Binary Orbital Dynamics from the Analysis of Spherical Harmonic Modes of Gravitational Waves


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1 Intro Motivations Binary Orbital Dynamics from the Analysis of Spherical Harmonic Modes of Gravitational Waves Dr. Jennifer Seiler (NASA GSFC) Gravity Theory Seminars University of Maryland College Park November 20, 2011
2 Outline Intro Motivations I show how to extract binary orbital dynamics from GWs by minimization of the asymmetric spherical harmonic modes of gravitational waveforms. 1 Introduction Gravitational Waves Binary Black Holes Numerical Relativity Wave Extraction Horizons GR Spins and Binary Dynamics 2 Motivations Astrophysics Data Analysis Waveform Data 3 Methods Wigner Rotation Deprecession 4 Results Phase and Orientation Precession Tracking 5 Conclusions 6 Future Work
3 Intro Motivations GW BH NR WE Horizons Spins General Relativity and Gravitational Waves Coaction between matter and curvature is described by the Einstein Equations: G µν = 8πT µν Black holes (BH) = Vacuum (T µν = 0) Gravitational Waves (GW) = finite deviation from Minkowski spacetime: Linearized field equations in GR g µν = η νµ + h νµ, h νµ 1. hµν = 16πT µν = ( 2 t + 2 ) hµν = 0.
4 Intro Motivations GW BH NR WE Horizons Spins Gravitational waves The coupling between matter and geometry is very weak. R αβ 1 2 Rg αβ = kt αβ k = 8πG c s 2 m kg Gravitational waves are small features, difficult to detect. Unobstructed by intervening matter Excellent probe into regions opaque to EM radiation.
5 Intro Motivations GW BH NR WE Horizons Spins Gravitational waves Currently there are many ground based detectors online which are designed to detect such passing gravitational waves (LIGO, VIRGO, TAMA, GEO). Even for binary black hole inspiral and merger, the signal strength is likely to be much less than the level of any detector noise. A technique used for this purpose is matched filtering, in which the detector output is crosscorrelated with a catalog of theoretically predicted waveforms. Therefore, chances of detecting a generic astrophysical signal depend on the size, scope, and accuracy of the theoretical signal template bank. The generation of such a template bank requires many models of the GW emitted from compact binary systems.
6 Binary black holes Intro Motivations GW BH NR WE Horizons Spins NGC2207 and IC2163 Optical, radio, and xray astronomy have provided us with abundant evidence that many galaxies contain SMBHs in their central nuclei. The loudest astrophysical signals in terms of SNR. Known examples among galactic binaries. Supermassive M. Low frequency sources spacebased detector (LISA) Formation processes for stellar mass binaries: Collapse within a binary neutron star system. Capture within a dense region, eg. globular cluster.
7 Binary black holes Intro Motivations GW BH NR WE Horizons Spins Chirp signal from a binary inspiral Black holes captured highly elliptical orbits. Radiation of gravitational energy circularisation of orbits. inspiral (PN) Decay of orbit leading to plunge (NR) merger (NR) Single perturbed BH remnant exponential ringdown to axisymmetric (Kerr) BH. q y ν = 0.1 horizon ISCO q x
8 Intro Motivations GW BH NR WE Horizons Spins Einstein equations in 3+1 form The Einstein equations are manifestly covariant Need to reformulate as a Cauchy problem We have ten equations and ten independent components of the four metric g µν, the same number of equations as unknowns. Only six of these ten equations involve second timederivatives of the metric. The other four equations, thus, are not evolutions equations. We call these our constraint equations. There are a number of nonunique aspects of the 3+1 decomposition Choice of evolution variables Choice of gauge Binary black hole codes currently use either a harmonic formulation, or a modified ( conformal traceless or BSSN ) ADM system.
