EMRIs, Kicks & Tails from Black Hole Perturbation Theory (using - PowerPoint PPT Presentation

EMRIs, Kicks & Tails from Black Hole Perturbation Theory (using GPUs) Gaurav Khanna Associate Professor, UMass Dartmouth 7/29/15 AEI Golm 1

Collaborators & Support • Alessandra Buonanno, AEI Golm • Lior Burko, Georgia G. College • Scott Hughes, MIT • Richard Price, UTexas / UMassD • P. Sundararajan, MIT, Morgan-Stanley • Andrea Taracchini, AEI Golm • Anil Zenginoglu, UMaryland Funding: National Science Foundation 7/29/15 AEI Golm 2

Talk Outline • EMRIs using BH perturbation theory • Teukolsky equation solver with particle-source in the time-domain • Waveforms and kicks & anti-kicks • Scalar-field “tails” in BH spacetimes • Kerr BH “tails” controversy • Computing with gaming devices • OpenCL / GPU computing • Summary and Future Outlook 7/29/15 AEI Golm 3

Kerr (rotating) black hole perturbation theory • Write Einstein’s GR field equations to linear order expanding about a BH solution • Teukolsky equation -- a wave-equation like PDE that describes how generic fields (scalar, vector, tensor) in the space-time of a Kerr BH behave (propagate/evolve) • In the gravitational field context -- describes the behavior of GWs emitted from Kerr black holes • Relatively simple: linear, hyperbolic, (3+1)D PDE .. (can be reduced down to (2+1)D) 7/29/15 AEI Golm 4

Teukolsky equation Here -- a: spin of the Kerr black hole M: mass of the Kerr black hole s: spin-weight of the field considered (s is -2 for outgoing GWs) 7/29/15 AEI Golm 5

EMRI: Extreme-Mass Ratio (binary) Inspiral EMRI 7/29/15 AEI Golm 6

Point-like compact object .. • Source of the GWs (perturbation) in EMRI is the inspiraling compact object • How to model a point-like compact object (technically a Dirac-delta function) on a numerical (finite-difference) grid? • Obvious approach would be to take a narrow Gaussian distribution; do several runs with successively narrower profiles; take some sort of a limit .. • Decent results, but very expensive 7/29/15 AEI Golm 7

Discrete Dirac-delta • Alternative approach to representing Dirac- delta on numerical finite-difference grid • Particularly well suited for numerical computation (time-domain, finite-difference) • Work done in collaboration with Pranesh Sundararajan & Scott Hughes (MIT) • Excellent (highly accurate) results • Extremely efficient (by over an order-of- magnitude) 7/29/15 AEI Golm 8

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Teukolsky Equation 7/29/15 AEI Golm 10

