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GWDAW - 10

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.

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GWDAW - 10

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  1. 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 GRAVITATIONAL WAVE ASTRONOMY

  2. GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Plan Template Placement Problem Tanaka Tagoshi Parameterization Tanaka-Tagoshi Style Placement for LIGO [preliminary results]

  3. GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY ML Detection of Chirps “Project” the (spectral data) over a certain set of templates of 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 parameters of signal detected from those of largest-projection template; False dismissal probability is a monotonically decreasing function of the signal-template match;

  4. GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Structure of (RedPN) Match Functional Maximises over extrinsic parameter Tc (through simple FFT) maximizes over extrinsic additive phase nntrinsic parameters

  5. GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Chirp Parameters (2PN) Free parameters: companion masses+ spin-spin & spin-orbit.

  6. 3 (mmin,mmax,) (mmin,mmin,) m1=m2 (mmax,mmax)  GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Spin-Free Search Manifold

  7. GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Minimal Match Prescription The template bank {k}, aka the set {Xk} in search manifold should be such that for any admissible signal h there is (at least) one template gsuch that M(h,g)=, viz. Corresponds roughly to allowing a fraction r = ( 1 - 3 ) of potentially observable CBS sources to be missed just as an effect of insufficient parameter-space sampling e.g., r = 10%   0.97

  8. The iso-match contour-line  (X0) =  S (X0), The span S (X0), GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Span of a Template X0 = par. space point (identifies template) X0

  9. (no holes) GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Building a Bank Translate MM prescription in terms of spans: 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)

  10. X0 X0 X’0 X’0 GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Span of a Template, contd. At relatively large MMs, iso-match contour lines turn out to be ellipses... You can always make them into circles via trivial coordinate transformations

  11. … vs. template lattices GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY … Regular Tilings of the Plane : Basic tile - Identify largest-area span-inscribed (regular) polygon : the basic tile ; - Tile the plane thereof; place templates-lattice nodes at tile vertexes. Effective spanVeff of tiled-templates:Voronoi dualof basic tile. Sparsest coverage obtained from triangular tiling (hexagonal effective span). (4) (3) (6) Area(Veff): Area(Veff) : Area(Veff) = 1.3:1:0.65

  12. GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Regular Tilings of the Plane: Triangular Triangular tile Voronoi conjugate (hexagon) of basic tile: effective span. R.P. Croce et al., Phys. Rev. D65 (2002) 102003.

  13. X0 GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Span of a Template, contd. ...At relatively large MMs, iso-match contour lines are ellipses : the match is well approximated by a quadratic form: 6D 2D metric has nonzero curvature! Iso-match ellipses stretch & rotate as one moves across  ! …and you cannot get rid of it via a global transformation… Template placement is tricky !

  14. 3  GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY 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 first stage of hierarchical strategies) The “quadratic” approximation may underestimates the span coverage e.g. by a factor ~1.45 at =0.95 ~6.13 at =0.8

  15. GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Template Placement : Mainstream 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] 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 which the node is external to; and the node is not on a vertexof.

  16. GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Template Placement, the Virgo way D. Buskulic et al, Class. Quantum Grav. 20, 789, 2003. 1) Triangulate par space coarsely, and compute exact (elliptical) spans at each vertex. 2) Compare actual span at triangle’s center with linear-interpolated one; 3) Sub-triangulate patches where interpolation fails (fifth iteration shown).

  17. time needed to compute bank; -Reduce number of templates needed; -Handle non-elliptical, low -  iso-match contours Our attempt : go through Tanaka Tagoshi transformation …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) ] 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 GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Progress (?) Directions

  18. Linear mapping  from “natural” search manifold  to a globally flat one such that: , << Regular grid of nodes, globally uniform lattice in GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Tanaka-Tagoshi Transformation T.Tanaka and H.Tagoshi, Ph. Rev. D62, 82001 2000.

  19. - Orthogonalize the ’ = ½   P and  obtained from Jordan decomposition G = PT P - 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. x = Q ’ such that …not unique Q ( p’3 – p’1 ) = 11x1 In practice, form (nonsingular) Z matrix out of col. vectors ^ ^ ^ (p’3 - p’1, p’2 - p’1, ’3, ’4, ’5) Q ( p’2 – p’1 ) = 21x1 + 22x2 The unique (straightforward) QR decomposition of Z , Z = Q T R Solves Q X = R , and hence the system. Hence it providesQ. GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Tanaka-Tagoshi Transformation, technical

  20. The Distance between and  100 x1 x2 500 GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY LIGO PSD (LAL) fin=40 Hz, foff=750 Hz 1M  M1,M2 3 M

  21. GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY TT Iso-match Contour Levels LIGO PSD (LAL) fin=40 Hz, foff=750 Hz  0.7 0.8 0.9 0.95 0.97 0.99

  22. GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY TT Iso-match Contour Level: Exact vs Quadratic Approximation =0.8 =0.95 Quadratic Quadratic Exact Exact Areaexact/Areaquadratic = 6.13 Areaexact/Areaquadratic = 1.43

  23. GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY TT-style Bank Generation Algorithm 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.

