits which would^ excite the fundamental2l 6 producesequations a relative error[34]. (in Our the nonresonant simplified regime) modelthe does not0:15 allowmodel than for it a does for the 0:10 and P v QNM of the gravastar for the givenA multipole.m! The Keplerianof:¼ order(2.25)p 5 v10 (but see [30] for a more careful dis- 0:20 models.¼ ¼ ¼ ð Þ¼ ð Þ N 2 10 lm K Àfinite¼ thickness of the shell, and a microscopic model of ð ÞE_ ¼ 128frequency ism! given in mHzv and rescaledj ð to a gravastarÞj cussion mass of the convergence properties of the post- In Fig. 2 the l 2 and l 3 peaks for 0:20 are lm 6 K Newtonian series).finite shellsWhen is* required0:166 the ISCO for a iscareful located analysiswell separated of this in¼ frequency prob- ¼ and an ‘‘antiresonance’’¼ is M6 10 M . X ¼ ð Þ outside the gravastarlem. However, and we plot for the energy the sake flux up of to argument, the visible let to the us right consider of the l 2 resonance. The nature of Here is the differential operator on the left-hand side ¼ LBPT ISCO velocity vISCO 0:408 (corresponding to R0 6M). this antiresonance can be explained by a simple harmonic " lM!QNM v M6=M #GW (mHz) 1=’2 " 0 as a ‘‘thickness¼ parameter’’ describing The normalized energy flux (2.25) can beð computedÞ For less compact up¼ gravastars,À plots! of the energy flux are oscillator model [16]. In the inset of the left panel of Fig. 2 of the BPT Eq. (2.14) and Élm !; a is the radial perturba- 0.499 97 2 0.1339 0.4061 4.328 truncated athow the velocity far the corresponding gravastar to the shell location can of bewe relative plot both tothe the resonance BH and antiresonance as functions 3 0.1508 0.3691 4.873 the shell. horizon. A power-law fit of the QNMsof the Keplerian of a thin-shell orbital frequency of the particle M!K for ð Þ to v 1=p6 0:408, which corresponds to the innermostThe complex structure of the spectrum for values of 0:2 and l 2 (dashed green line). A fit using the tion of the Weyl scalar, constructed according to Eq. (2.9) gravastar in the limit 0 yields f ¼ 3:828¼ 0.499 98 2 0.1276 0.3996 4.123 smaller than about 0.2 is best understood by! considering simpleGW harmonicÂð oscillatorÂ model of Ref. [16] (red line) ’ the real and10 imaginary5 0:1073 parts. The of the lower weakly frequency damped QNM sensitivityreproduces limit the for qualitative LISA features of both resonance and stable circular orbit (ISCO)3 0.1429 at R 0.36256M. 4.616 The post-À in terms of the perturbed metric functions in the interior 0 frequencies of a gravastarÞ (see Fig. 3). For clarity in Fig. 3 antiresonance: in this specific case the fit gives 2M!R ffiffiffi is dictated by acceleration noise. Assuming lower fre- 6 $ 0.499 99 2 0.1180 0.3893¼ 3.812 we only plot weakly damped QNMs, but our general5 argu- 0:072575 and 24M!I 2 10À , while QNM calculations Newtonian expansion of the energy flux P v for particlesquency cutoffs of flow 10À , 3 10À , 10À , we$ findÂ ments apply also to the second, ‘‘ordinary’’ family of using the resonance method yield 2M!R 0:07257 and and evaluated at the (exterior) surface of the gravastar. The 3 0.1310 0.3521 4.232 ¼ Â 6 $ ð Þ QNMs (cf.that paper the I). In peaks particular, will from sweep Fig. 2 out and of Fig. the 6 LISA2M!I band4 when10À . in paper I it should be clear48 that QNMs will43 be excited for 38 Modes$ withÂ l>2 are¼ typically harder to excite because in circular orbit around0.499 995 2 Schwarzschild 0.1096 0.3799 BHs 3.543 has been9:6 10À , 2:7 10À , 2:0 10À , respectively. This crucial point here is that the boundary conditions at the low values of theÂ compactness andÂwhen is very closeÂ to of their higher frequencies and lower quality factors. the BH valueestimate 1=2 of. Besides the ‘‘minimum these ‘‘ultracompact’’ measurableHowever, deviation because from of the complexa ‘‘selection rules’’ illus- studied by several0.499 authors 999 2 [21 0.0941,29,30 0.3610]. The instability 3.041 of¼ shell of a gravastar are drastically different from the modes, onlyBH’’ QNMs is whose admittedly real part very lies below sensitivethe hori- to thetrated fitting in Fig. function3 for l we2, sometimes only resonances with ¼ 2 zontal lineuse in the and left panel it may (corresponding change to when twice one the considersl>2 will be thick-shell visible. When vs 0:1 and > 0:21 only circular orbits with R0 < 6M sets an upper boundISCO on orbital the frequency for a particle in circular orbit) can modes with l>2 can be excited,¼ and only narrow l 3 ingoing-wave boundary conditions that must be imposed be excited. gravastars, but it suggests that LISAresonances has the potential can be seen to in Fig. 2 when the compactness¼ velocity of the pointQuite mass. interestingly, If the the radius gravastars of that the ‘‘try harder’’ gravastarFigure to 3 clarifiesreveal is that solid the range surfaces of over replacing which QNMs horizons even0:25 (cf. when Fig. 4 these). 2 ¼ 2 at the horizon of a black hole. As discussed in paper I, look like a BH (in the sense that their shell is closer tocan the be excited depends on vs. For vs 0:1 (the case When l 2 the imaginary part of one QNM with vs solid surfaces are very¼ close to the location¼ of the ¼ Schwarzschild event horizon) are also those that give awayconsidered to produce the energy fluxes of Fig. 2) QNM 0:1 crosses zero within our numerical accuracy at the perturbations near the shell will in general contain a larger than the ISCO (this typically occurs for Schwarzschild radius, so that the energy density and 0.4 correction of order p À . Roughly speaking, a truncation at -4 Resonances in the flux! 10 A ! m A^ ! ! m! : (2.20) where !QNM is the QNM frequency. Thus we expect0.2 sharp lm 0 lm K 5 -5 1.15 10 ð Þ¼ ð Þ ð À Þ 10 l=2 3 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 µ=0.10 l=310 µ=0.29 peaks to appear at the values of v corresponding2 Data to the 4 µ=0.15 10 Fit µµ=0.31 µ l=2 1 10 µ=0.20 10 µ=0.33 0 Then the time-averaged energy flux excitation of the gravastar QNMsµ=0.25 for different10 valuesFIG. 3 of (color the online). Real part (left)µ=0.35 and imaginary part (right) of gravastar QNMs with l 2 as a function of compactness for 3 Black hole -1 1.102 µ=0.37 ¼ 10 10 several fixed values of vs (as indicated in the legend). For clarity in illustrating the ‘‘selection rules’’ that determine QNM excitation -2 µ=0.49 2 l=2 10 during inspiral we only show the weakly damped part of the QNM spectrum (compare Fig. 2 and Fig. 6 in paper I). For vs > 0:8 the 2 real part of the frequency is plottedBlack down hole to the critical minimum compactness at which the imaginary part crosses zero within our l=3 l=5 0.032 0.036 0.04 10 M numerical accuracy. The horizontal line at 2M! 0:2722 corresponds to twice the orbital frequency of a particle in circular orbit at ωK R ’ l=4 l=6,the m=4 ISCO: only QNMs1.05 below this line can be excited during a quasicircular inspiral. 1 P(v) P(v) 10 l=3, m=1 0 l=4 l=6 10 084011-7 084011-4 l=5 l=6 l=3 1.00 -1 10 -2 10 0.25 0.30 0.35 0.40 0.32 0.34 0.36 0.38 0.40 v [Pani+, 0909.0287; 1001.3031]v

