gravitational-wave data...

Gravitational-wave Data Analysis

Patrick Brady

Bibliography• A few books/papers: your should also take a look at papers that

cite these.

• Basic Data Analysis: L A Wainstein and V D Zubakov, Extraction of signals from noise, Prentice-Hall, 1962

• Compact Binary Analysis: Finn, L.S. and Chernoff, D.F., Phys. Rev. D47, 2198-2219 (1993); Blanchet et al, Class.Quant.Grav.13:575-584,1996

• Burst Analysis: Anderson et al, Phys. Rev. D63:042003, 2001

• Continuous Waves Analysis: Jaranowski et al, Phys.Rev.D58:063001,1998; Brady et al, Phys.Rev.D57:2101-2116,1998

• Stochastic Background: Allen and Romano, Phys.Rev.D59:102001,1999.

Overview• Lecture 1: What the detector measures, noise, detection as

statistical process, detection of signals with known/unknown parameters. Basic exercise in data analysis.

• Lecture 2: Transient sources including compact binaries and unmodelled signals, detection in Gaussian noise, multi-detector, detection in real noise.

• Lecture 3: Introduction to gravitational-wave data and software. How is the data stored in files, how to read data, compute a power spectrum, generate a template bank for compact binary inspiral, filter the output, read the output data.

• Lecture 4: Other sources and discussion of measurement.


Lecture 1: From GR to signal analysisPatrick Brady

Review of gravitational waves

Spacetime interval can be written as

where is the Minkowski metric and is a metric perturbation

For weak gravitational fields

Solve the wave equation in vacuum

•

• Gravitational waves propagate at the speed of light

• Gravitational waves stretch and squeeze space

ds2 = (!!" + h!")dx!dx"

!! !2

!t2+"2

"h

!"= !16"T!"

h!"

= h!" ! 12!!"h

h!"

= A!" exp(ik#x#) , k!k! = 0

!!" h!"

h << 1

Physical Effects of the Waves

• As gravitational waves pass, they change the distance between neighboring bodies

• GR predicts two polarizations

• Fractional change in distance is the strain given by h = δL / L

t = 0 (period)/4 (period)/2 3(period)/4 (period)

L L+δL

Animations: Warren Anderson

Schematic DetectorAs a wave passes, one arm stretches

and the other shrinks ….

…causing the interference pattern to change at the photodiode

LIGO Observatories

Hanford: two interferometers in same vacuum envelope (4km, 2km)

Livingston: one interferometer (4km)

Probability and statistics• Real random variable: function X that maps events ω to

real numbers x such that the probability of {ω: X(ω)≤x }∈[0,1], in shorthand P[X≤x]∈[0,1]

• Example: coin toss experiment. The events are ω∈[heads, tails] and X(heads)=1, X(tails)=0. The probability density over the real numbers is

• Expectation value of a function of X is

• If two random variables are independent

pX(x) =

!"

#

0.5 if x = 00.5 if x = 10.0 otherwise

!f(X)" =!

f(x) pX(x) dx

!XY " = 0

Random processes• Random process is a sequence of random

variables

• Example: instrumental noise n(t) at the readout is a random process with sequence indexed by t

• A random process is stationary if its statistical properties do not change with t

• Correlation function:

• Define its inverse Q(τ) by

!n(t)n(t " !)# = R(t, !) != R(!)

!Q(t! t!!)R(t!! ! t!)dt!! = !(t! t!)

if !n" = 0

Gaussian Random Process

• The probability density for a Gaussian random variable is

• It generalizes to a random process as

• Example: consider a stationary process with

• then

pn(n) ! exp!"1

2

" "n(t)Q(t" t!)n(t!)dt dt!

#

R(!) = "2#(!) =! Q(!) = "!2#(!)

pn(n) ! exp!"

"n2(t)dt

2!2

##

$

t

exp!"n2(t)

2!2

#

pX [x] =1!

2!"2exp

!"x2

"2

"

Power spectrum

• The Fourier transform pair is

• If n(t) is stationary, then

n(f) =! !

"!n(t)e"2!iftdt

n(t) =! !

"!n(f)e2!iftdf

!n(f)n!(f ")" =! !

!n(t)n(t")" e#2!i(ft#f !t!)dtdt"

=!

R(!)e#2!if"d!

!e#2!i(f#f !)t!dt"

!n(f)n!(f ")" =12Sn(|f |)"(f # f ")

Power spectrum

• The Fourier transform pair is

• If n(t) is stationary, then

n(f) =! !