9 Intro Motivations GW BH NR WE Horizons Spins Numerical Relativity R αβ 1 2 g αβr = 8πT αβ The Einstein equations are a hyperbolic set of nonlinear wave equations for the geometry As such, they are most conveniently solved as an initialboundaryvalue problem: Assume the geometry is known at some initial time t 0. Evolve the data using the Einstein equations. Prescribe consistent boundary conditions at some finite radius r 0. Geometry specified on an initial data slice: metric g ab specifies the intrinsic geometry of the slice. extrinsic curvature determines the embedding in 4D space. Evolution equations are integrated using standard numerical methods, eg. RungeKutta. The equations are differentiated in space on a discrete computational grid using finite differencing methods
10 Wave extraction Intro Motivations GW BH NR WE Horizons Spins It has become standard to measure waves as expansions of the NewmanPenrose Ψ 4 scalar. An independent method measures gageinvariant perturbations of a Schwarzschild black hole. Observers are placed on a 2sphere at some large radius. Measure oddparity (Q lm ) and evenparity (Q + lm ) perturbations of the background metric. h + ih = 1 2r l=2 m=0 ( l t Q + lm i Q lm (t )dt ) 2Y lm,
11 Newman Penrose Intro Motivations GW BH NR WE Horizons Spins The Ψ s are defined as components of the Weyl tensor C abcd which is identical to the Riemann tensor R abcd. The complex Weyl scalars Ψ is given by Ψ 4 = C ab m a m b C ab = R ab K K ab + Ka c K cb iɛa cd d K bc In the chosen a tetrad that separates the radiation part of the Weyl tensor from the nonradiation content. Using spherical coordinates we obtain the tetrad: l 1 2 (ˆτ ˆr), n 1 2 (ˆτ + ˆr), m 1 2 (ˆθ i ˆφ), m 1 2 (ˆθ + i ˆφ), ( The spacetime metric can be described as g ab = 2m (a m b) 2n (a l b)
12 Newman Penrose Intro Motivations GW BH NR WE Horizons Spins Ψ 4 can be related to the gravitational radiation in the following way: in the transversetraceless (TT) gauge 1 4 (ḧtt ḧtt ˆθ ˆθ ˆφ ˆφ ) = Rˆτ ˆθˆτ ˆθ, 1 2 ḧtt ˆθ ˆφ = Rˆτ ˆθˆτ ˆφ (2) Finally, we can use R abcd = C abcd (G µν = 0), to yield the final relation between Ψ 4 and the radiation as a metric perturbation in terms of polarizations Ψ 4 = (ḧ + iḧ ). (3) Since two factors of m each carries a spinweight of 1, we decompose Ψ 4 in terms of spinweight 2 spherical harmonics 2Y lm (θ, φ), yielding Ψ 4 (t, r ) = 1 Mr l 2C lm (t) 2 Y lm (θ, φ), (4) m= l l=2
13 Zerilli Moncrief Intro Motivations GW BH NR WE Horizons Spins Regge and Wheeler formulated a gauge transformation that allows a radial/angular separation for even and odd parities for some perturbation of GR δg µν = δr µν = δγ ρ µν,ρ Γ ρ µρ,ν, δγ i jk = 1 2 gil (h jl,k + h kl,j h jk,l ), Perturbations γ ij can be decomposed using Y lm into γij lm (t, r) where γ ij (t, r, θ, φ) = γ ij (t, r, θ, φ) = l l=0 m= l γ lm ij (t, r) 6 p k (t, r)v k (θ, φ) k=0 where {V k } are basis for tensors on a twosphere. In Schwarzschild coordinates, the Regge Wheeler and Zerilli equations may be written 2 t Ψ (o/e) lm 2 r Ψ (o/e) lm + V (o/e) l Ψ (o/e) lm = S (o/e) lm where S (o) is the ReggeWheeler source function, S are the source function, V are the potential functions.
14 Zerilli Moncrief Intro Motivations GW BH NR WE Horizons Spins From Ψ (o) lm and Ψ(e) lm we obtain the gravitational wave amplitude h + ih = 1 r l,m (l + 2)! (l 2)! (Ψ(e) lm + Ψ(o) lm ) 2Y lm (θ, φ) + O( 1 r 2 ) The areal coordinate ˆr of each extraction sphere is calculated by [ ] 1/2 1 γθθγ ˆr = ˆr(r) = φφdθdφ S(ˆr) = 4π ( ) 2 ˆr r γ rr dθdφ We can calculate the six ReggeWheeler variables, on these spheres by integration of combinations of the metric components over each sphere. From here we can construct the gauge invariant quantities from these ReggeWheeler and Zerilli variables )] S Q lm = Q + lm = [ 2(l + 2)! c lm ( ˆr c lm 2 (l 2)! 2 2ˆr c lm 2 2(l 1)(l + 2) (4ˆrS 2 k 2 + l(l + 1)ˆrk 1 ) l(l + 1) (l 1)(l + 2) + 6M/ˆr. This formalism is convenient as it gives us h + and h already decomposed into l and m modes with no extra integration stages required. ˆr
15 Horizons Intro Motivations GW BH NR WE Horizons Spins An event horizon is defined nonlocally. Thus it can only be obtained as part of postprocessing for any simulation. An apparent horizon is defined locally in time on a spacelike slice, and can thus be calculated at each timestep in a simulation. It is the outermost smooth closed 2surface in a slice whose futurepointing outgoing null geodesics have zero expansion, Θ. Θ i n i + K ij n i n j K = 0 This outermost surface is coincident with the boundary of a trapped surface i.e. a surface whose futurepointing outgoing null geodesics have negative expansion. The existence of such a surface automatically implies the existence of a black hole (given certain technical assumptions are met, including energy conditions and a reasonable gauge).
