Preliminary Results of LIGO-ALLEGRO Stochastic Background Search - - PowerPoint PPT Presentation

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Whelan for LSC & ALLEGRO: prelim LIGO-ALLEGRO SB results LIGO-G060605-04-Z p.1 Preliminary Results of LIGO-ALLEGRO Stochastic Background Search John T. Whelan john.whelan@ligo.org on behalf of the LIGO Scientific Collaboration and the

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Whelan for LSC & ALLEGRO: prelim LIGO-ALLEGRO SB results LIGO-G060605-04-Z p.1

Preliminary Results of LIGO-ALLEGRO Stochastic Background Search

John T. Whelan john.whelan@ligo.org

  • n behalf of the LIGO Scientific Collaboration

and the ALLEGRO Group 10th Gravitational Wave Data Analysis Workshop 2006 December 20 LIGO-G060605-04-Z

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Outline

I Background/Motivation for LLO-ALLEGRO Search

  • LLO-ALLEGRO Pair (proximity, overlap modulation)
  • Technical Considerations (sampling, heterodyning)

II S4 Data Analysis

  • Data Volume by Orientation
  • Validation: Software & Hardware Injections
  • Preliminary Cross-Correlation Results
  • Statistical Interpretation: Upper Limit
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Sensitivity to Stochastic GW Backgrounds

  • Optimally filtered CC statistic

Y =

  • d

f s∗

1(f)

Q(f) s2(f)

  • Y (f)
  • Optimal filter

Q(f) ∝Sgw(f)γ12(f)

P1(f)P2(f)

(Initial analyses assume Sgw(f) or Ωgw(f) ∝ f3Sgw(f) constant across band)

  • Optimally filtered cross-correlation method has Ωgw sensitivity

σΩ ∝

  • T

d

f f6 γ2

12(f)

P1(f)P2(f)

−1/2

  • Significant contributions when

– detector noise power spectra P1(f), P2(f) small – overlap reduction function γ12(f) (geom correction) near ±1

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100 200 300 400 500 600 700 800 900 1000 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 Frequency (Hz)

Overlap Reduction Function

LLO−LHO LLO−ALLEGRO (N72°E) "XARM" LLO−ALLEGRO (E72°S) "YARM" LLO−ALLEGRO (N27°E) "NULL"

LLO-ALLEGRO only ∼ 40 km apart − → still sensitive @ 900 Hz Response different for XARM, YARM, NULL orientations

ALLEGRO ran in all 3 orientations during LIGO S4 Run (2005 Feb 22-Mar 23)

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LLO-ALLEGRO: Technical Considerations

  • LIGO data digitally downsampled 16384 Hz → 4096 Hz

ALLEGRO data heterodyned at 904 Hz & sampled at 250 Hz

  • Heterodyning means CC stat complex:

Y =

fmax

fmin

d f s∗

1(f)

Q(f) s2(f) real part Gaussian-distributed about SGWB strength; imag part Gaussian-distributed about 0.

  • Differently-sampled data correlated in freq domain

→ Method written up in CQG 22, S1087 (2005)

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LLO-ALLEGRO data from LIGO S4 Run

  • ∼ 10% of data set aside as “playground”;

co ¨ ınc Non-PG data surviving DQ vetoes divided into 60s segs; Incoherent stationarity cut applied to reject segs where sensitivity changing too rapidly (need stationarity for well-behaved optimal filter)

  • Non-playground data in 3 orientations:

– “NULL” (0.023 < γ(f) < 0.029): 88.2 hr after cuts “off-source” data useful for data quality & cross-checks – “YARM” (−0.89 > γ(f) > −0.91): 114.7 hr after cuts – “XARM” (0.95 < γ(f) < 0.96): 181.2 hr after cuts

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850 860 870 880 890 900 910 920 930 940 950 10

−23

10

−22

10

−21

10

−20

10

−19

10

−18

Avg Calibrated ASD from S4 non−NULL non−PG

Frequency (Hz) Strain (Hz−1/2) ALLEGRO ASD LLO ASD Spectrum for Ωgw(f)=1

Frequency band determined by ALLEGRO noise curve

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850 860 870 880 890 900 910 920 930 940 950 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Sensitivity Integrand from S4 non−playground data

Frequency (Hz) Sensitivity Integrand (arb units)

Most of sensitivity from 905–925 Hz

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Software Injections into S4 Playground

  • Combined 90% error bars for all playground data ∼ 2
  • Inject simulated signals of strength ΩR = 1.9, 3.9, 9.6, 19.
  • Note: individual jobs have error bars around 120.

SW injections only detectable over time.