"!n(t)e"2!iftdt

n(t) =! !

"!n(f)e2!iftdf

!n(f)n!(f ")" =! !

!n(t)n(t")" e#2!i(ft#f !t!)dtdt"

=!

R(!)e#2!if"d!

!e#2!i(f#f !)t!dt"

!n(f)n!(f ")" =12Sn(|f |)"(f # f ") 1-sided

Power Spectrum

LIGO Noise

• The noise in the LIGO interferometers is dominated by three different processes depending on the frequency band

LIGO Noise

• The noise in the LIGO interferometers is dominated by three different processes depending on the frequency band

Colored noise: power spectrum depends on fWhite noise: power spectrum is independent of f

Putting it together

• Rewrite

• using the fact that Q is inverse to R as

• where the real inner product is

pn(n) ! exp!"1

2

" "n(t)Q(t" t!)n(t!)dt dt!

#

pn(n) ! exp!"

" !

"!

n(f)n#(f)Sn(|f |) df

#

= exp!"1

2(n, n)

#

(a, b) = 2! !

"!

a(f)b#(f)Sn(|f |) df


Lecture 2: Detecting Signals in NoisePatrick Brady

Detection of signals• Gravitational-wave strain data s(t) consists

of noise n(t) and a possible signal h(t)

• Need to decide between

1. Signal is absent. Null hypothesis H0

2. Signal is present. Alternate hypothesis H1

• When the statistical properties of the noise are known, can use Bayes theorem to construct probability to distinguish these two distinct cases. Note: these are also complete

Bayes Theorem• Tells us the probability the signal is present given

s(t)

• Since either H1 or H0 must be true, then

• Plug that in and rearrange to get

• where

p[H1|s(t)] =p[H1] p[s(t)|H1]

p[s(t)]

p[s] = p[s|H0] p[H0] + p[s|H1] p[H1]

p[H1|s] =!(H1, s)

!(H1, s) + p[H0]/p[H1]

!�H1, s��p�s|H1�p�s|H0�

Probability is monotonicallyincreasing with likelihood

Detection: known signal• Let’s use the Gaussian noise and see what we

get.

• Under the alternate hypothesis (H1), s(t) is still a stationary, Gaussian process with non-zero mean

• Under the null hypothesis (H0), s(t) is a stationary, Gaussian with zero mean

• The likelihood is

p[s|H1] ! e!(s!h,s!h)/2

!(H1, s) =p[s|H1]p[s|H0]

= e(s,h)e!(h,h)/2

p[s|H0] ! e!(s,s)/2

Detection: known signal• Let’s use the Gaussian noise and see what we

get.

• Under the alternate hypothesis (H1), s(t) is still a stationary, Gaussian process with non-zero mean

• Under the null hypothesis (H0), s(t) is a stationary, Gaussian with zero mean

• The likelihood is

p[s|H1] ! e!(s!h,s!h)/2

!(H1, s) =p[s|H1]p[s|H0]

= e(s,h)e!(h,h)/2

Matched filter fora known signalp[s|H0] ! e!(s,s)/2

Decision Rules• The matched filter for a known signal is the

sum of Gaussian random variables, so it is Gaussian

• Set threshold on matched filter signal to noise ratio such that false positive is acceptably small

TP: true positiveFP: false positiveFN: false negativeTN: true negative

Detection: unknown parameters

• Suppose the signal depends on λ, then there is a set of alternate hypotheses Hλ and Bayes theorem tell us to marginalize over them

p[H1]p[s|H1]!!

p[!]p[s|H!]d!

!(H1, s) =!

p[!]p[s|H!]p[s|H0]

d! =!

p[!] ![H!, s]d!

It is often difficult to compute themarginalized likelihood. Examining the maximum

over λ can work well for detection and parameter estimation

Example 1

• Unknown amplitude A, such that h=A g, is easiest done in terms of log likelihood.

• Maximize over A to get

log !(H1, s) = (s, h)! (h, h)/2= A(s, g)!A2(g, g)/2

maxA

[log !(H!, s)] =(s, g)2

2(g, g)

Example 2• Unknown unknown time: h = h(t-tc)

• In this case, one must explicitly calculate the inner products for all tc and then maximize

• But there is a trick since

• Now since (h[tc],h[tc]) is independent of tc, one can compute (s,h[tc]) via the inverse Fourier transform and save computational effort

h(f, tc) = h(f, 0)e!2!ft0

(s, h[tc]) =!

s(f)h!(f, 0)Sn(|f |) e2!iftcdf

Example 3• Unknown phase

• Notice that (h,h) independent of Φ

• This time, one computes

h(t, !) = cos(2!ift + !)= cos(2!ift) cos(!) + sin(2!ift) sin(!)