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“source-term” T4 s4 = 1.D0/2.D0 s15 = -(r+cmplx(0.D0,1.D0)*a*ctheta)/(r**2+a**2*ctheta**2)*sqrt(2. s6 = 1/(r**2+a**2*ctheta**2) #D0)*(-a*stheta*nmu*(E*(a**2+rp**2)-a*lz)**2/dtdT/rp**6/pie**2/wr*e s9 = r**2+a**2 #xp(-(r-rp)**2/wr**2/2)/wt*exp(-ctheta**2/wt**2/2)*mm*dphidt*exp(cm s12 = (r+cmplx(0.D0,1.D0)*a*ctheta)/(r**2+a**2*ctheta**2)*sqrt(2.D #plx(0.D0,-1.D0)*mm*phip)/16+nmu*(E*(a**2+rp**2)-a*lz)**2/dtdT/rp** #0)*(a*stheta*nmu/rp**5*sqrt(2.D0)/dtdT*(E*(a**2+rp**2)-a*lz)*(a*E- #6/pie**2/wr*exp(-(r-rp)**2/wr**2/2)/wt**3*ctheta*stheta*exp(-cthet #lz)/pie**2/wr*exp(-(r-rp)**2/wr**2/2)/wt*exp(-ctheta**2/wt**2/2)*m #a**2/wt**2/2)*exp(cmplx(0.D0,-1.D0)*mm*phip)/16+1/stheta*nmu*(E*(a #m**2*dphidt**2*exp(cmplx(0.D0,-1.D0)*mm*phip)/16-nmu/rp**5*sqrt(2. #**2+rp**2)-a*lz)**2/dtdT/rp**6/pie**2/wr*exp(-(r-rp)**2/wr**2/2)/w #D0)/dtdT*(E*(a**2+rp**2)-a*lz)*(a*E-lz)/pie**2/wr*exp(-(r-rp)**2/w #t*exp(-ctheta**2/wt**2/2)*mm*exp(cmplx(0.D0,-1.D0)*mm*phip)/16)/2 #r**2/2)/wt**3*ctheta*stheta*exp(-ctheta**2/wt**2/2)*mm*dphidt*exp( s16 = -(cmplx(0.D0,-1.D0/2.D0)*a*(r+cmplx(0.D0,1.D0)*a*ctheta)/(r* #cmplx(0.D0,-1.D0)*mm*phip)/16-1/stheta*nmu/rp**5*sqrt(2.D0)/dtdT*( #*2+a**2*ctheta**2)**2*(r+cmplx(0.D0,-1.D0)*a*ctheta)*stheta*sqrt(2 #E*(a**2+rp**2)-a*lz)*(a*E-lz)/pie**2/wr*exp(-(r-rp)**2/wr**2/2)/wt #.D0)+cmplx(0.D0,1.D0)*a*(r+cmplx(0.D0,1.D0)*a*ctheta)**2/(r**2+a** #*exp(-ctheta**2/wt**2/2)*mm**2*dphidt*exp(cmplx(0.D0,-1.D0)*mm*phi #2*ctheta**2)**2*stheta*sqrt(2.D0))*nmu*(E*(a**2+rp**2)-a*lz)**2/dt #p)/16)/2 #dT/rp**6/pie**2/wr*exp(-(r-rp)**2/wr**2/2)/wt*exp(-ctheta**2/wt**2 s13 = cmplx(0.D0,1.D0/8.D0)*a*(r+cmplx(0.D0,1.D0)*a*ctheta)/(r**2+ #/2)*exp(cmplx(0.D0,-1.D0)*mm*phip)/16 #a**2*ctheta**2)**2*(r+cmplx(0.D0,-1.D0)*a*ctheta)*stheta*nmu/rp**5 s14 = s15+s16 #/dtdT*(E*(a**2+rp**2)-a*lz)*(a*E-lz)/pie**2/wr*exp(-(r-rp)**2/wr** s10 = s13+s14 #2/2)/wt*exp(-ctheta**2/wt**2/2)*mm*dphidt*exp(cmplx(0.D0,-1.D0)*mm s8 = s9*s10 #*phip)-(cmplx(0.D0,1.D0/2.D0)*a*(r+cmplx(0.D0,1.D0)*a*ctheta)**2/( s6 = s7*s8 #r**2+a**2*ctheta**2)**2*stheta*sqrt(2.D0)-(r+cmplx(0.D0,1.D0)*a*ct s8 = cmplx(0.D0,2.D0)*a*(r+cmplx(0.D0,1.D0)*a*ctheta)**2 #heta)/(r**2+a**2*ctheta**2)/tan(th)*sqrt(2.D0)/4)*nmu/rp**5*sqrt(2 s10 = 1/((r**2+a**2*ctheta**2)**2)*stheta #.D0)/dtdT*(E*(a**2+rp**2)-a*lz)*(a*E-lz)/pie**2/wr*exp(-(r-rp)**2/ s12 = sqrt(2.D0) #wr**2/2)/wt*exp(-ctheta**2/wt**2/2)*mm*dphidt*exp(cmplx(0.D0,-1.D0 s15 = 1/(r**2+a**2*ctheta**2)*(-(r**2+a**2)*nmu/rp**5*sqrt(2.D0)/d #)*mm*phip)/8 #tdT*(E*(a**2+rp**2)-a*lz)*(a*E-lz)/pie**2/wr*exp(-(r-rp)**2/wr**2/ s11 = s12+s13 #2)/wt*exp(-ctheta**2/wt**2/2)*mm*dphidt*exp(cmplx(0.D0,-1.D0)*mm*p s12 = s11-1/(r**2+a**2*ctheta**2)*((r**2+a**2)*nmu/rp**4/dtdT*(a*E #hip)/16+cmplx(0.D0,-1.D0/16.D0)*(r**2+a**2-2*M*r)*nmu/rp**5*sqrt(2 #-lz)**2/pie**2/wr*exp(-(r-rp)**2/wr**2/2)/wt*exp(-ctheta**2/wt**2/ #.D0)/dtdT*(E*(a**2+rp**2)-a*lz)*(a*E-lz)/pie**2/wr**3*(r-rp)*exp(- #2)*mm**2*dphidt**2*exp(cmplx(0.D0,-1.D0)*mm*phip)/8+cmplx(0.D0,1.D #(r-rp)**2/wr**2/2)/wt*exp(-ctheta**2/wt**2/2)*exp(cmplx(0.D0,-1.D0 #0/8.D0)*(r**2+a**2-2*M*r)*nmu/rp**4/dtdT*(a*E-lz)**2/pie**2/wr**3* #)*mm*phip)+a*nmu/rp**5*sqrt(2.D0)/dtdT*(E*(a**2+rp**2)-a*lz)*(a*E- #(r-rp)*exp(-(r-rp)**2/wr**2/2)/wt*exp(-ctheta**2/wt**2/2)*mm*dphid #lz)/pie**2/wr*exp(-(r-rp)**2/wr**2/2)/wt*exp(-ctheta**2/wt**2/2)*m #t*exp(cmplx(0.D0,-1.D0)*mm*phip)-a*nmu/rp**4/dtdT*(a*E-lz)**2/pie* #m*exp(cmplx(0.D0,-1.D0)*mm*phip)/16)/2 #*2/wr*exp(-(r-rp)**2/wr**2/2)/wt*exp(-ctheta**2/wt**2/2)*mm**2*dph s16 = cmplx(0.D0,-1.D0/8.D0)*(-(r+cmplx(0.D0,1.D0)*a*ctheta)**2/(r #idt*exp(cmplx(0.D0,-1.D0)*mm*phip)/8)/2 #**2+a**2*ctheta**2)**3*(r+cmplx(0.D0,-1.D0)*a*ctheta)*(r**2+a**2-2 s13 = s12+cmplx(0.D0,-1.D0/4.D0)*(-(r+cmplx(0.D0,1.D0)*a*ctheta)** #*M*r)/2+(r+cmplx(0.D0,1.D0)*a*ctheta)/(r**2+a**2*ctheta**2)**2*(r+ #2/(r**2+a**2*ctheta**2)**3*(r+cmplx(0.D0,-1.D0)*a*ctheta)*(r**2+a* #cmplx(0.D0,-1.D0)*a*ctheta)*(r-M)/2)*nmu/rp**5*sqrt(2.D0)/dtdT*(E* #*2-2*M*r)/2+(r+cmplx(0.D0,1.D0)*a*ctheta)/(r**2+a**2*ctheta**2)**2 #(a**2+rp**2)-a*lz)*(a*E-lz)/pie**2/wr*exp(-(r-rp)**2/wr**2/2)/wt*e #*(r+cmplx(0.D0,-1.D0)*a*ctheta)*(r-M)/2)*nmu/rp**4/dtdT*(a*E-lz)** #xp(-ctheta**2/wt**2/2)*exp(cmplx(0.D0,-1.D0)*mm*phip) #2/pie**2/wr*exp(-(r-rp)**2/wr**2/2)/wt*exp(-ctheta**2/wt**2/2)*mm* s14 = s15+s16 #dphidt*exp(cmplx(0.D0,-1.D0)*mm*phip) 7/29/15 AEI Golm 12