Keplerian-type parametrization : Its relevance for LISA &

SKA

Achamveedu Gopakumar

TPI, FSU-Jena, GermanyTPI, FSU-Jena, Germany

BirminghamBirmingham: : 30/3/06 30/3/06

The point to realizeThe point to realize

Post-NewtonianPost-Newtonian accurate accurate dynamicsdynamics of compact binaries of compact binaries along with the associated along with the associated Keplerian-type parametric Keplerian-type parametric solutionsolution will be required to will be required to

realize realize science potential of of Interferometric Interferometric GW DetectorsGW Detectors & & SKASKA

Plan Plan

► Symbolic introduction to Symbolic introduction to post-Newtonian post-Newtonian (PN) (PN) dynamicsdynamics, , relevant for compact binariesrelevant for compact binaries

► Efforts from Efforts from JenaJena relevant for relevant for VIRGOVIRGO,, A-LIGO A-LIGO, , LISA,…LISA,…

► Theoretical efforts useful for Theoretical efforts useful for SKASKA

IntroductionIntroduction

Introduction: IIntroduction: I

► Three stages for the dynamics of compactThree stages for the dynamics of compact binaries ( binaries (NS-NSNS-NS, , NS-BHNS-BH, , BH-BHBH-BH))

Early inspiralEarly inspiral

Late stages of inspiralLate stages of inspiral & & subsequent subsequent plungeplunge

Ring-down Ring-down

Introduction: IIIntroduction: II

► Till the early stages of late inspiral, Till the early stages of late inspiral, post-Newtonian (post-Newtonian (PNPN) approximation ) approximation holds goodholds good

►PNPN approximation gives corrections to approximation gives corrections to Newtonian gravitational theoryNewtonian gravitational theoryin terms of in terms of (v/c)(v/c)22 ~ ~ (G m /c(G m /c2 2 r)r)

Compact binaries are treated like Compact binaries are treated like point-massespoint-masses

Introduction : IIIIntroduction : III

► The dynamics of compact binaries in PN The dynamics of compact binaries in PN approximation is recently determined, both in approximation is recently determined, both in near-zone orbital dynamicsnear-zone orbital dynamics and in and in far-zone far-zone flux computationsflux computations, to , to third & halfthird & halfPN orderPN order : corrections up to : corrections up to (v/c)(v/c)77

► L. Blanchet, L. Blanchet, T. DamourT. Damour, G. Schäfer, G. Schäfer

(Esposito-Farese, Faye, Jaranowski, (Esposito-Farese, Faye, Jaranowski, Iyer, ..)Iyer, ..)

Introduction: IVIntroduction: IV

► H ( H ( rr, , pp))3.5PN= H= H0( ( r, pr, p)+H)+H11( ( r, pr, p))

+H+H22( ( r, pr, p)+)+HH2.52.5( ( r, pr, p)+)+HH33( ( r, pr, p)+)+HH3.53.5( ( r, pr, p))

[For the [For the near zone orbital dynamicsnear zone orbital dynamics]]

► Similar expansions for Similar expansions for hhx(t), h(t), h++(t), (t), ωω(t)(t) [ [ farfar zone zone measurable quantitiesmeasurable quantities]]

► Crucial to the Crucial to the templates for ICBs in quasi templates for ICBs in quasi circular orbitscircular orbits

Parametric solution: IParametric solution: I

► 3PN accurate conservative dynamics 3PN accurate conservative dynamics allows allows Keplerian typeKeplerian type parametric solution parametric solution

► H ( H ( rr, , pp)) PN= H= H0( ( r, pr, p)+H)+H11( ( r, pr, p))

+H+H22( ( r, pr, p)+)+ HH33( ( r, pr, p)+)+HHsoso( ( r, p,sr, p,s11,s,s22))

► Such solution exists even when Such solution exists even when compact objects compact objects spin spin

Keplerian parametrization :IKeplerian parametrization :I► Dynamics of point masses in Newtonian Dynamics of point masses in Newtonian

gravity allows Keplerian parametrization gravity allows Keplerian parametrization

Radial motionRadial motion► R = a ( 1 - e cos u )R = a ( 1 - e cos u )

Angular motionAngular motion

► φφ – – φφ00 = v = v = 2 tan= 2 tan-1-1 [ ( [ ( [ 1 + e[ 1 + e ]/ [ 1 - e ]]/ [ 1 - e ] ) )(1/2)(1/2) tan u/2] tan u/2]

