Analytical Relativityand
the First Direct Detections of Gravitational Waves
Piotr Jaranowski
Faculty of Physics, University of Białystok, Poland
The 2nd Workshop on
Singularities of General Relativity and Their Quantum Fate
Banach Mathematical Center, Warsaw, May 21–25, 2018
1 The First Direct Detections of Gravitational Waves
2 Indispensability of Matched Filtering in Data Analysis
3 Post-Newtonian Relativity of Two-Body Systems
4 Effective One-Body Approach
5 Conclusions
“The Weimar-triangle collaboration”: PN gravity/EOB approach
Thibault Damour, Institut des Hautes Etudes Scientifiques (IHES),Bures-sur-Yvette, France
Gerhard Schafer, Friedrich Schiller University, Jena, Germany
Piotr Jaranowski, University of Białystok, Poland
Virgo-POLGRAW group, member of Virgo Collaboration: analysis of LIGO/Virgo data
Scientists from Polish research institutes and universities:— Institute of Mathematics, Polish Academy of Sciences, Warszawa,— Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences,
Warszawa,— National Centre for Nuclear Research, Świerk,— University of Warsaw,— University of Białystok,— Zielona Góra University,— Jagiellonian University, Kraków,— Warsaw University of Technology.
Leaded by Andrzej Królak from Institute of Mathematics,Polish Academy of Sciences, Warszawa.
1 The First Direct Detections of Gravitational Waves
2 Indispensability of Matched Filtering in Data Analysis
3 Post-Newtonian Relativity of Two-Body Systems
4 Effective One-Body Approach
5 Conclusions
Gravitational-Wave Signal from Coalescing Black-Hole Binary
A laser-interferometric detector measures differential displacementalong the detector’s arms:
∆L(t) = δLx (t)− δLy (t) = h(t) L0,
where L0 = 4 km for LIGO detectors or L0 = 3 km for Virgo detector,h(t) is the dimensionless gravitational-wave strain.
The detected waveforms match the predictions of general relativityfor the inspiral and merger of a pair of stellar-mass black holes
and the ringdown of the resulting single Kerr black hole.
Very Sensitive Search Algorithms
The expected gravitational-wavesignals are weak compared toinstrumental noise,therefore the detection of signalsand identification of their sourcesrequires employing very sensitivesearch procedures.
Each search procedure defines a detection statistic that ranks likelihood ofpresence a gravitational-wave signal in the data.One then identifies candidate events of high enough value of the detectionstatistic that are detected at all observatories and their time of arrivals areconsistent with the intersite propagation times.
The significance of a candidate event is determined by false alarm rate (FAR):the rate at which detector noise produces events with a detection-statistic valueequal to or higher than the detection-statistic value of the candidate event.
This translates to a false alarm probability: a probability of observing one ormore noise events as strong as the candidate event during the analysis time.
Results of Advanced LIGO/Virgo Observing Runs O1 and O2
The 1st Advanced LIGO observing run O1(Sept 12, 2015—Jan 19, 2016):
— BH/BH signal GW150914 [FAR < 1/(1 670 000 yrs)],— BH/BH candidate event LVT151012 [FAR ∼= 1/(2.7 yrs)],— BH/BH signal GW151226 [FAR < 1/(1 670 000 yrs)].
The 2nd Advanced LIGO/Virgo observing run O2(Nov 30, 2016—Aug 25, 2017):
— BH/BH signal GW170104 [FAR < 1/(70 000 yrs)],— BH/BH signal GW170608 [FAR < 1/(3 000 yrs)],— BH/BH signal GW170814 [FAR < 1/(27 000 yrs)],— NS/NS signal GW170817 [FAR < 1/(80 000 yrs)].
1 The First Direct Detections of Gravitational Waves
2 Indispensability of Matched Filtering in Data Analysis
3 Post-Newtonian Relativity of Two-Body Systems
4 Effective One-Body Approach
5 Conclusions
Two Types of Search Algorithms
Two types of search algorithms are used in data analysis:
searches for generic gravitational-wave transients,which operate without a specific waveform modeland identify coincident excess powerin a time-frequency representation of the data;
matched-filtering searches using relativistic models ofcompact binary coalescence waveforms,they correlate the data with a copy of a waveform(i.e. a template) one expects to find in the data.
Results of Generic Transient Searches
GW150914
Significance of GW150914: FAR < 1/(22 500 yrs).
