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
various materials toward local topography on Earth,
movable
laboratory
masses,
produce
a narrow
resonance
line whose shift could be accurately determined. Other experiments
13
galaxy [249, 19], and have reached levels of 3since
⇥ 10 1960
[2].measured
The resulting
upperof spectral lines in the Sun’s gravitational field and the change in
the shift
ummarized in Figure 1 (TEGP 14.1 [281]; for a rate
bibliography
of clocks
experiments
up to aloft on aircraft, rockets and satellites. Figure 3 summarizes the
of atomic
transported
The founda3ons of general rela3vity important redshift experiments that have been performed since 1960 (TEGP 2.4 (c) [281]).
TESTS OF
40
LOCAL POSITION INVARIANCE
TESTS OF THE
WEAK EQUIVALENCE PRINCIPLE
10-8
10-1
Eötvös
Renner
Free-fall
10-9
η
Null
Redshift
10-2
Fifth-force
searches
10-10
Pound
Snider
Boulder
Saturn
Princeton
10-11
Eöt-Wash
Moscow
10-12
α
10-3
Hmaser
Eöt-Wash
LLR
THE PARAMETER (1+γ)/2
10-4
Cli↵ord M. Will
10-13
Null
Redshift
Solar spectra
η=
10-14
[Will, gr-qc/0510072]
Millisecond Pulsar
PoundRebka
a1 -a 2
10-5
(a1+a2)/2
1.10
Radio
Clocks in rockets
spacecraft & planes
DEFLECTION
OF LIGHT
Optical
00
20
90
19
80
19
60
70
19
19
00
20
90
19
80
19
70
19
60
19
40
19
20
19
00
19
YEAR OF EXPERIMENT
TESTS OF
ed tests of theLOCAL
weak equivalence
principle,
showing bounds on ⌘, which measures
LORENTZ
INVARIANCE
1.05
YEAR OF EXPERIMENT
Δν/ν = (1+α)ΔU/c2
2X10-4
VLBI
1.00
ence in acceleration of di↵erent materials or bodies. The free-fall and Eöt-Wash
e originally performed to search for a fifth force (green region, representing many
Figure 3: Selected
tests
of local
The blue band shows evolving bounds on ⌘ for gravitating
bodies from
lunar
laser position invariance via gravitational redshift experiments, showing
JPL
Michelson-Morley
Joos
TPA
10-6
Centrifuge
Living Reviews in Relativity
Cavities
http://www.livingreviews.org/lrr-2006-3
10-10
δ
Brillet-Hall
10-14
Hipparcos
PSR 1937+21
a Doppler shift caused by complex convective and turbulent motions in the
this e↵ect is actually
photosphere and lower chromosphere, and is expected to be minimized by observing at the solar
10-18
SHAPIRO
TIME
DELAY
Implies gravity is a metric theory: gravity is space3me curvature 1.00
Living Reviews in Relativity
Hughes Drever
NIST
Voyager
Harvard
http://www.livingreviews.org/lrr-2006-3
U. Washington
10-22
10-26
Weak Equivalence Principle+ Local Lorentz Invariance+ After almost 50 years of inconclusive or contradictory measurements, the gravitational redshift
Posi8on Invariance= of solar spectral Local lines was finally
measured reliably.
During the early years of GR, the failure
to measure this e↵ect in solar lines was siezed upon by some as reason to doubt the theory.
Unfortunately, the
measurement is not E
simple.
Solar spectral lines P
arerinciple subject to the “limb e↵ect”,
Einstein’s quivalence a variation of spectral line wavelengths between the center of the solar disk and its edge or1.05
“limb”;
bounds on ↵, which measures degree of deviation of redshift from the formula ⌫/⌫ = U/c2 . In
null redshift experiments, the bound is on the di↵erence in ↵ between di↵erent kinds of clocks.
