Physics  and  Astrophysics  with  gravitational  waves  from  

Binary  Neutron  star  Coalescences,  Black  Hole  Neutron  star  Coalescences,  

Pulsars  and  Magnetars

Hideyuki  Tagoshi(Osaka  City  University)

1新学術領域シンポジウム 2016/12/28  @  京大基研

B01"Physics  and  Astrophysics  with  

gravitational  waves  from  Binary  Neutron  star  Coalescences,  

Black  Hole  Neutron  star  Coalescences,  Pulsars  and  Magnetars"

2

Proposed  New  Innovative  Area  KAKENHI  

(GW  from  systems  which  include  at  least  a  neutron  star)

MembersData  AnalysisH.  Tagoshi：KAGRA  member,  CBC  AnalysisY.  Itoh： KAGRA&LSC,  Continuous  Wave  &  CBC  AnalysisK.  Cannon:  KAGRA&LSC

CBC  Analysis  with  gstlal (low  latency  and  offline)

Numerical  SimulationM.  Shibata,  K.  Kiuchi,  Kyutoku, Kawaguchi,  Hotokezaka

Neutron  star  Astrophysics: Y.Kojima

3

4

Plausibleworld-wideobservingscenarioforthenextdecade

July7,2016 DawnWorkshop2016-Atlanta 9

2017 2018 2019 2020 2021 2022 2023 2024 2025 20262016

LIGO-H1LIGO-L1VIRGOKAGRALIGO-INDIA

O2 O3 O4 O5

ObservingSmeCommissioningSmeDownSmeforupgrades

SiteconstrucSonanddetectorinstallaSon

bothsqueezingandcoaSng?bothLIGOsitessimultaneously?

A+Upgrade

By  Lisa  Barsotti

On  KAGRA and  LIGO-­‐VirgoProf.  Kajita,  PI  of  KAGRA,  confirmed  that  

「KAGRAは2019年から本格観測フェーズに入る予定である．LIGO-­‐Virgoとの共同観測とデータ共有もその後程なく開始される予定である．」

"KAGRA  is  planning  to  enter  the  observation  phase  in  2019.The  cooperative  observation  and  the  data  sharing  with  LIGO  and  Virgo  are  also  planned  to  start  soon  after  that"

Please  note,  however,  that  the  sensitivity  of  KAGRA  may  not  be  so  good  at  the  beginning  of  observation  in  2019.    Full  sensitivity  will  be  achieved  later  (〜2020?)

We  can  also  expect  to  observe  GWs  from  BBH,  BNS,  and    probably  NSBH,  if  KAGRA  achieve  the  sensitivity  good  enough  for  detection.  

5

Science  Target

6

Detection  of  GW  with  KAGRA  The  most  important  science  target  is

We  can  not  proceed  further  without  it.

(promised)  Signals:  BBH  (at  the  same  sensitivity  of  LIGO-­‐O1)

What  comes  next?NSBHBNS……

Expected  detection  rates

7

Class. Quantum Grav. 27 (2010) 173001 Topical Review

Table 4. Compact binary coalescence rates per Mpc3 per Myra.

Source Rlow Rre Rhigh Rmax

NS–NS (Mpc−3 Myr−1) 0.01 [1] 1 [1] 10 [1] 50 [16]NS–BH (Mpc−3 Myr−1) 6 × 10−4 [18] 0.03 [18] 1 [18]BH–BH (Mpc−3 Myr−1) 1 × 10−4 [14] 0.005 [14] 0.3 [14]

a See footnotes in table 2 for details on the sources of the values in this table.

Table 5. Detection rates for compact binary coalescence sources.

