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

5

10

15

20

0 5 10 15 20

NS-NS Detections

Time (weeks)

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2λ (Mpc-3 Myr-1)

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

0

5

10

15

20

0 5 10 15 20

NS-NS Detections

Time (weeks)

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2λ (Mpc-3 Myr-1)

Week 3

0

5

10

15

20

0 5 10 15 20

NS-NS Detections

Time (weeks)

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2λ (Mpc-3 Myr-1)

Simulated NS-NS Merger Rate Constraints

Week 4

0

5

10

15

20

0 5 10 15 20

NS-NS Detections

Time (weeks)

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2λ (Mpc-3 Myr-1)

Simulated NS-NS Merger Rate Constraints

Week 5

0

5

10

15

20

0 5 10 15 20

NS-NS Detections

Time (weeks)

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2λ (Mpc-3 Myr-1)

Simulated NS-NS Merger Rate Constraints

0

5

10

15

20

0 5 10 15 20

NS-NS Detections

Time (weeks)

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2λ (Mpc-3 Myr-1)

0

5

10

15

20

0 5 10 15 20

NS-NS Detections

Time (weeks)

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2λ (Mpc-3 Myr-1)

0

5

10

15

20

0 5 10 15 20

NS-NS Detections

Time (weeks)

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2λ (Mpc-3 Myr-1)

0

5

10

15

20

0 5 10 15 20

NS-NS Detections

Time (weeks)

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2λ (Mpc-3 Myr-1)

0

5

10

15

20

0 5 10 15 20

NS-NS Detections

Time (weeks)

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2λ (Mpc-3 Myr-1)

0

5

10

15

20

0 5 10 15 20

NS-NS Detections

Time (weeks)

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2λ (Mpc-3 Myr-1)

0

5

10

15

20

0 5 10 15 20

NS-NS Detections

Time (weeks)

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2λ (Mpc-3 Myr-1)

0

5

10

15

20

0 5 10 15 20

NS-NS Detections

Time (weeks)

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2λ (Mpc-3 Myr-1)

0

5

10

15

20

0 5 10 15 20

NS-NS Detections

Time (weeks)

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2λ (Mpc-3 Myr-1)

0

5

10

15

20

0 5 10 15 20

NS-NS Detections

Time (weeks)

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2λ (Mpc-3 Myr-1)

0

5

10

15

20

0 5 10 15 20

NS-NS Detections

Time (weeks)

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2λ (Mpc-3 Myr-1)

0

5

10

15

20

0 5 10 15 20

NS-NS Detections

Time (weeks)

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2λ (Mpc-3 Myr-1)

0

5

10

15

20

0 5 10 15 20

NS-NS Detections

Time (weeks)

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2λ (Mpc-3 Myr-1)

0

5

10

15

20

0 5 10 15 20

NS-NS Detections

Time (weeks)

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2λ (Mpc-3 Myr-1)

0

5

10

15

20

0 5 10 15 20

NS-NS Detections

Time (weeks)

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2λ (Mpc-3 Myr-1)

0

5

10

15

20

0 5 10 15 20

NS-NS Detections

Time (weeks)

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2λ (Mpc-3 Myr-1)

Simulated NS-NS Merger Rate Constraints