0.95
(1+γ)/2
10-2
δ=
1/c2
-1
10
20
00
20
90
19
80
19
70
19
60
19
40
19
20
19
00
19
YEAR OF EXPERIMENT
Best test of space8me curvature: 0.95
Cassini bound Cassini
(1X10-5)
Viking
1920
1940
1960
1970
1980
1990
YEAR OF EXPERIMENT
2000
Indirect detec3on of gravita3onal waves: the binary pulsar 1993 Nobel Prize to Hulse and Taylor: discovery of the binary pulsar 1913+16 Strongest test of GR: PSR J0348+0432, P=2.46hr, v/c=2x10-­‐3 [Antoniadis+, 1304.6875] [Weisberg+Taylor]
Strong field: gravita3onal field vs. curvature; probing vs. tes3ng -10
10
Neutron Stars
Black Holes
-13
10
-16
-19
10
-22
10
Grav Prob B
-25
AGN
10
Eclipse
Hulse-Taylor
-28
10
-31
10
Moon
-34
10
Over the Horizon
3 2
-2
ξ=GM/r c (cm )
10
Mercury
-37
10
-40
10
-15
10
-12
10
-9
10
-6
-3
0
10
10
10
2
ε=GM/rc [Psal8s, Living Reviews in Rela8vity] Gravita3onal-­‐wave tests Energy flux: E.g. scalar-­‐tensor theories predict dipole radia3on because (iner8al mass)≠(gravita8onal mass) Polariza3on: Up to six polariza3on states Propaga3on: If e.g. mgraviton≠0, gravita8onal waves would travel slower than EM waves Massive black hole inspirals can yield bounds mgraviton<∼10-­‐26eV (104-­‐106 beaer than Solar System) Direct detec3on: 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?
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 3me un3l the LISA Pathfinder mission (LPF) launch, and the need to maintain momentum in the U.S. community and guarantee a smooth transi3on to a joint NASA-­‐ESA mission. The panel recommends that NASA funding of LISA begin immediately, with con3nua3on beyond LPF con3ngent on the success of that mission. Timeline: pulsar 3ming arrays 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?
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 3me un3l the LISA Pathfinder mission (LPF) launch, and the need to maintain momentum in the U.S. community and guarantee a smooth transi3on to a joint NASA-­‐ESA mission. The panel recommends that NASA funding of LISA begin immediately, with con3nua3on beyond LPF con3ngent on the success of that mission. Timeline: space-­‐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?
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 3me un3l the LISA Pathfinder mission (LPF) launch, and the need to maintain momentum in the U.S. community and guarantee a smooth transi3on to a joint NASA-­‐ESA mission. The panel recommends that NASA funding of LISA begin immediately, with con3nua3on beyond LPF con3ngent on the success of that mission. Strong gravity tests:
“external” vs.“internal”!
“External” tests of general rela3vity – against what? (| |)@a
@
V (| |) + Lmat
, A (| |)gab
“External” t2ests of general rela3vity – against what? ⇤ abcd
|)RGB + f2 (|
1 (| principle §  +f
Ac8on §  Well-­‐posed §  Testable predic8ons §  Cosmologically viable L
= f0 ( )R
!( )@a @ a
|)Rabcd R
M ( ) + Lmat
⇥
2
, A ( )gab
⇤
+f1 ( )(R2 4Rab Rab + Rabcd Rabcd )
+f2 ( )Rabcd ⇤ Rabcd + Lorentz-­‐viola8ng vector fields… Alterna8ve theories usually: Introduce more corresponding
fields (scalars, vectors) r higher-­‐curvature Riemann
tensor
to othe
metric gab ,terms Rab
eed strong-­‐field tests! Challenge pillars of general rela8vity: and Ndenotes
matter
fields.
The
functions
fi (| |) (i =
§  Equivalence principle nciple § arbitrary,
but
they
are
not
all
independent
(in
Lorentz invariance (Einstein-­‐aether, TeVeS…) m can § beParity set equal
to one via field redefinitions without