GW151226
Normalized amplitude of a time-frequency representation of the strain data(left for LIGO Hanford, right for LIGO Livingston detector)
Advantages and Challenges of Matched Filtering
Advantages
Matched filtering has better sensitivity than unmodeled searches.
Constraining signal space decreases false alarm rate.
One can use signal-based vetoes to separate signals from transient noise.
ChallengesIt loses sensitivity if templates do not match signals:
accurate waveform models (templates) needed;
parameters of template must be close enough to signal: templates’ parametershave to cover the signal’s parameter space densly enough.
One has to construct a discrete bank of templates parametrized by possiblevalues of the gravitational-wave signal’s parameters:
h(t;m1,m2,S1,S2,Λ1,Λ2, . . .),
where m1, m2 and S1, S2 are masses and spins of binary components,Λ1, Λ2, . . . are tidal parameters.
Construction of accurate templates requiresaccurate enough solution of relativistic two-body problem
for each value of signal’s parameter present in the bank of templates.
Different Approaches for Solving Relativistic Two-Body Problem
Numerical relativity (breakthrough in 2005).
Approximate ‘analytical’ methods:
— post-Newtonian (PN) expansion,
— black-hole perturbation (self-force based) approach.
The adjective ”analytical” means here methods that rely onsolving explicit (that is analytically given) ordinary differential equations,contrary to full numerical relativity simulations.
Effective-one-body formalism (EOB): combines results of PN approach,black-hole perturbation theory, and numerical relativity.
PN expansion:0th order—Newtonian gravity;nPN order—corrections of order( v
c
)2n∼(Gm
rc2
)n
to the Newtonian gravity.
Perturbation approach:m1
m2� 1.
Computing Power Constraints
Construction of bank of templates requires multiple integration of
— partial differential equations in numerical relativity,
— ordinary differential equations in approximate analytical relativity.
For detection of gravitational-wave signals originated fromcoalescences of binaries made of spinning black holes/neutron starswith arbitrary mass ratios, due to limitations in available computing power,it will not be possible in the nearest futureto construct bank of templates based purely on numerical results.
Binary Coalescence Searches in Observing Run O1
The waveforms depend on the massesm1, m2 of the binary components andtheir dimensionless spins ~χ1 and ~χ2
[~χi := c ~Si/(Gm2i ), i = 1, 2].
The searches targeted binaries withindividual masses from 1 to 99M�,total mass not greater than 100M�,and dimensionless spins up to 0.9895.
Waveforms modeling systems withtotal mass less than 4M�:PN inspiral waveformsaccurate to 3.5PN order.
Waveforms modeling systems withtotal mass larger than 4M�:inspiral+merger+ringdown waveformsconstructed by means of theeffective-one-body formalism.
Around 250 000 templates were used.
The waveform model assumes that thespins of the merging objects are alignedwith the orbital angular momentum,but the resulting templates can recoversystems with misaligned spins (in theparameter region of detected signals).
1 The First Direct Detections of Gravitational Waves
2 Indispensability of Matched Filtering in Data Analysis
3 Post-Newtonian Relativity of Two-Body Systems
4 Effective One-Body Approach
5 Conclusions
Gravitational Waves from Inspiralling Binary on Circular OrbitsThe gravitational-wave strain measured by the laser-interferometric detector and induced by gravitational wavesfrom coalescing compact binary (made of nonspinning bodies) in circular orbits during inspiral phase:
h(t) =C
D
[φ(t)
]2/3 sin[2φ(t) + α
],
where φ(t) is the orbital phase of the binary [so φ(t) := dφ(t)/dt is the angular frequency],D is the luminosity distance of the binary to the Earth, C and α are some constants.
The orbital phase φ(t) is computed from the balance equation
dE
dt= −L =⇒ φ = φ(t),
which bothe sides have the following PN expansions:
E = EN +1
c2E1PN +
1
c4E2PN +
1
c6E3PN +
1
c8E4PN + O
((v/c)9)
,
L = LN +1
c2L1PN +
1
c3L1.5PN +
1
c4L2PN +
1
c5L2.5PN
+1
c6L3PN +
1
c7L3.5PN +
1
c8L4PN + O
((v/c)9)
.
Notation
Masses of the bodies: m1,m2, M := m1 + m2, µ :=m1m2
M, ν :=
µ
M=
m1m2
(m1 + m2)2, 0 ≤ ν ≤
1
4.
Dimensionless PN parameter introduced for circular orbits x :=1
c2(GMφ)2/3
.
