Astrophysical tests of
general relativity in the
strong-field regime
Emanuele Berti, University of Mississippi/Caltech
Texas Symposium, São Paulo, Dec 18 2012
1)
2)
3)
4)
What are “strong field” tests?
Alternatives to GR: massive scalars
BH dynamics and superradiance
GWs: SNR and event rates
(e)LISA and fundamental physics
5) BH spins and photon mass bounds
Coda: Advanced LIGO and astrophysics
Strong field: gravitational field vs. curvature; probing vs. testing
[Psaltis, Living Reviews in Relativity]
Testing general relativity – against what?
Finding a contender




Action principle
Well-posed initial-value problem
At most second-order equations of motion
Testable predictions!
[Clifton+, 1106.2476]
Generic scalar-tensor theory
Dynamical Chern-Simons
Einstein-dilaton-Gauss-Bonnet
A promising opponent: massive scalar fields
1) Phenomenology
 Modern equivalent of planets [Bertschinger]
 Well-posed, flexible (Damour & Esposito-Farése “spontaneous scalarization”)
 f(R) and other theories equivalent to scalar-tensor theories
2) High-energy physics
 Standard Model extensions predict massive scalar fields (dilaton, axions, moduli…)
 Not seen yet: dynamics must be frozen
 small coupling x - or equivalently large wBD~1/x
 large mass m>1/R (1AU~10-18eV!)
3) Cosmology
 “String axiverse”: light axions, 10-33eV < ms < 10-18eV [Arvanitaki++, 0905.4720]
Striking astrophysical implications: bosenovas, floating orbits
Are massive scalar fields viable?
Bounds from:
 Shapiro time delay: wBD>40,000 [Perivolaropoulos, 0911.3401]
 Lunar Laser Ranging
 Binary pulsars: wBD>25,000 [Freire++, 1205.1450]
[Alsing, EB, Will & Zaglauer, 1112.4903]
Wave scattering in rotating black holes
[Arvanitaki+Dubovsky, 1004.3558]
Quasinormal modes:
Massive scalar field:
 Ingoing waves at the horizon,
outgoing waves at infinity
 Discrete spectrum of damped
exponentials (“ringdown”)
[EB++, 0905.2975]
 Superradiance: black hole bomb
when 0 < w < mWH
 Hydrogen-like,
unstable bound states
[Detweiler, Zouros+Eardley…]
Quasinormal modes
 In GR, each mode determined
uniquely by mass and spin
[Visualization: NASA Goddard]
 One mode: (M,a)
Any other mode frequency:
No-hair theorem test
 Relative mode amplitudes:
pre-merger parameters
[Kamaretsos++,Gossan++]
 Feasibility depends on SNR:
Need SNR>30 [EB++, 2005/07] f = 1.2 x 10-2 (106Msun)/M Hz
1) Noise S(fQNM)
6M ) s
t
=
55
M/(10
sun
2) Signal h~E1/2, E=erdM
erd~0.01(4h)2 for comparable-mass mergers, h=m1m2/(m1+m2)2
(e)LISA vs. LIGO
f = 1.2 x 10-2 (106Msun)/M Hz
t = 55 M/(106Msun) s
[Schutz; see Sesana’s talk]
SNR=h/S: S comparable, h~hM1/2
Ringdown as a probe of SMBH formation
 LISA/eLISA studies:
merger-tree models of
SMBH formation
 Light or heavy seeds?
Coherent or chaotic accretion?
[Arun++, 0811.1011]
 eLISA can easily tell whether
seeds are light or heavy
[Sesana++, 1011.5893]
 Mergers: a~0.7
Chaotic accretion: a~0
Coherent accretion: a~1
[EB+Volonteri, 0802.0025]
 >10 binaries can be used for no-hair tests
 Spin observations constrain SMBH formation
[Sesana++, 2012]
alar flux
at infinity
be and
comput
ed in theinstabilities
lowMassive
bosoniccan
fields
superradiant
equency regime [25]. For r 0 / M
1 and l = m = 1,
Superradiance when w < mWH3/ 2
2
2 1 − µ2 r 3 / M
α
M
s 0
s
E˙Any
=
∞ light scalar can trigger
12π
r 04 a
m 2p Θ(Ωp − µs ) ,
(12)
black hole bomb (“bosenova”)
here
Θ(x) is the Heaviside
funct ion. For generic modes,
[Yoshino+Kodama,
1203.5070]
large distances and for ω = mΩp > µs , scalar raStrongest
instability:
ation
dominates
overmgravit
sM~1 at ional radiat ion: compare
0705.2880]
q. [Dolan,
(12) wit
h the standard quadrupole formula E˙ ∞g =
−5
/ 5 (r 0 / M ) m 2p / M 2 . This result is oblivious t o the
10
For
m
=1eV,
M=M
:
m
M~10
sun
s BH. In fact, for ω > µs , t he
esence s of the rotating
Need
(orare
primordial
black holes!)
uxes
at light
the scalars
horizon
negligible.
