Network analysis for coalescing binaries: coherent vs coincidence based strategies Andrea Viceré 19th December 2003 Università degli Studi di Urbino Andrea Viceré Milwaukee, December 19th 2001
Motivation and context ✔ It is well known that in gaussian noise a coherent network search of CB events is optimal ✔ A.Pai, S.Dhurandhar and S.Bose, Phys.Rev. D 64 , 042004 (2001). ✔ S.Dhurandhar and M.Tinto, Mon. Not. R. astr. Soc. 234 , 663-676 (1988).
Motivation and context ✔ It is well known that in gaussian noise a coherent network search of CB events is optimal ✔ A.Pai, S.Dhurandhar and S.Bose, Phys.Rev. D 64 , 042004 (2001). ✔ S.Dhurandhar and M.Tinto, Mon. Not. R. astr. Soc. 234 , 663-676 (1988). ✔ The resulting computational cost is however high: O(TFlops) for networks comprising more than 3 detectors ✔ A.Pai, S.Bose and S.Dhurandhar, Class.Quant.Grav. 19 , 1477-1484 (2002)
Motivation and context ✔ It is well known that in gaussian noise a coherent network search of CB events is optimal ✔ A.Pai, S.Dhurandhar and S.Bose, Phys.Rev. D 64 , 042004 (2001). ✔ S.Dhurandhar and M.Tinto, Mon. Not. R. astr. Soc. 234 , 663-676 (1988). ✔ The resulting computational cost is however high: O(TFlops) for networks comprising more than 3 detectors ✔ A.Pai, S.Bose and S.Dhurandhar, Class.Quant.Grav. 19 , 1477-1484 (2002) ✔ How does the coherent search compare with OR-based and AND-based strategies? Are there compromise solutions? ✔ I considered the case of NS-NS binaries for simplicity, and parameters of the existing network of ITFs Andrea Viceré Milwaukee, December 19th 2001 1
The global network Z Z T H T Y G L V G V Y H X L X Left: network from above US Right: from above EU
The global network Z Z T H T Y G L V G V Y H X L X Left: network from above US Right: from above EU Black lines represent the ITF axes.
The global network Z Z T H T Y G L V G V Y H X L X Left: network from above US Right: from above EU Black lines represent the ITF axes. Colored lines are the axes of the detector and Earth frames: Z crosses the North pole , X crosses the Greenwich meridian. Andrea Viceré Milwaukee, December 19th 2001 2
Design sensitivites of the individual detectors GEO600 H4K, L4K 1. · 10 - 19 H2K TAMA300 VIRGO 1. · 10 - 20 1. · 10 - 21 1. · 10 - 22 1. · 10 - 23 Hz 10 100 1000 10000 50 500 5000 They where used to estimate the sensitivity scale to NS-NS binaries � � f − 7 / 3 d f sens ∝ S n ( f ) . Andrea Viceré Milwaukee, December 19th 2001 3
The averaged response of the global network ✔ The network response depends on the source direction ϑ , ϕ , the binary inclination ε and the wave polarization ψ .
The averaged response of the global network ✔ The network response depends on the source direction ϑ , ϕ , the binary inclination ε and the wave polarization ψ . -0.5 0 0.5 1 1.5 -0.5 0 0.5 1 1.5 1.5 1.5 1 1 0.5 0.5 0 0 -0.5 -0.5 ✔ Averaging over ε and ψ one can plot the average cumulative SNR available to the network as a whole, as a function of the direction in the sky. Andrea Viceré Milwaukee, December 19th 2001 4
Individual contributions to the network SNR -0.5 0 0.5 1 1.5 -0.5 0 0.5 1 1.5 1.5 1.5 1 1 0.5 0.5 0 0 -0.5 -0.5
Individual contributions to the network SNR 0.2 0.4 0.6 -0.2 0 -0.25 0 -0.5 0 0.5 1 1.5 0.25 -0.5 0 0.5 1.5 0.5 1 1.5 1 1 1.5 1.5 1 1 0.5 0.5 0.5 0.5 0 0 0 0 -0.5 -0.5
Individual contributions to the network SNR 0.2 0.4 0.6 -0.2 0 -0.25 0 -0.5 0 0.5 1 1.5 0.25 -0.5 0 0.5 1.5 0.5 1 -0.005 -0.0025 0 0.0025 0.005 -0.005 -0.0025 0 1.5 1 1 0.0025 1.5 1.5 0.005 1 1 0.5 0.5 0.01 0.01 0.5 0.5 0.005 0.005 0 0 0 0 0 0 -0.5 -0.5 -0.005 -0.005 Left: LIGO network; center: GEO and Virgo network; right; TAMA
Individual contributions to the network SNR 0.2 0.4 0.6 -0.2 0 -0.25 0 -0.5 0 0.5 1 1.5 0.25 -0.5 0 0.5 1.5 0.5 1 -0.005 -0.0025 0 0.0025 0.005 -0.005 -0.0025 0 1.5 1 1 0.0025 1.5 1.5 0.005 1 1 0.5 0.5 0.01 0.01 0.5 0.5 0.005 0.005 0 0 0 0 0 0 -0.5 -0.5 -0.005 -0.005 Left: LIGO network; center: GEO and Virgo network; right; TAMA ✔ Note the different Virgo-GEO antenna pattern, which contributes to a more spherical overall pattern. Andrea Viceré Milwaukee, December 19th 2001 5
Rules for the comparison ✔ Set a false alarm rate of the network as a whole (1 event/year)
Rules for the comparison ✔ Set a false alarm rate of the network as a whole (1 event/year) ✔ Generate events with random direction ϑ , ϕ and source parameters ε , ψ , but the same network SNR. This means turning the response peanut into a sphere, to compare the strategies in a way independent from the source direction/polarization.
