LIGO-G060499-00-W The Search For Periodic Gravitational Waves Gregory Mendell, LIGO Hanford...
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Transcript of LIGO-G060499-00-W The Search For Periodic Gravitational Waves Gregory Mendell, LIGO Hanford...
LIGO-G060499-00-W
The Search For Periodic Gravitational WavesGregory Mendell, LIGO Hanford Observatory
on behalf of the LIGO Scientific Collaboration
The Laser Interferometer Gravitational-Wave Observatory
http://www.ligo.caltech.edu
Supported by the United States National Science Foundation
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Gravitational Waves• Gravitation = metric tensor:• Weak Field Limit:• Gauge choice:• Quadrupole field:• At the detector:
dxdxgds 2
01
2
2
22
htc
hg
)/(2
0000
00
00
0000
cztifTT ehh
hhh
]2)/(2cos[1
)2/(ˆˆ
crtfr
hTT
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suspended test masses
free masses
Laser Interferometer Gravitational-wave Detection
Gravitational-wave Strain: LLh /•LIGO’s arms: L = 4 km.
•Sensitivity: L nominally around 10-18 m. (Depending on frequency and duration of the signal this can be much larger or much smaller!)
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Inside LIGO
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Sources of Continuous Gravitational Waves
Mountain on neutron starPrecessing neutron star
Accreting neutron star
Oscillating neutron star
A B
C
D
Credits:
A. image by Jolien Creighton; LIGO Lab Document G030163-03-Z.
B. image by M. Kramer; Press Release PR0003, University of Manchester - Jodrell Bank Observatory, 2 August 2000.
C. image by Dana Berry/NASA; NASA News Release posted July 2, 2003 on Spaceflight Now.
D. image from a simulation by Chad Hanna and Benjamin Owen; B. J. Owen's research page, Penn State University.
Search methods can detected any type of periodic source.
Upper limits are set on gravitational-wave amplitude, h0, of rotating triaxial ellipsoid.
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Nature of gravitational wave signal• The GW signal from a triaxial pulsar can be modelled as
• The unknown parameters are
• h0 - amplitude of the gravitational wave signal
• - polarization angle of signal; embedded in Fx,+
• - inclination angle of the pulsar
• 0 - initial phase of pulsar (0)
• In the targeted searches we currently only look for signals at twice the rotation frequency of the pulsars
• For blind searches the location in the sky and the source’s frequency evolution are unknown.
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Figure: D. Sigg LIGO-P980007-00-D
);();( tFtF
Amplitude Modulation
Beam Pattern Response Functions:
Polarization Angle
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Periodic Signals
nc
trtT ˆ
)(
T = arrival time at SSBt = arrival time at detector
Solar SystemBarycenter
GW approaching from direction n
Relativistic corrections are included in the actual code.
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Phase and Frequency Modulation
0
100
0
100
)ˆ)(
()!1(
)()!1(
s
ss
s
ss
Tnc
trt
s
f
TTs
f
Phase at SSB:
Phase at detector:
...)ˆ)(
(!
ˆ)(
)(
)ˆ)(
(!
ˆ)(
1)(
010
100
s
ss
s
ss
Tnc
trt
s
ffn
c
tatf
Tnc
trt
s
ffn
c
tvtf
Frequency at detector:
df/dt :
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Analysis of Hardware Injections
RA (hours) RA (hours)
DE
C (
degr
ees)
DE
C (
degr
ees)
PRELIMINARY APS April 2006
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Fake gravitational-wave signals corresponding to rotating neutron stars with varying degrees of asymmetry were injected for parts of the S4 run by actuating on one end mirror. Sky maps for the search for an injected signal with h0 ~ 7.5e-24 are below. Black stars show the fake signal’s sky position.