IFO Sourcea Nlow yr−1 Nre yr−1 Nhigh yr−1 Nmax yr−1

NS–NS 2 × 10−4 0.02 0.2 0.6NS–BH 7 × 10−5 0.004 0.1

Initial BH–BH 2 × 10−4 0.007 0.5IMRI into IMBH <0.001b 0.01c

IMBH-IMBH 10−4 d 10−3 e

NS–NS 0.4 40 400 1000NS–BH 0.2 10 300

Advanced BH–BH 0.4 20 1000IMRI into IMBH 10b 300c

IMBH-IMBH 0.1d 1e

a To convert the rates per MWEG in table 2 into detection rates, optimal horizon distances of33 Mpc/445 Mpc are assumed for NS–NS inspirals in the Initial/Advanced LIGO–Virgo networks. ForNS–BH inspirals, horizon distances of 70 Mpc/927 Mpc are assumed. For BH–BH inspirals, horizondistances of 161 Mpc/2187 Mpc are assumed. These distances correspond to a choice of 1.4 M⊙ forNS mass and 10 M⊙ for BH mass. Rates for IMRIs into IMBHs and IMBH–IMBH coalescences arequoted directly from the relevant papers without conversion. See section 3 for more details.b Rate taken from the estimate of BH–IMBH IMRI rates quoted in [19] for the scenario of BH–IMBHbinary hardening via three-body interactions; the fraction of globular clusters containing suitableIMBHs is taken to be 10%, and no interferometer optimizations are assumed.c Rate taken from the optimistic upper limit rate quoted in [19] with the assumption that all globularclusters contain suitable IMBHs; for the advanced network rate, the interferometer is assumed to beoptimized for IMRI detections.d Rate taken from the estimate of IMBH-IMBH ringdown rates quoted in [20] assuming 10% of allyoung star clusters have sufficient mass, a sufficiently high binary fraction, and a short enough corecollapse time to form a pair of IMBHs.e Rate taken from the estimate of IMBH-IMBH ringdown rates quoted in [20] assuming all young starclusters have sufficient mass, a sufficiently high binary fraction, and a short enough core collapse timeto form a pair of IMBHs.

Where posterior probability density functions (PDFs) for rates are available, Rre refersto the PDF mean, Rlow and Rhigh are the 95% pessimistic and optimistic confidence intervals,respectively, and Rmax is the upper limit, quoted in the literature based on very basic limits setby other astrophysical knowledge (see table 1). However, many studies do not evaluate therate predictions in that way, and for some speculative sources even estimates of uncertaintiesmay not be available at present. In these cases, we assign the rate estimates available in theliterature to one of the four categories, as described in detail in section 4. The values in alltables in this section are rounded to a single significant figure; in some cases, the roundingmay have resulted in somewhat optimistic predictions.

9

Abadie et  al.  CQG27,  173001(2010)  

〜36

BNS,  NSBH  CoalescencesBNS,  NSBH• Waveform

High  precision  numerical  simulation  of  BNS  and  NSBH  and  construction  of  waveform  models  which  can  be  used  in  data  analysis.

• Detection  pipelineDevelopment  of  advanced  data  analysis  pipeline  by  using  state-­‐of-­‐the-­‐art  waveform  models.  

• Parameter  estimation  Development  of  data  analysis  method  to  measure  tidal  deformability  of  NS,  and  to  constrain  EOS with  BNS  and  NSBH  gravitational  wave  data.

8

Construction  of  templates  for  GWs  from  BNS  and  NSBH

This  would  be  essential  to  extract  physical  information  from  GW  signal  of  BNS  and  NSBH.  

High  precision  numerical  relativity  simulationsMany  simulationsHybrid  waveform  (e.g.,  EOB-­‐inspiral +  NR)Construction  of  waveform  models  (e.g.  in  freq.  domain)

Late  inspiral phasePost-­‐merger  phase

9

10-23

10-22

10-21

1

f hf (

r=10

0 M

pc)

f (kHz)

APR4ALF2

H4aLIGO

BH-BH

Overall spectrum by NR simulations�

BH-BHby SpEC�

SNR=0.3—1@ D=180Mpc�

0.5�

4hred − hblue( )

2

Sn f( )df

500Hz

1.2kHz

∫ ≈ 4 @ Deff =180Mpc

or SNR=20

$%&

'&

R=11.1km R=12.4km R=13.6km�Modeling by Effective-one-body

formalism is ongoing: Damour, Nagar, Buonanno, Hinderer ..�

Construction  of  templates  for  GWs  from  BNS  and  NSBH

10Slide  by  Shibata  ('15)

late  inspiral phase

Construction  of  templates  for  GWs  from  BNS  and  NSBH

-0.15-0.1

-0.05 0

0.05 0.1

0.15

0 10 20 30 40 50 60 70

h D

/ m

0

tret (ms)

APR4ALF2

H4

Latest Numerical Relativity Waveforms Last several phases are different �

Hotokezaka et al. 2015�

Mass: 1.35-1.35 solar mass�

SOFT###############11.1km#MODERATE###12.4km#STIFF###############13.6km�

Appreciable difference in phase� Slide  by  Shibata  ('15)11

Observable  range  of  post-­‐merger  signals

12

EOS Mass  [Msun] Inspiral range  [Mpc]  SNR=8

HMNS range  [Mpc]    SNR=5

APR

1.215 226 9.81.3 239 9.51.35 246 9.7

H4

1.215 226 141.3 239 151.35 246 171.4 254 15

With  Matched  filtering,  KAGRA  vRSE-­‐B,  Optimal  direction

Yuzurihara,  Hotokezaka,  Kyutoku,  Shibata,  Tagoshi,  Kanda  et  al.  in  preparation  

HHT  analysis  of  post  merger  phase  

Hilbert-Huang Transform• Time frequency analysis• It is possible to measure the

difference of frequency evolution • Distinguish between difference EOS

Hyperon EOS

Shen EOS

Instantane

ous  F

req.  [kHz

]

0-5 5 -10t-­‐t連星合体 (ms)

Hyper Massive NS

Time  from  merger   [ms]

Kaneyama et  al，PRD93,  123010  (2016)

Detailed  follow-­‐up  analysis  can  be  done(Same  analysis  can  be  done  with  NHA)

