Gravitational Wave Observatories III: Ground Based...
Transcript of Gravitational Wave Observatories III: Ground Based...
Gravitational Wave Observatories III: Ground Based Interferometers
Neil J. Cornish
Outline
• Interferometer design
• Sources of noise
• LIGO data analysis, theory and search results
• Astrophysical rates
Gravitational Wave Telescopes
T(t)Δ
end mirror 1
end mirror 2
^
^
t
^
beam splitteru
v^
ab
1
2
3
4
�(t) = 2⇡⌫0�T (t)
Interferometer Design
Consideration 1 �T / hL for fgw <c
L
Want L as large as possible up to L ⇠ c
103 Hz= 300 km
Expensive!
Solution: folded arms
Consideration 2 � / ⌫0�T
Fluctuations in Laser frequency can masquerade as GW signal
Partial Solution: Highly stable lasers
Solution: Michelson topology - cancel laser frequency noise
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)
|��FP| =✓⌫0hL
c
◆"8Fp
1 + (fgw/fp)2
#In the long wavelength limit, phase shift is
F =⇡pr1r2
1� r1r2Cavity Finesse
fp =1
4⇡⌧s⇡ c
4FLPole frequency
Single Fabry-Perot Cavity
Basic design |��FP| =✓⌫0hL
c
◆"8Fp
1 + (fgw/fp)2
#
F =⇡pr1r2
1� r1r2Cavity Finesse
fp =1
4⇡⌧s⇡ c
4FLPole frequency
Reflectivities are very close to unity
fp = 42Hze.g. advanced LIGO F = 450
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
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+ =L1 + L2
2L� =
L1 � L2
2
Differential mode contains the GW signal
f� =2
4⇡L+ln
✓1� rirs
reri � rers(t2i + r2i )
◆
e = end mirror
i = input mirror
s = signal recycling mirror
Advanced LIGO
1p1 + (fgw/f�)2
Transfer function
The gain from folding and recycling is now a complicated combination of terms. Comes out at a factor of ~1,100
f� = 350 Hz
Signal recycling and response shapingThe 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) =tse�i(2⇡f`s/c+�s)
1� rs⇣
ri�e�4⇡ifL/c
1�rie�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
Sources of Noise
Fundamental noise
Facility noise
Fundamental (quantum) noise
S1/2shot
/ 1pI0
S1/2rp /
pI0
SQL� x�p � ~
Shot noise: Digital camera, photons per pixel 10�3 10�2 10�1
1 10 100
103 104 105
S1/2shot
/ 1pI0
Radiation pressure noise
The test masses are essentially free (inertial) in the horizontal direction for frequencies above the pendulum frequency of the suspension
) x = � Frad
4⇡2f
2M
x =Frad
M
E[x] = 0
) S1/2rp /
pI0
f2M
E[xx⇤] / I0
f
4M
2
Fundamental (quantum) noise
The intensity of the laser light is frequency dependent due to the cavity response
SQ =~
⇡2f2ML2
✓I(f)
⇡2f2Mc+
⇡2f2Mc
I(f)
◆
I(f) ⇠ I01 + (f/f�)2
ShotRP
SQL is reached when the Shot and RP contributions are equal (minimum of )SQ
SSQL
=2~
⇡2f2
opt
ML2
fopt
Standard Quantum Limit
SQL� x�p � ~ SSQL
=2~
⇡2f2
opt
ML2
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)
Facility Noise
Facility noise
Seismic Noise
S1/2seis ⇠ 10�12
✓10Hz
f
◆2
Hz�1/2
Need one of these10
10too large
Seismic Noise
S1/2seis ⇠ 10�12
✓10Hz
f
◆2
Hz�1/2
Single pendulum suspension
S1/2seis,filt =
1
|1� (f/fpend)2|S1/2seis
aLIGO 5-stage pendulum suspension
S1/2seis,filt ⇡ 10�12
✓fpendf
◆2 ✓10Hz
f
◆2
Hz�1/2
fpend ⇠ 1Hz
S1/2seis,filt ⇡ 10�12
✓fpendf
◆10 ✓10Hz
f
◆2
Hz�1/2
Thermal Noise
Thermal noise contributions
Thermal NoiseFluctuation-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
ST (f) =4kBT
(2⇡f)2R[Y (f)]
For an anelastic spring with loss angle Hooke’s law becomes
ST (f) =kBT
2⇡3Mf
f2res�(f)
(f2res � f2)2 + f4
res�2(f)
F = �k x (1 + i�)�
Mirror Coating Thermal Noise
ST (f) =kBT
2⇡3Mf
f2res�(f)
(f2res � f2)2 + f4
res�2(f)
The resonant frequencies for the coatings are very high (tens of kHz), and the loss angle small (millionths)
SMC =
✓2kBT�
⇡3ML2f2MC
◆1
f
S1/2MC = 2.5⇥ 10�24
✓100Hz
f
◆1/2
Hz�1/2
For advanced LIGO
Suspension Thermal Noise
ST (f) =kBT
2⇡3Mf
f2res�(f)
(f2res � f2)2 + f4
res�2(f)
Pendulum mode
Violin modes
fpen = 9Hz
fv = 500, 1000, . . . Hz
Pendulum Suspension Thermal Noise
ST (f) =kBT
2⇡3Mf
f2res�(f)
(f2res � f2)2 + f4
res�2(f)
fres = fpen ⌧ f
For advanced LIGO
S1/2pen = 3.5⇥ 10�25
✓100Hz
f
◆5/2
Hz�1/2
Violin mode Thermal Noise
ST (f) =kBT
2⇡3Mf
f2res�(f)
(f2res � f2)2 + f4
res�2(f)
For advanced LIGO, first harmonic
fres = fv = 500Hz, 1000Hz, . . .
