Announcements Assignment 4 due today. Assignment 5 uploaded to website and Piazza. Will be due in two weeks, May 12th. No class next week, May 5th. Last lecture will be May 12th. Assignment 6 will be optional (see next slide). We will meet again for Final Assessment on May 29.
HW Adjustments Options for improving your grade: 1) On an assignment where you have done poorly, you can correct the problems, and receive half the points back. AND/OR 2) You can turn in the optional assignment 6 and its grade will replace your lowest grade.
Gravitational Wave Data Analysis Lecture 5: Gravitational Waves MSc Course
Crucial problem: How to extract a GW signal from the (typically) much larger detector noise? Key components: 1. Characterizing detector noise 2. Understanding detectors’ angular sensitivity 3. Applying filtering techniques
Output of GW detector Signal Input of detector has the form h ( t ) = D ij h ij ( t ) is the detector tensor; depends on detector geometry. D ij Signal Output of detector is related to input by ˜ h out = T ( f )˜ 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 to give the variance of the noise: = 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
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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 =+ , ×
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Detector Pattern Functions h ( t ) = h + ( t ) F + ( θ , φ ) + h × ( t ) F × ( θ , φ ) GWs have two polarizations 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 The detector output will be of the general form s ( t ) = h ( t ) + n ( t ) 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... Goal : Obtain the highest possible signal-to-noise ratio (SNR). Z ∞ s = ˆ dt s ( t ) K ( t ) −∞ Consider as a generic filter function. K ( t ) If we know the form of , what is the filter function h ( t ) K ( t ) that maximizes the SNR? Aka matched filtering
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