gwdaw - 10
DESCRIPTION
The Waves Group. Binary Chirp Template Banks: Tanaka-Tagoshi Parameterization for LIGO. R.P. Croce, Th. Demma, A. Fusco, V. Pierro*, I.M. Pinto, M. Principe. *workgroup coordinator. GWDAW - 10. CENTER FOR. Dec. 13-18, 2005. G RAVITATIONAL W AVE A STRONOMY. GWDAW - 10. CENTER FOR. - PowerPoint PPT PresentationTRANSCRIPT
GWDAW - 10Dec. 13-18, 2005
Binary Chirp Template Banks:Tanaka-Tagoshi Parameterization for LIGO
R.P. Croce, Th. Demma, A. Fusco, V. Pierro*,I.M. Pinto, M. Principe
The Waves Group
GGRAVITATIONAL RAVITATIONAL W WAVE AVE AASTRONOMYSTRONOMYCENTER FORCENTER FOR
*workgroup coordinator
GWDAW - 10Dec. 13-18, 2005GGRAVITATIONAL RAVITATIONAL W WAVE AVE AASTRONOMYSTRONOMY
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Plan
Template Placement Problem
Tanaka Tagoshi Parameterization
Tanaka-Tagoshi Style Placement for LIGO [preliminary results]
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ML Detection of Chirps
“Project” the (spectral data) over a certain set of templatesof the sought waveform;Take the largest projection as a detection-statistic;Compare to a false-alarm dictated detection-threshold;
If test is passed, declare detection & estimate parametersof signal detected from those of largest-projection template;
False dismissal probability is a monotonically decreasingfunction of the signal-template match;
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Structure of (RedPN) Match Functional
7 / 3
7 / 3
exp 2 ( ; , )( )sup
( , )
( )
off
in
off
in
f
c h g
f
fC
f
dff f T f
fM h g
T dff
f
( ) noise PSD; , spectral windowin offf f f
Maximises over extrinsicparameter Tc
(through simple FFT)
2 1
1
( ; , ) ( ) ( ) ( )N
h g n n h n kn
f f
maximizes over extrinsicadditive phase
nntrinsic parameters
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Chirp Parameters (2PN)
Free parameters: companion masses + spin-spin & spin-orbit.
GWDAW - 10Dec. 13-18, 2005GGRAVITATIONAL RAVITATIONAL W WAVE AVE AASTRONOMYSTRONOMY
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m1=m2
3
(mmin,mmin,)
(mmin,mmax,)
(mmax,mmax)
8 / 3 5/ 3 10 0
2 / 35/ 3 2 / 3 1
3 0
1 21 2 2
5
256
8
;
f M
f M
m mM m m
M
Spin-Free Search Manifold
GWDAW - 10Dec. 13-18, 2005GGRAVITATIONAL RAVITATIONAL W WAVE AVE AASTRONOMYSTRONOMY
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Minimal Match Prescription
The template bank {k}, aka the set {Xk} in search manifoldshould be such that for any admissible signal h there is (at least) one template g such that M(h,g)=, viz.
, : ( , )h g M h g
Corresponds roughly to allowing a fraction r = ( 1 - 3 )of potentially observable CBS sources to be missed justas an effect of insufficient parameter-space sampling
e.g., r = 10% 0.97
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X0
The span S (X0),
0 0( , ) , ( )C X X X S X
0 0( , ) , ( )C X X X X
The iso-match contour-line (X0) = S (X0),
Span of a Template
X0 = par. space point (identifies template)
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Building a Bank
Translate MM prescription in terms of spans:
( )i iS X (no holes)
Also try to cope with the following nice requirements: Use minimum number of templates (save on computation) Get minimum spillover across m1=m2 line (discard unphysical)
End up in a regular lattice (easier placement; interpolation)
GWDAW - 10Dec. 13-18, 2005GGRAVITATIONAL RAVITATIONAL W WAVE AVE AASTRONOMYSTRONOMY
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X0
Span of a Template, contd.
At relatively large MMs, iso-match contour lines turn out to be ellipses...
You can always make them into circlesvia trivial coordinate transformations
X0 X’0
X’0
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Regular Tilings of the Plane :
- Identify largest-area span-inscribed (regular) polygon : the basic tile ;- Tile the plane thereof; place templates-lattice nodes at tile vertexes. Effective span Veff of tiled-templates: Voronoi dual of basic tile.
