Calculating Accurate Waveforms for LIGO and LISA Data Analysis

Transcription

1 Calculating Accurate Waveforms for LIGO and LISA Data Analysis Lee Lindblom Theoretical Astrophysics, Caltech HEPL-KIPAC Seminar, Stanford 17 November 2009 Results from the Caltech/Cornell Numerical Relativity Collaboration. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

2 Motivation: Gravitational Wave Astronomy Recent work in numerical relativity is aimed at providing model waveforms for gravitational wave (GW) astronomy (LIGO, etc.). Binary black hole systems emit large amounts of GW as the holes inspiral and ultimately merge. These are expected to be among the strongest sources detectable by LIGO. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

3 Motivation: Gravitational Wave Astronomy Recent work in numerical relativity is aimed at providing model waveforms for gravitational wave (GW) astronomy (LIGO, etc.). Binary black hole systems emit large amounts of GW as the holes inspiral and ultimately merge. These are expected to be among the strongest sources detectable by LIGO. Numerical waveforms may be useful in detection (to construct better data filters), and/or in modeling detected signals n(t) h(t) t (sec.) Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

4 Motivation: Gravitational Wave Astronomy Recent work in numerical relativity is aimed at providing model waveforms for gravitational wave (GW) astronomy (LIGO, etc.). Binary black hole systems emit large amounts of GW as the holes inspiral and ultimately merge. These are expected to be among the strongest sources detectable by LIGO. Numerical waveforms may be useful in detection (to construct better data filters), and/or in modeling detected signals n(t) n(t) h(t) 0 h(t) h(t) t (sec.) t (sec.) Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

5 Gravitational Wave Data Analysis Signals h s (t) are detected in the noisy LIGO data by projecting them onto a model waveforms h m (λ, t). The measured signal-to-noise ratio, ρ m (λ), is maximized by adjusting the model waveform parameters λ. ρ m (λ) = 4 0 Re[h s (f )hm(f, λ)] df S n (f ) [ 4 0 h m (f, λ) 2 S n (f ) ] 1/2 df Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

6 Gravitational Wave Data Analysis Signals h s (t) are detected in the noisy LIGO data by projecting them onto a model waveforms h m (λ, t). The measured signal-to-noise ratio, ρ m (λ), is maximized by adjusting the model waveform parameters λ. ρ m (λ) = 4 0 Re[h s (f )hm(f, λ)] df S n (f ) [ 4 0 h m (f, λ) 2 S n (f ) ] 1/2 df A detection occurs whenever a signal is present that matches a model waveform with ρ m (λ) > ρ min. For LIGO searches ρ min 8. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

7 Gravitational Wave Data Analysis Signals h s (t) are detected in the noisy LIGO data by projecting them onto a model waveforms h m (λ, t). The measured signal-to-noise ratio, ρ m (λ), is maximized by adjusting the model waveform parameters λ. ρ m (λ) = 4 0 Re[h s (f )hm(f, λ)] df S n (f ) [ 4 0 h m (f, λ) 2 S n (f ) ] 1/2 df A detection occurs whenever a signal is present that matches a model waveform with ρ m (λ) > ρ min. For LIGO searches ρ min n(t) h(t) h(f) S 1/2 (f) t (sec.) f (Hz) Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

8 How Accurate Must Model Waveforms Be? Write the model waveform as h m (f ) = h s (f )e δχm(f )+iδφm(f ), where δχ m (f ) and δφ m (f ) represent errors in its amplitude and phase. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

9 How Accurate Must Model Waveforms Be? Write the model waveform as h m (f ) = h s (f )e δχm(f )+iδφm(f ), where δχ m (f ) and δφ m (f ) represent errors in its amplitude and phase. Inaccurate model waveforms degrade the measured signal-to-noise ratios ρ m (λ), resulting in missed detections. To ensure that the loss rate of detections does not exceed 10%, the waveform errors must not exceed: 0.01 ( ) 2 ( ) 2 [ δχ m + δφm (δχm ) 2 + (δφ m ) 2] 4 h s 2 df. ρ 2 S n 0 Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

