Ballistic orbits for Gravitational Waves

Transcription

1 for Gravitational Waves Giuseppe d'ambrosi Jan-Willem van Holten [arxiv: ] Kyoto th Capra meeting on Radiation Reaction in GR

2 1 2 3 Giuseppe d'ambrosi for Gravitational Waves 2

3 Black Hole perturbation Theory EMR binaries: µ M 1 PostNewtonian methods do not help much neglect eects of µ on metric use full GR (strong gravity) Giuseppe d'ambrosi for Gravitational Waves 3

4 Black Hole perturbation Theory Dierent stages of an EMR binary inspiral: eccentric orbit, adiabatic approximation plunge: inspiralling orbit merger and ringdown µ and M get closer to each other and the orbit circularizes (but [Cutler et al 1994], [Tanaka et al 1993]) The transition between inspiral and plunge takes place in the region of the last stable orbit (ISCO) Giuseppe d'ambrosi for Gravitational Waves 4

5 Schwarzschild Black Holes Black Hole perturbation Theory HORIZON: r = 2M LIGHT RING: r = 3M ISCO: r = 6M Schwarzschild's solution dτ 2 = e 2ν (dt) 2 e 2ψ (dφ ωdt q 2 dθ q 3 dr) 2 e µ2 (dθ) 2 e µ3 (dr) 2 Spherical symmetry: expand in spherical harmonics EM tensor and metric the Zerilli-Moncrief (even) and Regge-Wheeler (odd) equations respectively polar (ν, ψ, µ 2, µ 3 ) and axial (ω, q 2, q 3 ) perturbations [ 2 t lm V r 2 e/o Transform to RW gauge ] Ψ lm e/o (t, r ) = Fe/o lm r δ(r r p(t))+ge/o lm δ(r r p(t)) Giuseppe d'ambrosi for Gravitational Waves 5

6 Black Hole perturbation Theory From RW-ZM to the gravitational wave Two polarizations for the gravitational wave in TT gauge h + ih = 1 ( t ) Ψ lm e 2i Ψ lm o r dt VAB l m A m B lm T GW µν = 1 P = 1 32π 64π < D µh αβ D ν h αβ > ( ) Ψ lm e Ψ lm o 2 (l+2)! lm (l 2)! [ dl dt = Re i 64π lm m (l+2)! (l 2)! ( Ψlm e Ψ lm e + 4Ψ lm )] t o Ψ o(t )dt Giuseppe d'ambrosi for Gravitational Waves 6

7 Some waveforms in time domain Black Hole perturbation Theory Radial infall quadrupole wave for infall from r 0 = 30M [Lousto,Price 1997] Circular orbit r = 12M, (l, m) = (2, 2) wave [Martel 2005] Eccentric orbit from geodesic deviations, (l, m) = (2, 2) wave [Koekoek,van Holten 2011] Giuseppe d'ambrosi for Gravitational Waves 7

8 Black Hole perturbation Theory So.. Stages of an EMR binary inspiral: eccentric orbits, adiabatic waveforms, SF corrections, etc.. what about plunge? quasi-circular orbit/inspiralling orbit Giuseppe d'ambrosi for Gravitational Waves 8

9 Black Hole perturbation Theory So.. Stages of an EMR binary inspiral: eccentric orbits, adiabatic waveforms, SF corrections, etc.. what about plunge? quasi-circular orbit/inspiralling orbit Giuseppe d'ambrosi for Gravitational Waves 9

10 Circular orbits in Schwarzschild spacetime Ballistic Orbits Gravitational waveforms Lagrangian formulation of Schwarzschild metric [ (1 ) ] L = 1 2 2M r ṫ2 ṙ2 1 2M r 2 θ2 (r 2 sin 2 θ) ϕ 2 = const r energy and angular momentum rst integrals of the motion EMR binaries circularize until 6M ( ) 2 dr +( 1 2M dτ r ) ) (1 + L2 = E 2 r 2 intepret the terms as kinetic energy and potential energy Giuseppe d'ambrosi for Gravitational Waves 10

11 Circular orbits in Schwarzschild spacetime Ballistic Orbits Gravitational waveforms From the eective potential V l : V l = ( 1 2M ) ( ) r 1 + l2 r 2 for l 2 > 12M 2 there are two kinds of circular orbits ( ) V l extremized for R ± = l2 2M 1 ± 1 12M2 l 2 r = 2M r = 6M ξ = 1 12M 2 l 2 ε 2 ± = V l [R ± ] = 2 (2 ± ξ) ± ξ Introduce the ballistic orbit V r(ϕ) = 6M 1 + 2ξ + 3ξ cot 2 ( A 2 ϕ) U in 1 1 correspondence with stable circular orbits (r = R + )! Giuseppe d'ambrosi for Gravitational Waves 11

12 Circular orbits in Schwarzschild spacetime Ballistic Orbits Gravitational waveforms What for? Plunge phase beyond the ISCO, r < 6M µ starts at r 6M towards the horizon r(ϕ) = 6M(1 δ) 1 + e cot 2 ( A 2 ϕ) perturbative parameter δ = 2 3 e = 2ξ 1+2ξ, so 0 < δ < 2 3 solve geodesic equations as evolution equations r r rφ Giuseppe d'ambrosi for Gravitational Waves 12 M

13 Circular orbits in Schwarzschild spacetime Ballistic Orbits Gravitational waveforms Analytically solved evolution 1 e t t0 (2 e) 2(1 e) 4M = (3 2e) 2 Aϕ 2(2 e)(1 e)2 (11 11e + 2e2 ) e 2(1 e)2 1 e (2 e) 2(1 e) e(3 2e) cotan Aϕ + 2 2(1 e) Aϕ 1 + e cotan2 2 ( 1 arctan tan Aϕ ) e 2 2(1 e) arccotanh tan Aϕ. e 2 ν = µ M δ 10 3 γ = ν 2/5 δ r /M total turns turns after r universal geodesic behavior number of turns 4ν 1/5 [Buonanno 2000, Bernuzzi 2010] begin at r M = 6 αν2/5, [Buonanno 2000, Damour 2007] Giuseppe d'ambrosi for Gravitational Waves 13

