MONTANA STATE UNIVERSITY

Gravitational Wave Astronomy

An Introduction to Emission Sources and Data Analysis

John Pribyl

5/2/2014

1

Table of Contents

1 Introduction 2 2 Sources of Gravitational Wave Emission

2.1 High Frequency. . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Ligo Overview. . . . . . . . . . . . . . . . . . . 5 2.1.2 Compact Binaries. . . . . . . . . . . . . . . . . 7 2.1.3 Stellar Core Collapse. . . . . . . . . . . . . . . 8 2.1.4 Periodic Emitters. . . . . . . . . . . . . . . . . 8 2.1.5 Stochastic Background. . . . . . . . . . . . . . 9 2.2 Low Frequency. . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 LISA Overview. . . . . . . . . . . . . . . . . . . 10 2.2.2 Coalescing Binary Black Holes. . . . . . . . . . 12 2.2.3 Periodic Emitters. . . . . . . . . . . . . . . . . 12 2.2.4 Stochastic Background. . . . . . . . . . . . . . 13 2.3 Very Low Frequency and Ultra Low Frequency. . . . 13

2.3.1 VLF Pulsars. . . . . . . . . . . . . . . . . . . . . 13

2.3.2 ULF B-Mode Polarization. . . . . . . . . . . . . 14

3 Data Analysis

3.1 Detection Probability. . . . . . . . . . . . . . . . . . 15

3.1.1 Significance and Confidence. . . . . . . . . . . 15 3.1.2 Χ2 Likelihood and Goodness of Fit Test. . . . 15 3.1.3 Neyman-Pearson Approach. . . . . . . . . . . 16 3.2 Filtering. . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2.1 Constant Frequency Fourier Transform. . . . 17 3.2.2 Matched Filtering in 0-Mean Gaussian Noise 18 3.2.3 Templates. . . . . . . . . . . . . . . . . . . . . 19 3.2.4 Nyquist Theorem and Aliasing. . . . . . . . . 20 3.2.5 Maximum Likelihood Estimator. . . . . . . . 20 3.2.6 Bayesian Inference. . . . . . . . . . . . . . . . 21 3.2.7 MCMC Methods. . . . . . . . . . . . . . . . . 22

4 Summary 24 5 Acknowledgements 25

2

1 Introduction

Gravitational waves (GW) are incredibly small rippling oscillations

predicted by Einstein’s theory of General Relativity (GR) that transit the

fabric of space-time. They are the most energetic events in the known

universe, and are emitted by events as cataclysmic as the collision of two

black holes. However, these energetic ripples have proven to be

exceedingly elusive. Although GWs are prohibitively small (~10-21 or

smaller), a direct detection will likely occur in the next 5 years. By nature,

their interaction with matter is immensely feeble. The impact of this is that

GWs will eventually provide generous unobserved insights that pierce

deeply into the nature of their sources and out to the far reaches of the

universe. After analyzing the data and recovering all of the enclosed

information, GWs will measure the propagation speed (and consequently

the mass of) the graviton, provide the most stringent test of GR to date,

provide the equation of state of neutron stars, and provide insight into

countless other astrophysical phenomena. Additionally, there is a very

strong potential to discover new, inconceivable marvels that have hitherto

been buried somewhere in the recesses of the universe.

The tidal and quadrupolar elements of GWs cause them to stretch an

object in one direction while squeezing it in another. To envision this,

consider a piece of elastic. Now, grab the elastic with two hands before

pulling the hands apart horizontally. This will cause the elastic to stretch

horizontally while it compensates by squeezing together vertically in a

manner that roughly illustrates the field produced by GWs. The resulting

field may, in principle, be observed using Pulsar Timing Arrays (PTA) or

ground and space-based Laser Interferometers (LI).

Conservation of mass and linear momentum respectively prohibit the

monopole or dipole moments from radiating. Thus, the quadrupole

moment provides the lowest order source of GW emission. The

quadrupole formula [1] depends upon the density and may be

approximated by

∫ (1)

3

Where, omitting the prefactor, the amplitude of the gravitational wave

strain depends upon 1/r and not upon 1/r3

(2)

A further approximation using the virial theorem [2] provides the upper

bound for a non-spherical emission (spherical bodies will have no

quadrupole moment and as such will not radiate GWs)

(3)

This upper bound provides the highest possible strain amplitude that

ought to be examined. GWs are still undetectable as a result of their

prohibitive size. However, Joseph Taylor and Russel Hulse successfully

discovered the first binary pulsar system in 1974 [3]. They observed the

system over the two decades that followed before indirectly confirming the

emission of GWs. In 1993, they succeeded in showing that the period of

the binary was decreasing at precisely the rate that Einstein’s quadrupole

formula had predicted some 70 years earlier.