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All Sky Loudest Events 1000 SFTs, Fake Inst. Line & Noise:
DE
C (
rad
ian
s)
RA (radians)
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Maximum Likelihood
0,0,0cos
,0
),,cos,(
2
12
)(
...2
1
2
1
2
1)|(
2
0
22
0
2
00
222
22
2
)(
3
2
)(
2
2
)(
1
23
233
22
222
21
211
h
hfh
hhhxhx
eeehxP
j
j j
jj
j j
jj
j j
jj
hxhxhx
Likelihood of getting data x for model h for Gaussian Noise:
Chi-squared
Minimize Chi-squared = Maximize the Likelihood
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Coherent Matched Filtering
1
0
1
0
)0(M N
j
ijb
jbeFxX
1
0
/21 N
k
NijkSFTkj eX
Nx
1
0 )(2
)2cos1(2sin)0( 0
M
k b
bb
k
SFTki
b k
i
S
XeFX
Jaranowski, Krolak, & Schutz gr-qc/9804014; Schutz & Papa gr-qc/9905018; Williams and Schutz gr-qc/9912029; Berukoff and Papa
LAL Documentation
222
*2222 24
FFFF
XXeFFXFXF
MF
F-statistic
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Time Domain Coherent Search Using Bayesian Analysis
Ch
hxhx
dxxhPC
hxPxP
hPxhP
eehxP
baPbPabPaP
0
2
)(
2
2
)(
1
)|(
)|()(
)()|(
...2
1
2
1)|(
)|()()|()(
22
222
21
211
Bayes’ Theorem:
Likelihood:
Confidence Interval:
Posterior Probability:
Prior
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Semi-coherent Methods
Time
Fre
quen
cy
Time
Track Doppler shift and df/dt
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• Break up data into segments; FFT each segment.
• Track the frequency.
• StackSlide: add the power weighted by the noise inverse.
• Hough: add 1 or 0 if power is above/below a cutoff (advanced version includes weighted average of 1’s & 0’s).
• PowerFlux: add power using weights that maximize SNR
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Short Fourier Transforms (SFTs) & Periodograms:
Hough Number Count & Weighted Number Count:
StackSlide Power:
Power Flux:
kk S
FF22
1
0'
'24
'
221
0
1
0
1
0
1
0
1
0
21
0
2
'/)(
/)()()(
1
/1
//1
/0
~2~
M
k
k
k
M
kkSFT
M
kk
M
k
M
kk
M
kckk
ckkk
SFT
kk
n
j
tifjk
SF
SFwpw
T
SpM
P
pSp
pSp
T
xptexx jk
P
nNnNn w
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Frequentist Confidence Region
• Estimate parameters by maximizing likelihood or minimizing chi-squared
• Injected signal with estimated parameters into many synthetic sets of noise.
• Re-estimate the parameters from each injection
ftBftAts 2sin2cos)( Matlab simulation for signal:
Figure courtesy Anah Mourant, Caltech SURF student, 2003.
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Bayesian Confidence Region
x
Matlab simulation for signal: ftBftAts 2sin2cos)( Uniform Prior: )2/exp();,( 2xBAP
Figures courtesy Anah Mourant, Caltech SURF student, 2003.
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Frequentist Population-based Confidence Region From The Loudest Event.
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GWs from triaxial pulsar
zz
yyxx
I
II
r
fIh zz
20
4
2
0 c
G16
• For upper limits have to select a model. (This is not needed for detection!)• Ellipticity, , measures asymmetry in triaxially shaped pulsar
equatorial ellipticity
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The back of the envelope please…For a triaxial ellipsoid (or think of I as a way of scaling a the time varying quadrupole Q):
• Maximum expected ellipticities:
•Solid quark stars: 10-4
•Hybrid hyperon/heutron stars or buried B fields: 10-5
•Neutron stars with fluid core and solid crust: 10-6
• Estimated strain:
h = 42(G/c4) If 2/r
~ 22(vesc2/c2)(vR
2/c2)(R/r)
~10–25( /10-6)(I/1045 g.cm2) (f/300 Hz)2 (1kpc/r)
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Results1. S1 coherent time-domain and F-stat search targeting pulsar J1939+2134:
The LIGO Scientific Collaboration, Phys.Rev. D69 (2004) 082004; gr-qc/0308050
Best UL: h_0 < 1.4 10-22; < 2.9 10-4 (1045 g.cm2/I)
2. S2 coherent time-domain search targeting 28 pulsars: The LIGO Scientific Collaboration: B. Abbott, et al, M. Kramer, A.G. Lyne, Phys.Rev.Lett. 94 (2005) 181103;gr-qc/0410007
Best UL: h_0 < 1.7 10-24; Best UL < 4.5e 10-6
3. S2 semi-coherent all-sky, 200-400 Hz, Hough search: LIGO Scientific Collaboration, Phys.Rev. D72 (2005) 102004; gr-qc/0508065
Best UL: h_0 < 4.4 10-23
4. S3 coherent all-sky, 160-728.8 Hz and Sco X-1, 464-484 Hz & 604-624 Hz, F-stat search: LIGO Scientific Collaboration, submitted to Phys. Rev. D (2006); gr-qc/0605028
Best all-sky UL: h_0 < 6.6e-23 ; Best Sco X-1 UL: h_0 < 1.7e-22
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Preliminary Results and Plans5. S3 Einstein@Home search: preliminary S3 results posted at
http://einstein.phys.uwm.edu/. Final S3 and S4 results in preparation. Initial start of S5 search is running.
