1. Generalized quasi-Keplerian parametrization for compact binaries in hyperbolic orbits
Gihyuk Cho (SNU)
Collaborate with H.M. Lee (SNU), Achamveedu Gopakumar (TIFR)
The conference ICGAC13-IK15 on Gravitation

2. 1 Motivation

3. 1 Motivation
𝑠 𝑡 =ℎ 𝑡 +𝑛(𝑡)
in general
( ℎ <|𝑛|)

4. 1 Motivation 𝑠 𝑡 =ℎ 𝑡 +𝑛(𝑡) in general ( ℎ <|𝑛|) Go outside
Detection characterization
Matched Filtering

5. 1 Motivation 𝑠 𝑡 =ℎ 𝑡 +𝑛(𝑡) in general ( ℎ <|𝑛|)
To have more reliable Detection
We need more accurate Pre-knowledge
Go outside
Detection characterization
Matched Filtering

6. 2 Analytic Methodology Post-Newtonian theory
: Expand the Einstein field equation around 1 𝑐 ~0.
2. Gravitational self force theory
3. Blackhole perturbation theory
Blanchet (2014)
The relaxed Einstein’s field equation

7. 2 Analytic Methodology Post-Newtonian theory
: Expand the Einstein field equation around 1 𝑐 ~0.
Gravitational self force theory
: Expand the equation of motion of point mass on background around test mass limit 𝜈~0.
3. Blackhole perturbation theory

8. 2 Analytic Methodology 1. Post-Newtonian theory
: Expand the Einstein field equation around 1 𝑐 ~0.
Gravitational self force theory
: Expand the equation of motion of point mass on background around test mass limit 𝜈~0.
Blackhole perturbation theory
: Expand the metric field around the well-defined metric, such as Schwarzschild metric, or Kerr metric

9. 3 Bounded case Inspiral-Merger-Ringdown phase

10. 3 Bounded case Inspiral-Merger-Ringdown phase
Post-Newtonian Theory
Numerical Relativity
Blackhole perturbation theory

11. 4 Unbounded case 3.5PN accurate dynamics of Compact Binary
We have well-established 3.5PN Post-Newtonian equation of motion.. 𝑎 = 𝑎 𝑁 + 1 𝑐 𝑎 𝑐 2 𝑎 𝑐 3 𝑎 𝑐 4 𝑎 𝑐 5 𝑎 𝑐 6 𝑎 𝑐 7 𝑎 7

12. 4 Unbounded case 3.5PN accurate dynamics of Compact Binary
We have well-established 3.5PN Post-Newtonian equation of motion.. 𝑎 ( 𝑣 , 𝑟 )= 𝑎 𝑁 + 1 𝑐 𝑎 𝑐 2 𝑎 𝑐 3 𝑎 𝑐 4 𝑎 𝑐 5 𝑎 𝑐 6 𝑎 𝑐 7 𝑎 7
Vanishing!

13. 4 Unbounded case 3.5PN accurate dynamics of Compact Binary
We have well-established 3.5PN Post-Newtonian equation of motion.. 𝑎 = 𝑎 𝑁 + 1 𝑐 2 𝑎 𝑐 4 𝑎 𝑐 5 𝑎 𝑐 6 𝑎 𝑐 7 𝑎 7
Ignore Them!

14. 4 Unbounded case 3.5PN accurate dynamics of Compact Binary
We have well-established 3.5PN Post-Newtonian equation of motion.. 𝑎 = 𝑎 𝑁 + 1 𝑐 2 𝑎 𝑐 4 𝑎 𝑐 6 𝑎 6
Extremely Complicated

15. 4 Unbounded case Generalized quasi-Keplerian parametrization
𝑟− 𝑟 0 = 𝑎 𝑟 ( 𝑒 𝑟 cosh 𝑢 −1) 𝜙− 𝜙 0 = Φ 2𝜋 (𝜈+ 𝑓 𝜙 sin 2𝜈 + 𝑔 𝜙 sin 3𝜈 + ℎ 𝜙 sin 4𝜈 + 𝑖 𝜙 sin 5𝜈 ) 𝑡− 𝑡 0 = P 2𝜋 ( e t sinh 𝑢 −u+ 𝑓 𝑡 𝜈+ 𝑔 𝑡 sin 𝜈 + ℎ 𝑡 sin 2𝜈 + 𝑖 𝑡 sin 3𝜈 ) 𝜈=2arctan( 𝑒 𝜙 +1 𝑒 𝜙 −1 tanh( u 2 ))

