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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
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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.
60