Sparsest coverage obtained from triangular tiling (hexagonal effective span).
Basic tile
Area(Veff) : Area(Veff) : Area(Veff) = 1.3:1:0.65(6) (4) (3)
……vs. template lattices
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Regular Tilings of the Plane: Triangular
R.P. Croce et al., Phys. Rev. D65 (2002) 102003.
Triangular tile
Voronoi conjugate(hexagon) of basictile: effective span.
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X0
Span of a Template, contd.
...At relatively large MMs, iso-match contour lines are ellipses : the match is well approximated by a quadratic form:
Iso-match ellipses stretch & rotate as one moves across !
( , ) 1 1r s
r srs rs p q
p q
M h g G G m mm m
2D metric has nonzero curvature!
Template placement is tricky !
6D
…and you cannot get rid of it via a global transformation…
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Span of a Template, contd.
…as a further complication,iso-match contour lines are no-longer elliptical at relatively low MMs (e.g., = 0.9).
(as required, e.g. in the firststage of hierarchical strategies)
3
The “quadratic” approximationmay underestimates the span coverage e.g. by a factor
~1.45 at =0.95 ~6.13 at =0.8
GWDAW - 10Dec. 13-18, 2005GGRAVITATIONAL RAVITATIONAL W WAVE AVE AASTRONOMYSTRONOMY
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Template Placement : Mainstream
Place a string of templates along the equal mass line; [copes with minimal spillover requirement];
Lay a regular rectangular (hexagonal) tiling of the (0 ,3) plane; [pay in terms of redundancy here, to ignore iso-match contour line stretch-rotation pathology];
Do not put templates at nodes of the tiling for whichthe node is external to ; andthe node is not on a vertex of .
B. Owen and B.S. Sathyaprakash, Ph. Rev. D60 22002 (1999)
Content yourself with quadratic match approximation [OK for large , but bad underestimate of effective span at low , as in the first stage of hierarchical strategies]
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Template Placement, the Virgo wayD. Buskulic et al, Class. Quantum Grav. 20, 789, 2003.
1) Triangulate par space coarsely, and compute exact (elliptical) spans at each vertex.
3) Sub-triangulate patches where interpolation fails (fifth iteration shown).
2) Compare actual span at triangle’s center with linear-interpolated one;
GWDAW - 10Dec. 13-18, 2005GGRAVITATIONAL RAVITATIONAL W WAVE AVE AASTRONOMYSTRONOMY
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Progress (?) Directions
time needed to compute bank;-Reduce number of templates needed;
-Handle non-elliptical, low - iso-match contours
Our attempt : go through Tanaka Tagoshi transformation
Expected benefits:
- Work in flat manifold translation–invariant iso-math contours- Go workably beyond quadratic (elliptic contour) approximation…do not underestimated span-areas use lesser templates
…based on previous related work [ R.P. Croce et al., Ph. Rev. D64, 042005 (2001);R.P. Croce et al., Ph. Rev. D64, 042005 (2001) ]
GWDAW - 10Dec. 13-18, 2005GGRAVITATIONAL RAVITATIONAL W WAVE AVE AASTRONOMYSTRONOMY
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Tanaka-Tagoshi Transformation
T.Tanaka and H.Tagoshi, Ph. Rev. D62, 82001 2000.
Linear mapping from “natural” search manifold to a globally flat one such that:
( , ) ( , )1
( , )
M h g M h g
M h g
( , ) ( )ghM h g M X X
, ,h g h g ,<<
Regular grid of nodes, globally uniform lattice in
GWDAW - 10Dec. 13-18, 2005GGRAVITATIONAL RAVITATIONAL W WAVE AVE AASTRONOMYSTRONOMY
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Tanaka-Tagoshi Transformation, technical
i- Orthogonalize the
- Find a rotation which maps all three vertexes pi (i=1,2,3)
(mmin,mmin),(mmin,mmax),(mmax,mmax)
down into the (x1,x2) plane.
P and obtained from Jordan decomposition G = PT P ’ = ½
x = Q ’ such thatQ ( p’2 – p’1 ) = 21x1 + 22x2
Q ( p’3 – p’1 ) = 11x1 …not unique
In practice, form (nonsingular) Z matrix out of col. vectors
The unique (straightforward) QR decomposition of Z , Solves Q X = R , and hence the system. Hence it provides Q.
