Gravitational Wave Data Analysis: I. Detection Chris Van Den Broeck Kavli RISE Summer School on Gravitational Waves, Cambridge, UK, 23-27 September 2019
Advanced LIGO and Advanced Virgo
Detectable astrophysical sources? Fast-spinning neutron stars Merging neutron stars, black holes Primordial gravitational waves Supernovae
Inspiral-merger-ringdown
The first detections LIGO + Virgo, PRL 116 , 061102 (2016) Ten published binary black hole detections so far: • GW150914, GW151012, GW151226, GW170104, GW170608, GW170729, GW170809, GW170814, GW170818, GW170823 Binary neutron star: GW170817 •
Gravitational waves Ø Einstein field equations: R µ ⌫ − 1 2 R g µ ⌫ = 8 ⇡ G c 4 T µ ⌫ Ø Far away from matter/energy: Metric tensor is that of flat spacetime with a small perturbation g µ ν = η µ ν + h µ ν Ø Einstein equations reduce to a wave equation for the perturbation: � ∂ 2 c 2 ∂ t 2 + ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 + ∂ 2 ✓ ◆ h µ ν = 0 ∂ z 2 Ø In the “transverse-traceless” gauge, wave moving in z direction becomes:
The effect of gravitational waves on matter Ø Consider two point particles in free fall, small separation Ø Geodesics: Ø Take the difference of the two, expand to leading order in : Ø Local Lorentz frame: Ø Non-relativistic motion: Ø Relate to the Riemann tensor: Ø In terms of the TT metric perturbation: Ø Hence tidal effect
Gravitational waves Ø Gravitational waves have the effect of traveling tidal waves Ø Nearby point particles in free fall, small separation : tidal effect Ø Effect of “ plus ” and “ cross ” polarizations on ring of test particles: h + h × (
Detection of gravitational waves Ø Exploit tidal effect on matter Ø Resonant detectors Metal bar: resonant frequency § Measure tiny length changes § Also spherical resonant detectors: § equal sensitivity in all directions (e.g. miniGRAIL, Leiden)
Interferometric detectors Ø Interferometric detectors Laser beam through beam splitter § Power builds up in long (several km) § cavities At output: destructive interference § … unless gravitational wave present
Interferometric detectors Ø Motion of the end mirrors: x y x y … hence Ø Measured strain:
Interferometric detectors h ( t ) = 1 Ø For L-shaped detectors: 2( h xx � h yy ) Ø More generally: Ø Detector tensor:
Interferometric detectors h ( t ) = 1 2( h xx � h yy ) z x 0 Ø Signal from arbitrary direction In transverse-traceless frame: y 0 z 0 x y x 0 = x, y 0 = y, z 0 = z Ø If then x y Ø In general, need to apply linear transformation Ø Projection onto detector is also linear, hence
Interferometric detectors Ø Signal from an L-shaped interferometer: sky position, polarization angle Ø
Digging a signal out of noise Ø If there is a signal: measured strain is a sum of noise and signal : s ( t ) = n ( t ) + h ( t ) Ø If shape of signal approximately known: integrate against the output oscillatory positive definite period of the GW signal characteristic signal amplitude characteristic amplitude of the noise h 0 > ( τ 0 /T ) 1 / 2 n 0 Ø To detect a signal, don’t need but only h 0 > n 0 Binary coalescences: § τ 0 ⇠ 10 � 2 s , ( τ 0 /T ) 1 / 2 ⇠ 10 � 2 T ⇠ 100 s ! Millisecond pulsars: § ( τ 0 /T ) 1 / 2 ⇠ 10 � 5 τ 0 ⇠ 1 ms , T ⇠ 1 yr !
Characterizing the noise Ø Detector data comes in as a time series Ø If only noise: where t i +1 = t i + ∆ t ( n ( t 0 ) , n ( t 1 ) , . . . , n ( t N )) Ø Often convenient to take a (discrete) Fourier transform: where (˜ n ( f 0 ) , ˜ n ( f 1 ) , . . . , ˜ n ( f N )) f i +1 = f i + ∆ f Notation: § n ( f i ) = ˜ ˜ n i Ø Some noise realizations are more probable than others Probability distribution in each frequency bin: § p (˜ n i ) Ø We will assume that the noise is stationary and Gaussian : ni | 2 � | ˜ 2 σ 2 p (˜ n i ) / e i Z Z n i | 2 p (˜ Stationarity and Gaussianity: h | n i | 2 i = h n i i = ˜ n i p (˜ n i ) d ˜ n i = 0 , | ˜ n i ) d ˜ n i Ø Probability density for noise realization as a whole : N Y p [ n ] = p (˜ n 0 , ˜ n 1 , . . . , ˜ n N ) = p (˜ n i ) i =1
Characterizing the noise Ø Probability density for noise realization as a whole : N Y p [ n ] = p (˜ n 0 , ˜ n 1 , . . . , ˜ n N ) = p (˜ n i ) i =1 Ø For stationary, Gaussian noise: N ni | 2 | ˜ � 1 P N Y i =1 σ 2 n i ) = N e 2 p [ n ] = p (˜ i i =1 Ø For the purpose of this lecture, convenient to take continuum limit: ni | 2 | ˜ � 1 P N σ 2 i =1 2 p [ n ] = N e i ni | 2 | ˜ � 1 P N i ∆ f ∆ f i =1 σ 2 2 = N e n ( f ) | 2 R ∞ | ˜ Sn ( f ) d f ! N e � −∞ Ø Variance:
Characterizing the noise Ø Probability density and variance for noise realizations: n ( f ) | 2 R ∞ | ˜ Sn ( f ) d f p [ n ] = N e − −∞ Ø Could also have worked in the time domain Stationarity: § Gaussianity: completely determined by § where again denotes average over noise realizations Defining noise power spectral density as one finds
Characterizing the noise Ø (Square root of ) noise power spectral density in the first two observing runs of Advanced LIGO and Advanced Virgo:
Matched filtering Ø Instead of integrating the data against waveforms, use more optimal filter Z ∞ s = ˆ dt s ( t ) K ( t ) |{z} −∞ filter Ø Define to be the expected value of when a signal is present, and S N ˆ s S N S N h ( t ) let be the root-mean-square value when no signal is present: S N ⇤ 1 / 2 s 2 i h =0 � h ˆ s i 2 ⇥ S = h ˆ s i h N = h ˆ h =0 Ø Define signal-to-noise ratio : h ˆ s i h S/N = h =0 ] 1 / 2 s i 2 s 2 i h =0 � h ˆ [ h ˆ Ø Our task: find out which filter maximizes S/N K ( t ) Do this assuming particular signal shape § h Filter will be optimized for that particular signal! § Will deal later with case of many possible signal shapes §
Matched filtering Ø Write in the frequency domain: S N S = h ˆ s i h Z 1 = dt h s ( t ) i K ( t ) �1 Z 1 = dt h ( t ) K ( t ) �1 Z 1 f ˜ h ( f ) ˜ = d K ⇤ ( f ) �1 Ø Also : S N s i 2 ⇤ 1 / 2 s 2 i � h ˆ ⇥ N = h ˆ h =0 ⇤ 1 / 2 s 2 i ⇥ = h ˆ h =0 Z 1 Z 1 � 1 / 2 dt 0 h n ( t ) n ( t 0 ) i K ( t ) K ( t 0 ) = dt �1 �1 Z 1 � 1 / 2 f 1 2 S n ( f ) | ˜ K ⇤ ( f ) | 2 = d �1
Matched filtering Ø Then we arrive at: R 1 f ˜ h ( f ) ˜ �1 d K ⇤ ( f ) S N = i 1 / 2 hR 1 2 S n ( f ) | ˜ f 1 �1 d K ( f ) | 2 Ø Now define the noise-weighted inner product Z 1 A ⇤ ( f ) ˜ ˜ B ( f ) ( A | B ) ⌘ < d f 1 2 S n ( f ) �1 Z 1 A ⇤ ( f ) ˜ ˜ B ( f ) = 4 < d f S n ( f ) 0 … and rewrite as S/N S ( K | h ) K = 1 2 S n ( f ) ˜ N = K ( f ) ( K | K ) 1 / 2 … or S K N = ( ˆ ˆ K | h ) K = ( K | K ) 1 / 2 Maximizing is equivalent to making point in the same direction as ˆ h S/N K
Matched filtering S K = 1 K 2 S n ( f ) ˜ N = ( ˆ ˆ K | h ) K = K ( f ) ( K | K ) 1 / 2 Ø Maximizing is equivalent to making point in the same direction as ˆ S/N h K ˆ K / h K / h ! ˜ h ( f ) ˜ K ( f ) / ! S n ( f ) “Wiener filter” Ø Optimal signal-to-noise ratio ( K | h ) S N = p ( K | K ) ( h | h ) = p ( h | h ) p = ( h | h )
Matched filtering Ø So far we assumed that if signal present in the data, it would be of known shape: and S/N = ( ˆ K | h s i ) = ( ˆ h s i = h K | h ) Ø In practice we must consider many possible trial waveforms , and apply the / h i ˆ optimal filter for each of these to the data: K i / h i ✓ S ◆ = ( ˆ K i | h s i ) N i ( K i | h s i ) = p ( K i | K i ) ( h i | h s i ) = p ( h i | h i ) Ø In practice we only know the detector output , not its expectation value, so in s actual signal processing we approximate ✓ S ◆ ( h i | s ) ' p N ( h i | h i ) i
Matched filtering Ø For a given trial waveform : / h i ✓ S ◆ ( h i | s ) = N p ( h i | h i ) i Ø For e.g. compact binary coalescences, waveforms are characterized by source parameters ¯ θ = { θ 1 , θ 2 , . . . , θ N } Ø Lay out template banks over parameter space: points give waveforms ¯ h i = h (¯ θ i θ i ) Ø Find maximum of signal-to-noise ratio over the template bank: ✓ S ( h (¯ ◆ θ i ) | s ) = max ( h (¯ θ i ) | h (¯ p N i θ i )) max
Placement of template banks Ø Match between nearby templates : q h (¯ ˆ θ ) = h (¯ ( h (¯ θ ) | h (¯ θ ) / θ )) M = (ˆ h (¯ θ ) | ˆ h (¯ θ + ∆ θ )) Ø Expand in small quantities: ∂ 2 M θ )) + ∂ M ∂θ i ∆ θ i + 1 M ' (ˆ h (¯ θ ) | ˆ h (¯ ∂θ i ∂θ j ∆ θ i ∆ θ j 2 ∂ 2 M = 1 + 1 ∂θ i ∂θ j ∆ θ i ∆ θ j 2 Ø Define a metric tensor on parameter space: ∂ 2 M g ij ≡ − 1 2 ∂θ i ∂θ j Ø Mismatch between neighboring templates: 1 − M = g ij ∆ θ i ∆ θ j Place templates on parameter space such that metric distance never larger than a pre-specified mismatch, e.g. 0.03
Placement of template banks Ø Example of placement in ( m 1 , m 2 ) Spin dimensions not shown §
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