Gravitational Wave Memory

Revisited:Memory from binary black hole mergers

Marc FavataKavli Institute for Theoretical Physics

arXiv:0811.3451 [astro-ph] and arXiv:0812.0069 [gr-qc]

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

– Example: Hyperbolic orbits/gravitational two-body scattering/gravitational

bremsstrahlung (Turner ‘77; Turner & Will ‘78; Kovacs & Thorne ‘77, ’78)

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)• 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):

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

harmonic gauge

EFE…

…has a nonlinear source from

the GW stress-energy tensor…

solve EFE

[Wiseman & Will ‘91]

Types of GW memory:

Nonlinear memory can be related to the “linear” memory if we interpret the component

masses as the individual radiated gravitons (Thorne’92):

Nonlinear memory: (Payne ‘83; Christodoulou ‘91 ; see also Blanchet & Damour ‘92)

• Contribution to the distant GW field sourced by the emission of GWs

• 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 radiative mass multipoles, it

affects the waveform amplitude at leading (Newtonian) order.

Why is this interesting?:

affects the waveform amplitude at leading (Newtonian) order.

• The memory is hereditary: it depends on the entire past-history of

the source

Post-Newtonian calculation of the memory:

Motivation:• Little is known about the memory

– Wiseman & Will ‘92: computed leading order contribution to waveform for circular orbits and

small angle, two-body scattering (high-speed hyperbolic orbit)

– Kennefick ’94, Arun et al. ’04 confirmed this calculation …

– Blanchet et al. ‘08 showed that the 0.5PN correction vanishes.

– 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 • Since the memory enters the waveform at leading order, its not crazy to think

we could detect it.

• PN waveform now known to 3PN order (Blanchet et al. ‘08); memory waveform

to 0PN order. Higher-PN corrections needed to:

– Comparisons of numerical relativity (NR) calculations of the memory (more on this later)

– Improve estimates of memory’s detection

– Consistently compute waveform to 3PN order

Post-Newtonian calculation of the memory:

• Notation & brief review of PN wave-generation formalism:

– Radiative mass and current multipole moments describe the distant GW field:

– It is easier to work with the following mode decomposition of the polarizations:

– Other notation:

Post-Newtonian calculation of the memory:

• Notation & brief review of PN wave-generation formalism:

– Three types of moments enter the formalism:

• Radiative mass and current moments: Ulm and Vlm

• Canonical or algorithmic mass and current moments: Mlm and Slm

• Source mass and current moments: Ilm , Jlm + 4 more “gauge moments”

– The PN wave generation formalism relates the radiative moments (what the

observer measures) to the source moments (expressible as integrals over the

source stress-energy pseudotensor)

– Example: 3PN radiative mass quadrupole– Example: 3PN radiative mass quadrupole

– See Blanchet et al’08; Blanchet Living Review article; Blanchet gr-qc/9607025 for a shorter review

Post-Newtonian calculation of the memory:

• Why the 2.5PN memory enters the waveform at 0PN order:– Consider the leading-order piece of the memory waveform

– The integrand has oscillatory and non-oscillatory pieces:

Post-Newtonian calculation of the memory:

• Sketch of calculation of 3PN corrections to memory– Blanchet & Damour ‘92 give a general expression for the memory contribution to the radiative

mass multipole moments (which, after some manipulation, becomes):

– The energy flux can be computed from the 3PN waveform modes (Blanchet et al. ‘08):

– The angular integral over the triple harmonic product is expressible analytically as a complicated

sum over Gamma functions.

– To compute the time integral, one needs a 3PN-accurate model for the adiabatic evolution of the

binary frequency. This is straightforwardly derived from previous computations of the 3.5PN GW

luminosity (Blanchet , Faye, Iyer, Joguet):

Post-Newtonian calculation of the memory:• Result: expressions for waveform modes hlm and “+” polarization:

Post-Newtonian calculation of the memory:

Additional DC contributions to the waveform:

• Arun et al ’04 found a nonlinear, non-hereditary DC contribution to the “µ”

polarization at 2.5PN order; it originates from nonlinear corrections to the

radiative current octupole moment V3m:

• In addition, there are linear, zero-frequency terms that arise from the m=0

pieces of the source mass and current moments. For example:

• …which leads to a 5PN, non-oscillatory correction to the waveform

• Not clear if these effects produce a “true” memory (i.e., a permanent

displacement in a detector)

Memory from the merger:

• Inspiral memory now know to 3PN order; but this is only a small part of the

memory signal

– 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

• 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

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)

– A22n determined by matching derivatives at t=tm at Schwarzschild light-ring rm = 3M .