Analysis  methodSearch• gstlal pipeline  for  low  latency  analysis  (probably,  at  least  initially)

Already  connected  with  alert  system  of  LVC.  We  need  to  collaborate  with  LVC    (for  localization).Discussion  started.  To  do:  Implementation  of  new  waveforms……

• KAGALI  pipeline  for  offline  analysis  Many  developments  are  necessary

Challenging• Spin  precession  for  BBH/NSBH    (Large  #  of  parameters)• Low  mass  BBH  system  (Large  #  of  template  bank)At  least,  several  TFlops computing  power  for  simple  analysis(>103 cores).  Still  not  enough  for  above  cases.  LIGO  has  〜104 core  systems 14

KAGALIKAGRA  Algorithmic  LIbrarycurrent  version:  kagali-­‐v0r4a

• C  language• Version  control  by  git repository  

nightly  build,  code  check  by  cppchecker,  …• Autotools for  installation  in  various  platform• Own  error  handling  mechanism• ……

15

Analysis  method

16

⇢i[k] =M 0X

j=1

↵ij

N�1X

n=0

uj [n]x[k � n]

gstlal :  Time  domain,  SVD  for  templates,  down  sampling  

17

• We  represent  each  template  as  a  linear  combination  of  M’  (<<M)  basis  templates

hi[n] =M 0X

j=1

↵ijuj [n]

summation  (convolution)

basis

basis

basis

uj [n]

hi[n]↵ij

convolution

In  this  scheme,  we  need  C2=M’  N2    (convolution) +M  M’  N  (reconstruction)  operationsfor  N  data  points.  Thus,  C2/C1=M’/M  +  M’/N  <<  1,    if    M’  <<  M  &  M’  <<  N

(reconstruction)

hi0 [n]↵i0j

tn

⇢i[k] =N�1X

n=0

hi[n]x[k � n]

specific values a5 ¼ 0, a6 ¼ "20 (to which correspond,when ! ¼ 1=4, a1 ¼ "0:036 347, a2 ¼ 1:2468). Wehenceforth use M as a time unit.

Figure 1 compares (the real part of) our analyticalmetricquadrupolar waveform !EOB

22 =! to the corresponding(Caltech-Cornell) NR metric waveform !NR

22 =! (obtainedby a double time-integration, a la [20], from the originalNR curvature waveform c 22

4 ). [We used the ‘‘two-frequency pinching technique’’ of [19] with !1 ¼ 0:047and !2 ¼ 0:31.] The agreement between the analyticalprediction and the NR result is striking, even around themerger (see the close-up on the right). The phasing agree-ment is excellent over the full time span of the simulation(which covers 32 cycles of inspiral and about 6 cycles ofringdown), while the modulus agreement is excellent overthe full span, apart from two cycles after merger where onecan notice a difference. A more quantitative assessment ofthe phase agreement is given in Fig. 2, which plots the(!1-!2-pinched) phase difference"" ¼ "EOB

metric ""NRmetric.

"" remains remarkably small (#$0:02 radians) duringthe entire inspiral and plunge (!2 ¼ 0:31 being quite nearthe merger, see inset). By comparison, the root-sum of thevarious numerical errors on the phase (numerical trunca-tion, outer boundary, extrapolation to infinity) is about0.023 radians during the inspiral [6]. At the merger, andduring the ringdown, "" takes somewhat larger values(#$0:1 radians), but it oscillates around zero, so that, onaverage, it stays very well in phase with the NR waveform(as is clear on Fig. 1). By comparison, we note that [6]mentions that the phase error linked to the extrapolation toinfinity doubles during ringdown. We also found that theNR signal after merger is contaminated by unphysicaloscillations. We then note that the total ‘‘two-sigma’’ NRerror level estimated in [6] rises to 0.05 radians duringringdown, which is comparable to the EOB-NR phasedisagreement. Figure 3 compares the analytical and nu-merical metric moduli, j!22j=!. Again our (Pade-re-summed, NQC-corrected) analytical waveform yields aremarkably accurate description of the inspiral NR wave-form. During the early inspiral the fractional agreement

between the moduli is at the 3% 10"3 level; as late as timet ¼ 3900, which corresponds to 1.5 GW cycles beforemerger, the agreement is better than 1% 10"3. The dis-crepancy between the two moduli starts being visible onlyjust before and just after merger (where it remains at the2:5% 10"2 level). This very nice agreement should becompared with the previously considered EOB waveforms(which had a more primitive NQC factor, with a2 ¼ 0[19,20]) which led to large moduli disagreements(# 20%, see Fig. 9 in [20]) at merger. By contrast, thepresent moduli disagreement is comparable to the esti-mated NR modulus fractional error (whose two-sigmalevel is 2:2% 10"2 after merger [6]).We also explored another aspect of the physical sound-

ness of our analytical formalism: the triple comparisonbetween (i) the NR GW energy flux at infinity (whichwas computed in [21]); (ii) the corresponding analyticallypredicted GW energy flux at infinity (computed by sum-ming j _h‘mj2 over (‘, m)); and (iii) (minus) the mechanicalenergy loss of the system, as predicted by the general EOB

FIG. 1 (color online). Equal-mass case: agreement between NR (black online) and EOB-based (red online) ‘ ¼ m ¼ 2 metricwaveforms.