4PN-Accurate Bounding Energy in the Center-of-Mass Framefor Circular Orbits
E(x ; ν) = −µc2x
2
(1 + e1PN(ν) x + e2PN(ν) x2 + e3PN(ν) x3 +
(e4PN(ν) +
448
15ν ln x
)x4 + O
((v/c)10))
,
e1PN(ν) = −3
4−
1
12ν, e2PN(ν) = −
27
8+
19
8ν −
1
24ν
2,
e3PN(ν) = −675
64+
( 34445
576−
205
96π
2)ν −
155
96ν
2 −35
5184ν
3,
e4PN(ν) = −3969
128+
(−
123671
5760+
9037
1536π
2 +896
15(2 ln 2 + γ)
)ν +
(−
498449
3456+
3157
576π
2)ν
2 +301
1728ν
3 +77
31104ν
4
(γ is the Euler’s constant).
3.5PN-Accurate Gravitational-Wave Luminosity for Circular Orbits
L(x ; ν) =32c5
5Gν
2x5(
1 + `1PN(ν) x + 4π x3/2 + `2PN(ν) x2
+ `2.5PN(ν) x5/2 +(`3PN(ν) −
856
105ln(16x)
)x3 + `3.5PN(ν) x7/2 + O
((v/c)8))
,
`1PN(ν) = −1247
336−
35
12ν, `2PN(ν) = −
44711
9072+
9271
504ν +
65
18ν
2, `2.5PN(ν) =
(−
8191
672−
535
24ν
)π,
`3PN(ν) =6643739519
69854400+
16
3π
2 −1712
105γ +
(−
134543
7776+
41
48π
2)ν −
94403
3024ν
2 −775
324ν
3,
`3.5PN(ν) =
(−
16285
504+
214745
1728ν +
193385
3024ν
2)π.
Post-Newtonian Two-Body Problem
There are two sub-problems, usually analyzed separately:
problem of finding equations of motion (EOM),
problem of computing gravitational-wave luminosity.
EOM N 1PN 2PN 2.5PN 3PN 3.5PN 4PN 4.5PN 5PN 5.5PN 6PN 6.5PN
Luminosity — — — N — 1PN 1.5PN 2PN 2.5PN 3PN 3.5PN 4PN
Deriving equations of motion:after fixing the gauge (e.g. within ADM Hamiltonian formalismor harmonic coordinates) one perturbatively solves (in dimension D = 4− ε)Einstein’s equations using point masses (i.e. Dirac-delta sources);UV divergences linked to point-particle descriptionremoved by dimensional regularization;presence IR divergences on top of the UV divergenceslinked to nonlocality in time (related to tail effects).
Computing gravitational-wave luminosities:—Blanchet–Damour–Iyer formalism combines a multipolar post-Minkowskianexpansion in the exterior zone with a PN expansion in the near zone;—Will–Wiseman–Pati formalism uses a direct integrationof the relaxed Einstein equations.
Two-Body Post-Newtonian Equations of Motion (Without Spins)
1PN (including v2/c2):Lorentz–Droste 1917, Einstein–Infeld–Hoffmann 1938.
2PN (including v4/c4):Ohta–Okamura–Kimura–Hiida 1974, Damour–Deruelle 1981,Damour 1982, Schafer 1985, Kopeikin 1985.
2.5PN (including v5/c5):Damour–Deruelle 1981, Damour 1982, Schafer 1985, Kopeikin 1985.
3PN (including v6/c6):Jaranowski–Schafer 1998, Blanchet–Faye 2000,Damour–Jaranowski–Schafer 2001, Itoh–Futamase 2003,Blanchet–Damour–Esposito-Farese 2004, Foffa–Sturani 2011.
3.5PN (including v7/c7):Iyer–Will 1993, Jaranowski–Schafer 1997, Pati–Will 2002,Konigsdorffer–Faye–Schafer 2003, Nissanke–Blanchet 2005, Itoh 2009.
4PN (including v8/c8; new feature: nonlocality in time):Jaranowski–Schafer 2012–2013, Foffa–Sturani 2013, Bini–Damour 2013,Damour–Jaranowski–Schafer 2014–2015, Blanchet et al. 2016–2018,Foffa et al. 2016–2017.
Gravitational-Wave Energy Flux from Binary System (Without Spins)
Lowest order (quadrupole formula):Einstein 1918, Peters–Mathews 1963.
