Predictions for the last stages of inspiral and plungeusing analytical techniques
Alessandra Buonanno
Department of Physics, University of Maryland
Numerical Relativity 2005: Compact Binaries, NASA’s Goddard Space Flight Center
Alessandra Buonanno November 2, 2005
Content:
• How far we can push analytical calculations
• Original motivations of introducing resummation techniques which
use post-Newtonian calculations
• Transition from adiabatic inspiral to plunge for non-spinning
and spinning, precessing binaries: main features of the dynamics
and the waveforms
• Comparison between analytical and numerical predictions
• What would be needed for a successful detection and for an accurate
parameter estimation with ground- and space-based detectors
Numerical Relativity 2005: Compact Binaries, NASA’s Goddard Space Flight Center 1
Alessandra Buonanno November 2, 2005
Detectibility of inspiraling non-spinning binaries with LIGO-I
M = Mν3/5 M = m1 + m2 ν = m1m2/M2 SN ∝ M5/6
R
40 80 120 160 200M
0
3
6
9
S / N
no spins
101
102
103
f (Hz)
10-23
10-22
S n1/
2 (f)
( H
z
-1/2
)
BH/BH (30 Msun)
BH/BH (20 Msun)
NS/NS (2.8 Msun)
LIGOI
at 100 Mpc
at 100 Mpc
at 20 Mpc
Equal-mass binaries at 100 Mpc
Numerical Relativity 2005: Compact Binaries, NASA’s Goddard Space Flight Center 2
Alessandra Buonanno November 2, 2005
Reduction in signal power (with perfect match of the GW phase)
The significance of the last GW cycles
[AB, Chen & Vallisneri 02]
30 50 100 200f / Hz
0.1
0.2
0.5
1
| h(f
)/ f -
7/6 |
(orb. phase)0 = 0
(orb. phase)0 = pi/2
PN expanded model
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Alessandra Buonanno November 2, 2005
Reduction in signal power [continued]
[AB, Chen & Vallisneri 02]
PN expanded model
Number ofcycles left
frequencytime-domain
fractional sig. power
0 295.17 1.0001 132.18 0.7472 107.24 0.5623 93.45 0.4344 84.17 0.3445 77.32 0.2806 71.95 0.2317 67.59 0.194
Table 1: Instantaneous frequencies and fractional signal power (SNR squared) when
0,1,2,. . . 7 GW cycles are left.
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Alessandra Buonanno November 2, 2005
What determines the “adiabatic” waveforms
Inspiral: adiabatic sequence of circular orbits (quadrupole approximation)
h ∝ v2 cos 2ϕ
Keplerian velocity: v = (Mϕ)1/3 M = m1 + m2
Energy-balance equation:dE(v)
dt = −F (v)
E(v) and F (v) known as a Post-Newtonian expansion in v/c
Two crucial ingredients:
E(v) → center-of-mass energy F (v) → gravitational flux
Numerical Relativity 2005: Compact Binaries, NASA’s Goddard Space Flight Center 5
Alessandra Buonanno November 2, 2005
Initial motivations of introducing resummation techniques:PN-expanded circular-orbit energy
• Circular-orbit energy determined at 2PN order in 1995 by Blanchet, Damour, Iyer
Wiseman, Will and at 3PN order in 2001 by Damour, Jaranoswki & Schaefer
0.1 0.2 0.3 0.4 0.5 0.6v
-0.03
-0.02
-0.01
0.00
E(v
) / M
1PN2PN3PN
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Alessandra Buonanno November 2, 2005
Initial motivations of introducing resummation techniques:PN-expanded GW flux
• GW flux determined at 2.5PN order in 1996 by Blanchet and at 3PN and 3.5PN
order in 2004 by Blanchet, Damour, Esposio-Farese & Iyer
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5v
0.85
0.9
0.95
1
FT
n / F
N
2PN 2.5PN 3PN3.5PN exact
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5v
0.85
0.9
0.95
1
FT
n / F
N
1PN1.5PN 2PN2.5PN 3PN3.5PN
Test-mass limit case Equal-mass binaries
Numerical Relativity 2005: Compact Binaries, NASA’s Goddard Space Flight Center 7
Alessandra Buonanno November 2, 2005
Precessing versus non-precessing compact binaries
• Non spinning: Inspiral [fGW = 2forb, fend(m1, m2)], plunge, merger and ring down
1000 2000 3000 4000
-0.2
-0.1
0
0.1
0.2
h
time
Many more parameters!
