Analytic methods in the age of numerical relativity

Transcription

1 Analytic methods in the age of numerical relativity vs. Marc Favata Kavli Institute for Theoretical Physics University of California, Santa Barbara

2 Motivation: Modeling the emission of gravitational waves (GW) in preparation for their eventual detection LIGO LISA

3 Sources of gravitational waves: What will we learn? Compact binaries Inspiralling 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 spacetimearound a BH---test the validity of the mathematical description of BHs

4 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!

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

6 The challenge (experimental): Gravitational waves are very weak a typical source changes the LIGO arms by ~10-21 km ~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

7 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!)

8 Why this is hard: Newton vs. Einstein Equations are much more complex: 1 eq., 1 variable (F), simple differential operator 6 indep. eqs., 6 indep. variables (g mn ), complicated differential operator; many, many terms

9 Why this is hard: Newton vs. Einstein There are more sources of gravity: Only mass density density, velocity, kinetic energy, pressure, internal stress, EM fields,

10 Why this is hard: Newton vs. Einstein Gravity is a source for gravity (non-linearity) Highly non-linear differential operator Linear differential operator

11 Solving Einstein s equations: Exact solutions 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 Expand about a known, exact solution in the Perturbation limit that some quantity is small post-newtonian theory: expand about flat theory space assuming gravity is weak, speeds slow Black hole perturbation theory:expand about Kerr or Schwarzschild spacetime Numerical relativity Solve equations numerically on a computer No symmetries or approximation Round-off & truncation error Inexact initial conditions, gauge modes, junk radiation Computationally intensive

12 Stages of binary BH coalescence: BH perturbation theory Post-Newtonian theory Numerical relativity

13 Old picture of coalescence (Thorne):

14 New picture of coalescence: [slide adapted from Centrella]

15 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 Related to metric perturbations h (n) 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

16 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.

17 Post-Newtonian theory (example): The equations of motion for two point masses [Blanchet 06]:

18 Post-Newtonian theory (example): Gravitational waveform for a circularized binary [Blanchet et. al 08]:

19 PN/NR comparison: PN waveforms agree well with the NR simulations for most of the inspiral [Boyle, et al. 07]

20 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) 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

21 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 a i, b i known from 3PN order dynamics; 4PN flexibility parameter a 5 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), p r (t), p j (t) ] System must be supplemented by radiation-reaction force (based on 3.5PN de/dt but also contains several adjustable parameters)

22 Effective-one-body (EOB): EOB waveform: Inspiral+plunge piece: Re-summed PN corrections + additional flexibility parameters BH pert. piece: ringdown waveform Depends on final BH mass & spin [Bertiet. al 06] Match at some matching time near merger to determine A lmn

23 Effective-one-body (EOB): Comparison of EOB w/ Caltech/Cornell NR simulation [ Buonannoet. al 09]

24 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=2mode; 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)

25 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 +,

26 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.

27 What is the GW memory? Memory in a binary black hole merger

28 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)

29 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 ]

30 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 stressenergy 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.

31 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 ]

32 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 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. The memory is hereditary: it depends on the entire past-history of the source

33 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

34 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

35 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: arxiv: arxiv:

36 Detectability of the memory: will be difficult to observe w/ Advanced LIGO likely to be visible by LISA out to redshift z d2 Inspiral waves LISA noise memory Signal-to-noise ratio vs. total mass

37 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 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 2 nd order BH perturbation theory (hard!)

38 Future of semi-analytic perturbation theory: Studies of nonlinear mode-mode coupling in NR simulations (EXAMPLE) [ Schnittmanet. al 08 ] [ Baker et. al 08 ]

39 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 10M Ÿ ) inspiralling into a supermassive BH (m~ M Ÿ ) compact object executes ~10 5 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

40 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). additionally, may need to go to 2 nd 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

41 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