The event GW150914: Gravitational

waves and coalescing black-hole binaries

Gerhard Schafer

Theoretisch-Physikalisches Institut

Friedrich-Schiller-Universitat Jena

DESY Theory Seminar, Zeuthen, 17 March 2016

1

Outline

• Some History on Gravitational Waves

• Details on GW150914

• Interferometric Detection of GWs

• Gravitational Waves

• Hamiltonian for General Relativity

• Binary Black Hole Spacetimes

• Higher order post-Newtonian Hamiltonians

• Orbital Motion (ISCO)

• EOB Formalism

2

Some History on Gravitational Waves

3

1916, 1918 Einstein

1923 Eddington

1941 Landau/Lifshitz

1957 H. Bondi

1960, 1969 J. Weber, bar detector

1969 K. Thorne, radiation reaction

1972 R. Weiss (MIT), interferometric detector

1974 H. Billing (MPI), interferometric detector

1974, 1978 Hulse-Taylor pulsar, radiation damping

1989 LIGO (Caltech, MIT), Virgo (F-I), GEO600 (G-GB)

2002 LIGO, data aquisition

2015 aLIGO, data aquisition

4

Details on GW150914

B. P. Abbott et al., Phys. Rev. Lett. 116, 061102 (2016)

5

h(t)L = ∆L(t) = δLx − δLy

6

7

8

source power (3 solar masses radiated away in 0.2 seconds)

6×1030kg 9×1016( m

sec

)2

/200ms = 2.7×1048Watts = 0.75×10−4 c5

G

maximum power of a single process

c5

G=

Mc2

(GM/c2)/c= 3.6× 1052Watts

9

Interferometric Detection of GWs

10

11

line element ds2 = gµνdxµdxν

gravitational wave ds2 = ηµνdxµdxν + hTTij dxidxj

electromagnetic wave A = Aµdxµ = ATi dxi

Fermi normal coordinates

ds2 = −(1− 12c2

hTTij xixj)c2dt2 + dxidxi (|x| << c/f)

12

MTW: Gravitation (1973)

d2z/dt2 = 0

d2y/dt2 = 12 (−A+y + A×x)

d2x/dt2 = 12 (A+x + A×y)

propagating in z-direction

gravitational quadrupole wave

13

chirp mass M :=(m1m2)3/5

(m1 + m2)1/5=

c3

G

(596

f

π8/3f11/3

)3/5

velocityv

c=

(GMπf

c3

)1/3

gravitational quadrupole wave

hTTij (t,x) =

2r

G

c4Pijkm(n)Qkm

(t− r

c

)

Qkm =∑

a

ma(xakxa

m − 13δkmxa

l xal )

14

Gravitational Waves

15

Multipole expansion of far zone (FZ) field (e.g., Blanchet in LRR)

hTTij (t,x) =

G

c4

Pijkm(n)r

∞∑

l=2

(1c2

) l−22 4

l!M(l)

kmi3...il(t− r∗

c) Ni3...il

+(

1c2

) l−12 8l

(l + 1)!εpq(k S(l)

m)pi3...il(t− r∗

c) nq Ni3...il

Mij(t− r∗c

) = Mij

(t− r∗

c

)

+2Gm

c3

∫ ∞

0

dv ln( v

2b

)M

(2)

ij (t− r∗c− v) + O(1/c4),

r∗ = r +2Gm

c2ln

( r

cb

)+ O(1/c3)

16

Luminosity and energy loss

L(t) =c3

32πG

∮

FZ

(∂thTTij )2r2dΩ

L(t) =G

5c5

∞∑n=0

(1c2

)n

Ln(t)

=G

5c5

M(3)

ij M(3)ij +

1c2

[5

189M(4)

ijkM(4)ijk +

169

S(3)ij S(3)

ij

]

+1c4

[5

9072M(5)

ijkmM(5)ijkm +

584

S(4)ijkS(4)

ijk

]

