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Transcript of 1 Characterising the Space of Gravitational-Wave Bursts Patrick J. Sutton California Institute of...
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Characterising the Space of Characterising the Space of Gravitational-Wave BurstsGravitational-Wave Bursts
Patrick J. SuttonCalifornia Institute of Technology
Patrick Sutton, Caltech
GWDAW 10 UTB 2005.12.16 3
MotivationMotivation
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)!
Patrick Sutton, Caltech
GWDAW 10 UTB 2005.12.16 4
GoalsGoals
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
Patrick Sutton, Caltech
GWDAW 10 UTB 2005.12.16 5
The LogicThe Logic
Most important signal properties are distribution of signal energy in time, frequency.
– Not considered time-domain methods.
V
2
S(f)
(f)h~
df V
1
V = some time-frequencyvolume
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.
Patrick Sutton, Caltech
GWDAW 10 UTB 2005.12.16 6
Graphical ExampleGraphical Example(Heuristic)(Heuristic)
Characterise GWBs by these parameters:
– duration– central frequency– bandwidth
freq
uenc
ytimebest
match
Excess power search with various rectangular time-frequency tiles.
Overlap of signal with tile determined by signal duration, central frequency, bandwidth.
Patrick Sutton, Caltech
GWDAW 10 UTB 2005.12.16 7
Energy DistributionsEnergy Distributions
This is signal energy, not physical energy ( [th]2). Independent of polarization gauge choice.
Define duration, central frequency, bandwidth in terms of distribution of signal energy in time & frequency.
Energy distributions:
2
22
FDh
(f)h~
(f)h~
2 (f)ρ
frequency domain
2
22
TDh
(t)h(t)h (t)ρ time domain
Patrick Sutton, Caltech
GWDAW 10 UTB 2005.12.16 8
Energy Dist. (cont’d)Energy Dist. (cont’d)
0
22222
(f)h~
(f)h~
df(t)h(t)hdt h
1 (f)ρ df(t)ρdt FD
0
TD
Normalization factor is just hrss2 (Parseval’s theorem):
Normalized like probability distributions:
Patrick Sutton, Caltech
GWDAW 10 UTB 2005.12.16 9
Time-Domain MomentsTime-Domain Moments
Standard choice in signal-processing literature.
t(t)ρdt t TD
-
2TD
-
222 )-(t (t)ρdt tt
mean~ st dev
nTD
-
n t(t)ρdt t
Use lowest-order moments of energy distribution.
generalmoment
central time(not used)
duration
Patrick Sutton, Caltech
GWDAW 10 UTB 2005.12.16 10
Frequency-Domain MomentsFrequency-Domain Moments
Same thing:
f (f)ρ df fφ FD
0
nFD
0
n f (f)ρ df f
2FD
0
222 φ)-(f (f)ρ df ffΔφ
Can also consider higher moments of distributions.
bandwidth
centralfrequency
generalmoment
Patrick Sutton, Caltech
GWDAW 10 UTB 2005.12.16 11
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 PrincipleUncertainty Principle
.0.590106..Γ4π
ΓΔφ Δτ ,
Patrick Sutton, Caltech
GWDAW 10 UTB 2005.12.16 12
Uncertainty Principle(s)Uncertainty Principle(s)
Quantum-mechanics type argument implies second frequency-dependent uncertainty principle:
4π
1φΔφ Δτ 22
physical
not allowed
Patrick Sutton, Caltech
GWDAW 10 UTB 2005.12.16 13
The CheeseThe Cheese
duration
bandwidth
frequencyUncertainty relations:
From LIGO (approximate):
102 Hz < < 103 Hz 10-1 Hz < < 103 Hz 10-4 s < < 1 s
~ 0.1
Detectable physical signals live in here
“LIGO cheese”
= 103Hz
= 1s
(log-log-log plot)
= 103Hz
O(0.1)Δφ Δτ
Patrick Sutton, Caltech
GWDAW 10 UTB 2005.12.16 14
Spanning the CheeseSpanning 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.
Patrick Sutton, Caltech
GWDAW 10 UTB 2005.12.16 15
(LIGO cheese, top-down view)
S2 Sims: Bandwidth vs DurationS2 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 BHMergers
sine-Gaussians
~ 0.1
Patrick Sutton, Caltech
GWDAW 10 UTB 2005.12.16 16
SN: Bandwidth vs DurationSN: Bandwidth vs Duration
ZMDFMBO
(LIGO cheese, top-down view)
Patrick Sutton, Caltech
GWDAW 10 UTB 2005.12.16 17
SN: Frequency vs BandwidthSN: Frequency vs Bandwidth
ZMDFMBO
Patrick Sutton, Caltech
GWDAW 10 UTB 2005.12.16 18
Chirplets & Maximum EntropyChirplets & 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
Patrick Sutton, Caltech
GWDAW 10 UTB 2005.12.16 19
Waveform EntropyWaveform 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 .
ρ(x)ln ρ(x)dx entropy
Patrick Sutton, Caltech
GWDAW 10 UTB 2005.12.16 20
Maximum Entropy: Time DomainMaximum Entropy: Time Domain Impose constraints with Lagrange multipliers:
2210METD
TD
tλtλλ 1exp(t)ρδρ
δS0
Maximize action under variations of :
]Δτρdt t[λ 0]ρdt t [λ
1]ρdt [λ (t)ρln (t)ρdt S
2TD
22TD1
TD0TDTD
Apply constraint equations:
2
2
2
METD 2
texp
2
1(t)ρ
standard result:MaxEnt distribution
is a Gaussian
central time
duration
normalization
Patrick Sutton, Caltech
GWDAW 10 UTB 2005.12.16 21
Time-Domain SolutionTime-Domain Solution
High entropy waveforms: h+ and h× are phase-shifted versions of each other.
tih
2
2
4 2
MEME
4
texp
2)(t)ih(h
(t) is an arbitraryreal function of time
Corresponding waveform (general solution):
Patrick Sutton, Caltech
GWDAW 10 UTB 2005.12.16 22
i t 2 it
4
i1exp
2
h)(t)ih(h 2
24 2
MEME
Repeat maximum-entropy procedure in frequency domain to solve for (t). Approximate solution:
Solution: ChirpletsSolution: Chirplets
This is a chirplet: a Gaussian-modulated sinusoid with linearly sweeping frequency.
Chirp parameter related to bandwidth:
arbitrary phase
22
2
4
1
>0
Patrick Sutton, Caltech
GWDAW 10 UTB 2005.12.16 23
i t 2 it
4
i1exp
2
h)(t)ih(h 2
24 2
MEME
Chirplet contains sine-Gaussians, Gaussians as special cases:
Solution: ChirpletsSolution: Chirplets
= 0 Gaussian = 0 sine/cosine-Gaussian Setting =0 gives ~minimal-uncertainty waveforms.
Patrick Sutton, Caltech
GWDAW 10 UTB 2005.12.16 24
Metric in the CheeseMetric in the Cheese(in progress)(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).
Patrick Sutton, Caltech
GWDAW 10 UTB 2005.12.16 25
Line of ThoughtLine 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 = (,,).
S(f)
(f)h~
df V
1
2
Patrick Sutton, Caltech
GWDAW 10 UTB 2005.12.16 26
ExampleExample
2
22
2
2F/2F
F/2-F
T/2
T/2- 222
2exp
)(
1dfdt
FT
1
t
ft
fS
Detection statistic:
freq
uenc
y
time
F
T
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.
Patrick Sutton, Caltech
GWDAW 10 UTB 2005.12.16 27
SummarySummary
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).