重力波から広がる様々な物理
Takahiro Tanaka
Kyoto University

1Detected events is getting close to 100.
The next observation run O4 will start soon.

2

Detection of the first NS-NS binary (GW170817)
Understanding of NS-NS binary increased a lot by this single event.

• Follow-up observations• Short γ ray burst 1.7sec after merger• Macro(kilo)nova observation by
optical and infrared observations• r-process nucleosynthesis• Early blue component• Superluminal motion detected byradio observation• Existence of a relativistic jet• Constraint on the nuclear EOS• Propagation speed of GW ≃ c• Bulk of dark energy model was excluded.
• Possible independent determination of H0.

3

Also, various theoretical progresses
Many new models of gravity⇒ various tests of GR
Binary formation scenarios
Binary coalescence- Dynamics including various physics- Emission process, Nucleosynthesis- Nuclear physics
Supernova explosion⇒Exploding numerical models- understanding of various mechanism of GWs

New research areas are opening up.
-

Fundamental Physics
HE astrophysics
Binary formation4

Propagation speed of gravitational waves
1.7sec delay of gamma ray detection over 40Mpc implies2cGW− cγ2 < 10−15

Extension of gravity theory is tightly constrained.
Ex.) Horndeski theory: scalar-tensor theory with a single scalar field with only upto
2nd order time derivatives in EoM.

=
G4 G=
0
4 (φ ) , G5

Only rather traditional models survive.

K-essence, DGP braneworld, scalar-tensor theory5Motivation to consider
gravity theories beyond GR
1) Incompleteness of General relativity
GR is non-renormalizabile
Singularity formation after gravitational collapse
2) Dark energy problem3) To test General relativity
GR has been repeatedly tested since its first proposal.
The precision of the test is getting higher and higher.
⇒ Do we need to understand what kind of modificationis theoretically possible before experimental test?
Yes, especially in the era of gravitational wave observation!

Test of GR using coalescing binaries• Inspirals of binary BHs and NSs (Cutler et al, PRL 70 2984(1993))
Clean system: ～point particles

Waveform in Fourier space for quasi-circular inspiral
6 iΨ( f )Ah ( f ) ≈ A f −7 /=e=Ψ 2π f tc − ϕ c +

µ
5/6
35
25
,  µ=
=
M , η
3
D
M
20π
L
1

3
20  743 11  2
−5/3 
3

+
+
−
−
+

π
f
1
η
v
16
π
β
v
(
)
(
) 



128
9
331
4




1.5PN
1PN

v3 ≡ π M f7

Typical modification of GR

often discussed in the context of test by GWs

Scalar-tensor gravity

S=

1
−1
4
,α
−
−
− ∑ ∫ dτ a ma (φ )
d
x
g
φ
R
ω
φ
φ
φ
,α
BD
∫
16π
a
4 + 2ω BD scalar charge of self-gravitating body:G=φ (3 + 2ω BD )sa = −[∂ (ln ma ) / ∂ (ln G )]

(

)

0

G-dependence of thegravitational binding energyΨ = +

3(π M f )−5 / 3 αu −2 / 3 + 1 +  3715 + 55 η  u 2 / 3 − ( 16π −β )u + 
9 
128
 756



Dipole radiation＝－1 PN frequency dependence5(s − s )α =− 1 264ω BD2( )u =π M f = O v 3

2
(
)
s
−
s
≪1
For binaries composed of similar NSs, 1
2

9

Log|α 0|After conformal transformation, the action can be recast into the following form:ln A(ϕ ) = α 0 (ϕ − ϕ 0 ) +1
β 0 (ϕ − ϕ 0 )2 + 
2Cassini bound:
PSR J1738+0333(pulsar-WD)(Strong Equivalence Principle Test)
PSR J0337+1715

PSR J0348+0432(2M_sol pulsar-WD)

Constraint from future
GW observations:(Yagi & TT, arXiv:0908.3283)

LISA- 1.4M◎NS+1000M◎BH:
at 40Mpc
corresponding toSNR =log |α0| > -2

200

DECIGO-1.4M◎NS+10M◎BH：
collecting 104events
at cosmological distances
log |α0| > -4.1

β0(Berti et al., arXiv:1501.07274)

