Gravita(onal waves from black hole-‐neutron star binaries ... - LUTH
Gravita(onal waves from black hole-‐neutron star binaries ... - LUTH
Gravita(onal waves from black hole-‐neutron star binaries ... - LUTH
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<strong>Gravita</strong>'<strong>onal</strong> <strong>waves</strong> <strong>from</strong><br />
<strong>black</strong> <strong>hole</strong>-<strong>‐neutron</strong> <strong>star</strong> <strong>binaries</strong>:<br />
dependence on EOSs and BH spins<br />
Koutarou Kyutoku<br />
Yukawa Ins'tute for Theore'cal Physics,<br />
Kyoto University (Japan)<br />
Kyutoku, Shibata, Taniguchi Phys. Rev. D 82 (2010) 044049<br />
Kyutoku, Shibata, Taniguchi in prepara'on
Plan of the talk<br />
• 1. Introduc'on<br />
• 2. Numerical methods<br />
• 3. Results<br />
-‐ nonspinning BH case<br />
-‐ spinning BH case<br />
• 4. Summary
Introduc'on
Why do we inves'gate BH-‐NS binary?<br />
• <strong>Gravita</strong>'<strong>onal</strong> wave (GW) astronomy<br />
For ground-‐based laser-‐interferometric detectors,<br />
compact <strong>binaries</strong> are the most promising sources<br />
GWs as probes to GR and high density NS maYer<br />
• Short gamma-‐ray burst (SGRB)<br />
BH-‐hot, massive accre'on disk systems are<br />
the possible candidates of GRB progenitors<br />
Does compact <strong>binaries</strong> bring SGRBs or not?
<strong>Gravita</strong>'<strong>onal</strong>-‐wave detectors<br />
VIRGO at Cascina<br />
Sensitve in 10-‐1000Hz<br />
… astrophysical sources<br />
(Planned) LCGT at Kamioka<br />
LIGO at Hanford
Es'mated detec'on rate of BH-‐NS<br />
• No detec'on of<br />
the BH-‐NS binary<br />
• S'll unknown<br />
due to difficulty<br />
in popula'on<br />
synthesis<br />
• ~10 detec'on/yr<br />
may be possible<br />
with Adv. LIGO<br />
Ini'al LIGO<br />
BH-‐NS<br />
Advanced LIGO<br />
NS-‐NS<br />
BH-‐BH<br />
Kalogera+ (2007)
Unknown NS radius and EOS<br />
• The NS radius is determined by T=0 (cold) EOS<br />
• One main goal of GW astronomy is to determine<br />
or at least constrain the EOS at high density<br />
• NS radii are not<br />
constrained well<br />
by observa'ons<br />
• NS EOSs are not<br />
constrained by<br />
experiments<br />
Lagmer&Prakash (2007)
Why do we inves'gate BH-‐NS binary?<br />
• <strong>Gravita</strong>'<strong>onal</strong> wave (GW) astronomy<br />
For ground-‐based laser-‐interferometric detectors,<br />
compact <strong>binaries</strong> are the most promising sources<br />
GWs as probes to GR and high density NS maYer<br />
• Short gamma-‐ray burst (SGRB)<br />
BH-‐hot, massive accre'on disk systems are<br />
the possible candidates of GRB progenitors<br />
Does compact <strong>binaries</strong> bring SGRBs or not?
