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

- AdvLIGO/Virgo+ (~2015+)
- 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 ???
- ???

- Merging stellar-mass compact-object binaries (NS or BH)
- 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)

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