This is one of a series of notes taken during the 2015 meeting of the AAS Division on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.

Session: New Approaches to Classical Dynamical Problems I

Marc Favata (Montclair State University) (invited)

Abstract

Solving the relativistic two-body problem is difficult. Motivated by the construction, operation, and recent upgrades of interferometric gravitational-wave detectors, significant progress on this problem has been achieved over the past two decades. I will provide a summary of techniques that have been developed to solve the relativistic two-body problem, with an emphasis on semi-analytic approaches, their relevance to gravitational-wave astronomy, and remaining unsolved issues.

Notes

• Gravitational wave (GW) detector networks:
  • AdvLIGO/Virgo+ (~2015+)
    • Upgrades complete as of 1 April 2015!
    • ~3 yr to get to final design sensitivity
    • Upgrade: ~10 times more sensitive
  • Kagra (~2018)
  • LIGO-India (~2022)
  • Pulsar timing arrays (~now)
    • NANOgrav, EPTA, PPTA
  • Future: third-gen LIGO
• GW sources
  • Merging stellar-mass compact-object binaries (NS or BH)
    • measure masses and spins
    • determine merger rates
  • core-collapse SN
  • isolated neutron stars
  • cosmic strings, stochastic bg
  • unexpected
  • Low-freq sources (LISA):
    • merging SMBHs
    • extreme-mass ratio ???
    • ???
• Coalescing binaries
  • phases: inspiral (periodic, long), merger (frequency chirp and peak amplitude, short), and ringdown (damping)
  • During merger and ringdown, the two holes merge and the remnant undergoes damped oscillations
• Why two-body GR is hard
  • Einstein’s eqs. are just a lot more complicated
  • Newton: only mass density
  • E: density, vel., kinetic energy, etc.
  • Highly nonlinear
• Solutions to E equations
  • Exact solutions: Kerr and FrW
  • Perturbation theory: PN theory, BH pert. theory
  • Numerical relativity: finite resolution, inexact ICs, cpu time
• Numerical Relativity
  • Not really viable until ~2005, despite efforts from the 1960s
  • Mergers now routine
  • Future: detailed exploration of BH/BH param space
  • NS+BH, NS+NS: realistic EOS, mag. fields, neutrinos…
  • Computationally expensive beyond ~10 orbits
    • NS+NS: 8000 orbits, NS+BH: 1800 orbits, BH+BH: 300 orbits
    • Orbital and radiation-reaction timescales
    • small mass ratios < 1/10 very costly
    • Current best achievement: 176 orbits
• Need for phase accuracy
  • LIGO data is noisy $\rightarrow$ need good signal template
  • integral of an oscillating function
  • phase evol. of signal needs to be accurate to fraction of a cycle
  • Templates: >10 parameters
• PN approx.
  • write E eqs as perturbation on flat-space wave eqn
  • series expansions
  • plug expansions back into E eqs
  • iterate
  • gets very messy very quickly
  • radiative effects important
  • orbital phasing is where the information lives — need to get to as high an order as possible
    • need to get to 3.5PN ($v^7$)
  • high-order harmonics can be important
  • “memory modes”: non-oscillatory but time-varying modes (secular effects)
    • nonlinear effect
    • GWs themselves produce GWs(!)
  • Spin effects
    • aligned: minor correction
    • non-aligned: mess
    • eqs to describe spin evolution must also be solved
  • Eccentricity effects
    • GWs damp eccentricity, so often ignored
    • But eccentric signals possible from binaries
    • periastron precession
    • eccentricity-induced modulations to orbital phase & amplitude
    • corrections also need to be high-order
  • Tidal interactions
    • near end of inspiral
    • tidal distortionparameterized in terms of tidal Love number
      • Measuring tidal Love number provides useful constraints
    • types:
      • electric
      • magnetic
      • shape
    • electric Love number most observationally relevant
• BH pert. theory
  • EMRI orbits
  • very complicated — rich structure, resonant effects
    • produces interesting “jumps” in phasing and orbital elements
  • self-force approach
• Conclusion: $2^{nd}$ gen network of GW detectors is coming online now
  • Need good modeling
  • Need good control over systematic errors (hence high-order PN work)