Characterising the Space of Gravitational-Wave Bursts

1. Characterising the Space of Gravitational-Wave Bursts Patrick J. Sutton California Institute of Technology

2. The LIGO Cheese dedicated to MAB GWDAW 10 UTB 2005.12.16

3. GWB searches target signals from poorly modelled sources (mergers, GRB progenitors, SN, …) • We can test sensitivity to any particular waveform (e.g. DFM A2B4G1), but … Motivation • We don’t know what we’re doing. • … how do we establish that our search is sensitive to ``generic’’ GWBs? • Need to establish sensitivity empirically (injections)! GWDAW 10 UTB 2005.12.16

4. Goals • Study expected variation of sensitivity over this space (``metric’’). • Density of simulations needed. • In progress (no results yet). • Propose simple parametrization of the space of GWBs • “The cheese” • Based on lowest moments of energy distribution in time, frequency • Motivated by excess power detection technique (common in LIGO). • Find waveform family spanning this space • Chirplets as maximum-entropy waveforms GWDAW 10 UTB 2005.12.16

5. Parameterization based on excess power detection. • Anderson, Brady, Creighton, & Flanagan, PRD63 042003 (2001) • Excess power thresholds on: V = some time-frequency volume The Logic • Most important signal properties are distribution of signal energy in time, frequency. • Not considered time-domain methods. • Insensitive to details of waveform, most of the waveform information is irrelevant. GWDAW 10 UTB 2005.12.16

6. Characterise GWBs by these parameters: duration central frequency bandwidth frequency best match time Graphical Example(Heuristic) • Excess power search with various rectangular time-frequency tiles. • Overlap of signal with tile determined by signal duration, central frequency, bandwidth. GWDAW 10 UTB 2005.12.16

7. time domain frequency domain Energy Distributions • This is signal energy, not physical energy ( [thmn]2). • Independent of polarization gauge choice. • Define duration, central frequency, bandwidth in terms of distribution of signal energy in time & frequency. • Energy distributions: GWDAW 10 UTB 2005.12.16

8. Energy Dist. (cont’d) • Normalized like probability distributions: • Normalization factor is just hrss2 (Parseval’s theorem): GWDAW 10 UTB 2005.12.16

9. Use lowest-order moments of energy distribution. mean ~ st dev general moment Time-Domain Moments • Standard choice in signal-processing literature. central time (not used) duration GWDAW 10 UTB 2005.12.16

10. Frequency-Domain Moments • Same thing: general moment central frequency bandwidth • Can also consider higher moments of distributions. GWDAW 10 UTB 2005.12.16

11. , Uncertainty Principle • Signals cannot have both arbitrarily small bandwidth and duration simultaneously: • Not the quantum-mechanics result (G=1), because these are real signals • Hilberg & Rothe, Information and Control 18 103 (1971). GWDAW 10 UTB 2005.12.16

12. Uncertainty Principle(s) • Quantum-mechanics type argument implies second frequency-dependent uncertainty principle: physical not allowed GWDAW 10 UTB 2005.12.16

13. frequency bandwidth duration The Cheese f = 103Hz Uncertainty relations: From LIGO (approximate): 102 Hz < f < 103 Hz 10-1 Hz < Df < 103 Hz 10-4 s < Dt < 1 s Detectable physical signals live in here “LIGO cheese” Dt Df ~ 0.1 Df = 103Hz Dt = 1s (log-log-log plot) GWDAW 10 UTB 2005.12.16

14. Spanning the Cheese • Should be able to detect signals throughout the cheese. • Want to be able to test efficiency at any point in the cheese. • Astrophysical catalogs insufficient • So are sine-Gaussians and Gaussians used for most LIGO tuning. • Need waveform family that spans the cheese. • Current simulations (LIGO, LIGO-Virgo): band-limited white-noise bursts. GWDAW 10 UTB 2005.12.16