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Stats w/ & w/o SW Inj (19 60-sec segs)

2 4 6 8 10 12 14 16 18 20 −400 −200 200 400

no injection

Segment Number 1−minute ΩR estimate Re(Y/T) Im(Y/T) 2 4 6 8 10 12 14 16 18 20 −400 −200 200 400 Segment Number 1−minute ΩR estimate

ΩR=19.2901 injection

Re(Y/T) Im(Y/T)

Injecting Ω(f) = 19.3 has negligible impact on minute-by-minute correlations

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Stats w/ & w/o SW Inj (19 60-sec segs)

2 4 6 8 10 12 14 16 18 20 −400 −200 200 400

no injection

Segment Number 1−minute ΩR estimate Re(Y/T) Im(Y/T) 2 4 6 8 10 12 14 16 18 20 −400 −200 200 400 Segment Number 1−minute ΩR estimate

ΩR=192.9012 injection

Re(Y/T) Im(Y/T)

Compare Ω(f) = 193 injection, which is visible minute-by-minute

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−5 5 10 15 20 25 −5 5 10 15 20 25

ALLEGRO software injections (90% errorbars)

Ωinjected

Ωdetected

real imag

Ω(f) = 3.9, 9.6, 19 injections recovered from full PG (Ω(f) = 1.9 just at threshold of detectability) Note: injected same random signals w/different amplitudes into same noise

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S4 Hardware Injections

  • 1024-second simulated signals injected into LLO & ALLEGRO

hardware a total of nine times. Simulated all three orientations.

  • One “round” of three injections had non-const Ωgw(f)
  • Other two rounds (“A” & “B”) injected const Ωgw(f) = 8100

− → Focus on those

  • Sensitivity of cross-correlation to injections simulating

XARM (“plus”) and YARM (“minus”) is comparable

  • “null” injection less correlated b/c of simulated misalignment
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−4000 4000 8000 12000 16000 20000 24000 −8000 −4000 4000 8000 Re(Point Estimate) Im(Point Estimate)

Extraction of Hardware Injections

A−minus A−plus A−null B−minus B−plus B−null injected

Circles: 90% statistical uncertainty (null measurements less sensitive) 90% dashed calib uncertainty “teardrop” around ΩR = 8100 HW injections recovered consistent w/cal uncertainty

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−4000 4000 8000 12000 16000 20000 24000 −8000 −4000 4000 8000 Re(Point Estimate) Im(Point Estimate)

Extraction of Hardware Injections

A−minus A−plus A−null B−minus B−plus B−null injected

Circles: 90% statistical uncertainty (null measurements less sensitive) 90% dashed calib uncertainty “teardrop” around ΩR = 8100 HW injections recovered consistent w/cal uncertainty Zoom in on blue box . . .

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6800 7200 7600 8000 8400 8800 −800 −400 400 800 Re(Point Estimate) Im(Point Estimate)

Extraction of Hardware Injections

A−minus A−plus A−null B−minus B−plus B−null injected

Circles: 90% statistical uncertainty 90% dashed calib uncertainty “teardrop” around ΩR = 8100 Systematic offset < cal uncertainty

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S4 Preliminary Cross-Correlation Results

Optimally filter looking for const Ωgw(f) ≡ ΩR Assume H0 = 72 km/s/Mpc (so ΩR =h2

72Ωgw(f))

Analyzed non-playground data w/overlapping 60-sec Hann windows:

ΩR Type Teff (hrs) Point Estimate Error Bar XARM 181.2 0.61 + 0.25i 0.56 YARM 114.7 −0.47 + 0.47i 0.90 non-NULL 295.8 0.31 + 0.31i 0.48 NULL 88.2 10.96 − 43.89i 28.62 all 384.1 0.31 + 0.30i 0.48

No correlation observed − → Convert CC meas of 0.31 + 0.30i & theor errorbar of 0.48 into upper limit . . .

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Constructing Bayesian Posterior PDF

  • Formal prior on Ωgw(915 Hz)

from Explorer-Nautilus: uniform on [0, 115]

  • Marginalize likelihood fcn over calibration uncertainty:

L1 5% amp, 2◦ phase; A1 10% amp, 3◦ phase. (Assume Gaussian prior in ln(amp) and phase.)