FIG. 2 (color online). Phase difference between the analyticaland numerical (metric) waveforms of Fig. 1.

IMPROVED ANALYTICAL DESCRIPTION OF . . . PHYSICAL REVIEW D 79, 081503(R) (2009)

RAPID COMMUNICATIONS

081503-3

Down  Sampling

18

K.  Cannon  et  al.,    ApJ,  748,  136  (2012).  

Reduce  both  N  (data  points)  and  M’  (basis  templates)

low  frequency  parthigh  frequency  part

gstlal :  Time  domain,  SVD  for  templates,  down  sampling  

19

The Astrophysical Journal, 748:136 (14pp), 2012 April 1 Cannon et al.

ρ00

ρ01

x ρ02

::ρ0

M−1

↑ ↑ ↑ ·· ↑

ρ10

ρ11

↓ ρ12

::ρ1

M−1

↑ ↑ ↑ ·· ↑...

......

......

......

......

ρS−10

ρS−11

↓ ρS−12

::ρS−1

M−1

......

......

......

4096 Hz

512 Hz

32 Hz

z−t1f0 delay

z−(t2−t1)f1 delay

Decimationof input

Orthogonalfir filters

Reconstructionmatrices

Interpolation andsnr accumulation

Figure 3. Schematic of LLOID pipeline illustrating signal flow. Circles with arrows represent interpolation ↑ or decimation ↓ . Circles with plus signs representsumming junctions . Squares stand for FIR filters. Sample rate decreases from the top of the diagram to the bottom. In this diagram, each time slice contains threeFIR filters that are linearly combined to produce four output channels. In a typical pipeline, the number of FIR filters is much less than the number of output channels.

3.2.3. Early-warning Output

In the previous two sections, we described two transforma-tions that greatly reduce the computational burden of TD fil-tering. We are now prepared to define our detection statistic,the early-warning output, and to comment on the computationalcost of evaluating it.

First, the sample rate of the detector data must be decimated tomatch sample rates with each of the time slices. We will denotethe decimated detector data streams using a superscript “s” toindicate the time slices to which they correspond. The operatorH ↓ will represent the appropriate decimation filter that convertsbetween the base sample rate f 0 and the reduced sample rate f s:

xs[k] = (H ↓x0)[k].

We shall use the symbol H ↑ to represent an interpolation filterthat converts between sample rates f s+1 and f s of adjacent timeslices,

xs[k] = (H ↑xs+1)[k].

From the combination of the time slice decomposition inEquation (6) and the SVD defined in Equation (7), we define

the early-warning output accumulated up to time slice s usingthe recurrence relation,

ρsi [k] =

S/N from previous time slices! "# $%H ↑ρs+1

i

&[k] +

Ls−1'

l=0

vsilσ

sl

# $! "reconstruction

orthogonal fir filters! "# $Ns−1'

n=0

usl [n]xs[k − n] . (8)

Observe that the early-warning output for time slice 0, ρ0i [k],

approximates the S/N of the original templates. The signalflow diagram in Figure 3 illustrates this recursion relation as amultirate filter network with a number of early-warning outputs.

Ultimately, the latency of the entire LLOID algorithm isset by the decimation and interpolation filters because theyare generally time symmetric and slightly acausal. Fortunately,as long as the latency introduced by the decimation andinterpolation filters for any time slice s is less than that timeslice’s delay ts, the total latency of the LLOID algorithm will bezero. To be concrete, suppose that the first time slice, sampledat a rate f 0 = 4096 Hz, spans times [t0, t1) = [0 s, 0.5 s),and the second time slice, sampled at f 1 = 512 Hz, spans[t1, t2) = [0.5 s, 4.5 s). Then the second time slice’s output,

6

K.  Cannon  et  al.,    ApJ,  748,  136  (2012).  

Real  algorithm  is  complicated...