− <dE(tret)

dt> = < L(t) >

17

Hamiltonian for General Relativity

18

xitt+ dt

dxi = N i dtNdt

xi3+1 splitting of spacetime nµ = (1,−N i)/N

nµ = (−N, 0, 0, 0)

nµ

gij Kij

Kij = −NΓ0ij = −Ng0µ(giµ,j + gjµ,i − gij,µ)/2

ds2 = −(Ncdt)2 + gij(dxi + N icdt)(dxj + N jcdt)

19

ds2 = −(Ncdt)2 + gij(dxi + N icdt)(dxj + N jcdt)

H =∫

d3x(NH−N iHi) +c4

16πG

∮

i0d2si(gij,j − gjj,i)

N |i0 = 1 +O(1/r), N i|i0 = O(1/r)

If the constraints H = 0 and Hi = 0 are fulfilled and adaptedcoordinate conditions happen, then

H =c4

16πG

∮

i0d2si(gij,j − gjj,i) ≡ HADM

20

Binary Black Hole Spacetimes

21

independent field variables

gij =(

1 +18φ

)4

δij + hTTij

πii = 0, πij = −γ1/2(Kij − γijK), πii = πijhTT

ij , γ = det(γij), γij = gij

unique decomposition: πij = πij + πijTT

πij = ∂iπj + ∂jπ

i − 23δij∂kπk

πijTTc3/16πG: canonical conjugate to hTT

ij

22

Hamilton and momentum constraints

γ1/2R =1

γ1/2

(πi

jπji −

12πi

iπjj

)+

16πG

c3

∑a

(m2

ac2 + γijpaipaj

)1/2δa

G00 = 8πGc4 T 00

−2∂jπji + πkl∂iγkl =

16πG

c3

∑a

paiδa

G0i = 8πG

c4 T 0i

23

isolated BH

ds2 = −(

1− Gm2rc2

1 + Gm2rc2

)2

c2dt2 +(

1 +Gm

2rc2

)4

δijdxidxj

= −(

1− Gm2Rc2

1 + Gm2Rc2

)2

c2dt2 +(

1 +Gm

2Rc2

)4

δijdXidXj

symmetry transformation (inversion): Rr =(

Gm2c2

)2

R2 = XiXi, r2 = xixi

24

Brill/Lindquist, JMP 1963

25

MTW: Gravitation (1973)

26

Brill-Lindquist BHs

maximally sliced

ds2 = −(

1− β1G2r1c2 − β2G

2r2c2

1 + α1G2r1c2 + α2G

2r2c2

)2

c2dt2 +(

1 +α1G

2r1c2+

α2G

2r2c2

)4

dx2

total energy: EADM = − c4

2πG

∮

i0

dsi∂iΨ = (α1 + α2)c2

Ψ = 1 + α1G2r1c2 + α2G

2r2c2

inversion map of the three-metric at the throat of black hole 1

r′1 = r1α21G

2/4c4r21

r′1 = x′ − x1, r1 = x− x1, r1 = |x− x1|

27

dl2 = Ψ4dx2 =(1 + α1G

2r1c2 + α2G2r2c2

)4

dx2

dl2 = Ψ′4dx′2 =(1 + α1G

2r′1c2 + α1α2G2

4r2r′1c4

)4

dx′2

r2 = α21G2

4c4r′1r′21

+ r12, r12 = r1 − r2

m1 = − c2

2πG

∮i0

ds′i∂′iΨ′ = α1 + α1α2G

2r12c2

Ψ′ = 1 + α1G2r′1c2 + α1α2G2

4r2r′1c4

28

−(

1 +18

φ

)∆φ =

16πG

c2

∑a

maδa (hTTij = 0 = pai)

φ =4G

c2

(α1

r1+

α2

r2

)

