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

2. Patrick Sutton, CaltechGWDAW 10 UTB 2005.12.162 The LIGO Cheese dedicated to MAB

3. Patrick Sutton, CaltechGWDAW 10 UTB 2005.12.163 Motivation We don’t know what we’re doing. 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 … … how do we establish that our search is sensitive to ``generic’’ GWBs? – Need to establish sensitivity empirically (injections)!

4. Patrick Sutton, CaltechGWDAW 10 UTB 2005.12.164 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

5. Patrick Sutton, CaltechGWDAW 10 UTB 2005.12.165 The Logic Most important signal properties are distribution of signal energy in time, frequency. – Not considered time-domain methods. V = some time-frequency volume Parameterization based on excess power detection. – Anderson, Brady, Creighton, & Flanagan, PRD63 042003 (2001) Excess power thresholds on: Insensitive to details of waveform, most of the waveform information is irrelevant.

6. Patrick Sutton, CaltechGWDAW 10 UTB 2005.12.166 Graphical Example (Heuristic) Characterise GWBs by these parameters: – duration – central frequency – bandwidth frequency time best match Excess power search with various rectangular time- frequency tiles. Overlap of signal with tile determined by signal duration, central frequency, bandwidth.

7. Patrick Sutton, CaltechGWDAW 10 UTB 2005.12.167 Energy Distributions This is signal energy, not physical energy ( [  t h  ] 2 ). Independent of polarization gauge choice. Define duration, central frequency, bandwidth in terms of distribution of signal energy in time & frequency. Energy distributions: frequency domain time domain

8. Patrick Sutton, CaltechGWDAW 10 UTB 2005.12.168 Energy Dist. (cont’d) Normalization factor is just h rss 2 (Parseval’s theorem): Normalized like probability distributions:

9. Patrick Sutton, CaltechGWDAW 10 UTB 2005.12.169 Time-Domain Moments Standard choice in signal-processing literature. mean ~ st dev Use lowest-order moments of energy distribution. general moment central time (not used) duration

10. Patrick Sutton, CaltechGWDAW 10 UTB 2005.12.1610 Frequency-Domain Moments Same thing: Can also consider higher moments of distributions. bandwidth central frequency general moment

11. Patrick Sutton, CaltechGWDAW 10 UTB 2005.12.1611 Signals cannot have both arbitrarily small bandwidth and duration simultaneously: Not the quantum-mechanics result (  =1), because these are real signals – Hilberg & Rothe, Information and Control 18 103 (1971). Uncertainty Principle,

12. Patrick Sutton, CaltechGWDAW 10 UTB 2005.12.1612 Uncertainty Principle(s) Quantum-mechanics type argument implies second frequency-dependent uncertainty principle: physical not allowed

13. Patrick Sutton, CaltechGWDAW 10 UTB 2005.12.1613 The Cheese duration bandwidth frequency Uncertainty relations: From LIGO (approximate): 10 2 Hz <  < 10 3 Hz 10 -1 Hz <  < 10 3 Hz 10 -4 s <  < 1 s  ~ 0.1 Detectable physical signals live in here “LIGO cheese”  = 10 3 Hz  = 1s (log-log-log plot)  = 10 3 Hz

14. Patrick Sutton, CaltechGWDAW 10 UTB 2005.12.1614 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.

15. Patrick Sutton, CaltechGWDAW 10 UTB 2005.12.1615 (LIGO cheese, top-down view) 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. unphysical region physical region Gaussians Lazarus BH Mergers sine-Gaussians  ~ 0.1

16. Patrick Sutton, CaltechGWDAW 10 UTB 2005.12.1616 SN: Bandwidth vs Duration ZM DFM BO (LIGO cheese, top-down view)

17. Patrick Sutton, CaltechGWDAW 10 UTB 2005.12.1617 SN: Frequency vs Bandwidth ZM DFM BO

18. Patrick Sutton, CaltechGWDAW 10 UTB 2005.12.1618 Chirplets & Maximum Entropy Need family of waveforms that spans the cheese – Continuous parameters to get any  No physical principle to specify such a family. Use mathematical principle to motivate choice of waveform family: maximum entropy

19. Patrick Sutton, CaltechGWDAW 10 UTB 2005.12.1619 Waveform Entropy Derive  and h +, h × with maximum entropy subject to constraints on duration, bandwidth, frequency. – Solution turns out to be Gaussian-modulated chirps Shannon entropy for distribution  (x) (x can be time or frequency): – A measure of the “probability” of generating  by randomly distributing energy in time or frequency. – A measure of the amount of structure in .

20. Patrick Sutton, CaltechGWDAW 10 UTB 2005.12.1620 Maximum Entropy: Time Domain Impose constraints with Lagrange multipliers: Maximize action under variations of  : Apply constraint equations: standard result: MaxEnt distribution is a Gaussian central time duration normalization

21. Patrick Sutton, CaltechGWDAW 10 UTB 2005.12.1621 Time-Domain Solution High entropy waveforms: h + and h × are phase-shifted versions of each other.  (t) is an arbitrary real function of time Corresponding waveform (general solution):

22. Patrick Sutton, CaltechGWDAW 10 UTB 2005.12.1622 Repeat maximum-entropy procedure in frequency domain to solve for  (t). Approximate solution: Solution: Chirplets This is a chirplet: a Gaussian- modulated sinusoid with linearly sweeping frequency. Chirp parameter  related to bandwidth: arbitrary phase  >0

23. Patrick Sutton, CaltechGWDAW 10 UTB 2005.12.1623 Chirplet contains sine-Gaussians, Gaussians as special cases: Solution: Chirplets  = 0  Gaussian  = 0  sine/cosine-Gaussian Setting  =0 gives ~minimal-uncertainty waveforms.

24. Patrick Sutton, CaltechGWDAW 10 UTB 2005.12.1624 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).

25. Patrick Sutton, CaltechGWDAW 10 UTB 2005.12.1625 Line of Thought Study variation of detection statistic with changing parameters – Changing tile:    : Ambiguity function - density of tiling for guaranteed minimal sensitivity. – Changing signal:   : Variation in sensitivity over the cheese for fixed tiling (injection density). Excess power detection statistic: – Tile with parameters . – Signal with cheese parameters = ( , ,  ).

26. Patrick Sutton, CaltechGWDAW 10 UTB 2005.12.1626 Example Detection statistic: frequency time F TT FF Chirplet at = ( , ,  ) Rectangular tiles  = (F,  F,  T) – If tile is too large we include excess noise. – If tile is too small we miss signal power. Study behaviour numerically.

27. Patrick Sutton, CaltechGWDAW 10 UTB 2005.12.1627 Summary Future: Investigate sensitivity of various detection algorithms over the cheese. – How well do  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).

28. Patrick Sutton, CaltechGWDAW 10 UTB 2005.12.1628 Acknowledgements Albert Lazzarini, Shourov Chatterji, Duncan Brown for many fruitful conversations.