Short gamma-‐ray burst<br />
• Release huge energy<br />
about<br />
• In short 'mescale<br />
less than<br />
• BH-‐hot, massive disk<br />
-‐ LGRB: “collapsar” model<br />
-‐ SGRB: merger scenario?<br />
BH-‐NS or NS-‐NS<br />
From encyclopedia of science
Role of numerical rela'vity<br />
• Three different phases of binary coalescences<br />
1. inspiral phase … post Newtonian approxima'on<br />
2. merger phase … numerical rela'vity<br />
strong (nonlinear) gravity, hydrodynamic effect<br />
3. ringdown phase … BH perturba'on<br />
• Numerical rela'vity is the unique approach to<br />
inves'gate merger phases of binary coalescences<br />
• Especially, the 'dal disrup'on of the NS is interes'ng<br />
since they determine GWs / remnant disks
Main ques'ons for BH-‐NS<br />
• How does a BH-‐NS binary merge?<br />
-‐ the NS is 'dally disrupted, and disk is formed?<br />
-‐ just like a BH-‐BH binary, and no disk is leo?<br />
• What parameters are important?<br />
-‐ BH mass, BH spin, NS mass, NS radius or EOS<br />
• What do we know <strong>from</strong> GW signals?<br />
-‐ if the 'dal disrup'on occurs or not<br />
-‐ NS radius, and more informa'on about EOS
Onset of 'dal disrup'on (mass shedding)<br />
• BH 'dal force vs. NS self gravity, at NS surface<br />
(neglect NS deforma'on for simplicity)<br />
• BH Innermost stable circular orbit (ISCO)<br />
radius is propor'<strong>onal</strong> to the BH mass
When is the 'dal disrup'on important?<br />
• Two important dimensionless parameters<br />
-‐ mass ra'o of BH to NS<br />
-‐ compactness of the NS<br />
<strong>from</strong> the previous calcula'on,<br />
the ,dal disrup,on occurs outside the ISCO<br />
if the mass ra,o Q and/or compactness C is small<br />
(since the 'dal effect is a NS finite size effect)<br />
• The BH spin is also important since ISCO radius,<br />
equivalently depends strongly on the BH spin
Numerical<br />
methods
Nuclear-‐theory based EOS<br />
• Too many EOSs for systema'c calcula'ons<br />
• Tabulated EOS is heavy for<br />
the numerical simula'on,<br />
since the interpola'on is<br />
always required<br />
• Analy'c EOS with few<br />
parameters is preferable<br />
(also for observa'ons)<br />
Read+ (2009)
Piecewise polytrope (PWP)<br />
• PWP mimics many nuclear-‐theory based EOSs<br />
with 4 pieces, where 1 for crust and 3 for core<br />
• However, in this work we<br />
focus on 2-‐piece PWPs<br />
~crust<br />
~core<br />
(in a BH-‐NS case, we never<br />
have “hypermassive” object)<br />
• Sound velocity jumps…<br />
crust<br />
Read+ (2009)
PWP models in this work<br />
• Use 8 models of PWPs (crust EOS always fixed)<br />
• Core EOS is specified by 2 paremeters<br />
: s'ffness of the core<br />
: pressure at fiducial density<br />
• is storongly related<br />
to the NS radius and<br />
deformability<br />
(Lagmer&Prakash 2001)<br />
crust
M-‐R rela'on of PWP<br />
• We can reproduce some proper'es of a nuclear-‐<br />
theory based EOS even with a 2-‐piece PWP<br />
• Weak dependence<br />
of radius on mass<br />
= stellar structure<br />
• Cannot with<br />
a polytropic EOS<br />
Standard<br />
polytrope
Brief explana'on of procedure<br />
• First, we compute ini'al condi'ons<br />
-‐ solve the Einstein constraint equa'ons and some<br />
quasiequilibrium equa'ons<br />
-‐ solve equa'ons of hydrosta'c equilibrium<br />
• Next, we perform dynamical simula'ons<br />
-‐ solve the Einstein evolu'on equa'ons (free evol)<br />
-‐ solve hydrodynamic evolu'on equa'ons<br />
-‐ extract GWs, analyze remnant disks and BHs
Moving-‐puncture technique<br />
• Both for ini'al condi'on and 'me evolu'on<br />
(Campanelli+ 2006/Baker+ 2006 cf: Brandt&Brugmann 1997)<br />
• No physical singularity with appropriate slicing<br />
Horizon<br />
or throat<br />
Another spa'al infinity at “r=0”<br />
Our universe<br />
NS<br />
no inner boundary<br />
condi'ons at horizon
Ini'al condi'on<br />
• We solve constraint equa'ons for the metric within<br />
the moving-‐puncture framework assuming<br />
-‐ conformal flatness of 3-‐metric<br />
-‐ maximal slicing condi'on<br />
-‐ (quasi-‐)sta'onarity<br />
extended conformal thin-‐sandwich method<br />
• Extrinsic curvature is decomposed in a similar way<br />