1000 2000 3000 4000
-0.2
-0.1
0
0.1
0.2
h
time
• Precessing: Inspiral [fGW = (2forb, fprec), fend(S, m1, m2)], plunge (?) merger, ring down
Precession of the orbital plane modulates both amplitude and phase of gravity-wave
Numerical Relativity 2005: Compact Binaries, NASA’s Goddard Space Flight Center 8
Alessandra Buonanno November 2, 2005
Detectibility of inspiraling spinning binaries with GW interferometers
Spin-orbit coupling makes two-body gravitational interaction more (less)
repulsive when spins are aligned (anti-aligned)
V (r) = −mMr + L2
2mr2− L4
m3r4+ · · ·+ 2
r3L · S + · · ·
Duration of inspiral (and signal-to-noise ratio) modified by spin effects
2 6 10 14 18 22 26 300.8
0.9
1.0
radi
al p
oten
tial
radial separation
Innermost Stable Circular Orbit (ISCO)
40 80 120 160 200 240 280 320 360 400M
0
3
6
9
12
15
18
S / N
χL = 0
χL = + 0.25
χL = - 0.875
Equal-mass binaries at 100 Mpc
Numerical Relativity 2005: Compact Binaries, NASA’s Goddard Space Flight Center 9
Alessandra Buonanno November 2, 2005
Circular-orbit energy for PN expanded models
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2MΩ
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
E(Ω
)/Μ
2PN (no spins)2PN antialigned2PN aligned3PN (no spins)3PN antialigned3PN aligned
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2MΩ
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
E(Ω
)/Μ
2PN (no spins)2PN anti-aligned2PN aligned3PN (no spins)3PN anti-aligned2PN aligned
Numerically evaluated: Ω = ∂H(r,pr=0,pφ)
∂pφ
∂H(r,pr=0,pφ)
∂r = 0 ⇒ pφ = pφ(r)
Analytically evaluated: keeping only terms
until nPN order if working at nPN order
Numerical Relativity 2005: Compact Binaries, NASA’s Goddard Space Flight Center 10
Alessandra Buonanno November 2, 2005
Effective-one-body approach
1m
2m
1m 2m
µνg
realE Eeff
realJ Nreal effJ effN
Real description
µνg eff
??
Effective description
µ[AB & Damour 99]
µ = m1 m2/M
ν = m1 m2/M2
0 ≤ ν ≤ 1/4
Numerical Relativity 2005: Compact Binaries, NASA’s Goddard Space Flight Center 11
Alessandra Buonanno November 2, 2005
EOB approach: resummed Hamiltonian (non-spinning black holes)
[AB & Damour 99]
“Real” description“Effective” description
Hreal(Q, P ) ∼ M
1 + ν
»P 2
2 + MQ
–+ c4 P 4 + · · ·
ff
Hνeff(q, p) =
rAν(q)
h1 + p2 +
“Aν(q)Dν(q) − 1
”(n · p)2 + T4(p)
i
Himprovedreal (Q, P ) =
q1 + 2ν
`Hν
eff(q, p) − 1´
ds2eff = −Aν(q) dt2 + Dν(q)
Aν(r) dq2 + q2 dΩ2
• Canonical transf. (resummed dynamics): q = Q(Q,P ), p = P(Q,P )
• All dynamics condensed in Aν(q) and Dν(q)!
New resummed orbital energy function: Eimprreal (v)
Numerical Relativity 2005: Compact Binaries, NASA’s Goddard Space Flight Center 12
Alessandra Buonanno November 2, 2005
Effective one-body approach at 3PN
[Damour, Jaranowski & Schafer 00]
At 3PN order: one more equation to satisfy than number of unknowns
Higher order derivatives in the effective description0 = m2
0 + gαβeff pα pβ + Aαβγδ pα pβ pγ pδ + · · ·
Same matching between real and effective energy
Hνeff,3PN(q, p) =
sAν(q)
»· · · + z1
p4
q2+ z2
p2 (n · p)2
q2+ z3
(n · p)4
q2
–
Numerical Relativity 2005: Compact Binaries, NASA’s Goddard Space Flight Center 13
Alessandra Buonanno November 2, 2005
Result for effective metric at 2PN order and beyond it
ds2eff = −Aν(q) c2 dt2 +
Dν(q)Aν(q)
dq2 + q2 dΩ2
Aν(q) = 1 − 2GM
c2q+ 2ν
(GM
c2q
)3
Dν(q) = 1 − 6ν
(GM
c2q
)2
Effective potential: Wj(q) = Aν(q) [1 + j2
q2 ]
Location of the ISCO:∂Wj
∂q = 0 = ∂2Wj
∂q2
At higher PN orders: Aν(q) = 1 − 2 GMc2q
+ 2ν(
GMc2q
)3
+ 18.7ν(
GMc2q
)4
+ O(
GMc2q
)5
Possible resummation of Aν(q) to improve its behaviour