10 How Accurate Must Model Waveforms Be? Write the model waveform as h m (f ) = h s (f )e δχm(f )+iδφm(f ), where δχ m (f ) and δφ m (f ) represent errors in its amplitude and phase. Inaccurate model waveforms degrade the measured signal-to-noise ratios ρ m (λ), resulting in missed detections. To ensure that the loss rate of detections does not exceed 10%, the waveform errors must not exceed: 0.01 ( δχ m ) 2 + ( δφm ) 2 0 [ (δχm ) 2 + (δφ m ) 2] 4 h s 2 ρ 2 S n df. Physical properties of the GW source are measured by adjusting the model waveform parameters h m (f, λ) to achieve the largest measured signal-to-noise ρ m (λ). Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

11 How Accurate Must Model Waveforms Be? Write the model waveform as h m (f ) = h s (f )e δχm(f )+iδφm(f ), where δχ m (f ) and δφ m (f ) represent errors in its amplitude and phase. Inaccurate model waveforms degrade the measured signal-to-noise ratios ρ m (λ), resulting in missed detections. To ensure that the loss rate of detections does not exceed 10%, the waveform errors must not exceed: 0.01 ( δχ m ) 2 + ( δφm ) 2 0 [ (δχm ) 2 + (δφ m ) 2] 4 h s 2 ρ 2 S n df. Physical properties of the GW source are measured by adjusting the model waveform parameters h m (f, λ) to achieve the largest measured signal-to-noise ρ m (λ). To ensure the errors in the measured parameters λ are dominated by the intrinsic detector noise S n (f ) rather than model waveform error, the waveform errors must not exceed: 1 4ρ 2 max ( δχ m ) 2 + ( δφm ) 2. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

12 How Are Accurate Waveforms Calculated? Computational Challanges: Dynamics of binary black hole problem is driven by delicate adjustments to orbit due to emission of gravitational waves. Very big computational problem: Must evolve 50 dynamical fields (spacetime metric plus all first derivatives). Must accurately resolve features on many scales from black hole horizons r GM/c 2 to emitted waves r 100GM/c 2. Many grid points are required 10 6 even if points are located optimally. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

13 How Are Accurate Waveforms Calculated? Computational Challanges: Dynamics of binary black hole problem is driven by delicate adjustments to orbit due to emission of gravitational waves. Very big computational problem: Must evolve 50 dynamical fields (spacetime metric plus all first derivatives). Must accurately resolve features on many scales from black hole horizons r GM/c 2 to emitted waves r 100GM/c 2. Many grid points are required 10 6 even if points are located optimally. Most representations of the Einstein equations have mathematically ill-posed initial value problems. Constraint violating instabilities destroy stable numerical solutions in many well-posed forms of the equations. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

14 Outline of Remainder of Talk: Interesting Features of the Caltech/Cornell code: Constraint Damping. Pseudo-Spectral Methods. Horizon Tracking and Conforming Grid Structures.. Damped Harmonic Gauge Conditions. Results: Generic Mergers. Accurate BBH waveforms. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

15 Gauge and Constraints in Electromagnetism The usual representation of the vacuum Maxwell equations split into evolution equations and constraints: t E = B, E = 0, t B = E, B = 0. These equations are often written in the more compact 4-dimensional notation: a F ab = 0 and [a F bc] = 0, where F ab has components E and B. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

16 Gauge and Constraints in Electromagnetism The usual representation of the vacuum Maxwell equations split into evolution equations and constraints: t E = B, E = 0, t B = E, B = 0. These equations are often written in the more compact 4-dimensional notation: a F ab = 0 and [a F bc] = 0, where F ab has components E and B. Maxwell s equations are often re-expressed in terms of a vector potential F ab = a A b b A a : a a A b b a A a = 0. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

17 Gauge and Constraints in Electromagnetism The usual representation of the vacuum Maxwell equations split into evolution equations and constraints: t E = B, E = 0, t B = E, B = 0. These equations are often written in the more compact 4-dimensional notation: a F ab = 0 and [a F bc] = 0, where F ab has components E and B. Maxwell s equations are often re-expressed in terms of a vector potential F ab = a A b b A a : a a A b b a A a = 0. This form of Maxwell s equations is manifestly hyperbolic as long as the gauge is chosen correctly, e.g., let a A a = H(x, t), giving: a a A b ( 2 t + 2 x + 2 y + 2 z ) Ab = b H. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

