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Keplerian-type parametrization : Its relevance for LISA & SKA

This paper discusses the relevance of Keplerian-type parametrization in understanding the dynamics of compact binaries for interferometric GW detectors such as LISA and SKA. It explores the post-Newtonian accurate dynamics and its parametric solutions, highlighting its importance in realizing the science potential of these detectors.

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Keplerian-type parametrization : Its relevance for LISA & SKA

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  1. Keplerian-type parametrization : Its relevance for LISA & SKA Achamveedu Gopakumar TPI, FSU-Jena, GermanyBirmingham: 30/3/06

  2. The point to realize Post-Newtonianaccurate dynamics of compact binaries along with the associated Keplerian-type parametric solution will be required to realize science potential of Interferometric GW Detectors & SKA

  3. Plan • Symbolic introduction to post-Newtonian (PN) dynamics, relevant for compact binaries • Efforts from Jena relevant for VIRGO, A-LIGO, LISA,… • Theoretical efforts useful for SKA

  4. Introduction

  5. Introduction: I • Three stages for the dynamics of compact binaries (NS-NS, NS-BH, BH-BH) Early inspiral Late stages of inspiral & subsequent plunge Ring-down

  6. Introduction: II • Till the early stages of late inspiral, post-Newtonian (PN) approximation holds good • PN approximation gives corrections to Newtonian gravitational theoryin terms of (v/c)2 ~ (G m /c2 r) Compact binaries are treated like point-masses

  7. Introduction : III • The dynamics of compact binaries in PN approximation is recently determined, both in near-zone orbital dynamics and in far-zone flux computations, to third & halfPN order : corrections up to (v/c)7 • L. Blanchet, T. Damour, G. Schäfer(Esposito-Farese, Faye, Jaranowski, Iyer, ..)

  8. Introduction: IV • H ( r, p)3.5PN= H0( r, p)+H1( r, p)+H2( r, p)+H2.5( r, p)+H3( r, p)+H3.5( r, p) [For the near zone orbital dynamics] • Similar expansions for hx(t), h+(t), ω(t) [ farzone measurable quantities] • Crucial to the templates for ICBs in quasi circular orbits

  9. Parametric solution: I • 3PN accurate conservative dynamics allows Keplerian type parametric solution • H ( r, p) PN= H0( r, p)+H1( r, p)+H2( r, p)+H3( r, p)+Hso( r, p,s1,s2) • Such solution exists even when compact objects spin

  10. Keplerian parametrization :I • Dynamics of point masses in Newtonian gravity allows Keplerian parametrization Radial motion • R = a ( 1 - e cos u )Angular motion • φ –φ0 = v = 2 tan-1 [ ( [ 1 + e]/ [ 1 - e ] )(1/2) tan u/2]Kepler Equation • l = n ( t - t0 ) = u - e sin u

  11. Keplerian parametrization :II • u & v have geometrical meaning • 3PN accurate conservative dynamics of point masses also allows Keplerian type parametric solution

  12. Parametric solution: II • Radial motionr = ar ( 1 - er cos u ) • Angular motionφ –φ0 = (1 + k ) v + ( f 4 φ + f6φ) sin 2 v + ( g 4 φ + g 6 φ ) sin 3 v + i 6 φ sin 4 v+ h 6 φ sin 5 v v= 2 tan-1 [ ( [ 1 + eφ]/ [ 1 - e φ] )(1/2) tan u/2]

  13. Parametric solution: III • 3PN accurate Kepler Equationl = n ( t - t0 ) = u - et sin u+ ( g4t + g6t ) (v - u) + ( f4t+ f6t ) sin v + i6t sin 2 v+ h6t sin 3 v • n & k are gauge invariant quantities if expressed in terms of E, J, m 1 & m 2