as in conformal transverse-‐traceless decomposi'on<br />
-‐ Bowen-‐York extrinsic curvature is adopted for
Ini'al condi'on<br />
• Solve hydrosta'c equa'ons for maYer variables<br />
-‐ a perfect fluid as<br />
-‐ assume T=0 (cold) and adopt a PWP for the EOS<br />
-‐ an irrota'<strong>onal</strong> velocity field and<br />
• We obtain 9 ellip'c equa'ons for<br />
and 1 algebraic equa'on for (<strong>from</strong> EOS )<br />
• Solved by mul'-‐domain spectral-‐method library<br />
LORENE developed by Meudon rela'vity group<br />
-‐ We thank all the people developing LORENE
• Today we focus mainly on<br />
Models<br />
-‐ for a larger mass ra'o (or a heavier BH), almost<br />
no 'dal disrup'on if the BH is nonspinning<br />
(show some results for diffenrent mass ra'os)<br />
• We use 8 models of two-‐piece PWP (crust EOS fixed)<br />
-‐ for a spinning BH case, we fix the core s'ffness<br />
• We fix ini'al values of (PN parameter)<br />
-‐ typically ~5 orbits are tracked (dependent on BH spin)
Dynamical simula'on<br />
• BSSN formalism and moving-‐puncture gauge<br />
-‐ decompose<br />
-‐ introduce an auxiliary variable<br />
-‐ evolve and also<br />
• We never solve the Einstein constraints during<br />
the 'me evolu'on (“free evolu'on scheme”)<br />
• 4 th -‐order finite difference in both 'me and space,<br />
with non-‐centered difference for advec'on terms
Dynamical simula'on<br />
• Hydrodynamic equa'ons are solved with shock-‐<br />
capturing scheme, 3 rd -‐order difference in space<br />
• EOS: PWP for cold + ideal gas (rough) for thermal<br />
• <strong>Gravita</strong>'<strong>onal</strong> <strong>waves</strong> are extracted with<br />
• Adap've Mesh Refinement (AMR) code: SACRA<br />
(Yamamoto, Shibata, Taniguchi 2008)
AMR technique<br />
• WriYen by Masaru Shibata<br />
GW<br />
L >λ >> GM/c 2 l ∼ 4GM/c 2<br />
• Typically 113(x)*113(y)*57(z)*11(domain)<br />
• About 150 variables in SACRA<br />
• Required memory ~10 Gbytes<br />
• Feasible in desktop computers of ~2,000EURO
Results:<br />
nonspinning<br />
BH case
Anima'on: no 'dal disrup'on case<br />
• EOS is soo one<br />
(EOS is not so important<br />
in cases)<br />
• Disk mass is negligible
Anima'on: 'dal disrup'on case<br />
• EOS is typical one<br />
(EOS is important in<br />
cases)<br />
• Disk mass may be large<br />
enough for SGRB
GWs with different mass ra'os<br />
• EOS, NS mass fixed<br />
• (i.e. NS is the same)<br />
• -‐> 'dal disrup'on<br />
• -‐> ringdown waveform<br />
• (same as case)
GWs with different EOSs<br />
• only EOSs differ<br />
• soo EOS case<br />
• -‐> ringdown waveform<br />
(very) s'ff EOS case<br />
-‐> 'dal disrup'on
•<br />
GW spectrum<br />
quadrupole<br />
TaylorT4<br />
(PN)<br />
Larger radius
Extrac'ng “cutoff frequency”<br />
• Universal feature of the GW spectrum<br />
-‐ Low frequency: same as the PN result<br />
-‐ High frequency: “bump” is universal<br />
• Fit systema'cally with 7 free parameters<br />
Damped post-‐Newton<br />
+<br />
Bump component
Cutoff frequency -‐ compactness<br />
• Strong correla'on between the cutoff frequency<br />
('mes the total mass) and the NS compactness<br />
• Approximately,<br />
for<br />
• And changes<br />
for similar<br />
-‐ C is differ only by ~1%,<br />
fcut can differ by ~20%
Why is the core adiaba'c index?<br />
• Smaller = sooer EOS, more centrally condensed<br />
and survives longer aoer the mass shedding<br />
• Or “s'ff things are fragile.” : Large<br />
(-‐> imcompressible)<br />
: Small
Disk mass evolu'on<br />
• Mainly determined by the NS compactness<br />
• S'ffer EOS results in a more massive disk, since<br />
the 'dal disrup'on occurs at a more distant orbit<br />
(same as before)<br />
• long-‐lived disks<br />
Larger radius
The disk mass -‐ compactness<br />
• No'ce: not rescaleed by the ini'al NS mass<br />
• We again see<br />
the effect of
Results:<br />
spinning BH<br />
case
Effect of the BH spin<br />
• The BH spin changes the ISCO radius<br />
• For Schwarzschild BH,<br />
• For extreme Kerr, for a prograde orbit<br />
due to the spin-‐orbit interac'on<br />
(prograde = parallel, aligned)<br />
• For a prograde case, the 'dal disrup'on is “easy”<br />
even for a heavy BH, or large mass ra'o Q<br />