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Whelan for LSC & ALLEGRO: prelim LIGO-ALLEGRO SB results LIGO-G060605-04-Z p.19

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.5 1 1.5 2 2.5 ΩR Posterior PDF

Posterior PDF & 90% conf band from all non−PG data

prelim 90% CL UL: ΩR < 1.02 i.e.,

  • Sgw(915 Hz)< 1.5 × 10−23 Hz−1/2

100× improvement on Ωgw(907 Hz) < 115 [h2

100Ωgw(907 Hz) < 60]

from NAUTILUS-EXPLORER [Astone et al., A & A 351, 811 (1999)]

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LLO-ALLEGRO: Summary

  • First stochastic measurement correlating bar w/ifo data;

Probes higher frequency band than LLO-LHO: ∼ 850 − 950 Hz

  • Diff orientations of ALLEGRO −

→ different stochastic response (Data taken in 3 orientations during S4)

  • Preliminary S4 upper limit results from ∼ 370 hrs of data:
  • Sgw(915 Hz) < 1.5 × 10−23 Hz−1/2

I.e., Ωgw(915 Hz) < 1.02 [h2

100Ωgw(915 Hz) < 0.53],

100× better than EXPLORER-NAUTILUS (previous high freq UL)

  • Analysis extracts long-time, low-amplitude simulated signals

(software injections)

  • Hardware inj extracted consistent w/calibration uncertainty
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Extra Slides

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Overlap Reduction Function

γ12(f) = d1abdcd

2

5 4π

  • S2d2Ωˆ

n P TTab

cd(ˆ

n)ei2πfˆ

n·∆ r/c

Depends on alignment of detectors (polarization sensitivity) Frequency dependence from cancellations when λ distance → Widely separated detectors less sensitive at high frequencies

tLLO tLHO tGEO

max

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

zero min

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

This wave drives LHO & GEO out of phase

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Whelan for LSC & ALLEGRO: prelim LIGO-ALLEGRO SB results LIGO-G060605-04-Z p.23

Overlap Reduction Function

γ12(f) = d1abdcd

2

5 4π

  • S2d2Ωˆ

n P TTab

cd(ˆ

n)ei2πfˆ

n·∆ r/c

Depends on alignment of detectors (polarization sensitivity) Frequency dependence from cancellations when λ distance → Widely separated detectors less sensitive at high frequencies

tLLO tLHO tGEO

max

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

zero

This wave (same λ) drives LHO & GEO in phase

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Constructing Posterior PDF

  • Overall estimate

ΩR = x + iy has likelihood function (for given actual ΩR = Ωgw(915 Hz)) P(x, y|ΩR, σΩ) ∝ exp

  • −|x + iy − ΩR|2

2σΩ2

  • Bayes’s theorem gives posterior PDF

P(ΩR|x, y, σΩ) = P(x, y|ΩR, σΩ)P(ΩR) P(x, y|σΩ) ∝ e−(x−ΩR)2/2σΩ2P(ΩR) Note imag part y of pt est factors out

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Marginalization Over Calibration Uncertainty

  • Calibration of LLO & ALLEGRO uncertain in amp & phase

Marginalize over unknown correction factor eΛ+iφ: P(x, y|ΩR, σΩ, Λ, φ) ∝ exp

  −

  • x + iy − ΩReΛ+iφ
  • 2

2σΩ2

  

so the posterior after marginalizing the likelihood function is P(ΩR|x, y, σΩ) ∝

∞

−∞ dΛ

π

−π dφ exp

  −

  • x + iy − ΩReΛ+iφ
  • 2

2σΩ2

   P(Λ, φ) P(ΩR)

which does depend on imag part y

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0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.5 1 1.5 2 2.5 ΩR Posterior PDF

Posterior PDF from all non−PG data

no cal marg marg over cal

Cal marginalization doesn’t matter much @ low SNR

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20 40 60 80 100 120 0.1 0.2 0.3 0.4 0.5 ΩR Posterior PDF

Posterior PDF from ΩR=1.929 injection (no cal marg)

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20 40 60 80 100 120 0.1 0.2 0.3 0.4 0.5 ΩR Posterior PDF

Posterior PDF from ΩR=19.2901 injection (no cal marg)

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Time-Shift Analyses

  • Learned about timing issues via HW injections:

Time-shift analysis helped resolve issues w/ALLEGRO timing Also revealed sample-and-hold & other digital effects in injection system which introduce relative time shift of

1 2×4096 Hz−18 µs = 104 µs

  • Post-processing correction:

Simulate small timeshift w/freq-dependent phase shift Y (f) − → Y (f) ei2πfτ inv FT of CC integrand gives CC values as fcn of time-shift: Y (τ) =

fmax

fmin

d f Y (f) ei2πfτ

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0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10

4

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 ΩR Posterior PDF

Posterior PDF from all HW injections

no cal marg marg over cal

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SLIDE 31

Whelan for LSC & ALLEGRO: prelim LIGO-ALLEGRO SB results LIGO-G060605-04-Z p.31

2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 10

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ΩR Posterior PDF

Posterior PDF & 90% conf band from all HW injections