  14. Parametric solution: IV • Complicated parametric solution with Hso( r, p,s1,s2)Details in papers from Jena in 2004 & 2005 • There are 3 PN accurate (related) eccentricities e r , e t , eφ • Orbital elements & functions are PN accurate expressions in E, J, m 1 & m 2

  15. Parametric solution: V • Our efforts extends seminal works ofDamour, Deruelle, Schäfer & Wex • 1PN accurate Keplerian-type parametric solution is crucial for Damour-Deruelletiming formula

  16. LOTS OF APPLICATIONS Relevant for GW interferometrs & SKA

  17. Publications: I • T. Damour, A.G & B. Iyer Phys. Rev. D 70, 064028 (2004) • R. M. Memmesheimer, A.G & G. Schäfer Phys. Rev. D 70, 104011 (2004) • C. Königsdörffer & A. G Phys. Rev. D 71, 024039 (2005)

  18. Publications: II • A. G & C. Königsdörffer Phys. Rev. D 72 (Rapid Communications), 121501 (2005) • C. Königsdörffer & A. G Phys. Rev. D, 73, 044011 (2006) • C. Königsdörffer & A. G (gr-qc/0603056) • M. Tessmer& A.G, to be published(2006)

  19. Credits • DFG’s SFB/TR7Gravitationswellenastronomie(activities ofSchäfer’s group in Jena) • Collaborations with researchers in France, Germany, UK & USA

  20. Theoretical inputs useful for GW detectors

  21. Ready-to-use templates • Highly accurate & efficient ready-to-use GW templates for compact binaries of arbitrary mass ratio moving in inspiralling eccentric orbits • We adapted an approach of Damour that gave the heavily employed timing formula for relativistic binary pulsarsT. Damour, A.G, C.Königsdörffer, B.R. Iyer,..

  22. Our templates • Plots of h+(t) showing 3 relevant time scales Orbital evolution is NOT adiabatic (fully 3.5PN accurate)

  23. Our templates • Quasi-periodic variations in orbital elements • We can handle arbitrary eccentricities

  24. Applications… LISA will require these templates to hear GWs from • Galactic Stellar mass binaries • Intermediate mass BH binaries • SupermassiveBH binaries

  25. Templates for EMRIs ? • Our hx& h+should be employed to detect early stages of Extreme Mass Ratio Inspirals • Spin effects will be included soon.. • Current kludges are less accurate & inefficient • To benchmark self force computations

  26. hx& h+ for LIGO-VIRGO ? : I • Compact binaries in inspiralling eccentric orbits are plausible sources of GWs even for LIGO & VIRGO Lots of short period highly eccentric IBCs should exist • Scenarios for Short Gamma ray BurstsDavies, Leavan & King (2005), Page et.al (2006) Grindlay, Zwart & McMillan, Nature (2006) • Chaurasia & Bailes scenario (2005)

  27. hx& h+ for LIGO-VIRGO ? : II • Kozail Oscillations associated with hierarchical triplets • Interplay between GW induced dissipation & stellar scattering in the vicinity of an IMBHHopman & Alexander (2005) • Realistic dense star clusters simulations Talk by R. Spurzem

  28. SKA related investigations

  29. CreatingLISA-SKAlink: I • LISA & SKA(Square Kilometre Array)may become operational by2015+ • LISA & SKA can be used to answerWAS EINSTEIN 100 % RIGHT ?

  30. CreatingLISA-SKAlink: II • LISA & SKA will see simultaneously BinaryPulsars (What can we learn ?) • GWs from such binary pulsars will not display the effect of radiation reaction • They may be in eccentric orbits(We are working with S.Bose, M. Kramer,..)

  31. Riddle for LISA-SKA • The terminal recoilassociated with BH merger may leave signatures that may be observed with LISA & SKA Terminal recoil estimatesDamour & A.G (2006)

  32. Summary • PN accurate dynamics of spinning compact binaries of arbitrary mass ratio moving in inspiralling eccentric orbits will be required by GW & SKA communities • There are more avenues to explore

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