αa = ma − ma + mb

2+

c2rab

G

√1 +

ma + mb

c2rab/G+

(ma −mb

2c2rab/G

)2

− 1

HBL = (α1 + α2) c2 = (m1 + m2) c2 −G α1α2r12

29

Metric in d-dimensional conformally flat space:

gij =(

1 +14

d− 2d− 1

φ

) 4d−2

δij

φ =4G

c2

Γ(d−22 )

πd−22

(α1

rd−21

+α2

rd−22

)

Ψ = 1 +14

d− 2d− 1

φ

30

−∆−1δ =Γ((d− 2)/2)

4πd/2r2−d

Ψ = 1 +G(d− 2)Γ((d− 2)/2)

c2(d− 1)π(d−2)/2

(α1

rd−21

+α2

rd−22

)

(1 +

G(d− 2)Γ((d− 2)/2)c2(d− 1)π(d−2)/2

(α1

rd−21

+α2

rd−22

))α1δ1 = m1δ1

1 < d < 2(

1 +G(d− 2)Γ((d− 2)/2)

c2(d− 1)π(d−2)/2

α2

rd−212

)α1δ1 = m1δ1

31

Regularisation procedures for 4PN

Jaranowski/GS, PRD 92, 124043 (2015)3-dimensional Riesz-implemented Hadamard regularisation:local poles

IRH(3, ε1, ε2) :=∫

i(x)(

r1

s1

)ε1 (r2

s2

)ε2

d3x

= A + c1

(1ε1

+ lnr12

s1

)+ c2

(1ε2

+ lnr12

s2

)+ ...

pole at infinity

IRH(3, a, b, ε) :=∫

i(x)(

r1

r0

)aε (r2

r0

)bε

d3x

= A− c∞

(1

(a + b)ε+ ln

r12

r0

)+ ...

32

UV regularisation (small singularity-centered balls with radii la)

IRH(3, ε1, ε2) + ∆I1 + ∆I2

∆Ia := Ia(d)− IRHa (3; εa), a = 1, 2

IRHa (3; εa) = ca

(1εa

+ lnlasa

)+ ...

Ia(d) = −ca

2ε− 1

2c′a(3) + ca ln

lal0

+ ..., ε := d− 3

c′a(3) = c′aa(3) + c′ab(3) lnr12

l0, a 6= b

GD = GN ld−30 , D = d + 1

33

(IRH(3, ε1, ε2) + ∆I1 + ∆I2)|εa→0

= A− 12(c′11(3) + c′21(3))

− c1 + c2

2ε

+ (c1 + c2 − 12c′12(3)− 1

2c′22(3))ln

r12

l0+ ...

at the end:a unique total time derivative eliminates all poles and logarithms

34

IR regularisation 1 (outside large coordinates-origin-centered ballwith radius R)

IRH(3, a, b, ε) + ∆I1∞

∆I1∞ := FPI1

∞ − IRH∞ (3, a, b, ε)

IRH∞ (3, a, b, ε) = −c∞

(1

(a + b)ε+ ln

R

r0

)+ ...

FPI1∞ = − ∂c1

∞∂B

(3, 0) − c1∞(3, 0)ln

R

s

(IRH(3, a, b, ε) + ∆I1∞)|ε→0 = A− ∂c0

∞∂B

(0)− c∞lnr12

s

∆−1d hTT

(4)ij → ∆−1d

[(rs

)BhTT

(4)ij

]TT

35

IR regularisation 2 (outside large coordinates-origin-centered ballwith radius R)

FPI2∞ = −c2

∞(3)lnR

s(r1

s

)α1(r2

s

)α2

β = α1 + α2, ε = d− 3

I2∞(d, β) =

η(ε, β)β − 2ε

FPI2∞ = FPε→0limβ→0

η(ε, β)− η(ε, 2ε)β − 2ε

“IR regularisation 2 - IR regularisation 1” shows a final ambiguity

36

Higher Order Post-Newtonian Hamiltonians

¤−1sym =

(1 +

1c2

∆−1∂2t + ...

