The effect of f - modes on the gravitational waves during a binary inspiral

Transcription

1 The effect of f - modes on the gravitational waves during a binary inspiral Tanja Hinderer (AEI Potsdam) PRL 116, (2016), arxiv: and arxiv: ? A. Taracchini F. Foucart K. Hotokezaka A. Buonanno M. Duez K. Kyutoku J. Steinhoff L. E. Kidder M. Shibata H. P. Pfeiffer M. A. Scheel B. Szilagyi C. W. Carpenter Southampton Sept. 14, 2016

2 1 Overview Dynamical f-mode tides in General Relativity Inclusion in the Effective-One-Body (EOB) model Tests against results from Numerical Relativity simulations Outloook

3 2 Newtonian tidal excitation of f-modes NS companion s tidal field induces a fluid displacement ξ mc companion ξ is often decomposed into spherical harmonics (equivalent alternative: STF tensors) oscillator-like Lagrangian for the mode amplitudes Effect on GWs: NS-NS black hole binary tidal imprint in the phase evolution for models: convenient to work directly with the NS s induced multipole moments Qij,Qijk, (differ from mode amplitudes by a normalization factor)

4 3 Newtonian tidal excitation of f-modes NS s induced multipole moments Qij,Qijk, NS tidal Lagrangian (quadrupole only) mc L = 1 4! 2 f apple dqij dt dq ij dt! 2 f Q ij Q ij 1 2 E ijq ij + companion tidal deformability ` = 2(` 2) 2`+1 k`r (2` 1)!! f-mode frequency tidal field E ij j mc r contributions from other l=2 modes Love number NS radius for two NSs: simply add the same terms with the parameters of the other NS

5 4 Adiabatic limit f-mode frequency scales as! f p m NS /R 3 tidal forcing frequency: 2 2 p M/r 3 for 2Ω ω f : adiabatic tides (AT) Q AT ij = E ij adiabatic tidal Lagrangian: L AT = 4 E ij E ij involves only orbital variables & tidal deformability coefficient λ

6 5 Tidal deformability characterizes dominant EoS influence on GWs during inspiral log (pressure - density) λ- mass Einstein s Eqs: linear perturbations to equilibrium sol. [One 2nd order ODE] credit: B. Lackey

7 6 NS s tidal response during the inspiral dyn. tides end of the inspiral ` =2 ` =3 adiab. tides k 2 k 3 Γ = 2 polytropic EoS, mns=1.4m ` = 2(` 2) 2`+1 k`r (2` 1)!! Love number NS radius

8 6 Tidal excitation of f-modes in GR NS s worldline x ( ) u µ m, S µ, Q µ E µ = C µ recall Newtonian tidal Lagrangian: L = 1 4! 2 f apple dqij dt dq ij dt relativistic version: L DT = L z 4 z 2! 2 f redshift apple DQµ D d d! 2 f Q ij Q ij 1 2 E ijq ij DQ µ d z 2! 2 f Q µ Q µ other l=2 modes & other stuff e.g. due to incompleteness of modes z 2 E µ Q µ = covariant derivative along worldline u u z 2, C µ : Weyl curvature tensor (companion) +

9 7 Tidal excitation of f-modes in GR relativistic tidal Lagrangian: u µ L L DT = NS s worldline x ( ) m, S µ, Q µ redshift z 4 z 2! 2 f apple DQµ d DQ µ d z 2! 2 f Q µ Q µ z 2 E µ Q µ NS 3+1 decomposition to make explicit the physical degrees of freedom: frame-dragging interaction between orbital and Qij s angular momentum characterized by the tidal spin tensor: S ij Q = 2! 2 f Q k[i Q j]k

10 8 Explicit expressions in the weak field limit post-newtonian (PN) results for tidal effects without any new PN calculations: (i) tidal interaction terms vs. known PN results for spin-induced quadrupoles 1 2 E µ Q µ simply substitute C ES 2S j S i! mq ij C ES 2 2m E µ S µ S (ii) frame-dragging interaction betwen orbital angular momentum and tidal spin simply replace spin by SQ in known PN frame-dragging potentials the resulting 1PN accurate tidal Lagrangian agrees with Vines & Flanagan 2013