e.g., Pade approximants [Damour, Jaranowski & Schaefer 00]
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Alessandra Buonanno November 2, 2005
EOB approach with spins
• Approximate map of the conservative dynamics of two spinning black holes ofmass m1 and m2 onto the dynamics of a non-spinning particle of massµ = m1 m2/M moving in an effective metric [Damour 01]
• This metric can be viewed as a ν = µ/M deformation of a Kerr metric of massM = m1 + m2 and spin Seff
For simplicitly, we just added spin effects to the non-spinning EOB Hamiltonian[AB, Chen & Damour 05]
Himprreal (q,p,S1,S2) = Himpr
real (q,p)+HSO(q,p, S1, S2) + HSS(q,p,S1,S2)
HSO =2Seff · L
q3, Seff ≡
„1 +
3
4
m2
m1
«S1 +
„1 +
3
4
m1
m2
«S2
Numerical Relativity 2005: Compact Binaries, NASA’s Goddard Space Flight Center 15
Alessandra Buonanno November 2, 2005
Comparing EOB-resummed and PN-expanded binding energies forequal-mass binaries
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2MΩ
-0.04
-0.035
-0.03
-0.025
-0.02
-0.015
-0.01
-0.005
0
E(Ω
)/Μ
EOB 2PN (no spins)EOB 2PN antialignedEOB 2PN alignedEOB 3PN (no spins)EOB 3PN antialignedEOB 2PN aligned
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2MΩ
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
E(Ω
)/Μ
2PN (no spins)2PN anti-aligned2PN aligned3PN (no spins)3PN anti-aligned2PN aligned
Numerically evaluated: Ω = ∂Himpr(q,pq=0,pφ)
∂pφ
∂Himpr(q,pq=0,pφ)
∂q = 0 ⇒ pφ = pφ(q)
Analytically evaluated: keeping only terms
until nPN order if working at nPN order
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Alessandra Buonanno November 2, 2005
Comparing analytical and numerical results
0.08 0.12 0.16mΩ
0
-0.022
-0.02
-0.018
-0.016
-0.014
Eb /
m
CO: QECO: HKV-GGBCO: PN EOBCO: PN standard
IR: QEIR: IVP confIR: PN EOBIR: PN standard
0.03 0.06 0.09 0.12mΩ
0
-0.06
-0.05
-0.04
Eb/µ
CO: MS - d(αψ)/dr = (αψ)/2rCO: HKV - GGBCO: EOB - 3PNCO: EOB - 2PNCO: EOB - 1PN
[Damour, Gourgoulhon & Grandeclement 02; Cook & Pfeiffer 04]
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Alessandra Buonanno November 2, 2005
Comparing LSSO predictions for energy and frequency
[AB, Chen & Damour 05]Equal-mass and equal-spin binaries
−0.75 −0.5 −0.25 0 0.25 0.5 0.75χ
L
−0.10
−0.08
−0.06
−0.04
−0.02
0.00
EL
SSO
/M2
EOB 2PN (Damour [5])EOB 3PN (Damour [5])EOB 2PN (this paper)EOB 3PN (this paper)Adiabatic Taylor 2PNAdiabatic Taylor 3PN
−0.75 −0.5 −0.25 0 0.25 0.5 0.75χ
L
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
MΩ
LSS
O
EOB 2PN (Damour [5])EOB 3PN (Damour [5])EOB 2PN (this paper)EOB 3PN (this paper)Adiabatic Taylor 2PNAdiabatic Taylor 3PN
For spins aligned with angular momentum ⇒ non-linear effects dominate ⇒predictions differ, but for LIGO this would affect only binaries with mass >∼ 40M
Numerical Relativity 2005: Compact Binaries, NASA’s Goddard Space Flight Center 18
Alessandra Buonanno November 2, 2005
Comparing LSSO predictions using analytical calculations and oldresults from initial-value problem approach
[AB, Chen & Damour 05]Equal-mass and equal-spin binaries
−0.75 −0.5 −0.25 0 0.25 0.5 0.75χ
L
−0.10
−0.08
−0.06
−0.04
−0.02
0.00
EIS
CO
/M2
EOB 2PN EOB 3PN PN-expanded 2PNPN-expanded 3PNPTCGGB
−0.75 −0.5 −0.25 0 0.25 0.5 0.75χ
L
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
MΩ
ISC
O
EOB 2PN EOB 3PN PN-expanded 2PNPN-expanded 3PNPTCGGB
• “Effective potential”’, Initial-value-problem approach [Pfeiffer, Teukolsky & Cook 00]
• HKV-, QE-approach [Grandeclement, Gourgoulhon, Bonazzola 02; Cook 02; Cook & Pfeiffer 04]
Numerical Relativity 2005: Compact Binaries, NASA’s Goddard Space Flight Center 19