16 BH Data Intro Motivations GW BH NR WE Horizons Spins Horizon spins, relative velocities, and masses extracted from the shape and coordinate motion of those black hole horizons. The angular momentum of the horizon comes from the horizon φ Killing vector field L φ q ij = 0, where q ij := γ ij s i s j is the induced metric on the horizon for outwardpointing normals s i. We can then derive the magnitude of the angular momentum from J H = 1 8π S φ l s m K lm da. From the angular momentum and horizon surface we can then obtain the mass of the black hole from the equation where A H is the horizon area. M 2 H = A H 16π + 4πJ2 H A H,
17 Intro Motivations GW BH NR WE Horizons Spins Parameter studies with spinning black holes Aligned spin leads to an orbital hangup. r0 r1 r2 r3 r4 r5 r6 r7 r8 y (M) r x (M) r0 y (M) x (M) 0.4 r4 r8 y (M) r x (M) h+ 0 h time (M) time (M)
18 Kicks Intro Motivations GW BH NR WE Horizons Spins For an equalmass, nonspinning binary merger, the remnant will be a stationary, spinning black hole. If an asymmetry in the bodies is present, the emitted in gravitational waves will also have asymmetry. As a result, the remnant black hole will have momentum relative to distant stationary observers, called a recoil or kick. Asymmetries in the emitted gravitational wave energy are a result of: Unequal masses. Unequal spin magnitudes. Spins which are misaligned with each other or the orbital angular momentum.
19 Black hole kicks Intro Motivations GW BH NR WE Horizons Spins The recoil results from couplings of various wave modes, which are integrated over the entire inspiral time. F i Ṗ i = r 2 (ḣ2 ) dω n i 16π + + ḣ 2 PN (2.5) suggests a linear increase of recoil with spin ratio: ( q 2 (1 q) v kick = c 1 (1 + q) + c a 2 q 2 (1 qa 1 /a 2 ) 5 2 = c (1 + q) 5 2 a 2 1 a ) 1 a 2 In fact, the numerical data points to a quadratic dependence: v kick = a 2 (c 1 c 2 ( a 1 a 2 ) + c 3 ( a 1 a 2 ) 2 ) The maximum recoil for the antialigned case: v kick = 448 ± 5km/s
20 Spins Intro Motivations GW BH NR WE Horizons Spins Equation to predict final spin of merged black hole a fin = 1 [ a (1 + q) a 2 2 q a 2 a 1 q 2 cos α+ 2 ( a 1 cos β + a 2 q 2 cos γ ) l q + l 2 q 2 ] 1/2, where cos α â 1 â 2, cos β â 1 ˆl, cos γ â 2 ˆl. In order to obtain l we need to match this equation against general second order polynomial expansions for: Equal mass, unequal but aligned spin binaries Unequal mass, equal spin binaries s ( ) 4 l = a a 2 2 q a 1 a 2 q 2 cos α + (1 + q 2 ) 2 ( ) s5ν + t ( a 1 cos β + a 2 q 2 cos γ) q 2 Numerical simulations to obtain s 4, s 5, t 0, t 2, t 3. Test against generic misaligned spin binaries t 2ν + t 3ν 2.
21 Intro Motivations GW BH NR WE Horizons Spins Precession in General Relativity Spins are not aligned with orbital angular momentum will precess L = L orb + S 1 + S 2, L t J rad t causing whole system to precess, resulting in complex dynamics Even without GR binary precession incredible complicated. For EMRI limit for leading order (each term at least 1 page): H = Mc 2 + H 0 + V 1 + V 1 + V S1 + V S2 + V S1,S2 + V Q1 + V Q1 the EMRI approximations derived from 1978 plus conservations assumptions currently the only way spins are accounted for in PN
22 Motivations Intro Motivations DA WF Parameter studies with simple precession tracking will give us a phenomenological precession model could give us better formulae for SS and SO interaction for EOB and PN combining this model with Wigner rotation in the opposite direction gives us a tool to create generically precession waveforms... Phenomenological templates for precessing systems Cut down the parameter space by 4 degrees of freedom Challenges of NR: Parameter space is large and hard to model generically Simulations are computationally expensive and slow Length and accuracy requirements are not well known Setting up precessing simulations is especially challenging Numerically, this is also a tool to get gauge invariant measurements (as opposed to on the horizon) especially for waveforms extracted at null infinity (i.e. CCM/CCE)
23 Waveform Data Intro Motivations DA WF Parameter Horizon Waveform Masses MH 2 = AH 16π + 4πJ2 H A H A(Ψ 4 ) ˆL orb (Θ i n i + K ij n i n j K = 0) z θ(max(ψ 22 Ψ 20 )) L?? L 0 ˆL J dt Phase/ ˆraxis (Θ = 0) 1 (Θ = 0) 2 φ(min(ψ 20 )) Separation (Θ = 0) 1 (Θ = 0) 2 R 0 f (J Rad, φ) S fin J H = 1 8π S φl s m K lm da Quasinormal S JH,i = 1 8π S φl s m K lm da L J r 2 (ḣ2 Kick?? dω ni 16π + + ḣ ) 2 S 1 and S 2 J H = 1 8π S φl s m K lm da??