1PN correction (including v2/c2):Wagoner–Will 1976.
1.5PN correction (including v3/c3):Blanchet–Damour 1992, Wiseman 1993.
2PN correction (including v4/c4):Blanchet–Damour–Iyer 1995, Will–Wiseman 1995.
2.5PN correction (including v5/c5):Blanchet 1996.
3PN correction (including v6/c6):Blanchet–Damour–Esposito-Farese–Iyer 2004.
3.5PN correction (including v7/c7):Blanchet 1998.
4PN-Accurate 2-Point-Mass ADM Conservative Hamiltonian
H≤4PN [xa, pa ] = Hlocal≤4PN(xa, pa) + Hnonlocal4PN [xa, pa ] (a = 1, 2).
Local-in-time 4PN-accurate Hamiltonian
Hlocal≤4PN(xa, pa) = (m1 + m2)c2 + HN(xa, pa) +1
c2H1PN(xa, pa)
+1
c4H2PN(xa, pa) +
1
c6H3PN(xa, pa) +
1
c8Hlocal4PN (xa, pa).
Nonlocal-in-time 4PN Hamiltonian(Blanchet-Damour 1988, Damour-Jaranowski-Schafer 2014)
Hnonlocal4PN [xa, pa ] = −1
5
G2M
c8
...I ij × Pf2r12/c
(∫ +∞
−∞
dv
|v|
...I ij (t + v)
),
...I ij is a 3rd time derivative of the Newtonian quadrupole moment Iij of the system,
Iij :=∑a
ma
(xia xja −
1
3δij x2a
),
PfT denotes a Hadamard partie finie with time scale T ,
PfT
∫ +∞
0
dv
vg(v) :=
∫ T
0
dv
v
(g(v) − g(0)
)+
∫ +∞
T
dv
vg(v).
Newtonian/1PN/2PN Hamiltonians
The operation “+(
1 ↔ 2)
” used below denotes the addition for each term of another term obtained by the label permutation 1 ↔ 2.
HN(xa, pa) =p2
1
2m1−
Gm1m2
2r12+(1↔ 2
),
H1PN(xa, pa) = −(p2
1)2
8m31
+Gm1m2
4r12
(− 6p2
1
m21
+ 7(p1 · p2)
m1m2+
(n12 · p1)(n12 · p2)
m1m2
)+
G2m21m2
2r212
+(1↔ 2
),
H2PN(xa, pa) =1
16
(p21)3
m51
+1
8
Gm1m2
r12
(5
(p21)2
m41
−11
2
p21 p
22
m21m
22
−(p1 · p2)2
m21m
22
+ 5p2
1 (n12 · p2)2
m21m
22
− 6(p1 · p2) (n12 · p1)(n12 · p2)
m21m
22
−3
2
(n12 · p1)2(n12 · p2)2
m21m
22
)
+1
4
G2m1m2
r212
(m2
(10p2
1
m21
+ 19p2
2
m22
)−
1
2(m1 + m2)
27 (p1 · p2) + 6 (n12 · p1)(n12 · p2)
m1m2
)−
1
8
Gm1m2
r12
G2(m21 + 5m1m2 + m2
2)
r212
+(1↔ 2
).