)∆−1δ(t− t′)

Gret =(

1− 1c|r− r′|∂t +

12c2

|r− r′|2∂2t + ...

)1

4π|r− r′|δ(t− t′)

37

binary black holes to 4PN order

H(t) = m1c2 + m2c

2 + HN +1c2

H[1PN ]

+1c4

H[2PN ] +1c6

H[3PN ] +1c8

H[4PN ] + . . .

+1c5

H[2.5PN ](t) +1c7

H[3.5PN ](t) + . . .

H = (H −Mc2)/µ, µ = m1m2/M , M = m1 + m2

ν = µ/M , 0 ≤ ν ≤ 1/4

test particles: ν = 0, equal masses: ν = 1/4

CMF: p1 + p2 = 0, p = p1/µ, r = r12 = |x1 − x2|,pr = (n · p), q = (x1 − x2)/GM , n = n12 = q/|q|

38

HN =p2

2− 1

q

H[1PN ] =18(−1 + 3ν)p4 − 1

2[(3 + ν)p2 + νp2

r]1q

+1

2q2

H[2PN ] =116

(1− 5ν + 5ν2)p6

+18[(5− 20ν − 3ν2)p4 − 2ν2p2

rp2 − 3ν2p4

r]1q

+12[(5 + 8ν)p2 + 3νp2

r]1q2− 1

4(1 + 3ν)

1q3

39

2.5PN dissipative binary dynamics

H[2.5PN ](t) =25

d3Qij(t)dt3

(pipj − ninj

q

)

Qij(t) = ν(q′iq′j − 13q′2δij)

t = t/GM

40

H[3PN ] =1

128(−5 + 35ν − 70ν2 + 35ν3)p8

+116

[(−7 + 42ν − 53ν2 − 5ν3)p6 + (2− 3ν)ν2p2rp

4

+ 3(1− ν)ν2p4rp

2 − 5ν3p6r]

1q

+ [116

(−27 + 136ν + 109ν2)p4 +116

(17 + 30ν)νp2rp

2

+112

(5 + 43ν)νp4r]

1q2

+ [(−25

8+

(164

π2 − 33548

)ν − 23

8ν2

)p2

+(−85

16− 3

64π2 − 7

4ν

)νp2

r]1q3

+ [18

+(

10912

− 2132

π2

)ν]

1q4

41

H[4PN ] =(

7256

− 63256

ν +189256

ν2 − 105128

ν3 +63256

ν4

)p10

+

45128

p8 − 4516

p8ν +(

42364

p8 − 332

p2rp

6 − 964

p4rp

4

)ν2

+(−1013

256p8 +

2364

p2rp

6 +69128

p4rp

4 − 564

p6r +

35256

p8r

)ν3

+(− 35

128p8 − 5

32p2

rp6 − 9

64p4

rp4 − 5

32p6

r −35128

p8r

)ν4

1q

+

138

p6 +(−791

64p6 +

4916

p2rp

4 − 889192

p4r +

369160

p6r

)ν

+(

4857256

p6 − 54564

p2rp

4 +9475768

p4r −

1151128

p6r

)ν2

+(

2335256

p6 +1135256

p2rp

4 − 1649768

p4r +

103531280

p6r

)ν3

1q2

42

+

[10532

p4 + C41 ν + C42 ν2 +(− 553

128p4 − 225

64p2

r −381128

p4r

)ν3

]1q3

+

10532

+ C21 ν + C22 ν2

1q4

+

− 1

16+ c01 ν + c02 ν2

1q5

− 15I(3)ij (t)