10Spontaneous scalarization

ω BD (φ )14,α More general model


d
x
g
R
φ
φ
φ
−
−
,α

∫
φ
16π


1
1

4
,α 
(
)
φ
ϕ
ϕ
ϕ
d
x
g
R
=
−
−
 ϕ is canonically normalized

,α
16π ∫
2



S=

EOM

∆ϕ ≈ −8πφ ′(ϕ )TEffective potential for a star with radius R.

smaller radiuslarger radius
ϕ2/R2
ϕ2/R2
8πTφ (ϕ)ϕ

8πTφ (ϕ)ϕ

As two NS get closer, “spontaneous scalarization” may happen.
Sudden change of structure and starting scalar wave emission.
Most of parameter region will be excluded by the discovery of many pulsars.
Consistency with the cosmological evolution is difficult to achieve (1607.08888)
11

Ordinary scalar-tensor theory
BH no hairTuru-turu

NS can have a scalar hair

Einstein dilaton Gauss-Bonnet, Chern-Simons gravity[]

 RGB 1
2
4d x − g (∂θ ) + 2V (θ )−S⊃d x − g θ  *
∫
∫

2GN
GN
R
R


θ ×(higher curvature)α4RGB = R 2 − 4 Rµν R µν + Rαβ µν R µν αβ*

R R = ε αβ σχ Rσχ µν R µν αβ• For constant θ, these higher curvature terms are
topological invariant. Hence, no effect on EOM.
• Higher derivative becomes effective only in strong field.

Hairy BH - bold NS• NS in EDGB and CS do not have scalar monopole charge.

□θ ≈" R "2

1
Q = ∫ d x " R "= ∫ d 4 x " R2"T
3

2

topological invariant, which vanishes
on topologically trivial spacetime.

• By contrast, BH solutions in EDGB and CS have scalarmonopole and dipole, respectively.
EDGB： monopole charge
CS：dipole charge

dipole radiation (-1PN order)

2PN order corrections(Yagi, Stein, Yunes, Tanaka (2012))

Observational bounds• EDGB
1/ 2
12
α
<
×
.
3
10
cm
1
Cassini EDGB

(Amendola, Charmousis, Davis (2007))

Low mass X-ray binary, A0620-00/2(Yagi, arXiv:1204.4524)
α 1EDGB< 1.9 ×105 cm

Ground-based GW observation
GW200115+

• CS

1/2
α EDGB< 1×105 cm

Gravity Probe B, LAGEOS

1/ 2
α CS
< 1013 cm(arXiv:2104.11189, 2201.02543,
2302.10112)(Ali-Haimound, Chen (2011))

Ground GW observation with favorable spin alignment:
100Mpc, a～0.8M
1/ 2
α CS
< 106−7 cm

1/2
GW+NICER: α CS
< 8.5 ×105 cm(Yagi, Yunes, TT, arXiv:1208.5102)(Silva et al., arXiv:2004.01253)

Issue of strong coupling
Simplified model

2
2
2
2

Flat background S ≈ M ( ∇ϕ ) + ( ∇h ) + αϕ ( ∇ h ) 


h
2
pl

xϕ

ϕx

2 6


α
1
p
≈ ( M pl2 α p ) ×  2 2  × p 4 = 2
 M pl p 
M pl


4 2

h
xϕ

ϕ x

3

1/3

 M pl αstrong coupling scale ≈ With background curvature
ϕx
ϕ x

x h x

2
2

S ≈ M ( ∇ϕ ) + ( ∇h ) + αϕ ( ∇2h ) R 


2
pl

xϕ
xϕ

2

 1 
≈ ( M α p R) ×  2 2  =α 2R2
 M pl p 


2
pl

2

2

strong coupling when α R ≥ 1

15

Future bounds on EDGB from BH-pulsar system
1/ 2
α EDGBin the unit of km

Once a BH-pulsar system isfound, how precisely one
can measure the orbital
decay rate determines thestrength of the constraint on
αEDGB.

