Binary black hole initial data

Transcription

1 1 Binary black hole initial data Harald P. Pfeiffer California Institute of Technology Collaborators: Greg Cook (Wake Forest), Larry Kidder (Cornell), Mark Scheel (Caltech), Saul Teukolsky (Cornell), James York (Cornell) Relativistic Astrophysics Workshop, IPAM, UCLA, May 4, 2005

2 Black hole evolutions are important 2 See talks by Joan Centrella Frans Pretorius Luis Lehner Jerome Novak Oscar Reula Mark Scheel Lee Lindblom Matt Choptuik Tvsi Piran Saul Teukolsky evolutions need initial data

3 Initial data for GR 3 Initial data consists of metric g ij and extrinsic curvature K ij on one hypersurface Σ. It must satisfy the constraints R + K 2 K ij K ij = 0 j K ij g ij K = 0 Challenges: 1. How to solve the constraints 2. How to choose g ij and K ij

4 Outline of talk 4 1. Construction of any initial data Conformal method 2. Some surprising properties Non-uniqueness 3. Construction of BBH initial data Quasi-equilibrium method 4. Discussion & Thoughts How good is good enough?

5 Conformal method 5 Problem: Find solutions g ij, K ij of the constraint equations R + K 2 K ij K ij = 0 j K ij g ij K = 0 Strategy: Split g ij and K ij into smaller pieces, some freely specifiable, the rest completely determined. Specifically, g ij = ψ 4 g ij Choose free data Solve elliptic equations Assemble g ij, K ij

6 Numerical method: Spectral elliptic solver (HP, Kidder, Scheel, Teukolsky 2003) 6 Expand solution in basis-functions & solve for expansion-coefficients Smooth solutions exponential convergence Superior accuracy: Numerical errors physical effects Superior efficiency: Large parameter studies Domain decomposition: Multiple length-scales H 2 M 2 δe ADM δm K δj z N Binary black hole ID (Cook & HP, 2004) Bowen-York ID (HP, Kidder et al, 2003)

7 Standard and extended conformal thin sandwich 7 Standard CTS Extended CTS Free data g ij, t g ij, K, Ñ Free data g ij, t g ij, K, t K Four elliptic equations 2 ψ 1 Rψ K2 ψ à ij à ij = 0 8 «1 j 2Ñ Lβij 2 «1 3 ψ6 i K j 2Ñ ũij = 0 Five elliptic equations 2 ψ 1 Rψ K2 ψ à ij à ij =0 8 «1 j 2Ñ Lβij 2 «1 3 ψ6 i K j 2Ñ ũij =0 2`Ñψ 7 Ñψ7 1 8 R K2 ψ «Ã ij à ij = 8 + Quite a few existence and uniqueness Mathematical terra incognita results (especially for K = 0) ψ 5 ( t β k k )K

8 Some results on the standard system 8 Mathematics: 1. asymptotically flat 2. no inner boundaries 3. maximal slice K = 0 Yamabe constant Y[ g ij ]: Free data based on Teukolsky wave ingoing, M = 0, odd parity, centered at r = 20 (HP, Kidder, Scheel, Shoemaker 2005) g ij = δ ij + Ã h ij ũ ij = Ã th ij K = 0, Ñ = 1 Y[ g ij ] > 0 existence & uniqueness (Cantor 1977, Murray & Cantor 1981, Maxwell 2005) / max(ψ) max(ψ) A ~ A ~

9 Extended system (HP & York, gr-qc/ ) 9 g ij = δ ij + à h ij ũ ij = à th ij K = 0, t K = max(ψ) 0.03 ψ finite as à Ãc ψ crit - max(ψ) A ~ c,5 -A~ Parabolic behavior ψ ψ c const.(δã)1/ A ~

10 A more comprehensive look max(ψ) max(ψ) 0.80 min(n) min(n) Equatorial plane, Ã = max(β) 0.20 max(β) A ~

11 A second branch 11 So far p u (Ã) = u c v c δã ψ,, u = β i, N Two branches?? u ± (Ã) = u c ± v c pδã Problem: With simple initial guess, elliptic solver converges always to u ; need good guess to converge to u +. du (Ã) dã = 1 p 2 δã v c du (Ã) u + (Ã) u (Ã) + 4 δã dã Take two numeric solutions u of five coupled 3-D nonlinear elliptic equations, and finite-difference them!!

