Gravitational Wave Observatories III: Ground Based Interferometers. Neil J. Cornish

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1 Gravitational Wave Observatories III: Ground Based Interferometers Neil J. Cornish

2 Outline Interferometer design Sources of noise LIGO data analysis, theory and search results Astrophysical rates

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4 Gravitational Wave Telescopes

5 Interferometer Design (t) =2 0 T (t) Consideration 1 T / hl for f gw < c L t 4 Want L as large as possible up to L c 10 3 Hz = 300 km ^ Expensive! ^ 3 Solution: folded arms Consideration 2 / 0 T 2 ^ Fluctuations in Laser frequency can masquerade as GW signal ^u ^ â ΔT(t) beam splitter 1 ˆb ^v end mirror 2 Partial Solution: Highly stable lasers Solution: Michelson topology - cancel laser frequency noise end mirror 1

6 Interferometer Design Basic design The 4 km Fabry-Perot cavities effectively fold the arms Can calculate the response using basic E&M, electric field transmission and reflection coefficients at each mirror. (See Maggiore s text) In the long wavelength limit, phase shift is FP = 0 hl c " # 8F p 1+(fgw /f p ) 2 F = p r 1 r 2 1 r 1 r 2 Cavity Finesse f p = 1 4 s c 4 FL Pole frequency

7 Single Fabry-Perot Cavity Basic design FP = 0 hl c " # 8F p 1+(fgw /f p ) 2 F = p r 1 r 2 1 r 1 r 2 Cavity Finesse f p = 1 4 s c 4 FL Pole frequency Reflectivities are very close to unity e.g. advanced LIGO F = 450 f p = 42 Hz Note that increasing the Finesse improves the low frequency sensitivity (good), but lowers the pole frequency (bad) Advanced LIGO gets around this problem by using signal recycling

8 Michelson Interferometer with coupled Fabry-Perot Cavities The actual LIGO design is much more complicated. FP cavities are coupled by a power recycling and signal recycling mirrors There is a common mode and a differential mode L + = L 1 + L 2 2 L = L 1 L 2 2 Differential mode contains the GW signal f = 2 4 L + ln i = input mirror e = end mirror s = signal recycling mirror 1 r i r s r e r i r e r s (t 2 i + r2 i ) Advanced LIGO f = 350 Hz Transfer function 1 p 1+(fgw /f ) 2 The gain from folding and recycling is now a complicated combination of terms. Comes out at a factor of ~1,100

9 Signal recycling and response shaping The cavity transfer function for the Michelson-Fabry-Perot topology with signal recycling allows us to shape the response and target particular signals by changing the distance to the signal recycling mirror C(f) = 1 r s t s e i(2 f`s/c+ s ) r i e 4 ifl/c 1 r i e 4 ifl/c e 2i(2 f`s/c+ s ) s =2 0 `s/c The pole frequency quoted on the last slide was for the zero detuned configuration

10 Sources of Noise Fundamental noise Facility noise

11 Fundamental (quantum) noise S 1/2 shot / 1 p I0 S 1/2 rp / p I 0 SQL x p ~

12 Shot noise: Digital camera, photons per pixel S 1/2 shot / 1 p I

13 Radiation pressure noise The test masses are essentially free (inertial) in the horizontal direction for frequencies above the pendulum frequency of the suspension ẍ = F rad M F rad ) x = 4 2 f 2 M E[ x] =0 E[ x x ] / I 0 f 4 M 2 p I0 ) S 1/2 rp / f 2 M

14 Fundamental (quantum) noise The intensity of the laser light is frequency dependent due to the cavity response S Q = ~ 2 f 2 ML 2 RP Shot I(f) 2 f 2 Mc + 2 f 2 Mc I(f) I(f) I 0 1+(f/f ) 2 f opt SQL is reached when the Shot and RP S Q contributions are equal (minimum of ) S SQL = 2~ 2 f 2 optml 2

