Introduction to Gravitational Wave

Data AnalysisLarry Price

2010 International School on Numerical Relativity and Gravitational Waves

July 26-30, APCTP

References

• 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: (LIGO) Allen and Romano, Phys.Rev.D59:102001,1999. (Pulsar Timing) Anholm et al, Phys.Rev.D79: 084030, 2009

Overview

• Lecture 1: Brief introduction to the instrument and what it measures. Introduction to time series analysis.

• Lecture 2: Frequentist vs. Bayesian. Bayes’s Theorem. Decision Rules. The likelihood function.

• Lecture 3: Optimal statistics for detecting signals in noise.

Introduction to Gravitational Wave

Data AnalysisLecture 1: The basics

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 = (ηαβ + hαβ)dxαdxβ

�− ∂2

∂t2+∇2

�h

αβ= −16πTαβ

hαβ

= hαβ − 12ηαβh

hαβ

= Aαβ exp(ikδxδ) , kαkα = 0

ηαβ hαβ

h << 1

• Quadrupolar in nature

• Like EM field, there are two polarizations

GW direction x+GW direction Credit: Warren Anderson

h =δL

L

L

δL h ∼ 10−21

L ∼ 4 ly

δL ∼ 10−5 mcharacterizes GW signal

Gravitational Waves

Schematic DetectorAs a wave passes, one arm stretches

and the other shrinks ….

…causing the interference pattern to change at the photodiode

Some IFOs I knows

Time series analysis

• A time series, x(t), is some (continuous or discrete) function of time.

• In practice they are discrete:

xi(t) ≡ x(ti)

tj = t0 + j∆t

Time series analysis

• The Fourier transform:

n(f) =� ∞

−∞n(t)e−2πiftdt

n(t) =� ∞

−∞n(f)e2πiftdf

Parseval’s theorem

• Relates a function and it’s Fourier transform

• Energy spectral density

� ∞

−∞dt |x(t)|2 =

� ∞

−∞df |x(f)|2

Ex(f) ≡ |x(f)|2

Power spectral density• Use Parseval to define the (one-sided)

power spectral density

Power = limT→∞

1T

� T/2

−T/2dt |x(t)|2

= limT→∞

1T

� ∞

−∞df

�����

� T/2

−T/2dtx(t)e−2πift

�����

2

= limT→∞

1T

� ∞

−∞df |x(f)|2

≡ 12

� ∞

−∞dfSx(|f |)

Convolution

• Definition

• Describes the effect of linear systems

• Essentially all we need for GW data analysis

(x � K)(t) ≡� ∞

−∞dt� x(t�)K(t− t�)

signal filter

Convolution

• Theorem

• is called the impulse response

• is called the frequency response

K(t)

K(f)

(x � K)(t)↔ x(f)K(f)

The correlation theorem

• In the time domain

• Correlation measures how well two time series match up when shifted in time.

• Correlation theorem (generalization of Parseval)

Rxy(t)↔ x∗(f)y(f)

Rx,y(t) ≡� ∞

−∞dt� x(t�)y(t� + t)

Noise

• Noise is anything that isn’t the signal we’re interested in.

• Usually instrumental, but can be other signals! (cf. white dwarfs in LISA)

• Typically random in nature (If we knew what it looked like we’d take it out!)

• Characterized by its statistical properties

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

Stationarity

• A random process is stationary if its statistical properties are independent of time.

• According to the ergodic theorem time averages over a single realization are equivalent to ensemble averages over many realizations, for stationary processes. E.g.

µ = �x� = limT→∞

1T

� T/2

−T/2dt x(t)

• Auto-correlation function

• For a stationary process

Auto-correlation function

R(t) ≡ limT→∞

� T/2

−T/2dt� x(t�)x(t� + t)

R(t) = �x(t�)x(t� + t)�

• For a random variable with zero mean

• For a periodic process

• For a white process

Auto-correlation function

R(0) = σ2 = �x2� − �x�2

R(0) = R(nT )

R(t) = σ2δ(t)

max value!

Back to PSD

• The PSD and auto-correlation function are related by

• It also follows that the frequency components are statistically independent of each other for different frequencies

�x∗(f)x(f �)� =12Sx(|f |)δ(f − f �)

Sx(f) = 2� ∞

−∞dt R(t)e−2πift

Simple PSD estimate

• The periodogram

• The mean of the periodogram is

• And the variance is

Px(f) =1T

|x(f)|2

�Px(f)� = Sx(f)

�P 2x (f)� − �Px(f)�2 = S2

x(f)

Probability and random variables• 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

Gaussian distribtuion

• Some reasons for assuming a Gaussian distribution:

• It might actually be Gaussian

• The central limit theorem

• It only requires a mean and a variance

Gaussian distribution

• The probability density for a Gaussian random variable is

• It generalizes to a random process as

• Where Q is inverse to R

pn(n) ∝ exp�−1

2

� �n(t)Q(t− t�)n(t�)dt dt�

�

pX [x] =1√

2πσ2exp

�−x2

2σ2

�

�Q(t− t��)R(t�� − t�)dt�� = δ(t− t�)

Gaussian Random Process

• E.g. Consider

• Then

R(τ) = σ2δ(τ) =⇒ Q(τ) = σ−2δ(τ)

pn(n) ∝ exp�−

�n2(t)dt

2σ2

�∼

�

t

exp�−n2(t)

2σ2

�

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