Stochastic Background Searches with Interferometers and Bars John - PowerPoint PPT Presentation

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 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

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 )

Stoch GW Response of Detector Pair h 2 ( f ′ ) � = δ ( f − f ′ ) 3 H 2 20 π 2 | f | − 3 Ω GW ( f ) γ ( f ) • � � h ∗ 1 ( f ) � 0 • Sensitivity, e.g., possible upper limit for Ω GW ( f ) = const: � � − 1 / 2 � γ 2 ( f ) Ω UL ∼ T d f f 6 P 1 ( f ) P 2 ( f ) • Both depend on Overlap Reduction Function �� γ ( f ) = 5 � S 2 d 2 Ω e i 2 πf ˆ x /c F 1 A (ˆ Ω · ∆ � Ω ) F 2 A (ˆ Ω ) 8 π A =+ , × where F { 1 , 2 } A (ˆ Ω ) are detector beam pattern fcns

Example: Overlap Reduction Function (LLO and other detectors) 1 LHO 0.8 GEO−600 Virgo ALLEGRO 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0 100 200 300 400 500 600 700 800 900 1000 Frequency (Hz)

Example: Overlap Reduction Function (LLO and other detectors) 1 LHO 0.8 GEO−600 Virgo ALLEGRO 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0 50 100 150 200 250 300 Frequency (Hz)

Example: Overlap Reduction Function (European bar detectors) AURIGA−NAUTILUS AURIGA−EXPLORER 1.2 NAUTILUS−EXPLORER 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 0 100 200 300 400 500 600 700 800 900 1000 Frequency (Hz)

Overlap Dependence on Bar Orientation (LLO−ALLEGRO) 1 0.8 IGEC Orientation (N40 ° W) 0.6 Aligned (N72.08 ° E) Antialigned (N17.92 ° W) 0.4 Misaligned (N27.08 ° E) 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0 100 200 300 400 500 600 700 800 900 1000 Frequency (Hz)

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 Geodetic North f = 0 Hz LLO/ALLEGRO Bearing S66.68 W 1 γ max : S72.08 W 0.5 γ null : S26.15 W γ min : S17.92 E γ ( σ ;f) σ A (degrees) - 25 0 25 50 75 100 125 - 0.5 f = 921Hz - 1

Overlap Reduction Function • Can write �� 5 Ω ) e i 2 πf ˆ γ ( f ) = d 1 ab d cd S 2 d 2 Ω P TT ab Ω · ∆ � x /c cd (ˆ 2 4 π � Ω ) = 1 where P TT ab A =+ , × e ab cd (ˆ A (ˆ Ω ) e Acd (ˆ Ω ) is a projection 2 operator onto traceless symmetric tensors transverse to ˆ Ω . ab = 1 • Detector response tensors d ifo y b ) & d bar 2 (ˆ x a ˆ x b − ˆ y a ˆ = ˆ u a ˆ u b ab • Note we can replace each d ab with its traceless part D ab = d ab − 1 3 δ ab d c c

Co ¨ ıncident Overlap Reduction Function • In f → 0 limit, or for co ¨ ıncident detectors, get �� 5 γ (0) = d 1 ab d cd S 2 d 2 Ω P TT ab Ω ) = 2 D 1 ab D ab cd (ˆ 2 2 4 π �� S 2 d 2 Ω P TT ab Ω ) ∝ P T ab cd (ˆ • Result comes from cd (by symmetry); proportionality constant from P T ab ab = 5 & P TT ab ab (ˆ Ω )= 2 • For two IFOs or bar-IFO (in same plane) γ (0) = cos 2( ζ 1 − ζ 2 ) For two bars, γ (0) = cos 2( ζ 1 − ζ 2 ) + 1 3 ✓ ✓ ❩ ❩ ✓ ❩ ✓ ✓ ✓ . | ζ 1 − ζ 2 | . | ζ 1 − ζ 2 | . | ζ 1 − ζ 2 | ❩ . . . . . . . . . . . . . . . . . . ✓ . . . . ✓ . . . . ✓ . . . . . . . . . . . . . ❩ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ❩ ✓ . . . ✓ . . . ✓ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ✓ ✓ ✓ ✓ ✓ ✓

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

General Problem of Azimuth Dependence • Want γ ( d 1 ,� x 1 , d 2 ,� x 2 , f ) where detector 1 is arbitrary & detector 2 is a bar w/azimuth ζ (CW of local North) u along bar axis i.t.o. local North ˆ N & East ˆ • Unit vector ˆ E : ζ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˆ . . . . . . . . . . . . . . . . . . . . . . . . . u = ˆ N cos ζ + ˆ N . . . . . . . . . . . . . . . . . ˆ E sin ζ . . . . . . . . . . . . . . . . . . ✚✚✚✚✚✚✚✚❙ ❃ ˆ u ❙ ✻ ✚ ✚ ✚✚ ✚ ✚ ✲ ✚ • Bar response tensor: ˆ ✚ E ✚ ✚ ❙ ❙ ✚ N a cos ζ + ˆ E a sin ζ )( ˆ N b cos ζ + ˆ E b sin ζ ) d ab ( ˆ = 2 d ab 0 + d ab C cos 2 ζ + d ab = S sin 2 ζ N a ˆ E a ˆ N a ˆ E a ˆ N a ˆ E a ˆ N b + ˆ E b E b + ˆ N b 0 = ˆ C = ˆ N b − ˆ E b S = ˆ where d ab ; d ab ; d ab 2 2 2