18 Constraint Damping Where are the constraints: a a A b = b H? Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

19 Constraint Damping Where are the constraints: a a A b = b H? Gauge condition becomes a constraint: 0 = C a A a H. Maxwell s equations imply that this constraint is preserved: a a C = 0. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

20 Constraint Damping Where are the constraints: a a A b = b H? Gauge condition becomes a constraint: 0 = C a A a H. Maxwell s equations imply that this constraint is preserved: a a C = 0. Modify evolution equations by adding multiples of the constraints: a a A b = b H+γ 0 t b C = b H+γ 0 t b ( a A a H). These changes also affect the constraint evolution equation, a a C γ 0 t b b C = 0, so constraint violations are damped when γ 0 > 0. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

21 Constraint Damped Einstein System Generalized Harmonic form of Einstein s equations have properties similar to Maxwell s equations: Gauge (coordinate) conditions are imposed by specifying the divergence of the spacetime metric: a g ab = H b +... Evolution equations become manifestly hyperbolic: g ab =... Gauge conditions become constraints. Constraint damping terms can be added which make numerical evolutions stable C Without Constraint Damping 10-8 With Constraint Damping t/m Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

22 Numerical Solution of Evolution Equations t u = F (u, x u, x, t). Choose a grid of spatial points, x n. x n 1 x n x n+1 Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

23 Numerical Solution of Evolution Equations t u = F (u, x u, x, t). Choose a grid of spatial points, x n. Evaluate the function u on this grid: u n (t) = u(x n, t). u n 1 u n u n+1 x n 1 x n x n+1 Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

24 Numerical Solution of Evolution Equations t u = F (u, x u, x, t). Choose a grid of spatial points, x n. Evaluate the function u on this grid: u n (t) = u(x n, t). u n 1 u n u n+1 x n 1 x n x n+1 Approximate the spatial derivatives at the grid points x u(x n ) = k D n ku k. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

25 Numerical Solution of Evolution Equations t u = F (u, x u, x, t). Choose a grid of spatial points, x n. Evaluate the function u on this grid: u n (t) = u(x n, t). u n 1 u n u n+1 x n 1 x n x n+1 Approximate the spatial derivatives at the grid points x u(x n ) = k D n ku k. Evaluate F at the grid points x n in terms of the u k : F(u k, x n, t). Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

26 Numerical Solution of Evolution Equations t u = F (u, x u, x, t). Choose a grid of spatial points, x n. Evaluate the function u on this grid: u n (t) = u(x n, t). u n 1 u n u n+1 x n 1 x n x n+1 Approximate the spatial derivatives at the grid points x u(x n ) = k D n ku k. Evaluate F at the grid points x n in terms of the u k : F(u k, x n, t). Solve the coupled system of ordinary differential equations, du n (t) dt = F[u k (t), x n, t], using standard numerical methods (e.g. Runge-Kutta). Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

27 Basic Numerical Methods Different numerical methods use different ways of choosing the grid points x n, and different expressions for the spatial derivatives x u(u n ) = k D n k u k. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

28 Basic Numerical Methods Different numerical methods use different ways of choosing the grid points x n, and different expressions for the spatial derivatives x u(u n ) = k D n k u k. Most numerical groups use finite difference methods: Uniformly spaced grids: x n x n 1 = x = constant. Use Taylor expansions to obtain the needed expressions for x u: x u(x n ) = u n+1 u n 1 2 x + O( x 2 ). Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

29 Basic Numerical Methods Different numerical methods use different ways of choosing the grid points x n, and different expressions for the spatial derivatives x u(u n ) = k D n k u k. Most numerical groups use finite difference methods: Uniformly spaced grids: x n x n 1 = x = constant. Use Taylor expansions to obtain the needed expressions for x u: x u(x n ) = u n+1 u n 1 + O( x 2 ). 2 x Grid spacing decreases as the number of grid points N increases, x 1/N. Errors in finite difference methods scale as N p. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