Pb (days)(Yagi, Stein, Yunes, arXiv:1510.02152)
16

Parametrized test of gravitational wave formh ( f ) ≈ A f −7 / 6 ei Ψ ( f )GW waveform from a binary in a quasi-circular orbitA( f ) → 1 + ∑ α i u ai  AGR ( f )iΨ ( f ) → Ψ GR ( f ) + ∑ β i u bii(Yunes & Pretorius (2009))

Calculable leading order corrections are described by theparameter b and β in many cases.
Parametrized Post Einstein (PPE) formalism
17Constraint on the relative amplitude of deviation from GR for each PN
coefficient of the phase evolution

-1PN

0PN

1PN

2PN

(LIGO/Virgo 2010.14529)

Constraints on the modification to gravitational waves during the inspiral phase.

18But these are not the whole story.
There are also many examples of possible extension fromGR that go beyond the scope of PPE formalism.
1) Identifying theoretically well-motivated models of extended gravity
2) Theoretical prediction on the modification in the GW waveform
3) If the prediction is beyond the scope of the generic test of gravity, weneed to implement the analysis code and applied.

19

Challenges on testing gravity
Constructing gravitational wave templates inextended gravity scenarios
 Higher dimensional operator corrections(EDGB, CS).
 Low energy extra degrees of freedom thatevade all the weak gravity tests (Screening
mechanism, Parity odd, Very weaklyinteracting with matter).
 Modified nature of black hole horizon (ECO).

Massive scalar dipole radiation

Scalar-tensor theory is already strongly constrained by thepulsar timing observations.
Constraint on dipole radiation from GW170817 (1811.00364)
is to be compared with pulsar constraint

However, the scalar field may have a small mass.
The scalar field should be stabilized for Tuniv<10MeV, since theeffective gravitational coupling GN must be unchanged during BBN.
mscalar > 10−16 eV,
which is marginally sufficient to evade the constraint from pulsar
timing (PSR J0348+0432).
LIGO-Virgo-KAGRA band corresponds to the mass range(Ramazanoglu et al. 1601.07475)
10−13 eV > mscalar > 10−14 eV

21

Result of application of the test to O1 and O2 data
K. Yamada, T. Narikawa and TT arXiv:1905.11859

90% confidence level upper boundscalar mass
Even we introduce the scalar mass, GR is more preferred.
22

Einstein dilaton Gauss-BonnetS⊃

α EdGB
GN

−

µν
αβ
µν
4
2
−
−
+
θ
d
x
g
R
R
R
R
R
4
(
µν
µν
αβ )
∫

1
4
( ∂θ )2 − m 2θ 2 
−
d
x
g


2GN ∫

Turu-turu

α EdGB = 0
α EdGB ≠ 0
Hairly=ScalarchargeGauss-Bonnet coupling with massive scalar field in this mass range
was constrained for the first time.
23

Parity violating dispersion relation
Modified dispersion relation of GWs
has been tested by LIGO/Virgo collaboration. (1903.04467)

α=0 (Aα > 0) corresponds to the graviton
mass(although non-covariant theory)⇒ mgraviton < 4.7 ×10-23 eV

How about the parity violating cases?
Right and left-handed gravitons propagate differently.

h(L,R )

1 (+ )=
h + ih (× )
2

(

)
24

Result of application of the test to the real data
(K. Yamada and TT, PTEP 2020 093E01,
arXiv:2006.11086)

Ψ ( f ) → Ψ GR ( f ) + λL,R ∑ β i u biiGeneralized modelin the previous slide

γ − δ < 10−11 km
Constraint from |cT -1| < 10-15

The obtained
upper boundmagnitude issimilar to nonparity violatingcase.

γ − δ < 10−18 km(Nishizawa, Kobayashi 1809.00815)

25

Generalized action including a term with the factor
ε αβµν Rαβγδ Rµνρσ and a scalar field.