conserva8on… [Gair+,1212.5575; Clifon+, 1106.2476] “Internal” tests: the black-­‐hole paradigm Compact massive objects in galac8c centers coevolve with galaxies Evidence based on correla8ons: MBH – σ, MBH – Lbulge Are these Kerr black holes? Good reasons for theore3cal bias Alterna3ves are ugly/indis3nguishable •  dense star clusters: unstable •  fermion stars: ruled out •  exo8c maaer: violate energy condi8ons [McConnell+, 1112.1078] •  naked singulari8es: unstable, violate causality •  boson stars/gravastars: forma8on? •  black holes in alterna8ve theories: ü  in scalar-­‐tensor theories (and many other theories), solu8ons are the same [Thorne-­‐Dykla, Hawking, So8riou-­‐Faraoni…] ü  when solu8ons are not the same (EDGB, Chern-­‐Simons) devia8ons are (most likely) astrophysically unmeasurable Figure 3 | Correlations of dynamically measured black hole masses and bulk properties of host galaxies. (a) Black hole mass, MBH , versus ste
dispersion, , for 65 galaxies with direct dynamical measurements of MBH . For galaxies with spatially resolved stellar kinematics, is the luminos
average within one effective radius (Supplementary Information). (b) Black hole mass versus V -band bulge luminosity, LV (L ,V , solar value),
type galaxies with direct dynamical measurements of MBH . Our sample of 65 galaxies consists of 32 measurements from a 2009 compilation 9 ,
with masses updated since 2009, 15 new galaxies with MBH measurements, and the two galaxies reported here. A complete list of the galaxie
Supplementary Table 4. BCGs (defined here as the most luminous galaxy in a cluster) are plotted in green, other elliptical and S0 galaxies are p
and late-type spiral galaxies are plotted in blue. The black hole masses are measured using dynamics of masers (triangles), stars (stars), or g
Error bars, 68% confidence intervals. For most of the maser galaxies, the error bars in MBH are smaller than the plotted symbol. The solid bla
shows the best-fitting power law for the entire sample: log10 (MBH / M ) = 8.29 + 5.12 log10 [ /(200 km s 1 )]. When early-type and late-ty
are fit separately, the resulting power laws are log10 (MBH / M ) = 8.38 + 4.53 log10 [ /(200 km s 1 )] for elliptical and S0 galaxies (dashed r
log10 (MBH / M ) = 7.97 + 4.58 log10 [ /(200 km s 1 )] for spiral galaxies (dotted blue line). The solid black line in (b) shows the best-fitting
log10 (MBH / M ) = 9.16 + 1.16 log10 (LV /1011 L ). We do not label Messier 87 as a BCG, as is commonly done, as NGC 4472 in the Vir
0.2 mag brighter.
The (massive) black-­‐hole paradigm: posi3ve evidence? Tests based on electromagne3c observa3ons: [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 accre3on: Radio flux implies that brightness temperature TB > 1010 K Radia8ng 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 accre8on cannot come from a surface – evidence for event horizon Proposed tests based on electromagne3c observa3ons: ü  Measurement of quadrupole moment/higher moments of Sag A*: [Will, Merria…] 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 solu3on in most alterna3ve theories We need to test the dynamics of the theory – gravita3onal waves! Dynamics: wave scadering in rota3ng black holes Ergo-region
Barrier
region
Potential Well
Exponential
growth region
Potential
“Mirror”
at r~1/µ
A
up
la
Fi
re
ce
of
de
Black Hole Horizon
[Arvanitaki+Dubovsky, 1004.3558] r*
th
Quasinormal modes: Massive scalar field: FIG. 7 (color online). The shape of the radial Schroedinger
ho
q Ingoing aves at eigenvalue
the horizon, problem
q Superradiance: lack hhole
ole bomb potentialwfor
the
in the rotating bblack
lin
outgoing w
aves a
t i
nfinity when 0
<
ω
< m
Ω
H
background. Superradiant modes are localized in a potential
well
in
q Discrete spectrum of damped q Hydrogen-­‐like, region created by the mass ‘‘mirror’’ from the spatial infinity on
exponen8als “ringdown”) unstable bound states and
the
right, and (by
the centrifugal barrier
from the
ergo-region
[EB++, 0905.2975] [Detweiler, Zouros+Eardley…] horizon
on the left.