∫ +∞

−∞dw ln

( |w|c2q

)I(4)ij (t− w) ν

43

C42 =(−1189789

28800+

1849116384

π2

)p4 +

(−127

3− 4035

2048π2

)p2

rp2

+(

575631920

− 3865516384

π2

)p4

r

C22 =(

67281119200

− 15817749152

π2

)p2 +

(−21827

3840+

11009949152

π2

)p2

r

c02 = −125645

+74033072

π2

44

C41 =(−589189

19200+

27498192

π2

)p4 +

(633471600

− 10591024

π2

)p2

rp2

+(−23533

1280+

3758192

π2

)p4

r

C21 =(

18576119200

− 218378192

π2

)p2 +

(340177957600

− 2869124576

π2

)p2

r

c01 = −1691992400

+62371024

π2

4PN

Jaranowski/GS (’12,’13)[in part], Damour/Jaranowski/GS (’14)

45

Hnear−zone (s)4PN [xa,pa] = H loc 0

4PN [xa,pa]

+25

G2M

c8

(I(3)ij

)2(ln

r12

s+ C

)

+d

dtG[xa,pa]

Htailsym (s)4PN (t) = −1

5G2M

c8I(3)ij (t)

×∫ +∞

−∞dv ln

( |v|c2s

)I(4)ij (t− v)

46

Matching to results by Bini/Damour for perturbed Schwarzschildmetric from particle in circular motion yields C = − 1681

1536 .

Htailsym4PN (t) = −1

5G2M

c8I(3)ij (t) Pf2r12/c

∫ +∞

−∞

dv

|v| I(3)ij (t− v)

Classical gravitational “Lamb Shift” (orbital average):

∆E = − G

5c5

GM

c3P

∫ P

0

dt

[I(3)ij (t) Pf2r12/c

∫ +∞

−∞

dv

|v| I(3)ij (t− v)

]

<dE

dt> = − G

5c5

1P

∫ P

0

dt

[I(3)ij (t)I(3)

ij (t)]

(Einstein’s quad. formula)

47

∆EcircOrb = 325 µc2

(µ

Mj2

)(1j2

)4 [ln

(1j2

)+ 2ln4 + 2γE

]

j ≡ cJGMµ , 1

j2 = GMrc2 < 1

Lamb Shift: δEnS = − 83πn3 mc2(α)(Zα)4

[ln(Zα) + 1

2 lnKn − 1148

]

α ≡ e2

~c , Zα < 1, L.S. Brown, QFT, CUP 1992 [“orb” only]

ln4 ' 1.39, γE ' 0.58, lnK2 ' 2.81

48

Orbital Motion (ISCO)

49

ISCO

H = H(p, r), p2 = p2r + j2/r2, pr = (p · r)/r

circular orbits: pr = 0, p2 = j2/r2, H = H(j, r)

circular motion: ∂∂r H(j, r) = 0 → H(j)

orbital frequency: ω = dH(j)dj → H(ω)

ISCO: dH(ω)dω = 0 or, alternatively ∂2

∂r2 H(j, r) = 0

SBH: E(x) =1− 2x

(1− 3x)1/2− 1

= −12x +

38x2 +

2716

x3 +675128

x4 +3969256

x5 + ...

E(x) ≡ H(x)−mc2

mc2, x =

(GMω

c3

)2/3

50

circular orbits:

c2E4PN ≡ HN + c−2H[1PN ] + c−4H[2PN ] + c−6H[3PN ] + c−8H[4PN ]

E4PN (x) = −x

2+

(38

+124

ν

)x2 +

(2716− 19

16ν +

148

ν2

)x3

+(

675128

+(−34445

1152+

205192

π2

)ν +

155192

ν2 +35

10368ν3

)x4

− 12

(− 3960128 + [c1 + 448

15 lnx]ν +(− 498449

3456 + 3157576 π2

)ν2 + 301

1728ν3 + 7731104ν4

)x5

Damour (’10)[lnx], Blanchet/Detweiler/Le Tiec/Whiting (’10)[lnx]

Jaranowski/GS (’12)[lnx, ν3, ν4], (’13)[ν2], Foffa/Sturani (’13) [lnx, ν3, ν4]

c1 = − 1236715760 + 9037

1536π2 + 179215 ln 2 + 896

15 γE = 153.88...