Computing  cost

20

The Astrophysical Journal, 748:136 (14pp), 2012 April 1 Cannon et al.

interpolation filter. The length of the decimation filter affectsmismatch as well, but has very little impact on performance.Effect of SVD tolerance. We studied how the SVD toleranceaffected S/N loss by holding N↓ = N↑ = 192 fixed aswe varied the SVD tolerance from (1–10−1) to (1–10−6).The minimum, maximum, and median mismatch are shownas functions of SVD tolerance in Figure 5(a). As the SVDtolerance increases toward 1, the SVD becomes an exact matrixfactorization, but the computational cost increases as the numberof basis filters increases. The conditions presented here aremore complicated than in the original work (Cannon et al.2010) due to the inclusion of the time-sliced templates andinterpolation, though we still see that the average mismatchis approximately proportional to the SVD tolerance downto (1–10−4). However, as the SVD tolerance becomes evenhigher, the median mismatch seems to saturate around 2 ×10−4. This could be the effect of the interpolation, or anunintended technical imperfection that we did not model orexpect. However, this is still an order of magnitude below ourtarget mismatch of 0.003. We find that an SVD tolerance of(1–10−4) is adequate to achieve our target S/N loss.Effect of interpolation filter length. Next, keeping the SVD tol-erance fixed at (1–10−6) and the length of the decimation filterfixed at N↓ = 192, we studied the impact of the length N↑

of the interpolation filter on mismatch. We use GStreamer’sstock audioresample element, which provides an FIR deci-mation filter with a Kaiser-windowed sinc function kernel. Themismatch as a function of N↑ is shown in Figure 5(b). Themismatch saturates at ∼2 × 10−4 with N↑ = 64. We find thata filter length of 16 is sufficient to meet our target mismatch of0.003.

Having selected an SVD tolerance of (1–10−4) and N↑ = 16,we found that we could reduce N↓ to 48 without exceeding amedian mismatch of 0.003.

We found that careful design of the decimation and interpola-tion stages made a crucial difference in terms of computationaloverhead. Connecting the interpolation filters in cascade fashionrather than in parallel resulted in a significant speedup. Also,only the shortest interpolation filters that met our maximum mis-match constraint resulted in a sub-dominant contribution to theoverall cost. There is possibly further room for optimization be-yond minimizing N↑. We could design custom decimation andinterpolation filters, or we could tune these filters separately foreach time slice.

5.3. Other Potential Sources of S/N Loss

One possible source of S/N loss for which we have notaccounted is the leakage of sharp spectral features in thedetector’s noise spectrum due to the short durations of thetime slices. In the LLOID algorithm, as with many other GWsearch methods, whitening is treated as an entirely separate dataconditioning stage. In this paper, we assume that the input tothe filter bank is already whitened, having been passed througha filter that flattens and normalizes its spectrum. We elected toomit a detailed description of the whitening procedure since thefocus here is on the implementation of a scalable inspiral filterbank.

However, the inspiral templates themselves consist of theGW time series convolved with the impulse response of thewhitening filter. As a consequence, the LLOID algorithm mustfaithfully replicate the effect of the whitening filter. Sincein practice the noise spectra of ground-based GW detectorscontain both high-Q lines at mechanical, electronic, and control

Table 4Computational Cost in flop s−1 and Latency in Seconds of the Direct TD

Method, the Overlap-save FD Method, and LLOID

Flop s−1 Latency Flop s−1 Number ofMethod (Sub-bank) (s) (NS–NS) Machines

Direct (TD) 4.9 × 1013 0 3.8 × 1015 ∼3.8 × 105

Overlap-save (FD) 5.2 × 108 2 × 103 5.9 × 1010 ∼5.9LLOID (theory) 6.6 × 108 0 1.1 × 1011 ∼11LLOID (prototype) (0.9 cores) 0.5 . . . !10

Notes. Cost is given for both the sub-bank described in Section 5.1 and a full1–3 M⊙ NS–NS search. The last column gives the approximate number ofmachines per detector required for a full Advanced LIGO NS–NS search.

resonances and a very sharp rolloff at the seismic wall, thefrequency response of the LLOID filter bank must contain bothhigh-Q notches and a very abrupt high-pass filter. FIR filterswith rapidly varying frequency responses tend to have longimpulse responses and many coefficients. Since the LLOIDbasis filters have, by design, short impulse responses and veryfew coefficients, one might be concerned about spectral leakagecontaminating the frequency response of the LLOID filter bank.

The usual statement of the famous Nyquist–Shannon theo-rem, stated below as Theorem 1, has a natural dual, Theorem 2,that addresses the frequency resolution that can be achieved withan FIR filter of a given length.

Theorem 1 (After Oppenheim et al. 1997, p. 518) Let x(t) bea band-limited signal with continuous Fourier transform x(f )such that x(f ′) = 0 ∀ f ′ : |f ′| > fM . Then, x(t) is uniquelydetermined by its discrete samples x(n/f 0), n = 0,±1,±2, . . .,if f 0 > 2fM .

Theorem 2 Let x(t) be a compactly supported signal suchthat x(t ′) = 0 ∀ t ′ : |t ′| > tM . Then its continuous Fouriertransform x(f ) is uniquely determined by the discrete frequencycomponents x(n ∆f ), n = 0,±1,±2, . . ., if ∆f < 1/(2tM ).

Another way of stating Theorem 2 is that, provided x(t)is nonzero only for |t | < 1/(2 ∆f ), the continuous Fouriertransform can be reconstructed at any frequency f from aweighted sum of sinc functions centered at each of the discretefrequency components, namely,

x(f ) ∝∞!

n=−∞x (n ∆f ) sinc [π (f − n ∆f )/∆f ] .