30 Basic Numerical Methods II A few groups (including ours) use pseudo-spectral methods. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

31 Basic Numerical Methods II A few groups (including ours) use pseudo-spectral methods. Represent functions as finite sums: u(x, t) = N 1 k=0 ũk(t)e ikx. Choose grid points x n to allow exact (and efficient) inversion of the series: ũ k (t) = N 1 n=0 w n u(x n, t)e ikxn. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

32 Basic Numerical Methods II A few groups (including ours) use pseudo-spectral methods. Represent functions as finite sums: u(x, t) = N 1 k=0 ũk(t)e ikx. Choose grid points x n to allow exact (and efficient) inversion of the series: ũ k (t) = N 1 n=0 w n u(x n, t)e ikxn. Obtain derivative formulas by differentiating the series: x u(x n, t) = N 1 k=0 ũk(t) x e ikxn = N 1 m=0 D n m u(x m, t). Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

33 Basic Numerical Methods II A few groups (including ours) use pseudo-spectral methods. Represent functions as finite sums: u(x, t) = N 1 k=0 ũk(t)e ikx. Choose grid points x n to allow exact (and efficient) inversion of the series: ũ k (t) = N 1 n=0 w n u(x n, t)e ikxn. Obtain derivative formulas by differentiating the series: x u(x n, t) = N 1 k=0 ũk(t) x e ikxn = N 1 m=0 D n m u(x m, t). Errors in spectral methods are dominated by the size of ũ N. Estimate the errors (for Fourier series of smooth functions): π ũ N = 1 u(x)e inx dx 2π π 1 N p max d p u(x) dx p. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

34 Basic Numerical Methods II A few groups (including ours) use pseudo-spectral methods. Represent functions as finite sums: u(x, t) = N 1 k=0 ũk(t)e ikx. Choose grid points x n to allow exact (and efficient) inversion of the series: ũ k (t) = N 1 n=0 w n u(x n, t)e ikxn. Obtain derivative formulas by differentiating the series: x u(x n, t) = N 1 k=0 ũk(t) x e ikxn = N 1 m=0 D n m u(x m, t). Errors in spectral methods are dominated by the size of ũ N. Estimate the errors (for Fourier series of smooth functions): π ũ N = 1 u(x)e inx dx 2π π 1 N p max d p u(x) dx p. Errors in spectral methods decrease faster than any power of N. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

35 Comparing Different Numerical Methods Wave propagation with second-order finite difference method: N=200 N=200 N= t= t= t= U(x,t) 1.5 U(x,t) 1.5 U(x,t) π/2 π 3π/2 2π x π/2 π 3π/2 2π x π/2 π 3π/2 2π x Figures from Hesthaven, Gottlieb, & Gottlieb (2007). Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

36 Comparing Different Numerical Methods Wave propagation with second-order finite difference method: N=200 N=200 N= t= t= t= U(x,t) 1.5 U(x,t) 1.5 U(x,t) π/2 π 3π/2 2π x π/2 π 3π/2 2π x Wave propagation with spectral method: π/2 π 3π/2 2π x 3.0 N=10 N=10 N= t= t= t= U(x,t) 1.5 U(x,t) 1.5 U(x,t) π/2 π 3π/2 2π x π/2 π 3π/2 2π x Figures from Hesthaven, Gottlieb, & Gottlieb (2007) π/2 π 3π/2 2π x Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

37 Caltech/Cornell Spectral Einstein Code (SpEC): Multi-domain pseudo-spectral method. Constraint damped generalized harmonic Einstein equations: g ab = F ab (g, g). Constraint-preserving, physical and gauge boundary conditions. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

38 Moving Black Holes Black hole interior is not in causal contact with exterior. Interior is removed, introducing an excision boundary. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

39 Moving Black Holes Black hole interior is not in causal contact with exterior. Interior is removed, introducing an excision boundary. Numerical grid must be moved when black holes move too far. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

