This document summarizes a lecture on using gravitational wave waveform models to test general relativity and probe the nature of compact objects through gravitational wave observations. It discusses how waveform models can be used to bound post-Newtonian coefficients, constrain phenomenological merger-ringdown parameters, and probe the quasi-normal modes of black hole ringdowns. Measuring multiple modes could verify the no-hair theorem and black hole uniqueness properties. Future observations from LIGO and Virgo at design sensitivity may allow high-precision black hole spectroscopy and tests of general relativity in the strong, dynamical gravity regime.

1. Alessandra Buonanno
Max Planck Institute for Gravitational Physics
(Albert Einstein Institute)
Department of Physics, University of Maryland
“Waves on the Lake: The Astrophysics behind Gravitational Waves”
Lake Como School of Gravitational Waves, May 28 - June 1, 2018
The Analytical/Numerical Relativity Interface
behind Gravitational Waves: Lecture III

2. Outline
•Lecture I: Motivations and actual development of inspiral, merger
and ringdown waveforms
•Lecture II: Using waveform models to infer astrophysics and
cosmology with gravitational-wave observations
•Lecture III: Using waveform models to probe dynamical gravity and
extreme matter with gravitational-wave observations
(visualization credit: Benger @ Airborne Hydro
Mapping Software & Haas @AEI)
(NR simulation: Ossokine, AB & SXS @AEI)

3. • UMD/AEI graduate course on GW Physics & Astrophysics taught
in Winter-Spring 2017: http://www.aei.mpg.de/2000472.
References:
• AB’s Les Houches School Proceedings: arXiv:0709.4682.
• E.E. Flanagan & S.A. Hughes’ review: arXiv:0501041.
• M. Maggiore’s books: “Gravitational WavesVolume 1:Theory and
Experiments” (2007) & “Gravitational WavesVolume II: Astrophysics
and Cosmology” (2018).
• E. Poisson & C. Will’s book:“Gravity” (2015).
• AB & B. Sathyaprakash’s review: arXiv:1410.7832.

4. •Given current tight constraints
on GR (e.g., Solar system, binary
pulsars), can any GR deviation be
observed with GW detectors?
highly-dynamical
strong-field
10 5 10 4 10 3 10 2 10 1 10010 8
10 7
10 6
10 5
10 4
10 3
10 2
10 1
100
Solar
System
Binary
Pulsars
Gravitational
Waves
v/c
Newton
(credit: Sennett)
Solar system:
Binary pulsars:
LIGO/Virgo:
Extreme gravity, dynamical spacetime: tests of General Relativity

5. PN templates in stationary phase approximation: TaylorF2
i =
Si
m2
i
1PN 1.5PN
2PN
spin-orbit
1.5PN
spin-spin
2PN
0PN
graviton with
non zero mass
1PN
dipole
radiation
-1PN

6. Waveforms encode plethora of physical effects
BH absorptiontail effects
merger
ringdown
•Binary black hole
Spin effects
inspiral
(credit: Hinderer)
•Compact-object binary with matter or in modified theory to GR?
quasi-normal modes
echoes?
(credit: Hinderer)
mergerinspiral

7. •GW150914/GW122615’s rapidly varying orbital periods allow us to bound
higher-order PN coefficients in gravitational phase.
0PN 0.5PN 1PN 1.5PN 2PN 2.5PN 3PN 3.5PN
PN order
10 1
100
101
|ˆ'|
GW150914
GW151226
GW151226+GW150914
(Arun et al. 06 , Mishra et al. 10, Yunes &
Pretorius 09, Li et al. 12)
•PN parameters describe: tails of
radiation due to backscattering,
spin-orbit and spin-spin couplings.
(Abbott et al. PRX6 (2016))
•PN parameters take different values
in modified theories to GR.
'(f) ='ref + 2⇡ftref + 'Newt(Mf) 5/3
+ '0.5PN(Mf) 4/3
+ '1PN(Mf) 3/3
+ '1.5PN(Mf) 2/3
+ · · ·
˜h(f) = A(f)ei'(f)
90% upper bounds
Bounding PN parameters: inspiral