S1/2v =
3⇥ 10�24
1 + (f2v � f2)2/�f 4
Hz�1/2 �f = fv�1/2 ⇡ 2Hz
Gravity Gradient Noise
This is why we need LISA!
x = G
Z�⇢(x0, t)
|x� x
0|3 (x� x
0) d3x
We can’t escape Newton
Real aLIGO noise spectra from O1
Suspension Thermal noise Violin modes
Real aLIGO noise spectra from O1
60 Hz power line and harmonicsElectronic noise
Real aLIGO noise spectra from O1/S6
Calibration lines
Calibration and the control system is a whole other topic…
Recall: Likelihood for Stationary Gaussian Noise
(a|b) = 2
Z 1
0
a(f)b⇤(f) + a⇤(f)b(f)
Sn(f)df
p(d|~�) = const. e��2(~�)
2
�2(~�) = (d� h(~�)|d� h(~�))
LIGO searches for binary inspired signals
Suppose that we have two hypotheses: H1 : A signal with parameters ~� is present
H0 : No signal is present
⇤(~�) =p(d|h(~�), H1)
p(d, H0)Likelihood ratio: For Gaussian noise: ⇤(~�) = e�(d|h)+ 1
2 (h|h)
Frequentist Hypothesis Testing
p(Λ|H0)
ΛΛ∗
p(Λ|H1)
Set threshold such that favors hypothesis �� � > �� H1
Λ
Type II error Type I error
Type 1 error - False Alarm
Type 1I error - False Dismissal
� �Detection Statistic H0 �Noise Hypothesis H1 �Noise + Signal Hypothesis
Neyman-Pearson Theorem
For a fixed false alarm rate, the false dismissal rateis minimized by the likelihood ratio statistic
The likelihood ratio is maximized over the signal parameters.
⇤(~�) =p(d|h(~�), H1)
p(d, H0)
The rho statistic is often used in place of the likelihood ratio
Writing h = ⇢h where (h|h) = 1 ⇤(~�) = e ⇢(d|h)� 12⇢
2
@⇤(~�)@⇢ = 0 ) ⇢(~�) = (d|h(~�))Maximizing wrt rho
log⇤(
~�) =1
2
⇢2(~�)
The rho statistic and SNR
The signal-to-noise ratio (SNR) is defined:
SNR =Expected value when signal present
RMS value when signal absent
=E[�]�
E[�20]� E[�0]2
= (h|h)
=�
(h|h)
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) 062001
SNR2 = 4
Z 1
0
|h(f)|2
Sn(f)df
Matched Filtering = Maximum Likelihood
Convolve the data with a filter (template) K: Y =�
dt
�du K(t� u)d(u)
Maximizing the SNR yields K(f) =h(f)S(f)
SNR =E[Y ]�
E[Y 20 ]� E[Y0]2
=2
��0 df(h�(f)K(f) + h(f)K�(f))
�4
��0 df |K(f)|2S(f)
) Y = (d|h) = ⇢p(h|h)
Frequentist Detection Threshold For stationary, Gaussian noise the detection statistic is Gaussian distributed.
p0(�) =1�2�
e��2/2For the null hypothesis we have
p1(�) =1�2�
e�(�2�SNR2)/2For the detection hypothesis we have
�
Setting a threshold of gives the false alarm and false dismissal probabilities��
PFA =12erfc(��/
�2)
PFD =12erfc((�� � SNR)/
�2)
FAR =PFA
Tobs
LIGO/Virgo analyses do not use SNR thresholds, but rather use False Alarm Rate thresholds
e.g. FAR = One in million years and an observation time of one year
PFA = 10�6 aka 4.9� ⇢⇤ = 4.8
Grid Based SearchesGoal 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
M(�x, �y) =(h(�x)|h(�y))�
(h(�x)|h(�x))(h(�y)|h(�y))
with a template and defines a natural metric on parameter space:
(Owen Metric)gij =(h,i|h,j)(h|h)
� (h|h,i)(h|h,j)(h|h)2where
Taylor expanding M(�x, �x + ��x) = 1� gij�xi�xj + . . .
