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06 May 2003 Lazzarini - LIGO Science Seminar 1LIGO-G030242-01-E
Search implementation for the gravitational wave stochastic background applied to the S1 LIGO I Science Run
Albert Lazzarinifor the LIGO Scientific Collaboration
06 May 2003LIGO Science Seminar at Caltech
SGWB Working Group WWW SITE: http://feynman.utb.edu/~joe/research/stochastic/upperlimits/
06 May 2003 Lazzarini - LIGO Science Seminar 2LIGO-G030242-01-E
Outline of Talk
• Stochastic GW background
• LIGO S1 run summary
• Search technique, implementation
• Details of analysis
• Results
• Conclusions
06 May 2003 Lazzarini - LIGO Science Seminar 3LIGO-G030242-01-E
• The stochastic GW background arises from an incoherent superposition of unresolved sources of gravitational radiation bathing Earth. » Measure: GW -- energy density in Universe associated with GWs gw(f) -- frequency distribution of energy
• Cosmological sources» GW can probe the very early universe» Inhomogeneities near Planck time, inflation» Phase transitions» Cosmological defects
• Astrophysical sources» NS/NS, WD/WD, periodic sources, SNe
• gw(f) < 10-8 in LIGO band (Maggiore, gr-qc/0008027)
The Stochastic GW Background
06 May 2003 Lazzarini - LIGO Science Seminar 4LIGO-G030242-01-E
Relationship between cosmological quantities and measurable quantities
• Power spectrum, Sgw(f):
• gw(f) in terms of Sgw(f):
• Strain for constant :
06 May 2003 Lazzarini - LIGO Science Seminar 5LIGO-G030242-01-E
2
Allen & Koranda, PhysRevD
Lommen, astro-ph/0208572
Kolb & Turner, TheEarlyUniverseAddisonWesley1990
What is known about the stochastic background?
06 May 2003 Lazzarini - LIGO Science Seminar 6LIGO-G030242-01-E
Stochastic GW Background Detection
s1( f)s2( f )≠0, {f}∉∅
€
L < λGW ( f )⇒ 2πfLc
<1
• Cross-correlate the output of two (independent) detectors with a suitable filter kernel:
• Requires:(i) Two detectors must have overlapping frequency response functions i.e.,(ii) Detectors sensitive to same polarization state (+, x) of radiation field,
hgw.(iii) Baseline separation must be suitably “short”:
• Limits of detection (1 year integration):» LIGO I: gw< 10-5
» Advanced LIGO: gw< 5 x 10-9
06 May 2003 Lazzarini - LIGO Science Seminar 7LIGO-G030242-01-E
• Correlation kernel weighting function -> optimal filter
• SNR is maximized for:
Stochastic GW Background Detection
06 May 2003 Lazzarini - LIGO Science Seminar 8LIGO-G030242-01-E
Overlap Reduction Factor, (f)
• Overlap reduction function, (f), is a function of detector geometries, orientations and detector separations
€
( f ) = 58π
k= +,×
∑ d ˆ Ω ∫ e2πif ˆ Ω ⋅Δr x / c d1 : e1
k ˆ Ω ( ) ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟d2 : e2
k ˆ Ω ( ) ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟;
eab+ ˆ Ω ( ) = ˆ φ a ˆ φ b − ˆ θ a ˆ θ b
eab× ˆ Ω ( ) = ˆ φ a ˆ θ b − ˆ θ a ˆ φ b