Kepler EquationKepler Equation► l = l = nn ( t - t ( t - t00 ) = u - e sin u ) = u - e sin u

Keplerian parametrization :IIKeplerian parametrization :II

► uu & & vv have have geometrical meaninggeometrical meaning

► 3PN 3PN accurate accurate conservative conservative dynamics of point dynamics of point masses also allows masses also allows Keplerian typeKeplerian type parametric solution parametric solution

Parametric solution: IIParametric solution: II

► Radial motionRadial motion r = ar = arr ( 1 - e ( 1 - err cos u ) cos u )

► Angular motionAngular motion φφ – – φφ00 = (1 + = (1 + k k ) v + ( ) v + ( f f 4 4 φφ + + ff66φφ) sin 2 v ) sin 2 v + ( + ( g g 4 4 φφ + + g g 6 6 φφ ) sin 3 v ) sin 3 v + + i i 6 6 φφ sin 4 v+ sin 4 v+ h h 6 6 φφ sin 5 v sin 5 v

v= 2 tanv= 2 tan-1-1 [ ( [ ( [ 1 + e[ 1 + e φφ]/ [ 1 - e ]/ [ 1 - e φφ]] ) )(1/2)(1/2) tan u/2 tan u/2]]

Parametric solution: IIIParametric solution: III

► 3PN accurate 3PN accurate Kepler EquationKepler Equation

l = l = nn ( t - t ( t - t00 ) = u - e ) = u - et t sin u+ sin u+ ( ( gg4t4t + + gg6t6t ) (v - u) + ( ) (v - u) + ( ff4t4t+ + ff6t6t ) sin v ) sin v + + ii6t6t sin 2 v+ sin 2 v+ hh6t6t sin 3 v sin 3 v

► n & kn & k are gauge invariant quantities are gauge invariant quantities if expressed in terms of if expressed in terms of E, JE, J, , m m 11 & & m m 22

Parametric solution: IVParametric solution: IV

Complicated parametric solution Complicated parametric solution with with HHsoso( ( r, p,sr, p,s11,s,s22)) Details in papers from Details in papers from JenaJena in in 2004 2004 & & 20052005

There are 3 PN accurate (related) There are 3 PN accurate (related)

eccentricities e eccentricities e r , r , ee t , t , ee φφ

Orbital elements & functions are PN Orbital elements & functions are PN accurate expressions in accurate expressions in E, JE, J, , m m 11 & & m m 22

Parametric solution: VParametric solution: V

► Our efforts extends seminal works ofOur efforts extends seminal works of DamourDamour, Deruelle, Schäfer & Wex , Deruelle, Schäfer & Wex

► 1PN accurate Keplerian-type 1PN accurate Keplerian-type parametric solution is crucial for parametric solution is crucial for

Damour-DeruelleDamour-Deruelle timing formulatiming formula

LOTS OF APPLICATIONSLOTS OF APPLICATIONS

Relevant for Relevant for GW interferometrsGW interferometrs & & SKASKA

Publications: I► T. Damour, T. Damour, A.GA.G & B. Iyer & B. Iyer

Phys. Rev. D 70, 064028 ( Phys. Rev. D 70, 064028 (20042004))

► R. M. Memmesheimer, R. M. Memmesheimer, A.GA.G & G. Schäfer & G. Schäfer

Phys. Rev. D 70, 104011 (Phys. Rev. D 70, 104011 (20042004))

C. Königsdörffer & C. Königsdörffer & A. GA. G Phys. Rev. D 71, 024039 ( Phys. Rev. D 71, 024039 (20052005))

Publications: II► A. GA. G & C. Königsdörffer & C. Königsdörffer Phys. Rev. D 72 (Rapid Communications), 121501 Phys. Rev. D 72 (Rapid Communications), 121501

((20052005))

► C. Königsdörffer & C. Königsdörffer & A. GA. G Phys. Rev. D, 73, 044011 (Phys. Rev. D, 73, 044011 (20062006))

C. Königsdörffer & C. Königsdörffer & A. G (A. G (gr-qc/0603056gr-qc/0603056))

M. TessmerM. Tessmer && A.G, A.G, to be publishedto be published (2006)(2006)

Credits►

DFGDFG’s ’s SFB/TR7SFB/TR7 GravitationswellenastronomieGravitationswellenastronomie ((activities ofactivities of Schäfer’sSchäfer’s group in group in JenaJena))

