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Stochastic Background Searches with Interferometers and Bars John T. Whelan Loyola University New Orleans jtwhelan@loyno.edu 8th Gravitational Wave Data Analysis Workshop 2003 December 19 G030692-00-Z Stochastic Background Searches with

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SLIDE 1

Stochastic Background Searches with Interferometers and Bars

John T. Whelan Loyola University New Orleans jtwhelan@loyno.edu 8th Gravitational Wave Data Analysis Workshop 2003 December 19 G030692-00-Z

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SLIDE 2

Stochastic Background Searches with Interferometers and Bars

(Dependence of Response on Bar Azimuth)

John T. Whelan Loyola University New Orleans jtwhelan@loyno.edu 8th Gravitational Wave Data Analysis Workshop 2003 December 19 G030692-00-Z

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SLIDE 3

Motivation/Background

  • Bar-IFO correlation searches for stochastic BG:

– LLO-ALLEGRO underway w/S2 data – Virgo-{AURIGA/NAUTILUS/EXPLORER} planned

  • Features of Bar-IFO searches (response to isotropic BG)

– Geographically close detector pairs (needed for high freq) – Changing orientation of bar changes GW response – Both explained by overlap reduction function γ(f)

  • This talk examines dependence of γ(f) on bar azimith

– Geometrical explanation of f = 0 behavior – General form of γ(f, ζ)

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SLIDE 4

Stoch GW Response of Detector Pair

h∗

1(f)

h2(f′) = δ(f − f′) 3H2

20π2|f|−3 ΩGW(f)γ(f)

  • Sensitivity, e.g., possible upper limit for ΩGW(f) = const:

ΩUL ∼

  • T
  • d

f γ2(f) f6P1(f)P2(f)

−1/2

  • Both depend on Overlap Reduction Function

γ(f) = 5 8π

  • A=+,×
  • S2d2Ω ei2πf ˆ

Ω·∆ x/c F1A(ˆ

Ω) F2A(ˆ Ω)

where F{1,2}A(ˆ

Ω) are detector beam pattern fcns

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SLIDE 5

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)

Example: Overlap Reduction Function (LLO and other detectors)

LHO GEO−600 Virgo ALLEGRO

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SLIDE 6

50 100 150 200 250 300 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 Frequency (Hz)

Example: Overlap Reduction Function (LLO and other detectors)

LHO GEO−600 Virgo ALLEGRO

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SLIDE 7

100 200 300 400 500 600 700 800 900 1000 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 1.2 Frequency (Hz)

Example: Overlap Reduction Function (European bar detectors)

AURIGA−NAUTILUS AURIGA−EXPLORER NAUTILUS−EXPLORER

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SLIDE 8

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 Dependence on Bar Orientation (LLO−ALLEGRO)

IGEC Orientation (N40°W) Aligned (N72.08°E) Antialigned (N17.92°W) Misaligned (N27.08°E)

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SLIDE 9

Signal Modulation w/LLO & ALLEGRO

  • Proposed by Finn & Lazzarini (gr-qc/0104040)
  • Combine measurements in diff orientations

to cancel CC noise & add GW background

  • Empirical sinusoidal dependence of γ(f) on bar azimuth

γ(σ;f) f = 0 Hz σA (degrees) f = 921Hz

  • 25

25 50 75 100 125

  • 1
  • 0.5

0.5 1

Geodetic North LLO/ALLEGRO Bearing S66.68 W γmax: S72.08 W γnull: S26.15 W γmin: S17.92 E

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SLIDE 10

Overlap Reduction Function

  • Can write

γ(f) = d1abdcd

2

5 4π

  • S2d2Ω P TTab

cd(ˆ

Ω)ei2πf ˆ

Ω·∆ x/c

where P TTab

cd(ˆ

Ω) = 1

2

  • A=+,×eab

A (ˆ

Ω)eAcd(ˆ Ω) is a projection

  • perator onto traceless symmetric tensors transverse to ˆ

Ω.