6. S3/S4 Targested search including pulsars: 32 isolated and 44 in binaries; results in preparation.
April 2005 APS Best UL: h_0 < few 10-25, and < 10-6 for one pulsar
7. S4 all-sky, 50-1000 Hz, PowerFlux, StackSlide, Hough search: results in preparation; S5 preliminary all-sky PowerFlux search
April 2005 APS Best UL: h_0 < few 10-24.
8. Start of S5 preliminary coherent time-domain targeted pulsar search:
Best UL h0 < few 10-25; Best UL: < few x 10-7
9. S4/S5 F-stat targeted isolated x-ray sources: RX J1856.5-3754 (nearest known NS) and Cas A (youngest known NS) search started.
10. S5 Multi-IFO PowerFlux & Multi-IFO Hierarchical Einstein@Home Search under development.
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S5 targeted h0 Results
• Spin-down upper limit calculated with intrinsic spin-down value if available i.e. corrected for Shklovskii transverse velocity effect
• Closest to spin-down upper limit
Crab pulsar ~ 2.1 times greater than spin-down (fgw = 59.6 Hz, dist = 2.0 kpc)
h0 = 3.0x10-24, = 1.6x10-3
Assumes I = 1038 kgm2
• Sensitivity curves use:
Crab pulsar
Frequency (Hz)
h 0
PRELIMINARY APS April 2006
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Einstein@Home S3 ResultsPRELIMINARY
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Many searches are underway. You can join via Einstein@home:
http://einstein.phys.uwm.edu/• Like SETI@home, but for LIGO/GEO matched filter search for GWs from rotating compact stars.
• Support for Windows, Mac OSX, and Linux clients
• Our own clusters have thousands of CPUs.
• Einstein@home has many times more computing power at low cost.
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End
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S5 And Future Prospects
• PowerFlux will give quick views of S5 data, and provide candidates for follow-up if any unexplained signals turn up.
• Hierarchical searches, with coincidence, other vetoes, and follow-up searches, will improve sensitivity.
Divide data into ~30 hr segments and generate coherent F-statistic for each.
Apply StackSlide/Hough of the F-statistic segments. Automatic follow-up of candidates will be done. The code will run under Einstein@Home.
• Advanced LIGO will have 10x sensitivity, better a lower frequencies. Using signal recycling can further improve sensitivity in a narrow frequency band.
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Einstein@Home S3 ResultsPRELIMINARY
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Line Avoiding: Doppler Skybands (e.g., for no-spindown)
Skyband 0 (good)
Skyband 10 (worst)
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All Sky Loudest Events 1000 SFTs, Fake Pulsar & Noise:
DE
C (
rad
ian
s)
RA (radians)
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Dealing With Instrument Lines: Cleaning
Frequency (Hz)
Spe
ctra
l Den
sity
[1/
Sqr
t(H
z)]
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Astrophysical Sources
Black Holes Dense Stars
Supernovae Stochastic Background
Photos: http://antwrp.gsfc.nasa.gov; http://imagine.gsfc.nasa.gov
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Neutron and Strange-quark Stars
http://chandra.harvard.edu/resources/illustrations/neutronstars_4.html
NASA/CXC/SAO
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Calibration• Measure open loop gain G,
input unity gain to model• Extract gain of sensing
function C=G/AD from model
• Produce response function at time of the calibration, R=(1+G)/C
• Now, to extrapolate for future times, monitor single calibration line in AS_Q error signal, plus any changes in gain beta, and form alphas
• Can then produce R at any later time t, given alpha and beta at t
• A photon calibrator, using radiation pressure, gives results consistent with the standard calibration.
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S4 StackSlide “Loudest Events” 50-225 Hz
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•Searched 450 freq. per .25 Hz band, 51 values of df/dt, between 0 & -1e-8 Hz/s, up to 82,120 sky positions (up to 2e9 templates). The expected loudest StackSlide Power was ~ 1.22 (SNR ~ 7)
•Veto bands affected by harmonics of 60 Hz.
