GWDAW - 10

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

CENTER FORCENTER FOR

Template Placement Problem

Tanaka Tagoshi Parameterization

Tanaka-Tagoshi Style Placement for LIGO [preliminary results]

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;

Structure of (RedPN) Match Functional

exp 2 ( ; , )( )sup

dff f T f

fM h g

( ) noise PSD; , spectral windowin offf f f

Maximises over extrinsicparameter Tc

(through simple FFT)

( ; , ) ( ) ( ) ( )N

h g n n h n kn

maximizes over extrinsicadditive phase

nntrinsic parameters

Chirp Parameters (2PN)

Free parameters: companion masses + spin-spin & spin-orbit.

(mmin,mmin,)

(mmin,mmax,)

(mmax,mmax)

8 / 3 5/ 3 10 0

2 / 35/ 3 2 / 3 1

1 21 2 2

m mM m m

Spin-Free Search Manifold

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

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)

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)

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

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

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.

...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

M h g G G m mm m

2D metric has nonzero curvature!

Template placement is tricky !

…and you cannot get rid of it via a global transformation…

…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)

The “quadratic” approximationmay underestimates the span coverage e.g. by a factor

~1.45 at =0.95 ~6.13 at =0.8

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]

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;

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) ]

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

( , ) ( )ghM h g M X X

, ,h g h g ,<<

Regular grid of nodes, globally uniform lattice in

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

LIGO200

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

TT Iso-match Contour Levels

2. 1. 0. 1. 2.x1

0.70.80.90.950.970.99

LIGO PSD (LAL)fin=40 Hz,foff=750 Hz

TT Iso-match Contour Level: Exact vs Quadratic Approximation

2. 1. 0. 1. 2.x1

0.2 0.1 0. 0.1 0.2x1

Quadratic

Areaexact/Areaquadratic = 6.13 Areaexact/Areaquadratic = 1.43

=0.8 =0.95

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

Optimum Tiling: Maximum Span

Draw convex hull& identify butterflyshaped region

iso-match contour in TT plane

Maximal-Span Tiling

Chords through centerwithin butterfly-shaped region parameterized in angle;

Take one

Maximal-Span Tiling

Chords through centerwithin butterfly-shaped region parameterize inan angle;

Take one

Slide chord to bottom (touch w/our intersect)

Maximal-Span Tiling

Take one

Slide chord to bottom (touch w/out intersect)

Maximal-Span Tiling

Take one

Slide chord to bottom (touch w/our intersect)

Find 3rd vertex formax triangle areaTwo-parameter (two angles) optimization

Maximal-Span Tilingx2

… coverage mechanism …

Maximal-Span Tiling, contd.

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.

Quasi-Maximal-Span Reduced-Spillover Tiling

Take the one parallelto the line connecting the vertexes of thesearch-manifold (TT)equal mass line

One-parameter (upper vertex angle) search only

Span Reduction after Tile Baseline Rotation

0. 0.5 1. 1.5 2. 2.5 3.

AA tpo

0.2 0.1 0. 0.1 0.2x1

LIGO I Optimum triangular tiling

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.

search manifold

Vertex-joining line

Iso-matchcontour line

Draw associated iso-match contour line. This identifies location of first template.

Case # 1

search manifold

Vertex-joining line

Tilinglattice

Deploy tiling lattice thereof.

Case # 1

search manifold

Vertex-joining line

Deploy tiling lattice thereof.

Case # 1

search manifold

Vertex-joining line

Discard un-needed lattice points (null intersection between span and )

Case # 1

100. 200. 300. 400. 500.x1

Discarding Un-needed Templates

…see this at workin a practical case…

LIGOfin=40Hz, foff=730 Hz,1 M M1,M2 3 M

100. 200. 300. 400. 500.x1

B’ B

…simple constructionyields fiducial template-lineendpoints, to berefined using asimple algorithm…

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 = 1S = 0

Discard template at X Move X one lattice node backward (if started from A,B,C) or forward (if started from B’, C’)

Span of template at X Search manifold

fin = 40Hz,

foff=730 Hz,

1 M M1,M2 3 M

= 0.97

# templates = 2331

100. 200. 300. 400. 500.x1

(1416)

TT-Style Template Placement for LIGO

100. 200. 300. 400. 500.x1

1M M1,M2 3 M

# templates = 2331

354. 355. 356. 357. 358. 359.0.6

235. 236. 237. 238. 239. 240.0.6

90. 92. 94. 96. 98. 100.0.6

158.5 159. 159.5 160. 160.5 161. 161.5 162.0.6

LIGOfin=40Hz, foff=740 Hz

Close-ups

0.2 0.4 0.6 0.8 1.0 1.20

LIGO PSD (LAL)

fin=40Hz,

foff=730 Hz,

1 M M1,M2 3 M

= 0.97

# templates = 2331

1. 1.5 2. 2.5 3. 3.5 4.m1

FF 0.97

fin=40Hz,

foff=730 Hz,

1 M M1,M2 3 M

= 0.97

# templates = 2331

fin=40Hz,

foff=730 Hz,

1 M M1,M2 3 M

= 0.80

# templates = 410

100. 200. 300. 400. 500.x1

0.2 0.4 0.6 0.8 1.0 1.20

fin=40Hz,

foff=730 Hz,

1 M M1,M2 3 M

= 0.80

# templates = 410

2000. 4000. 6000. 8000.x1

The search manifoldbaseline may have a different concavity…

LIGOfin=40Hz, foff=2200 Hz,0.2 M M1,M2 1 M

2000. 4000. 6000. 8000.x1

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…

2000. 4000. 6000. 8000.x1

…one draws lines midway between thetemplate-lines., andprojecting the inter-sections with …

2000. 4000. 6000. 8000.x1

…one obtains naïve first estimates of theTemplate-line end-points, to be refinedas already shown…

2000. 4000. 6000. 8000.x1

fin=40Hz,

foff=2200 Hz,

0.2 M M1,M2 1 M

= 0.95

# templates = 56328(17088)

(19674)(8499)

(5587)

(3468)

(1721)

fin=40Hz,

foff=2200 Hz,

0.2 M M1,M2 1 M

= 0.80

# templates = 7199

2000. 4000. 6000. 8000.x1

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

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.

Future Work

Write LAL compliant code & documentation

Perform blind injection benchmarks

… special thanks to Duncan Brown for providing comparative figures in “real time” !

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