  • Detector response tensors difo

ab = 1 2(ˆ

xaˆ xb − ˆ yaˆ yb) & dbar

ab

= ˆ uaˆ ub

  • Note we can replace each dab with its traceless part

Dab = dab−1 3 δab dc

c

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SLIDE 11

Co ¨ ıncident Overlap Reduction Function

  • In f → 0 limit, or for co

¨ ıncident detectors, get γ(0) = d1abdcd

2

5 4π

  • S2d2Ω P TTab

cd(ˆ

Ω) = 2D1abDab

2

  • Result comes from
  • S2d2Ω P TTab

cd(ˆ

Ω) ∝ P Tab

cd (by symmetry);

proportionality constant from P Tab

ab= 5 & P TTab ab(ˆ

Ω)= 2

  • For two IFOs or bar-IFO (in same plane) γ(0) = cos 2(ζ1−ζ2)

For two bars, γ(0) = cos 2(ζ1 − ζ2) + 1

3

✓ ✓ ✓ ✓ ✓ ✓ ❩ ❩ ❩ ❩ ❩ ❩

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |ζ1 − ζ2|

✓ ✓ ✓ ✓ ✓ ✓

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |ζ1 − ζ2|

✓ ✓ ✓ ✓ ✓ ✓

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |ζ1 − ζ2|

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SLIDE 12

Overlap For Co ¨ ıncident Bars

  • Note for parallel bars, γ(0) = 4

3

but for perpendicular bars γ(0) = −2

3

  • Some authors use different normalization for bars

so that max(γ(f)) remains unity

  • 4

3 vs 1 represents geometry: bars “more omnidirectional”

bars have whole plane of “optimal” propagation directions ⊥ bars have only one “optimal” propagation direction

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SLIDE 13

General Problem of Azimuth Dependence

  • Want γ(d1,

x1,d2, x2, f) where detector 1 is arbitrary

& detector 2 is a bar w/azimuth ζ (CW of local North)

  • Unit vector ˆ

u along bar axis i.t.o. local North ˆ N & East ˆ E:

ˆ

u = ˆ N cos ζ + ˆ E sin ζ

✻

ˆ

N

✚✚ ✚ ❃ˆ

u

✲

ˆ

E

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

ζ

✚✚✚✚✚✚✚✚❙ ❙ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ✚ ❙ ❙

  • Bar response tensor:

dab

2

= ( ˆ Na cos ζ + ˆ Ea sin ζ)( ˆ Nb cos ζ + ˆ Eb sin ζ) = dab

0 + dab C cos 2ζ + dab S sin 2ζ

where dab

0 = ˆ Na ˆ Nb+ ˆ Ea ˆ Eb 2

; dab

C = ˆ Na ˆ Nb− ˆ Ea ˆ Eb 2

; dab

S = ˆ Na ˆ Eb+ ˆ Ea ˆ Nb 2

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SLIDE 14

Dependence of γ(f) on Bar Azimuth

  • Because γ(f) linear in both response tensors,

γ(f) = γ0(f) + γC(f) cos 2ζ + γS(f) sin 2ζ where γ0,C,S(f)= γ(d1,

x1,d0,C,S, x2, f)

  • Can also write

γ(f) = γ0(f) + γA(f) cos 2(ζ−ζmax(f)) Note: if γ0(f)< 0, optimal azimuth is ζmax(f)+90◦

  • Offset γ0(f), amplitude γA(f), “aligned” azimuth ζmax(f)

determined by properties of detector 1 & location of bar. Look at some real-world examples . . .