40 Moving Black Holes Black hole interior is not in causal contact with exterior. Interior is removed, introducing an excision boundary. Numerical grid must be moved when black holes move too far. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

41 Moving Black Holes Black hole interior is not in causal contact with exterior. Interior is removed, introducing an excision boundary. Numerical grid must be moved when black holes move too far. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

42 Moving Black Holes Black hole interior is not in causal contact with exterior. Interior is removed, introducing an excision boundary. Numerical grid must be moved when black holes move too far. Problems: Difficult to get smooth extrapolation at trailing edge of horizon. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

43 Moving Black Holes Black hole interior is not in causal contact with exterior. Interior is removed, introducing an excision boundary. Numerical grid must be moved when black holes move too far. Problems: Difficult to get smooth extrapolation at trailing edge of horizon. Causality trouble at leading edge of horizon. t Horizon Outside Horizon x Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

44 Outside Horizon Moving Black Holes Black hole interior is not in causal contact with exterior. Interior is removed, introducing an excision boundary. Numerical grid must be moved when black holes move too far. Problems: Difficult to get smooth extrapolation at trailing edge of horizon. Causality trouble at leading edge of horizon. Solution: t Choose coordinates that smoothly track the motions of the centers of the black holes. Horizon x Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

45 Horizon Tracking Coordinates Coordinates must be used that track the motions of the holes. A coordinate transformation from inertial coordinates, ( x, ȳ, z), to co-moving coordinates (x, y, z), consisting of a rotation followed by an expansion, x y z = e a( t) cos ϕ( t) sin ϕ( t) 0 sin ϕ( t) cos ϕ( t) is general enough to keep the holes fixed in co-moving coordinates for suitably chosen functions a( t) and ϕ( t). Since the motions of the holes are not known a priori, the functions a( t) and ϕ( t) must be chosen dynamically and adaptively as the system evolves. x ȳ z, Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

46 Horizon Tracking Coordinates II δϕ y c Measure the co-moving centers of the holes: x c (t) and y c (t), or equivalently Q x (t) = x c(t) x c (0), x c (0) Q y (t) = y c(t) x c (t). x c Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

47 Horizon Tracking Coordinates II δϕ y c Measure the co-moving centers of the holes: x c (t) and y c (t), or equivalently Q x (t) = x c(t) x c (0), x c (0) Q y (t) = y c(t) x c (t). Use a feedback-control system to adjust the map parameters a(t) and ϕ(t) in such a way that Q x (t) and Q y (t) remain small, thus keeping the positions of the black holes at fixed coordinate locations along the x axis. x c Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

48 Horizon Distortion Maps Tidal deformation, along with kinematic and gauge effects cause the shapes of the black holes to deform: Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

49 Horizon Distortion Maps Tidal deformation, along with kinematic and gauge effects cause the shapes of the black holes to deform: If the holes become significantly distorted relative to the spherical excision surface bad things happen: Some points on the excision boundary are much deeper inside the singular black hole interior. Numerical errors and constraint violations are largest there, sometimes leading to instabilities. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

50 Horizon Distortion Maps Tidal deformation, along with kinematic and gauge effects cause the shapes of the black holes to deform: If the holes become significantly distorted relative to the spherical excision surface bad things happen: Some points on the excision boundary are much deeper inside the singular black hole interior. Numerical errors and constraint violations are largest there, sometimes leading to instabilities. When the horizons move relative to the excision boundary points, the excision boundary can become timelike, and boundary conditions are then needed there. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

51 Horizon Distortion Maps II Adjust the placement of grid points near each black hole using a horizon distortion map that moves grid coordinates x i into points in the black hole rest frame x i : θ A = θ A, ϕ A = ϕ A, r A = r A f A (r A, θ A, ϕ A ) L l l=0 m= l Adjust the coefficients λ lm A (t) using a feedback control system to keep the excision surface the same shape and slightly smaller than the horizon. Choose f A to scale linearly from f A = 1 on the excision boundary, to f A = 0 on the surrounding cube. λ lm A (t)y lm (θ a, ϕ A ). Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