φε αβµν Rαβγδ Rµν γδ : Chern-Simons gravity is a typical example
Quadratic action(Nishizawa, Kobayashi 1809.00815)

δ ( t ) ijk 2
1
ijk 
l
l

LPV
=
γ ( t ) ε hil ∂ j hk + 2 ε ∂ hil ∂ j hk a (t )
4
Action for the Chern-Simons: γ ( t ) = δ ( t )− g ( LGR + LPV =) a 2 ( t ) 1 − k λL,Rγ ( t ) hL′2,R − k 2hL2,R 

Large modification of amplitude occurs when this factor
change in time: a ( t ) 1 − k λL,Rγ ( t ) hL,R ~ const.
Remark) At a slightly higher frequency,
negative energy states appear.
In more general models dispersion relation becomesω 2 + ( 2aH − λL,R kγ ′) iω − (1 + λL,R k (γ − δ ) ) k 2 ≈ 0
In general, gravitational wave phase is also modified.
26

Why do we need many similar detectors in the world？

KAGRA LIGO

Difference in the arrival time
Direction of the source
EM follow-up observations

Sensitivity to each polarization mode
depends on the antenna direction

Search for extra-polarization modes

Can we detect extra-polarization modes,
even if we assume some exotic extended gravity models?
Propagation process:(arXiv:2304.14430)

Case with the propagation direction k is in z-direction)

independently propagating modes, assuming
slowly varying non-trivial background.

Quadratic action take this standard form by choosing
hIA appropriately under the condition that the
propagation speed is unity.
Although the amplitude in terms of hij may change,
the energy flux is not changed at all as long as the
waveform is not significantly modified.

Detection process:Case with an extra polarization exists)
We assume1) no background anisotropies in solar system in gravitational sector and2) the presence of an extra scalar propagating mode.

Contribution of the tensor modes
Adding scalar contribution
γ −1
2

:=

(δ ij hij / 3) − h00
h00< 1.2 ×10−5

Presence of scalar mode is observationally allowed only for fine tuned case.
Even in that case PN correction to the Newtonian potential will constrain theamplitude of the scalar mode. (|β −1|<3×10−3)

Detection process:Case with background anisotropies)
We assume 1) no extra propagating mode and2) the background spatial vector V i and tensor W ij to the linear order(a) V k ek(T(i) k j ) , V k kk eij(T ) (b) W kl ekl(T ) ki k j , W kl kk kl eij(T ) , kkW k(i e(jT)l) k l (c) k 2δ ijW kl ekl(T )eij(T ) = eij( + ) or eij(×)+ mode and x mode should be modified by the same operation.

(a)-terms give correction to the Newtonian potential
Shift of theM
V kGW < 1mm × 200Hz ≈ 10−8h00 ≈ V k ∂ kcenter of massr(b)-terms give correction to the Newtonian potentialM
h00 ≈ W kl ∂ k ∂ l
The earth has fixed quadrupole moments Q
r
δr
r

≈

W
δφ
Q
W
≈
≈
, δ lmax ≈ δ vP ≈ δ r vP =,22
φ M ⊕ R⊕ R⊕
r
R⊕
r

W r

R⊕

R⊕

sep
δ l ≈ δ lmax sep =GRACE(d) kW k( i e(jT) k)

W k

2
GW

<

δl2
rsep

2

k

2
GW

1mm  200Hz −7R ≈ ≈ 10220km  50Hz 2⊕(c)-term gives no correction to the Newtonian
potential, but not detected by GW detectors.

Eöt-Wash(a) V k ek(T(i) k j ) , V k kk eij(T ) (b) W kl ekl(T ) ki k j , W kl kk kl eij(T ) , kkW k(i e(jT)l) k l (c) k 2δ ijW kl ekl(T )(d) kW k(i e(jT) k)(d)-term gives no correction to the Newtonian potential and can bedetected as the breathing mode by GW detectors.
h0i ≈ Wi j ∂ j

M
r

δr
r

≈

δφ v W v≈,
φ c R⊕ cr

δ lmax ≈ δ vP ≈

δr

v
vP =
W ,
r
c

r

v
R⊕ c

sep
δ l ≈ δ lmax sep =W

R⊕

W kGW <

δl2
rsep

−1

1mm  200Hz  4v
−3
kGW R⊕   ≈
10 ≈ 10220km  50Hz 
c

The constraint is weaker owing to the reduction factor v/c≃10-4.
To conclude, extra GW polarizations induced by the hypothetical hidden local
anisotropic background are hardly detected, as long as simultaneous detection
is assumed using GR templates.

Black Hole Echoes
There was a claim that echoes might have been detected in the LIGO GW events.