12 Constructing the upper branch u max(ψ) 1.12 max(ψ) 1.0 min(n) min(n) Parabolic close to Ãc u + and u meet at à c 0.1 max(β) max(β) u + deviates strongly from Minkowski at small à Two solutions for arbitrarily small Ã!! A ~

13 Energy & Apparent horizon 13 ADM energy apparent horizon u+ u- standard standard system system Apparent horizons for à = 0.02 z ψ=1.8 ψ=2.0 ψ= A ~ x

14 Unique solutions, nevertheless? 14 Physics is determined by g ij, K ij. For example g ij = ψ 4 g ij = ψ 4 δ ij + ψ 4 Ã h ij Physical amplitude of perturbation is A = ψ 4 Ã Question: For given physical amplitude A, how many solutions exist? 3.0 max(ψ) 1.12 max(ψ) 100 ADM energy u max(β) min(n) min(n) max(β) 10 1 u- standard system A max =(max ψ) 4 A ~ A max =(max ψ) 4 A ~

15 Lessons from journey into mathematical wonderland 15 Life is more interesting than expected (HP & York, gr-qc/ ). Similar issues possible in any system containing the extended conformal thin sandwich equations ( Jerome Novak s talk?). Non-unique data sets are very different from BBH initial data. Just solving works, so let s go!

16 Astrophysically realistic BBH initial data 16 Question: How to choose free data and boundary conditions for a binary black hole in a circular orbit? Traditional approach (conformally flat Bowen-York) has many problems: ISCO wrong, BHs plunge too quickly,... Need better method!

17 Quasi-equilibrium method 17 Basic idea: Approx. time-independence in corotating frame Approx. helical Killing vector (both concepts essentially equivalent, both useful depending on context) History: Wilson & Matthews 1985: Binary neutron stars Gourgoulhon, Grandclement & Bonazzola, 2002a,b BBH ID with inner boundary conditions (basically right, but various details deficient) Cook & HP, 2002, 2003, 2004 General quasi-equilibrium method with isolated horizon BCs

18 Quasi-equilibrium method (the easy pieces) 18 Time-independence in corotating frame natural choice: vanishing time derivatives Extented conformal thin sandwich formalism 1. t g ij = 0 = t K 2. g ij and K still undetermined Boundary conditions at infinity from asymptotic flatness & corotation: ψ = 1 β i = ( Ω orbital r) i N = 1 New contribution: inner boundary conditions (next slide...)

19 Quasi-equilibrium excision boundary conditions 19 k µ s i S Σ k µ outward pointing null normal to S s i (conformal) spatial normal to S Excise topological sphere(s) S Require 1. S be apparent horizon(s) 2. When evolved, the coordinate locations of the AH s remain stationary 3. The shear σ µν of k µ vanishes Item 3 is an isolated horizon condition. It implies for the expansion θ L k θ = 1 2 θ2 σ µν σ µν = 0 on S AH moves along k µ, and its area is constant (initially)

20 Quasi-equilibrium excision boundary conditions cont d 20 Rewrite in variables of conformal thin sandwich Boundary conditions for ψ and β i r ψ =... on S (1) β i = ψ 2 N s i + β i on S (2) Rotating black holes Vanishing shear restricts β i. Freedom to specify spin of BH remains. Lapse boundary condition not fixed by IH (also Jaramillo et al, 2004). Determine orbital frequency by E ADM = M K.

21 Numerical solutions: Single black holes I 21 g ij, K, S not determined. and lapse-bc. For now arbitrary choices: Conformal flatness, S =sphere Do not use knowledge of single BH solutions use single BHs to test method Spherical symmetry 1. w.l.o.g. conformally flat 2. Try different choices for K and lapse boundary condition 3. any spherically symmetric K and any spherically symmetric lapse-bc yield: exact slice through Schwarzschild totally vanishing time-derivatives t g ij = t K ij = 0 4. Full success: Recover Schwarzschild independent of arbitrary choices.