  24. x2 (TT) x1 iso-match contour  in TT plane GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Optimum Tiling: Maximum Span Draw convex hull & identify butterfly shaped region

  25. x2 (TT) x1 GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Maximal-Span Tiling Draw convex hull & identify butterfly shaped region Chords through center within butterfly-shaped region parameterized in angle; Take one

  26. x2 (TT) x1 GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Maximal-Span Tiling Draw convex hull & identify butterfly shaped region Chords through center within butterfly-shaped region parameterize in an angle; Take one Slide chord to bottom (touch w/our intersect)

  27. x2 (TT) x1 GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Maximal-Span Tiling Draw convex hull & identify butterfly shaped region Chords through center within butterfly-shaped region parameterize in an angle; Take one Slide chord to bottom (touch w/out intersect)

  28. x2 (TT) x1 Two-parameter (two angles) optimization GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Maximal-Span Tiling Draw convex hull & identify butterfly shaped region Chords through center within butterfly-shaped region parameterize in an angle; Take one Slide chord to bottom (touch w/our intersect) Find 3rd vertex for max triangle area

  29. x2 (TT) x1 GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Maximal-Span Tiling … coverage mechanism …

  30. x2 (TT) x1 GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Maximal-Span Tiling, contd.  2

  31. GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY 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 minimized if 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.

  32. x2 (TT) x1 One-parameter (upper vertex angle) search only GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Quasi-Maximal-Span Reduced-Spillover Tiling Draw convex hull & identify butterfly shaped region Chords through center within butterfly-shaped region parameterize in an angle; Take the one parallel to the line connecting the vertexes of the search-manifold (TT) equal mass line

  33. .97 Area reduction factor  GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Span Reduction after Tile Baseline Rotation

  34. base-tile shape search manifold equal-mass line (TT) Vertex-joining line GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Template Placement Start (downward-concave equal-mass line) Draw basic-tile with lower-left vertex coincident with (leftmost) m1=m2=mmax TT search-manifold vertex.

  35. search manifold equal-mass line (TT) Iso-match contour line Vertex-joining line GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Template Placement Start Case # 1 Draw associated iso-match contour line. This identifies location of first template.

  36. Tiling lattice search manifold equal-mass line (TT) Vertex-joining line GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Template Placement Start Case # 1 Deploy tiling lattice thereof.

  37. search manifold equal-mass line (TT) Vertex-joining line GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Template Placement Start Case # 1 Deploy tiling lattice thereof.

  38. search manifold equal-mass line (TT) Vertex-joining line GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Template Placement Start Case # 1 Discard un-needed lattice points (null intersection between span and )

  39. 1.4 (201) 1.3 1.2 1.1 x 2 1. 0.9 0.8 100. 200. 300. 400. 500. x 1 GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Discarding Un-needed Templates …see this at work in a practical case… LIGO fin=40Hz, foff=730 Hz, 1MM1,M23 M

  40. 1.4 (201) C C’ 1.3 1.2 B’ B 1.1 x 2 1. A 0.9 0.8 100. 200. 300. 400. 500. x 1 GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Discarding Un-needed Templates …simple construction yields fiducial template-line endpoints, to be refined using a simple algorithm…

  41. N S = 0 Exit GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY Discarding Un-needed Templates Starting from X = A, B, C, B’, C’ S=0 Search manifold Span of template at X N Y S = 1 Y Accept template at X. Move X one lattice node forward (if started from A,B,C) or backward (if started from B’, C’) Discard template at X Move X one lattice node backward (if started from A,B,C) or forward (if started from B’, C’)

  42. (201) (109) (606) (1416) GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY TT-Style Template Placement for LIGO LIGO fin = 40Hz, foff=730 Hz, 1M M1,M23 M   = 0.97 # templates = 2331

  43. Close-ups GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY 1MM1,M2 3 M  = .97 # templates = 2331 LIGO fin=40Hz, foff=740 Hz

  44. GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY TT-Style Template Placement for LIGO LIGO PSD (LAL) fin=40Hz, foff=730 Hz, 1M M1,M23 M  = 0.97 # templates = 2331

  45. GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY TT-Style Template Placement for LIGO LIGO fin=40Hz, foff=730 Hz, 1M  M1,M2 3 M = 0.97 # templates = 2331

  46. GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY TT-Style Template Placement for LIGO (401) LIGO fin=40Hz, foff=730 Hz, 1 M M1,M2 3 M  = 0.80 # templates = 410

  47. GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY TT-Style Template Placement for LIGO LIGO fin=40Hz, foff=730 Hz, 1 M M1,M2 3 M  = 0.80 # templates = 410

  48. 4. 3.5 3. 2.5 x 2 2. 1.5 1. 0.5 2000. 4000. 6000. 8000. x 1 GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY TT-Style Template Placement for LIGO The search manifold baseline may have a different concavity… LIGO fin=40Hz, foff=2200 Hz, 0.2 M M1,M2 1 M

  49. 4. 3.5 3. 2.5 x 2 2. 1.5 1. 0.5 2000. 4000. 6000. 8000. x 1 GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY TT-Style Template Placement for LIGO Here one starts placing basic-tile with lower-left vertex at the contact point of  with a par- allel to the line joining the lower  vertexes. The subsequent steps are the same…

  50. 4. 3.5 3. 2.5 x 2 2. 1.5 1. 0.5 2000. 4000. 6000. 8000. x 1 GWDAW - 10 CENTER FOR Dec. 13-18, 2005 GRAVITATIONAL WAVE ASTRONOMY TT-Style Template Placement for LIGO …one draws lines midway between the template-lines., and projecting the inter- sections with …

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