8. •GW150914/GW122615’s rapidly varying orbital periods allow us to bound
higher-order PN coefficients in gravitational phase.
First tests of General Relativity in dynamical, strong field
0PN 0.5PN 1PN 1.5PN 2PN 2.5PN 3PN 3.5PN
PN order
10 1
100
101
|ˆ'|
GW150914
GW151226
GW151226+GW150914
(Arun et al. 06 , Mishra et al. 10, Yunes &
Pretorius 09, Li et al. 12)
•PN parameters describe: tails of
radiation due to backscattering,
spin-orbit and spin-spin couplings.
(Abbott et al. PRX6 (2016))
•PN parameters take different values
in modified theories to GR.
'(f) ='ref + 2⇡ftref + ˜'Newtv 5
⇥
1 + ˜'0.5PN v + ˜'1PN v2
+ ˜'1.5PN v3
+ · · ·
⇤
˜h(f) = A(f)ei'(f)
v = (2Mf)1/3
90% upper bounds

9. Some modified theories to General Relativity
(Yunes & Siemens 2013)

10. 20 50 100 150 200 250 300
Frequency (Hz)
1.00
0.10
0.01
|hGW(f)|/1022(Hz)
inspiral intermediate
merger
ringdown
low frequency high frequency
• Merger-ringdown phenomenological parameters
(βi and αi) not yet expressed in terms of relevant
parameters in GR and modified theories of GR.
Bounding phenom parameters: intermediate/merger-RD
(Abbott et al. PRL 116 (2016) 221101 )
GW150914 + GW151226 + GW170104
(Abbott et al. PRL 118 (2017) 221101)

11. How to test GR and probe nature of compact objects:
building deviations from GR & BHs/NSs
• Will GR deviations be fully captured in perturbative-like descriptions
during merger-ringdown stage? Likely not. (e.g., Yunes & Pretorius 09, Li et al. 12,
Endlich et al. 17)
• Need NRAR waveforms of binaries composed of exotic objects (BH &
NS mimickers), such as boson stars, gravastar, etc. (e.g., Palenzuela et al. 17)
•Need NRAR waveforms in modified theories of GR: scalar-tensor theories,
Einstein-Aether theory, dynamical Chern-Simons, Einstein-dilaton Gauss-
Bonnet theory, massive gravity theories, etc. (e.g., Stein et al. 17, Cayuso et al.17,
Hirschmann et al. 17)
• Do current GR waveform models include all physical effects? Not yet.
• Including deviations from GR in EOB formalism.
(Julie & Deruelle 17, Julie 17, Khalil et al. in prep 18)

12. Q`mn = !`mn ⌧`mn/2
(Berti,Cardoso&Will06)
zero overtone
Kerr BH
zero overtone
Kerr BH
Probing nature of remnant through quasi-normal modes
•One frequency and damping time (or quality factor) of ringdown signal cannot
determine values of (l,m,n) corresponding to mode detected, because there are
several values of parameters (M, j, l, m, n) that yield same frequencies and
damping times.
•Multipole frequencies and decay times will be smoking gun that Nature’s black
holes are black holes of GR. We can test no-hair conjecture and second-law black-
hole mechanics. (Israel 69, Carter 71; Hawking 71, Bardeen 73)
•Note that those conjectures refer to isolated, stationary black holes, not to
dynamical back holes (in a binary, merging).

13. Black-hole spectroscopy using damped sinusoids
•BH spectroscopy: measuring multipole QNMs.
(Dreyer et al. 2004, Berti et al. 2006, Gossan et al. 2012, Meidam et al. 2014, Bhagwat et al. 2017,
Yang et al. 2017, Brito,AB, Raymond 18, Carullo et al. 18)
(Gossan et al. 2012)
!`mn = !GR
`mn (1 + !`mn)
⌧`mn = ⌧GR
`mn (1 + ⌧`mn)
non-GR mock signalGR mock signal
•Plausible assumptions: likely we detect zero
overtone & .
90% CL
` = 2, 3
(using Einstein Telescope or third-generation ground-based detectors)
(Gossan et al. 2012)

