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

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

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

Introduction

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

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

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 & half PN order : corrections up to (v/c) 7 ► 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 & half PN order : corrections up to (v/c) 7 ► L. Blanchet, T. Damour, G. Schäfer (Esposito-Farese, Faye, Jaranowski, Iyer,..)

Introduction: IV ► H ( r, p)= H( r, p)+H 1 ( r, p) +H 2 ( r, p)+H 2.5 ( r, p)+H 3 ( r, p)+H 3.5 ( r, p) ► H ( r, p) 3.5PN = H 0 ( r, p)+H 1 ( r, p) +H 2 ( r, p)+H 2.5 ( r, p)+H 3 ( r, p)+H 3.5 ( r, p) [For the near zone orbital dynamics ] [For the near zone orbital dynamics ] ► Similar expansions for h(t), h + (t), ω(t) [ far zone measurable quantities ] ► Similar expansions for h x (t), h + (t), ω(t) [ far zone measurable quantities ] ► Crucial to the templates for ICBs in quasi circular orbits

Parametric solution: I ► 3PN accurate conservative dynamics allows Keplerian type parametric solution ► H ( r, p)= H( r, p)+H 1 ( r, p) +H 2 ( r, p)+ H 3 ( r, p)+H so ( r, p,s 1,s 2 ) ► H ( r, p) PN = H 0 ( r, p)+H 1 ( r, p) +H 2 ( r, p)+ H 3 ( r, p)+H so ( r, p,s 1,s 2 ) ► Such solution exists even when compact objects spin

Keplerian parametrization :I ► Dynamics of point masses in Newtonian gravity allows Keplerian parametrization Radial motion ► R = a ( 1 - e cos u ) Angular motion ► φ – φ 0 = v = 2 tan -1 [ ( [ 1 + e ]/ [ 1 - e ] ) (1/2) tan u/2] Kepler Equation ► l = n ( t - t 0 ) = u - e sin u ► l = n ( t - t 0 ) = u - e sin u

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

Parametric solution: II ► Radial motion r = a r ( 1 - e r cos u ) ► Angular motion φ – φ 0 = (1 + k ) v + ( f 4 φ + f 6φ ) 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 + e φ ]/ [ 1 - e φ ] ) (1/2) tan u/2 ] v= 2 tan -1 [ ( [ 1 + e φ ]/ [ 1 - e φ ] ) (1/2) tan u/2 ]

Parametric solution: III ► 3PN accurate Kepler Equation l = n ( t - t 0 ) = u - e t sin u+ ( g 4t + g 6t ) (v - u) + ( f 4t + f 6t ) sin v + i 6t sin 2 v+ h 6t sin 3 v ► n & k are gauge invariant quantities if expressed in terms of E, J, m 1 & m 2

Parametric solution: IV  Complicated parametric solution with H so ( r, p,s 1,s 2 ) Details in papers from Jena in 2004 & 2005  There are 3 PN accurate (related) eccentricities e r, e t, e φ  Orbital elements & functions are PN accurate expressions in E, J, m 1 & m 2

Parametric solution: V ► Our efforts extends seminal works of Damour, Deruelle, Schäfer & Wex ► 1PN accurate Keplerian-type parametric solution is crucial for Damour-Deruelle timing formula Damour-Deruelle timing formula

LOTS OF APPLICATIONS Relevant for GW interferometrs & SKA

Publications: I ► T. Damour, A.G & B. Iyer Phys. Rev. D 70, (2004) ► R. M. Memmesheimer, A.G & G. Schäfer Phys. Rev. D 70, (2004) Phys. Rev. D 70, (2004)  C. Königsdörffer & A. G Phys. Rev. D 71, (2005 )

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

Credits ► DFG ’s SFB/TR7 Gravitationswellenastronomie (activities of Schäfer’s group in Jena) ► DFG ’s SFB/TR7 Gravitationswellenastronomie (activities of Schäfer’s group in Jena) ► Collaborations with researchers in France, Germany, UK & USA

Theoretical inputs useful for GW detectors

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 pulsars T. Damour, A.G, C.Königsdörffer, B.R. Iyer,..

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

Our templates Quasi-periodic variations in orbital elements ► We can handle arbitrary eccentricities

Applications… LISA will require these templates to hear GWs from LISA will require these templates to hear GWs from Galactic Stellar mass binaries Galactic Stellar mass binaries Intermediate mass BH binaries Intermediate mass BH binaries Supermassive BH binaries Supermassive BH binaries

Templates for EMRIs ?  Our h x & 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

h x & h + h x & 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 Bursts Davies, Leavan & King (2005), Page et.al (2006) Grindlay, Zwart & McMillan, Nature (2006) Grindlay, Zwart & McMillan, Nature (2006)  Chaurasia & Bailes scenario (2005)

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

SKA related investigations

Creating LISA-SKA link: I ► LISA & SKA (Square Kilometre Array) may become operational by  LISA & SKA can be used to answer  LISA & SKA can be used to answer WAS EINSTEIN 100 % RIGHT ?

Creating LISA-SKA link: II ► LISA & SKA will see simultaneously Binary Pulsars ( 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,..)

Riddle for LISA-SKA  The terminal recoil associated with BH merger may leave signatures that may be observed with LISA & SKA Terminal recoil estimates Damour & A.G (2006)

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