Analytic methods in the age of

numerical relativity

vs.

Marc Favata

Kavli Institute for Theoretical Physics

University of California, Santa Barbara

Motivation:

• Modeling the emission of gravitational waves (GW) in preparation

for their eventual detection

LIGO

LISA

Sources of gravitational waves: What will we learn?

Compact binariesInspiralling stellar-mass compact objects (white dwarfs, neutron stars, black holes)

• Determine merger rates, measure compact object masses & spins

• Constrain models of stellar evolution and globular cluster dynamics

• Probe central engine of gamma-ray bursts (NS/NS, NS/BH)

• Study the equation of state of nuclear matter at ultra-high densities (NS/NS, NS/BH)

• Test the validity of GR in the strong-field, highly non-linear regime (BH/BH); constrain alternative theories of gravity

Supermassive black hole (SMBH) binaries

• Learn how SMBH’s grow---mergers vs. accretion

• Probe GR with high precision

• Study radiation-recoil of BHs and its consequences

• Constrain the nature of dark energy

Extreme-mass-ratio inspirals

• Determine number density of compact objects in galactic centers

• Precision map of the spacetime around a BH---test the validity of the mathematical description of BHs

Sources of gravitational waves: What will we learn?

Individual compact objects

Core-collapse supernova

• Probe the inner-workings of the explosion mechanism

• Study the nuclear physics at high densities

Continuous sources

• Pulsars w/ small mountains

• Accreting neutron stars

• Fluid instabilities in rotating neutron stars

Exotic sources?

• Gravitational wave remnants from the big bang

• Phase transitions in the early universe

• Cosmic strings

• Signatures of extra dimensions

THE UNEXPECTED!

?

Sources of gravitational waves: What will we learn?

Mathematical-physics questions

• What is the full solution to the relativistic two-body problem?

• To what extent and in what regimes are analytic approximations accurate?analytic approximations accurate?

• What types of interesting nonlinear effects manifest themselves? Are they observable?

The challenge (experimental):

Gravitational waves are very weak

• a typical source changes the LIGO arms by ~10-21km ~ 10-18 m ~ 10-9 nm

• Need very high precision measurements

Noisy environment---background noise obscures the signals

• Seismic noise: including…ocean waves, logging, traffic…

• Gravity gradient noise: including people, cars, wind, tumbleweed…

• Suspension noise: modes of test-mass suspension

• Shot noise, laser noise

• Radiation pressure from laser on the test masses

• Light scattering

• Residual gas in vacuum tube

• Cosmic rays

The challenge (theoretical):

Need a template gravitational wave to extract the physical parameters (masses, spins) from the detected signals

Compute gravitational waves from a given type of source (eg., compact binary)

Compute the motion of the source

Solve the Einstein field equations (hard!)

Newton vs. Einstein

Equations are much more complex:

1 eq., 1 variable (F), simple

differential operator6 indep. eqs., 6 indep. variables

(�mn), complicated differential

operator; many, many terms…

Why this is hard:

operator; many, many terms…

Newton vs. Einstein

Why this is hard:

There are more sources of gravity:

Only mass density

density, velocity, kinetic energy,

pressure, internal stress, EM

fields, …

Newton vs. Einstein

Why this is hard:

Gravity is a source for gravity (non-linearity)

Highly non-linear differential operator

Linear differential

operator

Solving Einstein’s equations:

• Only known for situations with special symmetry

• Only 2 astrophysically relevant exact solutions:

• Kerr metric (rotating black hole)

• Friedman-Robertson-Walker metric(homogenous & isotropic universe

Exact solutions

• Expand about a known, exact solution in the limit that some quantity is small

Perturbation limit that some quantity is small

• post-Newtonian theory: expand about flat space assuming gravity is weak, speeds slow

• Black hole perturbation theory: expand about Kerr or Schwarzschild spacetime

Perturbation theory

• Solve equations numerically on a computer

• No symmetries or approximation

• Round-off & truncation error

• Inexact initial conditions, gauge modes, junk radiation

• Computationally intensive

Numerical relativity

Stages of binary BH coalescence:

Post-Newtonian theory

BH perturbation theory

Numerical relativity

Old picture of coalescence (Thorne):

New picture of coalescence:

[slide adapted from Centrella ]

Black hole perturbation theory in a nutshell:

Expand the metric and stress-energy tensor as:

plug into Einstein’s equations and solve at each order:

Linear differential

wave operator

Linear order perturbation theory used to study:

• the oscillation modes of BHs (quasinormal-modes)

• gravitational waves from a point-particle source moving on a geodesic

wave operator

Related to metric

perturbations h(n)

Post-Newtonian (PN) theory in a nutshell:

Can (schematically) write the full Einstein’s equations as :

Solution procedure is complex and has been developed over the last 30+

years [see Blanchet (2006) for a review].

