Gravitational Wave Memory Revisited:

Transcription

1 Gravitational Wave Memory Revisited: Memory from binary black hole mergers Marc Favata Kavli Institute for Theoretical Physics arxiv: [astro-ph] and arxiv: [gr-qc]

2 What is the GW memory? Generally think of GW s as oscillating functions w/ zero initial and final values: But some sources exhibit differences in the initial & final values of h +, An ideal GW detector would experience a permanent displacement after the GW has passed---leaving a memory of the signal.

3 Types of GW memory: Linear memory:(zel dovich& Polnarev 74; Braginsky& Grishchuk 78; Braginsky & Thorne 87) due to changes in the initial and final values of the masses and velocities of the components of a gravitating system Example: Hyperbolic orbits/gravitational two-body scattering/gravitational bremsstrahlung(turner 77; Turner & Will 78; Kovacs & Thorne 77, 78)

4 Types of GW memory: Linear memory:(zel dovich& Polnarev 74; Braginsky& Grishchuk 78; Braginsky & Thorne 87) due to changes in the initial and final values of the masses and velocities of the components of a gravitating system Examples: Hyperbolic orbits/gravitational two-body scattering/gravitational bremsstrahlung(turner 77; Turner & Will 78; Kovacs & Thorne 77, 78) Binary that becomes unbound (eg., due to mass loss) Anisotropic neutrino emission (Epstein 78) Asymmetric supernova explosions (see Ott 08 for a review) GRB jets (Sago et al., 04) A general formula for the linear memory is given by (Thorne 92, Braginsky& Thorne 87):

5 Types of GW memory: Nonlinear memory:(payne 83; Christodoulou 91 ; see also Blanchet & Damour 92) Contribution to the distant GW field sourced by the stress-energy of GWs harmonic gauge EFE has a nonlinear source from the GW stress-energy tensor solve EFE [Wiseman & Will 91]

6 Types of GW memory: Nonlinear memory:(payne 83; Christodoulou 91 ; see also Blanchet & Damour 92) Contribution to the distant GW field sourced by the emission of GWs Nonlinear memory can be related to the linear memory if we interpret the component masses as the individual radiated gravitons (Thorne 92):

7 Why is this interesting?: The Christodoulou memory is a unique, nonlinear effect of general relativity The memory is non-oscillatory and only affects the + polarization (for quasi-circular orbits) Although it is a 2.5PN correction to the mass multipole moments, the memory affects the waveform amplitude at leading (Newtonian) order. The memory is hereditary: it depends on the entire past-history of the source

8 calculation of the memory: motivation Little is known about the memory For quasi-circular orbits, inspiral memory has only been computed to leading PN order [Wiseman & Will 92; Kennefick 94; Arunet al. 04; Blanchet et al. 08 ] Zero knowledge about how the memory grows and saturates to its final value post-inspiral Since the memory enters the waveform at leading order, its not crazy to think we could detect it. Two calculations: Compute all PN corrections to the memory up to 3PN order Use effective-one-body formalism to compute evolution of memory during the inspiral, merger, and ringdown.

9 Post-Newtonian calculation of the memory: Result: expressions for waveform modes h lm and + polarization:

10 Post-Newtonian calculation of the memory:

11 Memory from the merger: Expectation is that the memory will slowly accumulate during the inspiral, then rapidly rise during the merger/ringdown, and finally asymptote to some final, non-zero value. Objective: calculate the evolution of the memory through all stages of coalescence Difficult to do w/ numerical relativity: memory modes of the waveform are numerically suppressed relative to oscillatory modes Procedure:(1) use a low-order multipole expansion for the memory waveform; focus only on the l=m=2 mode (2) Use ``effective-one-body (EOB) approach (Buonanno& Damour) to model the multipole moments during inspiral, merger, ringdown Free parameters in EOB formalism calibrated to Caltech/Cornell & Jena simulations Also introduce minimal-waveform toy model: bare-bones, fully-analytic EOB

12 Memory from the merger: results ( equal-mass, non-spinning ) minimal waveform model EOB w/o amplitude corrections F 22 f QC 22 full-eob minimal waveform X 0.77 t m = 3522 M full-eob w/ memory full-eob w/o memory

