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Gravitational Wave Data Analysis Lecture 4: Gravitational Waves MSc Course
Output of GW detector Input of detector has the form h ( t ) = D ij h ij ( t ) is the detector tensor; depends on detector geometry. D ij h out = T ( f )˜ ˜ Output of detector is related to input by h ( f ) is the transfer function. T ( f ) However, output of any real detector will have noise, so output will really be given by s out ( t ) = h out ( t ) + n out ( t ) s ( t ) = h ( t ) + n ( t )
Detector noise To compare detector performances, we use the detectors’ noise . n ( t ) If noise is stationary, then the different Fourier components are uncorrelated. Then the ensemble average of the Fourier components of the noise is of the form: n ( f 0 ) i = δ ( f � f 0 )1 n ⇤ ( f )˜ h ˜ 2 S n ( f ) Ensemble average can be replaced by time average: = 1 ∆ f = 1 D n ( f ) | 2 E | ˜ 2 S n ( f ) T T T is time of experiment.
Noise spectral density is the noise spectral density (aka noise spectral S n ( f ) sensitivity or noise power spectrum): Z ∞ n 2 ( t ) ⌦ ↵ = f S n ( f ) d 0
Detector Pattern Functions Polarization tensors where are orthogonal to u , ˆ ˆ v propagation direction e + ij (ˆ n ) = ˆ u i ˆ u j − ˆ v i ˆ v j e × ij (ˆ n ) = ˆ u i ˆ v j + ˆ v i ˆ u j In the frame where is along the direction, we ˆ ˆ n z can choose v = ˆ ˆ u = ˆ ˆ y x 0 � 1 � 1 0 e + e × ab = ab = 1 0 0 − 1 ab ab
Detector Pattern Functions GW with given propagation direction is given by ˆ n Z ∞ f ˜ X e A h A ( f ) e − 2 π if ( t − ˆ n · x /c ) h ij ( t, x ) = ij (ˆ n ) d −∞ A =+ , × Z ∞ f ˜ X e A h A ( f ) e − 2 π ift h ij ( t ) = ij (ˆ n ) d −∞ A =+ , × X e A = ij (ˆ n ) h A ( t ) A =+ , × Fold in the detector tensor: h ( t ) = D ij h ij ( t ) X D ij e A h ( t ) = ij (ˆ n ) h A ( t ) A =+ , ×
Detector Pattern Functions Detector pattern functions depend on the direction of propagation of the wave. n = ( θ , φ ) ˆ n ) = D ij e A F A (ˆ ij (ˆ n ) h ( t ) = h + ( t ) F + ( θ , φ ) + h × ( t ) F × ( θ , φ )
Detector Pattern Functions h ( t ) = h + ( t ) F + ( θ , φ ) + h × ( t ) F × ( θ , φ ) GWs have two polarizations, usually called + and x. F x F rms F + Detectors are not omnidirectional but exhibit an antenna pattern. Credit: T. Fricke
Interferometric GW Detector Pattern Functions F + ( θ , φ ; ψ = 0) = 1 1 + cos 2 θ � � cos 2 φ 2 F × ( θ , φ ; ψ = 0) = cos θ sin 2 φ Thus GW interferometers have blind directions. For instance, for a GW with plus polarization, y and φ = π / 4 F + = 0 x This wave produces the same displacement in the and arm. y x Differential phase shift vanishes!
Finding signals Naively, we can detect a GW signal only when is larger than | h ( t ) | | n ( t ) | But we will rather be in the situation | h ( t ) | ⌧ | n ( t ) |
Fundamental question: How can we dig out the GW signal from a much larger noise? Answer: We can detect values of much smaller h ( t ) than the floor of the noise if we know, at least to some level of accuracy, the form of . h ( t ) s ( t ) = h ( t ) + n ( t ) Z T Z T Z T 1 dt s ( t ) h ( t ) = 1 dt h 2 ( t ) + 1 dt n ( t ) h ( t ) T T T 0 0 0 Z T Z T ⌘ 1 / 2 1 1 ⇣ τ 0 dt h 2 ( t ) ∼ h 2 dt n ( t ) h ( t ) ∼ n 0 h 0 0 T T T 0 0
Fundamental question: How can we dig out the GW signal from a much larger noise? Z T ⌘ 1 / 2 1 ⇣ τ 0 dt n ( t ) h ( t ) ∼ n 0 h 0 T T 0 Goes to zero for large T! Thus, it is not necessary to have . h 0 > n 0 It is sufficient to have . h 0 > ( τ 0 /T ) 1 / 2 n 0
Let’s turn the previous procedure into a method we can use... Z ∞ s = ˆ dt s ( t ) K ( t ) −∞ Consider as a generic filter function. K ( t ) Goal : Obtain the highest possible signal-to-noise ratio (SNR). If we know the form of , what is the filter function h ( t ) K ( t ) that maximizes the SNR?