Angular momentum barrier

Hypothetical barrier near EH

time(Abedi et al, arXiv:1612.00266)r*∆ techot - tmerger

1) Reflecting boundary at the Planck distance from the horizon
2) Slowly decaying echoes of the same waveform with overall phase shift π.
3) Echo period ∆ techo is the one predicted by the angular momentum barrier
and the reflection wall at the Planck distance from the horizon.
→ 3σ detection of signal
32

We considered more appropriate template
will increase the significance (arXiv:1704.07175)

Perturbation Equation of Kerr BH (Sasaki-Nakamura) [l=m=2 mode only]
d 2Y(r* → ∞ )exp(− iωr* )
0
+
V
Y
=
Y
=
SN

dr*2
Yup exp(− ikr* ) + Ydown exp(ikr* ) (r* → −∞ )Reflection rate :reflection rateY
R( f ) = C

1.0

up

Ydown

C ≠1
since VSN is not real
Phase shift : Yup  ?
φ ( f ) = arg
 Ydown 

0.5
-1.0

-0.5

0.5

1.0

f / Re( f R )phase shift/π

depends on the frequency f.
φ can depend on the nature of the near-horizon boundary even if we
assume complete reflection.
However, φ ( f ) can be approximated by a linear fn.
for a narrow range of f close to the QNM frequency fR.
⇒ φ ( f ) can be absorbed by
the time interval between echoes and the overall phase shift φ0 33

0.012
0.010
0.008
0.006

Power spectrum changes at each reflection• Low frequency modes cannot penetrate
the angular momentum barrier.

1st echo

Initial ingoing
waveform isassumed to be thesame as theoutgoing one

2nd echo

There is a different opinion. (1907.03091)

The waveform is totally different from thesimple repetition of initial outgoing waveform
used by the analyses by other groups.

4th echoa / M = 0.7

8th echo

0.004

16th echo

0.002
0.5

1.0

1.5

f / Re( fQNM )2.0

Echo spectrum excited by a small
particle falling into Kerr black hole.

=a M 0.7,=
Lz 0.9 Lmarginal
Early phase of echo can be much louder,especially at high frequencies.
(Sago and T.T. arXiv 2009.08086)

How does the significance of the echo
signal change by taking into account thismore physically motivated waveform?

No Signal!

34

Analysis method

N. Uchikata, H. Nakano, T. Narikawa, N. Sago, H. Tagoshi, TT arXiv:1906.00838
N. Uchikata, et al., in preparation

We follow the basic strategy taken by Abedi et al. (2017）
Search the maximum signal-to-noise ratio (SNR) in the interval of 0.99 ≤ x ≤ 1.01,
varying the other parameters.

x≡

techo
∆techotecho

Background estimation
Perform the same analysis forthe neighboring 4096sec data,
where no signal is expected.
p-value# of reference data with SNR > SNR event# of all reference data
If we find p << 1, there might be some true signal.
p≡

35

Results of our re-analysis
We performed reanalysis using our new template, which we think is more
physically motivated, expecting the increase of significance.
List of p-values
SimpleOur
repetition template
GW150914

0.157

0.984

GW151012

0.047

0.882

GW170104

0.071

0.677

GW170608

0.079

0.488

GW170729

0.567

0.575

GW170814

0.024

0.472

GW170818

0.929

0.976

GW170823

0.055

0.315

average

0.241

0.671

2.75σ

The analysis using the template assuming
simple repetition of the same waveform gives
relatively small p-value, although p-value islarge for a few events.

Expected improvement of thesignificance was not achieved.

36Challenges on testing gravity
Motivation to consider gravity theories beyond GR
 Cosmological observations (DE/DM, Hubble/S8 tension, etc.)
 Black hole information paradox
 Quantum gravity

Constructing gravitational wave templates inextended gravity scenariosHigher dimensional operator corrections (EDGB, CS).
Low energy extra degrees of freedom that evades all the weakgravity tests (Screening mechanism, Parity odd, Very weaklyinteracting with matter).
Modified nature of black hole horizon (ECO).

Challenges


Theoretical predictions of the waveform for binary coalescence inextended gravity theories are very limited.
Optimizing data analyses based on the theoretical inputs.
For extreme mass-ratio inspirals, even GR templates are not ready yet.