GR requires these waves to propagate in either a plus (+) or a cross (×)

polarization state. However, several alternative theories of gravity have

presented four additional polarization states. If GR were modified slightly,

it would be theoretically possible to observe GWs in two helicity-0 modes

(Transverse breathing mode, and Longitudinal mode) and two helicity-1

shear modes in addition to the + and × states. [4] The basis tensors for

these six states are:

(4)

Where the first two are those predicted by GR. In this way (among others),

a direct observation of GWs and subsequent characterization of the waves

4

will enable a test of the current theory of gravity under extreme density

and pressure. Their study will provide a method for listening to events that

could never possibly be seen.

5

2 Sources of Gravitational Wave Emission

The sources that are known (or theorized) to emit gravitational waves are

so variegated that it is absolutely crucial to categorize them. For example,

the frequency of the waves emitted may vary from 10-18 Hz to 104 Hz

between sources. Thus, frequency becomes a natural classification

attribute. There are four main frequency subsets, aptly named the High

(HF), Low (LF), Very Low (VLF), and Ultra Low Frequency (ULF) bands.

The sources within each frequency band are further subdivided by the

nature of their origin. Specifically, there are gravitational waves triggered

by coalescing compact binary systems, stellar core collapse, periodic

emitters, and the stochastic background of radiation. Additionally, there is

a special case in the ULF band. This year, the existence of primordial GWs

was indirectly verified by observing B-mode polarization in the Cosmic

Microwave Background (CMB)

2.1 High Frequency Sources

2.1.1 LIGO Overview

High frequency emitters fall in the 1 Hz – 104 Hz range and are the major

focus of ground based interferometry. There are several interferometers

located around the world; however, the Laser Interferometer Gravitational

Wave Observatory (LIGO) is perhaps the most prominent and will be the

primary focus of this paper.

LIGO is a kilometer-scale interferometer project founded and constructed

in the 1990s by MIT and Caltech (among others). The project employs

three Michelson Interferometers in two different locations. In Louisiana,

there is a 4-km L1 detector, while another 4-km H1 and a 2-km H2

detector both reside in Washington. The project aims for a sensitive range

between 40 and 7000 Hz. [5]

6

“Figure 1: Optical sensing configuration of the LIGO 4 km

interferometers. The IO block includes laser frequency and

amplitude stabilization, and electro-optic modulators. The power

recycling cavity is formed between the PRM and the two ITMs, and

contains the BS. The inset photo shows an end test mass mirror in

its pendulum suspension. The near face is the high-reflecting

surface, through which one can see mirror actuators arranged in a

square pattern near the mirror perimeter, and optics for handling

the transmitted beam behind the mirror”[Image and caption

taken from 5].

On a conceptual level, the tidal polarization of GWs will cause extremely

small fluctuations in the length of an interferometer’s arms. Specifically,

one arm will be stretched while the other will be squeezed. Over time, the

difference in arm length will be constantly recorded and ultimately provide

a direct GW detection. Unfortunately, it is impossible to make

measurements on such a small scale without encountering noise. Noise

may be caused by any number of factors including seismic disturbances

(wind, earthquakes, human activity, etc) and mechanical thermal noise. It

is possible to filter the data (see section III) in a manner that will minimize

the effect of noisy data, but it is not possible to entirely remove noise.

LIGO is also able to mitigate the effect of seismic noise by comparing the

data collected at their two locations. It is very unlikely that the human

activity surrounding one site will produce noise that is a perfect

counterpart to the noise resulting from human activity thousands of miles

away.

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2.1.2 Compact Binaries

A compact binary consists of a two-star system with highly compact

bodies. The system may contain a pair of Neutron Stars (NS-NS), a

Neutron Star and a Black Hole (NS-BH), or two Black Holes (BH-BH).

These binaries will be locked in a cosmic dance of sorts as they slowly

converge. Compact binary coalescence occurs in three stages: inspiral,

merger, and ringdown.