Are massive scalar fields viable? Bounds from: ü  Shapiro 8me delay: ωBD>40,000 [Perivolaropoulos, 0911.3401] ü  Lunar Laser Ranging ü  Binary pulsars: ωBD>25,000 [Freire++, 1205.1450] 5
10
0
10
3
Upper bound on ξ
Lower bound on (ωBD+3/2)
4
10
10
2
10
1
10
0
10
10
-1
Cassini
J1141-6545
J1012+5307
LLR
10
10
10
-1
-2
-3
-4
-5
-2
10 -21
10
10
10
-20
10
-19
-18
-17
-16
-15
10 -21
10
10
10
10
10
ms(eV) [Alsing, EB, Will & Zaglauer, 1112.4903] Massive black hole mergers: black hole spectroscopy q In GR, black holes oscillate in a set of complex-­‐frequency modes determined only by mass and spin [Visualiza8on: NASA Goddard]
q One mode: (M,a) Any other mode frequency: No-­‐hair theorem test Rela8ve mode amplitudes: pre-­‐merger parameters [Kamaretsos+,Gossan+] q Feasibility depends on SNR: f = 1.2 x 10-­‐2 (106Msun)/M Hz Need SNR>30 [EB+, 2005/07] τ = 55 M/(106Msun) s 1) Noise S(fQNM) 2) Signal h∼E1/2, E=εrdM εrd∼0.01(4η)2 for comparable-­‐mass mergers, η=m1m2/(m1+m2)2 (e/u)LISA vs. AdLIGO: strongest tests from space f = 1.2 x 10-­‐2 (106Msun)/M Hz τ = 55 M/(106Msun) s [Schutz] SNR=h/S: S comparable, h∼ηM1/2 SNR for eLISA/NGO 20
50
18
10
20
16
20
Redshift z
50
50
14
20
10
12
100
10
20
50
8
10
6
10
100
20
10
50
200
100
4
50
2
10
2
3
20
100
4
20
300
200
500
10
100 50
200
300
1000
2500
5
6
7
log(M/M )
2500
8
9
10
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
Ringdown: no-­‐hair tests, merger dynamics and black hole forma3on q  LISA parameter es8ma8on studies: merger-­‐tree models Light or heavy seeds? Accre8on mode? [Arun+, 0811.1011] [Sesana+, 1011.5893] q  based on these models, >10 binaries can be used for no-­‐hair tests q  Accurate spin observa3ons will constrain SMBH forma3on [Barausse’s talk] q  Ringdown could provide consistency tests with inspiral [Kamaretsos+, Gossan+] q  Possible evidence for nonlineari3es in general rela8vity [London+, 1404.3197] [EB, 1302.5702] Constraint maps eLISA measures redshifed masses and spins Angular mo8on 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 solu8on: redshif 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 localiza8on within 10 deg2 or even 1 deg2 ü  Distance determina8on within ∼10% [see Sathya’s talk, Colpi’s talk; Sesana 1209.4671] text, a dagger (†) denotes static bounds; a number sign (#)
dynamical bounds; an asterisk (*) bounds that couldcould
be achieved
we set
by comparing GW and electromagnetic observations.
by gravitational-wave
Black hole mergers and graviton mass bounds (GW) obse
!g ½km(
Current bounds
Binary pulsars#
Solar system†
Clusters†
Weak lensing†
1:6 % 1010
2:8 % 1012
6:2 % 1019 h0
1:8 % 1022
inspiralling compact binaries with future space
mg ½eV( tectors?
Reference
Using !g ¼ h=ðmg cÞ, upper limits on t
7:6 % 10&20 mass[8]
mg (in eV) can be expressed as lower li
&22
4:4 % 10
[9,10]
Compton
wavelength !g (in km), since
&29
&1
2:0 % 10 h
[11]
6:9 %
!g ½km(
Proposed bounds
0
&32
10
[12]
mg ½eV(
Reference
!g ½km( % mg ½eV( ¼ 1:24 % 10&9 :
Pulsar timing#
4:1 % 1013
3:0 % 10&23
[13]
14
&24
RAPID
COMMUNICATIONS
8:8 % 10
[14]
White dwarfs*
1:4 % 10
*berti@phy.olemiss.edu
15
16
&24
&25
–10 D 84,10101501(R)
–10 (2011)† GRAVITON
[15]
EM counterparts*
10REVIEW
MASS BOUNDS FROM SPACE-BASED . . .
ANA
PHYSICAL
One Michelson
LISA
New LISA C2
New LISA C5
5000
4000
1-1
3000
‡
Two Michelsons
SE
jgair@ast.cam.ca.uk
One Michelson
alberto@aei.mpg.de
400
Two Michelsons
SE
LISA
New LISA C2
New LISA C5
300
! 2011 American Physical Society
200
2000
100
1550-7998=
2011=84(10)=101501(6)
1000
0
LE
5000
4000
0
400
LE
300
3000
200
2000
100
1000
0
15
16
10
10
λg
15
16
10
10
λg
0
16
10
17
10
λg
16
17
10
10
λg
ing to $25# behind the Earth in five years.
lower than this.
Black hole mergers and gmass
raviton 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
Classic LISA
New LISA C2
New LISA C5
SE, 1 Mich.
LE, 1 Mich.
SE, 2 Mich.
LE, 2 Mich.
4.26(2.60)
3.03(2.44)
3.41(2.53)
6.83(5.77)
3.62(3.27)
4.63(4.13)
4.87(2.72)
3.60(2.80)
4.02(2.84)
9.13(7.72)
4.76(4.29)
6.15(5.48)
Mean (median) of combined bound (1016 km)
Detector
Classic LISA
New LISA C2
New LISA C5
SE, 1 Mich.