24 Methods Results Conclusions Future Wigner Codes Wigner Rotation In precessing systems the dominant l = 2, m = ±2 quadrupole mode mixes into nonsymmetric modes as the obsever rotates off the L orb axis This ruins existing phenomenological waveform formulae Rotate reference frame to return to quadrupole dominant system Maximize Ψ 4,22 and minimize Ψ 4,20 Tracks the direction of L orb relative to observes and precession of the system
25 Methods Results Conclusions Future Wigner Codes Implementation We consider SWSHs of spin weight s = 2: Ψ 0 4 = e iωu 2ψ lm 2 Y lm (θ, φ), lm The transformation of the Weyl scalar Ψ 0 4 Ψ 0 4(θ, φ ) = e 2iχ(θ(θ ),φ(φ )) Ψ 0 4(θ(θ ), φ(φ )). Given the transformation of the harmonics 2Y lm (θ (θ, φ), φ (θ, φ)) = l m= l where D (l) 1 m;m is the Wigner rotation matrix given by D (l) 1 m;m 2 Y lm (θ, φ), ψ lm = l D (l) m;m (θ, φ) = e imφ (l + m)!(l m)!(l + m )!(l m )! k 1 k+m m k!(l + m k)!(l m k)!(m m + k)! m= l ( 2ψ lm D (l) 1 m;m, sin θ 2 ) 2k+m m ( cos θ 2 ) 2l 2k m +m
26 Methods Results Conclusions Future Wigner Codes Wigner Rotation Use iterative numerical solver to find values of θ, and φ for each time that minimize the amplitude of the asymmetric mode ψ 20 and maximize ψ 22 Find rotation angles that place the observer back on the axis of the orbital angular momentum of the system. The modes m = 1 will vanish only when the two black holes can be exchanged by symmetry The m = 0 optimally is reduced to a nonoscillating mode related to memory effects. This complicates minimization. Maximizing only the l = 2, m = 2 modes for equal mass systems is perfectly sound. Unequal mass systems have a natural asymmetry that will show in the l = 2, m = 1 modes.
27 Methods Results Conclusions Future Tilt Precession Results for Tilted Simulation e06 Transformed Ψ 4,22 Original Ψ 4,22 Transformed Ψ 4,20 Original Ψ 4,20 1e08 1e10 Ψ lm Amplitude 1e12 1e14 1e16 1e18 z (M) e x (M) Time (M)
28 Methods Results Conclusions Future Tilt Precession Results for Tilted Simulation max( Ψ 4,22 Ψ 4,20 ) max(ψ 4,22 ) Puncture Tracks max( Ψ 4,22 Ψ 4,20 ) Puncture tracks θ (degrees) φ (degrees) Time (M)
29 Methods Results Conclusions Future Tilt Precession Results For Precessing Simulations 1e06 Transformed Ψ 4,22 Original Ψ 4,22 Transformed Ψ 4,20 Original Ψ 4,20 Ψ lm Amplitude 1e08 1e10 1e e e Time (M)
30 Methods Results Conclusions Future Tilt Precession Results For Precessing Simulations max( Ψ 4,22 Ψ 4,20 ) Puncture Track max( Ψ 4,22 Ψ 4,20 ) Puncture tracks θ (degrees) 0 φ (degrees) Time (M) Time (M)
31 Methods Results Conclusions Future Conclusions Accurate method of tracking orbital precession evolution from only observed/extracted waveforms. Gauge invariant measures of angular momentum, and spin direction. Precession reconstruction from only waveforms data. Significant simplification of the emitted GW signal. Deprecessed WFs have high overlap with nonprecessing equivalents. Potential for method for the generation of generic precessing binaries.
32 Methods Results Conclusions Future Future Work Hybrid model of precession rates Wigner rotation of nonprecessing runs to cover parameter space Better understanding of SS, SO contributions Test of momentum measures accuracy Better understanding of mode contributions, for better phenomenological waveform predictions Final spin direction predictions
33 Methods Results Conclusions Future Thank You.
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