15 Standard Quantum Limit SQL x p ~ S SQL = 2~ 2 f 2 optml 2 The SQL is not a fundamental limit to GW detector sensitivity 1) Measure momentum change rather than position change (speedmeter) 2) The RP and Shot noise can be made to be correlated. Allows us to reshape uncertainty ellipse using squeezed light (Quantum no-demolition measurement)

16 Facility Noise Facility noise

17 Seismic Noise Need one of these too large 1/2 Sseis Hz f 2 Hz 1/2

18 Seismic Noise 1/2 Sseis Hz f 2 Hz 1/2 Single pendulum suspension 1/2 Sseis,filt 1/2 Sseis,filt = /2 S seis 2 (f /fpend ) fpend f 2 10 Hz f 2 fpend 1 Hz Hz 1/2 aligo 5-stage pendulum suspension 1/2 Sseis,filt fpend f Hz f 2 Hz 1/2

19 Thermal Noise Thermal noise contributions

20 Thermal Noise Fluctuation-dissipation theorem: PSD of fluctuations of a system in equilibrium at temperature T is determined by the dissipative terms that return the system to equilibrium S T (f) = 4k BT R[Y (f)] (2 f) 2 For an anelastic spring with loss angle Hooke s law becomes F = kx(1 + i ) S T (f) = k BT f 2 res (f) 2 3 Mf (f 2 res f 2 ) 2 + f 4 res 2 (f)

21 Mirror Coating Thermal Noise S T (f) = k BT f 2 res (f) 2 3 Mf (f 2 res f 2 ) 2 + f 4 res 2 (f) The resonant frequencies for the coatings are very high (tens of khz), and the loss angle small (millionths) 1 2kB T S MC = 3 ML 2 f 2 MC f For advanced LIGO S 1/2 MC = Hz f 1/2 Hz 1/2

22 Suspension Thermal Noise S T (f) = k BT 2 3 Mf f 2 res (f) (f 2 res f 2 ) 2 + f 4 res 2 (f) Pendulum mode f pen = 9 Hz Violin modes f v = 500, 1000,... Hz

23 Pendulum Suspension Thermal Noise S T (f) = k BT f 2 res (f) 2 3 Mf (f 2 res f 2 ) 2 + f 4 res 2 (f) f res = f pen f For advanced LIGO S 1/2 pen = Hz f 5/2 Hz 1/2

24 Violin mode Thermal Noise S T (f) = k BT f 2 res (f) 2 3 Mf (f 2 res f 2 ) 2 + f 4 res 2 (f) f res = f v = 500 Hz, 1000 Hz,... For advanced LIGO, first harmonic S 1/2 v = (f 2 v f 2 ) 2 / f 4 Hz 1/2 f = f v 1/2 2 Hz

25 Gravity Gradient Noise We can t escape Newton ẍ = G Z (x 0,t) x x 0 3 (x x0 ) d 3 x This is why we need LISA!

26 Real aligo noise spectra from O1 Suspension Thermal noise Violin modes

27 Real aligo noise spectra from O1 Electronic noise 60 Hz power line and harmonics

28 Real aligo noise spectra from O1/S6 Calibration lines Calibration and the control system is a whole other topic

29 LIGO searches for binary inspired signals Recall: Likelihood for Stationary Gaussian Noise p(d ~ ) = const. e 2 ( ~ ) 2 2 ( ~ )=(d h( ~ ) d h( ~ )) (a b) =2 Z 1 0 ã(f) b (f)+ã (f) b(f) S n (f) df Suppose that we have two hypotheses: H 1 : A signal with parameters ~ is present H 0 : No signal is present Likelihood ratio: ( ~ )= p(d h(~ ),H 1 ) p(d,h 0 ) For Gaussian noise: ( ~ )=e (d h)+ 1 2 (h h)

30 Frequentist Hypothesis Testing Detection Statistic H 0 Noise Hypothesis H 1 Noise + Signal Hypothesis p(λ H 0 ) p(λ H 1 ) Λ Λ Set threshold such that favors hypothesis > H 1 Type 1 error - False Alarm Type II error Type I error Type 1I error - False Dismissal Λ