The Inspiral is the first stage, occurring when the two masses commence

rotating about each other. The signal from this stage will slowly increase in

frequency as the distance between the masses decreases. The frequency

emitted by low-mass bodies during this stage may be approximated as a

factor of the system mass and cosmological redshift [6].

[

] (5)

During an inspiral, a NS-NS system will produce waves in the high

frequency band that may be detected to a distance of 300Mpc. A NS-BH

inspiral may be detected out to 650Mpc, and a BH-BH inspiral to a

redshift of z = .4. Once the new LIGO-II system is fully functional, these

sources are expected to provide hundreds of detections per year [7].

Similar to the inspiral, the ringdown of a low-mass coalescing binaries will

have a frequency of [6]:

[ (

)

] (6)

[

]

Naturally, the frequency in this stage is expected to be a factor of 3-8 times

higher than the frequency during the inspiral. In fact, during the final

moments of the ringdown, the frequency is actually expected to jump even

higher and produce a “chirp.” The waves generated by the merger and

ringdown phases of BH-BH binaries will both be substantially stronger

than the waves generated by inspirals. However, despite a larger detection

range (up to z ~ 1-2 for M~2000M ), it is impossible to say with any

8

degree of certainty that these events will generate more detections than

inspirals. They are not as common and very little is known about the

merger stage of a BH-BH binary system.

Unfortunately, it is quite unlikely for the merger/ringdown of a NS-NS

system to fall within the sensitive range of laser interferometers. As such,

NS-NS systems will probably not provide much insight into the as yet

unknown NS equation of state.

However, it is possible for the immense tidal forces of a black hole in a NS-

BH binary to rip the NS apart before it even reaches the ringdown. This

phenomenon is known as the tidal disruption and will generate GWs out to

140Mpc that will enable the equation of state to be recovered!

2.1.3 Stellar Core Collapse

The formation of a type II supernovae and accretion induced collapse

(AIC) both provide prime candidates for the generation of detectable GWs.

However, the mechanism of these collapses is not well enough understood

to guarantee waves. It is very difficult to accurately model the degree of

asymmetry during a collapse. As detailed in section I, a symmetric body

lacks a quadrupole moment and will fail to produce GWs.

If the stars are spinning rapidly enough to generate an instability that

leads to asymmetry and GW emission, then the detection of waves

resulting from core collapse will provide insight into the mechanism of

collapse. Moreover, even if GWs from core collapse are not detected, their

absence will eventually provide insight into the nature of collapse as well.

EG once the lack of emission is not attributed to a lack of sensitivity, the

collapse may be assumed symmetric.

2.1.4 Periodic Emitters

The term Periodic Emitter most frequently refers to a Pulsar. Pulsars are

NS’s that rotate very rapidly with extreme phase precision. A non-

axisymmetric pulsar will have a quadrupole moment that is guaranteed to

generate gravitational waves. These waves will impact the PTA (EG Hulse-

Taylor, see section I).

9

The frequency of these pulsars is ~ 2(rotation frequency of NS). Therefore,

a rapidly rotating millisecond pulsar has the potential to be readily

detectable in the high frequency range. The amplitude of the waves

generated by pulsars may be derived from equation (2) after including the

prefactor. It will depend upon the inertia, I, and ellipticity, . [8]

(7)

The upper limit that has been proposed for ellipticity suggests that the

amplitude , which remains considerably beneath the current

threshold for detection. This limit raises a question. Why are pulsars

considered one of the most promising sources for GW detection in the next

5 years?

If a number of wave cycles may be coherently tracked, the signal may be

amplified by a factor ~ √ , where N represents the number of wave cycles

tracked. Although tracking cycles is extremely computationally expensive,

the process has been an ongoing project over the past 20 years. The

amalgamation of data is rapidly approaching a critical mass on the

imminent path to detection.

2.1.5 Stochastic Background

Waves from the stochastic background (SB) span a vast variety of

frequencies, but the amplitude of the GW background is decidedly smaller

than current ground-based detection capacities. Thus, the search for a

stochastic background of GWs involves merging the data sets of several

independent detectors and looking for recurring patterns in the noise.

Noise ought to be entirely random. As such, there should not be any

predictable correlation between the noise of data collected at different

locations. Any lapse in unpredictability (EG any noise correlation) that

may not be attributed to a global event (like a massive earthquake) would

provide evidence supporting the detection of a GW background.