Alessandra Buonanno November 2, 2005
EOB approach: incorporating radiation reaction effects
[AB & Damour 00; AB, Chen & Damour 05]
dqi
dt = ∂Himpr
∂pi
dpidt = −∂Himpr
∂qi + Fi
• Assumptions: quasi-circular orbits and leading spin-dependent terms
• Radiation-reaction force matches known rates of energy and angular
momentum loss for quasi-adiabatic orbits
Fi = 1
Ω |L|dEdt pi + 8
15 ν2 v8
L2q
(61 + 48 m2
m1
)p · S1 +
(61 + 48 m1
m2
)p · S2
Li
• Pade resummation of the GW flux including spin effects
[Damour, Sathyaprakash & Iyer 98; Porter & Sathyaprakash 04; AB, Chen & Damour 05]
Numerical Relativity 2005: Compact Binaries, NASA’s Goddard Space Flight Center 20
Alessandra Buonanno November 2, 2005
Evaluation of waveform and energy released
[AB & Damour 00; AB, Chen & Damour 05]Quadrupole approximation:
hij = Hij
D ≡ 2µD
d2
dt2(qi qj), qk = −M qk/q3 ⇒ Hij = 4µ
(Vi Vj − M
qi qj
q3
)
• δH
• The time integral of EI withdEIdt = 1
5
d3Iij
dt3
d3Iij
dt3
Iij = µ`qi qj − 1
3δij qk qk
´• The time integral of Eh
withdEhdt = 1
20
R Pij HTF
ij HTFij
0 100 200 300 400f (Hz)
0
0.01
0.02
0.03
0.04
δE/M
δEI
δEh
δEH
0 100 200 300 400f (Hz)
0
0.01
0.02
0.03
0.04
δE/M
δEI
δEh
δEH
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Alessandra Buonanno November 2, 2005
Energy and angular-momentum released during inspiral and plunge
[AB, Chen & Damour 05]• Maximal spins and (15 + 15)M
• Energy release before 40 Hz is ∼ 0.008/M
0 50 100 150 200 250 300 350 400 450 500f (Hz)
0
0.01
0.02
0.03
0.04
0.05
δE/M
alignedanti-alignedgeneric-upgeneric-downnon-spinning
0 50 100 150 200 250 300 350 400 450 500f (Hz)
0
0.25
0.5
0.75
1
1.25
1.5
|J|/E
2
alignedanti-alignedgeneric-upgeneric-downnon-spinning
Rotation parameter J/E2 smaller than one at the end of inspiral
⇒ Kerr black hole could already form
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Alessandra Buonanno November 2, 2005
Energy and angular-momentum released until 40 Hz and from 40 Hzto the LSSO
(θS1, φS1, θS2, φS2) [δEH ]f<40 Hz/M fLSSO (Hz) [δEH ]40 HzLSSO/M
h|J|/E2
iLSSO
(15 + 15)M, 3PN
nospin 0.0082 190 0.0107 0.82
(0,0,0,0) 0.0086 (1430) − −(180,0,180,0) 0.0077 97 0.0033 0.51
(60,90,60,0) 0.0084 (767) − −(120,90,120,0) 0.0079 123 0.0054 0.74
(15 + 5)M, 3PN
nospin 0.0048 265 0.0084 0.62
(0,0,0,0) 0.0049 (1442) − −(180,0,180,0) 0.0046 140 0.0034 0.14
(60,90,60,0) 0.0049 (798) − −(120,90,120,0) 0.0047 177 0.0049 0.62
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Alessandra Buonanno November 2, 2005
Energy and angular-momentum released until 40 Hz and from 40 Hzup to the end of evolution
(θS1, φS1, θS2, φS2) [δEH ]f<40 Hz/M ffinˆδEH
˜40,Hzfin
/Mh|J|/E2
ifin
(15 + 15)M, 3PN
nospin 0.0082 325 0.0183 0.77
(0,0,0,0) 0.0086 474 0.0528 0.96
(180,0,180,0) 0.0077 194 0.0064 0.47
(60,90,60,0) 0.0084 440 0.0353 0.91
(120,90,120,0) 0.0079 242 0.0101 0.70
(15 + 5)M, 3PN
nospin 0.0048 484 0.0141 0.58
(0,0,0,0) 0.0049 817 0.0495 0.95
(180,0,180,0) 0.0046 289 0.0054 0.11
(60,90,60,0) 0.0049 706 0.0292 0.91
(120,90,120,0) 0.0047 354 0.0080 0.60
No a priori obstacles at having a Kerr black hole form right after the end of thenon-adiabatic “plunge” ⇒ no ground for expecting a large emission of GWsbetween plunge and merger
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Alessandra Buonanno November 2, 2005
Energy released from the LSSO up to the end of the evolution
(θS1, φS1, θS2, φS2) fLSSO (Hz) ffinˆδEH
˜LSSOfin /M
(15 + 15)M, 3PN
nospin 190 325 0.0075
(0,0,0,0) 1430 474 0.0527
(180,0,180,0) 97 194 0.0031
(60,90,60,0) 760 440 0.0353
(120,90,120,0) 123 242 0.0047
(15 + 5)M, 3PN