Failure to meet the conditions of this dual of the samplingtheorem results in spectral leakage. For a TD signal to capturespectral features that are the size of the central lobe of the sincfunction, the signal must have a duration greater than 1/∆f . Ifthe signal x(t) is truncated by sampling it for a shorter duration,then its Fourier transform becomes smeared out; conceptually,power “leaks” out into the side lobes of the sinc functions andwashes away sharp spectral features. In the GW data analysisliterature, the synthesis of inspiral matched filters involves astep called inverse spectrum truncation (see Allen et al. 2011,Section VII) that fixes the number of coefficients based on thedesired frequency resolution.

In order to effectively flatten a line in the detector’s noisepower spectrum, the timescale of the templates must be at leastas long as the damping time τ of the line, τ = 2Q/ω0, whereQ is the quality factor of the line and w0 is the central angularfrequency. To put this into the context of the sampling theorem,in order to resolve a notch with a particular Q and f0, an FIR

10

K.  Cannon  et  al.,    ApJ,  748,  136  (2012).  Can  comparable  with  Freq.  domain  analysisStill  latency  is  very  low

Analysis  method

21

Parameter  estimation• Accurate  waveform  models  (NR  simulation)

• Bayesian  methods  :  time  consuming• Development  for  efficient  code  for  speed-­‐up

KAGALI  MCMC  (under  development):Korean  group:  Hyung Won  Lee,  Chunglee Kim

LAL  codes:  Narikawa (with  help  of  UK-­‐Birmingham)

Ongoing  iKAGRA analysis

22

Projects MembersCBC  offline Yuzurihara, Tagoshi,  (Ueno)

CBC-­‐PE  in  KAGALI H.W.  Lee,  Jeongcho Kim, Chunglee Kim  

Burst Hayama

CW Eda,  Itoh

Radiometry  for  CW   K.Tanaka,  Kanda,  Itoh

CBC-­‐PE  on  injected signals Narikawa, (Tagoshi)

HHT  on  injected  signals Ueki, Takahashi,  Oohara,  Kanda,  Yokozawa

HW  injection  signals Yokozawa,  ……

Gaussianity,  CBC  non-­‐Gaussian  triggers,…

Kitaoka,  Sasaki,  Kanda,  Tagoshi,  …

Analysis  pipeline  of  KAGRA

23

CBC  (original  code  developed  in  KAGALI)• Offline  pipeline  (one  detector)  • (Low  latency  pipeliine)  (stopped)CBC,  gstlal (LAL)  network  =>  one  detector  (done)

Burst  (original  code)(will  be  included  in  KAGALI)  • Online/Offline  pipeline  (one  detector,  excess  power)

Continuous  Wave  (LAL  code)• Known  pulsar• Unknown  pulsar

Appendix

24

HHT  analysis  of  post  merger  phase

25

Using the results in Sec. IV B, we perform the fitting toevaluate the coefficient β2 from 1000 samples. Figure 7shows the distribution of β2 for H135 (purple region) andS15 (green region). The mean and the standard deviation ofβ2 become ð40.3" 18.7Þ Hz=ms for H135 and ð5.4"12.2Þ Hz=ms for S15. From these results, we can confirmthat Finst;3 increases with time for H135, while Finst;3 isapproximately constant for S15. This fact suggests thepossibility that we can distinguish the EOS from theevolution of the IF of MNS by using the HHT analysis.Note, however, that since the distribution of β2 for H135and S15 overlap each other in Fig. 7, there is a possibility

that we cannot distinguish the frequency evolution clearlyat this distance.

V. ACCURACY OF THE FREQUENCYEVOLUTION

Now we focus on the GWs emitted in the merger andpost-merger phases. A MNS is formed after the merger forthe models considered in this paper. We now examine howaccurately we can measure the GW frequency from MNSwith the HHT.In order to evaluate the measurement accuracy of the IF

of GWs from MNS, we define [23]

δ ¼ 100 ×WTSS½Finst;iðtÞ − FsignalðtÞ&

WTSS½FsignalðtÞ&; ð9Þ

WTSS½FinstðtÞ& ¼X

j

A2instðtjÞF2

instðtjÞ; ð10Þ

where FsignalðtÞ is the frequency of the injected signal.We evaluate δ by using the IF and IA of IMF c2 or c3 for

each model listed in Table I. In Table III, we show the meanvalue of δ and the standard deviation for an event at adistance of 10 Mpc. The mean values and the standarddeviation of δ are plotted as functions of the source distancein Fig. 8.

