Gravitational Wave Memory Revisited: Memories from the merger and recoil Marc Favata Kavli Institute for Theoretical Physics
Metals have memory too
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.
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 & Will 78; Kovacs & Thorne 77, 78) Binary that becomes unbound (eg., due to mass loss) Anisotropic neutrino emission (Epstein 78) Asymmetric supernova explosions GRB jets (Sago et al., 04) A general formula for the linear memory is given by (Thorne 92, Braginsky& Thorne 87):
Types of GW memory: Linear memory:(braginsky& Thorne 87 + earlier refs.) due to changes in the initial and final values of the masses and velocities of the components of a gravitating system Example: Hyperbolic orbit
Types of GW memory: Nonlinear memory:(christodoulou 91 ; see also Blanchet & Damour 92) due to change in the mass of a binary caused by the emission of GWs Nonlinear piece of Einstein s equations solve [Wiseman & Will 91] Nonlinear memory can be related to the linear memory if we interpret the component masses as the individual radiated gravitons (Thorne 92):
Types of GW memory: Nonlinear memory:(christodoulou 91; see also Blanchet & Damour 92) Christodoulou memory comes from a correction to the radiative mass multipoles: In terms of leading-order scalar source multipoles:
Types of GW memory: Nonlinear memory:(christodoulou 91; see also Blanchet & Damour 92) In terms of leading-order scalar source multipoles: During the inspiral, the result is well known: [Wiseman & Will 91, Kennefick 94, Arunet. al 04] Christodoulou memory
Why is this interesting?: The Christodoulou memory is a unique, nonlinear effect of general relativity Although it is a 2.5PN correction to the radiativemass multipoles, it affects the waveform amplitude at leading (Newtonian) order. The memory depends on the entire history of the source, and is nonoscillatory
Motivation and objective: The size of the Christodoulou memory has only been calculated for the inspiral (Wiseman & Will 91), so we have no clear understanding of its final value. Even with numerical relativity, the memory is difficult to calculate (Bertiet al. 08). Numerical relativity and its calibration of EOB (effective-one-body) techniques now allow simple analytic calculations of the memory: an example of the synergism between numerical and analytic techniques. Previous estimates of the detectabilityof the memory (Thorne 92, Kennefick 94) used outdated noise curves and focused on ground-based detectors. I ll use two analytic models to compute the memory: Minimal waveform model = barebones EOB, fully analytic and simple. Full EOB formalism Estimate signal-to-noise ratios for LIGO, Adv. LIGO, LISA, esp. SMBH mergers.
Memory calculation: minimal waveform model We focus only on non-spinning, circularized BH/BH mergers Expand waveforms in terms of leading-order scalar mass multipole Model multipolesduring the inspiral, and ringdown and match multipoles & 2 derivatives at the light-ring (r/m º3) to get A 22n. r(t) and j(t) evolve via the standard leading-order formulas: Use NR results (Baker et al., 08) to determine the final mass and spin (as a function of h) that enter the quasi-normal mode (QNM) frequencies and damping times [Berti, Cardoso, & Will 06 ]
Memory calculation: minimal waveform model The result for the final saturation value of the memory is easily derived (choosing the matching at t=0) and likewise for A 221 and A 222.
Memory calculation: full EOB model Model is similar to minimal-waveform version except: We use the full EOB formalism (see series of papers by Damour, Nagar and collaborators) to model the orbit during the inspiral & transition to merger. We match to a ringdown multipole with 5 QNMs, matching at 5 points near the deformed light-ring. The EOB model depends on several adjustable inputs. These are chosen as indicated by Damour et al. 08, which are calibrated to the Caltech/Cornell and Jena NR results.
Memory calculation: results Minimal waveform model Full EOB model
Memory calculation: EOB memory waveform
Memory calculation: comparison Minimal model EOB
Signal-to-noise ratios: DETECTOR SOURCE S/N LIGO 10M Ÿ + 10M Ÿ @ 20Mpc 0.82 50M Ÿ + 10M Ÿ @ 20Mpc 4.1 Adv. LIGO 10M Ÿ + 10M Ÿ @ 20Mpc 9.4 50M Ÿ + 50M Ÿ @ 20Mpc 47 50M Ÿ + 50M Ÿ @ 200Mpc 4.9 500M Ÿ + 500M Ÿ @ 200Mpc 49 10M Ÿ + 10M Ÿ @ 1Gpc 0.23 100M Ÿ + 100M Ÿ @ 1Gpc 2.3 LISA 10 6 M Ÿ + 10 6 M Ÿ @ 1Gpc 2080 10 5 M Ÿ + 10 6 M Ÿ @ 1Gpc 120 10 6 M Ÿ + 10 6 M Ÿ @ z=1 520 10 5 M Ÿ + 10 6 M Ÿ @ z=1 30 10 6 M Ÿ + 10 6 M Ÿ @ z=3 270
Measurement in numerical simulations? Numerical simulations compute... so a constant piece is not directly observable in the simulations. However, there is a small, slowly-growing correction to the amplitude as the memory accumulates [ at 5PN order!]:
Effects of recoil on the gravitational waveform Like the memory, GW kick is a single, physically-interpretable number that depends sensitively on the nonlinearity of BH mergers Unlike memory, kicks have dramatic astrophysical importance Both kick and memory calculations started out as crude estimates, focused on the inspiral, but with no clear idea of a cut-off radius or final saturation value. Later kludge calculations attempted to address the merger NR ultimately will tell us the exact answer in both cases: For kicks, the numerical values turned out to be: V max ~ 175 km/s for non-spinning holes V max ~ 450 km/s for maximally spinning, anti-aligned holes V max ~ 2000-4000 km/s for superkick configuration Kicks have many important electromagnetically observable effects What are the effects of kicks on the gravitational waveform?
Effects of recoil on the gravitational waveform During the inspiral, recoil corrections enter at 7 th post-newtonian order: really small
Effects of recoil on the gravitational waveform Most of the recoil accumulates very rapidly during the plunge/merger/ringdown: Two other potentially detectable effects: (1) direction-dependent Doppler shift of the ringdown waves: in simulations, should see a dependence of the QNM frequencies on extraction direction 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 for which the line-of-sight recoil can be measured: [Berti, Cardoso, Will 06]
Effects of recoil on the gravitational waveform (2) linear memory due to recoil of the remnant: Christodoulou memory kick memory
Summary: The Christodoulou memory is a unique manifestation of the nonlinearity of GR that effects the waveform amplitude at Newtonian order. Via EOB techniques, numerical simulations are now helping to guide and calibrate analytic studies of BH mergers. 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: not good Advanced LIGO: we might get lucky LISA: should be detectable from SMBH mergers with primary wave SNR >>100 Binaries that recoil after merger also show a memory effect, as well as a dopplershift of the QNM frequencies. Both effects rapidly accumulate during the merger.
Future work: A challenge to the numerical relativists: compute the memory accurately! A challenge to the post-newtonians: compute the memory correction to the 1PN waveform amplitudes (partly completed) A challenge to who ever has extra time on their hands: compute the Christodoulou memory in other theories of gravity