11 9 Approaches to computing waveforms orbital separation r/m Newtonian dynamics post-newtonian theory path to merger Numerical relativity gravitational self-force test particle limit LIGO band mass ratio M/μ

12 10 Approaches to computing waveforms orbital separation r/m Newtonian dynamics post-newtonian theory Effective path One-Body (EOB) model: test combines all to information into a complete waveform particle model merger for LIGO searches limit Numerical relativity black hole perturbation theory [Buonanno, Damour 1999, 2000] LIGO band mass ratio M/μ

13 11 Effective-One-Body (EOB) approach Binary problem Effective description MAP effective particle lengthy PN description ν=µ/m effective spacetime ds 2 e = A(M,,r)dt2 + B(M,,r)dr 2 + r 2 d 2 Hamiltonian for the dynamics: s H EOB (r, p r,p ; M, ) =M 1+2 radiation reaction forces factorized waveforms He µ 1 A =1 2M r + APN (r; M, ) effective Hamiltonian Heff

14 12 Performance of EOB waveforms Calibrated numerical relativity m1=m2, S1=S2=0.98 Smax EOB no tuning GW cycles tuned Calibrated recent extension to precessing spins GW cycles [Babak, Taracchini+ 2016] [courtesy A. Taracchini]

15 Adiabatic tides in the EOB Hamiltonian MAP μ ds 2 e = Adt2 + Bdr 2 + r 2 d 2 A = A pp (M,,r) `A AT (M,,r) different possibilities for A AT : [Damour, Nagar, Bini, Faye, Bernuzzi ] post-newtonian: A AT PN = 3q r 6 apple 1+ p 1( ) r + p 2( ) r 2 + O 1 r 3 q=mc/mns self-force (GSF): A AT GSF ( = 3q r 6 " 1+ r r LR r + q 1 g 1 (r) r LR r 7/2 + O 1 q 2 # rlr=light ring tuned GSF: A AT tgsf ( = 3q r 6 " 1+ r r LR r + q 1 g 1 (r) r LR r 7/2 + p 00 2 ( )/2 q 2 1 r LR r p # adjustable: 4 p 6 13

16 14 EOB Hamiltonian with dynamic tides full evolutions: H EOB (r, p r,p, Q`m, P`m ; M,, `,! f ) conjugate momentum to Qlm tidal modifications to the effective spacetime, the mass of the effective particle, etc. derived from PN results for the Lagrangian no additional approximations many extra variables (24 if up to hexadecapole tides are included) no problems in evolving the dynamics but inefficient for data analysis implementations, no direct control of the effects

17 15 EOB with approximate dynamic tides effective description from an approximate solution for Qlm: A = A pp (M,,r) e ` AAT PN (M,,r) φ e ` `! 2 f! 2 f (m ) 2 &!2 f ( f ) & cos ( f ) 2 FresnelS( f ) k 2 before resonance common term near resonance where φ ~ φf 0 k all fns. of {M,,! f,r} using a Newtonian inspiral requires evolving only the orbital variables tidal effects encoded in λ eff

18 Performance of the tidal model: NS-BH Re(D L h 22 /M) NS-BH q=2 EOB NR (t r * )/M NR data from Francois Foucart, SXS collaboration 0.5 tuned GSF (p=6) φ 22 [rad] NR error BH-BH A AT PN A AT GSF dynamic tides tuned GSF (p=4) -1.5 NS-BH q=2, CNS= GW cycles NS disrupts 16

19 17 Performance of the tidal model: NS-NS Re(D L h 22 /M) φ 22 (rads) NSNS H4 ad. tides 2PN ad. tides Bernuzzi+ ad. tides GSF dyn. tides A AT GSF NR data from M. Shibata and K. Hotokezaka A AT PN (t r * )/M dynamic tides tuned GSF (p=4) merger (peak in h22 )

20 18 Conclusions Main imprint of NS microphysics in the GWs from NS inspirals: tidal effects Dynamic f-mode tides can be significant, now included in EOB Further work needed to test/improve the models and measurement potential quantify and reduce systematics include more realistic physics optimize data analysis strategies (e.g. parameterization) Accurate NR simulations are crucial to inform model developments

21 Thank you