nospin 265 484 0.0057
(0,0,0,0) 1442 819 0.0493
(180,0,180,0) 140 289 0.0024
(60,90,60,0) 793 719 0.0294
(120,90,120,0) 177 351 0.0031
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Alessandra Buonanno November 2, 2005
Energy released from the LSSO up to the end of the evolution[continued]
Comparison with Flanagan & Hughes 97; Baker, Bruegmann, Campanelli, Lousto &
Takahashi 00; Baker, Campanelli, Lousto & Takahashi 04:
• No spin: 1.4% of M against 3 − 4% of M by BBCLT
• Small spins: differences of few percent with BCLT but theyinclude also ring-down
However BBCL and BCLT use IVP formulation for initial data
• With spins: energy released not as large as predicted by a roughestimate of Flanagan & Hughes
Numerical Relativity 2005: Compact Binaries, NASA’s Goddard Space Flight Center 26
Alessandra Buonanno November 2, 2005
Comparable-mass case: several possible definitions of ISCO crossing
-200 -100 0t/M-0.48
-0.38
-0.28
-0.18
0.08
0.02
0.12
0.22
h GW
(t)
inspiral + plunge
Schwar. ISCOr-ISCOj-ISCOE-ISCO
w-LSO
equal-mass case
Radiation reaction effects rather large ⇒ transition to the plunge blurred
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Alessandra Buonanno November 2, 2005
Gravity-wave signal from inspiral-plunge(–ring-down)
[AB, Chen & Damour 05]
• χ1 = χ2 = 0.5
and (15 + 15)M
• MBH = Efin
and aBH = [J/E2]fin
-150 -135-120 -105 -90 -75 -60 -45 -30 -15 0 15 30 45 60t/M
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
h GW
generic-upnon-spinninggeneric-down
• When spins are present
the ring-down part is just an example: we restricted to l = m = 2, assuming
that the total angular momentum is dominated by the orbital angular momentum
Numerical Relativity 2005: Compact Binaries, NASA’s Goddard Space Flight Center 28
Alessandra Buonanno November 2, 2005
Summary
• Within analytical calculations the EOB is the only approach which
can describe the dynamics and the gravity-wave signal beyond
the adiabatic approximation
• It can provide initial data (q,p, gij, kij) for black holes close to
the plunge to be used by numerical relativity
•It can be used as a diagnostic for (or to fit) numerical relativity results
• Current results indicate good agreement between numerical and
analytical estimate of the binding energy without spin effects.
Predictions (using, e.g., HKV and QE methods) which include spin
couplings are needed
Numerical Relativity 2005: Compact Binaries, NASA’s Goddard Space Flight Center 29
Alessandra Buonanno November 2, 2005
Summary [continued]• Waveforms generated from initial data compatible with analytical
calculations are needed
• Extension of EOB to NS-NS and NS-BH [AB, Damour & Gourgoulhon]
• Detection with LIGO/VIRGO: phenomenological templates or extensions
of EOB templates can cover possible differences between analytical
and numerical waveforms for the last stages of inspiral and plunge
• Accurate parameter estimation and tests of GR with LIGO and LISA:
we would need more accurate waveforms for late inspiral and plunge
•Only the detection (and coalescence waves from NR!) will reveal us if
the two-body problem is a smooth deformation of a one-body problem,
at least from the point of view of the gravitational-wave emission
Numerical Relativity 2005: Compact Binaries, NASA’s Goddard Space Flight Center 30
Alessandra Buonanno November 2, 2005
Where the waveforms from NR will be?
[AB, Chen & Vallisneri 02]
0 50000 100000 150000 200000 250000
−2000
−1000
0
1000
(20+20)(20+15)
(15+15)
(20+10)(15+10) (10+10)(20+5)(15+5) (10+5) (5+5)
ψ0
ψ3/2
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