−10 −5 0 5 10 15t − tmerge (ms)

0

1

2

3

4

Fin

st (k

Hz)

0.5 × 10−21

1.0 × 10−21

1.5 × 10−21

2.0 × 10−21

2.5 × 10−21

3.0 × 10−21

−10 −5 0 5 10 15t − tmerge (ms)

0

1

2

3

4

Fin

st (k

Hz)

0.5 × 10−21

1.0 × 10−21

1.5 × 10−21

2.0 × 10−21

2.5 × 10−21

3.0 × 10−21

2 4 6 8 10 12t − tmerge (ms)

1.8

2

2.2

2.4

2.6

2.8

3

Fin

st (k

Hz)

0.5 × 10−21

1.0 × 10−21

1.5 × 10−21

2.0 × 10−21

2.5 × 10−21

3.0 × 10−21

2 4 6 8 10 12t − tmerge (ms)

1.8

2

2.2

2.4

2.6

2.8

3

Fin

st (k

Hz)

0.5 × 10−21

1.0 × 10−21

1.5 × 10−21

2.0 × 10−21

2.5 × 10−21

3.0 × 10−21

FIG. 4. Time-frequency-amplitude map (HHT map). The color bar denotes the value of the IA. The GW signal H135 (left top) and S15(right top) at a hypothetical distance of 5 Mpc are used. The left and right bottom figures are the enlargement of left and right top figures,respectively.

2.0

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2 4 6 8 10 12

Fin

st, 3

(kH

z)

t − tmerge (ms)

H135S15

FIG. 5. The Finst;3 for H135 (solid red curve) and S15 (dashedblue curve), as functions of time in post-merger phase.

MASATO KANEYAMA et al. PHYSICAL REVIEW D 93, 123010 (2016)

123010-8

Kaneyama et  al，PRD93,  123010  (2016)Hyperon EOS Shen EOS

Current  computer  system  at  OCU

So  called"Shingakujutsu system"

Computing  nodes28  nodes    (760  cores)Storage:    304  TiB

This  was  first  introduced  in  2013,  and  has  been  upgraded  gradually.  This  will  not  be  enough  for  observation  era  of  bKAGRA from  2019.Especially,  CPUs  are  not  enough.  Further  upgrade  or  new  system  is  necessary.  

26

Expected  event  rate

27

Class. Quantum Grav. 27 (2010) 173001 Topical Review

Table 4. Compact binary coalescence rates per Mpc3 per Myra.

Source Rlow Rre Rhigh Rmax

NS–NS (Mpc−3 Myr−1) 0.01 [1] 1 [1] 10 [1] 50 [16]NS–BH (Mpc−3 Myr−1) 6 × 10−4 [18] 0.03 [18] 1 [18]BH–BH (Mpc−3 Myr−1) 1 × 10−4 [14] 0.005 [14] 0.3 [14]

a See footnotes in table 2 for details on the sources of the values in this table.

Table 5. Detection rates for compact binary coalescence sources.

IFO Sourcea Nlow yr−1 Nre yr−1 Nhigh yr−1 Nmax yr−1

NS–NS 2 × 10−4 0.02 0.2 0.6NS–BH 7 × 10−5 0.004 0.1

Initial BH–BH 2 × 10−4 0.007 0.5IMRI into IMBH <0.001b 0.01c

IMBH-IMBH 10−4 d 10−3 e

NS–NS 0.4 40 400 1000NS–BH 0.2 10 300

Advanced BH–BH 0.4 20 1000IMRI into IMBH 10b 300c

IMBH-IMBH 0.1d 1e

a To convert the rates per MWEG in table 2 into detection rates, optimal horizon distances of33 Mpc/445 Mpc are assumed for NS–NS inspirals in the Initial/Advanced LIGO–Virgo networks. ForNS–BH inspirals, horizon distances of 70 Mpc/927 Mpc are assumed. For BH–BH inspirals, horizondistances of 161 Mpc/2187 Mpc are assumed. These distances correspond to a choice of 1.4 M⊙ forNS mass and 10 M⊙ for BH mass. Rates for IMRIs into IMBHs and IMBH–IMBH coalescences arequoted directly from the relevant papers without conversion. See section 3 for more details.b Rate taken from the estimate of BH–IMBH IMRI rates quoted in [19] for the scenario of BH–IMBHbinary hardening via three-body interactions; the fraction of globular clusters containing suitableIMBHs is taken to be 10%, and no interferometer optimizations are assumed.c Rate taken from the optimistic upper limit rate quoted in [19] with the assumption that all globularclusters contain suitable IMBHs; for the advanced network rate, the interferometer is assumed to beoptimized for IMRI detections.d Rate taken from the estimate of IMBH-IMBH ringdown rates quoted in [20] assuming 10% of allyoung star clusters have sufficient mass, a sufficiently high binary fraction, and a short enough corecollapse time to form a pair of IMBHs.e Rate taken from the estimate of IMBH-IMBH ringdown rates quoted in [20] assuming all young starclusters have sufficient mass, a sufficiently high binary fraction, and a short enough core collapse timeto form a pair of IMBHs.