Result is the metric h in terms of a sum of mass and current (electric and

magnetic) multipole moments that are related to integrals over the matter

stress-energy tensor.

Post-Newtonian theory (example):

The equations of motion for two point masses [Blanchet ‘06]:

Post-Newtonian theory (example):

Gravitational waveform for a circularized binary [Blanchet et. al ‘08]:

PN/NR comparison:

PN waveforms agree

well with the NR

simulations for most of

the inspiral

[Boyle, et al. ’07]

Effective-one-body (EOB): A hybridization of PN, NR, and BH perturbation theory techniques

(see Buonanno & Damour ’99, ‘00; lecture notes by Damour ’08)

Motivation: provide a quick, semi-analytic way to generate waveform

templates that include the inspiral + merger + ringdown

Contains the following features:

• PN piece: An extension of the PN two-body dynamics to the non-adiabatic region

(the transition from inspiral to plunge)(the transition from inspiral to plunge)

• BH pert. piece: A matching of the “inspiral +

plunge” waveform to a ringdown waveform

• NR piece: A variety of “flexibility” parameters

that can be fit to NR simulations

Effective-one-body (EOB):

PN piece (inspiral + plunge):

• Maps the 2-body PN Hamiltonian to a Hamiltonian equivalent to a point particle

with mass equal to the reduced mass moving on a “deformed” Schwarzschild

metric

• ai , bi known from 3PN order dynamics; 4PN “flexibility” parameter a5 introduced

and adjusted to match phasing of NR waveforms

• A Hamiltonian is constructed from this metric. Solving Hamilton’s

equations gives the conservative dynamics [ r(t), j(t), pr(t), pj(t) ]

– System must be supplemented by radiation-reaction force

(based on 3.5PN dE/dt but also contains several adjustable parameters)

Effective-one-body (EOB):

EOB waveform:

Re-summed PN corrections +

additional “flexibility” parameters

Inspiral+plunge piece:

additional “flexibility” parameters

BH pert. piece: ringdown waveform

Match at some matching time near merger to determine Almn

Depends on final BH mass

& spin [Berti et. al ‘06]

Effective-one-body (EOB):

Comparison of EOB w/

Caltech/Cornell NR simulation

[ Buonanno et. al ’09]

Limitations of numerical relativity (NR):

• NR can best compute the modes of the curvature perturbation

Need to integrate twice to get the observable waveform:

• NR simulations can most accurately compute the l=m=2 mode; higher-order modes

are smaller and resolved with less resolution

•While one of the main purposes of NR is to fully compute all the nonlinear effects

involved in BH mergers, some effects are obscured

• One especially interesting (and possibly observable) such effect is the “memory”

• (late-time ringdown tails would be another, but are probably unobservable)

What is “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+,×values of h+,×

What is the GW memory?

• An “ideal” (freely-falling) GW detector would experience a permanent displacement after the GW has passed---leaving a “memory” of the signal.

• The late-time constant displacement is not directly measureable,but its buildup is.

• While the memory’s buildup is in principle measureable in both LIGO and LISA, in LIGO the mirror displacement would not be truly permanent, but it would be in LISA.

What is the GW memory?

Memory in a binary

black hole merger

Origin of the 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: unbound (hyperbolic) orbits (Turner ’77)

Origin of the 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:

• unbound (hyperbolic) orbits (Turner ’77)

• 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)

[Burrows &

Hayes ‘96 ]

Origin of the memory?Nonlinear memory: (Christodoulou ‘91 ; see also Blanchet & Damour ‘92)

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

• Recall previous form of the Einstein’s equations:

Grav’l wave stress-

energy tensor…energy tensor…

…contributes to the changing

multipole moments…

…which determines the GW field...

…which has a slowly-growing, non-oscillatory

piece related to the radiated GW energy.