31 Neyman-Pearson Theorem For a fixed false alarm rate, the false dismissal rate is minimized by the likelihood ratio statistic ( ~ )= p(d h(~ ),H 1 ) p(d,h 0 ) The likelihood ratio is maximized over the signal parameters. The rho statistic is often used in place of the likelihood ratio Writing h = ĥ where (ĥ ĥ) =1 ( ~ )=e (d ĥ) Maximizing wrt ( ~ =0 ) ( ~ )=(d ĥ(~ )) log ( ~ )= ( ~ )

32 The rho statistic and SNR The signal-to-noise ratio (SNR) is defined: SNR = Expected value when signal present RMS value when signal absent In practice, the detector noise is not perfectly Gaussian, and variants of the rho statistic are now used, notably the new SNR statistic, introduced by B. Allen Phys.Rev. D71 (2005) E[ ] = E[ 2 0 ] E[ 0] 2 = (h ĥ) = (h h) SNR 2 =4 Z 1 0 h(f) 2 S n (f) df

33 Matched Filtering = Maximum Likelihood Convolve the data with a filter (template) K: Y = dt du K(t u)d(u) SNR = E[Y ] E[Y 2 0 ] E[Y 0] 2 = 2 0 df ( h (f) K(f)+ h(f) K (f)) 4 0 df K(f) 2 S(f) Maximizing the SNR yields K(f) = h(f) S(f) ) Y =(d h) = p (h h)

34 Frequentist Detection Threshold For stationary, Gaussian noise the detection statistic is Gaussian distributed. For the null hypothesis we have p 0 ( )= 1 2 e 2 /2 For the detection hypothesis we have p 1 ( )= 1 2 e ( 2 SNR 2 )/2 Setting a threshold of gives the false alarm and false dismissal probabilities P FA = 1 erfc( / 2) 2 P FD = 1 erfc(( SNR)/ 2) 2 LIGO/Virgo analyses do not use SNR thresholds, but rather use False Alarm Rate thresholds FAR = P FA T obs e.g. FAR = One in million years and an observation time of one year P FA = 10 6 aka 4.9 =4.8

35 Grid Based Searches Goal is to lay out a grid in parameter space that is fine enough to catch any signal with some good fraction of the maximum matched filter SNR The match measures the fractional loss in SNR in recovering a signal with a template and defines a natural metric on parameter space: M(x, y) = (h(x) h(y)) (h(x) h(x))(h(y) h(y)) Taylor expanding M(x, x + x) =1 g ij x i x j +... where g ij = (h,i h,j ) (h h) (h h,i )(h h,j ) (h h) 2 (Owen Metric) Number of templates (for a hypercube lattice in D dimensions) N = V V = d D x g (2 (1 M min )/D) D Cost grows geometrically with D for any lattice

36 LIGO Style Grid Searches Typically 2-3 dimensional, 1000 s points

37 Reducing the cost of a search In most cases it is possible to analytically maximize over 3 or more parameters Distance: The unit normalized template ĥ defines a reference distance D Scaling this template to distance D gives h = D D ĥ The distance is then estimated from the data as D = D (d ĥ)

38 Reducing the cost of a search Phase Offset: Generate two templates h( = 0) and h( = /2) Then (d h) max = (d h(0)) 2 +(d h( /2)) 2 Easy to see this in the Fourier domain. Suppose d = h 0 e i, then (d h(0)) = (h 0 h 0 ) cos (d h( /2)) = (h 0 h 0 ) sin

39 Reducing the cost of a search 1e-20 8e-21 6e-21 4e-21 Time Offset: h 2e e-21-4e-21-6e-21-8e-21-1e t Fourier transform treats time as periodic - use this to our advantage Compute the inverse Fourier transform of the product of the Fourier transforms: (d h)( t) =4 d (f) h(f) S(f) e 2 if t df Then if the template and data differ by a time shift: d(t) =h(t t 0 ) (d h) max t =(d h)( t = t 0 )

40 Workflow for pycbc search Template bank constructed Matched filtering is done per-detector (not coherent) Detection statistic computed ( new SNR ) Coincidence in time/mass enforced Data quality vetoes applied Monte Carlo background to compute FAR vs new SNR