Rules for the comparison ✔ Set a false alarm rate of the network as a whole (1 event/year) ✔ Generate events with random direction ϑ , ϕ and source parameters ε , ψ , but the same network SNR. This means turning the response peanut into a sphere, to compare the strategies in a way independent from the source direction/polarization. ✔ Set false alarm rates R FA on the individual detectors, and rules to combine the events that lead to the same overall R FA as the “coherent network”.
Rules for the comparison ✔ Set a false alarm rate of the network as a whole (1 event/year) ✔ Generate events with random direction ϑ , ϕ and source parameters ε , ψ , but the same network SNR. This means turning the response peanut into a sphere, to compare the strategies in a way independent from the source direction/polarization. ✔ Set false alarm rates R FA on the individual detectors, and rules to combine the events that lead to the same overall R FA as the “coherent network”. ✔ Compute the SNR seen by each detector, hence local detection probabili- ties P DET for each sampled direction/polarization.
Rules for the comparison ✔ Set a false alarm rate of the network as a whole (1 event/year) ✔ Generate events with random direction ϑ , ϕ and source parameters ε , ψ , but the same network SNR. This means turning the response peanut into a sphere, to compare the strategies in a way independent from the source direction/polarization. ✔ Set false alarm rates R FA on the individual detectors, and rules to combine the events that lead to the same overall R FA as the “coherent network”. ✔ Compute the SNR seen by each detector, hence local detection probabili- ties P DET for each sampled direction/polarization. ✔ Combine with various strategies (OR, AND); obtain the average P DET
Rules for the comparison ✔ Set a false alarm rate of the network as a whole (1 event/year) ✔ Generate events with random direction ϑ , ϕ and source parameters ε , ψ , but the same network SNR. This means turning the response peanut into a sphere, to compare the strategies in a way independent from the source direction/polarization. ✔ Set false alarm rates R FA on the individual detectors, and rules to combine the events that lead to the same overall R FA as the “coherent network”. ✔ Compute the SNR seen by each detector, hence local detection probabili- ties P DET for each sampled direction/polarization. ✔ Combine with various strategies (OR, AND); obtain the average P DET ✔ Compare with the coherent case, and vary the SNR available to the net- work. Andrea Viceré Milwaukee, December 19th 2001 6
Statistics It is worth recalling that the SNR 2 seen by the individual detectors and by the network obey to different statistics
Statistics It is worth recalling that the SNR 2 seen by the individual detectors and by the network obey to different statistics ✔ On a single detector the SNR 2 is a χ 2 with 2 DOF, hence if ξ is a threshold � ∞ P FA ( ξ ) = e − ξ ; P DET ( ξ , E sig ) = 2 ξ e − E − E sig I 0 � � � E ∗ E sig dE
Statistics It is worth recalling that the SNR 2 seen by the individual detectors and by the network obey to different statistics ✔ On a single detector the SNR 2 is a χ 2 with 2 DOF, hence if ξ is a threshold � ∞ P FA ( ξ ) = e − ξ ; P DET ( ξ , E sig ) = 2 ξ e − E − E sig I 0 � � � E ∗ E sig dE ✔ On the network, the corresponding quantity is a χ 2 with 4 DOF, hence � ∞ � E P FA ( ξ ) = ( 1 + ξ ) e − ξ ; 2 P DET ( ξ , E sig ) = e − E − E sig I 1 � � � E ∗ E sig dE E sig ξ
Statistics It is worth recalling that the SNR 2 seen by the individual detectors and by the network obey to different statistics ✔ On a single detector the SNR 2 is a χ 2 with 2 DOF, hence if ξ is a threshold � ∞ P FA ( ξ ) = e − ξ ; P DET ( ξ , E sig ) = 2 ξ e − E − E sig I 0 � � � E ∗ E sig dE ✔ On the network, the corresponding quantity is a χ 2 with 4 DOF, hence � ∞ � E P FA ( ξ ) = ( 1 + ξ ) e − ξ ; 2 P DET ( ξ , E sig ) = e − E − E sig I 1 � � � E ∗ E sig dE E sig ξ This is just to remind that the interpretation of the SNR clearly depends on the kind of statistic, and we have to refer to P DET , P FA for a meaningful comparison. Andrea Viceré Milwaukee, December 19th 2001 7
Coherent vs OR network 1 0,9 0,8 0,7 P DETECTION 0,6 0,5 0,4 0,3 0,2 Coherent search (R FA ~1/year) 0,1 OR search (R FA ~ 1/year) 0 4 5 6 7 8 9 10 SNR NETWORK ✔ Errors represent the RMS spread due to the non-uniform antenna patterns.
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