14. Probing remnant of GW150914 through quasi-normal modes
(Abbott et al. PRL 116 (2016) 221101 )
• Bayesian analysis with damped-sinusoid
template to extract frequency and
decay time, starting at different times
after merger.
200 220 240 260 280 300
QNM frequency (Hz)
0
2
4
6
8
10
12
14
QNMdecaytime(ms)
1.0ms
3.0 ms
5.0 ms
7.0 ms7.0
m
s
IMR (l = 2,m = 2,n = 0)
• Starting from 5 msec after merger,
posterior distributions of frequencies
and decay times from damped
sinusoid and IMR waveform are
consistent.
•First (low-accuracy) verification of
black hole uniqueness properties (?)
SNRtot ' 23 SNRRD ' 7

15. (Abbott et al. PRL 116 (2016) 221101 )
• Bayesian analysis with damped-sinusoid
template to extract frequency and
decay time, starting at different times
after merger.
•IMR (l = 2, m=2) posterior obtained
from full Bayesian analysis of
GW150914, plus information from
NR to obtain final mass and spin from
component masses and spins.
SNRtot ' 23 SNRRD ' 7
•First (low-accuracy) verification of
black hole uniqueness properties (?)
Probing remnant of GW150914 through quasi-normal modes
(Abbott et al. PRL 116 (2016) 241102 )

16. Black-hole spectroscopy by making full use of GW modeling
•We build parametrized inspiral-
merger-ringdown waveforms
(pEOBNR):
- QNM’s frequencies and decay
times are free parameters;
- (2,2), (2,1), (3,3), (4,4) & (5,5)
modes are present.
mass ratio = 6(Brito, AB & Raymond 18)
•Merger-ringdown EOBNR model
reproduces time & phase shifts
between NR modes’ at peak.
200 220 240 260 280 300
f220
(Hz)
0
1
2
3
4
5
6
7
8
9
10
τ220
(ms)
GW150914
pEOBNR
3ms
5ms
1m
s
•GW150914’s frequency and decay
time recovered “without ambiguity”
on a priori unknown starting time of
QNM ringing.
dashed curves NR
cont. curves EOBNR
(Pan,ABetal.12)

17. Trying to extract 2 quasi-normal modes from GW150914 event
(Brito, AB & Raymond 18)
•(2,2) mode resolved,
but not (3,3) mode.
GW150914
•Posterior distributions of
frequency and decay times
of two QNMs employing
pEOBNR against data.

18. 0 10 20 30 40 50 60 70 80 90 100
Number of events
0.1
1
10
errorat2σ(%)
δf220
δf330
δτ220
•We can bound quasi-normal mode frequencies & decay times by combining
several BH observations.
one event GW150914-like
with LIGO & Virgo @ design
sensitivity
(Brito,AB & Raymond 18)
•Let us assume we had GW observations and did not find deviations from GR.
GW150914-like events with LIGO
& Virgo @ design sensitivity
•About 30 GW150914-like events
are needed to achieve errors of 5%.
We will soon verify more accurately
black hole uniqueness properties.
!`mn = !GR
`mn (1 + !`mn) ⌧`mn = ⌧GR
`mn (1 + ⌧`mn)
Measuring at least 2 QNMs with LIGO & Virgo
errors scale/ 1/
p
N

19. Testing no-hair conjecture with several events @ design sensitivity
(O1 run) (@ LIGO/Virgo design sensitivity)
(Brito, AB & Raymond 18)
using pEOBNR
(@ LIGO/Virgo design sensitivity)

20. (Cardosoetal.16)
same
ringdown
signal
different QNM signals
t
(Damour & Solodukhin 07, Cardoso, Franzin & Pani 16)
• If remnant is horizonless, and/or
horizon is replaced by “surface”, new
modes in the spectrum, and ringdown
signal is modified: echoes signals
emitted after merger.
Remnant: black hole or exotic compact object (ECO)?
horizonless objects
black hole
(Cardoso et al. 16)
wormhole
boson stars, fermion stars, etc.
(e.g., Giudice et al. 16)