€
( f ) = ρ1( 2πLf
c) d1 :d2 + ρ2( 2πLf
c ) ˆ n 12⋅d1( ) ⋅ d2⋅ˆ n 12( ) + ρ3(2πLf
c) ˆ n 12⋅d1⋅ˆ n 12( ) ˆ n 12⋅d2⋅ˆ n 12( )
€
1(α )ρ2(α )ρ3(α )
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟=
5 −10α
5α 2
−10 40α
−50α 2
52
−25α
1752α 2
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟
⋅j0(α )j1(α )j2(α )
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
LIGO-G030242-01-E
Overlap Reduction Factor, (f)
• WA - WA == 1
• LA - WA shown
• (0) ~ -1 because of WA-LA interferometer orientations:
LHO LLO
06 May 2003 Lazzarini - LIGO Science Seminar 10LIGO-G030242-01-E
In-Lock Data Summary from S1In-Lock Data Summary from S1Red lines: integrated up time Green bands (w/ black borders): epochs of lock
•August 23 – September 9, 2002: 408 hrs (17 days).•Individual interferometers:
•H1 (4km): duty cycle 57.6% ; Total Locked time: 235 hrs •H2 (2km): duty cycle 73.1% ; Total Locked time: 298 hrs •L1 (4km): duty cycle 41.7% ; Total Locked time: 170 hrs
•Double coincidences: •L1 & H1 : duty cycle 28.4%; Total coincident time: 116 hrs •L1 & H2 : duty cycle 32.1%; Total coincident time: 131 hrs •H1 & H2 : duty cycle 46.1%; Total coincident time: 188 hrs
•Triple Coincidence: L1, H1, and H2 : duty cycle 23.4% ;•Total coincident time: 95.7 hrs
H1: 235 hrs H2: 298 hrs L1: 170 hrs
For this analysis:L1-H1:Valid data: 75 hrsQuiet Data: 75 hrsCalibrated Data: 64 hrsNet uptime: 15.7%L1-H2:Valid data: 81 hrsQuiet Data: 66 hrsCalibrated Data: 51 hrsNet uptime: 12.5%H1-H2:Valid data: 134 hrsQuiet Data: 119 hrsCalibrated Data: 100 hrsNet uptime: 24.5%
06 May 2003 Lazzarini - LIGO Science Seminar 11LIGO-G030242-01-E
S1 Sensitivities
• Spectra taken just before run
• Cross correlation technique allows one to “dig” signal below noise floor in individual instruments
• Dashed lines show expected 90% confidence bounds one could set:
»100 hrs of observation with H2km +L4km (=10)
»150 hrs of observation with H2km + H4km (=1)
»Limits from theoretical SNR equation
06 May 2003 Lazzarini - LIGO Science Seminar 12LIGO-G030242-01-E
• Analysis performed in data intervals of 900s (15 min)»For each 900s interval, I :
–Average power spectrum, mid point calibration used for entire interval
–Ten 90s segments are analyzed separately–Provides 10 independent estimates, statistics of estimates
»For each 90s segment J, estimate:–resample data to 1024 samples/s (512 Hz Nyquist)
• 90% of SNR comes below f~300Hz –FFT, window data–Calculate estimate YIJ
–Average n = 45 frequency bins to obtain cross-correlation spectra with f = 0.25 Hz
»Average 10 values of YIJ to obtain interval average, YI, sample variance, sI
2
Implementation of analysis
IJ
06 May 2003 Lazzarini - LIGO Science Seminar 13LIGO-G030242-01-E
Implementation of analysis• Results for 900s intervals combined to obtain run averaged
answer (“point estimate”).