The strength of any SB depends upon the energy density it contributes to

the universe as a fraction of the critical energy density required to close the

universe and halt expansion [9].

10

(8)

With

and Ho corresponding to the Hubble constant. When

comparing data across different detectors, the waves will be expected to

have strain amplitudes that depend upon frequency and energy density [9]

√ (9)

Upper bounds on the stochastic background strain indicate that they will

not be likely to provide the first direct detection of GWs; however, they will

remain a field of future research for quite some time. Whenever the SB is

detected, it will provide insight into the very utmost origins of the

universe. GWs will be the first time that humans have been able to peer (or

perhaps more accurately, listen) into the period of time preceding the

moment of recombination and the genesis of electromagnetic waves.

2.2 Low Frequency Sources

2.2.1 LISA Overview

Low frequency sources emit in the range between 10-5 Hz – 1 Hz. The

seismic noise levels at this frequency range are prohibitive. They prevent a

ground based detector from being a feasible method of data collection.

However, the sources that emit in this band are still incredibly interesting.

GWs in this band would be readily observable if an observatory could be

constructed in a quiet location removed from low frequency noise. Space

provides an ideal environment: ample room for construction without

people, planets or any other jiggling things.

The Laser Interferometer Space Antenna (LISA) will eventually put an

interferometer into orbit around the earth. LISA is a cooperative global

project that involves the European Space Agency (ESA) and the National

Aeronautics and Space Administration (NASA). On its most basic

conceptual level, LISA will operate in a similar fashion to LIGO insofar as

both detectors are composed of laser interferometers that monitor test

masses. LISA will consist of three spacecraft that are launched into a solar

11

orbit following the earth with a triangular configuration. The arm length

will change slowly and predictably throughout the course of orbit, but will

always remain approximately 5 million kilometers.

Figure 2: Artistic representation of LISA Mission (Left) and

orbital configuration of LISA (Right) Images taken from sources [3]

(left) and [9] (right).

A 5 million kilometer distance between spacecraft will cause diffraction to

spread the lasers. By the time that the laser reaches the detector, it will

have a diameter of ~20km. A small portion of this light is captured by the

detecting spacecraft. This light is then “interfered with a sample of light

from the on-board laser. Each spacecraft thus generates two interference

data streams.” [Excerpt taken from 9] The six resulting interference

streams are then combined in order to “construct the time variations of

LISA's armlengths and then build both gravitational-wave polarizations.”

[9] Modulating the amplitude of GW strain throughout the course of

LISA’s orbit around the sun will also supply the position of LF sources in

the sky.

Once launched, LISA will be able to verify the Kerr metric and No Hair

theory of BHs. It will facilitate a new degree of scrutiny into the nature and

development of massive BHs and has the potential to specify Dark Energy

(DE) parameters and discover new astronomical phenomena.

12

2.2.2 Coalescing Binary BHs

Although many coalescing binary systems will emit in the HF band, the

mergers of massive black holes (MBH) with roughly equal masses and

extreme mass ratio inspirals (EMRI) will emit at a lower frequency. LISA

will be able to detect MBH mergers out to a redshift z ~ 5-10. This is one

of the most exciting prospects of the mission

MBH mergers are only poorly understood. Observing their coalescence

will offer a wealth of information regarding the nature of black holes as

well as another opportunity to test gravity in its most extreme conditions.

Rate estimates for these events are estimated to be hundreds per year!

EMRI events will also be quite enlightening. The GWs from an inspiral of a

relatively small body (WD, NS, BH with mass 10 M ) into a MBH (mass

~ 105 – 107 M ) contain information regarding the geometry of space-time

during the event. Recovering this information will map the Kerr geometry

and enable the verification of the Kerr metric and No Hair theorem (EG

the theory stating that BHs are perfectly categorized by the mass and spin

because these are the only parameters that are not lost inside the void).

2.2.3 Periodic Emitters

Unlike the LIGO, LISA is actually guaranteed to detect GWs after its

launch. There are binary pulsar systems which are known to emit GWs

that fall well within LISA’s sensitivity band. In fact, at a frequency below ~

0.002 Hz, there will actually be too many binary systems to resolve. The

result will be a type of “noise” that is actually caused by GWs!