FIG. 2 (color online). Left: The energy ﬂux (summed up to l 6) of GWs emitted by a small mass orbiting a thin-shell gravastar 2 ¼ with vs 0:1 and different values of " (plotted as a function of the particle orbital velocity v) is compared with the ﬂux for a Schwarzschild¼ BH. All peaks (with the exception of the last two peaks on the right) are due to the excitation of QNMs with l m. Right: same for v2 0:1 and selected values of " 0:29; 0:49 . No QNMs are excited in this range. ¼ s ¼ 2 ½

084011-6 Conclusions Space-based gravitational-wave detectors are a theorist’s dream for testing strong-field general relativity: 1) Large SNR (indisputable probe of Kerr dynamics) 2) Large redshift (constraint maps – probe strong-field GR at all z!)

Constraint maps need EM counterparts, i.e. source localization, i.e. three arms!

Was Einstein right? (…and about what?)

1) Theory of gravity ü Gravitational-wave detectors in space can probe polarization states, speed of propagation, energy flux… ü Alternative theories may have striking signatures (e.g. floating orbits)

2) Nature of black holes ü Massive black-hole mergers: black hole spectroscopy probes no-hair theorem throughout the Universe ü Extreme mass-ratio inspirals: map of Kerr spacetime in nearby Universe – relativistic version of Sgr A*