(s, h[!]) = (s, cos[2!ift]) cos(!) + (s, sin[2!ift]) sin(!)

=! max!

(s, h[!]) =!

(s, cos[2!ift])2 + (s, sin[2!ift])2

Compact Binaries• Pairs of black holes,

neutron stars, or a black hole and neutron star

• As they orbit one another, they emit gravitational waves causing the objects get closer together, eventually merging

• LIGO is sensitive to last few minutes before the merger

Weak-field waveform• Even in Newtonian gravity, there are many parameters to

describe a binary system:

• Neglecting spins and eccentricity, one can show that the waveform to leading post-Newtonian order is given by

m1,m2, tc,!c, D,

i,", #,$,%s1,%s2, &,

f(t) = F (m1,m2)T (m1,m2)3/8(tc ! t)!3/8 + . . .

h+(t) = A(m1,m2, i)1Mpc

D[f(t)/F (m1,m2)]2/3 cos[2!(t)! 2!0]

h"(t) = B(m1,m2, i)1Mpc

D[f(t)/F (m1,m2)]2/3 sin[2!(t)! 2!0]


Lecture 3: The real dealPatrick Brady

Waveform• Most binaries are expected to circularize before

reaching this frequency band

• Spin is most important for higher mass systems with unequal masses, modulates the waveform

Inspiral Matched Filter• The signal h(t) at the detector is a linear combination

of the two polarization states

• The coefficients F+ and Fx are called the antenna pattern functions

• Analytically maximizing the log-likelihood over the constant amplitude and phase one gets1/D and !

h(t) =1MpcD [ hc(t! tc;m1,m2) cos ! + hs(t! tc;m1,m2) sin ! ]

D =D!

F 2+(1 + cos2 i)2 + F 2

! 4 cos2 i

!2(tc;m1,m2) =(s, hc)2 + (s, hs)2

(hc, hc)using (hc, hs) = 0, (hc, hc) ! (hs, hs)

• Discrete set of templates labeled by (m1, m2)

• Low-mass (< 35 Msun): use post-Newtonian templates, ignore merger-ringdown

• High-mass (>25Msun): use hybrid templates including merger-ringdown motivated by numerical relativity

• Place the templates so that there is some maximum loss in expected signal to noise, typically 3%.

Signal→Template

Filter to suppress high/low freq

Coalescence TimeSN

R

Gaussian noise + simulated inspiral

Matched filtering

Real data• .... is non-stationary and non-Gaussian

• Matched filter will be large for any noise with power in the time-frequency track of the waveform

• For example, a delta-function impulse gives a time-reversed chirp as the SNR output

• Can think of real data as Gaussian noise plus nuisance signals that leak into h(t) from the environment and instrumental subsystems (plus possible gravitational-wave signals)

• Need vetoes and independent instruments ....

Real, non-stationary noiseSN

RC

HIS

Q

Time: tc

Dealing with real data• Discriminants in single instruments include χ2

tests and environmental vetoes

• Anything that separates signal from background

SNR

CH

ISQ




SNR

CH

ISQ

SNR threshold : lots of background survives




SNR

CH

ISQ

Effective SNR threshold: less background, same

signals. It’s a win.

Most-powerful• Multiple detectors provide one of the most powerful

discriminants: require coincidence and coherence

Parameter coincidence• Can condense coincidence in multiple parameters

using variant of the inner product introduced earlier

Introduction to Frames• All the major projects (LIGO, GEO, TAMA, Virgo)

have used a common data format to store gravitational-wave data: Frame format

• Provides the ability to store time, frequency, and event data in a compressed binary format

• Time: data is stamped with a GPS time, i.e. seconds since Sun Jan 06 00:00:00 UTC 1980, stored as two integers seconds, nanoseconds since last whole second

• Time series data is organized into channels. A channel is a time series that contains information recorded from the detector.

Introduction to Frames• Channels come in different types:

• ADC - raw data recorded from the instrument

• Proc - processed data, e.g. the strain data which is produced by combining information from raw channels and other measurements

• Sim - simulated data

• The numerical values in a channel have standard types: float, int, .....

• A frame is basic building block for the time-series data.