•Simple cut: if SNR > 7 in only one IFO veto; if in both IFOs, veto if abs(fH1-fL1) > 1.1e-4*f0
Sta
ckS
lide
Pow
erS
tack
Slid
e P
ower
Frequency (Hz)
APS April 2006
PRELIMINARY
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S4 StackSlide “Loudest Events” 50-225 Hz
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Sta
ckS
lide
Pow
erS
tack
Slid
e P
ower
Frequency (Hz)
APS April 2006
PRELIMINARY
FakePulsar3=>
FakePulsar3=>
FakePulsar8=>
FakePulsar8=>
<=Follow-up Indicates Instrument Line
<=Follow-up Indicates Instrument Line
•Searched 450 freq. per .25 Hz band, 51 values of df/dt, between 0 & -1e-8 Hz/s, up to 82,120 sky positions (up to 2e9 templates). The expected loudest StackSlide Power was ~ 1.22 (SNR ~ 7)
•Veto bands affected by harmonics of 60 Hz.
•Simple cut: if SNR > 7 in only one IFO veto; if in both IFOs, veto if abs(fH1-fL1) > 1.1e-4*f0
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Best: 4.4e-24 For 139.5-139.75 Hz Band
PRELIMINARY
Frequency (Hz)
S4 StackSlide h0 95% Confidence All Sky Upper Limits 50-225 Hz
APS April 2006
Note that h0 is the gravitational-wave amplitude for a rotating triaxial ellipsoid. Monte Carlo injections of triaxial sources over the search parameter space & source orientations set the ULs.
Best: 5.4e-24 For 140.75-141.0 Hz Band
h0 U
pper
Lim
it h
0 Upp
er L
imit
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Linear amplitude (=0.5*h0
worst-pulsar-orient.)
Doppler Skyband 0
Color coding
BLUE – Non-Gaussian
Diamond – Wand. Line
Green – Upper Limit
Red point – Loose Candidate (SNR > 7)
PRELIMINARYPowerFlux 95% CL limits – S4 H1 (50-1000 Hz)
APS April 2006
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PowerFlux S5 H1 40-800 Hz (no spindown)
PRELIMINARY APS April 2006
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Linear amplitude (=0.5*h0
worst-pulsar-orient.)
Doppler Skyband 0
Color coding
BLUE – Non-Gaussian
Diamond – Wand. Line
Green – Upper Limit
Red point – Loose Candidate (SNR > 7)
Detector artifacts worse at low frequencies for L1 39.87 Hz comb
(RF oscillator)
*For APS
PRELIMINARY95% CL limits – L1 (50-1000 Hz)
APS April 2006
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Sensitivity of PeriodicSearch Techniques• False alarm & false
dismissal rates determine SNR of detectable signal.
• Coherent matched filtering tracks phase.
• Incoherent power averaging tracks frequency only.
• Optimal search needs 1023 templates per 1 Hz for 1 yr of data; Hierarchical approach needed.
4/17
2/12325
7
2/12326
101800
10102
10
10103
/
obscoh
n
incohc
obs
n
cohc
nobsc
T
s
T
s
Hz
Sh
T
s
Hz
Sh
SThSNR
These are for 1% false alarm & 10% false dismissal rates, an average sky position and source orientation. A hierarchical search employs a much large false alarm rate, then use coincidence and other vetoes to maximize sensitivity and narrow follow-up searches to the most promising candidates.
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S2 Time Domain Search Results
Best ho UL = 1.7 x10-24. Best ellipticity UL = 4.5 x 10-6 (I = 1045 gcm2)
LIGO Scientific Collaboration & The Pulsar Group, Jodrell Bank Observatory, gr-qc/0410007, Phys. Rev. Lett. 94 (2005) 181103.
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S2 All-sky Hough Search Results
Best ho Upper Limit = 4.43 x10-23
LIGO Scientific Collaboration, gr-qc/0508065, submitted to Phys. Rev. D (2005).
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Estimate for population of objects spinning down due to gravitational waves
birthratebirthrategalaxy
galaxyage
birthrate
age
yrs
c
GI
Rh
Rr
fc
fGI
rh
ffKffQIE
10010
8
51
~
8
||201
||4/||
243
min
3
52
Blanford (1984) as cited by Thorne in 300 Years of Gravitation; see also LIGO Scientific Collaboration, gr-qc/0508065, submitted to Phys. Rev. D (2005); and LIGO Scientific Collaboration, S2 Maximum Likelihood Search, in prepartion.
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Simulation:What might detection sound or look like?
Play Me
(AM & FM modulation greatly exaggerated!)