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SLIDE 15

100 200 300 400 500 600 700 800 900 1000 0.2 0.4 0.6 0.8 1

Overlap Modulation for LLO/ALLEGRO (amplitudes)

100 200 300 400 500 600 700 800 900 1000 30 60 90 120 150 180 f (Hz) ζmax(f) (degrees)

Overlap Modulation for LLO/ALLEGRO (optimal azimuth)

γ0(f) γA(f) ζmax(f)

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SLIDE 16

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 Dependence on Bar Orientation (LLO−ALLEGRO)

IGEC Orientation (N40°W) Aligned (N72.08°E) Antialigned (N17.92°W) Misaligned (N27.08°E)

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SLIDE 17

100 200 300 400 500 600 700 800 900 1000 0.2 0.4 0.6 0.8 1

Overlap Modulation for VIRGO/AURIGA (amplitudes)

100 200 300 400 500 600 700 800 900 1000 30 60 90 120 150 180 f (Hz) ζmax(f) (degrees)

Overlap Modulation for VIRGO/AURIGA (optimal azimuth)

γ0(f) γA(f) ζmax(f)

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SLIDE 18

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 Dependence on Bar Orientation (VIRGO−AURIGA)

IGEC Orientation (N44°E) Aligned (N20.46°E) Antialigned (N69.54°W) Misaligned (N24.54°W)

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SLIDE 19

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

Overlap Modulation for VIRGO/NAUTILUS (amplitudes)

100 200 300 400 500 600 700 800 900 1000 30 60 90 120 150 180 f (Hz) ζmax(f) (degrees)

Overlap Modulation for VIRGO/NAUTILUS (optimal azimuth)

γ0(f) γA(f) ζmax(f)

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SLIDE 20

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 Dependence on Bar Orientation (VIRGO−NAUTILUS)

IGEC Orientation (N44°E) Aligned (N20.90°E) Antialigned (N69.10°W) Misaligned (N24.10°W)

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SLIDE 21

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

Overlap Modulation for VIRGO/EXPLORER (amplitudes)

100 200 300 400 500 600 700 800 900 1000 −60 −30 30 60 90 120 f (Hz) ζmax(f) (degrees)

Overlap Modulation for VIRGO/EXPLORER (optimal azimuth)

γ0(f) γA(f) ζmax(f)

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SLIDE 22

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 Dependence on Bar Orientation (VIRGO−EXPLORER)

IGEC Orientation (N44°E) Aligned (N16.39°E) Antialigned (N73.61°W) Misaligned (N24.10°W)

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SLIDE 23

100 200 300 400 500 600 700 800 900 1000 −1 −0.5 0.5 1

Overlap Modulation for EXPLORER/AURIGA (amplitudes)

100 200 300 400 500 600 700 800 900 1000 −90 −60 −30 30 60 90 f (Hz) ζmax(f) (degrees)

Overlap Modulation for EXPLORER/AURIGA (optimal azimuth)

γ0(f) γA(f) ζmax(f)

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SLIDE 24

100 200 300 400 500 600 700 800 900 1000 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 1.2 Frequency (Hz)

Overlap Dependence on AURIGA Orientation (EXPLORER−AURIGA)

IGEC Orientation (N44°E) Aligned (N43.07°E) Antialigned (N46.93°W) Misaligned (N1.93°W)

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SLIDE 25

100 200 300 400 500 600 700 800 900 1000 −1 −0.5 0.5 1

Overlap Modulation for EXPLORER/NAUTILUS (amplitudes)

100 200 300 400 500 600 700 800 900 1000 −90 −60 −30 30 60 90 f (Hz) ζmax(f) (degrees)

Overlap Modulation for EXPLORER/NAUTILUS (optimal azimuth)

γ0(f) γA(f) ζmax(f)

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SLIDE 26

100 200 300 400 500 600 700 800 900 1000 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 1.2 Frequency (Hz)

Overlap Dependence on NAUTILUS Orientation (EXPLORER−NAUTILUS)

IGEC Orientation (N44°E) Aligned (N43.54°E) Antialigned (N46.46°W) Misaligned (N1.46°W)

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SLIDE 27

Summary

  • Stoch GW Response (∝ γ(f)) dep on Detector Orientation
  • Co

¨ ıncident/low-freq limit γ(0) = 2D1abDab

2

  • General Dependence on Bar Azimuth ζ:

γ(f) = γ0(f) + γA(f) cos 2(ζ−ζmax(f)) Need γA(fbar) > |γ0(fbar)| for full modulation