Define the signal-to-noise ratio... h s ( t ) i = h n ( t ) i + h h ( t ) i (if noise is stationary) h n ( t ) i = 0 Signal: expected value if signal is present Z ∞ Z ∞ Z ∞ f ˜ h ( f ) ˜ S = dt h s ( t ) i K ( t ) = dt h ( t ) K ( t ) = K ∗ ( f ) d −∞ −∞ −∞ Noise: root-mean-square value if no signal present Z 1 N 2 = h⌦ s ( t ) i 2 i s 2 ( t ) ↵ dt dt 0 K ( t ) K ( t 0 ) h n ( t ) n ( t 0 ) i ˆ � h ˆ h=0 = �1 0 Z ∞ ✓ 1 ◆ 2 � � � ˜ = S n ( f ) K ( f ) d f � � 2 � −∞
Define the signal-to-noise ratio... R ∞ f ˜ h ( f ) ˜ K ∗ ( f ) −∞ d S N = 2 � 1 / 2 R ∞ � 1 � � � ˜ � S n ( f ) K ( f ) −∞ d f � � 2 � Convenient definition: noise-weighted scalar product between two real functions and A ( t ) B ( t ) Z ∞ A ∗ ( f ) ˜ ˜ B ( f ) ( A | B ) = Re d f (1 / 2) S n ( f ) −∞ Z ∞ A ∗ ( f ) ˜ ˜ B ( f ) = 4Re d f S n ( f ) 0
Define the signal-to-noise ratio... Using this scalar product definition, we have: ( u | h ) S u ( f ) = 1 2 S n ( f ) ˜ where N = ˜ K ( f ) ( u | u ) 1 / 2 u/ ( u | u ) 1 / 2 We are searching for vector such that its scalar product with vector h is maximum. ˜ h ( f ) They should be parallel (i.e. proportional): ˜ K ( f ) = const . S n ( f ) This is the Wiener filter (aka matched filter).
Define the signal-to-noise ratio... Optimal value of signal-to-noise ratio: ✓ S ◆ = ( h | h ) 1 / 2 N ✓ S f | ˜ ◆ 2 Z ∞ h ( f ) | 2 = 4 d S n ( f ) N 0 Completely generic and independent of form of ˜ h ( f )
Data Quality Vetoes Smith, et al. CQG 28 235005 (2011) If a particularly noisy period of data is identified, and if it is correlated with a known instrument problem, then the data may be removed.
Coincidence Window Light travel time between detectors plus k standard deviations. � 1 / 2 σ 2 1 + σ 2 � | t 1 − t 2 | ≤ ( ∆ t ) light + k 2 σ 2 1 , σ 2 - variances on arrival times. 2 Coincidence must be within a few tens of milliseconds.
Gravitational-wave Sources
Do we really need a matched filter search? GW150914 GW151226
Formal Definition of Matched Filter Cross-correlate data with template, weighted by detector noise: Z ∞ s ( f )˜ h ∗ ( f ) e 2 π ift ˜ Z ( t ) = A d f S n ( f ) ∞ Cross-correlate template with template, weighted by detector noise: Z ∞ ˜ h ( f )˜ h ∗ ( f ) σ 2 = 2 d f S n ( f ) 0 Normalize output of optimal filter: ρ ( t ) = | Z ( t ) | σ
Matched Filter Search ˆ L ~ S 1 ~ S 1 , 2 · ˆ � 1 , 2 ∝ L m 2 m 1 ~ S 2 Template Bank Matched filter signal-to-noise ratio
The O1 UberBank ~200,000 templates, low frequency cutoff = 30 Hz
Signal-based Vetoes 25 Time-frequency Spectrograms: Measured ρ ( t ) Measured Glitches versus Signals 20 Predicted ρ ( t ) Predicted 15 10 H1 ρ ( t ) 5 0 − 5 − 10 − 0 . 10 − 0 . 05 0 . 00 0 . 05 0 . 10 15 Measured ρ ( t ) Measured Predicted ρ ( t ) Predicted 10 L1 ρ ( t ) 5 0 − 5 − 10 − 0 . 10 − 0 . 05 0 . 00 0 . 05 0 . 10 Time from peak ( s ) Messick, et al, arXiv 1604.04324
Which glitch is not like the others? × × ×
Coincident Triggers Coincident GW candidate Likelihood-based ranking statistic P ( d H , d L | signal) L = P ( d H | noise) P ( d L | noise) Numerator: semi-analytic signal model Denominator: background noise model Background noise model
Likelihood Parameters 𝛙 𝞇 Signal-to-noise ratio A multivariate ranking Signal-based veto 25 statistic Measured ρ ( t ) 20 Predicted ρ ( t ) 15 P ( d H , d L | signal) L = H1 ρ ( t ) 10 P ( d H | noise) P ( d L | noise) 5 0 − 5 − 10 − 0 . 10 − 0 . 05 0 . 00 0 . 05 0 . 10 Detector sensitivity dt
Signal and Background Models Background Model Signal Model + - GW150914 + - GW151226 + - LVT151012
Evolution of Noise Model with Mass The search background is calculated separately for each mass and chi bin. 36
Estimating Significance LVC, PRL 116, 241103 (2016) Only when we leave coincident triggers in our noise model, i.e. gravitational waves, do we form accidental coincidences at the same level of significance or higher.
Gravitational-wave Sources
Burst Signals • GW searches with minimal assumptions • Do not assume accurate source models - affected by noise • Limited ability to measure waveforms, sky positions, polarizations • Can detect the unexpected Cosmic string cusps Heavy stellar binary Supernovae Pulsar glitching black holes Eccentric binary black holes Credit: NASA, ESA, Sanskrit, Credit: NASA, CXC, PSU, Credit: B. Allen & E. P . Shellard Blair Pavlov
Burst Analysis with Wavelets • Wavelets are waveforms of limited duration and bandwidth • GW bursts can be described as superposition of wavelets Credit: MathWorks
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