Frequencies above ~ 0.002 Hz will be scarce enough to be resolvable,

which is to say that the number of sources will be downscaled by a factor of

104-5 to ~ 104 detectable binaries. These binaries will facilitate the

generation of a 3D map of the binary systems in the galaxy that is not

impeded by dust.

13

2.2.4 Stochastic Background

The GWs present in the SB in the LF band are expected to be emitted by a

combination of primordial (EG Origin of the Universe) and astrophysical

events. One possible source of SB radiation is a phase change. The

frequency of GWs resulting from phase transitions are temperature

dependent and typically peak around

(

) (10)

Where T represents the temperature value and f is the peak frequency.

Using this, the waves resulting from the electroweak phase transition (EG

the moment at which the electric and weak forces began to differentiate) at

temperatures between T ~ 100-1000 could generate waves within the

sensitive band! [10]

Unlike the LIGO however, LISA will not be able to compare data collected

from multiple operation sites to the extent that there will only be one

spaced-based LF band detector. Although the primordial waves resulting

from inflation would be detectable in theory, they are likely to remain

below the noise levels. In light of the recent BICEP2 findings (higher than

anticipated amplitude of polarization), however, it does seem plausible

that LISA would find direct evidence of inflationary GWs.

2.3 VLF and ULF Sources

2.3.1 Very Low Frequency Pulsars

The VLF band covers frequencies from 10-9 Hz - 10-7 Hz and has not

received the amount of attention that the LF and HF bands have. This is

primarily because it proves much more difficult to study sources that emit

in this range. Neither ground-based nor space-based detectors prove

sufficient. Millisecond pulsars timing arrays actually constitute the main

detectors. VLF GWs resulting from SMBH inspirals and early-universe

phase transitions will cause small fluctuations in the arrival times of the

14

pulses. Once these fluctuations have been studied for long enough, then

information from the GWs may be recovered in a process somewhat

reminiscent of the Taylor-Hulse binary pulsar timing array described in

section I. VLF waves will help set limits on the stochastic background .

2.3.2 Ultra Low Frequency B-Mode Polarization

Recently, the publication of the BICEP2 findings has lent a lot of attention

to the ULF band. Waves in this range have frequencies between 10-18 Hz

and 10-15 Hz. They are the result of an inflationary epoch and oscillate on

“scales comparable to the size of the universe.” [Excerpt taken from 9]

ULF waves are studied indirectly through their polarization on the Cosmic

Microwave Background (CMB). Specifically, inflationary GWs are (roughly

speaking) the only source of helical B-mode polarizations on the CMB.

These waves would have been created (and amplified) during an

inflationary epoch such that the root mean square amplitude is

proportional to the energy scale of inflation

(

)

(11)

The BICEP2 findings claim to have detected B-mode polarization of light

in the CMB. More surprisingly, they hold that the tensor-scalar ratio r =

0.20. This is substantially larger than was anticipated. Assuming their

methodology withstands the scrutiny of the scientific community, the

scientists involved have provided the basis for detecting the “smoking gun

of inflation.”

15

3 Data Analysis

What is perhaps the most crucial piece of Gravitational Wave Astronomy does

not occur until after the data has all been collected. The process of data analysis

takes a series of noisy oscillations and turns it into information about the past

and present universe. Unfortunately, the random nature of noise makes it

impossible to ever declare detection with 100% certainty. There will always

remain some possibility that random noise has conspired to look like a very

convincing signal. When studying GWs, it is therefore crucial for the scientific

community to agree upon some degree of significance. Eventually, with a high

enough degree of significance the uncertainty will become small enough to be

considered negligible. Consider, for example the hypothetical case wherein a

signal has been generated which would only be duplicated by noise once every

10 million years. It is so incredibly unlikely that any particular instance is noise

that detection would be almost certain!