• Weights I2 are obtained from the power spectra for each
interval,I.» Measures data quality -- how quiet the interferometer pair was during interval I
• Overall statistical error for the estimate derived from individual interval variances
06 May 2003 Lazzarini - LIGO Science Seminar 14LIGO-G030242-01-E
Pipeline analysis
flow
06 May 2003 Lazzarini - LIGO Science Seminar 15LIGO-G030242-01-E
• Sources of spectral leakage:
» Analysis of data in finite segments
» Redness of spectrum (seismic wall)
» Narrow band features– Will be removed by notching,
need to make sure their effect is contained in a few frequency bins
• Studied different window widths
» Used a Tukey window» Flat top plus smooth fall-off at
ends:» 0.5s + 89s + 0.5s
Windowing Effects
Window
Data
T = 90s
1/2 THann
06 May 2003 Lazzarini - LIGO Science Seminar 16LIGO-G030242-01-E
End-to-end pipeline validationSW and HW injection of simulated
stochastic backgrounds
• Generate time series of correlated random noise with same properties as SGWB
• Injected in hardware during S1 run at several amplitudes• Inject post run for further verification
06 May 2003 Lazzarini - LIGO Science Seminar 17LIGO-G030242-01-E
Extraction of injected signals
• 2 with 2 D.O.F: time offset, amplitude• Theoretical curve determined by
power spectra, P1(f), P2(f)• Points obtained by performing a time-
shift analysis of data
Extractions consistent with
injections
• Analysis sensitive to relative timing of data streams
• 270 s offset consistent with post-run GPS measurements
06 May 2003 Lazzarini - LIGO Science Seminar 18LIGO-G030242-01-E
Survey of cross-spectral coherences, 12(f)
• Narrowband coherences persist over long integration times» n x 60 Hz (H1-H2) line harmonics» n x16 Hz (all pairs) GPS-synchronized clocking electronics for data acquisition system» 250 Hz (?)
• Lines excised by excluding them from the integration over frequency to obtain estimates, YIJ
06 May 2003 Lazzarini - LIGO Science Seminar 19LIGO-G030242-01-E
Time-frequency map of cross-correlationstatistics YIJ for entire S1 run H1-H2
IJ
06 May 2003 Lazzarini - LIGO Science Seminar 20LIGO-G030242-01-E
Data Quality --
Variability in noise floor during S1
• Variations in statistical weights, I2, with
which individual measurements are combined
• Horizontal lines correspond to “representative” power spectra shown earlier
• Variability shown is on 900s scale, taken into account by weights
• Variability on <900s leads to source of error in estimate beyond statistical errors
•H1-H2: PSD nonstationarity~ 0.1•H1-L1: PSD nonstationarity ~ 0.3•H2-L1: PSD nonstationarity ~ 9.3
• Effect estimated by rerunning analysis with a finer granularity
06 May 2003 Lazzarini - LIGO Science Seminar 21LIGO-G030242-01-E
Data Quality --Variations in calibrations during S1Example :H2-L1 (H2 interferometer)
• 900s midpoint calibrations used» 60s trends in calibration were acquired
• Variability on <900s leads to source of error in estimate beyond statistical errors
• H1-H2: calibration~ 0.2• H1-L1: calibration ~ 0.4• H2-L1: calibration ~ 1.2
• Re-ran analysis on finer granularity to assess effect of varying R
€
R( f : t) = α (t) ⋅C0( f )1+ α (t) ⋅β ⋅H0( f )
˜ X ( f ) = R( f ) ⋅ ˜ g ADC ( f )
06 May 2003 Lazzarini - LIGO Science Seminar 22LIGO-G030242-01-E
Data Quality --Variations in timing during S1
Example :H2-L1
•Timing offsets of acquisition system changed during S1»Hardware reboots
•Variability over run leads to source of systematic error in estimate beyond statistical errors•Scale factor, time, in estimate
• H1-H2: time~ 1 (not signifcant)• H1-L1: time ~ 1 (not signifcant)• H2-L1: time ~ 1.05
• +200 s shift
•Include in final result as a scaling up of estimate
06 May 2003 Lazzarini - LIGO Science Seminar 23LIGO-G030242-01-E
< >
Frequency, time dependence ofcross correlation kernels
• Run-averaged kernels• Integrals correspond to estimates of eff
06 May 2003 Lazzarini - LIGO Science Seminar 24LIGO-G030242-01-E
Time evolution over S1 of estimates
• Running estimates of eff over run
• End points correspond to estimates of eff:• Bottom panels show probability of observing running estimate at each
time, T, if underlying process is zero-mean Gaussian noise
+/- 1.65
06 May 2003 Lazzarini - LIGO Science Seminar 25LIGO-G030242-01-E
€
X IJ = YIJ −Y σ I
Statistics of estimates
• Normal deviates of 90s estimates from average values are Gaussian RVs:
» <XIJ> = 0 » = 1
06 May 2003 Lazzarini - LIGO Science Seminar 26LIGO-G030242-01-E
• H2-L1, H1-L1 consistent with random excursions from point to point
• H1-H2 exhibits time variations not consistent with random noise» Influence of residual instrumental
correlations present» Time series not consistent with
SNR=10 signal due to =const.