Bini/Damour (’13), Le Tiec/Blanchet/Whiting (’12) [numerical value]

51

0.00 0.05 0.10 0.15 0.20 0.250.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Ν

x LSC

O1PN

2PN3PN

4PN* * test particle exact

Jaranowski/GS (’13)

52

spin-gravity interaction

leading order spin orbit

HLOSO =

G

c2

∑a

∑

b 6=a

1r2ab

(Sa × nab) ·[

3mb

2mapa − 2pb

]

leading order spin(1)-spin(2)

HLOS1S2

=G

c2

∑a

∑

b6=a

12r3

ab

[3(Sa · nab)(Sb · nab)− (Sa · Sb)]

leading order spin(1) spin(1)

HLOS1S1

=G

c2

12r3

12

[3(S1 · n12)(S1 · n12)− (S1 · S1)]

53

Hcon = HN + H1PN + H2PN + H3PN + H4PN

+ HLOSO + HLO

S1S2+ HLO

S21

+ HLOS2

2

+ HNLOSO + HNLO

S1S2+ HNLO

S21

+ HNLOS2

2

+ HNNLOSO + HNNLO

S1S2

+ HLOS2

1+ HLO

S22

+ HNLOS2

1+ HNLO

S22

+ HLOS4

1+ HLO

S41

+ HLOp1S3

2+ HLO

p2S31

+ HLOp1S1S2

2+ HLO

p2S2S21

Hdiss(t) = H2.5PN (t) + H3.5PN (t)

+ HDLOSO (t) + HDLO

S1S2(t)

Steinhoff (2011), Hartung/Steinhoff/GS (2013), Levi/Steinhoff (2015)

54

EOB Formalism

Damour/Jaranowski/GS, arxiv:0803.0915

Damour, arXiv:1312.3505

55

56

Standard representation

Heff

µc2≡ H2 −m2

1c4 −m2

2c4

2m1m2c4= 1 +

HNR

µc2+

ν

2

(HNR

µc2

)2

HNR ≡ H −Mc2

H = Mc2

√1 + 2ν

(Heff

µc2− 1

)

EOB representation

gµνeff PµPν + Q4(Pi) = −µ2c2, H

EOB(1)eff ≡ −P0c

HEOBeff = N i

effPic + Neffc

√µ2c2 + γij

effPiPj + Q4(Pi)

HEOB = Mc2

√1 + 2ν

(HEOB

eff

µc2− 1

)

57

Canonical transformation to connect:

HEOBeff ≡ H

EOB(1)eff (X,P, S0) + H

EOB(2)eff (X, P, S0, σ) = Heff(x, p, S1, S2)

gµνeff PµPν =

1R2 + a2cos2θ

(∆R(R)P 2

R + P 2θ

+1

sin2θ

(Pφ + a sin2θ

Pt

c

)2

− 1∆t(R)

((R2 + a2)

Pt

c+ aPφ

)2

∆t(R) = R2Pnm

[A(R) +

a2

R2

], ∆R(R) = ∆t(R)D−1(R)

a = S0/Mc, Pnm: (n,m)-Pade approximation

58

A(R) = 1− 2u + 2νu3 +(

943− 41

32π2

)νu4

D−1(R) = 1 + 6νu2 + 2(26− 3ν)νu3

u = GMRc2

possible improvement through (n,m)-Pade approximant:

1 + c1u + c2u2 + ... + cn+mun+m ∼ 1 + a1u + a2u

2 + ... + anun

1 + b1u + b2u2 + ... + bmum

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Augmented through radiation reaction, the equations of motion forinspiral are easily solvable ordinary differential equations.

The connection to gravitational waveforms is given in, e.g. Damour,arXiv:1312.3505.

Therein, the generalisation to the merger of binary black holes and ofthe ringdown of the final black hole can be found too, includingcomparison with numerical relativity.

EOB played a crucial role in the identification of GW150914.

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