41 Things that go bump in the night (Bandpass filtered, whitened, time domain) T Samples from the Syracuse Audio Study of Glitches

42 Contending with non-stationary, non-gaussian noise Adiabatic drifts in the PSD - work with short data segments Non-stationary: Glitches - vetoes and time-slides Non-Gaussian:

43 Analysis of 16 days of data from September 2015 Divide by 16 days to get FAR new SNR

44 Early Morning, September Hanford Livingston time (seconds)

45 BayesWave reconstruction of the signal Hanford Livingston time (seconds)

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50 LIGO detections to-date GW LVT GW GW150914

51 Triangulating the Source Hanford

52 Triangulating the Source Hanford + Livingston

53 Triangulating the Source Hanford + Livingston + Virgo

54 Parameter estimation GW170104

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56 Detection without templates BayesWave Cornish & Littenberg 2015 Bayesian model selection Three part model (signal, glitches, gaussian noise) Trans-dimensional Markov Chain Monte Carlo Wavelet decomposition Glitch & GW modeled by wavelets Number, amplitude, quality and TF location of wavelets varies Continuous Morlet/Gabor Wavelets

57 Lines and a drifting noise floor h(f) 2 Cubic spline fit Lorentzian fit Model it Power spectral density (Hz -1 ) Data Lorentzian fit Cubic spline fit frequency (Hz)

58 Glitches Time-Frequency Scalograms of LIGO data Model these too X

59 Gravitational Waves X and model these

60 Caedite eos. Novit enim Dominus qui sunt eius Arnaud Amalric (Kill them all. For the Lord knoweth them that are His.)

61 Example from LIGO s S5 science run

62 Reconstructing GW with wavelets

63 Reconstructing GW with wavelets

64 Match GR prediction Inferred Signal Predicted Signal GW GW170104

65 Astrophysical Inference Would like to know merger rate to constrain population synthesis models. Even better, would like to know merger rate as a function of mass, spin, redshift etc Many cool techniques being developed to do this using things like Gaussian processes Only have time to discuss the total merger rate

66 Astrophysical Rate Limits Even without a detection we can produce interesting astrophysical results such as bounds on the binary merger rate for NS-NS Expect binary mergers to be a Poisson process. If the expected number of events is, then the probability of detecting k events is p(k )= k e /k! If the event rate is R [Mpc 3 year 1 ], and the observable 4-volume is VT [Mpc 3 year] = RVT The probability of observing zero events (k=0) is then p(r) =VTe RVT Follows from p(0 )=p(r) dr = e RVT

67 Astrophysical Rate Limits p(r) =VTe RVT The probability distribution is peaked at a rate of zero. A 90% rate upper limit can be computed: R 0 p(r)dr =1 e R VT =0.9 R VT = ln(0.1) R = 2.3 VT

68 e.g. NS-NS Merger Rate, aligo at design sensitivity V = 4 3 (200 Mpc) NS-NS Detections Time (weeks) λ (Mpc -3 Myr -1 ) Week 1 Truth 90% upper limit

69 Astrophysical Rate Limits When a signal is detected the probability distribution for the rate is no longer peaked at zero. For example, with a single detection (k=1) we have p(r) =R(VT) 2 e RVT This distribution is peaked at R = 1 VT The 90% confidence interval now sets upper and lower limits on the merger rate.

70 Simulated NS-NS Merger Rate Constraints NS-NS Detections Time (weeks) λ (Mpc -3 Myr -1 ) Week 2

71 Simulated NS-NS Merger Rate Constraints NS-NS Detections Time (weeks) λ (Mpc -3 Myr -1 ) Week 3

72 Simulated NS-NS Merger Rate Constraints NS-NS Detections Time (weeks) λ (Mpc -3 Myr -1 ) Week 4

73 Simulated NS-NS Merger Rate Constraints NS-NS Detections Time (weeks) λ (Mpc -3 Myr -1 ) Week 5

74 Simulated NS-NS Merger Rate Constraints NS-NS Detections Time (weeks) λ (Mpc -3 Myr -1 )