21. GW polarizations in gravity
•Generic metric theories of gravity have 6 geometrically distinct
polarizations:
tensor vector scalar
6 polarization tensors6 amplitudes
(Will 72)
hij(x) =
X
A
hA(x) eA
ij A = 1, · · · , 6
GR

22. Detector antenna patterns for different GW polarizations
detector tensor
Photodetector
Beam
Splitter
Power
Recycling
Laser
Source
100 kW Circulating Power
b)
a)
Signal
Recycling
Test
Mass
Test
Mass
Test
Mass
Test
Mass
Lx = 4 km
20 W
H1
L1
10 ms light
travel time
Ly=4km
depends on
GW source
depends only on
geometry,
• Angular responses of a
differential-arm detector
differential-arm
detector
FA
|FA|
hI(t, xI) = Dij
I hij(t, xI)
=
X
A
hA(t, xI) (Dij
I eA
ij)
Dij
=
1
2
(di
x dj
x di
y dj
y)

23. Testing extra GW polarizations with two LIGOs & Virgo
detector tensor
depends on
GW source
depends only on
geometry, FA
•Two LIGOs are nearly co-aligned, approximately sensitive to the
same linear combination of polarizations.
•Five (differential-arm) detectors would allow to extract all
the 5 polarizations (2 scalar polarization are degenerate).
•Observation of GW170814 with two LIGOs plus Virgo allowed to test
pure tensor polarizations against pure scalar (vector), finding Bayes
factors 1000 (200). (Abbott et al. PRL 119 (2017) 141101)
examples of antenna patterns:
tensor
scalar
hI(t, xI) = Dij
I hij(t, xI)
=
X
A
hA(t, xI) (Dij
I eA
ij)

24. Tests of Lorentz Invariance/Bounding Graviton Mass
(Will 94, Mirshekari,Yunes & Will 12)
vg
c
= 1 + (↵ 1)
A
2
E↵ 2
E2
= p2
c2
+ Ap↵
c↵
↵ 0
mg  7.7 ⇥ 10 23
eV/c2
↵ = 0, A > 0
(Abbott et al. PRL118 (2017))
•Phenomenological approach: modified dispersion relation. GWs travel at
speed different from speed of light.

25. Constraints on speed of GWs & test of equivalence principle
(Abbott et al. APJ 848 (2017) L12)
• Strong constraints on scalar-tensor and vector-tensor theories of gravity.
• Combining GW and GRB observations:
(Creminelli et al. 17, Ezquiaga et al. 17, Sakstein et al. 17, Baker et al. 17)
c
c
' c
t
D
t = tEM tGW
c = cGW c
4 ⇥ 10 15

c
c
 7 ⇥ 10 16
assuming GRB is
emitted 10 s after
GW signal
assuming observed
time delay is entirely
due to different speed
t ' 1.7s
•EM waves & GWs follow same geodesic. Metric perturbations (e.g., due to
potential between source and Earth) affect their propagation in same way.
gravitational
potential of Milky
Way outside
sphere of 100 kpc
(Abbott et al. APJ 848 (2017) L12)(Shapiro 1964)

26. •GR is non-linear theory.
Complexity similar to QCD.
- approximately, but analytically
(fast way)
- exactly, but numerically on
supercomputers (slow way)
•Einstein’s field equations can
be solved:
•Synergy between analytical and numerical relativity is crucial.
•GW170817: SNR=32 (strong),
3000 cycles (from 30 Hz), one
minute.
last 0.07sec
modeled by NRlast minutes
modeled by AR
(Abbott et al. PRL 119 (2017) 161101)
Solving two-body problem in General Relativity (including radiation)

27. Numerical-relativity simulation of GW170817
(visualization: Dietrich, Ossokine, Pfeiffer & Buonanno @ AEI)
(numerical simulation: Dietrich @ AEI and BAM collaboration)
Minerva:
High-Performance Computer Cluster
@ AEI Potsdam (~10,000 cores)