3PN Hamiltonian (Damour-Jaranowski-Schafer 1998–2001)
H3PN(xa, pa) = −5
128
(p21)4
m71
+1
32
Gm1m2
r12
(− 14
(p21)3
m61
+ 4
((p1 · p2)2 + 4 p2
1 p22
)p2
1
m41m
22
+ 6p2
1 (n12 · p1)2(n12 · p2)2
m41m
22
− 10
(p2
1 (n12 · p2)2 + p22 (n12 · p1)2
)p2
1
m41m
22
+ 24p2
1 (p1 · p2)(n12 · p1)(n12 · p2)
m41m
22
+ 2p2
1 (p1 · p2)(n12 · p2)2
m31m
32
+
(7 p2
1 p22 − 10 (p1 · p2)2
)(n12 · p1)(n12 · p2)
m31m
32
+
(p2
1 p22 − 2 (p1 · p2)2
)(p1 · p2)
m31m
32
+ 15(p1 · p2)(n12 · p1)2(n12 · p2)2
m31m
32
− 18p2
1 (n12 · p1)(n12 · p2)3
m31m
32
+ 5(n12 · p1)3(n12 · p2)3
m31m
32
)+
G2m1m2
r212
(1
16(m1 − 27m2)
(p21)2
m41
−115
16m1p2
1 (p1 · p2)
m31m2
+1
48m2
25 (p1 · p2)2 + 371 p21 p
22
m21m
22
+17
16
p21(n12 · p1)2
m31
+5
12
(n12 · p1)4
m31
−3
2m1
(n12 · p1)3(n12 · p2)
m31m2
−1
8m1
(15 p2
1 (n12 · p2) + 11 (p1 · p2) (n12 · p1))(n12 · p1)
m31m2
+125
12m2
(p1 · p2) (n12 · p1)(n12 · p2)
m21m
22
+10
3m2
(n12 · p1)2(n12 · p2)2
m21m
22
−1
48(220m1 + 193m2)
p21(n12 · p2)2
m21m
22
)+
G3m1m2
r312
(−
1
48
(425m2
1 +(
473−3
4π2)m1m2 + 150m2
2
)p2
1
m21
+1
16
(77(m2
1 + m22) +
(143−
1
4π2)m1m2
)(p1 · p2)
m1m2+
1
16
(20m2
1 −(
43 +3
4π2)m1m2
)(n12 · p1)2
m21
+1
16
(21(m2
1 + m22) +
(119 +
3
4π2)m1m2
)(n12 · p1)(n12 · p2)
m1m2
)+
1
8
G4m1m32
r412
((227
3−
21
4π2)m1 + m2
)+(1↔ 2
).
4PN Local Hamiltonian (Damour-Jaranowski-Schafer 2012–2015)
H local4PN (xa, pa) =
7(p21)5
256m91
+Gm1m2
r12
(45(p2
1)4
128m81
−9(n12 · p1)2(n12 · p2)2(p2
1)2
64m61m
22
+15(n12 · p2)2(p2
1)3
64m61m
22
−9(n12 · p1)(n12 · p2)(p2
1)2(p1 · p2)
16m61m
22
−3(p2
1)2(p1 · p2)2
32m61m
22
+15(n12 · p1)2(p2
1)2p22
64m61m
22
−21(p2
1)3p22
64m61m
22
−35(n12 · p1)5(n12 · p2)3
256m51m
32
+25(n12 · p1)3(n12 · p2)3p2
1
128m51m
32
+33(n12 · p1)(n12 · p2)3(p2
1)2
256m51m
32
−85(n12 · p1)4(n12 · p2)2(p1 · p2)
256m51m
32
−45(n12 · p1)2(n12 · p2)2p2
1(p1 · p2)
128m51m
32
−(n12 · p2)2(p2
1)2(p1 · p2)
256m51m
32
+25(n12 · p1)3(n12 · p2)(p1 · p2)2
64m51m
32
+7(n12 · p1)(n12 · p2)p2
1(p1 · p2)2
64m51m
32
−3(n12 · p1)2(p1 · p2)3
64m51m
32
+3p2
1(p1 · p2)3
64m51m
32
+55(n12 · p1)5(n12 · p2)p2
2
256m51m
32
−7(n12 · p1)3(n12 · p2)p2
1p22
128m51m
32
−25(n12 · p1)(n12 · p2)(p2
1)2p22
256m51m
32
−23(n12 · p1)4(p1 · p2)p2
2
256m51m
32
+7(n12 · p1)2p2
1(p1 · p2)p22
128m51m
32
−7(p2
1)2(p1 · p2)p22
256m51m
32
−5(n12 · p1)2(n12 · p2)4p2
1
64m41m
42
+7(n12 · p2)4(p2
1)2
64m41m
42
−(n12 · p1)(n12 · p2)3p2
1(p1 · p2)
4m41m
42
+(n12 · p2)2p2
1(p1 · p2)2
16m41m
42
−5(n12 · p1)4(n12 · p2)2p2
2
64m41m
42
+21(n12 · p1)2(n12 · p2)2p2
1p22
64m41m
42
−3(n12 · p2)2(p2
1)2p22
32m41m
42
−(n12 · p1)3(n12 · p2)(p1 · p2)p2
2
4m41m
42
+(n12 · p1)(n12 · p2)p2
1(p1 · p2)p22
16m41m
42
+(n12 · p1)2(p1 · p2)2p2
2
16m41m
42
−p2
1(p1 · p2)2p22
32m41m
42
+7(n12 · p1)4(p2
2)2
64m41m
42
−3(n12 · p1)2p2
1(p22)2
32m41m
42
−7(p2
1)2(p22)2
128m41m
42
)+
G2m1m2
r212
m1
(369(n12 · p1)6
160m61
−889(n12 · p1)4p2
1
192m61
+49(n12 · p1)2(p2
1)2
16m61
−63(p2
1)3
64m61
−549(n12 · p1)5(n12 · p2)