► Collaborations with Collaborations with researchersresearchers in in FranceFrance, , GermanyGermany,, UK UK & & USAUSA

Theoretical inputs useful Theoretical inputs useful for GW detectorsfor GW detectors

Ready-to-use templates

► Highly accurate & efficient Highly accurate & efficient ready-to-use ready-to-use GW GW templatestemplates for compact binaries of arbitrary mass for compact binaries of arbitrary mass ratio moving in inspiralling eccentric orbitsratio moving in inspiralling eccentric orbits

► We adapted an approach of We adapted an approach of DamourDamour that gave that gave the heavily employed the heavily employed timing formulatiming formula for for relativistic relativistic binary pulsarsbinary pulsars

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

Our templates

Plots of hPlots of h++(t) showing 3 relevant time scales

Orbital evolution is NOT adiabatic (fully 3.5PN accurate)

Our templates

Quasi-periodic Quasi-periodic variations in orbital elementsvariations in orbital elements

► We can handle We can handle arbitrary eccentricitiesarbitrary eccentricities

Applications…Applications…

LISA LISA will require these templates to will require these templates to hearhear GWs from GWs from

Galactic StellarGalactic Stellar mass binaries mass binaries

Intermediate massIntermediate mass BH binaries BH binaries

SupermassiveSupermassive BH binariesBH binaries

Templates for EMRIs ?

Our Our hhxx && h h++

should be employed to detect early stages of should be employed to detect early stages of

Extreme Mass Ratio InspiralsExtreme Mass Ratio Inspirals

Spin effectsSpin effects will be included soon.. will be included soon..

Current Current kludgeskludges are less accurate & are less accurate & inefficientinefficient

To benchmark To benchmark self forceself force computations computations

hhxx && h h++ for LIGO-VIRGO ? : I

► Compact binaries in inspiralling eccentricCompact binaries in inspiralling eccentric orbits are orbits are plausible sourcesplausible sources of GWs even of GWs even for for LIGO & LIGO & VIRGOVIRGO

Lots of short period highly eccentric IBCs should Lots of short period highly eccentric IBCs should existexist

Scenarios for Scenarios for Short Gamma ray BurstsShort Gamma ray Bursts Davies, Leavan & King (Davies, Leavan & King (20052005), Page ), Page et.alet.al ( (20062006))

Grindlay, Zwart & McMillan, Grindlay, Zwart & McMillan, NatureNature ( (20062006))

Chaurasia & BailesChaurasia & Bailes scenario ( scenario (20052005))

hhxx && h h++ for LIGO-VIRGO ? : II

Kozail Oscillations associated with Kozail Oscillations associated with hierarchical triplets hierarchical triplets

Interplay between GW induced Interplay between GW induced

dissipationdissipation & stellar scattering in the vicinity of an & stellar scattering in the vicinity of an IMBHIMBH Hopman & Alexander (Hopman & Alexander (20052005))

Realistic Realistic dense star clusters dense star clusters simulations simulations Talk by Talk by R. SpurzemR. Spurzem

SKA related investigations

Creating LISA-SKA link: I

►LISA LISA && SKA SKA (Square Kilometre Array)(Square Kilometre Array) may become operational bymay become operational by 2015+2015+

LISA LISA && SKA SKA can be used to answercan be used to answer

WAS EINSTEIN 100 % RIGHT ?

Creating LISA-SKA link: II

►LISA LISA && SKA SKA will will see simultaneouslysee simultaneously Binary BinaryPulsars (Pulsars (What can we learn ?)What can we learn ?)

► GWs from such binary pulsars will not GWs from such binary pulsars will not displaydisplay the effect of radiation reaction the effect of radiation reaction

► They may be in eccentric orbitsThey may be in eccentric orbits

(We are working with (We are working with S.Bose, S.Bose, M. M. Kramer,..Kramer,..))

Riddle for LISA-SKA

The The terminal recoilterminal recoil associated with BH associated with BH merger may leave merger may leave signaturessignatures that may that may be be observed with observed with LISALISA & & SKASKA

Terminal recoil Terminal recoil estimatesestimates Damour & A.G (Damour & A.G (20062006))

SummarySummary

PN accurate dynamics of PN accurate dynamics of spinning spinning compact binariescompact binaries of arbitrary mass of arbitrary mass ratio ratio moving moving in inspiralling eccentric orbits in inspiralling eccentric orbits will be will be required by required by GW GW && SKA SKA communities communities

There are more avenues to There are more avenues to explore explore