Key ideas on how inspiral-merger-ringdown waveforms are built within the effective-one-body formalism

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1 Key ideas on how inspiral-merger-ringdown waveforms are built within the effective-one-body formalism Alessandra Buonanno Maryland Center for Fundamental Physics & Joint Space-Science Institute Department of Physics, University of Maryland Department of Physics, University of Maryland

2 The significance of merger and ringdown signals for LIGO/Virgo [Pan, AB, Pretorius & NASA-Goddard 7] (5+5) (15+15) (3+3) Original Whitened Time (1M) Time (M) Time (M) (1+1) (2+2) (5+5) Time (1M) Time (M) Time (M) Department of Physics, University of Maryland 1

3 Combining post-newtonian, perturbation theory and numerical-relativity results: the effective-one-body (EOB) approach EOB approach introduced before the NR breakthrough [AB & Damour 99, ] many papers since then! PN Theory conservative dynamics and GW emission computed as a Taylor expansion Effective one body conservative dynamics and GW emission re written in summed and/or factorized form Numerical relativity two body dynamics and GW emission computed with all non linearities The EOB formalism uses the best information available in PN theory, but sums it in a suitable way to be able to describe accurately the full evolution: inspiral, merger and ringdown. The EOB formalism provides us with a moment in time when to switch from the from the two-body to the one-body description. Department of Physics, University of Maryland 2

4 Effective-one-body approach in a nutshell [AB & Damour 99] ν = µ/m = m 1 m 2 /M 2 ν 1/4 Real description m 2 Effective description µ Resum so that known testmass limit results are recovered m 1 Resum the PN expansion assuming that the equal-mass limit is a ν-deformation of E real g µν m m 1 2 E eff g eff µν the test-mass limit J real N J real eff N eff E real (N, J) = f[e eff (N, J)] Department of Physics, University of Maryland 3

5 Finding the energy for comparable-mass black holes Thinking quantum mechanically: the classical Hamiltonian and bound orbits are replaced by the Hamiltonian operator and quantum bound states. Real description: [ E real (N, J) = M c 2 1 µ α 2 ( N 2 α2 6 c 2 N J ν N 2 ) + ], α = G M µ Effective description: [ ( ) E eff (N, J) = µ c 2 1 µ α N 2 α2 C3,1 c 2 N J + C 4, + N 2 ] Allow transformation of energy axis: [ ( ) E NR eff = ENR E real 1 + α real NR 1 µ c 2 + α E real NR 2 ] 2 µ c 2 + α 1 = ν 2, α 2 = Department of Physics, University of Maryland 4

6 Energy for comparable-mass bodies Classical gravity [AB & Damour 99] (up to 3PN order) E 2 real = m2 1 + m m 1 m 2 ( Eeff µ ) Quantum electrodynamics (eikonal approximation) [Brézin, Itzykson & Zinn-Justin 7] Ereal 2 = m2 1 + m m 1 m 2 1+Z 2 α 2 /(n ϵ j ) 2 Department of Physics, University of Maryland 5

7 In summary, here is the summed PN conservative dynamics [AB & Damour 99] Real description H PN real = H Newt + 1 c 2 H 1PN + 1 c 4H 2PN + Effective description H ν eff = µ A ν (r) [ ( ) 1 + p2 µ Bν(r) 1 p 2 r µ 2 ] H EOB real = M 1 + 2ν ( H ν eff µ 1 ) ds 2 eff = A ν(r) dt 2 + B ν (r) dr 2 + r 2 dω 2 Dynamic condensed in A ν (r) and B ν (r) A ν (r), which encodes the energetics for circular orbits, is rather simple A ν (r) = 1 2M r + 2M 3 ν + ( 94 r π2) M 4 ν r 4 + a 5(ν) r 5 + a 6(ν) r 6 + Department of Physics, University of Maryland 6

8 EOB inspiral-plunge waveform 646 waveform - inspiral plunge GW frequency inspiral plunge least-damped QNM f (Hz) for binary with M = 3 M sun t/m t/m The plunge is a smooth continuation of the adiabatic inspiral [AB & Damour ] Department of Physics, University of Maryland 7

9 EOB inspiral-merger-ringdown waveforms waveform - inspiral plunge ringdown GW frequency inspiral superposition of QNMs plunge least-damped QNM f (Hz) for binary with M = 3 M sun t/m t/m Very short transition merger ringdown Energy quickly released during merger E rad 2% 12% M c 2 1M c erg 1 56 GeV! Department of Physics, University of Maryland 8

10 Full waveform as predicted by the EOB model The plunge ( 1.5 GW cycles) is a smooth continuation of the inspiral phase The transition merger to ringdown was assumed very short One single QNM matched using M BH =.976 M, a BH /M BH = r LSO ν = 1/4.1 y 5 5 h(t) inspiral-plunge merger-ring-down x [AB & Damour 99, ] t/m Department of Physics, University of Maryland 9

11 First comparisons/calibrations between NR and EOB model [AB, Cook & Pretorius 6] [AB, Pan & NASA-Goddard 7].1 numerical relativity EOB inspiral-plunge waveform EOB merger-ringdown waveform NR waveform EOB waveform (t - t CAH Uncalibrated EOB model at 3PN order t/m Calibrated EOB model at 4PN order Department of Physics, University of Maryland 1

12 [AB, Cook & Pretorius 6] The (plunge and) merger 6 4 initial pre-merger AH shapes final pre-merger AH shapes initial enveloping AH shape late time AH shape est. late time co-rotating light ring.3 5 de/dt x 1 ω c ω λ 99% energy radiated peak energy flux (67% E, 85% J z radiated) y i /M 2.15 common AH horizon detected 5% energy radiated coordinate separation reaches light ring % angular momentum radiated x i /M Short transition merger ringdown (t-t peak [AB, Cook & Pretorius 6] Energy and angular-momentum quickly released during merger Department of Physics, University of Maryland 11

13 Calibrating highly accurate waveforms for several mass-ratio binaries [Pan, AB, Boyle, Buchman, Kidder, Pfeiffer & Scheel 11] q=2 EOB waveform NR waveform q=3 EOB waveform NR waveform φ h (rad).2 A / A φ h (rad) A / A q=4.1.1 q= EOB waveform NR waveform φ h (rad) A / A EOB waveform NR waveform φ h (rad) A / A Department of Physics, University of Maryland 12

14 The EOB waveforms were used in LIGO searches high-mass compact binaries Horizon distance (Mpc) [Aasi et al. 12 (The LSC/Virgo Collaboration)] S6, H1 S6, L1 VSR2, V1 VSR3, V Binary total mass (M ) m2(m ) Sensitive distance (Mpc) m 1 (M ) Department of Physics, University of Maryland 13

15 Calibrating waveforms of spinning, non-precessing black holes EOB models with spins [Damour 1, Damour, Jaranowski & Schaefer 8, Barausse & AB 9,11] The PN Hamiltonian of two BHs of masses m 1,2 and spins S 1,2 is mapped into the effective Hamiltonian of a spinning test-particle of mass µ and spin S moving in a deformed-kerr spacetime with mass M and spin S Kerr. [Pan et al. 9; Taracchini, Pan, AB, Barausse, Chu, Boyle, Pfeiffer & Scheel 12] q = 1 q = EOB waveform - NR waveform EOB waveform NR waveform φ Α/Α.1.1 φ Α/Α Department of Physics, University of Maryland 14