22 Numerical solutions: Single black holes II 22 Spinning/Boosted black holes Compute quantities that vanish for Kerr: E rad qe 2 ADM P 2 ADM q M 2 irr + J2 ADM /(4M 2 irr ) (3) Ω Ω r J ADM /EADM 3 q, v v P ADM (4) J ADM 2 /E4 E ADM ADM 10-1 ISCO orbital frequency 10-1 ISCO velocities E rad / E ADM Ω E ADM t lnψ M irr Ω r E rad / E ADM v t lnψ v

23 Binary black hole solutions (corotating, K = 0) 23 Lapse positive through horizon

24 Sequences of quasi-circular orbits (corotating, K = 0) 24 Three different lapse boundary conditions Compare to GGB and post-newtonian results CO: MS - d(αψ)/dr = 0 CO: MS - αψ = 1/2 CO: MS - d(αψ)/dr = (αψ)/2r CO: MS - d(αψ)/dr = (αψ)/2r CO: HKV - GGB CO: EOB - 3PN CO: EOB - 2PN CO: EOB - 1PN E b /µ E b /µ J/µm J/µm No difference solution robust Excellent agreement

25 Testing the 2nd law 25 Normalize sequences such that de ADM = Ω 0 dj ADM Irreducible mass along these sequences M irr CO: MS - d(αψ)/dr = 0 CO: MS - αψ = 1/2 CO: MS - d(αψ)/dr = (αψ)/2r corotating sequences (three different lapse BC s) M irr slightly increasing during inspiral ok M irr IR: MS - d(αψ)/dr = 0 IR: MS - αψ = 1/2 IR: MS - d(αψ)/dr = (αψ)/2r mω 0 irrotational sequences (three different lapse BC s) M irr decreasing during inspiral Normalization of sequences wrong? Remaining free data insufficient? Rigidity theorem?

26 ISCO location 26 Caution: ISCO is not a sharp, well-defined concept! Anyway ,2,3 PN (EOB) 2,3 PN (standard) Color: Corotating BH s Grey: Irrotational BH s E b / m PN (EOB) GGB 02 Cook&Pfeiffer 04 (3 data points) Cook&Pfeiffer 04 (3 data points) PN (standard) Cook mω 0 Excellent agreement between NR and PN Superior to Bowen-York w/ effective potential (cf. Cook 1994)

27 Qualitity of todays QE-BBH initial data 27 Contains two black holes ISCO agrees with PN predictions ID-solve results in lapse & shift BHs at rest early in evolution Time-derivatives small Spins incorporated exactly? Tidal distortion of BHs correct? Satisfies testmass limit? Correct ṙ (slightly negative)? Contains wavetrain of earlier insipral? of course yes yes yes ψ/ t ψ 50T orbit 50T orbit T inspiral at large separation! no no no no no superior to Bowen-York, but room for further improvement

28 28 How good is good enough? Just some thoughts and conjectures further opinions welcome!!

29 How good is good enough? 29 Answer depends on application and desired accuracy. Short term improving evolution codes: Testing evolution codes ID doesn t matter. (But smaller initial transient may well be advantageous for dynamic gauge conditions) Qualitative features of plunge wave-forms ( Frans Pretorius talk) My feeling is that ID won t matter (must be demonstrated, though!). The first evolution of two orbits and plunge ID must contain BHs in circular-ish orbits. I believe this is the case (no proof). Such evolutions must quantify initial transient caused by initial data.

30 Long term gravitational wave science If ID contains significant initial transients, then one must wait until they are beyond wave-extraction radius R extract T wait 2R extract 100M CPU-hours Expensive. But let s assume we re willing to pay Initial transient changes coordinatesystem, masses, spins, orbital elements. Accumulated phase-error very sensitive to (e.g.) eccentricity ε. How to measure ε in a dynamical spacetime in an unknown coordinatesystem? This matters for Testing PN-results Parameter extraction May be limited by our knowledge of evolution-parameters. Coherent template PN-inspiral+numerical evolution+ringdown Most stringent test of strong field GR. Total phase error must be (much) smaller than QN-period. (LISA SMBH removal!)

31 Summary 31 Non-unique solutions exist (and may bite us) Quasi-equilibrium initial data vast improvement over Bowen-York and contains handles for next round of improvements ( g ij, S) Further improvements essential, especially when evolutions mature QE-initial data publicly available soon.

32 Summary of QE method 32 Framework for BBH initial data in a kinematical setting (helical Killing vector) Explicitly displays the remaining choices g ij, K, S, Lapse-BC Close in spirit to GGB, but greatly improved: 1. Constraints are satisfied 2. Incorporates isolated horizon boundary conditions 3. General spins possible 4. Retains freedom to choose any g ij, K, S. 5. Lapse is positive on horizon Agrees very well with PN (even with simple choices) Data sets will be publicly available soon.