• Result is the following simple analytic function:

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, pr* , pj• Ringdown same as minimal-waveform model, but with 5

QNMs

• Matching done for I22(2) at 5 points near the EOB-deformed

light ring

• Construct I22(3) ; plug into previous leading-order expression

for the memory; numerically integrate from r0=15M using

leading-PN order memory for the initial condition

Memory from the merger: results

full-EOB

EOB w/o amplitude corrections F22 f22�QC

minimal waveform model

( equal-mass, non-spinning )

minimal waveform X 0.77

tm = 3522 M

full-EOB w/ memory

full-EOB w/o memory

Detectability of the memory:

• Previous estimates of the memory

– Thorne ‘92: Assumed an un-modeled burst source w/ memory

– Kennefick ‘94: used leading-order inspiral memory, truncated near r = 5M

– Used dated noise curves

• Build-up during the merger/ringdown potentially important:

– Characteristic rise time:

Signal-to-noise ratios:

DETECTOR SOURCE S/N S/N (Kennefick model)

LIGO 50MŸ + 50MŸ @ 20Mpc 3.2 (21)

500MŸ + 500MŸ @ 200Mpc 4.5 (36)

Adv. LIGO 10MŸ + 10MŸ @ 20Mpc 4.1 (34)

50MŸ + 50MŸ @ 20Mpc 31 (226)

50MŸ + 50MŸ @ 200Mpc 3.1 (2.3)Ÿ Ÿ

500MŸ + 500MŸ @ 200Mpc 48 (~400)

100MŸ + 100MŸ @ 1Gpc 1.5 (~10)

LISA 106MŸ + 106MŸ @ 1Gpc 427 (2890)

106MŸ + 106MŸ @ z=1 67 (482)

106MŸ + 106MŸ @ z=3 18 (146)

105MŸ + 105MŸ @ z=3 1.1 (9)

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

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 r0

– Errors from gauge effects and finite extraction radius can further contaminate

NR waveforms and swamp a small memory signal

Memory in numerical relativity simulations:

• Partial solution: hybrid PN + NR calculation

• Use NR energy flux in PN expression for the memory, combined with

PN initial conditions

Signature of GW recoil on the GW signal:

• Numerical relativity has shown a large range of possible kick speeds:

– Vmax ~ 175 km/s for non-spinning BHs

– Vmax ~ 450 km/s for equal-mass BHs w/ spins maximal, anti-aligned along orbital angular

momentum

– Vmax ~ 4000 km/s for equal-mass, maximal spinning BHs, in “super-kick” configuration

• There have been many papers that consider EM signatures of the recoil

• Are signatures of the recoil present in the GW signal?

(1) linear memory due to recoil of

the remnant:

Effects of recoil on the gravitational waveform

Christodoulou memory kick memory

Effects of recoil on the gravitational waveform

(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-of-

sight recoil. One can then cross check this measurement with the simulationssight 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 GW signal.

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 that cannot yet be easily calculated with NRquantities that cannot yet be easily calculated with NR

• 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 large SNR SMBH mergers

• Binaries that recoil after merger also show a memory effect, as well as a doppler shift of the QNM frequencies.

Future work:

• Improve detectability estimates; examine unequal-mass case

• Compute PN memory for eccentric and spinning binaries

• Improvements to EOB calculations of waveforms (and memory)

• include memory in signal in Mock LISA Data Challenge

• Use hybrid NR+PN approach to compute memory

• Examine the memory w/ BH perturbation theory (probably need to go to 2nd order)

• compute the Christodoulou memory in other theories of gravity (how generic is this effect?)