(p’3 - p’1, p’2 - p’1, ’3, ’4, ’5)^ ^ ^
Z = Q T R
GWDAW - 10Dec. 13-18, 2005GGRAVITATIONAL RAVITATIONAL W WAVE AVE AASTRONOMYSTRONOMY
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LIGO200
400
x1
0.8
1
1.2
x2
0.02
0.025
0.03
0.02
0.025
0.03
x1x2
100
500
1/ 22 2 23 4 5x x x
LIGO PSD (LAL)fin=40 Hz,foff=750 Hz
1 M M1,M2 3 M
The Distance between and
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TT Iso-match Contour Levels
2. 1. 0. 1. 2.x1
2.
1.
0.
1.
2.
x 2
0.70.80.90.950.970.99
LIGO PSD (LAL)fin=40 Hz,foff=750 Hz
GWDAW - 10Dec. 13-18, 2005GGRAVITATIONAL RAVITATIONAL W WAVE AVE AASTRONOMYSTRONOMY
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TT Iso-match Contour Level: Exact vs Quadratic Approximation
2. 1. 0. 1. 2.x1
2.
1.
0.
1.
2.
x 2
0.2 0.1 0. 0.1 0.2x1
0.2
0.1
0.
0.1
0.2
x 2
Quadratic
Quadratic
Exact
Exact
Areaexact/Areaquadratic = 6.13 Areaexact/Areaquadratic = 1.43
=0.8 =0.95
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1) Compute Tanaka-Tagoshi transformation [ from antenna PSD, (fin, foff), & PN phasing formula ] ;
2) Compute iso-match contour in TT plane [ from prescribed ] ; 3) Construct search-manifold in TT plane [ from prescribed mass-range] ;
4) Build-up “optimal” triangular-tiling [trade maximal sparsity for minimal spillover] ;
5) Create regular template lattice covering ;
6) Map TT lattice back to (0,3) or (m 1,m 2) using -1.
TT-style Bank Generation Algorithm
GWDAW - 10Dec. 13-18, 2005GGRAVITATIONAL RAVITATIONAL W WAVE AVE AASTRONOMYSTRONOMY
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Optimum Tiling: Maximum Span
Draw convex hull& identify butterflyshaped region
x2
x1
(TT)
iso-match contour in TT plane
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Maximal-Span Tiling
Chords through centerwithin butterfly-shaped region parameterized in angle;
Take one
Draw convex hull& identify butterflyshaped region
x2
x1
(TT)
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Maximal-Span Tiling
Chords through centerwithin butterfly-shaped region parameterize inan angle;
Take one
Draw convex hull& identify butterflyshaped region
Slide chord to bottom (touch w/our intersect)
x2
x1
(TT)
GWDAW - 10Dec. 13-18, 2005GGRAVITATIONAL RAVITATIONAL W WAVE AVE AASTRONOMYSTRONOMY
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Maximal-Span Tiling
Chords through centerwithin butterfly-shaped region parameterize inan angle;
Take one
Draw convex hull& identify butterflyshaped region
Slide chord to bottom (touch w/out intersect)
x2
x1
(TT)
GWDAW - 10Dec. 13-18, 2005GGRAVITATIONAL RAVITATIONAL W WAVE AVE AASTRONOMYSTRONOMY
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Maximal-Span Tiling
Chords through centerwithin butterfly-shaped region parameterize inan angle;
Take one
Draw convex hull& identify butterflyshaped region
Slide chord to bottom (touch w/our intersect)
Find 3rd vertex formax triangle areaTwo-parameter (two angles) optimization
x2
x1
(TT)
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Maximal-Span Tilingx2
x1
(TT)
… coverage mechanism …
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Maximal-Span Tiling, contd.
x2
x1
(TT)
2
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Maximal-Span vs. Minimal Spillover
Using maximal – span tilings does not guarantee Minimal template spillover across equal mass line;
Spillover across equal mass line efficiently minimizedif tile-baseline is parallel to the straight-line connecting the endpoints of the equal mass line;
This choice entails a negligible span-reduction (less than 1% at =.97), resulting in nearly-maximal sparsity, and minimal-spillover;
Resulting template-placement algorithm straightforward.