15. (LIGO cheese, top-down view) Lazarus BH Mergers Gaussians sine-Gaussians S2 Sims: Bandwidth vs Duration S2 simulations hug the “minimum uncertainty” side of the cheese. Cover only a small portion of the signal space accessible with LIGO. physical region Dt Df ~ 0.1 unphysical region GWDAW 10 UTB 2005.12.16

16. ZM DFM BO SN: Bandwidth vs Duration (LIGO cheese, top-down view) GWDAW 10 UTB 2005.12.16

17. SN: Frequency vs Bandwidth ZM DFM BO GWDAW 10 UTB 2005.12.16

18. Chirplets & Maximum Entropy • Need family of waveforms that spans the cheese • Continuous parameters to get any Dt, Df, f. • No physical principle to specify such a family. • Use mathematical principle to motivate choice of waveform family: maximum entropy GWDAW 10 UTB 2005.12.16

19. Shannon entropy for distribution r(x) (x can be time or frequency): • A measure of the “probability” of generating r by randomly distributing energy in time or frequency. • A measure of the amount of structure in r. Waveform Entropy • Derive r and h+, h× with maximum entropy subject to constraints on duration, bandwidth, frequency. • Solution turns out to be Gaussian-modulated chirps GWDAW 10 UTB 2005.12.16

20. Maximum Entropy: Time Domain normalization • Impose constraints with Lagrange multipliers: central time duration • Maximize action under variations of r: • Apply constraint equations: standard result: MaxEnt distribution is a Gaussian GWDAW 10 UTB 2005.12.16

21. Time-Domain Solution • Corresponding waveform (general solution): • High entropy waveforms: h+ and h× are phase-shifted versions of each other. F(t) is an arbitrary real function of time GWDAW 10 UTB 2005.12.16

22. a>0 Solution: Chirplets • Repeat maximum-entropy procedure in frequency domain to solve for F(t). Approximate solution: • This is a chirplet: a Gaussian-modulated sinusoid with linearly sweeping frequency. • Chirp parameter a related to bandwidth: arbitrary phase GWDAW 10 UTB 2005.12.16

23. Solution: Chirplets • Chirplet contains sine-Gaussians, Gaussians as special cases: • a, f, g = 0 Gaussian • a, g = 0  sine/cosine-Gaussian • Setting a=0 gives ~minimal-uncertainty waveforms. GWDAW 10 UTB 2005.12.16

24. Metric in the Cheese(in progress) • Last ingredient: a measure of distance in the cheese (a metric). • Are two signals are close or far apart? • How closely should we space injections in frequency, bandwidth, and duration? • How rapidly should sensitivity vary? • We have a simple parametrization of the space of GWBs (the cheese). • We have a waveform family that spans the cheese (chirplets). GWDAW 10 UTB 2005.12.16

25. Line of Thought • Excess power detection statistic: • Tile with parameters L. • Signal with cheese parameters l = (f,Df,Dt). • Study variation of detection statistic with changing parameters • Changing tile: Lx : Ambiguity function - density of tiling for guaranteed minimal sensitivity. • Changing signal: lx : Variation in sensitivity over the cheese for fixed tiling (injection density). GWDAW 10 UTB 2005.12.16

26. Chirplet at l = (f,Df,Dt) • Rectangular tiles L = (F,DF,DT) • If tile is too large we include excess noise. • If tile is too small we miss signal power. DF F frequency DT • Detection statistic: time Example • Study behaviour numerically. GWDAW 10 UTB 2005.12.16

27. Summary Future: • Investigate sensitivity of various detection algorithms over the cheese. • How well do f, Df, Dt predict sensitivity? • Are additional parameters needed? • Does it work for time-domain methods? • Derive ``metric.’’ • Proposed a simple parametrization of the space of GWBs. • Based on excess power searches. • Derived maximum-entropy waveform family that spans the cheese (chirplets). GWDAW 10 UTB 2005.12.16

28. Acknowledgements • Albert Lazzarini, Shourov Chatterji, Duncan Brown for many fruitful conversations. GWDAW 10 UTB 2005.12.16