ブラックホール・中性子星連星からの重力波と状態方程式の影響
Yukawa Institute of Theoretical Physics,
Kyoto University D3
Koutarou Kyutoku (久徳 浩太郎)
Kyutoku, Shibata, Taniguchi PRD 82 (2010) 044049
Kyutoku, Okawa, Shibata, Taniguchi PRD 84 (2011) 064018

Black hole-neutron star binaries• Ones of the most promising GW sources- emit 1-a few kHz GWs just before the merger
fairly promising for ground-based detectors- a robust test of GR / a new prove for the NS• Possible candidates of the short GRB engine- merger scenario <-> accretion disk formation• A candidate of the r-process environment(do not go into detail today)Gravitational-wave detectors
LCGT at Kamioka

LIGO at Hanford

Sensitive in 10-1000Hz… astrophysical sources

VIRGO at Cascina

Neutron star equations of state• The EOS at high density is not confirmed yet• NS radii are determined by the cold (T=0) EOS,
but not constrained accurately by (X-ray) observ.
• Can we determine
the NS radius by 1.97 M sun
GW observation?
• We can obtain
the EOS if we know
the M-R relation

Lattimer, Prakash (2007)

Typical GWs from BH-NS binaries• Classified as inspiral-merger-ringdown (if any)

Red: numerical relativity
Black: post-NewtonianImportance of numerical relativity• Appropriate methods differ among the three phases
1. inspiral phase … post-Newtonian approx.
2. merger phase … numerical relativity
strong (=nonlinear) gravity, hydrodynamics
3. ringdown phase … BH perturbation technique• Numerical relativity is the unique approach for themerger phase of compact binary coalescences• Tidal disruption of the NS determines the GWs andproperties of the remnant disk, such as the mass

Advantages of binary GW observation• Less systematic errors than EM observation- the largest model is the EOS and GR itself• The dynamics before the merger is nearly free ofthe finite-temperature effects, magnetic fields…
- in BH-NS, all GWs are determined by T=0 EOSs(- in NS-NS, hypermassive NSs also emits GWs)
• The early inspiral gives us different information- masses of each component, the BH spin

When the tidal disruption occurs?
• Two nondimensional parameters are important- Mass ratio of BH/NS Q = M BH / M NS ≥ 1
C = M NS / RNS ~ 0.15 − 0.2- NS compactness
tidal disruption occurs outside the ISCO if themass ratio Q and/or compactness Q are small= the small BH mass and/or the large NS radius∵ tidal effect is a finite size effect of the NS2• The large BH spin a ≡ S BH / M BH is also favored∵ The ISCO radius is small for a large BH spin

The formulation and methods• Solve Gab = 8πTab , ∇ bT = 0, ∇ a ( ρu ) = 0• Compute initial conditions- the spectral method library LORENEbaa

http://www.lorene.obspm.fr/- developed new code for the BH spin / NS EOS• Perform simulations of time evolution- AMR code SACRA developed in our group
Yamamoto, Shibata, Taniguchi PRD (2008) 78 064054• EOSs closes the system of equations

The piecewise polytrope (PWP)• T=0, nuclear-theory based EOSs areapproximated by analytic broken-polytropes κ1 ρ Γ1 (ρ < ρ1 ) ~crustP= κ 2 ρ Γ2 (ρ > ρ1 ) ~core- less computational costs- only a few parameters• Thermal effects areincorporated by ideal gases

crustRead+ (2009)Comparison of tidal disruption• (only the BH is different in this example)
M NS = 1.35M sol , M BH = 2.7 M sol (Q = 2)
No tidal disruption

Tidal disruptionlog ρ (g / cm 3 )(a = 0.75)(a = −0.5)

GWs: differences due to the EOS
M BH = 2.7 M sol
M NS = 1.35M sol
a=0•
••
•
•
•

RNS = 11km

RNS = 15km

Top: soft EOS
the small NS radius
weak disruption
Bottom: stiff EOSthe large NS radius
strong disruption

The GW spectrum• M NS = 1.35M sol , M BH = 2.7 M sol (Q = 2), a = 0quadrupole

PNLarger radiusCompactness vs cutoff frequency• a = 0 cases• Strong correlations= The NS radius
can be read offfrom GW observ.

Quasinormal mode of BH

M NS / RNS• f cut m0 ∝ C when tidal disruption occurs• Q ≥ 3 ( M BH ≥ 4.05M sol ) seems to bring notidal disruption -> an unrealistically story?
4GWs: difference due to the BH spin
• Strong if the BH spin is parallel to orbiｔ.ang.mom.

a = 0.75

tidal disruption

ringdown

a = −0.5Changes of spectra due to the spin
• The large amplitude at low frequency• Cutoff occurs at lower frequency

spin increases

Compactness – cutoff frequency• BH spins lower the cutoff frequency- the low frequency is preferred from observ.
quasinormal modes
for different BH spins

we have to know
the BH spin using
inspiral GWs•••
M NS / RNSThe GW spectrum for Q=5• In this case, M BH = 6.75M sol -> maybe realistic• Adv. LIGO/LCGT will observe the spectrum cutoff

Cutoff frequencies for a=0.75• Massive (realistic) BH-NS will tell us the NS EOSM NS / RNSFormed accretion disks• The mass of the disk can exceed 0.1M sol
when the tidal disruption is stronglog ρlog ρM disk = 0.2 M sol

M disk = 0.1M sol

M BH = 2.7 M sol

M BH = 6.75M sol

Summary• We computed gravitational waves from black
hole-neutron star binaries by numerical relativity.
• Gravitational waves from black hole-neutron starbinaries will tell us the equation of state ofneutron star matter, especially when tidal
disruption occurs.
• Astrophysically realistic black hole-neutron starbinaries will also tell us the equation of state if
the BH spin is moderately strong.
• The mass of the accretion disk can be sufficiently
high to bring the short-hard gamma-ray burst.

Future work
• More waveforms for generic configurations- an urgent task for the GW astronomy• Incorporate neutrinos and magnetic fields- essential for GRBs, multi-messenger astron.
• Analyze how accurately we can know the EOS- the Fisher analysis using numerical relativity
Lackey, KK, Shibata, Brady, Friedman PRD submitted

appendix

Short-hard gamma-ray burst• Release 10 49−51 erg
within ~ 2 s- jet opening angle -> ?
• BH-accretion disks?
- LGRB: “collapsar” model- SGRB: merger scenario?
BH-NS or NS-NS
Is a massive disk formed?
From encyclopedia of science

The mass of the neutron star
• EOSs are constrained by
the NS maximum mass,recently 1.97Mo found• Radius measurement isfairly difficult- degeneracy w/ distance- rad./atmosphere models
Lattimer, Prakash (2010)

Lattimer & Prakash (2007)