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Quasi-Maximal-Span Reduced-Spillover Tiling
Chords through centerwithin butterfly-shaped region parameterize inan angle;
Take the one parallelto the line connecting the vertexes of thesearch-manifold (TT)equal mass line
Draw convex hull& identify butterflyshaped region
One-parameter (upper vertex angle) search only
x2
x1
(TT)
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Span Reduction after Tile Baseline Rotation
0. 0.5 1. 1.5 2. 2.5 3.
0.99
0.992
0.994
0.996
0.998
1.
AA tpo
0.2 0.1 0. 0.1 0.2x1
0.2
0.1
0.
0.1
0.2
x2
LIGO I Optimum triangular tiling
.97
Are
a re
duct
ion
fact
or
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Template Placement Start
search manifold
equal-mass line (TT)
Vertex-joining line
base-tileshape
(downward-concave equal-mass line)
Draw basic-tile with lower-left vertex coincident with (leftmost) m1=m2=mmax TT search-manifold vertex.
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Template Placement Start
search manifold
equal-mass line (TT)
Vertex-joining line
Iso-matchcontour line
Draw associated iso-match contour line. This identifies location of first template.
Case # 1
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Template Placement Start
search manifold
equal-mass line (TT)
Vertex-joining line
Tilinglattice
Deploy tiling lattice thereof.
Case # 1
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Template Placement Start
search manifold
equal-mass line (TT)
Vertex-joining line
Deploy tiling lattice thereof.
Case # 1
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Template Placement Start
search manifold
equal-mass line (TT)
Vertex-joining line
Discard un-needed lattice points (null intersection between span and )
Case # 1
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(201)
100. 200. 300. 400. 500.x1
0.8
0.9
1.
1.1
1.2
1.3
1.4
x2
Discarding Un-needed Templates
…see this at workin a practical case…
LIGOfin=40Hz, foff=730 Hz,1 M M1,M2 3 M
GWDAW - 10Dec. 13-18, 2005GGRAVITATIONAL RAVITATIONAL W WAVE AVE AASTRONOMYSTRONOMY
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(201)
100. 200. 300. 400. 500.x1
0.8
0.9
1.
1.1
1.2
1.3
1.4
x2
CC’
B’ B
A
Discarding Un-needed Templates
…simple constructionyields fiducial template-lineendpoints, to berefined using asimple algorithm…
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Discarding Un-needed Templates
Starting from X = A, B, C, B’, C’
Accept template at X. Move X one lattice node forward (if started from A,B,C)or backward (if started from B’, C’)
( )S X Y
S=0
S = 1S = 0
N
Exit
Y
Discard template at X Move X one lattice node backward (if started from A,B,C) or forward (if started from B’, C’)
N
Span of template at X Search manifold
GWDAW - 10Dec. 13-18, 2005GGRAVITATIONAL RAVITATIONAL W WAVE AVE AASTRONOMYSTRONOMY
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LIGO
fin = 40Hz,
foff=730 Hz,
1 M M1,M2 3 M
= 0.97
# templates = 2331
(201)
100. 200. 300. 400. 500.x1
0.8
0.9
1.
1.1
1.2
1.3
1.4
x2
(1416)
(606)
(109)
TT-Style Template Placement for LIGO
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100. 200. 300. 400. 500.x1
0.8
0.9
1.
1.1
1.2
1.3
1.4
x2
1M M1,M2 3 M
= .97
# templates = 2331
354. 355. 356. 357. 358. 359.0.6
0.7
0.8
0.9
1.
1.1
1.2
1.3
235. 236. 237. 238. 239. 240.0.6
0.8
1.
1.2
1.4
1.6
90. 92. 94. 96. 98. 100.0.6
0.7
0.8
0.9
1.
1.1
158.5 159. 159.5 160. 160.5 161. 161.5 162.0.6
0.8
1.
1.2
LIGOfin=40Hz, foff=740 Hz
Close-ups
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0.2 0.4 0.6 0.8 1.0 1.20
0.06
0.08
0.10
0.12
3
LIGO PSD (LAL)
fin=40Hz,
foff=730 Hz,
1 M M1,M2 3 M
= 0.97
# templates = 2331
TT-Style Template Placement for LIGO
GWDAW - 10Dec. 13-18, 2005GGRAVITATIONAL RAVITATIONAL W WAVE AVE AASTRONOMYSTRONOMY
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1. 1.5 2. 2.5 3. 3.5 4.m1
1.