128m51m2
+67(n12 · p1)3(n12 · p2)p2
1
16m51m2
−167(n12 · p1)(n12 · p2)(p2
1)2
128m51m2
+1547(n12 · p1)4(p1 · p2)
256m51m2
−851(n12 · p1)2p2
1(p1 · p2)
128m51m2
+1099(p2
1)2(p1 · p2)
256m51m2
+3263(n12 · p1)4(n12 · p2)2
1280m41m
22
+1067(n12 · p1)2(n12 · p2)2p2
1
480m41m
22
−4567(n12 · p2)2(p2
1)2
3840m41m
22
−3571(n12 · p1)3(n12 · p2)(p1 · p2)
320m41m
22
+3073(n12 · p1)(n12 · p2)p2
1(p1 · p2)
480m41m
22
+4349(n12 · p1)2(p1 · p2)2
1280m41m
22
−3461p2
1(p1 · p2)2
3840m41m
22
+1673(n12 · p1)4p2
2
1920m41m
22
−1999(n12 · p1)2p2
1p22
3840m41m
22
+2081(p2
1)2p22
3840m41m
22
−13(n12 · p1)3(n12 · p2)3
8m31m
32
+191(n12 · p1)(n12 · p2)3p2
1
192m31m
32
−19(n12 · p1)2(n12 · p2)2(p1 · p2)
384m31m
32
−5(n12 · p2)2p2
1(p1 · p2)
384m31m
32
+11(n12 · p1)(n12 · p2)(p1 · p2)2
192m31m
32
+77(p1 · p2)3
96m31m
32
+233(n12 · p1)3(n12 · p2)p2
2
96m31m
32
−47(n12 · p1)(n12 · p2)p2
1p22
32m31m
32
+(n12 · p1)2(p1 · p2)p2
2
384m31m
32
−185p2
1(p1 · p2)p22
384m31m
32
−7(n12 · p1)2(n12 · p2)4
4m21m
42
+7(n12 · p2)4p2
1
4m21m
42
−7(n12 · p1)(n12 · p2)3(p1 · p2)
2m21m
42
+21(n12 · p2)2(p1 · p2)2
16m21m
42
+7(n12 · p1)2(n12 · p2)2p2
2
6m21m
42
+49(n12 · p2)2p2
1p22
48m21m
42
−133(n12 · p1)(n12 · p2)(p1 · p2)p2
2
24m21m
42
−77(p1 · p2)2p2
2
96m21m
42
+197(n12 · p1)2(p2
2)2
96m21m
42
−173p2
1(p22)2
48m21m
42
+13(p2
2)3
8m62
)+
G3m1m2
r312
(m2
1
(5027(n12 · p1)4
384m41
−22993(n12 · p1)2p2
1
960m41
−6695(p2
1)2
1152m41
−3191(n12 · p1)3(n12 · p2)
640m31m2
+28561(n12 · p1)(n12 · p2)p2
1
1920m31m2
+8777(n12 · p1)2(p1 · p2)
384m31m2
+752969p2
1(p1 · p2)
28800m31m2
−16481(n12 · p1)2(n12 · p2)2
960m21m
22
+94433(n12 · p2)2p2
1
4800m21m
22
−103957(n12 · p1)(n12 · p2)(p1 · p2)
2400m21m
22
+791(p1 · p2)2
400m21m
22
+26627(n12 · p1)2p2
2
1600m21m
22
−118261p2
1p22
4800m21m
22
+105(p2
2)2
32m42
)+ m1m2
((2749π2
8192−
211189
19200
)(p2
1)2
m41
+
(63347
1600−
1059π2
1024
)(n12 · p1)2p2
1
m41
+
(375π2
8192−
23533
1280
)(n12 · p1)4
m41
+
(10631π2
8192−
1918349
57600
)(p1 · p2)2
m21m
22
+
(13723π2
16384−
2492417
57600
)p2
1p22
m21m
22
+
(1411429
19200−
1059π2
512
)(n12 · p2)2p2
1
m21m
22
+
(248991
6400−
6153π2
2048
)(n12 · p1)(n12 · p2)(p1 · p2)
m21m
22
−(
30383
960+
36405π2
16384
)(n12 · p1)2(n12 · p2)2
m21m
22
+
(1243717
14400−
40483π2
16384
)p2
1(p1 · p2)
m31m2
+
(2369
60+
35655π2
16384
)(n12 · p1)3(n12 · p2)
m31m2
+
(43101π2
16384−
391711
6400
)(n12 · p1)(n12 · p2)p2
1
m31m2
+
(56955π2
16384−
1646983
19200
)(n12 · p1)2(p1 · p2)
m31m2
))+
G4m1m2
r412
(m3
1
(64861p2
1
4800m21
−91(p1 · p2)
8m1m2+
105p22
32m22
−9841(n12 · p1)2
1600m21
−7(n12 · p1)(n12 · p2)
2m1m2
)+ m2
1m2
((1937033
57600−
199177π2
49152
)p2
1
m21
+
(176033π2
24576−
2864917
57600
)(p1 · p2)
m1m2+
(282361
19200−
21837π2
8192
)p2
2
m22
+
(698723
19200+
21745π2
16384
)(n12 · p1)2
m21
+
(63641π2
24576−
2712013
19200
)(n12 · p1)(n12 · p2)
m1m2+
(3200179
57600−
28691π2
24576
)(n12 · p2)2
m22
))
+G5m1m2
r512
(−
m41
16+
(6237π2
