DDA 2015 – Modeling relativistic orbits and gravitational waves

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)