14 Lousto-Price algorithm ψrw lm /ZM (t + 2, r ) = ψrw lm /ZM (t, r ) 1 + A A3 4 Circular orbits in Schwarzschild spacetime Ballistic Orbits Gravitational waveforms V lm RW /ZM (r ) V lm RW /ZM (r ) + 1 A1 V l 4 RW /ZM (r ) 1 + A3 V l ψ 4 RW /ZM (r RW lm /ZM ) (t +, r ) + 1 A4 V l 4 RW /ZM (r + ) ψ 1 + A3 V lm 4 RW /ZM (r RW lm /ZM ) (t +, r + ) cell S lm RW /ZM(t, r)dtdr 4 + V l (r )A 4 Giuseppe d'ambrosi for Gravitational Waves 14

15 Universal geodesic behavior Circular orbits in Schwarzschild spacetime Ballistic Orbits Gravitational waveforms smaller values of δ imply longer quasi-circular phase quasi-circular phase gives quasi-periodic emission nal burst independent of δ ν Giuseppe d'ambrosi for Gravitational Waves 15

16 Waveforms obtained with this orbit Circular orbits in Schwarzschild spacetime Ballistic Orbits Gravitational waveforms (2,2)-(3,3)-(4,4) waves amplitude decreases fast with increasing l-mode Giuseppe d'ambrosi for Gravitational Waves 16

17 How to help?! Energy and angular momentum of ballistic orbits bigger than ISCO (ξ = 0) ε 2 = 2 (2 + ξ) 2 2(2 e)2 = l 2 = 12M ξ 9 9e + 2e 2 1 ξ = 4M 2 (3 2e) e + e 2 Knowing a simple geodesic in analytical form in time, one can try to turn it into a more general orbit: x µ = x µ + σn µ σ2 (k µ Γ µ λν nλ n ν ) + with n µ = x µ σ σ=0 and k µ = nµ σ σ=0 + Γ µ λν nλ n ν with the geodesic deviation equation to be solved, to rst-order: D 2 n µ Dτ 2 Rµ λνκ ū µ ū λ n ν = 0 Giuseppe d'ambrosi for Gravitational Waves 17

18 Energy and angular momentum get corrections as well ( 1 2M ) dt r dτ = ε = ε 0 + σε 1 + o(σ 2 ) r 2 dϕ dτ = l = l 0 + σl 1 + o(σ 2 ) ε 1 and l 1 tunable to any matching orbit: this case r = 6M n r n µ obtained analytically corrections decrease along plunge in preparation... Giuseppe d'ambrosi for Gravitational Waves 18

19 Outline Extreme Mass Ballistic Ratio binaries orbits Conclusions....and further directions explored a sector of geodesic motion developed a way to investigate GW during plunge use ballistic orbit for geodesic deviations obtain a con rmation of the universal behaviour? use bal.orbits for GWs thank you for your attention! Giuseppe d'ambrosi for Gravitational Waves 19

20 Outline Extreme Mass Ballistic Ratio binaries orbits Conclusions....and further directions explored a sector of geodesic motion developed a way to investigate GW during plunge use ballistic orbit for geodesic deviations obtain a con rmation of the universal behaviour? use bal.orbits for GWs thank you for your attention! Giuseppe d'ambrosi for Gravitational Waves 20

21 Outline Extreme Mass Ballistic Ratio binaries orbits Conclusions....and further directions explored a sector of geodesic motion developed a way to investigate GW during plunge use ballistic orbit for geodesic deviations obtain a con rmation of the universal behaviour? use bal.orbits for GWs thank you for your attention! Giuseppe d'ambrosi for Gravitational Waves 21

22 Outline Extreme Mass Ballistic Ratio binaries orbits Conclusions....and further directions explored a sector of geodesic motion developed a way to investigate GW during plunge use ballistic orbit for geodesic deviations obtain a con rmation of the universal behaviour? use bal.orbits for GWs thank you for your attention! Giuseppe d'ambrosi for Gravitational Waves 22

23 Outline Extreme Mass Ballistic Ratio binaries orbits Conclusions....and further directions explored a sector of geodesic motion developed a way to investigate GW during plunge use ballistic orbit for geodesic deviations obtain a con rmation of the universal behaviour? use bal.orbits for GWs thank you for your attention! Giuseppe d'ambrosi for Gravitational Waves 23

24 Outline Extreme Mass Ballistic Ratio binaries orbits Conclusions....and further directions explored a sector of geodesic motion developed a way to investigate GW during plunge use ballistic orbit for geodesic deviations obtain a con rmation of the universal behaviour? use bal.orbits for GWs thank you for your attention! Giuseppe d'ambrosi for Gravitational Waves 24

25 Outline Extreme Mass Ballistic Ratio binaries orbits Conclusions....and further directions explored a sector of geodesic motion developed a way to investigate GW during plunge use ballistic orbit for geodesic deviations obtain a con rmation of the universal behaviour? use bal.orbits for GWs thank you for your attention! Giuseppe d'ambrosi for Gravitational Waves 25

26 Outline Extreme Mass Ballistic Ratio binaries orbits Conclusions....and further directions explored a sector of geodesic motion developed a way to investigate GW during plunge use ballistic orbit for geodesic deviations obtain a con rmation of the universal behaviour? use bal.orbits for GWs thank you for your attention! Giuseppe d'ambrosi for Gravitational Waves 26