52 Dynamical Gauge Conditions The spacetime coordinates x b are fixed in the generalized harmonic Einstein equations by specifying H b : a a x b H b. The generalized harmonic Einstein equations remain hyperbolic as long as the gauge source functions H b are taken to be functions of the coordinates x b and the spacetime metric g ab. The simplest choice H b = 0 (harmonic gauge) fails for very dynamical spacetimes, like binary black hole mergers. We think this failure occurs because the coordinates themselves become very dynamical solutions of the wave equation a a x b = 0 in these situations. Another simple choice keeping H b fixed in the co-moving frame of the black holes works well during the long inspiral phase, but fails when the black holes begin to merge. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

53 Dynamical Gauge Conditions II Some of the extraneous gauge dynamics could be removed by adding a damping term to the harmonic gauge condition: a a x b = H b = µt a a x b = µt b = µg bt / g tt. This works well for the spatial coordinates x i, driving them toward solutions of the spatial Laplace equation on the timescale 1/µ. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

54 Dynamical Gauge Conditions II Some of the extraneous gauge dynamics could be removed by adding a damping term to the harmonic gauge condition: a a x b = H b = µt a a x b = µt b = µg bt / g tt. This works well for the spatial coordinates x i, driving them toward solutions of the spatial Laplace equation on the timescale 1/µ. For the time coordinate t, this damped wave condition drives t to a time independent constant, which is not a good coordinate. A better choice sets H t proportional to µ log det g ij /g tt. This time coordinate condition keeps the ratio det g ij /g tt close to unity, even during binary black hole mergers where it becomes of order 100 using our simpler gauge conditions. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

55 Generic Mergers Recent improvements now allow the Caltech/Cornell code SpEC to perform inspiral merger and ringdown simulations robustly for a wide range of black hole binary systems. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

56 Numerical BBH Gravitational Waveforms r Re Ψ 4 22 / M ω2 The Caltech/Cornell collaboration has computed high precision numerical inspiral-merger-ringdown waveforms for several simple equal-mass BBH systems t / M Do these waveforms meet the required accuracy standards for LIGO data analysis? Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

57 Determining Numerical Waveform Accuracy 0.01 δφ (radians) N5 - N t/m Numerical convergence of gravitational waveform. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

58 Determining Numerical Waveform Accuracy 0.01 δφ (radians) R outer =722M - R outer =462M R outer =202M - R outer =462M N5 - N t/m Numerical convergence of gravitational waveform. Phase dependence on outer boundary location. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

59 Determining Numerical Waveform Accuracy 0.01 δφ (radians) R outer =722M - R outer =462M R outer =202M - R outer =462M N5 - N t/m Numerical convergence of gravitational waveform. Phase dependence on outer boundary location. M(t)-M /M N2 N1 N3 N4 N5 N t/m Constancy of the black hole masses. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

60 Summary of Numerical Waveform Phase Errors: Effect δφ (radians) Numerical truncation error Finite outer boundary Drift of mass M Extrapolation r Wave extraction at r areal =const? Coordinate time = proper time? Lapse spherically symmetric? 0.01 root-mean-square sum 0.01 Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

61 Summary of Numerical Waveform Phase Errors (Including Physical Parameter Errors): Effect δφ (radians) Numerical truncation error Finite outer boundary Drift of mass M Extrapolation r Wave extraction at r areal =const? Coordinate time = proper time? Lapse spherically symmetric? 0.01 residual orbital eccentricity 0.02 residual black hole spin 0.03 root-mean-square sum 0.04 Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26

62 Summary of Numerical Waveform Phase Errors (Including Physical Parameter Errors): Effect δφ (radians) Numerical truncation error Finite outer boundary Drift of mass M Extrapolation r Wave extraction at r areal =const? Coordinate time = proper time? Lapse spherically symmetric? 0.01 residual orbital eccentricity 0.02 residual black hole spin 0.03 root-mean-square sum 0.04 It isn t known yet whether these waveforms meet the real frequency domain waveform accuracy standards required for LIGO data analysis. Lee Lindblom (Caltech) Calculating Accurate Waveforms Stanford 17 November / 26