Number of templates (for a hypercube lattice in D dimensions)
Cost grows geometrically with D for any lattice
N =V
�V=
�dDx
�g
(2�
(1�Mmin)/D)D
LIGO Style Grid Searches
Typically 2-3 dimensional, 1000’s points
Reducing the cost of a searchIn most cases it is possible to analytically maximize over 3 or more parameters
Distance:
The unit normalized template defines a reference distance Dh
Scaling this template to distance givesD
h =D
Dh
The distance is then estimated from the data as
D =D
(d|h)
Reducing the cost of a search
Phase Offset:
Generate two templates and h(� = 0) h(� = �/2)
Then (d|h)max � =�
(d|h(0))2 + (d|h(�/2))2
Easy to see this in the Fourier domain.
d = h0 ei�Suppose , then
(d|h(0)) = (h0|h0) cos �
(d|h(�/2)) = (h0|h0) sin�
Reducing the cost of a search
Time Offset:
-1e-20
-8e-21
-6e-21
-4e-21
-2e-21
0
2e-21
4e-21
6e-21
8e-21
1e-20
0 0.2 0.4 0.6 0.8 1
h
t
Fourier transform treats time as periodic - use this to our advantage
(d|h)(�t) = 4�
d�(f)h(f)S(f)
e2�if�t df
Compute the inverse Fourier transform of the product of the Fourier transforms:
d(t) = h(t� t0)Then if the template and data differ by a time shift:
(d|h)max t = (d|h)(�t = t0)
Workflow for pyCBC search
Matched filtering is done per-detector (not coherent)
Template bank constructed
Detection statistic computed (“new SNR”)
Coincidence in time/mass enforced Data quality vetoes applied
Monte Carlo background to compute FAR vs new SNR
T
(Bandpass filtered, whitened, time domain)
Samples from the Syracuse Audio Study of Glitches
Things that go bump in the night
Contending with non-stationary, non-Gaussian noise
Non-stationary:
Adiabatic drifts in the PSD - work with short data segments
Non-Gaussian:
Glitches - vetoes and time-slides
Analysis of 16 days of data from September 2015
Divide by 16 days to get FAR
“new SNR”
-8
-6
-4
-2
0
2
4
6
8
0 0.5 1 1.5 2
Han
ford
-8-6-4-2 0 2 4 6 8
0 0.5 1 1.5 2
Livi
ngst
on
time (seconds)
Early Morning, September 14 2015
BayesWave reconstruction of the signal
-8
-6
-4
-2
0
2
4
6
8
0 0.5 1 1.5 2
Han
ford
-8-6-4-2 0 2 4 6 8
0 0.5 1 1.5 2
Livi
ngst
on
time (seconds)
GW150914
GW151226
LVT151012
GW170104
LIGO detections to-date
Triangulating the Source
Hanford
Triangulating the Source
Hanford + Livingston
Triangulating the Source
Hanford + Livingston + Virgo
GW170104
Parameter estimation
• 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
BayesWave
Continuous Morlet/Gabor Wavelets
Cornish & Littenberg 2015
Detection without templates
Lines and a drifting noise floor
10-48
10-46
10-44
10-42
10-40
10-38
10-36
50 100 150 200 250 300 350 400 450
Pow
er s
pect
ral d
ensi
ty (H
z-1)
frequency (Hz)
|h(f)|2
Cubic spline fit
Lorentzian fit
Data
Cubic spline fit
Lorentzian fit
Model it
Time-Frequency Scalograms of LIGO data
Glitches
Model these too
X
Gravitational Waves
and model theseX
“Caedite eos. Novit enim Dominus qui sunt eius” Arnaud Amalric (Kill them all. For the Lord knoweth them that are His.)
Example from LIGO’s S5 science run
Reconstructing GW150914 with wavelets
Reconstructing GW170104 with wavelets
Match GR prediction
Inferred SignalPredicted Signal
GW150914 GW170104
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
Astrophysical Rate Limits
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|�) = �ke��/k!
�
If the event rate is , and the observable 4-volume is R [Mpc�3 year�1] V T [Mpc3 year]
� = R V T
The probability of observing zero events (k=0) is then
p(R) = V T e�R V T
p(0|�) = p(R) dR = e�R V TFollows from
Even without a detection we can produce interesting astrophysical results such as bounds on the binary merger rate for NS-NS
Astrophysical Rate Limits
p(R) = V T e�R V T
The probability distribution is peaked at a rate of zero. A 90% rate upper limit can be computed:
� R�
0p(R)dR = 1� e�R�V T = 0.9
� R�V T = ln(0.1)
� R� =2.3V T
0
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Time (weeks)
0
0.5
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1.5
2
2.5
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e.g. NS-NS Merger Rate, aLIGO at design sensitivity
Week 1
V =4⇡
3(200Mpc)3
Truth 90% upper limit
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(V T )2 e�R V T
This distribution is peaked at
R =1
V T
The 90% confidence interval now sets upper and lower limits on the merger rate.
Simulated NS-NS Merger Rate Constraints
Week 2
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Simulated NS-NS Merger Rate Constraints