13 Detectability of the memory: Previous estimates of the memory Assumed either unmodeledbursts (Thorne 92), or only considered the inspiral and focused on NS binaries with LIGO (Kennefick 94) Used dated noise curves; little discussion of SMBH mergers including build-up during the merger/ringdown is potentially important: Characteristic rise memory time: DETECTOR SOURCE S/N S/N (ZFL) (preliminary) Adv. LIGO 50M Ÿ + 50M 20Mpc 8 (28) LISA 10 6 M Ÿ M 1Gpc 130 (1200) 10 6 M Ÿ M z=1 8.6 (180) 10 6 M Ÿ M z=2 2 (77)

14 Signature of GW recoil on the GW signal: Summary of kick speeds computed in numerical relativity simulations V max ~ 175 km/s for non-spinning BHs V max ~ 450 km/s for equal-mass BHs w/ spins maximal, anti-aligned along orbital angular momentum V max ~ 4000 km/s for equal-mass, maximal spinning BHs, in super-kick configuration There have been manypapers that consider EM signatures of merger and recoil See talks in next session by Kocsis, Schnittman, and Komossa Are signatures of the recoil present in the GW signal?

15 Signature of GW recoil on the GW signal: (1) linear memory due to recoil of the remnant: Christodoulou memory kick memory

16 Signature of GW recoil on the GW signal: (2) direction-dependent Doppler shift of the ringdown waves: for high signal-to-noise SMBH mergers w/ LISA, parameters measured during the inspiral could be fed into NR simulations, predicting the rest-frame QNM frequencies. If the observed QNM frequencies are measured accurately, should be able to extract line-ofsight recoil. One can then cross check this measurement with the simulations rough estimate for the accuracy with which the line-of-sight recoil can be measured: [Berti, Cardoso, Will 06] *if*doppler shift and kick-memory could both be measured, kick magnitude and direction could be reconstructed from the GW signal.

17 Summary: The Christodoulou memory is a unique manifestation of the nonlinearity of GR that affects the waveform amplitude at Newtonian order. PN corrections to the inspiral memory have been computed; this completes the waveform to 3PN order Via EOB techniques, numerical simulations are now helping to guide and calibrate analytic studies of BH mergers---including of quantities are not yet easily calculated with numerical relativity Using full EOB and a much simpler, fully analytic minimal-waveform model, I ve computed the final saturation value of the memory. Prospects for detecting the Christodoulou memory: Initial LIGO: forget it Advanced LIGO: only if we get very lucky LISA: should be detectable from SMBH mergers out to z ~1 2. Binaries that recoil after merger also show a memory effect, as well as a Doppler shift of the QNM frequencies.

18 Extra slides

19 Memory from the merger: minimal waveform model First use analytic toy model: bare-bones EOB Inspiral moments given by leading-order 0PN expansion: Ringdown given by quasi-normal mode (QNM) sum: QNMs given by Berti, Cardoso, Will 06; final BH mass and spin determined from numerical relativity fits (Baker et al. 08) A 22n determined by matching derivatives at t=t m at Schwarzschild light-ring r m = 3M. Result is the following simple analytic function:

20 Memory from the merger: full-eob model Use EOB method applied in Damour, Nagar, Hannam, Husa, Brugmann 08 ``flexibility parameters in EOB formalism were calibrated to Caltech/Cornell inspiral waveforms and Jena merger waveforms Inspiral moments modeled as: Solve EOB equations for r, j, p r*, p j Ringdown same as minimal-waveform model, but with 5 QNMs Matching done for I 22 (2) at 5 points near the EOB-deformed light ring Construct I 22 (3) ; plug into previous leading-order expression for the memory; numerically integrate from r 0 =15M using leading-pn order memory for the initial condition

21 Memory in numerical relativity simulations: Extracting the memory from NR simulations faces several challenges: Physical memory only present in m=0 modes, which are numerically suppressed

22 Memory in numerical relativity simulations: Other problems with NR computations of the memory: Need to choose two integration constants to go from curvature to metric perturbation Choosing these incorrectly leads to artificial memory (Berti et al. 07) Memory sensitive to past-history of the source ( depends on initial separation) Consider leading-order (2,0) memory mode, with a finite separation r 0 Errors from gauge effects and finite extraction radius can further contaminate NR waveforms and swamp a small memory signal