Dependence of γ ( f ) on Bar Azimuth • Because γ ( f ) linear in both response tensors, γ ( f ) = γ 0 ( f ) + γ C ( f ) cos 2 ζ + γ S ( f ) sin 2 ζ where γ 0 ,C,S ( f )= γ ( d 1 ,� x 1 , d 0 ,C,S ,� x 2 , f ) • Can also write γ ( f ) = γ 0 ( f ) + γ A ( f ) cos 2( ζ − ζ max ( f )) Note: if γ 0 ( f ) < 0, optimal azimuth is ζ max ( f )+90 ◦ • Offset γ 0 ( f ), amplitude γ A ( f ), “aligned” azimuth ζ max ( f ) determined by properties of detector 1 & location of bar. Look at some real-world examples . . .

Overlap Modulation for LLO/ALLEGRO (amplitudes) 1 0.8 γ 0 (f) 0.6 γ A (f) 0.4 0.2 0 0 100 200 300 400 500 600 700 800 900 1000 Overlap Modulation for LLO/ALLEGRO (optimal azimuth) 180 ζ max (f) 150 ζ max (f) (degrees) 120 90 60 30 0 0 100 200 300 400 500 600 700 800 900 1000 f (Hz)

Overlap Dependence on Bar Orientation (LLO−ALLEGRO) 1 0.8 IGEC Orientation (N40 ° W) 0.6 Aligned (N72.08 ° E) Antialigned (N17.92 ° W) 0.4 Misaligned (N27.08 ° E) 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0 100 200 300 400 500 600 700 800 900 1000 Frequency (Hz)

Overlap Modulation for VIRGO/AURIGA (amplitudes) γ 0 (f) 1 γ A (f) 0.8 0.6 0.4 0.2 0 0 100 200 300 400 500 600 700 800 900 1000 Overlap Modulation for VIRGO/AURIGA (optimal azimuth) 180 ζ max (f) 150 ζ max (f) (degrees) 120 90 60 30 0 0 100 200 300 400 500 600 700 800 900 1000 f (Hz)

Overlap Dependence on Bar Orientation (VIRGO−AURIGA) IGEC Orientation (N44 ° E) 1 Aligned (N20.46 ° E) Antialigned (N69.54 ° W) 0.8 Misaligned (N24.54 ° W) 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0 100 200 300 400 500 600 700 800 900 1000 Frequency (Hz)

Overlap Modulation for VIRGO/NAUTILUS (amplitudes) γ 0 (f) 1 γ A (f) 0.8 0.6 0.4 0.2 0 −0.2 0 100 200 300 400 500 600 700 800 900 1000 Overlap Modulation for VIRGO/NAUTILUS (optimal azimuth) 180 ζ max (f) 150 ζ max (f) (degrees) 120 90 60 30 0 0 100 200 300 400 500 600 700 800 900 1000 f (Hz)

Overlap Dependence on Bar Orientation (VIRGO−NAUTILUS) IGEC Orientation (N44 ° E) 1 Aligned (N20.90 ° E) Antialigned (N69.10 ° W) 0.8 Misaligned (N24.10 ° W) 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0 100 200 300 400 500 600 700 800 900 1000 Frequency (Hz)

Overlap Modulation for VIRGO/EXPLORER (amplitudes) γ 0 (f) 1 γ A (f) 0.8 0.6 0.4 0.2 0 −0.2 0 100 200 300 400 500 600 700 800 900 1000 Overlap Modulation for VIRGO/EXPLORER (optimal azimuth) 120 ζ max (f) 90 ζ max (f) (degrees) 60 30 0 −30 −60 0 100 200 300 400 500 600 700 800 900 1000 f (Hz)

Overlap Dependence on Bar Orientation (VIRGO−EXPLORER) IGEC Orientation (N44 ° E) 1 Aligned (N16.39 ° E) Antialigned (N73.61 ° W) 0.8 Misaligned (N24.10 ° W) 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0 100 200 300 400 500 600 700 800 900 1000 Frequency (Hz)

Overlap Modulation for EXPLORER/AURIGA (amplitudes) 1 γ 0 (f) γ A (f) 0.5 0 −0.5 −1 0 100 200 300 400 500 600 700 800 900 1000 Overlap Modulation for EXPLORER/AURIGA (optimal azimuth) 90 ζ max (f) 60 ζ max (f) (degrees) 30 0 −30 −60 −90 0 100 200 300 400 500 600 700 800 900 1000 f (Hz)