3.1 Detection Probability

3.1.1 Significance and Confidence

Statistically speaking, the process of testing for a signal is very similar to setting

significance (α-value) or constructing a confidence interval during hypothesis

testing. The false alarm probability is synonymous with significance and

corresponds to the probability that random noise has generated a passable

signal. The Probability Density Function (PDF) of a false detection is given by

[11]

∫

(12)

While confidence may be calculated quite simply by considering 1-Pf(R) where R

Rcomplement spans all possible values of x. Similarly, the probability of a

detection is given by [11]

∫

(13)

3.1.2 Χ2 Likelihood and Goodness of Fit Test

Typically, datasets from ground based interferometers like LIGO may be

approximated as having stationary, normally distributed noise such that the

16

expectation values for the mean and mean-square levels of noise in the absence

of a signal are given respectively by [12]

[ ] [ ] (14)

[

[ ] (15)

Where the expectation value is defined according to the standard definition in

both the discrete and continuous cases

[ ] ∑ (16)

[ ] ∫

(17)

E[|n|2] depends upon “the one-sided noise spectral density at frequency f, Sn(f),

and the observation time, T” [Excerpt taken from 12]. If a signal were in fact

present, the signal stream s(t)=R(t,τ)h(τ) + n(t) where the first term no longer

vanishes. The likelihood test (19) in this case simply becomes an expression of

the Χ2 test (17) that describes the goodness of fit of a model [12, 13,16]

∑

(17)

Where the likelihood test is given by the inner product [12]

∑

(18)

Such that the likelihood may be found after accounting for the prefactor by the

exponential of a chi-squared test [12]

(19)

With the residual r = s - R h.

3.1.3 – Neyman-Pearson Approach

Statistically, the Neyman-Pearson Approach (NPA) may be interpreted as a kind

of trade-off between type I and type II error. Error occurs in hypothesis testing

17

when the wrong conclusion regarding the null hypothesis (H0) is reached.

Specifically, type I error occurs when H0 is wrongly rejected while type II error

occurs when H0 is wrongly favored. NPA typically examines the ratio between

probability of detection and significance [13]

(20)

A threshold value will be set using this ratio to determine the minimum criteria

for detection.

3.2 Filtering

3.2.1 Constant Frequency Fourier Transform Filter

Before performing statistical detection testing, the data must be filtered in a

manner that will remove noise without disturbing the desired signal. This is

predictably tricky and depends heavily upon the distribution of the noise and

the nature of the signal. It is, therefore, especially difficult to filter adequately

when the signal has never been detected before! If the data is expected to

contain a signal of constant albeit unknown frequency, then a simple Fourier

Transform (FT) of the signal will suffice [11,13]

∫

(21)

With:

The Fourier domain will rearrange a signal of constant frequency such that the

noise is evenly distributed across the entire spectrum while the power of the

signal is all stacked upon a single frequency. After detecting the source in the

Fourier domain, it may be reverted by applying the standard inverse FT

∫

(22)

Unfortunately, this filter is largely insufficient. Although most sources will emit

at a constant frequency, the orbital motion of detectors will produce a Doppler

shift on the arriving waves. It was included here as a simple introduction into

the conceptual realm of filtering.

18

3.2.2 Matched Filtering in Zero-Mean Gaussian Noise

In reality, noise will not be a perfectly zero-mean Gaussian; however, the

approximation is not terrible and the assumption will suffice for the purposes of

this section. The predominant method for categorizing noise (especially

Gaussian noise) is defined simply by taking the expectation value of the noise at

two distinct times. It is referred to as the autocorrelation function [11, 13]

[ ] (23)

The matched filter (MF) q(t’) for an expected signal s(t) depends upon the time

evolution of the signal and may be calculated by solving [11, 13]

∫

(24)

for a given set of parameter values. The expected value of s(t) that is input into

this equation is generally referred to as the template of a filter. This will be

discussed further in section 3.2.3.

If the data is observed continuously over some time interval then the logarithm

of the likelihood function will depend upon the time evolution of the matched

filter, expected signal, and observed data set [11, 13]

[ ] ∫

∫

(25)

With the total observed data set occurring as the additive superposition of the

noise and the signal [11, 12, 13]

(26)

Equation (25) may be simplified by noticing that only the first term actually

depends upon the collected data set. As such, the second term may be

successfully incorporated as a constant given a set of parameters and an

observational window. The important ratio depends only upon a the detection

statistic [11, 13]

∫

(27)

If the detection statistic surpasses a given threshold G0 according to the process

described in section 3.1, then detection may be claimed. Although the

19

calculations are outside the scope of this paper, it ought to be mentioned that in

the case of stationary noise such that K(t, t’) depends only upon time differences

and the matched filter calculations are simplified

enough to be exactly analytically calculable by the use of FT techniques. Refer to

[13] for a review of this process. The final solution that falls out for the FT of the

matched filter depends upon the one-sided noise spectral density and the FT of

the template selected

(28)

Where Sn is only defined for the set of non-negative frequencies. Increasing the

noise density within a filter will directly decrease the filter’s contribution to the

detection statistic. This resonates intuitively. Higher noise levels make detection

less probable.

