LISA Aperture Synthesis for LISA Aperture Synthesis for Searching Binary Compact ObjectsSearching Binary Compact Objects

Aaron RoganAaron RoganWashington State University Washington State University

roganelli@wsu.eduroganelli@wsu.edu

Collaborator: Sukanta BoseCollaborator: Sukanta Bose

GWDAW 2003GWDAW 2003 Space-Based Detectors II Analysis Methods Space-Based Detectors II Analysis Methods

Supported by NASA: NASA-NAG5-12837Supported by NASA: NASA-NAG5-12837

IntroductionIntroduction LISA is a network of LISA is a network of

three detectorsthree detectors 2 are independent 2 are independent Total of 6 elementary Total of 6 elementary

data streamsdata streams Main Sources of Noise:Main Sources of Noise:

• Laser Frequency Laser Frequency Fluctuation Fluctuation

• Relative Craft Motion Relative Craft Motion Time delay Time delay

interferometry can interferometry can eliminate much of the eliminate much of the dominating noisedominating noise

Introduction Cont’dIntroduction Cont’d Lasers aborad LISA will Lasers aborad LISA will

have a frequency have a frequency stability of a few parts stability of a few parts in 10in 10-13-13. .

Desired sensitivity Desired sensitivity range at least a few range at least a few parts in 10parts in 10-20-20

Time delay Time delay interferometry uses interferometry uses time shift operators, Etime shift operators, Eii. .

The time shift operator The time shift operator described by:described by:

EEii (t) = (t) = (t-L(t-Lii/c)/c) Using these generators Using these generators

or pseudo-strains LISA or pseudo-strains LISA can achieve the desired can achieve the desired levels of sensitivitylevels of sensitivity

Introduction Cont’dIntroduction Cont’d The pseudo-strains do not The pseudo-strains do not

span a vector spacespan a vector space They use data from all six data They use data from all six data

streams to cancel noisestreams to cancel noise Act as a network of 3 Act as a network of 3

independent detectors independent detectors The pseudo-strains have The pseudo-strains have

different sensitivities to the different sensitivities to the same sky position same sky position

An optimal combination of the An optimal combination of the pseudo-strains is needed to:pseudo-strains is needed to:• Maintain the highest signal-to-Maintain the highest signal-to-

noise ratio possible over the noise ratio possible over the entire orbitentire orbit

• Maintain the highest level of Maintain the highest level of sensitivity for all sky positionssensitivity for all sky positions

• Maintain the most efficient Maintain the most efficient search over all sky positionssearch over all sky positions

The ProblemThe Problem To obtain the optimal combination of the To obtain the optimal combination of the

data streams one must consider:data streams one must consider:• The pseudo-strains are a function of the orbital The pseudo-strains are a function of the orbital

position of LISAposition of LISA• How to weight each pseudo-strain for a given How to weight each pseudo-strain for a given

orbital position orbital position The advantages of an optimal combination The advantages of an optimal combination

are:are:• Maintaining the maximum sensitivity to a wider Maintaining the maximum sensitivity to a wider

range of range of {{θθ,,ΦΦ}} values values• Maintaining the maximum sensitivity for all Maintaining the maximum sensitivity for all

points on LISA’s orbitpoints on LISA’s orbit

How to Approach the Problem?How to Approach the Problem? Identify the time domain polarization Identify the time domain polarization

amplitudes, amplitudes, ++(t) and (t) and xx(t).(t). Derive the appropriate Fourier domain Derive the appropriate Fourier domain

polarization amplitudes polarization amplitudes Combine the 3 weighted pseudo-strains to Combine the 3 weighted pseudo-strains to

obtain the complete signal, obtain the complete signal, AA((ΩΩ).). Analytically maximize over the following Analytically maximize over the following

parameters: parameters: {{ΨΨ,,εε,,δδ}} Obtain the optimal statistic, Obtain the optimal statistic, λλ||ΨΨ,,ЄЄ,,δδ . . Develop a template bank over remaining Develop a template bank over remaining

parameters, namely parameters, namely {{θθ,,ФФ}.}. Determine the computational feasibility of Determine the computational feasibility of

a searcha search

The Optimal StatisticThe Optimal Statistic The matched filter is used to obtain the optimal The matched filter is used to obtain the optimal

detection statistic. Before maximization it takes detection statistic. Before maximization it takes the following form:the following form:

Now maximizing over the source polarization Now maximizing over the source polarization and inclination angles can be achievedand inclination angles can be achieved

3

1

3

1

3

1)(

,,A i

Ai

Ai

i

AA

AA xSFeNxh

Signal-To-Noise RatioSignal-To-Noise Ratio The SNR for a each The SNR for a each

pseudo-strain is plotted pseudo-strain is plotted to the right. to the right.

The holes indicate The holes indicate directions that minimize directions that minimize the SNRthe SNR

Compare the optimal SNR Compare the optimal SNR to the SNR for a given to the SNR for a given pseudo-strainpseudo-strain

The optimal statistic The optimal statistic improves the SNR for all improves the SNR for all orbital positionsorbital positions

Network SensitivityNetwork Sensitivity The optimal statistic also The optimal statistic also

greatly improves the greatly improves the sensitivity of LISA sensitivity of LISA

Although a single pseudo-Although a single pseudo-strain spans all {strain spans all {θθ,,ФФ} values} values• It does not obtain a maximum It does not obtain a maximum

sensitivity to all {sensitivity to all {θθ,,ФФ}}• At any given point in the orbit, At any given point in the orbit,

the sensitivity is very limited the sensitivity is very limited The optimal statistic The optimal statistic

advantages are:advantages are:• All {All {θθ,,ФФ} values are maximized } values are maximized

at some point in the orbitat some point in the orbit• The likelihood of finding a source The likelihood of finding a source

is increasedis increased

Developing the Template BankDeveloping the Template Bank Develop a metric on the parameter space Develop a metric on the parameter space

{{ΩΩ,,θθ,,ФФ} as outlined by Owen} as outlined by Owen Project out Project out ΩΩ from the 3-dimensional metric from the 3-dimensional metric This new metric will define the overall volume of This new metric will define the overall volume of

your parameter spaceyour parameter space Decide on a Minimal Mismatch (MM)Decide on a Minimal Mismatch (MM) The Minimal Mismatch will fix the number density The Minimal Mismatch will fix the number density

of the templatesof the templates Determine the grid spacing within this volumeDetermine the grid spacing within this volume Finally determine the number of templatesFinally determine the number of templates

I Would Like to Thank The Following I Would Like to Thank The Following Individuals and Organizations for Their Direct Individuals and Organizations for Their Direct

Contributions to My Research:Contributions to My Research:

Shawn SeaderShawn Seader Rajesh Kumble Nayak Rajesh Kumble Nayak Washington State University Physics DepartmentWashington State University Physics Department National Aeronautics and Space AdministrationNational Aeronautics and Space Administration

If not for the previous work by the following If not for the previous work by the following individuals I would not be here todayindividuals I would not be here today

S. BoseS. Bose S. DhurandharS. Dhurandhar K. R NayakK. R Nayak J-Y VinetJ-Y Vinet A. PaiA. Pai M. TintoM. Tinto B. Owen B. Owen B. SchutzB. Schutz T. PriceT. Price S. LarsonS. Larson J.W. ArmstrongJ.W. Armstrong

A. EastabrookA. Eastabrook

Signal-To-Noise Ratio for the Signal-To-Noise Ratio for the Optimal Statistic and a single Optimal Statistic and a single

Pseudo-StrainPseudo-Strain

Sensitivity of the a single Pseudo-Sensitivity of the a single Pseudo-Strain and the Optimal Statistic for Strain and the Optimal Statistic for

the Entire Orbitthe Entire Orbit