1.5
2.
2.5
3.
3.5
4.
m2
FF 0.97
LIGO
fin=40Hz,
foff=730 Hz,
1 M M1,M2 3 M
= 0.97
# templates = 2331
TT-Style Template Placement for LIGO
GWDAW - 10Dec. 13-18, 2005GGRAVITATIONAL RAVITATIONAL W WAVE AVE AASTRONOMYSTRONOMY
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LIGO
fin=40Hz,
foff=730 Hz,
1 M M1,M2 3 M
= 0.80
# templates = 410
TT-Style Template Placement for LIGO
100. 200. 300. 400. 500.x1
0.8
1.
1.2
1.4
x2
(401)
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TT-Style Template Placement for LIGO
0.2 0.4 0.6 0.8 1.0 1.20
0.06
0.08
0.10
0.12
0.14
3
LIGO
fin=40Hz,
foff=730 Hz,
1 M M1,M2 3 M
= 0.80
# templates = 410
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TT-Style Template Placement for LIGO
2000. 4000. 6000. 8000.x1
0.5
1.
1.5
2.
2.5
3.
3.5
4.
x2
The search manifoldbaseline may have a different concavity…
LIGOfin=40Hz, foff=2200 Hz,0.2 M M1,M2 1 M
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TT-Style Template Placement for LIGO
2000. 4000. 6000. 8000.x1
0.5
1.
1.5
2.
2.5
3.
3.5
4.
x2
Here one starts placingbasic-tile with lower-leftvertex at the contactpoint of with a par-allel to the line joiningthe lower vertexes.
The subsequent stepsare the same…
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TT-Style Template Placement for LIGO
2000. 4000. 6000. 8000.x1
0.5
1.
1.5
2.
2.5
3.
3.5
4.
x2
…one draws lines midway between thetemplate-lines., andprojecting the inter-sections with …
GWDAW - 10Dec. 13-18, 2005GGRAVITATIONAL RAVITATIONAL W WAVE AVE AASTRONOMYSTRONOMY
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TT-Style Template Placement for LIGO
2000. 4000. 6000. 8000.x1
0.5
1.
1.5
2.
2.5
3.
3.5
4.
x2
…one obtains naïve first estimates of theTemplate-line end-points, to be refinedas already shown…
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2000. 4000. 6000. 8000.x1
0.5
1.
1.5
2.
2.5
3.
3.5
4.
x2
LIGO
LIGO
fin=40Hz,
foff=2200 Hz,
0.2 M M1,M2 1 M
= 0.95
# templates = 56328(17088)
(19674)(8499)
(5587)
(3468)
(1721)
(201)
TT-Style Template Placement for LIGO
GWDAW - 10Dec. 13-18, 2005GGRAVITATIONAL RAVITATIONAL W WAVE AVE AASTRONOMYSTRONOMY
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LIGO
fin=40Hz,
foff=2200 Hz,
0.2 M M1,M2 1 M
= 0.80
# templates = 7199
2000. 4000. 6000. 8000.x1
1.
2.
3.
4.
x2
LIGO
TT-Style Template Placement for LIGO
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0.80 410
0.90 754
0.95 1203+332 = 1535
0.97 1566+646+119 = 2331
1 - 3M
40-730 Hz
# templates
0.2 - 1M
40-2200 Hz
# templates
0.80 6504+695 = 7199
0.95 17088+19674+8499+5587+3468+1721+201= 56328
TT-Style Template Budget for LIGO
98936
3624
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From TT to Companion Masses or Chirp-times
X1k= 1ii(M,)X2k= 2ii(M,)
TT operator (direct)
Solve formally for (algebraic,2nd degree)
yields higher order algebraic equation in M.(solve numerically, e.g. Cholewsky)Has only one real root for which acceptable.
Pretty fast, even in MATHEMATICA pre-code.
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Future Work
Write LAL compliant code & documentation
Perform blind injection benchmarks
… special thanks to Duncan Brown for providing comparative figures in “real time” !