Where posterior probability density functions (PDFs) for rates are available, Rre refersto the PDF mean, Rlow and Rhigh are the 95% pessimistic and optimistic confidence intervals,respectively, and Rmax is the upper limit, quoted in the literature based on very basic limits setby other astrophysical knowledge (see table 1). However, many studies do not evaluate therate predictions in that way, and for some speculative sources even estimates of uncertaintiesmay not be available at present. In these cases, we assign the rate estimates available in theliterature to one of the four categories, as described in detail in section 4. The values in alltables in this section are rounded to a single significant figure; in some cases, the roundingmay have resulted in somewhat optimistic predictions.

9

Class. Quantum Grav. 27 (2010) 173001

0.1  – 300  /Gpc3/yr12

FIG. 8. The cumulative (right to left) distribution of observed trig-gers in the GstLAL analysis as a function of the log likelihood. Thebest fit signal + noise distribution, and the contributions from signaland noise are also shown. The shaded regions show 1s uncertain-ties. The observations are in good agreement with the model. Atlow likelihood, the distribution matches the noise model, while athigh likelihood it follows the signal model. Three triggers are clearlyidentified as being more likely to be signal than noise. GW150914stands somewhat above the expected distribution, as it is an unusu-ally significant event – only 6% of the astrophysical distribution ofsources appearing in our search with a false rate of less than one percentury will be more significant than GW150914.

than was achieved in [42], due to the longer duration of datacontaining a larger number of detected signals.

To do so, we consider two classes of triggers: those whoseorigin is astrophysical and those whose origin is terrestrial.Terrestrial triggers are the result of either instrumental or en-vironmental effects in the detector, and their distribution iscalculated from the search background estimated by the anal-yses (as shown in Fig. 3). The distribution of astrophysicalevents is determined by performing large-scale simulations ofsignals drawn from astrophysical populations and added to thedata set. We then use our observations to fit for the number oftriggers of terrestrial and astrophysical origin, as discussed indetail in Appendix C. Figure 8 shows the inferred distributionsof signal and noise triggers, as well as the combined distribu-tion. The observations are in good agreement with the model.

It is clear from the figure that three triggers are more likelyto be signal (i.e. astrophysical) than noise (terrestrial). Weevaluate this probability and find that, for GW150914 andGW151226, the probability of astrophysical origin is unityto within one part in 106. Meanwhile for LVT151012, it iscalculated to be 0.87 and 0.86, for the PyCBC and GstLALanalyses respectively.

Given uncertainty in the formation channels of the various

Mass distribution R/(Gpc�3yr�1)

PyCBC GstLAL CombinedEvent based

GW150914 3.2+8.3�2.7 3.6+9.1

�3.0 3.4+8.6�2.8

LVT151012 9.2+30.3�8.5 9.2+31.4

�8.5 9.4+30.4�8.7

GW151226 35+92�29 37+94

�31 37+92�31

All 53+100�40 56+105

�42 55+99�41

AstrophysicalFlat in log mass 31+43

�21 30+43�21 30+43

�21Power Law (�2.35) 100+136

�69 95+138�67 99+138

�70

TABLE II. Rates of BBH mergers based on populations with massesmatching the observed events, and astrophysically motivated massdistributions. Rates inferred from the PyCBC and GstLAL analysesindependently as well as combined rates are shown. The table showsmedian values with 90% credible intervals.

BBH events, we calculate the inferred rates using a variety ofsource population parametrizations. For a given population,the rate is calculated as R = L/hV T i where L is the numberof triggers of astrophysical origin and hV T i is the population-averaged sensitive space-time volume of the search. We usetwo canonical distributions for BBH masses:

i a distribution uniform over the logarithm of componentmasses, p(m1,m2) µ m1

�1m2�1 and

ii assuming a power-law distribution in the primary mass,p(m1) µ m�2.35

1 with a uniform distribution on the sec-ond mass.

We require 5M� m2 m1 and m1 +m2 100M�. The firstdistribution probably overestimates the fraction of high-massblack holes and therefore overestimates hV T i resulting in anunderestimate the true rate while the second probably over-estimates the fraction of low-mass black holes and thereforeunderestimating hV T i and overestimating the true rate. Theinferred rates for these two populations are shown in Table IIand the rate distributions are plotted in Figure 10.

In addition, we calculate rates based upon the inferred prop-erties of the three significant events observed in the data:GW150914, GW151226 and LVT151012 [140]. Since theseclasses are distinct, the total event rate is the sum of the indi-vidual rates: R ⌘ RGW150914 + RLVT151012 + RGW151226. Notethat the total rate estimate is dominated by GW151226, as itis the least massive of the three likely signals and is thereforeobservable over the smallest space-time volume. The resultsfor these population assumptions also are shown in Table II,and in Figure 9. The inferred overall rate is shown in Fig. 10.As expected, the population-based rate estimates bracket theone obtained by using the masses of the observed black holebinaries.

The inferred rates of BBH mergers are consistent withthe results obtained in [42] following the observation ofGW150914. The median values of the rates have decreasedby approximately a factor of two, as we now have three likely

9 – 240  /Gpc3/yr