1024−
169799
2400
)m3
1m2 +
(44825π2
6144−
609427
7200
)m2
1m22
)+(1↔ 2
).
1 The First Direct Detections of Gravitational Waves
2 Indispensability of Matched Filtering in Data Analysis
3 Post-Newtonian Relativity of Two-Body Systems
4 Effective One-Body Approach
5 Conclusions
Effective One-Body Formalism
Effective one-body (EOB) formalism provides accurate templatesneeded for detection of gravitational-wave signals(and estimation of their parameters) of coalescing BH/BH binaries.
EOB templates correspond to the full coalescence process of BH/BH systemsfrom early inspiral to ringdown.
EOB formalism is based on approximate results and it allows to modelanalytically motion and radiation of BH/BH system from its adiabatic inspiral,through merger, up to vibrations of the resultant Kerr BH.
After incorporating tidal interactions EOB formalismalso describes inspiral phase of BH/NS and NS/NS systems.
EOB formalism was initiated in 1999–2000by T.Damour and A.Buonannoat the 2PN level and is being developedsince then by them and their collaborators.
T.Damour, PJ, G.Schafer, 2000: incorporating3PN-level orbital dynamics;T.Damour, PJ, G.Schafer, 2008: incorporatingnext-to-leading order spin-orbit corrections;T.Damour, PJ, G.Schafer, 2015: incorporating4PN-level orbital dynamics.
(A.Buonanno and T.Damour, PRD, 2000)
Main Idea and Structure of EOB Approach
Main idea of EOB approach is based on two observations:—Waveforms computed numerically and by means of thePN approximation of high enough order agree very well in the region,where the objects are sufficiently far away.—Gravitational waves emitted in the last stage of the BH/BH evolutionare accurately describable as a superpositionof several quasi-normal modes of the Kerr BH.
Main idea of EOB approach: extend the domain of validity of PN and BHperturbation theories up to merger and define EOB waveform as:
hEOB(t) = θ(tm − t) hinsplunge(t) + θ(t − tm) hringdown(t),
θ(t) denotes Heaviside’s step function, tm is the time at which the twowaveforms hinsplunge and hringdown are matched.
Ringdown waveform hringdown(t) is computed from BH perturbation theory.Computation of inspiral + plunge waveform hinsplunge(t) requires usage ofresummation techniques, which include translation of real two-body probleminto effective one and usage of Pade approximants:
— PN conservative Hamiltonian −→ EOB-improved Hamiltonian;
— PN gravitational-wave luminosities −→ EOB radiation-reaction force;
— PN waveform.
Why Does It Work?
The merging phase could be verycomplicated...
...but it is not!
Templates used for GW150914.
Real Two-Body Problem vs Effective One-Body Problem
M := m1 + m2, µ :=m1m2
m1 + m2, ν :=
µ
M=
m1m2
(m1 + m2)2, 0 ≤ ν ≤
1
4
At the Newtonian level the two-body problem can be reduced to motion ofa test particle of mass µ orbiting around an external mass M.
The EOB approach is a general relativistic generalization of this fact.