3.2.3 Templates

Template construction is an extremely important part of the process of data

analysis. The process of generating a template is typically a combination of

analytic and numerical approximations based on assumptions about parameters.

Searching data via templates can be a very effective method of raising signals

out of the noise. Generally, a single template will prove to be insufficient, but

after constructing the number of templates required, a given search will be

orders of magnitude more efficient.

The efficiency of a search depends largely upon the number of parameters that

the analysis must track. Using a well-constructed template-based model allows

the search to account for far fewer parameters.

Numerically, consider the example of the search for a binary NS-NS inspiral.

Evidence has shown that the signal may be coherently tracked over N ~ 10,000

cycles. This will amplify the signal by a factor of ~ 100 (see section 2.1.4: EG √

dependence of amplification). Detection by the use of a general model requires

an integrated SNR > 100 [12] which implies an absolute S/N ratio > 1. The use

of a template-based model only requires an integrated SNR > 7 for detection [12]

This means that an absolute S/N ratio > 7/100 ~ S/N > 0.07 will suffice!

Although the use of templates has the potential to greatly increase the

confidence of a given data set, it is absolutely essential to select an accurate

template. Template-based models force a search to select between a signal that

20

matches the given amplitude no signal. Thus, a faulty template would

automatically eliminate the possibility of detection. Moreover, if the template

s(t) in a matched filter has miscalculated the phase of a signal, then the template

will cause a destructive interference that proactively reduces the sensitivity of

the filter. Even if the amplitude is accurate, phase error makes detection

exceedingly unlikely. When employing the use of a template-based model, note

that the model will be much more sensitive to phase error than amplitude error.

3.2.4 Nyquist Theorem and Aliasing

The Nyquist theorem is actually quite straightforward. It sets a lower bound on

the required sampling interval. If data is sampled at intervals of Δt, then the

sampling rate will naturally be 1/Δt. The maximum frequency that can

physically be observed in this time is bounded by f = 1/2Δt. In other words, the

sampling rate may be truncated at a value of 2f. This truncation will prevent the

noise from higher frequencies from inadvertently being mapped down and

contributing to the noise levels of the observation band. Refer to [11] for the

formal development of this concept.

3.2.5 Maximum Likelihood Estimator

When a signal is analyzed, the process frequently seems to resemble

orienteering without a map on a foggy night. An eventually successful pioneer

must tolerate repeated experimentation and, frequently, some degree of

guesswork that has its basis in past failures. More concretely, the waveform of a

signal is often only known as a function of unknown parameters. Signal

amplitude and time of arrival (TOA), chirp mass, and arrival phase are all rarely

known parameters. The Maximum Likelihood Estimator (MLE) for a given set

of parameters is found by choosing the parameters in a manner that maximizes

the parameter-dependent likelihood ratio. This is given by an equation that is

easily motivated by the likelihood ratio presented in the Neyman-Pearson

Approach (C.f. equation 20).

(29)

Similarly, the filter and detection statistic (C.f. equations 27, 28) will also have

parameter-dependent analogs:

(30)

21

∫

(31)

In order to find the MLE for m unknown parameters, an m-dimensional grid

populated with discrete parameter values must be solved. Effectively, this will

constitute a standard optimization problem from calculus in which the

derivatives of m equations are simultaneously set to zero in order to solve for

the critical points

[ ] (32)

The process of solving an m-dimensional grid very quickly becomes expensive

in multiparameter problems.

3.2.6 Bayesian Inference

The development of Bayesian data analysis algorithms is a rapidly emerging

field of study. At present, there is not a fully developed procedure detailing the

Bayesian approach to data analysis. Currently, Bayesian Inference (BI) is

primarily used as a supplement to improve upon the well-established search

algorithms (Notably MFs and MLEs). The approach functions by prescribing a

cost to every decision that is made and then discerning the posterior

distribution (EG the conditional probability of an event after accounting for

relevant factors) of a model from the prior probability of the model and the

marginal likelihood of the model.