-‐ for a retrograde(an'-‐parallel) case, very difficult
Ini'al data of spinning BH-‐NS<br />
• Control by Bowen-‐York extrinsic curvature<br />
• The BH spin is calculated within a so-‐called<br />
isolated-‐horizon framework with an approximate<br />
Killing vector (Cook&Whi'ng 2007)<br />
-‐ thank a lot for Eric Gourgoulhon/Nicolas Vasset<br />
• Maximum spin is (drawback!)<br />
-‐ Kerr BH cannot be conformal flat in 3-‐space
Anima'on: retrograde spin case<br />
• Spin-‐orbit aYrac'on<br />
• Less orbits than<br />
nonspinning case<br />
• Disk mass is<br />
negligible
Anima'on: prograde spin case<br />
• Spin-‐orbit repulsion<br />
• More orbit than<br />
nonspinning case<br />
• Disk mass is large
GWs for different BH spin (Q=3)<br />
• Long, strong emissions <strong>from</strong> prograde BH spins
GW spectra for different BH spins<br />
• Amplitude is high at low frequency<br />
• “Cutoff” occurs also at lower frequency
Cutoff frequency -‐ compactness<br />
• Slight change in defini'on<br />
• Cutoff frequency becomes lower with BH spin<br />
QNM for<br />
different<br />
BH spin
Massive disks are easily formed<br />
• Upper panels are with prograde BH spins
Disk mass -‐ compactness<br />
• Rescaled to the ini'al NS mass<br />
• Heavy disks are universal for large BH spins
And even for a heavy BH<br />
• Test run with<br />
a low resolu'on<br />
• Scale changes<br />
during evolu'on<br />
• Disk mass is
Q=4, a=0.75 cases (preliminary)<br />
• Astrophysically more realis'c as the BH mass?<br />
• beYer for<br />
• GW observa'on<br />
SGRB sources?
Summary<br />
and future<br />
work
Summary<br />
• <strong>Gravita</strong>'<strong>onal</strong> <strong>waves</strong> <strong>from</strong> BH-‐NS <strong>binaries</strong> tell<br />
us the informa'on of EOS at high density, such<br />
as NS compactness and possibly the s'ffness<br />
of the core, through the cutoff frequency.<br />
• The mass of the remnant disk is also sensi've<br />
to the EOS.<br />
• The BH spin significantly changes the final fate<br />
of the BH-‐NS binary and the 'dal disrup'on of<br />
NS by a heavy BH becomes possible.
Future work<br />
• How accurately the EOS parameter can be<br />
determined? … with UWM group<br />
• More detailed treatment of finite temperature<br />
effects and microphysics … with Y. Sekiguchi<br />
• More rapidly spinning BH case, e.g.<br />
(overspinning?) … Now I am working on here<br />
• Effect of magne'c fields, e.g. Blandford-‐Znajek<br />
process … just a plan
Merci !<br />
Arigato-‐<br />
gozaimasu !
appendix
Forma'on paths of BH-‐NS <strong>binaries</strong><br />
• No observa'<strong>onal</strong> constraints on BH-‐NS<br />
Kalogera(2007)<br />
• Some possibility of “dynamical capture” in<br />
globular clusters (three-‐body interac'on)
Gamma-‐ray bursts a<br />
• Short 'mescale<br />
• Hard spectral index<br />
• Early type galaxies<br />
• SGRBs are rela'vely<br />
unknown than LGRBs<br />
Nakar (2007)
EOS Models<br />
• We use 8 models of two-‐piece PWP (crust fixed)<br />
: s'ffness of the core<br />
: pressure at fiducial density
Binary simula'on @ home (office)<br />
• Core i7x, 3.33 GHz, 4 cores, 12 or 24 GB memory<br />
May be better than supercomp. 10 yrs ago<br />
• 111*111*56 *7 AMR levels 10 GB memory<br />
• About 50 days for 7 orbits by SACRA code<br />
• Parameter parallel by ~ 30 machines now
Baker et al. PRD, 2007<br />
BH-BH
Comment on cutoff frequency<br />
• Combina'on is useful irrespec've of<br />
mass of NS<br />
-‐ not so for, e.g., combina'on<br />
• Quasinormal-‐mode frequency or spin of the<br />
remnant BH is almost independent on the EOS<br />
• Approximately
Comment on rest-‐mass density<br />
• Maximum density is correlated with the disk mass<br />
• If the massive disk is formed, -‐trapping may occur<br />
• NDAF?
BH spin angular momentum<br />
• Surface term of the Hamiltonian (similar to ADM)<br />
• Approximate Killing vector: Cook&Whi'ng (2007)<br />
-‐ minimize , (non-‐Killing aspects)<br />
-‐ two linear, ellip'c equa'ons on two-‐surface<br />
-‐ assured to be divergence free by construc'on<br />
(to be invariant under the boost transforma'on)
GW spectra for Q=2, a=0.75<br />
Large compactness
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