Real two-body problem:two black holes of masses m1, m2 and spins S1, S2
orbiting around each other
lEffective one-body problem:
one test particle (with additional nongeodesic corrections)of mass µ and spin S∗
moving in some background metric g effectiveαβ
The effective metric g effectiveαβ is a ν-deformed Kerr metric
of mass M and spin SKerr := S1 + S2.
The spin of the effective particle reads
S∗ :=m2
m1S1 +
m1
m2S2 + (spin-orbit terms).
The Mapping Rules Between the Two Problems(Motivated by Quantum Considerations)
The adiabatic invariants (the action variables) Ii =∮pi dqi
are identified in the two problems.
The energies are mapped through a function f :
Eeffective = f (Ereal),
f is determined in the process of matching.
One looks for a metric g effectiveαβ such that
the energies of the bound states of a particle moving in g effectiveαβ
are in one-to-one correspondencewith the energies of the real two-body bound states:
Eeffective(Ii ) = f (Ereal(Ii )) .
The identification of the action variables guaranteesthat the two problems are mapped by a canonical transformation.
Pade Approximants
Pade approximant of (k, l)-type (with k + l = n)for series wn(x) = c0 + c1 x + · · · + cn x
n (with c0 6= 0):
Pkl [w(x)] :=
Nk (x)
Dl (x),
where the polynomials Nk (of degree k) and Dl (of degree l) are suchthat the Taylor expansion of Pk
l [wn(x)] coincides with wn(x) up to O(xn+1) terms:
Nk (x)
Dl (x)= c0 + c1 x + · · · + cn x
n +O(xn+1).
Pade-Improved EOB Potential A(u; ν)(for nonspinning bh/bh systems)
The EOB potential A(u; ν) = −g effective00 (u; ν) [with u := GM/(c2rEOB)]
has the following 4PN-accurate Taylor expansion:
A(u; ν) = 1− 2u + 2νu3 + a4(ν)u4 + (a51(ν) + a52(ν) ln u)u5 +O(u6).
By continuity with the test-mass case ν = 0, one expects that A(u; ν) will exhibita simple zero defining an EOB “effective horizon” that is smoothly connected,
when ν → 0, to the Schwarzschild event horizon at u = 1/2.
Therefore it is reasonable to factor a zero of A(u; ν) by introducing the Pade-improvedAP4 (u; ν) defined at the 4PN level as
AP4 (u; ν) := P14
[1− 2u + 2νu3 + a4(ν)u4 + (a51(ν) + a52(ν) ln u)u5
].
Synergy Between EOB Formalism and Numerical Relativity
Select sample of NR waveforms
↓Introduce EOB flexibility parametersand calibrate them to NR wavefroms
↓Define NR-improved EOB waveforms(used in analysis of LIGO/Virgo data)
Flexibility Parameters in the EOB Potential A(u; ν)(for nonspinning bh/bh systems)
Instead of using the 4PN-accurate truncated Taylor expansion(maybe in Pade-improved form),
A4PN(u; ν) = 1− 2u + 2νu3 + a4(ν)u4 + (a51(ν) + a52(ν) ln u)u5,
one considers (maybe in Pade-improved form) 3-parameter class ofextensions of A4PN(u; ν) defined by
A(u; ν, b61, b62, b63) := A4PN(u; ν) + ν(b61 + b62ν + b63ν2)u6 + a62(ν)u6 ln u.
1 The First Direct Detections of Gravitational Waves
2 Indispensability of Matched Filtering in Data Analysis
3 Post-Newtonian Relativity of Two-Body Systems
4 Effective One-Body Approach
5 Conclusions
Conclusions
Analytical results of PN relativity for two-body systems:point particles together with dimensional regularizationgive unique equations of motion up to the 4PN orderand gravitational-wave luminosities up to the 3.5PN order.
Analytical PN results for equations of motion and gravitational-wave luminositiesare key ingredients of the EOB approach to relativistic two-body problem.
Direct PN results and the EOB formalism improved by numerical relativitywere employed for constructing banks of templates
h(t;m1,m2,S1,S2,Λ1,Λ2, . . . ),
which has been used to detect, by matched filtering, gravitational-wave signalsfrom compact binary coalescenses, in the observing runs O1/O2of the Advanced LIGO/Virgo detectors.
Next observing runs of the Advanced LIGO/Virgo detectors will also employbanks of templates constructed by using direct PN results and the EOB approach.