28. mergerinspiral
post-merger
•PN waveform model was used for:
- template bank: to observe GW170817
- Bayesian analyses: to infer astrophysical,
fundamental physics information of
GW170817
Analytical waveform modeling for GW170817
(DalCanton&Harry16)
50,000 PN
templates
tail effects tidal effectsspin effects

29. Probing equation of state of neutron stars
(Antoniadisetal.2016)
tidal interactions (credit: Hinderer)
Neutron Star:
- mass: 1-3 Msun
- radius: 9-15 km
- core density > 1014g/cm3
• NS equation of state (EOS) affects
gravitational waveform during
late inspiral, merger and post-
merger.

30. 10 50 100 500 1000 5000
10 25
10 24
10 23
10 22
10 21
f Hz
BH BH
Initial LIGO
AdvancedLIGO
Einstein Telescope
10 50 100 500 1000 5000
10 25
10 24
10 23
10 22
10 21
f Hz
NS NS EOS HB
Initial LIGO
AdvancedLIGO
Einstein Telescope
NS-NS
post
merger
effectively point-particle tidal effects
BH-BH
Probing equation of state of neutron stars
(credit:Read)
• measures star’s quadrupole
deformation in response to
companion perturbing tidal field:
•Tidal effects imprinted on
gravitational waveform during
inspiral through parameter .
Qij = Eij

31. NS deformation in external tidal field
⇢(t, x0
) = ⇢(r0
) + ⇢(t, x0
)
1
|x x0|
=
1
r
+
x · x0
r3
+
(3 ni nj ij)
2r3
x0
i x0
j + . . . ni =
xi
r
•Gravitational potential generated by perturbed NS
•In presence of external potential, (non-rotating) NS acquires a deformation:
self-gravitating fluid is perturbed from equilibrium configuration
•Quadrupolar tidal field:
equilibrium configuration
perturbations
•Multipole expansion around CM:
r > r0
outside NS
Qij =
Z
d3
x0
⇢(t, x0
) (x0
i x0
j
1
3
r02
ij)
Newtonian tidal
deformationsEij = @i@jUext
U(t, x) = G
Z
d3
x0 ⇢(t, x0
)
|x x0|
U(t, x) =
G mNS
r
G(3ni nj ij)
2r3
Qij + . . .

32. NS deformation in external tidal field (contd.)
•Total gravitational potential outside NS:
•Considering quasi-static perturbations (tidal force frequency much smaller
than NS’s eigenfrequency of normal mode of oscillation, i.e., f modes):
Qij = Eij k2 =
3
2
G
R5
NS
U(t, x) =
GmNS
r
+
1
2
Eij xi xj
"
1 + 2k2
✓
RNS
r
◆5
#
+ O
✓
1
r4
◆
+ O(x3
)
g00 = 1
2GmNS
r
+ Eij xi xj
"
1 + 2k2
✓
RNS
r
◆5
#
+ O
✓
1
r4
◆
+ O(x3
)
U(t, x) =
GmNS
r
3G
2r3
ni nj Qij + O
✓
1
r4
◆
+
1
2
xi xj Eij + O(x3
)

33. PN templates in stationary phase approximation: TaylorF2
i =
Si
m2
i
1PN 1.5PN
2PN
spin-orbit
1.5PN
0PN
graviton with
non zero mass
1PN
dipole
radiation
-1PN
· · ·
39
2
⌫ 2 ˜⇤ (⇡Mf)10/3
spin-spin
2PN
tidal
5PN
⇤ =
m5
NS
=
2
3
k2
✓
RNSc2
GmNS
◆5
it can be
large
Depends on EOS
& compactness

34. Probing equation of state of neutron stars
•Where in frequency the information about (intrinsic) binary parameters
predominantly comes from.
(Harry & Hinderer 17, see also Damour et al . 12)
•Tidal effects typically change overall number of GW cycles from 30 Hz
(about 3000) by one single cycle!