LE, 1 Mich.
SE, 2 Mich.
LE, 2 Mich.
4.93(4.87)
2.29(2.25)
3.10(3.07)
5.67(5.59)
2.73(2.71)
3.64(3.62)
6.51(6.45)
3.09(3.04)
4.16(4.12)
7.50(7.37)
3.66(3.64)
4.85(4.82)
101501-4
[Will, gr-­‐qc/9709011; many itera8ons… EB+,1107.3528] Extreme mass-­‐ra3o inspirals By the no-­‐hair theorem of GR, a black-­‐hole space8me has (mass and current) mul8poles determined only by mass and spin: Ml+iSl=(ia)lMl+1 eLISA-­‐like detectors may observe ∼tens of 1-­‐10Msun BHs (or NSs) spiralling into ∼106Msun BHs. ∼104-­‐105 cycles, periapsis/orbital plane precession. Payoff: ü  map Kerr space8me from gravita8onal 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 radia8on in scalar-­‐tensor theories) ü  probe astrophysical perturba8ons (e.g. by a second SMBH) ü  ergodicity of orbits, resonances ü  smoking gun of alterna8ve theories? [Amaro-­‐Seoane+, astro-­‐ph/0703495] xpected interferometer constraints on scalar-tensor gravi
LISA tests vs. binary pulsar matter
tests of scalar-­‐tensor gravity 0|
ϕ
[Esposito-­‐Farése]
LLR
100
LIGO/VIRGO
NS-BH
LIGO/VIRGO
NS-NS
B1534+12
SEP
10
B1913+16
LISA NS-BH
PSR-BH
J0737–3039
J1141–6545
10
LLR
Cassini
J1738+0333
10
ALLOWED
THEORIES
10
−6
−4
−2
matter
0
2
4
6
0
ϕ
ϕ
occurs
in allfrequencies.
theories of gravity The
with Kerr BHs as ba
C.
Analytic
solution
at low
ngular
velocity of the
horizon
ground
solutions
and ainscalar
of mass @!s couple
Massive osonic fields nd superradiant instabili3es elow we
set at
G ¼infinity
cb¼
1). Acan
alar
flux
be acomputed
thefield
low!H incidentregime
on the BH
(where
equency
[25].
For r0 /M ≫ 1 and l = m = 1,
m number)
is
amplified
in
Superradiance when ωa < mΩH %
ess energy coming$from the
2
2 1 − µ2 r3 /M 3/2
αis M
erradiance
responsible for
s 0
2
s
m
(12)
Ė
=
Any l
ight s
calar c
an t
rigger a
p Θ(Ωp − µs ) ,
16].∞Here we
explore
the
in4
12πbomb (“bosenova”) r0
black h
ole bject in orbit around a rotating
modes
to appreciable ampli[Yoshino+Kodama, 1203.5070] here
Θ(x)
is
the
Heaviside
function. For generic modes,
ound
the
BH
it
loses
energy
in
large
distances
and for ω = mΩp > µs , scalar raspiraling
in, as shown experiStrongest instability: µgravitational
sM∼1 ation
dominates
radiation: compare
or
binary
pulsar.
This over
follows
[Dolan, 0705.2880] g
energy
of the
is
q.rbital
(12)
with
theparticle
standard
quadrupole formula Ė∞
=
nal plus scalar)
−5energy flux is
/5 (r0 /M ) m2p /M 2 . This result
is oblivious to the
10 For µs=1eV, M=Msun : µsM∼10
esence
of the rotating BH. In fact, for ω > µs , the
s
_
Need ight scalars (or primordial black holes!)
þE
¼
uxes
at0:lthe
horizon (1)
are
negligible.
However, for freFIG. 1 (color online). Pictorial description of floating or
erefore
the
orbit
shrinks
encies
close
to
µflsux , awith
occurs
orbiting
body excites
superradiant scalar
modes close
Nega3ve scalar aresonance
t the An
horizon close tat
o s[24]:
uperradiant resonances at to
e that, due to superradiance,
BH horizon. Since the scalar field is massive, the flux at infi
)
*2 solely of gravitational radiation.
E_ p ¼ 0, and the particle can
consists
µs M
2
2
2
[Detweiler 1980]
ωres = µs − µs
, n = 0, 1, ...