Time shift analysis of final results
06 May 2003 Lazzarini - LIGO Science Seminar 27LIGO-G030242-01-E
S1 results
• H1-H2: (h100)2 + instrumental = (-11 +/- 2) + 20%
• H1-L1: (h100)2 < 70 + 20%
• H2-L1: (h100)2 < 23 + 20%
H1-H2 H1-L1 H2-L1Point Estimates, e -
Statitialtat Calibationvaiational
NontationaityPS Totaleoquadatue 8
Tiineoaleato Finaleult -
Syeti%CL:+/-6 Uppe%CLe+8 Calibationunetainty+/- % % %
06 May 2003 Lazzarini - LIGO Science Seminar 28LIGO-G030242-01-E
Stochastic Gravitational Wave Background“Landscape”
S2 ->
LIGO-G030242-01-E
Summary and conclusions
• H2-L1 provides the best upper limit from S1Measurement BW f = 274 Hz: 40Hz , f < 314 HzWithin 2x of expectation at beginning of run (~10 vs ~23)
64 hrs vs 100 hrs -> 1.25xcalibration variation, overall uncertaintynon-stationarity of noise floorstiming offsets
2x - 3x better than previous (narrowband, f = 1 Hz) direct measurement with bars @ 1 kHz
LIGO-G030242-01-E
Summary and conclusions
• H1-H2 was not usable, even though it would have had 10x sensitivity
» Statistical error is as expected, ~ 1» Bias (-10) was not foreseen (could have been expected)
• Correlations exhibit instrumental features» WA-LA: narrowband
– GPS synchronization of data acquisition systems– 250 Hz feature of presently unknown nature
» WA-WA : narrowband and broadband– 60 Hz mains and harmonics, upconversion broadening– Acoustic couplings within corner station between detection systems
• Broadband 200 - 300 Hz
06 May 2003 Lazzarini - LIGO Science Seminar 31LIGO-G030242-01-E
Summary and conclusions (2)
• PlansforS2 andbeyond» Take calibration variability, non-stationarity into account on the finest
possible time scales» Improve on calibration uncertainty» Identifyandeliminate or remove correlatednoisesourcesforH-H2» Review, improvedataanalysispipelineeghigh-
passfilteringlineremovalfrequency rangewindowing» SetupinfrastructureforALLEGRO-LLOGEO-LIGOcorrelations
– ALLEGRO - LLO correlation may allow identification of instrumental biases vs signal• ALLEGRO can be rotated, allowing for an improved analysis (Finn & Lazzarini,
PRD 15 October 2001)» Restructureanalysiscodeformoreefficienttime-shiftanalysis,
simulations» ExpectedS2 upper-limit:(h100)2 <10-2 for H2-L1» Ultimate LIGO I (1 yr integration): (h100)2 <10-5 H2-L1
06 May 2003 Lazzarini - LIGO Science Seminar 32LIGO-G030242-01-E
FINIS
06 May 2003 Lazzarini - LIGO Science Seminar 33LIGO-G030242-01-E
06 May 2003 Lazzarini - LIGO Science Seminar 34LIGO-G030242-01-E
06 May 2003 Lazzarini - LIGO Science Seminar 35LIGO-G030242-01-E
H1 - H2
06 May 2003 Lazzarini - LIGO Science Seminar 36LIGO-G030242-01-E
H2 - L1