The first step in Bayesian analysis is to compute/approximate the marginal

likelihood and the prior. This is done by assessing and calculating various Bayes

Risk factors. While important, the process is tedious and rather lengthy and

may be seen in section 3 of [13].

After calculating these values, they posterior probability of a model is given by

[14, 15]

(33)

Where pr(s) is the normalization factor that could theoretically be obtained by

integrating [15]

22

∫

(34)

However, this integration is problematic and expensive at best, and unsolvable

at worst. Fortunately, it may be skirted by considering the relative probability of

two models. Let Mi denote the model claiming the presence of a GW signal

within the noise. Similarly, let Mj refer to the model claiming the absence of any

GW signal. The ratio of the two is given by [14]

(35)

Defining the product of prior belief [14]

(36)

And the Bayes factor [14]

(37)

The ratio may be abbreviated as the product of prior belief and the Bayes factor

[14]

(38)

3.2.7 A (Very) Brief Introduction to MCMC Methods

Although Markov Chain Monte Carlo Methods (MCMC) are too intricate to

sufficiently summarize in this paper, their relation to Bayesian Inference is

fairly straightforward. MCMC methods require a likelihood, prior, and proposal.

Defining a likelihood function is not often an easy task. In the case of Gaussian

noise, it may be derived by applying Bayesian analysis to a matched filter. The

result is an application of equation (19).

First, recognize that “the conditional probability of measuring [data set] s(f)

when a signal h(f) is present, is equal to the conditional probability of

measuring [data set] s0 = s(f)−h(f) assuming that no signal is present” [excerpt

taken from 14] such that

(39)

23

In other words, subtracting the signal from a data set with said signal present

ought to give a data set with no signal present. Although it is easy to get turned

around in the words, the end result of substituting conditional probabilities and

applying the definition of an inner product from (18) will look very similar to (19)

(40)

Values for the priors required are typically well understood and may be

approximated by well-accepted literature values. The proposal distribution

must be selected carefully in a way that does not take a prohibitively long

amount of time to produce a stationary distribution that provides an accurate

approximation of the posterior distribution.

24

4 Summary

Gravitational Wave Astronomy is one of the most exciting fields of study to

emerge in the past century. In the imminent future, their detection will enable a

galactic array of insight into astrophysical phenomena including: the nature of

gravity and relativity, the process by which compact binary systems (NS-NS,

NS-BH, and BH-BH) coalesce, the nature of dark energy, the opening seconds of

the universe before recombination, the process of early-universe phase

transitions, the nature of black holes, the NS equation of state, and possibly the

process by which stellar bodies collapse, not to mention the potential for

discovering entirely new, unobserved phenomena. Despite abundant, nearly

overbearing obstacles complicating data collection and analysis, a global

coalition of the world’s brightest minds is rapidly overcoming every

complication and racing toward a new era of astrophysics.

The detection and characterization of gravitational waves will be comparable

to a “baby using ultrasound from inside the womb to observe the world

outside” (N. J. Cornish, 2014).

25

Acknowledgements:

[1] B. F. Schutz (1997), gr-qc/9710079

[2] T. Creighton, Quadrupole moment, http://www.tapir.caltech.edu

[3] J. Baker et al (1997), LISA-LIST-RP-436, vol. 1, pp 1-117.

[4]K. J. Lee (2014), arXiv:1404.2090

[5] B. Abbot et al (2007), arXiv:0711.3041v1 [gr-qc]

[6] E. E. Flanagan and S. A. Hughes, Phys. Rev. D 57 (1998), 4535.

[7] C. Cutler and K. S. Throne (2002), gr-qc/0204090

[8]N. J. Cornish and J. B. Camp (2004). Annu. Rev. Nucl. Part. Sci. 54:525–77

[9] S. A. Hughes (2003), Annals Phys. 303 (2003) 142-178

[10] D. Sticlet (2009). Phase Transitions in the Early Universe

[11] B. F. Shutz (1997), gr-qc/9710080

[12] N. J. Cornish (2013), Phil. Trans. R. Soc. A. 371, 20110540

[13] P. Jaranowski and A. Krolak (2009), Cambridge University Press

[14] N. J. Cornish (2009), Cambridge University Press

[15] C. Andrieu, N. D. Freitas, A. Doucet, and M. I. Jordan (2003). Machine

Learning, 50, 5–43

[16] Chi-Squared Goodness of Fit Test (1997), http://www.stat.yale.edu