Origin of the memory?Nonlinear memory: (Christodoulou ‘91 ; see also Blanchet & Damour ‘92)

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

In analogy to the linear memory, the nonlinear memory can be interpreted as arising from changes in the mass quadrupole moment due to the radiated gravitons (Thorne ‘92) [ just as radiated neutrinos cause linear memory in supernovae ]

• 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 with the standard choices for e+ij e×

ij )

• Although it is a 2.5PN correction to the mass multipole moments, 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

Memory in numerical relativity simulations:

• Extracting the memory from NR simulations faces several challenges:

– Physical memory only present in m=0 modes (for quasi-circular orbits),

which are numerically suppressed

(2,2), (4,4), (3,2), (4,2) modes much larger than the

memory modes (2,0), (4, 0), etc..

Orbital separation decreasing � Orbital separation decreasing �

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

EOB calculation of memory from BH mergers:

• Use EOB formalism calibrated to NR simulations to compute (2,2) mode.

• Feed this into post-Newtonian calculation of the memory modes in terms

of the (2,2) mode

For details see:For details see:

arXiv:0811.3451

arXiv:0812.0069

arXiv:0902.3660

Detectability of the memory:

Inspiral

waves memoryLISA noise

Signal-to-noise ratio vs. total mass

• will be difficult to observe w/ Advanced LIGO

• likely to be visible by LISA out to redshift z d 2

Future of semi-analytic perturbation theory:

• Further PN/NR comparison studies

– Non-spinning case is well explored

– Much work remains for spinning binaries

• Extension of EOB formalism to spinning binaries

– Also more work needed to treat eccentric binaries (likely relevant only for

SMBH mergers)

• Extension of EOB to eccentric binaries not yet attempted

• Various improvements/extensions to memory calculations• Various improvements/extensions to memory calculations

• Comparisons of NR with linear BH perturbation theory

– NR codes approaching “smallish” mass ratios (1:10) [ see Gonzalez, et al ’08]

– Meaningful comparisons might be possible for the inspiral waves

• Studies of nonlinear mode-mode coupling in NR simulations

– Explain features in the NR waveform mode amplitude

– Need to develop a theoretical framework for doing this using either PN theory

or 2nd order BH perturbation theory (hard!)

Future of semi-analytic perturbation theory:

• Studies of nonlinear mode-mode coupling in NR simulations

(EXAMPLE)

[ Schnittman et. al ’08 ][ Baker et. al ’08 ]

Future of semi-analytic perturbation theory: EMRIs

(extreme-mass-ratio inspirals)

The last unsolved regime of the

relativistic 2-body problem

• stellar-mass compact object (m~1 – 10MŸ) inspiralling into

a supermassive BH (m~106 – 109 MŸ)

• compact object executes ~105 orbits in the last year of inspiral close to the horizon• compact object executes ~105 orbits in the last year of inspiral close to the horizon

• too many cycles for NR

• too relativistic for PN

• GW signal will encode a precise map of the spacetime---allowing us to extract the

multipole moments of the spacetime: “holiodesy”

Future of semi-analytic perturbation theory: EMRIs

(extreme-mass-ratio inspirals)

Challenges in modeling EMRIs:

• radiation reaction problem is difficult to solve for EMRIs

• current techniques rely on linear BH pert. theory +

“adiabatic approximation:” slow evolution of conserved constants of the motion

• this is not accurate enough: need to evaluate the full “self force” acting on the

particle---(several technically difficulties, but much recent progress). particle---(several technically difficulties, but much recent progress).

• additionally, may need to go to 2nd order in BH perturbation theory to compute

EMRI phasing to the required accuracy

• other effects on EMRI orbits:

• effect of viscous torques on the compact object from the BH’s accretion disk

• effects of tidal distortions if the EMRI is a WD

• effects from the perturbations of a distant third body

Conclusions and summary:

• After 30+ years, numerical relativity is successful

• But analytic theory studies of binary inspiral + merger still have

important roles to play

• Post-Newtonian shows good agreement with NR simulations

where expected

• Memory effect: an example of the synergy between PN theory and •

NR

• Open problems in analytic studies of the relativistic 2-body

problem:

– PN/NR comparisons in the spinning case

– Comparisons to BH perturbation theory

– Analysis of mode-coupling in the NR simulations

– EMRI studies: a variety of unsolved problems