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Exploring Gravitational Waves and Self-Force: m-mode Regularization in EMRIs

This talk presented by Sam Dolan and Leor Barack from the University of Southampton delves into the intriguing realm of gravitational waves, particularly focusing on Extreme Mass Ratio Inspirals (EMRIs). It covers the motivation behind studying gravitational waves, introduces the concept of self-force, and discusses the m-mode regularization scheme. By implementing models in Schwarzschild spacetime and analyzing results, the presentation identifies future directions for research in gravitational wave physics. This work is vital for understanding the dynamics of EMRIs, with implications for gravitational wave detection.

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Exploring Gravitational Waves and Self-Force: m-mode Regularization in EMRIs

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Presentation Transcript

  1. BriXGrav, Dublin 2010 Self-Force and the m-mode regularization scheme Sam Dolan (with Leor Barack) University of Southampton

  2. TALK OVERVIEW • Motivation: Gravitational Waves and EMRIs • Introduction to Self-Force • The m-mode scheme • Implementation: scalar charge, circular orbit, Schwarzschild spacetime • Results + Future Directions

  3. 1. MOTIVATION Gravitational Waves and Extreme Mass Ratio Inspirals

  4. Evidence for Gravitational Waves:The Hulse-Taylor Binary Pulsar • Doppler shift in pulsar period (59ms) • => Orbital period (7.75hr) • Cumulative shift of 40 sec over 30 yr • Energy loss due to GW emission • => Inspiral Figure from Weisberg & Taylor (2004)

  5. Electromagnetic Waves Gravitational Waves • Coherent • Bulk dynamics • Amplitude: h ~ 1 / r • Not scattered or absorbed • λ > diameter • “Hearing” • All-sky, stereo • Incoherent, 1023 emitters • Thermodynamic state • Intensity: I ~ 1 / r2 • Scattered and absorbed • λ << diameter • Imaging, focussing • Narrow field vs

  6. Sources of Gravitational Waves • Binary neutron star mergers, ~ 1.5 Ms • Stellar mass black hole mergers, ~ 10 Ms • Supermassive black hole mergers, ~ 107 Ms • Extreme Mass Ratio Inspirals (EMRI), ~ 10 Ms and ~ 106 Ms

  7. EMRI dynamics and LISA • LISA will see ~10to 1000s inspirals simultaneously • Scattering of NS or BH into highly eccentric orbits • Eccentric until plunge • ~105 wave cycles over final year • Waveforms => Physics (e.g. “map” of near-horizon geometry, no-hair theorem, parameter estimation) t=-106 yr t=-1yr t=0

  8. EMRI Timescales • e.g. M ~ 106 Ms and m ~ 10 Ms • To track orbital phase over Tobs, need high accuracy:

  9. log(r/M) Post-Newtonian Theory 4 3 2 Numerical Relativity Perturbation Theory 1 log(M/m) 1 2 3 4 Two-Body Problem in GR: regimes

  10. 2. SELF-FORCE Radiation Reaction in Curved Spacetime

  11. Radiation Reaction in Classical Electromagnetism • Accelerated charge => radiation • Loss of energy => force acting on charge • Self-Force: charge interacts with own field • Point charge  infinite field … interpretation? • Regularization method needed

  12. Dirac’s approach: • “S” is infinite but symmetric on worldline => mass renormalisation • “R” is regular on worldline => self-force

  13. EM Self-Force in Flat Spacetime

  14. Flat Curved Self-Force in Curved Spacetime: Problem of Regularization In flat spacetime, Green’s function has support on light-cone only In curved spacetime, Green’s function also has a tail within the lightcone Difficulty: Local Radiative potential becomes non-causal in curved space!

  15. EM Self-Force in Curved Spacetime Local “instantaneous” terms tail integral over past history of motion DeWitt & Brehme (1960)

  16. Self-Force Derivations E.M.: DeWitt & Brehme (1960) Gravitational: Mino, Sasaki & Tanaka (1997) Scalar: Quinn (2000) Example: Matched Asymptotic Expansions Near zone: Far zone: Match in buffer zone M >> r >> m, to obtain equation of motion

  17. 3. SF CALCULATIONS: m-mode regularization in 2+1D

  18. l = 1 l = 2 l = 3 l = 0 + + + + … + + + + … m = 0 m = 1 m = 2 m = 3 Two Mode-Sum Methods • “l-mode” regularization (spher. sym, e.g. Schw.) Decompose field into spherical harmonics, then regularize mode sum over l • “m-mode” regularization (axisymmetric, e.g. Kerr) Decompose field into exp(imϕ), introduce puncture field, then sum over modes.

  19. m-mode decomposition • Kerr perturbation not separable in 1+1D • 2+1D decomposition • Evolve in time domain using finite difference scheme. • Problem: each m-mode diverges logarithmically at particle position • Resolution: analytically expand divergence; introduce a puncture function, leaving a regular residual.

  20. Puncture Scheme • Idea: The divergence at the particle has a simple logarithmic form; subtract it out and evolve the regular residual field delta-function source on particle worldline Extended source, without distributional component regular on worldline

  21. Puncture Function Construction (Barack, Golbourn & Sago 2007) Puncture field: Choose such that: where Σ: t = t0 r=r0 worldline

  22. Puncture Function (II) • Some freedom in choice of • We choose • where • m-mode decomposition: Functions of orbital parameters only Elliptic Integrals

  23. Puncture and World-tube • Construct world-tube around particle • Inside the tube, solve for • with extended source • Outside the tube, solve vacuum eqns for • The self-force is found from derivative of residual field at particle position: world-tube t θ world-line r mode sum converges

  24. First Implementation: Scalar Field, Circular Orbits, Schwarzschild • 2+1D time evolution on 2+1 grid: u = t + r*, v = t - r*, θ • 2nd order accurate finite difference scheme

  25. ImplementationDetails • Causal grid, arbitrary i.c. • Boundary condition at poles • No radial boundary condition • High-res runs on Iridis3 HPC (in Southampton). • Stability constraint => max. angular resolution • Resolution x2 => runtime x8, memory x4 u v

  26. 4. FIRST RESULTS

  27. 1. Puncture Field Evolution m=0 m=1 m=2 r*/M q (Barack & Golbourn 2007)

  28. 2. Extrapolate To Infinite Resolution • Extrapolate from results of simulations at range of resolutions: where h = grid spacing.

  29. 3. Sum over m-modes Fit tail: Monopole and dipole are negative

  30. Scheme of Work: • Scalar SF, circular orbits, Schw. • Scalar SF, circular orbits, Kerr (equatorial). • Gravitational SF, circ. orbits, Kerr. • Eccentric orbits (elliptic orbits, zoom-whirl) References: m-mode regularization • L Barack & D Golbourn, PRD 76 (2007) 044020 [arXiv:0705.3620]. • L Barack, D Golbourn &N Sago, PRD 76 (2007) 124026 [arXiv:0709.4588]. • S Dolan & L Barack, (2010) in progress.

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