16. 4 Unbounded case Generalized quasi-Keplerian parametrization
𝑟− 𝑟 0 = 𝑎 𝑟 ( 𝑒 𝑟 cosh 𝑢 −1) 𝜙− 𝜙 0 = Φ 2𝜋 (𝜈+ 𝑓 𝜙 sin 2𝜈 + 𝑔 𝜙 sin 3𝜈 + ℎ 𝜙 sin 4𝜈 + 𝑖 𝜙 sin 5𝜈 ) 𝑡− 𝑡 0 = P 2𝜋 ( e t sinh 𝑢 −u+ 𝑓 𝑡 𝜈+ 𝑔 𝑡 sin 𝜈 + ℎ 𝑡 sin 2𝜈 + 𝑖 𝑡 sin 3𝜈 ) 𝜈=2arctan( 𝑒 𝜙 +1 𝑒 𝜙 −1 tanh( u 2 ))

17. 4 Unbounded case Generalized quasi-Keplerian parametrization
𝑟− 𝑟 0 = 𝑎 𝑟 ( 𝑒 𝑟 cosh 𝑢 −1) 𝜙− 𝜙 0 = Φ 2𝜋 (𝜈+ 𝑓 𝜙 sin 2𝜈 + 𝑔 𝜙 sin 3𝜈 + ℎ 𝜙 sin 4𝜈 + 𝑖 𝜙 sin 5𝜈 ) 𝑡− 𝑡 0 = P 2𝜋 ( e t sinh 𝑢 −u+ 𝑓 𝑡 𝜈+ 𝑔 𝑡 sin 𝜈 + ℎ 𝑡 sin 2𝜈 + 𝑖 𝑡 sin 3𝜈 ) 𝜈=2arctan( 𝑒 𝜙 +1 𝑒 𝜙 −1 tanh( u 2 ))

18. 4 Unbounded case Incorporating the Radiation Reaction
𝑛 ≔ 2𝜋 𝑃 = 𝑛 (𝑟,𝜙, 𝑟 , 𝜙 )
e t = 𝑒 𝑡 (𝑟,𝜙, 𝑟 , 𝜙 )

19. 4 Unbounded case Incorporating the Radiation Reaction
𝑛 ≔ 2𝜋 𝑃 = 𝑛 (𝑟,𝜙, 𝑟 , 𝜙 )
𝑑 𝑛 𝑑𝑡 = 𝛻 𝑣 𝑛 ⋅ 𝑎 𝑟𝑟
𝑑 𝑒 𝑡 𝑑𝑡 = 𝛻 𝑣 𝑒 𝑡 ⋅ 𝑎 𝑟𝑟
e t = 𝑒 𝑡 (𝑟,𝜙, 𝑟 , 𝜙 )

20. 4 Unbounded case Incorporating the Radiation Reaction
𝑛 ≔ 2𝜋 𝑃 = 𝑛 (𝑟,𝜙, 𝑟 , 𝜙 )
𝑑 𝑛 𝑑𝑡 = 𝛻 𝑣 𝑛 ⋅ 𝑎 𝑟𝑟
𝑛 (𝑡)
𝑑 𝑒 𝑡 𝑑𝑡 = 𝛻 𝑣 𝑒 𝑡 ⋅ 𝑎 𝑟𝑟
e t = 𝑒 𝑡 (𝑟,𝜙, 𝑟 , 𝜙 )
𝑒 𝑡 (𝑡)

21. Energy : 0.01
Angular Momentum : 10

22. Energy : 0.01
Angular Momentum : 10

23. Energy : 0.01
Angular Momentum : 4

24. Energy : 0.01
Angular Momentum :4
Non-oscillating and Persisting Effect
: Linear Memory Effect
: Soft bremsstrahlung
: SSB of BMS super-translation

25. 4 Unbounded case Ongoing projects on Hyperbolic case
Computation of tail effect
(Yannick(UZH), Gopakumar(TIFR), Cho(SNU))
2. Preparing future usages for Data analysis
(Shubhanshu Tiwari(INFN), Gopakumar(TIFR),Cho(SNU)
3. More accurate Capturing process

26. 5 Non-Local effect at 4PN order
Hyperbolicity : Finite Speed of propagation
Non Linearity : Graviton-Graviton interaction
Speed of propagation slower than speed of light

27. 5 Non-Local effect at 4PN order
Future
past
Gravitational Interaction with yourself in the past.

28. 5 Non-Local effect at 4PN order
Damour, Jaranowski, Schafer (2014)

29. 6 Spin dynamics
Spin effects in the phasing of gravitational waves from binaries on eccentric orbit. (Antoine Klein, Philippe Jetzer, 2014)

30. 6 Summary
I derived 3PN accurate generalized quasi-Keplerian parametrization of Hyperbolic orbit.
This derivation allow us extend accuracy of hyperbolic passage waveform from hyperbolic passage to 3.5PN accurate.
I am now working on 4.5PN accurate Eccentric waveform which includes the spin effect and non-local effect.

31. Thank you