• A frame can contain multiple channels, frequency series, etc

• A frame covers an interval of time

• A frame contains metadata about the instruments that took the data and other information relevant to describe the data

• A frame file may contain one or more frames

Software tools• There are numerous software tools in use for gravitational-wave data

analysis

• Some used in LIGO data analysis:

• FrameL - Developed by Virgo. Provide low-level functions and data types to manipulate frames and frame files

• LAL - Algorithm libraries written in C licensed under GPL. Provides data types, a high-level API to frames, simulation tools, analysis tools, ...covers all major source types

• PyLAL - Python wrappers for many LAL codes and standalone codes for manipulating data and triggers

• LALApps - Applications that rely on LAL. Provides tools to read and manipulate the data ranging from simple processing to full scale searches for gravitational waves

• MatApps - Applications that use Matlab as the primary language. Provides tools to do things ranging from simple processing to full scale searches

• GLUE - Grid LSC User environment. Provides tools to enable running searches on the LIGO Data Grid

More about LIGO software

• Data Analysis Software Working Group https://www.lsc-group.phys.uwm.edu/daswg/

• LIGO Data Grid https://www.lsc-group.phys.uwm.edu/lscdatagrid/

https://www.lsc-group.phys.uwm.edu/daswg/

https://www.lsc-group.phys.uwm.edu/daswg/

https://www.lsc-group.phys.uwm.edu/lscdatagrid

https://www.lsc-group.phys.uwm.edu/lscdatagrid


Lecture 4: Overview of Search Results Patrick Brady

GRB 070201• Short gamma-ray burst

• IPN error box included M31!

• Exclude any compact binary progenitor in our simulation space at the distance of M31 at > 99% confidence level

• Exclude compact binary progenitor with masses 1 M⊙ < m1< 3 M⊙ and 1 M⊙ < m2 < 40 M⊙ with D < 3.5 Mpc away at 90% CL

No plausible gravitational waves found

Abbott et al [LIGO and Virgo Collaboration], Astrophys.J.681:1419-1428,2008.

Searches for compact binaries

• Most likely rate for binary neutron stars is ~ 5x10-5 / yr / L10

• L10 is unit of luminosity. Milky Way has ~1.7 L10

• Neutron star black hole rates are ~1.5x10-6 / yr / L10

• Black hole binaries are ~ 2x10-7 / yr / L10

Abbott et al [LIGO and Virgo Collaboration], Phys. Rev. D 80 (2009) 047101

http://link.aps.org/abstract/PRD/v80/e047101

http://link.aps.org/abstract/PRD/v80/e047101

Searches for Bursts

typical Galactic distance

Virgo cluster

Q =8.9 sine-Gaussians, 50% detection probability:

For a 153 Hz, Q =8.9 sine-Gaussian, the S5 search can see with 50% probability: ∼ 2 × 10–8 M c2 at 10 kpc (typical Galactic distance) ∼ 0.05 M c2 at 16 Mpc (Virgo cluster)

Cou

rtes

y: La

ura

Cad

onat

i

Bursts from cosmic strings (S4)

Abb

ott

et a

l [LI

GO

Sci

entifi

c C

olla

bora

tion]

, Ph

ys R

ev D

80

(200

9) 0

6200

2

http://link.aps.org/doi/10.1103/PhysRevD.80.062002

http://link.aps.org/doi/10.1103/PhysRevD.80.062002

Continuous Signals

• Signals lasting as long as, or longer than, the obervation time

• Known radio pulsars could also emit gravitational waves

• Unknown radio pulsars that are not beamed toward earth

Cre

dit:

Dan

a Be

rry/

NA

SAC

redi

t: M

. Kra

mer

Searches for continuous waves

• Strength of gravitational waves depends on gravitational ellipticity

• Crab pulsar:

• observed spindown allows maximum gravitational ellipticity around 10-3

• observations < 10-4

Abbott et al [LIGO and Virgo Collaboration], arXiv:0909.3583

http://arxiv.org/abs/0909.3583

http://arxiv.org/abs/0909.3583

Stochastic Background

Credit: Jolien Creighton

Energy density in gravitational wavesdivided by critical density

Searches for stochastic waves

Abbott et al [LIGO Scientific Collaboration] Nature 460 (2009) 990

http://www.nature.com/nature/journal/v460/n7258/pdf/nature08278.pdf

http://www.nature.com/nature/journal/v460/n7258/pdf/nature08278.pdf

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