35. (Dietrich & Hinderer 17) time
State-of-art waveform models for binary neutron stars
•Synergy between analytical and numerical work is crucial.
(Damour 1983, Flanagan & Hinderer 08, Binnington & Poisson 09, Vines et al. 11, Damour & Nagar 09,
12, Bernuzzi et al. 15, Hinderer et al. 16, Steinhoff et al. 16, Dietrich et al. 17, Dietrich et al. 18)
NR
EOBNR

36. Strong-field effects in presence of matter in EOB theory
(Hindereretal.2016,Steinhoffetal.2016,
seealsoBernuzzietal.15)
Tides make gravitational interaction more attractive
1 2 3 4 5 6 7
r/M
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
A(r)
EOBNR
TEOBNR
Schwarzschild
ν
EOBNR
Schwarzschild
Schwarzschild
λ
light ring
light ring
ISCO
A(r) = A⌫(r) + Atides(r)
⇤

37. (Abbott et al. PRL 119 (2017) 161101)
•Observation of binary pulsars
in our galaxy indicate spins are
not larger than ~0.04.
•Fastest-spinning neutron star
has (dimensionless) spin ~0.4.
Unveiling binary neutron star properties: masses
•Degeneracy between masses
and spins.

38. Constraining Love numbers with GW170817
(Abbott et al. PRL 119 (2017) 161101)
black hole
⇤ =
m5
NS
=
2
3
k2
✓
RNSc2
GmNS
◆5
Depends on EOS & compactness
NS’s Love number
M
S1
M
S1b
H
4
M
PA1
APR4SLy
less compact
more compact
•Effective tidal deformability
enters GW phase at 5PN
order:
•With state-of-art waveform
models, tides are reduced by
~20%. More analyses are
ongoing.

39. 0.04 0.05 0.06 0.07 0.08 0.09
MΩ
0.00
0.05
0.10
0.15
8.6 7.5 6.5 5.8 5.3 4.8
r / M
k2
eff
k2
k3
eff k3
k4
eff
k4
NSBH mass ratio 2
Γ=2 polytropic
CNS=0.14444
•Dynamical tides: NS’s f-modes can be excited toward merger.
(Kokkotas et al. 1995, Flanagan et al. 08, Hinderer, … AB et al. 16, Steinhoff, … AB et al. 16)
(Hinderer, …,AB et al. 16)
NS’s effective response to
dynamical tidal effects
Including dynamical tidal effects in EOB model
•Tidal force frequency approaches eigenfrequency of NS’s normal
modes of oscillation, resulting in an enhanced, more complex tidal response.

40. Boson stars as black-hole/neutron-star mimickers
(Sennett…AB et al. 17)
(see also Cardoso et al. 17, Johnson-
Mcdaniel 18)
•Boson stars are self-
gravitating configurations
of a complex scalar field
•Black holes:
•Boson stars:
⇤ = 0
⇤min ⇠ 1
•Neutron stars:
⇤ = /M5
(credit: Sennett)
0 2 4 6 8 10
100
101
102
103
104
C =
GM
Rc2
⇤min ⇠ 10
Boson star
0.08
0.158
0.3
0.349
0.5
CompactnessV (| |2) Mmax
Mini BS µ2 2
⇣
85peV
µ
⌘
M
Massive BS µ2 2 + 2 | |4
p ⇣
270MeV
µ
⌘2
M
Neutron star 2 4 M
Solitonic BS µ2 2
⇣
1 2| |2
2
0
⌘2 ⇣
µ
0
⌘2 ⇣
700TeV
µ
⌘3
M
Black hole 1

41. The new era of precision gravitational-wave astrophysics
• We can now learn about gravity in the
genuinely highly dynamical, strong field
regime.
• Theoretical groundwork in analytical and
numerical relativity has allowed us to build
faithful waveform models to search for
signals, infer properties and test GR.
• We have new ways to explore relationships between gravity, light ,
particles and matter.
• As for any new observational tool, gravitational (astro)physics will likely
unveil phenomena and objects never imagined before.
(visualization: Benger @ Airborne
Hydro Mapping Software & Haas @AEI)
(NR simulation: Ossokine, AB, SXS)
•We can probe matter under extreme pressure and density.