(13)
l+1+n
101(5)
241101-1
! 2011 American Physical Soc
g
s
tational
plus
scalar)
energy
flux
is
Ė
=
Ė
+
Ė
T
Light scalars: floa3ng orbits (Press & Teukolsky ,1then
972) Ėp + Ė g + Ė s = 0 .
3
(1)
-2
-2
(mp/M) (dElm/dt)
10
-3
g
s
10
Usually
Ė
+
Ė
> 0, and therefore the orbit shrinks
-4
with10time.
However it is possible that, due to superradi-5
10 g
s
ance, -6Ė + Ė-(dE
=/dt)
0.s In this case Ėp = 0, and the particle
-4
11
r+
10
10
can10
hover
in(dE
a “floating
orbit”
[9,
10].
Here
we
show
that
-7
g
-5
/dt) T
10
22
-6 net gravitational energy loss
-8 orbits, for which the
floating
10
10
-7
-9
10
at infinity
is
entirely
provided
by the BH’s rotational en10
-8
10
-10can exist for a wide
ergy,
-9 range of scalar-field masses.
10
10
-16 0
-16
-11
-2×10
2×10
Orbiting
bodies
will
float
until
they
extract
sufficient an10
Δr
/r
-12
0 res
gular
10 momentum from the BH or until disruptive (per-13
10 nonlinear) effects stop the process. When the BH
haps
-14
10
rotates slowly5 the condition
for
at
10
15 superradiance
20
25 these
resonances is not met, butr0/M
we show that resonances at
[Cardoso++ 1109.6021; Yunes++, 1112.3351]
! # 0:15. Correspondingly,
the l ¼ 2 QNM
peak
visible
can leave a signature
on the
gravitati
in the energy
flux
of
Fig.
2
occurs
later
in
the
inspiral
for
small, inspiraling compact objects.
see that, for each assigned l, a mode
when the orbital frequency satisfies the
m! ¼ !
2MωR
gravastars.
Quasicircular xtreme ass-­‐ra3o nspiral around a gflux
ravastar One may worry thatethe
resonancem
will
eventually iget
normalized
energy
PðvÞ as a f
out of the LISA band for gravastars having ! extremely
velocity for gravastar models with v
1.4
l=2
Gravastar m
odel d
epends o
n (
v
, µ
) close to the Schwarzschild value. The following
naive
s
ness in the range 0:1 & ! & 0:4
argument suggests that this is not the case. The ‘‘thick- 1.2 Schwarzschild BH. The total flux wa
Modes of tmodel
he shell excited hen predicts a 1.0 all multipoles (jmj ( l) and by tru
shell
gravastar’’
by Mazur
andw
Mottola
pffiffiffiffiffiffiffiffiffiffiffi
microscopic
but finite shell thickness ‘ ! LPl rS ’ 3 " 0.8 expansion at l ¼ 6. As discussed in
K% Þ1=2 cm, QNM
10#14
where LPl is the Planck scale and rS 0.6 multipole of order l contributes
to
ðM=M
ISCO
s the Schwarzschild radius, so that the energy density and 0.4 correction of order p2#l . Roughly sp
;
0.10
0.20
0.30
0.50
0.70
0.80
0.90
1.00
(2.26)
Resonances in the flux! 5
4
10
10
3
10
l=2
Data
Fit
2
10
1
10
0
10
-1
10
-2
10
l=3 l=5 0.032
1
l=4
0
l=4 l=6
10
10
l=3 103
l=2
2
10
l=2
µ=0.10
µ=0.15
µ=0.20
µ=0.25
Black hole
0.3
0.4
0.5
µ=0.29
µ=0.31
µ=0.33
µ=0.35
FIG. 3 (color online). Real part (left)
and imaginary part (right
2
µ
=0.37
in the legend). For clarity
several fixed values of vs (as indicated
µ=0.49
during inspiral we only show the weakly
damped part of the QNM
Black hole
real
part
of
the
frequency
is
plotted
down
to the critical minimum
0.04
µ
numerical accuracy. The horizontal line at 2M!R ’ 0:2722 corres
ISCO: only QNMs
1.05below this line can be excited during a qu
l=6,the
m=4
l=3, m=1
l=5
l=3
-1
-2
0.036
MωK
0.2
P(v)
P(v)
QNM frequency. Thus we expect 0.2sharp
1.15
10
0.0
the values of v corresponding to the0.1
10
vastar QNMs for different values of the
1.10
0.25
0.30
v
0.35
l=6
0.40
084
1.00
0.32
0.34
0.36
[Pani+, 0909.0287; 1001.3031]
v
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*