However, for freencies
close
t o µflux
resonance
at superradiant
[24]:
s , aat
Negative
scalar
the horizonoccurs
close to
resonances at
ωr2es
=
µ2s
−
µ2s
µs M
l + 1+ n
2
,
n = 0, 1, ...
[Detweiler
1980]
(13)
Light scalars: floating orbits (Press & Teukolsky 1972)
3
-2
10
-3
10
-2
(mp/M) (dElm/dt)
-4
10
-5
10
-6
10
-7
10
-8
s
r+
g
(dE22/dt) T
-(dE11/dt)
-4
10
-5
10
-6
10
-7
10
-8
10
-9
10
10
-9
10
-10
10
-11
-2´10
10
-12
10
-13
10
-16
0
Dr0/rres
2´10
-16
-14
10
5
10
15
r0/M
20
25
[Cardoso++ 1109.6021; Yunes++, 1112.3351]
complex frequency ω = ωR + i ωI . Physically mo
number m are still
boundary
correspond
t o quasinormal
Photon
bound condit
from ions
rotating
black holes
fferent values
of mass
(pert urbations having ingoing-wave boundary con
rotat ion there is a
the horizon and outgoing-wave condit ions at
rstProca
orderperturbations
in a
˜ , per- inatKerr
[Data points: Brenneman++, 1104.1132]
finity
[31])
and
bound
states (perturbations t hat a
only
coupled
wit
h
do not decouple
t ially localized wit hin t he vicinity of the BH a
milarly t o the case
cay exponent ially at null infinity). Here we fo
rturbations
wit h a
Use Kojima’s
bound modes, which are expected (based on a
oseslow-rotation
wit h ± 2 and
field analogy) t o become superradiantly unsta
approximation
ωR < mΩH [10], where ΩH = a
˜ / (2r + ). Our result
ons can always be
for the first time, that massive vect or fields do ind
Stronger instability
come unstable when ωR < mΩH .
than for massive scalars
The bound state modes of t he system (4)–(5)
Maximum (again) for found by standard numerical met hods [29]. When
0.1, our numerical results for the fundamental mo
msM~1
consistent wit h a hydrogenic spectrum, ωR ∼ µ a
A˘ + 2 + O(˜a3 ) , (2)
mg<10-20 (or 4x10-21) eV
4 + 5+ 2S
M
ω
∼
γ
(˜
a
m
−
2r
µ)
(M
µ)
,
I
S
+
PDG: mg<10-18 eV
[Pani++, 1209.0465; 1209.0773]
Spin-orbit resonances and spin alignment
[Schnittman 04; Kesden++; Lousto’s talk]
Can Advanced LIGO reconstruct binary evolution?
[Gerosa++, in preparation]
Summary
Tests within GR
1) (e)LISA: Tens of events could allow us to test the no-hair theorem
Advanced LIGO/ET can also test no-hair theorem - if IMBHs exist!
2) Spin measurements constrain SMBH merger/accretion history
[EB++, 0905.2975; EB+Volonteri, 0802.0025]
Massive bosons and superradiant instabilities
3) Weak-field: Solar System, binary pulsars
Cassini: wBD>40,000 for ms<2.5x10-20 eV
Binary pulsars will do better in a few years
[Alsing++, 1112.4903; Horbatsch++, in preparation]
4) Massive scalars: floating orbits
[Cardoso++, 1109.6021; Yunes++, 1112.3351]
5) Massive vectors and SMBH spins: best bounds on photon mass
mg<10-20 (4x10-21eV) (Particle Data Group: mg<10-18eV)
[Pani++, 1209.0465; 1209.0773]
Advanced LIGO
6) Spin alignment may encode formation history of the binary
Effect of tides? Reverse mass ratio?